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# Number System Aptitude Questions and Answers:
#### Overview:
Questions and Answers Type: MCQ (Multiple Choice Questions). Main Topic: Quantitative Aptitude. Quantitative Aptitude Sub-topic: Number System Aptitude Questions and Answers. Number of Questions: 10 Questions with Solutions.
1. If $$\frac{3}{4}$$ th of $$\frac{2}{7}$$ th of a number is 34 then find the number?
1. 156.22
2. 158.67
3. 159.32
4. 162.53
Solution: Let the number is x then $$x \times \frac{3}{4} \times \frac{2}{7} = 34$$ $$\frac{3x}{14} = 34$$ $$x = \frac{34 \times 14}{3}$$ $$x = \frac{476}{3}$$ $$x = 158.67$$
1. If $$\frac{2}{5}$$ th of 80 is x then find the value of x?
1. 32
2. 23
3. 33
4. 22
Solution: According to the question $$\frac{2}{5} \times 80 = x$$ $$x = 32$$
1. If the fractions $$\frac{2}{5}$$, $$\frac{3}{5}$$, $$\frac{6}{7}$$, and $$\frac{2}{3}$$ are arranged in ascending order of their values. Which one will be in the second place?
1. $$\frac{2}{5}$$
2. $$\frac{3}{5}$$
3. $$\frac{6}{7}$$
4. $$\frac{2}{3}$$
Answer: (b) $$\frac{3}{5}$$
Solution: $$\frac{2}{5} = 0.4$$ $$\frac{3}{5} = 0.6$$ $$\frac{6}{7} = 0.857$$ $$\frac{2}{3} = 0.67$$ By writing the values in ascending order $$\frac{2}{5}, \frac{3}{5}, \frac{2}{3}, \frac{6}{7}$$ Hence $$\frac{3}{5}$$ will be on second place.
1. If the fractions $$\frac{3}{8}$$, $$\frac{2}{9}$$, $$\frac{4}{7}$$, and $$\frac{5}{12}$$ are arranged in descending order of their values. Which one will be in the third place?
1. $$\frac{3}{8}$$
2. $$\frac{2}{9}$$
3. $$\frac{4}{7}$$
4. $$\frac{5}{12}$$
Answer: (a) $$\frac{3}{8}$$
Solution: $$\frac{3}{8} = 0.375$$ $$\frac{2}{9} = 0.23$$ $$\frac{4}{7} = 0.57$$ $$\frac{5}{12} = 0.4167$$ By writing the values in descending order $$\frac{4}{7}, \frac{5}{12}, \frac{3}{8}, \frac{2}{9}$$ Hence $$\frac{3}{8}$$ will be on third place.
1. I have some horses and pigeons. If the total number of animal-heads is 51 and the total number of feet is 178 then find how many pigeons I have?
1. 15
2. 14
3. 13
4. 12
Solution: Let I have x number of pigeons and y number of horses then $$x + y = 51....(1)$$ As pigeons have two feet each and horses have four feet each then $$2x + 4y = 178....(2)$$ by multiplying 4 with equation (1) $$4x + 4y = 204....(3)$$ by subtracting equation (2) from equation (3) $$4x + 4y - 2x - 4y = 26$$ $$2x = 26$$ $$x = 13$$ Hence I have 13 pigeons.
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# Periodic function evaluation
1. Dec 18, 2007
### dr3vil704
1. The problem statement, all variables and given/known data
Suppose that f(x) is a periodic function with period 1/2 and that f(2)=5, f(9/4)=2, and f(11/8)=3. Evaluate f(1/4), f(-3), f(1,000) and F(x) - f(x+3)
(I'm not sure on this one, the teacher never really taught us this, we are on Derivative right now, but this is just one of his AP challenge problem)
2. Relevant equations
But Ok, I don't know much about period, the only thing I know about it is the trig function, which is a periodic function too, i think. But I read some where it stated that period function is
F(x + P)= f(x)
3. The attempt at a solution
so I try to set it up as
f(x)=f(x+1/2), since we know P is 1/2. so I try to find the X of the F(x + P), So that it make sense that f(2)=f(2+ 1/2) and also the f(2)=f(2-1/2). So
I begin to start subtracting 2 by .5 to get f(-3), which mean f(-3)=5, because f(-3)=f(2) because of the continuous of the period. I kind of ran into problem with the others.
so I'm stuck right here
I know that f(x+1/2) is the equation. but I have no clue as how make it a general equation to find f(1/4), f(100) and the others. I might be able to guess and check, but I really want to find out the general equation for this. Could some one help me?
Last edited: Dec 18, 2007
2. Dec 18, 2007
### Sleek
If a function is periodic, then f(x)=f(x+p), as you said. Also notice that, f(x)=f(x+np) where n is an integer. Watch it like this. If you have an angle on the unit circle, for every 2*pi rotation, you arrive at the same point. Hence, it doesn't matter how many times you rotate, until it is an integral multiple of 2*pi.
Similarly,
Consider f(1/4+n1/2). We know that n has to be an integer. We have to choose such an integer.
f(1/4+n1/2)=f(1/4) {which is what you have to find}
But if you can find a suitable integer which gives you a value of x whose f(x) is know to you, then you can equate them.
1/4+2n/4=(2n+1)/4 {for sake of simplicity}
For n = 4, I notice that the numerator becomes 9, thus the fraction becomes 9/4, the functional value of which is known to you.
Thus, f(1/4)=f(1/4+4*1/2)=f(9/4)=2.
Can you now manage with others?
Regards,
Sleek.
Last edited: Dec 18, 2007
3. Dec 19, 2007
### dr3vil704
oh yes, Thank you very much.
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# How to correctly model a small helix antenna
I am getting weird results trying to model a small 433 MHz helix antenna similar to this one with 4nec2:
In particular I am not able to obtain a resonable SWR value. This is the antenna an example of the nec file.
CE
SY L=0.0205 'Length of helix
SY S=0.001 'Pitch
SY N=(L-2*R)/S 'Turns
SY D=0.005 'Helix diameter
SY L1=0.008 'Length of base wire
SY L2=0.0055 'Length wire to be soldered
GW 1 3 0 0 -L1 0 L2 -L1 R
GW 2 1 0 0 0 0 0 -L1 R
GW 3 1 D/2 0 0 0 0 0 R
GH 4 N*12 S L D/2 D/2 D/2 D/2 R
GM 0 0 -90 0 0 0 L1 L2 0
GE 1
GN 1
EK
EX 0 1 2 0 1 0 0
FR 0 0 0 0 433 0
EN
I tried also adding a reflector to the model but still the results waren't right. I think that the problem could be the segments length, near the $$0.001\lambda$$ limit.
Here an example of the outputs I am getting:
In my experience this can't be modelled in NEC. It doesn't handle tight corners, close segments, short segments like this. Your geometry input file looks technically correct, but you're asking too much of the kernel.
Your best chance would be to specify the helix so that it uses only 4 or 5 segments per turn, ensuring that they overlap perfectly, i.e. it forms a rectangular box, not a staggered cylinder.
GH 4 **N*4** S L D/2 D/2 D/2 D/2 R
And make the wire thinner so that the spacing is about 2 * Diameter. The self-resonant frequency will be wrong, but it's wrong anyway.
Use the model checker to catch obvious errors first.
Use the 3D radiation pattern integration to find total radiated power, and check this against input power for sanity.
I really think think the only way to use NEC here would be to calculate the inductance per metre of the cylinder, and use three fat segments to model the entire antenna, loaded appropriately. Be careful of length/diameter and diameter/diameter rules, and don't try to turn corners. It won't help you choose the exact number of turns or the thickness of the wire, but you'll get some indication of the impedance, the bandwidth, radiation pattern, etc.
After several years of near-daily modelling in NEC, I tried an inductor in FEKO and was amazed to find that it just simulated it correctly, without a fuss.
• Thanks for the answer. My intent is to determine if a combination of these helix antennas in different arrangements like a dipole, rhombic, etc. can improve the gain in respect to a single one and how the radiation pattern would be affected. Given the fact this is a normal mode helix antenna, I am wondering if I can model it in 4nec2 as a simple wire (monopole) and expected that the overall behaviour of a combination of them can be used as an effective qualitative indication. Nov 9 '21 at 8:02
• For that case I think it would work well. Just make a wire of 3 or 4 segments, 3 mm diameter, 20 mm long, load in the middle and source on one end. This will be a reasonable model of the helical antenna for the purpose of radiation, coupling etc. Nov 9 '21 at 13:56
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# Is the time-derivative Hermitian?
I want to know why the time-derivative acts as though it's Hermitian under conjugation. I have read elsewhere that the time-derivative isn't a true operator in the quantum mechanical sense but I don't understand why that's the case, and if that's the case I still don't understand why $\partial_t^\dagger = \partial_t$. From what I do understand, in quantum mechanics a operator acts on a state vector which then gives us a new vector. This seems to be the case for the time-derivative, at least in the Schrodinger picture. For spatial derivatives we can use integration by parts to deal with the conjugate operators of the derivatives (i.e. in a scalar product) but since we're not integrating through time I can't find such a method to deal with the time derivative. So can anyone show me explicitly why the time-derivative is Hermitian?
Thanks
vanhees71
Gold Member
2021 Award
The time derivative cannot be hermitian or self-adjoint, because time is not an observable in quantum theory. It's clear, why this must be so, as was explained by Pauli in the early 1930ies already: If time was an observable it should be described by a self-adjoint operator in Hilbert space and, due to the definition of observables through their association with space-time symmetries a la Noether (the mathematical satisfactory form of the "correspondence principle" as a way to heuristically "derive" a quantum model from a classical model), the canonically conjugate momentum to time would have to be the Hamiltonian (representing energy). But then you'd have a commutation relation as for position and momentum, implying that necessarily the entire real axis is the energy spectrum, which in conclusion what not be bounded from below.
This contradicts the most evident empirical finding for the correctness of quantum theory: that matter is pretty stable. At least it doesn't "decay" into lower and lower energy states, which is because in Nature obviously there is a state of lowest energy (called "the vacuum" in quantum field theory). So the idea that time may be an observable leads to a contradiction with basic empirical evidence about the stability of the matter around us, and thus time is a parameter in quantum mechanics as it is one in classical physics.
dextercioby
Homework Helper
vanhees71, your argument in the first paragraph is mathematically flawed. It's not necessary for both operators to have unbounded spectra.
[Mentor's note: some follow-on discussion moved to https://www.physicsforums.com/threads/follow-on-to-is-the-time-derivative-hermitian.792270/] [Broken]
Let's turn to the OP. The differentiation with respect to time is not a true operator and if it was, then going from Psi(t) to Psi'(t) is equal to H applied to Psi(t), for all Psi (t) in the Hamiltonian's domain.
Last edited by a moderator:
Thank you for the explanations, and while your argument is physically intuitive I still don't understand why the time-derivative should be treated as anything other than an operator. From what I understand, mathematically a linear operator is a device that associates a new vector with every vector in the Hilbert space with a linear correspondence. The derivative has the linearity properties associated with it, and it assigns a new vector to every vector in the Hilbert space as far as I can tell. So while I know there is no time operator, I still don't understand why the time-derivative doesn't satisfy the properties of a linear operator.
Nevertheless, this is all a bit more abstract than I need. If $\partial_t$ is not an operator in quantum mechanics then I don't understand the meaning of the following operation: $\left[ \partial_t |\psi\rangle \right]^\dagger$. The time-derivative does not appear to be a scalar, ket, or bra. If it's not an operator, then what is it exactly? These are more or less the only quantum mechanical mathematical entities I know.
More specifically, I know that the Hermitian conjugate of the Schrodinger equation is
$$\left[ \partial_t |\psi\rangle = \frac{1}{i \hbar} H(t) |\psi\rangle\right]^\dagger \rightarrow \partial_t \langle \psi | = - \frac{1}{i \hbar} H(t) \langle \psi |.$$
Why does the time derivative in the conjugated Schrodinger equation remain the simple time derivative?
dextercioby
Homework Helper
I'm afraid the use of the bra-ket formalism is not clearing up things, instead this use does what it usually does: hides some delicate mathematical issues.
Let's get it from the beginning. One of the key (not explicit) assumptions of the so-called 'orthodox' formulation of QM (due largely to Dirac and von Neumann through their 1930/1932 books) is that 'time is a parameter', not an observable. This folds nicely with classical dynamics in the Hamiltonian formulation: there, too the 't' is nothing but a parameter, not a classical observable (the phase space will not include it, instead we're interested in further parametrizing a classical state (point in the phase space) using 'time', time is then the intrinsic parameter of curves in phase space: evolution parameter of classical Hamiltonian states). So the nice Schrödinger building of a Hilbert space (ignore for now rigged Hilbert spaces or the notions of pure versus mixed states) will take 't' as a parameter (think actually of the Heisenberg picture, where the Hilbert space is <still>/<frozen>, while the observables such as momentum & position depend on time) in the following sense: At each moment in time, ## \psi (x)## is a normalized vector in a Hilbert space which is then (by the uniqueness theorem of von Neumann) safely to be taken as ##\mathcal{L}^2 \left(\mathbb{R},dx\right)## (assuming infinite motion).
So, in what sense do we interpret the LHS of the Schrödinger equation: ##\frac{\partial \psi (x,t)}{\partial t} ##? Well, in the very old way of treating partial derivatives of independent variables in an ordinary function of 2 (3,4, etc.) variables, i.e. using limits. The Schrödinger eqn. asserts that provided that
## \lim_{\delta t \rightarrow 0} \frac{1}{\delta t} \left[\psi (x, t+\delta t) - \psi (x, t)\right] ## exists, then it is equal to the vector in the codomain of the Hamiltonian operator at the same moment of time.
One can surely see the difference of 2 vectors taken before the limit as another vector whose norm then tends to 0 so that the limit is still finite (by postulation) and the result can even be normalized to unity. But one must understand that this limiting process is extraneous to the standard Hilbert space per se, i.e. this limit is not a weak, nor a strong one. Just as in classical dynamics, time acts as a parameter for curves of normalized unit vectors in the (complex, inf-dim., separable) Hilbert space. These curves are assumed to be at least of class $C^1$ in the parameter (differentiable of 1st order and the 1st derivative to be continuous).
As I wrote above, if you're willing to take the process of 1-time differentiation of a Hilbert space vector-valued function as a linear operator in the abstract sense, then, by virtue of the Schrödinger's equation, you can always compute the range of this 'linear operator': it's the vector you're getting by applying the Hamiltonian onto it.
So (as people call it nowadays here) FAPP, this linear operator is the Hamiltonian, a genuine operator, in the sense it's a function of the usual fundamental quantum observables: position, momentum, spin. You can even consider that this linear operator which takes a vector-valued parametric function to its derivative (also vector-valued) to have a hermitian adjoint in the regular sense. This adjoint is of course the adjoint of the Hamiltonian, which is of course the Hamiltonian, because as, again by postulating, the Hamiltonian is self-adjoint. The formal manipulation in the bra-ket formulation trivially follows.
Last edited:
strangerep
(I think some of the answers here might be too advanced for the OP, so I offer a simpler approach...)
Nevertheless, this is all a bit more abstract than I need. If $\partial_t$ is not an operator in quantum mechanics then I don't understand the meaning of the following operation: $\left[ \partial_t |\psi\rangle \right]^\dagger$. The time-derivative does not appear to be a scalar, ket, or bra. If it's not an operator, then what is it exactly? These are more or less the only quantum mechanical mathematical entities I know.
It's an operator, (loosely speaking). There are subtleties involved, as others have said, but let's just think of it as an operator that's densely defined (for present purposes, you could think of this as meaning: well-defined almost everywhere, for physically reasonable wave functions).
Next, do you know the correct definition for "adjoint of an operator" in Hilbert space? (Simply applying a dagger or complex conjugate to everything in sight is not the correct answer.) If you're unsure, try this Wikipedia page, in particular the section on "adjoint of densely defined operators". (If you replace the x,y vectors there by wavefunctions ##\phi,\psi##, the notation might become more familiar.)
If you understand that, then: what is the momentum operator in ordinary QM? (I hope you know this -- if not then you really do need a textbook.) Then prove from the above definitions that the momentum operator is self-adjoint. (Hint: use an explicit representation of the inner product as a spatial integral, and use integration by parts).
If you can do that, you're well on your way to understanding the rest. (Hint: think in terms of ##i\partial_t## rather than just ##\partial_t##.) If not, say what you do/don't understand from the above.
I've proved that the momentum operator is self-adjoint by showing $\langle \phi | P | \psi \rangle = \langle \psi | P | \phi \rangle^*$ by projecting everything into position space and integrating, however there was a very nifty integration by parts trick that allowed me to prove that. Since I never integrate over time, I don't really see how to do this for the time-derivative. I can provide more details if necessary.
bhobba
Mentor
Well, as others have pointed out, there are mathematical niceties involved that is only really resolved in so called Rigged Hilbert Spaces but I will leave that issue aside.
You said you never integrate over time - I don't know what you mean by that - its done all the time - pun intended
My answer though is a bit different. Have a look at Ballentine - Quantum Mechanics - A Modern Development chapter 3 where Schroedinger's equation is derived - not postulated - but derived (it assumes the symmetries of Galelaean relativity). Since the RHS is Hermitian - so must the LHS.
Also in that chapter you will encounter the very important Wigners theorem:
http://arxiv.org/abs/0808.0779
Hopefully you will see what happens when you take the derivative of a unitary operator - I think its Stones theorem or something like that - but non rigorously its not hard - for a small parameter t U(t) = 1 + t dU/dt and apply UU(conjugate) = 1.
I just realised - of course you don't need the full machinery of deriving Schroedingers equation - you simply need Wigners theorem - but read chapter 3 anyway.
Thanks
Bill
Last edited:
atyy
strangerep
I've proved that the momentum operator is self-adjoint by showing $\langle \phi | P | \psi \rangle = \langle \psi | P | \phi \rangle^*$ by projecting everything into position space and integrating, however there was a very nifty integration by parts trick that allowed me to prove that. Since I never integrate over time, I don't really see how to do this for the time-derivative.
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Volume 27
, Issue 1
1991
This article is a revised version of a talk given to the Mathematics Club at the Technion, Israel's Technological University and subsequently printed in Etgar-Gilianot Mathematica, the Israeli version of Parabola.
In everyday life we tend to trust the numbers which come out of a computer or calculator.
The development of quantum mechanics earlier this century was a joint effort by a number of physicists, of whom E. Schrödinger and W. Heisenberg figure prominently.
Here is a puzzle to end all puzzles.
Q.817 Find all integers $x,y$ such that $x(3y-5) = y^2+1$.
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## Spring Stretch
Consider a 10 kg object stretches a spring 0.25 m. If the object were swapped out with a larger 15 kg one, how many meters would the spring stretch?
Hint
A spring’s deflection and force are related by:
$$F=kx$$$where $$k$$ is the spring constant and $$x$$ is the deflection. Hint 2 First solve for the spring constant with the lower weight. Then, use the spring constant to determine the deflection via the same equation. A spring’s deflection and force are related by: $$F=kx$$$
where $$k$$ is the spring constant and $$x$$ is the deflection.
Since $$Force=mass \times a$$ , we need to multiply the mass by acceleration due to gravity for both objects:
$$F_1=10kg\times 9.8m/s^2=98\:N$$$$$F_2=15kg\times 9.8m/s^2=147\:N$$$
The spring constant is the same for both scenarios, so let’s calculate $$k$$ for the lower weight because there is only one unknown variable:
$$k=\frac{F_1}{x_1}=\frac{98N}{0.25m}=392\:N/m$$$Next, let’s solve for the spring stretch with the larger weight using the spring constant we just determined: $$x_2=\frac{F_2}{k}=\frac{147N}{392N/m}=0.375\:m$$$
Alternatively, we could have solved this problem more quickly using a proportional relationship since the spring constant and gravity are the same in both scenarios:
$$\frac{m_1}{x_1}=\frac{k}{g}=\frac{m_2}{x_2}$$$$$\frac{10kg}{0.25m}=\frac{15kg}{x_2}$$$
$$x_2=\frac{15kg(0.25m)}{10kg}=\frac{3.75m}{10}=0.375\:m$$\$
0.375 m
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Let $ABCD$ be a quadrilateral. Let $\{F\}=AB\bigcap CD$ and $\{E\}=BC\bigcap AD$. Then $AFCE$ is a complete quadrilateral.
The complete quadrilateral has four sides : $\overline{ABF}$, $\overline{ADE}$, $\overline{BCE}$, $\overline{DCF}$, and six angles: $A$, $B$, $C$, $D$, $E$, $F$.
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How do you write an equation of a line given slope 2 and passes through (0,-4)?
Apr 12, 2017
$y = 2 x - 4$
Explanation:
For this, we use the point-gradient formula:
$y - {y}_{1} = m \left(x - {x}_{1}\right)$
Where (x_1,y_1) is a specific point that lies on the line (in this case, we are given $\left(0 , - 4\right)$
and $m$ is the gradient/slope of the line, in this case $2$
By putting these values in, we get:
$y - \left(- 4\right) = 2 \left(x - 0\right)$
$y + 4 = 2 x$
$y = 2 x - 4$
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# Effective mass approximation
• I
I have noticed that in a lot of theoretical modelling of semiconductors you assume that the electrons living in the bottom of the conduction band obey a free particle Hamiltonian:
H = p^2/2m*
, where m* is the effective mass in the conduction band and p^2 is the usual differential operator. I am not sure how this is derived rigourously. I suppose you solve the band structure and show that as a function of k the band structure is parabolic in k about the minimum of the conduction band:
E ≈ E0 + ħ^2k^2/2m*
But how do you rigorously go from this expression, which contains the wave numbers k = (kx,ky,kz) back to differential operators? I hope you understand my question.
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# Publications
### Additive energies on discrete cubes
We prove that for $d\geq 0$ and $k\geq 2$, for any subset $A$ of a discrete cube ${0,1}^d$, the $k-$higher energy of $A$ (i.e., the …
### On classical inequalities for autocorrelations and autoconvolutions
In this paper we study an autocorrelation inequality proposed by Barnard and Steinerberger. The study of these problems is motivated by …
### Decoupling for fractal subsets of the parabola
We consider decoupling for a fractal subset of the parabola. We reduce studying l2Lp decoupling for a fractal subset on the parabola …
### On Sparsity in Overparametrised Shallow ReLU Networks
The analysis of neural network training beyond their linearization regime remains an outstanding open question, even in the simplest …
### A geometric lemma for complex polynomial curves with applications in Fourier restriction theory
The aim of this paper is to prove a uniform Fourier restriction estimate for certain 2−dimensional surfaces in R2n. These surfaces are …
### Role Detection in Bicycle-Sharing Networks Using Multilayer Stochastic Block Models
Urban spatial networks are complex systems with interdependent roles of neighborhoods and methods of transportation between them. In …
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### phantom11's blog
By phantom11, history, 6 months ago,
## What is this for?
It pains to see so many people struggling due to covid and even dying because of lack of help in our country. I find myself guilty and helpless every night thinking about it, being so far away and not being able to contribute in any way. People need help, they really do. If you are a student and looking for job opportunities, I can hear you out, provide some counselling and give you tips that might be useful in your career.
## Who am I?
My name is Lokesh Khandelwal and I am a software engineer at Facebook. I was born and brought up in a small town, and did my undergrad in not so reputed university. Therefore, I can understand your situation if you come from a similar background. I have cracked interviews of famous companies like Amazon, Facebook, Google and have also been an interviewee myself. I have also gone offbeat and completed my Masters in Astronomy. I am a good listener and have traveled and lived in multiple countries, so I am aware of different and diverse cultures and work environments.
## What can you ask?
Anything. It could be as simple as helping you decide your future goals or choosing career paths. If you are preparing for an interview, I could give you tips or could even do a mock round for you. Or if you want to know about what is it like to work in these big tech companies, I can reveal some inside setups, team structures, compensations to expect etc. If you are stuck in some problem, I can help solve that, especially if it is related to mathematics, physics or computer science. If you are thinking to moving to a place and wondering what is life like in Tokyo, or Singapore or Amsterdam or London, I can help you there too. In short, I will give you my 30 minutes, and you can Ask Me Anything.
## What will I get in return?
In exchange for my time and service, all I am asking from you is to help with Rs.1000 or its equivalent to any LOCAL charity or club or organisation in your city. Local is important here, because looking at the situation, it feels like the help does not reach at the root level in proper and timely manner. There are many people around you who need help. If you are medically aware and equipped, you can help / donate oxygen cylinders or concentrators or medicines. People have lost jobs in covid and cannot get treatments or feed themselves, you could help them financially. Or you could give away that old gaming laptop / mobile phone to a child who cannot attend school anymore because of digital education. Look around yourself, you will surely find someone who needs help in your neighbourhood itself.
## How does it work?
Once you have helped someone, book a time slot in my calendar here [ https://doodle.com/mm/lokeshkhandelwal/ask-me-anything ]. Send me a personal message with the proof of your help (receipt of local charity / picture of you helping someone needy) and I will then give you a zoom link and password to connect with me. In your message also mention what you are looking out for from me, so that I am prepared. The world needs us right now, and it is time we step up, help each other. Looking forward to meet you guys.
Read more »
• +228
By phantom11, 16 months ago,
My roommates and me are planning a house party. Because of covid, the Dutch rules says that we can have a party as long as social distancing is maintained — each individual has to be at least 1.5m away from each other. This brings me to a cool real life problem — what is the minimum area required to fit N people such that each one maintains social distancing with everyone else. N > 2, since we are already 3 roommates :)
My guessed solution is to make everyone stand on the vertices of equilateral triangle, and then join these triangles to form hexagonal lattice (just like a cellular network). Hence the answer would be (N — 2) * area of equilateral triangle of side 1.5m. Is this correct?
If yes, a follow up question would be : What if I wanted to minimise the distance between the farthest people as well? Would I try to fit these triangles inside a square?
Will be nice to hear out solutions to both these problems, for any number of people (N).
Read more »
• +83
By phantom11, history, 20 months ago,
Most probably, smart people here might have already thought about this, but still..
During the system test phase of Codecraft' 20, I observed many submissions failing around the 100th test case or so. To me, this seems like waste of processing power, to let the codes run till 100 submission and then figure out its failing.
Isn't it possible to sort the test cases by: - decreasing order of number of non-ac submissions - in case of tie, increasing order of average time spent in judging it.
There could be an evaluator thread (for example), which does this after every 100 submissions or so (or something better)?
Is this something too difficult to achieve with the current codeforces infrastructure? One possible difficulty I can imagine is the scenario to show test cases to users. Are there more issues?
Read more »
• +238
By phantom11, 7 years ago,
Hello All,
I invite you to participate in monthly medium contest of HackerEarth — November Rain (this time). There will be 5 problems in total and you will have 2.5 hours to solve them. The problems are comparable to Div-2 difficulty, though you won't find any cake walk problem. All you need is a HackerEarth account to participate in the contest.
Date & Time
Contest Link
Prizes — Top 3 winners to get HackerEarth T-shirts!
The problems have been prepared by me, and I thank the testers and editorialist for their contribution.
Problem Testers — Bidhan(4 problems), akashdeep(1 problem).
Problem Editorialist — ashish1610
Read more »
• +5
By phantom11, 7 years ago,
There are so many blogs coming up on codeforces everyday and we sometimes loose the blog which we are following which have got new comments. One of my friends, xorfire wanted something like this in his recent blog yesterday, in which I have also commented my idea, and so I decided to write a script to automate the procedure. Firstly thanks to xorfire for coming up with this idea.
You can download the zipped folder from here. Extract it to some folder name of your choice. Inside it you will find many files which are actually dependencies, but you need to open just notification.exe file. I will describe the the functionality now.
• Username / Password fields -> Update them once in the beginning or whenever you want to change your username password. You do not need to write it every time.
• Add Favorite Blog -> Add the complete URL of the blog you want to follow. Example -> "http://codeforces.com/blog/entry/12727" (quotes for clarity). You can add as many as you want one by one.
• Manage stored blogs -> If you want to delete some blogs, click this button and it will show your complete list in a new window. There will be delete option beneath every URL. You can delete any number of them, and then close the popped up window.
• Run Notifier -> This is the main thing. It will search all your favorite blogs and notify you for those which have new comments. It takes around 10 second for each blog (depends on your internet) . So run it and forget it for a minute (I assume you will not have more than 6-7 fav blogs, otherwise it will take more time). A new window will open where you will find the blogs which have new comments. These are click-able links and on clicking will redirect you to the concerned blogs.
I have made this in 8-9 hours and have given absolutely no time to beautify it. Also it was my first time to use GUI in python. So you may find it ugly, but it works well and that is what is important. Also I have made it for personal use of everyone. So don't try to be smart and do unnecessary things to hack it like writing some blogs which do not exist at all (It might fail & I don't mind it). I made it for the welfare and personal use of the community and also in quick time, so use it in the right spirit. There are no security measures , (your username and password are stored in a file in the same directory) , so do not use it in public systems. I have tested in on Windows 7 and Windows XP and it works fine on both.
I hope you all will like it!!!
Read more »
• +14
By phantom11, 7 years ago,
UPDATE : Version 2.0
Rewrote whole code (frontend + parser) from scratch. Here are the changes made(in order of importance in my view) :
- Fixed bugs in parser.
- Div 2 Rounds Unofficial find their place in the standings. (this and this. You can now see Div 2 standings of ONLY Officials, ONLY UnOfficals, as well as both.
- Added some internal securities
- Now have a about us page too. :)
- Much neater and a modular code.
Very often we are interested in knowing our position in our country for a contest. Unfortunately, Codeforces does not provide this feature. An ugly way to do it is to make friends with all your country people. But ofcourse that is ugly, and programmers are supposed to be smart.
So I decided to write a scraper and do it on my own. When I graduated this year, and had some time in my pocket, franky and I made this feature. Now that the script is ready, you can find all the contest's country wise standings here.
EDIT: I forgot to mention that those who have not filled in their countries here on codeforces, come under "Alien" in the country-wise standings. I request these people to fill in their correct nationality.
Read more »
• +160
By phantom11, 8 years ago,
I saw no blog on TCO Round 1A in codeforces.
So here are the information :
EDIT : UPDATED INFORMATION
• Start time of the contest
• Registrations begins 3 hours prior to start time and closes 5 minutes before.
• Top 750 from this round will advance to round 2. There will 2 more rounds like this 1B and 1C.
• Registrations are limited to first 2500 participants only.
Lets discuss problems here once the contest is completely finished.
UPDATE — The editorial of this round has been released. link
Read more »
• +12
By phantom11, 9 years ago,
TOP CAREER Virtual Study Abroad Fair -2013 Spring-” From Jan.23-Mar.31!
http://vf.topcareer.jp/?referrer=2712
Register and Participate from the URL above and win a great prize such as flight tickets to/from Japan, hotel coupons, online shopping cards, etc.!
You can also win an opportunity of internship offer from Fourth Valley Concierge Corporation, after you pass the screening process.
*Prize winners are chosen randomly from registered participants.
Participate in an online fair where you can find information about scholarship, admission, and etc.), virtually visit schools all around the world, and directly chat with schools’ representatives without actually flying all the way.
Join anytime (24hrs/day!), anywhere!
[Fair Information]
Name: TOP CAREER Virtual Study Abroad Fair -2013 spring-
Date: January 23, 2013 (Wed) – March 31, 2013 (Fri)
Read more »
• +6
By phantom11, 9 years ago,
Firstly a very happy new year to all coders here :) Hope this year brings lots of Accepted to you.
I have written this blog to help people who ask questions regarding how to change the handle. If you decide to change the handle, (and you thought twice about it, because it is irreversible and can be done only once) , this blog might help you.
This is a new year gift from codeforces and only valid for the first ten days.
Follow the mouse pointer and you are done!!!
Read more »
• +13
By phantom11, 9 years ago,
This is to remind you that the USACO December 2012 Contest is going to take place from tomorrow.The duration of the contest is 4 days.You can appear in the maximum of any of the 4 hour window during the contest.
Link to Contest Page .(to be updated before the contest starts )
You can use this blog space to discuss problems NOT during the contest but ONLY after the contest is complete.
UPD -> [Results]
Read more »
• +27
By phantom11, 9 years ago,
I am going to participate in the IIT, Kharagpur regional finals next week. Due to my semester exams , I could not devote much time to programming in the last 2 weeks. So I seek your help and advice.
How to go for it from now. Should I keep solving problems like I did for the whole year , or learn some new algorithms and practice them , or sit for 5 hours in an ICPC like environment and take practice tests ?? I personally want to learn new algorithms but I dont know if I am too late now.Your take ??
Also I have to make a 25 side reference material. I dont want to waste my time building it from scratch .So if any one has it ready , please share it with me .I promise not to duplicate it (in case you dont want it too , you can send me a message here .)I will make suitable edits to make it good for myself so it not wastes time.
Read more »
• +3
By phantom11, 9 years ago,
I was trying to solve the Problem H of the last contest.[problem link]
But I keep on getting Memory Limit Exceeded.Please some one tell me why this is happening .I am bringing here 3 codes. One from which I have learnt which gets accepted and the other two are mine. Please tell me why I get MLE in both cases and why 'B' got passed in test case 5 by making small change in 'A' (only change is made during the last for loop where printing is done) .
mfv code -> [link] accepted
case A ->[Link] fails on test case 5
case B ->[Link] fails on test 15
Please clear my concepts so that I dont face these issues in future.Thank You
Read more »
• -9
By phantom11, 9 years ago,
Topcoder SRM 559 is going to be held on Tuesday , 30th October 2012 at 21:00 EDT (your timezone)
The registration starts 3 hours prior to the match in the arena and is restricted to the first 2500 contestants .
You can discuss about the SRM here once it is over.(and also on the topcoder forum here).
Also , there is going to be a test round 2 from 2 hours from now in the arena (your timezone)
Read more »
• +25
By phantom11, 9 years ago,
Hi friends, During my practice I face this problem regularly. Some of the SRM's (and I practice on old SRM's) does not open for practice .It keeps on loading and loading , until I have to close it manually. For eg. SRM 355Div2,SRM 356Div 2 although SRM 354Div 2 opens up quickly. Is this problem that only I face?? How to get through it ?? Any other alternatives ??
This has also happened a lot in previous SRM's . But today I got really frustated and thought of asking help from you
Read more »
• -3
By phantom11, 9 years ago,
Here is the topcoder account of the founder and CEO of Facebook.[Link] .(courtesy red_coder ) .So all those rated above him, you cannot imagine how much potential you have . So never lose hope :)
Read more »
• +34
By phantom11, 9 years ago,
From Two days I am facing a lot of difficulties..Please help me out..
• As usual I was training myself @topcoder problems but I was not able to enter the practice rooms although I was able to login into the chat arena.Is anyone facing the same problem/What is the solution then??.I posted in the topcoder forums but got no replies[link]..So I thought of posting it here in codeforces which I feel has more active community.
• I was solving the USACO problem Mother's Milk[link].I solved it but I am getting TLE in the test case 20 10 9 which passes in my system in 2ms .I checked the number of times the recursive function is executed and it is only 24 for that case..Link to code..Please tell me why this is happeneing??
UPD: I had also reported the problem to the USACO team and after a few e-mail exchanges they came to the conclusion that the java grader was broken.They have fixed it now.And the above solution passes now :)
UPD2: Contacted service@topcoder.com and mystic_tc fixed the issue.Thanks to him :)
Read more »
• +6
By phantom11, 9 years ago,
After this round my rating increased and I was back to blue with a rating of 1568 I think.. But in my profile it is showing the old rating of 1470 although they made me blue .. Is this only with me .??Or is this a bug in the system?? You can see the screenshot here
EDIT : I think it is with all ..They have shifted back to the old ratings for unknown reasons but forgot to change the color back..First time with less than 1500 and blue ;)
Read more »
• -6
By phantom11, 9 years ago,
I am organising a multiplayer event where each player has a rating (new player is assigned a rating of 1500) ..I want to have a good formula for calculating the new ratings after a competition..I searched the net and found the Elo rating system.This is what I have worked out thus far from the internet resources..:
From The ELo formula let P(B,A[i]) refer to the probablity that B defeats A[i] for all i.
Then total[B]=summation of all P(B,A[i]) for all i's
Let n=number of participants..
Then Expected Rank[B]=n-total[B]
new rating=old rating+ k* (ExpectedRank[B] - ActualRank[B])
Now the issues:
1)I want to know what value of k should be employed .
2)I want to have a regualrity factor also so that a player who is consistently performing well and if some day performs badly his rating does not go too down .ie.ratings dont change drastically on good /bad performance in one match..
3)If this is not the best method to calclate ratigs in a multiplayer competition then which one should be used??Please give details , formulas also
Please help me out .Statistics is not my subject yet I need to get this thing done to complete my project
Read more »
• -24
By phantom11, 9 years ago,
The TopCoder Open 2013 registrations have begun.. The final will be held in Washington DC next year. You can register yourselves here.The logo and website designing competitions is going to start soon..
Also I take this blog to remind you of the SRM 548 whose registeration closes 15 mins from now..So those who have not registered please do it ASAP.
Read more »
• -15
By phantom11, 9 years ago,
Wishing Sir Alan Turing , the father of Computer Science and Artificial Intelligence a very happy birthday... And what a doodle in this warm occassion ..
Read more »
• +1
By phantom11, 9 years ago,
Got a very inspirational quote today which I wish to share with all my fellow coders:
"**Darkness is just the absence of light , it is not that light does not exist.**
Similarly a problem is just an absence of an idea , it is not that its solution does not exist."
Read more »
• -20
By phantom11, 9 years ago,
Hi!!I am not able to go to the handle option in the settings tab...I want to check that out Just for curiosity ,but its blocked by the system admin since a week or so...Or perhaps they have forgotten to unblock it....Anyone else facing the same problem??
Read more »
• -4
By phantom11, 10 years ago,
I was the lucky one to win the T-shirt of the lucky draw of SRM-528.I contacted topcoder services team and they told me that they had released it on 6th Jan 2012 and that the t-shirt would come to me within 10 weeks.But I have not received it till now...Could anyone tell me as to :
1) how will it reach me??(eg- registered post,courier ,or some contracted person etc.)
2)How do I claim for it??(For eg it comes through post then I have to show some docket number or some other document to claim it directly from the post office )Then how do I claim it??
I am an Indian Citizen...Apart from this ,I am also due for the NASA USPTO challenge T-shirt and the Accenture Mobile Survey T-shirt from topcoder...I wrote this in the topcoder forum but got no reply and I dont want to lose my special gifts ...Someone Please help???
Read more »
• -18
By phantom11, 10 years ago,
I submitted this code and the verdict was TLE. I just removed my printing statement which ran O(len(a)) times and it was accepted .Code . Strange behaviour and very strict time constarints. Could anyone explain why such thing happened...
Read more »
• -8
By phantom11, 10 years ago,
This is a gentle reminder to all coders that SRM 537 is going to happen on the coming Saturday i.e. 17th March.This round is sponsered by CITI ,so there is a total purse of 5000 USD.So gear up and get ready for the challenge.You can find the round details and timings here. You can get the exact prize divisions and a bit of discussion on the SRM on vexorian blog here.
A couple of SRM's back ,the registeration limit of 2500 members was reached some 5 minutes before the closing time.So dont be late or else you may miss a chance to grab some money. Good Luck and Best wishes to all :-)
Read more »
• +12
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+0
# Helllllppppp !! :(((
0
148
9
A baseball is hit so that it travels straight upward after being struck by the bat. If the ball takes 4.5 s to reach its maximum height, then the ball's initial velocity is
Sep 22, 2020
#1
+10840
+1
A baseball is hit so that it travels straight upward after being struck by the bat. If the ball takes 4.5 s to reach its maximum height, then the ball's initial velocity is
Hello Guest!
$$h=v_0\cdot t-\frac{1}{2}gt^2\\ v_0\cdot t=\frac{1}{2}gt^2\\ v_0= \frac{1}{2}gt\\ v_0=\frac{1}{2}\cdot9.81\frac{m}{sec^2}\cdot4.5sec$$ That is the wrong approach. sorry.
$$v_0=22.0725m/sec$$
!
Sep 22, 2020
edited by asinus Sep 22, 2020
#2
+2148
0
Asinus, this is NOT the correct equation for this question.
--. .-
GingerAle Sep 22, 2020
#3
+28026
+1
v = vo - 1/2 at v = 0 a = 9.8 m/s2 t = 4.5 s
0 = v - 1/2 (9.8)(4.5)
vo = 1/2 (9.8)(4.5) = 22.05 m/s (Just like asinus found !)
Sep 22, 2020
edited by ElectricPavlov Sep 22, 2020
#5
+2148
0
vo = 1/2 (9.8)(4.5) = 22.05 m/s (Just like asinus found !)
Yes, and this is demonstrable evidence that dumbness is contagious.
This equation will work if the total time of flight is used. 4.5 seconds is half (½) of the total flight time.
Or take the direct approach and use the equation Alan presented, below ...
--. .-
GingerAle Sep 22, 2020
#6
+28026
+1
Agree with Alan..... I made the same mistake Asinus did
v = v- at where v = 0 v0 = initial velocity a = 9.81 m/s^2 t = 4/5 sec (there is no 1/2 in the equation!!)
Oops ! THANX , Alan !
ElectricPavlov Sep 22, 2020
#7
+28026
+1
I think it is more an example of the power of suggestion .....
I do not think either answer is 'dumb'......just incorrect because of a mistake....
ElectricPavlov Sep 22, 2020
#8
+2148
0
I think it is more an example of the power of suggestion ....
Ok... But why didn’t the suggestion of “...this is NOT the correct equation for this question,” have any power?
...Id est, why did you believe that re-solving the equation would confirm its validity?
I do not think either answer is 'dumb'......just incorrect because of a mistake...
I do not think the answers are dumb either, but I do think ...dumbness is contagious.
--. .-
GingerAle Sep 22, 2020
#4
+31530
+5
The correct equation is $$v = v_0 + at$$
Here $$v = 0m/s; a = -9.81m/s^2;t=4.5s$$
Sep 22, 2020
#9
+10840
+1
Hello ElectricPavlov, hello Alan!
Thanks for the correction.
I'm trying differential calculus. Is that OK?
$$\color{blue}h(v)=vt-\frac{v^2}{2g}\\ \frac{dh(v)}{dv}=t-\frac{v_0}{g}=0\\ v_0=gt=\frac{9.81m\cdot4.5s}{s^2}$$
$$v_0=44.145m/s$$
$$h_{max}=v_0\cdot t-\frac{v_0^2}{2g}\\ h_{max}=44.145m/s\cdot4.5s-\frac{44.145^2}{2\cdot 9.81}m$$
$$h_{max}=99.33m$$
!
Sep 22, 2020
edited by asinus Sep 22, 2020
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• The Inria's Research Teams produce an annual Activity Report presenting their activities and their results of the year. These reports include the team members, the scientific program, the software developed by the team and the new results of the year. The report also describes the grants, contracts and the activities of dissemination and teaching. Finally, the report gives the list of publications of the year.
• Legal notice
• Personal data
## Section: New Results
### Stabilization of MIMO fractional systems with delays
Participants : Catherine Bonnet, Le Ha Vy Nguyen, Alban Quadrat.
In order to yield the set of all stabilizing controllers of a large class of MIMO fractional time-delay systems, we may look for coprime factorizations of the transfer function and their corresponding Bézout factors. As primary results, in considering ${H}_{\infty }$ stability, left coprime factorizations and left Bézout factors have been determined analytically from the transfer function. Then a particular stabilizing controller has been derived. We also proved the existence of right coprime factorizations.
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# 004 Sample Final A, Problem 2
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search
a) Find the vertex, standard graphing form, and x-intercepts for ${\displaystyle f(x)={\frac {1}{3}}x^{2}+2x-3}$
b) Sketch the graph. Provide the y-intercept.
Foundations
1) What is the standard graphing form of a parabola?
2) What is the vertex of a parabola?
3) What is the ${\displaystyle y}$-intercept?
Answer:
1) Standard graphing form is ${\displaystyle y-h=a(x-k)^{2}}$.
2) Using the standard graphing form, the vertex is ${\displaystyle (h,k)}$.
3) The ${\displaystyle y}$-intercept is the point ${\displaystyle (0,y)}$ where ${\displaystyle f(0)=y}$.
Solution:
Step 1:
First, we put the equation into standard graphing form. Multiplying the equation ${\displaystyle y={\frac {1}{3}}x^{2}+2x-3}$ by 3, we get
${\displaystyle 3y=x^{2}+6x-9}$.
Step 2:
Completing the square, we get ${\displaystyle 3y=(x+3)^{2}-18}$. Dividing by 3 and subtracting 6 on both sides, we have
${\displaystyle y+6={\frac {1}{3}}(x+3)^{2}}$.
Step 3:
From standard graphing form, we see that the vertex is (-3,-6). Also, to find the ${\displaystyle x}$ intercept, we let ${\displaystyle y=0}$. So,
${\displaystyle 18=(x+3)^{2}}$. Solving, we get ${\displaystyle x=-3\pm 3{\sqrt {2}}}$.
Thus, the two ${\displaystyle x}$ intercepts occur at ${\displaystyle (-3+3{\sqrt {2}},0)}$ and ${\displaystyle (-3-3{\sqrt {2}},0)}$.
Step 4:
To find the ${\displaystyle y}$ intercept, we let ${\displaystyle x=0}$. So, we get ${\displaystyle y=-3}$.
Thus, the ${\displaystyle y}$ intercept is (0,-3).
Final Answer:
The vertex is (-3,-6). The equation in standard graphing form is ${\displaystyle y+6={\frac {1}{3}}(x+3)^{2}}$.
The two ${\displaystyle x}$ intercepts are ${\displaystyle (-3+3{\sqrt {2}},0)}$ and ${\displaystyle (-3-3{\sqrt {2}},0)}$.
The ${\displaystyle y}$ intercept is (0,-3)
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## Basic College Mathematics (9th Edition)
Published by Pearson
# Chapter 2 - Multiplying and Dividing Fractions - Test: 25
#### Answer
estimated 36 grams exact : 30$\frac{5}{8}$ grams
#### Work Step by Step
Step 1: Read the problem. The problem asks for the grams that can be synthesized in 12$\frac{1}{4}$ days Step 2 Work out a plan. Multiply the 2$\frac{1}{2}$ grams per day and grams can be synthesized in 12$\frac{1}{4}$ days. Step 3 Estimate a reasonable answer. Round 2$\frac{1}{2}$ grams to 3 grams and 12$\frac{1}{4}$ days to 12 days Multiply 3 grams by 12 days = 36 grams. Step 4 Solve the problem. Find the exact answer 2$\frac{1}{2}$ * 12$\frac{1}{4}$ Convert mixed fractions into improper fractions $\frac{5}{2}$ * $\frac{49}{4}$ = $\frac{245}{8}$ = 30$\frac{5}{8}$ grams Step 5 State the answer. 30$\frac{5}{8}$ grams can be synthesized in 12$\frac{1}{4}$ days Step 6 Check. The exact answer, 30$\frac{5}{8}$ grams , is close to our estimate of 36 grams.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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# Electronic – How to use a transformer as an inductor
ideasinductanceinductortransformer
Lp: Self inductance of the primary winding.
Ls: Self inductance of the secondary winding.
Lm: Mutual inductance between the primary and secondary windings.
Assume that I need an iron core inductor with large inductance to use under 50Hz or 60Hz.
How do I obtain an inductor from the given transformer in the image? I don't want to use any other circuit elements unless it is absolutely required. The dot convention of the transformer is given in the image; terminal connections must be done so that the inductance of the resulting inductor must be maximum (I think that happens when the fluxes generated by the primary and secondary windings happen to be in the same direction inside the transformer core).
I'm expecting an answer like "Connect \$P_2 \$ and \$S_2 \$ to together, \$P_1 \$ will be \$L_1 \$ and \$S_1 \$ will be \$L_2 \$ of the resulting inductor.".
I understand that I can use the primary and secondary windings separately by making the unused winding open, but I'm looking for a smart way of connecting the windings so that the resulting inductance will maximize.
What will be the inductance of the inducter in terms of \$L_p \$, \$L_s \$ and \$L_m \$ ?
What will be the frequency behavior of the resulting inductor? Will it have a good performance at frequencies other than the original transformer was rated to run in.
How do I obtain an inductor from the given transformer in the image? ... So that the inductance of the resulting inductor must be maximum.
• Connect the undotted end of one winding to the dotted end of the other.
eg P2 to S1 (or P1 to S2) and use the pair as if they were a single winding.
(As per example in diagram below)
• Using just one winding does NOT produce the required maximum inductance result.
• The resulting inductance is greater than the sum of the two individual inductances.
Call the resultant inductance Lt,
• Lt > Lp
• Lt > Ls
• Lt > (Lp + Ls) !!! <- this may not be intuitive
• \$L_t = ( \sqrt{L_p} + \sqrt{L_s}) ^ 2 \$ <- also unlikely to be intuitive.
• \$\dots = L_p + L_s + 2 \times \sqrt{L_p} \times \sqrt{L_s} \$
Note that IF the windings were NOT magnetically linked (eg were on two separate cores) then the two inductances simply add and Lsepsum = Ls + Lp.
What will be the frequency behavior of the resulting inductor? Will it have a good performance at frequencies other than the original transformer was rated to run in.
"Frequency behavior" of the final inductor is not a meaningful term without further explanation of what is meant by the question and depends on how the inductor is to be used.
Note that "frequency behavior" is a good term as it can mean more than the normal term "frequency response" in this case.
For example, applying mains voltage to a primary and secondary in series, where the primary is rated for mains voltage use in normal operation will have various implications depending on how the inductor is to be used.Impedance is higher so magnetising current is lower so core is less heavily saturated. Implications then depend on application - so interesting. Will need discussing.
Connecting the two windings together so that their magnetic fields support each other will give you the maximum inductance.
When this is done
• the field from current in winding P will now also affect winding S
• and the field in winding S will now also affect winding P
so the resultant inductance will be greater than the linear sum of the two inductances.
The requirement to get the inductances to add where there 2 or more windings is that the current flows into (or out of) all dotted winding ends at the same time.
• \$L_{effective} = L_{eff} = (\sqrt{L_p} + \sqrt{L_s})^2 \dots (1) \$
Because:
Where windings are mutually coupled on the same magnetic core so that all turns in either winding are linked by the same magnetic flux then when the windings are connected together they act like a single winding whose number of turns = the sum of the turns in the two windings.
ie \$N_{total} = N_t = N_p + N_s \dots (2) \$
Now: L is proportional to turns^2 = \$N^2 \$
So for constant of proportionality k,
\$L = k.N^2 \dots (3) \$
So \$N = \sqrt{\frac{L}{k}} \dots (4) \$
k can be set to 1 for this purpose as we have no exact values for L.
So
From (2) above: \$N_{total} = N_t = (N_p + N_s) \$
But : \$N_p = \sqrt{k.L_p} = \sqrt{Lp} \dots (5) \$
And : \$N_s = \sqrt{k.L_s} = \sqrt{L_s} \dots (6) \$
But \$L_t = (k.N_p + k.N_s)^2 = (N_p + N_s)^2 \dots (7) \$
So
\$\mathbf{L_t = (\sqrt{L_p} + \sqrt{L_s})^2} \dots (8) \$
Which expands to: \$L_t = L_p + L_s + 2 \times \sqrt{L_p} \times \sqrt{L_s} \$
In words:
The inductance of the two windings in series is the square of the sum of the square roots of their individual inductances.
Lm is not relevant to this calculation as a separate value - it is part of the above workings and is the effective gain from crosslinking the two magnetic fields.
[[Unlike Ghost Busters - In this case you are allowed to cross the beams.]].
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# Cube in a sphere
The cube is inscribed in a sphere with volume 9067 cm3. Determine the length of the edges of a cube.
Result
a = 14.9 cm
#### Solution:
$V = 9067 \ cm^3 \ \\ V = \dfrac{ 4 }{ 3 } \pi r^3 \ \\ \ \\ r = \sqrt[3]{ \dfrac{ 3 \cdot \ V }{ 4 \pi } } = \sqrt[3]{ \dfrac{ 3 \cdot \ 9067 }{ 4 \cdot \ 3.1416 } } \doteq 12.9358 \ cm \ \\ \ \\ D = 2 \cdot \ r = 2 \cdot \ 12.9358 \doteq 25.8715 \ cm \ \\ \ \\ u = D = 25.8715 \doteq 25.8715 \ cm \ \\ \ \\ u = \sqrt{ 3 } a \ \\ \ \\ a = u/\sqrt{ 3 } = 25.8715/\sqrt{ 3 } \doteq 14.9369 = 14.9 \ \text { cm }$
Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):
Showing 1 comment:
Math student
i am good at this
1 year ago 1 Like
#### Following knowledge from mathematics are needed to solve this word math problem:
Need help calculate sum, simplify or multiply fractions? Try our fraction calculator. Do you want to convert length units? Tip: Our volume units converter will help you with the conversion of volume units. Pythagorean theorem is the base for the right triangle calculator.
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Two boxes cube with edges a=38 cm and b = 81 cm is to be replaced by one cube-shaped box (same overall volume). How long will be its edge?
8. Cube corners
From cube of edge 14 cm cut off all vertices so that each cutting plane intersects the edges 1 cm from the nearest vertice. How many edges will have this body?
9. The volume 2
The volume of a cube is 27 cubic meters. Find the height of the cube.
10. Root
Use law of square roots roots: ?
11. Gasholder
The gasholder has spherical shape with a diameter 20 m. How many m3 can hold in?
12. Spherical segment
Spherical segment with height h=7 has a volume V=198. Calculate the radius of the sphere of which is cut this segment.
13. Tetrahedron
Calculate height and volume of a regular tetrahedron whose edge has a length 18 cm.
14. Gasoline tank cylindrical
What is the inner diameter of the tank, which is 8 m long and contains 40 cubic cubic meters of gasoline?
15. Holidays - on pool
Children's tickets to the swimming pool stands x € for an adult is € 2 more expensive. There was m children in the swimming pool and adults three times less. How many euros make treasurer for pool entry?
16. Calculation
How much is sum of square root of six and the square root of 225?
17. Theorem prove
We want to prove the sentence: If the natural number n is divisible by six, then n is divisible by three. From what assumption we started?
|
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# Math Help - Integrating IVP
1. ## Integrating IVP
dx/dt = (x^3).(e^t)
intial condition : x(1) = -4
this looks v.simple but trust me, its not!
help me out! aaaah...
thank you
2. Originally Posted by matlabnoob
dx/dt = (x^3).(e^t)
intial condition : x(1) = -4
this looks v.simple but trust me, its not!
help me out! aaaah...
thank you
$\frac{dx}{dt}=x^3e^t$
$\frac{dx}{x^3}=e^tdt$
Now integrate both sides
$-\frac{1}{2x^2}=e^t+c$
can you handle it from here by plugging in your intial value to find c
3. Originally Posted by artvandalay11
$\frac{dx}{dt}=x^3e^t$
$\frac{dx}{x^3}=e^tdt$
Now integrate both sides
$-\frac{1}{2x^2}=e^t+c$
can you handle it from here by plugging in your intial value to find c
thank you!
i have gotten that far and then i ended up with =S...
after i substituted 1 into t..
x^2 = -1/2e+2c
?
that looks v.v.wrong to me. or is it just me =S
4. Originally Posted by matlabnoob
thank you!
i have gotten that far and then i ended up with =S...
after i substituted 1 into t..
x^2 = -1/2e+2c
?
that looks v.v.wrong to me. or is it just me =S
$x(1)=-4$ so when t=1, x=-4
$
-\frac{1}{2(-4)^2}=e^1+c
$
$-\frac{1}{32}=e+c$
So $c=-e-\frac{1}{32}$
And the solution is
$-\frac{1}{2x^2}=e^t-e-\frac{1}{32}$
which can be combined or manipulated or whatever you wanna do but I'll leave it like that
5. thank you!=] woaah its that easy.... i spent 2 hourson this!!
|
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## How do I mass convert mp3 to an open format without losing quality?
1
I have some mp3 files - stems from a track ready for remix. The originals in a non-lossy format are not available. I would like to open them in Ardour, on linux, but Ardour doesn't open mp3s, because of licensing problems. I would like to convert to an open format, with no data loss, and with file sizes as small as possible (I know converting from mp3 to a lossless format doesn't gain me any quality. I just don't want to lose any more quality). I was thinking that flac would be the obvious solution, but am open to other suggestions.
Suggestions for other OSs are welcome too, as long as the result is an open format.
Clarification: it'd be nice to be able to maintain meta-data too.
Just purchase the song you are going to remix instead of trying to find it as an MP3. Convert the song to AAC, which is lossless uncompressed. – Cole Johnson – 2012-05-27T00:13:06.443
@ColeJohnson the stems are not available as anything other than mp3. The are also not available for purchase, but were free as part of a promotion. Also, AAC is not a completely open format. – None – 2012-05-27T06:40:44.043
@Cole AAC is not lossless, it's just a better lossy codec than MP3. – None – 2013-01-25T17:17:47.120
1
Hrm. Looks like SoundKonverter can handle this, I was getting an error
soundkonverter(2255): couldn't create slave: "Unable to create io-slave:
klauncher said: Unknown protocol ''.
"
But that's fixed by installing mpg321.
Interesting to note that some of these flac files are significantly smaller than the mp3s (which were stereo 320kbps), although that is probably because those files consist largely of silence :)
Lesson: Use the appropriate file format to start with, children!
2Just a side note, granted that the question is already answered, remember that using FLACs on your DAW might be a rather useless situation. Since the program will decode the FLAC to a uncompressed format, it might save the samples as WAVs or AIFFs. At least I imagine that it would do that, but I'm not completely sure.
If it doesn't save as WAVs or AIFFs it means that your processor is decoding the audio in real time, and that is something that should be avoided, since you'd like to save your CPU for other things than decoding audio. – None – 2012-03-05T16:33:59.717
Ardour uses non-destructive editing, as far as I know, it doesn't change the files at all, just saves time codes for where to cut in and out (ie. when you've cropped a segment out of a file). But yeah, for destructive editing you do have a point, saving in WAV would be much more sensible :) – None – 2012-03-06T00:46:50.520
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# FEM solution becoming wider as number of nodes increase
My FEM scheme uses a 4-node quadrilateral element with bilinear shape functions. The simple problem I'm solving is. $$\nabla ^2 f = 5$$ But as I increase the number of nodes, the plot of the solution becomes flatter. I'm stumped as to why this might happen.Two plots below show how when I change the size of the mesh from a 7x7 to a approx. 30x30, the plot becomes much more spread out. Also the boundary condition is that $$f = 0$$ on the boundary
FIXED, had some typos in my code
• The result is obviously wrong. I think there is a mistake in your code but it's impossible to say what the mistake is without seeing the code. – knl Sep 22 '20 at 4:52
• The fact that the solution changes significantly when you refine your grid hints in the direction, that some h-dependent scaling might have been forgotten. Can you give us a bit more information? What code are you using, do you have a minimal working example? Do you have a 1D example that you might check first? – MPIchael Sep 24 '20 at 13:42
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Four fair dice are marked on their six faces, using the mathematical constants $e$, $\pi$ and $\phi$ as follows:
A: 4 4 4 4 0 0 B: $\pi \pi \pi \pi \pi \pi$ where $\pi$ is approximately 3.142 C: e e e e 7 7 where e is approximately 2.718 D: 5 5 5 $\phi \phi \phi$ where $\phi$ is approximately 1.618
The game is that we each have one die, we throw the dice once and the highest number wins. I invite you to choose first ANY one of the dice. Then I can always choose another one so that I will have a better chance of winning than you. You may think this is unfair and decide you want to play with the die I chose. In that case I can always chose another one so that I still have a better chance of winning than you. Investigate the probabilities and explain the choices I make in all possible cases.
Does it make any difference if the dice are marked with 3 instead of $\pi$, 2 instead of $e$ and 1 instead of $\phi$?
|
{}
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# source:doc/papers/concurrency/Paper.tex@f1f8e55
aaron-thesisarm-ehcleanup-dtorsdeferred_resndemanglerenumforall-pointer-decayjacob/cs343-translationjenkins-sandboxnew-astnew-ast-unique-exprnew-envno_listpersistent-indexerresolv-newwith_gc
Last change on this file since f1f8e55 was 5ff188f, checked in by Peter A. Buhr <pabuhr@…>, 4 years ago
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9\documentclass[10pt]{article}
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48%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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50% Names used in the document.
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67%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
68
69\setcounter{secnumdepth}{2} % number subsubsections
71% \linenumbers % comment out to turn off line numbering
72
73\title{Concurrency in \CFA}
74\author{Thierry Delisle and Peter A. Buhr, Waterloo, Ontario, Canada}
75
76
77\begin{document}
78\maketitle
79
80\begin{abstract}
81\CFA is a modern, \emph{non-object-oriented} extension of the C programming language.
82This paper serves as a definition and an implementation for the concurrency and parallelism \CFA offers. These features are created from scratch due to the lack of concurrency in ISO C. Lightweight threads are introduced into the language. In addition, monitors are introduced as a high-level tool for control-flow based synchronization and mutual-exclusion. The main contributions of this paper are two-fold: it extends the existing semantics of monitors introduce by~\cite{Hoare74} to handle monitors in groups and also details the engineering effort needed to introduce these features as core language features. Indeed, these features are added with respect to expectations of C programmers, and integrate with the \CFA type-system and other language features.
83\end{abstract}
84
85%----------------------------------------------------------------------
86% MAIN BODY
87%----------------------------------------------------------------------
88
89% ======================================================================
90\section{Introduction}
91% ======================================================================
92
93This paper provides a minimal concurrency \textbf{api} that is simple, efficient and can be reused to build higher-level features. The simplest possible concurrency system is a thread and a lock but this low-level approach is hard to master. An easier approach for users is to support higher-level constructs as the basis of concurrency. Indeed, for highly productive concurrent programming, high-level approaches are much more popular~\cite{HPP:Study}. Examples are task based, message passing and implicit threading. The high-level approach and its minimal \textbf{api} are tested in a dialect of C, called \CFA. Furthermore, the proposed \textbf{api} doubles as an early definition of the \CFA language and library. This paper also provides an implementation of the concurrency library for \CFA as well as all the required language features added to the source-to-source translator.
94
95There are actually two problems that need to be solved in the design of concurrency for a programming language: which concurrency and which parallelism tools are available to the programmer. While these two concepts are often combined, they are in fact distinct, requiring different tools~\cite{Buhr05a}. Concurrency tools need to handle mutual exclusion and synchronization, while parallelism tools are about performance, cost and resource utilization.
96
97In the context of this paper, a \textbf{thread} is a fundamental unit of execution that runs a sequence of code, generally on a program stack. Having multiple simultaneous threads gives rise to concurrency and generally requires some kind of locking mechanism to ensure proper execution. Correspondingly, \textbf{concurrency} is defined as the concepts and challenges that occur when multiple independent (sharing memory, timing dependencies, etc.) concurrent threads are introduced. Accordingly, \textbf{locking} (and by extension locks) are defined as a mechanism that prevents the progress of certain threads in order to avoid problems due to concurrency. Finally, in this paper \textbf{parallelism} is distinct from concurrency and is defined as running multiple threads simultaneously. More precisely, parallelism implies \emph{actual} simultaneous execution as opposed to concurrency which only requires \emph{apparent} simultaneous execution. As such, parallelism is only observable in the differences in performance or, more generally, differences in timing.
98
99% ======================================================================
100% ======================================================================
101\section{\CFA Overview}
102% ======================================================================
103% ======================================================================
104
105The following is a quick introduction to the \CFA language, specifically tailored to the features needed to support concurrency.
106
107\CFA is an extension of ISO-C and therefore supports all of the same paradigms as C. It is a non-object-oriented system-language, meaning most of the major abstractions have either no runtime overhead or can be opted out easily. Like C, the basics of \CFA revolve around structures and routines, which are thin abstractions over machine code. The vast majority of the code produced by the \CFA translator respects memory layouts and calling conventions laid out by C. Interestingly, while \CFA is not an object-oriented language, lacking the concept of a receiver (e.g., {\tt this}), it does have some notion of objects\footnote{C defines the term objects as : region of data storage in the execution environment, the contents of which can represent
108values''~\cite[3.15]{C11}}, most importantly construction and destruction of objects. Most of the following code examples can be found on the \CFA website~\cite{www-cfa}.
109
110% ======================================================================
111\subsection{References}
112
113Like \CC, \CFA introduces rebind-able references providing multiple dereferencing as an alternative to pointers. In regards to concurrency, the semantic difference between pointers and references are not particularly relevant, but since this document uses mostly references, here is a quick overview of the semantics:
114\begin{cfacode}
115int x, *p1 = &x, **p2 = &p1, ***p3 = &p2,
116 &r1 = x, &&r2 = r1, &&&r3 = r2;
117***p3 = 3; //change x
118r3 = 3; //change x, ***r3
119**p3 = ...; //change p1
120*p3 = ...; //change p2
121int y, z, & ar[3] = {x, y, z}; //initialize array of references
122typeof( ar[1]) p; //is int, referenced object type
123typeof(&ar[1]) q; //is int &, reference type
124sizeof( ar[1]) == sizeof(int); //is true, referenced object size
125sizeof(&ar[1]) == sizeof(int *); //is true, reference size
126\end{cfacode}
127The important take away from this code example is that a reference offers a handle to an object, much like a pointer, but which is automatically dereferenced for convenience.
128
129% ======================================================================
131
132Another important feature of \CFA is function overloading as in Java and \CC, where routines with the same name are selected based on the number and type of the arguments. As well, \CFA uses the return type as part of the selection criteria, as in Ada~\cite{Ada}. For routines with multiple parameters and returns, the selection is complex.
133\begin{cfacode}
134//selection based on type and number of parameters
135void f(void); //(1)
136void f(char); //(2)
137void f(int, double); //(3)
138f(); //select (1)
139f('a'); //select (2)
140f(3, 5.2); //select (3)
141
142//selection based on type and number of returns
143char f(int); //(1)
144double f(int); //(2)
145char c = f(3); //select (1)
146double d = f(4); //select (2)
147\end{cfacode}
148This feature is particularly important for concurrency since the runtime system relies on creating different types to represent concurrency objects. Therefore, overloading is necessary to prevent the need for long prefixes and other naming conventions that prevent name clashes. As seen in section \ref{basics}, routine \code{main} is an example that benefits from overloading.
149
150% ======================================================================
151\subsection{Operators}
152Overloading also extends to operators. The syntax for denoting operator-overloading is to name a routine with the symbol of the operator and question marks where the arguments of the operation appear, e.g.:
153\begin{cfacode}
154int ++? (int op); //unary prefix increment
155int ?++ (int op); //unary postfix increment
156int ?+? (int op1, int op2); //binary plus
157int ?<=?(int op1, int op2); //binary less than
158int ?=? (int & op1, int op2); //binary assignment
159int ?+=?(int & op1, int op2); //binary plus-assignment
160
161struct S {int i, j;};
162S ?+?(S op1, S op2) { //add two structures
163 return (S){op1.i + op2.i, op1.j + op2.j};
164}
165S s1 = {1, 2}, s2 = {2, 3}, s3;
166s3 = s1 + s2; //compute sum: s3 == {2, 5}
167\end{cfacode}
168While concurrency does not use operator overloading directly, this feature is more important as an introduction for the syntax of constructors.
169
170% ======================================================================
171\subsection{Constructors/Destructors}
172Object lifetime is often a challenge in concurrency. \CFA uses the approach of giving concurrent meaning to object lifetime as a means of synchronization and/or mutual exclusion. Since \CFA relies heavily on the lifetime of objects, constructors and destructors is a core feature required for concurrency and parallelism. \CFA uses the following syntax for constructors and destructors:
173\begin{cfacode}
174struct S {
175 size_t size;
176 int * ia;
177};
178void ?{}(S & s, int asize) { //constructor operator
179 s.size = asize; //initialize fields
180 s.ia = calloc(size, sizeof(S));
181}
182void ^?{}(S & s) { //destructor operator
183 free(ia); //de-initialization fields
184}
185int main() {
186 S x = {10}, y = {100}; //implicit calls: ?{}(x, 10), ?{}(y, 100)
187 ... //use x and y
188 ^x{}^y{}; //explicit calls to de-initialize
189 x{20}; y{200}; //explicit calls to reinitialize
190 ... //reuse x and y
191} //implicit calls: ^?{}(y), ^?{}(x)
192\end{cfacode}
193The language guarantees that every object and all their fields are constructed. Like \CC, construction of an object is automatically done on allocation and destruction of the object is done on deallocation. Allocation and deallocation can occur on the stack or on the heap.
194\begin{cfacode}
195{
196 struct S s = {10}; //allocation, call constructor
197 ...
198} //deallocation, call destructor
199struct S * s = new(); //allocation, call constructor
200...
201delete(s); //deallocation, call destructor
202\end{cfacode}
203Note that like \CC, \CFA introduces \code{new} and \code{delete}, which behave like \code{malloc} and \code{free} in addition to constructing and destructing objects, after calling \code{malloc} and before calling \code{free}, respectively.
204
205% ======================================================================
206\subsection{Parametric Polymorphism}
207\label{s:ParametricPolymorphism}
208Routines in \CFA can also be reused for multiple types. This capability is done using the \code{forall} clauses, which allow separately compiled routines to support generic usage over multiple types. For example, the following sum function works for any type that supports construction from 0 and addition:
209\begin{cfacode}
210//constraint type, 0 and +
211forall(otype T | { void ?{}(T *, zero_t); T ?+?(T, T); })
212T sum(T a[ ], size_t size) {
213 T total = 0; //construct T from 0
214 for(size_t i = 0; i < size; i++)
215 total = total + a[i]; //select appropriate +
217}
218
219S sa[5];
220int i = sum(sa, 5); //use S's 0 construction and +
221\end{cfacode}
222
223Since writing constraints on types can become cumbersome for more constrained functions, \CFA also has the concept of traits. Traits are named collection of constraints that can be used both instead and in addition to regular constraints:
224\begin{cfacode}
225trait summable( otype T ) {
226 void ?{}(T *, zero_t); //constructor from 0 literal
227 T ?+?(T, T); //assortment of additions
228 T ?+=?(T *, T);
229 T ++?(T *);
230 T ?++(T *);
231};
232forall( otype T | summable(T) ) //use trait
233T sum(T a[], size_t size);
234\end{cfacode}
235
236Note that the type use for assertions can be either an \code{otype} or a \code{dtype}. Types declared as \code{otype} refer to complete'' objects, i.e., objects with a size, a default constructor, a copy constructor, a destructor and an assignment operator. Using \code{dtype,} on the other hand, has none of these assumptions but is extremely restrictive, it only guarantees the object is addressable.
237
238% ======================================================================
239\subsection{with Clause/Statement}
240Since \CFA lacks the concept of a receiver, certain functions end up needing to repeat variable names often. To remove this inconvenience, \CFA provides the \code{with} statement, which opens an aggregate scope making its fields directly accessible (like Pascal).
241\begin{cfacode}
242struct S { int i, j; };
243int mem(S & this) with (this) //with clause
244 i = 1; //this->i
245 j = 2; //this->j
246}
247int foo() {
248 struct S1 { ... } s1;
249 struct S2 { ... } s2;
250 with (s1) //with statement
251 {
252 //access fields of s1 without qualification
253 with (s2) //nesting
254 {
255 //access fields of s1 and s2 without qualification
256 }
257 }
258 with (s1, s2) //scopes open in parallel
259 {
260 //access fields of s1 and s2 without qualification
261 }
262}
263\end{cfacode}
264
266
267% ======================================================================
268% ======================================================================
269\section{Concurrency Basics}\label{basics}
270% ======================================================================
271% ======================================================================
272Before any detailed discussion of the concurrency and parallelism in \CFA, it is important to describe the basics of concurrency and how they are expressed in \CFA user code.
273
274\section{Basics of concurrency}
275At its core, concurrency is based on having multiple call-stacks and scheduling among threads of execution executing on these stacks. Concurrency without parallelism only requires having multiple call stacks (or contexts) for a single thread of execution.
276
277Execution with a single thread and multiple stacks where the thread is self-scheduling deterministically across the stacks is called coroutining. Execution with a single and multiple stacks but where the thread is scheduled by an oracle (non-deterministic from the thread's perspective) across the stacks is called concurrency.
278
279Therefore, a minimal concurrency system can be achieved by creating coroutines (see Section \ref{coroutine}), which instead of context-switching among each other, always ask an oracle where to context-switch next. While coroutines can execute on the caller's stack-frame, stack-full coroutines allow full generality and are sufficient as the basis for concurrency. The aforementioned oracle is a scheduler and the whole system now follows a cooperative threading-model (a.k.a., non-preemptive scheduling). The oracle/scheduler can either be a stack-less or stack-full entity and correspondingly require one or two context-switches to run a different coroutine. In any case, a subset of concurrency related challenges start to appear. For the complete set of concurrency challenges to occur, the only feature missing is preemption.
280
281A scheduler introduces order of execution uncertainty, while preemption introduces uncertainty about where context switches occur. Mutual exclusion and synchronization are ways of limiting non-determinism in a concurrent system. Now it is important to understand that uncertainty is desirable; uncertainty can be used by runtime systems to significantly increase performance and is often the basis of giving a user the illusion that tasks are running in parallel. Optimal performance in concurrent applications is often obtained by having as much non-determinism as correctness allows.
282
284One of the important features that are missing in C is threading\footnote{While the C11 standard defines a threads.h'' header, it is minimal and defined as optional. As such, library support for threading is far from widespread. At the time of writing the paper, neither \texttt{gcc} nor \texttt{clang} support threads.h'' in their respective standard libraries.}. On modern architectures, a lack of threading is unacceptable~\cite{Sutter05, Sutter05b}, and therefore modern programming languages must have the proper tools to allow users to write efficient concurrent programs to take advantage of parallelism. As an extension of C, \CFA needs to express these concepts in a way that is as natural as possible to programmers familiar with imperative languages. And being a system-level language means programmers expect to choose precisely which features they need and which cost they are willing to pay.
285
286\section{Coroutines: A Stepping Stone}\label{coroutine}
287While the main focus of this proposal is concurrency and parallelism, it is important to address coroutines, which are actually a significant building block of a concurrency system. \textbf{Coroutine}s are generalized routines which have predefined points where execution is suspended and can be resumed at a later time. Therefore, they need to deal with context switches and other context-management operations. This proposal includes coroutines both as an intermediate step for the implementation of threads, and a first-class feature of \CFA. Furthermore, many design challenges of threads are at least partially present in designing coroutines, which makes the design effort that much more relevant. The core \textbf{api} of coroutines revolves around two features: independent call-stacks and \code{suspend}/\code{resume}.
288
289\begin{table}
290\begin{center}
291\begin{tabular}{c @{\hskip 0.025in}|@{\hskip 0.025in} c @{\hskip 0.025in}|@{\hskip 0.025in} c}
292\begin{ccode}[tabsize=2]
293//Using callbacks
294void fibonacci_func(
295 int n,
296 void (*callback)(int)
297) {
298 int first = 0;
299 int second = 1;
300 int next, i;
301 for(i = 0; i < n; i++)
302 {
303 if(i <= 1)
304 next = i;
305 else {
306 next = f1 + f2;
307 f1 = f2;
308 f2 = next;
309 }
310 callback(next);
311 }
312}
313
314int main() {
315 void print_fib(int n) {
316 printf("%d\n", n);
317 }
318
319 fibonacci_func(
320 10, print_fib
321 );
322
323
324
325}
326\end{ccode}&\begin{ccode}[tabsize=2]
327//Using output array
328void fibonacci_array(
329 int n,
330 int* array
331) {
332 int f1 = 0; int f2 = 1;
333 int next, i;
334 for(i = 0; i < n; i++)
335 {
336 if(i <= 1)
337 next = i;
338 else {
339 next = f1 + f2;
340 f1 = f2;
341 f2 = next;
342 }
343 array[i] = next;
344 }
345}
346
347
348int main() {
349 int a[10];
350
351 fibonacci_func(
352 10, a
353 );
354
355 for(int i=0;i<10;i++){
356 printf("%d\n", a[i]);
357 }
358
359}
360\end{ccode}&\begin{ccode}[tabsize=2]
361//Using external state
362typedef struct {
363 int f1, f2;
364} Iterator_t;
365
366int fibonacci_state(
367 Iterator_t* it
368) {
369 int f;
370 f = it->f1 + it->f2;
371 it->f2 = it->f1;
372 it->f1 = max(f,1);
373 return f;
374}
375
376
377
378
379
380
381
382int main() {
383 Iterator_t it={0,0};
384
385 for(int i=0;i<10;i++){
386 printf("%d\n",
387 fibonacci_state(
388 &it
389 );
390 );
391 }
392
393}
394\end{ccode}
395\end{tabular}
396\end{center}
397\caption{Different implementations of a Fibonacci sequence generator in C.}
398\label{lst:fibonacci-c}
399\end{table}
400
401A good example of a problem made easier with coroutines is generators, e.g., generating the Fibonacci sequence. This problem comes with the challenge of decoupling how a sequence is generated and how it is used. Listing \ref{lst:fibonacci-c} shows conventional approaches to writing generators in C. All three of these approach suffer from strong coupling. The left and centre approaches require that the generator have knowledge of how the sequence is used, while the rightmost approach requires holding internal state between calls on behalf of the generator and makes it much harder to handle corner cases like the Fibonacci seed.
402
403Listing \ref{lst:fibonacci-cfa} is an example of a solution to the Fibonacci problem using \CFA coroutines, where the coroutine stack holds sufficient state for the next generation. This solution has the advantage of having very strong decoupling between how the sequence is generated and how it is used. Indeed, this version is as easy to use as the \code{fibonacci_state} solution, while the implementation is very similar to the \code{fibonacci_func} example.
404
405\begin{figure}
406\begin{cfacode}[caption={Implementation of Fibonacci using coroutines},label={lst:fibonacci-cfa}]
407coroutine Fibonacci {
408 int fn; //used for communication
409};
410
411void ?{}(Fibonacci& this) { //constructor
412 this.fn = 0;
413}
414
415//main automatically called on first resume
416void main(Fibonacci& this) with (this) {
417 int fn1, fn2; //retained between resumes
418 fn = 0;
419 fn1 = fn;
421
422 fn = 1;
423 fn2 = fn1;
424 fn1 = fn;
426
427 for ( ;; ) {
428 fn = fn1 + fn2;
429 fn2 = fn1;
430 fn1 = fn;
432 }
433}
434
435int next(Fibonacci& this) {
436 resume(this); //transfer to last suspend
437 return this.fn;
438}
439
440void main() { //regular program main
441 Fibonacci f1, f2;
442 for ( int i = 1; i <= 10; i += 1 ) {
443 sout | next( f1 ) | next( f2 ) | endl;
444 }
445}
446\end{cfacode}
447\end{figure}
448
449Listing \ref{lst:fmt-line} shows the \code{Format} coroutine for restructuring text into groups of character blocks of fixed size. The example takes advantage of resuming coroutines in the constructor to simplify the code and highlights the idea that interesting control flow can occur in the constructor.
450
451\begin{figure}
452\begin{cfacode}[tabsize=3,caption={Formatting text into lines of 5 blocks of 4 characters.},label={lst:fmt-line}]
453//format characters into blocks of 4 and groups of 5 blocks per line
454coroutine Format {
455 char ch; //used for communication
456 int g, b; //global because used in destructor
457};
458
459void ?{}(Format& fmt) {
460 resume( fmt ); //prime (start) coroutine
461}
462
463void ^?{}(Format& fmt) with fmt {
464 if ( fmt.g != 0 || fmt.b != 0 )
465 sout | endl;
466}
467
468void main(Format& fmt) with fmt {
469 for ( ;; ) { //for as many characters
470 for(g = 0; g < 5; g++) { //groups of 5 blocks
471 for(b = 0; b < 4; fb++) { //blocks of 4 characters
472 suspend();
473 sout | ch; //print character
474 }
475 sout | " "; //print block separator
476 }
477 sout | endl; //print group separator
478 }
479}
480
481void prt(Format & fmt, char ch) {
482 fmt.ch = ch;
483 resume(fmt);
484}
485
486int main() {
487 Format fmt;
488 char ch;
489 Eof: for ( ;; ) { //read until end of file
490 sin | ch; //read one character
491 if(eof(sin)) break Eof; //eof ?
492 prt(fmt, ch); //push character for formatting
493 }
494}
495\end{cfacode}
496\end{figure}
497
498\subsection{Construction}
499One important design challenge for implementing coroutines and threads (shown in section \ref{threads}) is that the runtime system needs to run code after the user-constructor runs to connect the fully constructed object into the system. In the case of coroutines, this challenge is simpler since there is no non-determinism from preemption or scheduling. However, the underlying challenge remains the same for coroutines and threads.
500
501The runtime system needs to create the coroutine's stack and, more importantly, prepare it for the first resumption. The timing of the creation is non-trivial since users expect both to have fully constructed objects once execution enters the coroutine main and to be able to resume the coroutine from the constructor. There are several solutions to this problem but the chosen option effectively forces the design of the coroutine.
502
503Furthermore, \CFA faces an extra challenge as polymorphic routines create invisible thunks when cast to non-polymorphic routines and these thunks have function scope. For example, the following code, while looking benign, can run into undefined behaviour because of thunks:
504
505\begin{cfacode}
506//async: Runs function asynchronously on another thread
507forall(otype T)
508extern void async(void (*func)(T*), T* obj);
509
510forall(otype T)
511void noop(T*) {}
512
513void bar() {
514 int a;
515 async(noop, &a); //start thread running noop with argument a
516}
517\end{cfacode}
518
519The generated C code\footnote{Code trimmed down for brevity} creates a local thunk to hold type information:
520
521\begin{ccode}
522extern void async(/* omitted */, void (*func)(void*), void* obj);
523
524void noop(/* omitted */, void* obj){}
525
526void bar(){
527 int a;
528 void _thunk0(int* _p0){
529 /* omitted */
530 noop(/* omitted */, _p0);
531 }
532 /* omitted */
533 async(/* omitted */, ((void (*)(void*))(&_thunk0)), (&a));
534}
535\end{ccode}
536The problem in this example is a storage management issue, the function pointer \code{_thunk0} is only valid until the end of the block, which limits the viable solutions because storing the function pointer for too long causes undefined behaviour; i.e., the stack-based thunk being destroyed before it can be used. This challenge is an extension of challenges that come with second-class routines. Indeed, GCC nested routines also have the limitation that nested routine cannot be passed outside of the declaration scope. The case of coroutines and threads is simply an extension of this problem to multiple call stacks.
537
538\subsection{Alternative: Composition}
539One solution to this challenge is to use composition/containment, where coroutine fields are added to manage the coroutine.
540
541\begin{cfacode}
542struct Fibonacci {
543 int fn; //used for communication
544 coroutine c; //composition
545};
546
547void FibMain(void*) {
548 //...
549}
550
551void ?{}(Fibonacci& this) {
552 this.fn = 0;
553 //Call constructor to initialize coroutine
554 (this.c){myMain};
555}
556\end{cfacode}
557The downside of this approach is that users need to correctly construct the coroutine handle before using it. Like any other objects, the user must carefully choose construction order to prevent usage of objects not yet constructed. However, in the case of coroutines, users must also pass to the coroutine information about the coroutine main, like in the previous example. This opens the door for user errors and requires extra runtime storage to pass at runtime information that can be known statically.
558
559\subsection{Alternative: Reserved keyword}
560The next alternative is to use language support to annotate coroutines as follows:
561
562\begin{cfacode}
563coroutine Fibonacci {
564 int fn; //used for communication
565};
566\end{cfacode}
567The \code{coroutine} keyword means the compiler can find and inject code where needed. The downside of this approach is that it makes coroutine a special case in the language. Users wanting to extend coroutines or build their own for various reasons can only do so in ways offered by the language. Furthermore, implementing coroutines without language supports also displays the power of the programming language used. While this is ultimately the option used for idiomatic \CFA code, coroutines and threads can still be constructed by users without using the language support. The reserved keywords are only present to improve ease of use for the common cases.
568
569\subsection{Alternative: Lambda Objects}
570
571For coroutines as for threads, many implementations are based on routine pointers or function objects~\cite{Butenhof97, C++14, MS:VisualC++, BoostCoroutines15}. For example, Boost implements coroutines in terms of four functor object types:
572\begin{cfacode}
573asymmetric_coroutine<>::pull_type
574asymmetric_coroutine<>::push_type
575symmetric_coroutine<>::call_type
576symmetric_coroutine<>::yield_type
577\end{cfacode}
578Often, the canonical threading paradigm in languages is based on function pointers, \texttt{pthread} being one of the most well-known examples. The main problem of this approach is that the thread usage is limited to a generic handle that must otherwise be wrapped in a custom type. Since the custom type is simple to write in \CFA and solves several issues, added support for routine/lambda based coroutines adds very little.
579
580A variation of this would be to use a simple function pointer in the same way \texttt{pthread} does for threads:
581\begin{cfacode}
582void foo( coroutine_t cid, void* arg ) {
583 int* value = (int*)arg;
584 //Coroutine body
585}
586
587int main() {
588 int value = 0;
589 coroutine_t cid = coroutine_create( &foo, (void*)&value );
590 coroutine_resume( &cid );
591}
592\end{cfacode}
593This semantics is more common for thread interfaces but coroutines work equally well. As discussed in section \ref{threads}, this approach is superseded by static approaches in terms of expressivity.
594
595\subsection{Alternative: Trait-Based Coroutines}
596
597Finally, the underlying approach, which is the one closest to \CFA idioms, is to use trait-based lazy coroutines. This approach defines a coroutine as anything that satisfies the trait \code{is_coroutine} (as defined below) and is used as a coroutine.
598
599\begin{cfacode}
600trait is_coroutine(dtype T) {
601 void main(T& this);
602 coroutine_desc* get_coroutine(T& this);
603};
604
605forall( dtype T | is_coroutine(T) ) void suspend(T&);
606forall( dtype T | is_coroutine(T) ) void resume (T&);
607\end{cfacode}
608This ensures that an object is not a coroutine until \code{resume} is called on the object. Correspondingly, any object that is passed to \code{resume} is a coroutine since it must satisfy the \code{is_coroutine} trait to compile. The advantage of this approach is that users can easily create different types of coroutines, for example, changing the memory layout of a coroutine is trivial when implementing the \code{get_coroutine} routine. The \CFA keyword \code{coroutine} simply has the effect of implementing the getter and forward declarations required for users to implement the main routine.
609
610\begin{center}
611\begin{tabular}{c c c}
612\begin{cfacode}[tabsize=3]
613coroutine MyCoroutine {
614 int someValue;
615};
616\end{cfacode} & == & \begin{cfacode}[tabsize=3]
617struct MyCoroutine {
618 int someValue;
619 coroutine_desc __cor;
620};
621
622static inline
623coroutine_desc* get_coroutine(
624 struct MyCoroutine& this
625) {
626 return &this.__cor;
627}
628
629void main(struct MyCoroutine* this);
630\end{cfacode}
631\end{tabular}
632\end{center}
633
634The combination of these two approaches allows users new to coroutining and concurrency to have an easy and concise specification, while more advanced users have tighter control on memory layout and initialization.
635
637The basic building blocks of multithreading in \CFA are \textbf{cfathread}. Both user and kernel threads are supported, where user threads are the concurrency mechanism and kernel threads are the parallel mechanism. User threads offer a flexible and lightweight interface. A thread can be declared using a struct declaration \code{thread} as follows:
638
639\begin{cfacode}
641\end{cfacode}
642
643As for coroutines, the keyword is a thin wrapper around a \CFA trait:
644
645\begin{cfacode}
647 void ^?{}(T & mutex this);
648 void main(T & this);
650};
651\end{cfacode}
652
653Obviously, for this thread implementation to be useful it must run some user code. Several other threading interfaces use a function-pointer representation as the interface of threads (for example \Csharp~\cite{Csharp} and Scala~\cite{Scala}). However, this proposal considers that statically tying a \code{main} routine to a thread supersedes this approach. Since the \code{main} routine is already a special routine in \CFA (where the program begins), it is a natural extension of the semantics to use overloading to declare mains for different threads (the normal main being the main of the initial thread). As such the \code{main} routine of a thread can be defined as
654\begin{cfacode}
656
657void main(foo & this) {
658 sout | "Hello World!" | endl;
659}
660\end{cfacode}
661
662In this example, threads of type \code{foo} start execution in the \code{void main(foo &)} routine, which prints \code{"Hello World!".} While this paper encourages this approach to enforce strongly typed programming, users may prefer to use the routine-based thread semantics for the sake of simplicity. With the static semantics it is trivial to write a thread type that takes a function pointer as a parameter and executes it on its stack asynchronously.
663\begin{cfacode}
664typedef void (*voidFunc)(int);
665
667 voidFunc func;
668 int arg;
669};
670
671void ?{}(FuncRunner & this, voidFunc inFunc, int arg) {
672 this.func = inFunc;
673 this.arg = arg;
674}
675
676void main(FuncRunner & this) {
677 //thread starts here and runs the function
678 this.func( this.arg );
679}
680
681void hello(/*unused*/ int) {
682 sout | "Hello World!" | endl;
683}
684
685int main() {
686 FuncRunner f = {hello, 42};
687 return 0?
688}
689\end{cfacode}
690
691A consequence of the strongly typed approach to main is that memory layout of parameters and return values to/from a thread are now explicitly specified in the \textbf{api}.
692
693Of course, for threads to be useful, it must be possible to start and stop threads and wait for them to complete execution. While using an \textbf{api} such as \code{fork} and \code{join} is relatively common in the literature, such an interface is unnecessary. Indeed, the simplest approach is to use \textbf{raii} principles and have threads \code{fork} after the constructor has completed and \code{join} before the destructor runs.
694\begin{cfacode}
696
697void main(World & this) {
698 sout | "World!" | endl;
699}
700
701void main() {
702 World w;
704
705 //Printing "Hello " and "World!" are run concurrently
706 sout | "Hello " | endl;
707
708 //Implicit join at end of scope
709}
710\end{cfacode}
711
712This semantic has several advantages over explicit semantics: a thread is always started and stopped exactly once, users cannot make any programming errors, and it naturally scales to multiple threads meaning basic synchronization is very simple.
713
714\begin{cfacode}
716 //...
717};
718
719//main
721 //...
722}
723
724void foo() {
726 //Start 10 threads at the beginning of the scope
727
728 DoStuff();
729
730 //Wait for the 10 threads to finish
731}
732\end{cfacode}
733
734However, one of the drawbacks of this approach is that threads always form a tree where nodes must always outlive their children, i.e., they are always destroyed in the opposite order of construction because of C scoping rules. This restriction is relaxed by using dynamic allocation, so threads can outlive the scope in which they are created, much like dynamically allocating memory lets objects outlive the scope in which they are created.
735
736\begin{cfacode}
738 //...
739};
740
742 //...
743}
744
745void foo() {
747 {
748 //Start a thread at the beginning of the scope
750
751 //create another thread that will outlive the thread in this scope
753
754 DoStuff();
755
756 //Wait for the thread short_lived to finish
757 }
758 DoMoreStuff();
759
760 //Now wait for the long_lived to finish
761 delete long_lived;
762}
763\end{cfacode}
764
765
766% ======================================================================
767% ======================================================================
768\section{Concurrency}
769% ======================================================================
770% ======================================================================
771Several tools can be used to solve concurrency challenges. Since many of these challenges appear with the use of mutable shared state, some languages and libraries simply disallow mutable shared state (Erlang~\cite{Erlang}, Haskell~\cite{Haskell}, Akka (Scala)~\cite{Akka}). In these paradigms, interaction among concurrent objects relies on message passing~\cite{Thoth,Harmony,V-Kernel} or other paradigms closely relate to networking concepts (channels~\cite{CSP,Go} for example). However, in languages that use routine calls as their core abstraction mechanism, these approaches force a clear distinction between concurrent and non-concurrent paradigms (i.e., message passing versus routine calls). This distinction in turn means that, in order to be effective, programmers need to learn two sets of design patterns. While this distinction can be hidden away in library code, effective use of the library still has to take both paradigms into account.
772
773Approaches based on shared memory are more closely related to non-concurrent paradigms since they often rely on basic constructs like routine calls and shared objects. At the lowest level, concurrent paradigms are implemented as atomic operations and locks. Many such mechanisms have been proposed, including semaphores~\cite{Dijkstra68b} and path expressions~\cite{Campbell74}. However, for productivity reasons it is desirable to have a higher-level construct be the core concurrency paradigm~\cite{HPP:Study}.
774
775An approach that is worth mentioning because it is gaining in popularity is transactional memory~\cite{Herlihy93}. While this approach is even pursued by system languages like \CC~\cite{Cpp-Transactions}, the performance and feature set is currently too restrictive to be the main concurrency paradigm for system languages, which is why it was rejected as the core paradigm for concurrency in \CFA.
776
777One of the most natural, elegant, and efficient mechanisms for synchronization and communication, especially for shared-memory systems, is the \emph{monitor}. Monitors were first proposed by Brinch Hansen~\cite{Hansen73} and later described and extended by C.A.R.~Hoare~\cite{Hoare74}. Many programming languages---e.g., Concurrent Pascal~\cite{ConcurrentPascal}, Mesa~\cite{Mesa}, Modula~\cite{Modula-2}, Turing~\cite{Turing:old}, Modula-3~\cite{Modula-3}, NeWS~\cite{NeWS}, Emerald~\cite{Emerald}, \uC~\cite{Buhr92a} and Java~\cite{Java}---provide monitors as explicit language constructs. In addition, operating-system kernels and device drivers have a monitor-like structure, although they often use lower-level primitives such as semaphores or locks to simulate monitors. For these reasons, this project proposes monitors as the core concurrency construct.
778
779\section{Basics}
780Non-determinism requires concurrent systems to offer support for mutual-exclusion and synchronization. Mutual-exclusion is the concept that only a fixed number of threads can access a critical section at any given time, where a critical section is a group of instructions on an associated portion of data that requires the restricted access. On the other hand, synchronization enforces relative ordering of execution and synchronization tools provide numerous mechanisms to establish timing relationships among threads.
781
782\subsection{Mutual-Exclusion}
783As mentioned above, mutual-exclusion is the guarantee that only a fix number of threads can enter a critical section at once. However, many solutions exist for mutual exclusion, which vary in terms of performance, flexibility and ease of use. Methods range from low-level locks, which are fast and flexible but require significant attention to be correct, to higher-level concurrency techniques, which sacrifice some performance in order to improve ease of use. Ease of use comes by either guaranteeing some problems cannot occur (e.g., being deadlock free) or by offering a more explicit coupling between data and corresponding critical section. For example, the \CC \code{std::atomic<T>} offers an easy way to express mutual-exclusion on a restricted set of operations (e.g., reading/writing large types atomically). Another challenge with low-level locks is composability. Locks have restricted composability because it takes careful organizing for multiple locks to be used while preventing deadlocks. Easing composability is another feature higher-level mutual-exclusion mechanisms often offer.
784
785\subsection{Synchronization}
786As with mutual-exclusion, low-level synchronization primitives often offer good performance and good flexibility at the cost of ease of use. Again, higher-level mechanisms often simplify usage by adding either better coupling between synchronization and data (e.g., message passing) or offering a simpler solution to otherwise involved challenges. As mentioned above, synchronization can be expressed as guaranteeing that event \textit{X} always happens before \textit{Y}. Most of the time, synchronization happens within a critical section, where threads must acquire mutual-exclusion in a certain order. However, it may also be desirable to guarantee that event \textit{Z} does not occur between \textit{X} and \textit{Y}. Not satisfying this property is called \textbf{barging}. For example, where event \textit{X} tries to effect event \textit{Y} but another thread acquires the critical section and emits \textit{Z} before \textit{Y}. The classic example is the thread that finishes using a resource and unblocks a thread waiting to use the resource, but the unblocked thread must compete to acquire the resource. Preventing or detecting barging is an involved challenge with low-level locks, which can be made much easier by higher-level constructs. This challenge is often split into two different methods, barging avoidance and barging prevention. Algorithms that use flag variables to detect barging threads are said to be using barging avoidance, while algorithms that baton-pass locks~\cite{Andrews89} between threads instead of releasing the locks are said to be using barging prevention.
787
788% ======================================================================
789% ======================================================================
790\section{Monitors}
791% ======================================================================
792% ======================================================================
793A \textbf{monitor} is a set of routines that ensure mutual-exclusion when accessing shared state. More precisely, a monitor is a programming technique that associates mutual-exclusion to routine scopes, as opposed to mutex locks, where mutual-exclusion is defined by lock/release calls independently of any scoping of the calling routine. This strong association eases readability and maintainability, at the cost of flexibility. Note that both monitors and mutex locks, require an abstract handle to identify them. This concept is generally associated with object-oriented languages like Java~\cite{Java} or \uC~\cite{uC++book} but does not strictly require OO semantics. The only requirement is the ability to declare a handle to a shared object and a set of routines that act on it:
794\begin{cfacode}
795typedef /*some monitor type*/ monitor;
796int f(monitor & m);
797
798int main() {
799 monitor m; //Handle m
800 f(m); //Routine using handle
801}
802\end{cfacode}
803
804% ======================================================================
805% ======================================================================
806\subsection{Call Semantics} \label{call}
807% ======================================================================
808% ======================================================================
809The above monitor example displays some of the intrinsic characteristics. First, it is necessary to use pass-by-reference over pass-by-value for monitor routines. This semantics is important, because at their core, monitors are implicit mutual-exclusion objects (locks), and these objects cannot be copied. Therefore, monitors are non-copy-able objects (\code{dtype}).
810
811Another aspect to consider is when a monitor acquires its mutual exclusion. For example, a monitor may need to be passed through multiple helper routines that do not acquire the monitor mutual-exclusion on entry. Passthrough can occur for generic helper routines (\code{swap}, \code{sort}, etc.) or specific helper routines like the following to implement an atomic counter:
812
813\begin{cfacode}
814monitor counter_t { /*...see section $\ref{data}$...*/ };
815
816void ?{}(counter_t & nomutex this); //constructor
817size_t ++?(counter_t & mutex this); //increment
818
819//need for mutex is platform dependent
820void ?{}(size_t * this, counter_t & mutex cnt); //conversion
821\end{cfacode}
822This counter is used as follows:
823\begin{center}
824\begin{tabular}{c @{\hskip 0.35in} c @{\hskip 0.35in} c}
825\begin{cfacode}
826//shared counter
827counter_t cnt1, cnt2;
828
833 ...
835\end{cfacode}
836\end{tabular}
837\end{center}
838Notice how the counter is used without any explicit synchronization and yet supports thread-safe semantics for both reading and writing, which is similar in usage to the \CC template \code{std::atomic}.
839
840Here, the constructor (\code{?\{\}}) uses the \code{nomutex} keyword to signify that it does not acquire the monitor mutual-exclusion when constructing. This semantics is because an object not yet con\-structed should never be shared and therefore does not require mutual exclusion. Furthermore, it allows the implementation greater freedom when it initializes the monitor locking. The prefix increment operator uses \code{mutex} to protect the incrementing process from race conditions. Finally, there is a conversion operator from \code{counter_t} to \code{size_t}. This conversion may or may not require the \code{mutex} keyword depending on whether or not reading a \code{size_t} is an atomic operation.
841
842For maximum usability, monitors use \textbf{multi-acq} semantics, which means a single thread can acquire the same monitor multiple times without deadlock. For example, listing \ref{fig:search} uses recursion and \textbf{multi-acq} to print values inside a binary tree.
843\begin{figure}
844\begin{cfacode}[caption={Recursive printing algorithm using \textbf{multi-acq}.},label={fig:search}]
845monitor printer { ... };
846struct tree {
847 tree * left, right;
848 char * value;
849};
850void print(printer & mutex p, char * v);
851
852void print(printer & mutex p, tree * t) {
853 print(p, t->value);
854 print(p, t->left );
855 print(p, t->right);
856}
857\end{cfacode}
858\end{figure}
859
860Having both \code{mutex} and \code{nomutex} keywords can be redundant, depending on the meaning of a routine having neither of these keywords. For example, it is reasonable that it should default to the safest option (\code{mutex}) when given a routine without qualifiers \code{void foo(counter_t & this)}, whereas assuming \code{nomutex} is unsafe and may cause subtle errors. On the other hand, \code{nomutex} is the normal'' parameter behaviour, it effectively states explicitly that this routine is not special''. Another alternative is making exactly one of these keywords mandatory, which provides the same semantics but without the ambiguity of supporting routines with neither keyword. Mandatory keywords would also have the added benefit of being self-documented but at the cost of extra typing. While there are several benefits to mandatory keywords, they do bring a few challenges. Mandatory keywords in \CFA would imply that the compiler must know without doubt whether or not a parameter is a monitor or not. Since \CFA relies heavily on traits as an abstraction mechanism, the distinction between a type that is a monitor and a type that looks like a monitor can become blurred. For this reason, \CFA only has the \code{mutex} keyword and uses no keyword to mean \code{nomutex}.
861
862The next semantic decision is to establish when \code{mutex} may be used as a type qualifier. Consider the following declarations:
863\begin{cfacode}
864int f1(monitor & mutex m);
865int f2(const monitor & mutex m);
866int f3(monitor ** mutex m);
867int f4(monitor * mutex m []);
868int f5(graph(monitor *) & mutex m);
869\end{cfacode}
870The problem is to identify which object(s) should be acquired. Furthermore, each object needs to be acquired only once. In the case of simple routines like \code{f1} and \code{f2} it is easy to identify an exhaustive list of objects to acquire on entry. Adding indirections (\code{f3}) still allows the compiler and programmer to identify which object is acquired. However, adding in arrays (\code{f4}) makes it much harder. Array lengths are not necessarily known in C, and even then, making sure objects are only acquired once becomes none-trivial. This problem can be extended to absurd limits like \code{f5}, which uses a graph of monitors. To make the issue tractable, this project imposes the requirement that a routine may only acquire one monitor per parameter and it must be the type of the parameter with at most one level of indirection (ignoring potential qualifiers). Also note that while routine \code{f3} can be supported, meaning that monitor \code{**m} is acquired, passing an array to this routine would be type-safe and yet result in undefined behaviour because only the first element of the array is acquired. However, this ambiguity is part of the C type-system with respects to arrays. For this reason, \code{mutex} is disallowed in the context where arrays may be passed:
871\begin{cfacode}
872int f1(monitor & mutex m); //Okay : recommended case
873int f2(monitor * mutex m); //Not Okay : Could be an array
874int f3(monitor mutex m []); //Not Okay : Array of unknown length
875int f4(monitor ** mutex m); //Not Okay : Could be an array
876int f5(monitor * mutex m []); //Not Okay : Array of unknown length
877\end{cfacode}
878Note that not all array functions are actually distinct in the type system. However, even if the code generation could tell the difference, the extra information is still not sufficient to extend meaningfully the monitor call semantic.
879
880Unlike object-oriented monitors, where calling a mutex member \emph{implicitly} acquires mutual-exclusion of the receiver object, \CFA uses an explicit mechanism to specify the object that acquires mutual-exclusion. A consequence of this approach is that it extends naturally to multi-monitor calls.
881\begin{cfacode}
882int f(MonitorA & mutex a, MonitorB & mutex b);
883
884MonitorA a;
885MonitorB b;
886f(a,b);
887\end{cfacode}
888While OO monitors could be extended with a mutex qualifier for multiple-monitor calls, no example of this feature could be found. The capability to acquire multiple locks before entering a critical section is called \emph{\textbf{bulk-acq}}. In practice, writing multi-locking routines that do not lead to deadlocks is tricky. Having language support for such a feature is therefore a significant asset for \CFA. In the case presented above, \CFA guarantees that the order of acquisition is consistent across calls to different routines using the same monitors as arguments. This consistent ordering means acquiring multiple monitors is safe from deadlock when using \textbf{bulk-acq}. However, users can still force the acquiring order. For example, notice which routines use \code{mutex}/\code{nomutex} and how this affects acquiring order:
889\begin{cfacode}
890void foo(A& mutex a, B& mutex b) { //acquire a & b
891 ...
892}
893
894void bar(A& mutex a, B& /*nomutex*/ b) { //acquire a
895 ... foo(a, b); ... //acquire b
896}
897
898void baz(A& /*nomutex*/ a, B& mutex b) { //acquire b
899 ... foo(a, b); ... //acquire a
900}
901\end{cfacode}
902The \textbf{multi-acq} monitor lock allows a monitor lock to be acquired by both \code{bar} or \code{baz} and acquired again in \code{foo}. In the calls to \code{bar} and \code{baz} the monitors are acquired in opposite order.
903
904However, such use leads to lock acquiring order problems. In the example above, the user uses implicit ordering in the case of function \code{foo} but explicit ordering in the case of \code{bar} and \code{baz}. This subtle difference means that calling these routines concurrently may lead to deadlock and is therefore undefined behaviour. As shown~\cite{Lister77}, solving this problem requires:
905\begin{enumerate}
906 \item Dynamically tracking the monitor-call order.
907 \item Implement rollback semantics.
908\end{enumerate}
909While the first requirement is already a significant constraint on the system, implementing a general rollback semantics in a C-like language is still prohibitively complex~\cite{Dice10}. In \CFA, users simply need to be careful when acquiring multiple monitors at the same time or only use \textbf{bulk-acq} of all the monitors. While \CFA provides only a partial solution, most systems provide no solution and the \CFA partial solution handles many useful cases.
910
911For example, \textbf{multi-acq} and \textbf{bulk-acq} can be used together in interesting ways:
912\begin{cfacode}
913monitor bank { ... };
914
915void deposit( bank & mutex b, int deposit );
916
917void transfer( bank & mutex mybank, bank & mutex yourbank, int me2you) {
918 deposit( mybank, -me2you );
919 deposit( yourbank, me2you );
920}
921\end{cfacode}
922This example shows a trivial solution to the bank-account transfer problem~\cite{BankTransfer}. Without \textbf{multi-acq} and \textbf{bulk-acq}, the solution to this problem is much more involved and requires careful engineering.
923
924\subsection{\code{mutex} statement} \label{mutex-stmt}
925
926The call semantics discussed above have one software engineering issue: only a routine can acquire the mutual-exclusion of a set of monitor. \CFA offers the \code{mutex} statement to work around the need for unnecessary names, avoiding a major software engineering problem~\cite{2FTwoHardThings}. Table \ref{lst:mutex-stmt} shows an example of the \code{mutex} statement, which introduces a new scope in which the mutual-exclusion of a set of monitor is acquired. Beyond naming, the \code{mutex} statement has no semantic difference from a routine call with \code{mutex} parameters.
927
928\begin{table}
929\begin{center}
930\begin{tabular}{|c|c|}
931function call & \code{mutex} statement \\
932\hline
933\begin{cfacode}[tabsize=3]
934monitor M {};
935void foo( M & mutex m1, M & mutex m2 ) {
936 //critical section
937}
938
939void bar( M & m1, M & m2 ) {
940 foo( m1, m2 );
941}
942\end{cfacode}&\begin{cfacode}[tabsize=3]
943monitor M {};
944void bar( M & m1, M & m2 ) {
945 mutex(m1, m2) {
946 //critical section
947 }
948}
949
950
951\end{cfacode}
952\end{tabular}
953\end{center}
954\caption{Regular call semantics vs. \code{mutex} statement}
955\label{lst:mutex-stmt}
956\end{table}
957
958% ======================================================================
959% ======================================================================
960\subsection{Data semantics} \label{data}
961% ======================================================================
962% ======================================================================
963Once the call semantics are established, the next step is to establish data semantics. Indeed, until now a monitor is used simply as a generic handle but in most cases monitors contain shared data. This data should be intrinsic to the monitor declaration to prevent any accidental use of data without its appropriate protection. For example, here is a complete version of the counter shown in section \ref{call}:
964\begin{cfacode}
965monitor counter_t {
966 int value;
967};
968
969void ?{}(counter_t & this) {
970 this.cnt = 0;
971}
972
973int ?++(counter_t & mutex this) {
974 return ++this.value;
975}
976
977//need for mutex is platform dependent here
978void ?{}(int * this, counter_t & mutex cnt) {
979 *this = (int)cnt;
980}
981\end{cfacode}
982
983Like threads and coroutines, monitors are defined in terms of traits with some additional language support in the form of the \code{monitor} keyword. The monitor trait is:
984\begin{cfacode}
985trait is_monitor(dtype T) {
986 monitor_desc * get_monitor( T & );
987 void ^?{}( T & mutex );
988};
989\end{cfacode}
990Note that the destructor of a monitor must be a \code{mutex} routine to prevent deallocation while a thread is accessing the monitor. As with any object, calls to a monitor, using \code{mutex} or otherwise, is undefined behaviour after the destructor has run.
991
992% ======================================================================
993% ======================================================================
994\section{Internal Scheduling} \label{intsched}
995% ======================================================================
996% ======================================================================
997In addition to mutual exclusion, the monitors at the core of \CFA's concurrency can also be used to achieve synchronization. With monitors, this capability is generally achieved with internal or external scheduling as in~\cite{Hoare74}. With \textbf{scheduling} loosely defined as deciding which thread acquires the critical section next, \textbf{internal scheduling} means making the decision from inside the critical section (i.e., with access to the shared state), while \textbf{external scheduling} means making the decision when entering the critical section (i.e., without access to the shared state). Since internal scheduling within a single monitor is mostly a solved problem, this paper concentrates on extending internal scheduling to multiple monitors. Indeed, like the \textbf{bulk-acq} semantics, internal scheduling extends to multiple monitors in a way that is natural to the user but requires additional complexity on the implementation side.
998
999First, here is a simple example of internal scheduling:
1000
1001\begin{cfacode}
1002monitor A {
1003 condition e;
1004}
1005
1006void foo(A& mutex a1, A& mutex a2) {
1007 ...
1008 //Wait for cooperation from bar()
1009 wait(a1.e);
1010 ...
1011}
1012
1013void bar(A& mutex a1, A& mutex a2) {
1014 //Provide cooperation for foo()
1015 ...
1016 //Unblock foo
1017 signal(a1.e);
1018}
1019\end{cfacode}
1020There are two details to note here. First, \code{signal} is a delayed operation; it only unblocks the waiting thread when it reaches the end of the critical section. This semantics is needed to respect mutual-exclusion, i.e., the signaller and signalled thread cannot be in the monitor simultaneously. The alternative is to return immediately after the call to \code{signal}, which is significantly more restrictive. Second, in \CFA, while it is common to store a \code{condition} as a field of the monitor, a \code{condition} variable can be stored/created independently of a monitor. Here routine \code{foo} waits for the \code{signal} from \code{bar} before making further progress, ensuring a basic ordering.
1021
1022An important aspect of the implementation is that \CFA does not allow barging, which means that once function \code{bar} releases the monitor, \code{foo} is guaranteed to be the next thread to acquire the monitor (unless some other thread waited on the same condition). This guarantee offers the benefit of not having to loop around waits to recheck that a condition is met. The main reason \CFA offers this guarantee is that users can easily introduce barging if it becomes a necessity but adding barging prevention or barging avoidance is more involved without language support. Supporting barging prevention as well as extending internal scheduling to multiple monitors is the main source of complexity in the design and implementation of \CFA concurrency.
1023
1024% ======================================================================
1025% ======================================================================
1026\subsection{Internal Scheduling - Multi-Monitor}
1027% ======================================================================
1028% ======================================================================
1029It is easy to understand the problem of multi-monitor scheduling using a series of pseudo-code examples. Note that for simplicity in the following snippets of pseudo-code, waiting and signalling is done using an implicit condition variable, like Java built-in monitors. Indeed, \code{wait} statements always use the implicit condition variable as parameters and explicitly name the monitors (A and B) associated with the condition. Note that in \CFA, condition variables are tied to a \emph{group} of monitors on first use (called branding), which means that using internal scheduling with distinct sets of monitors requires one condition variable per set of monitors. The example below shows the simple case of having two threads (one for each column) and a single monitor A.
1030
1031\begin{multicols}{2}
1033\begin{pseudo}
1034acquire A
1035 wait A
1036release A
1037\end{pseudo}
1038
1039\columnbreak
1040
1042\begin{pseudo}
1043acquire A
1044 signal A
1045release A
1046\end{pseudo}
1047\end{multicols}
1048One thread acquires before waiting (atomically blocking and releasing A) and the other acquires before signalling. It is important to note here that both \code{wait} and \code{signal} must be called with the proper monitor(s) already acquired. This semantic is a logical requirement for barging prevention.
1049
1050A direct extension of the previous example is a \textbf{bulk-acq} version:
1051\begin{multicols}{2}
1052\begin{pseudo}
1053acquire A & B
1054 wait A & B
1055release A & B
1056\end{pseudo}
1057\columnbreak
1058\begin{pseudo}
1059acquire A & B
1060 signal A & B
1061release A & B
1062\end{pseudo}
1063\end{multicols}
1064\noindent This version uses \textbf{bulk-acq} (denoted using the {\sf\&} symbol), but the presence of multiple monitors does not add a particularly new meaning. Synchronization happens between the two threads in exactly the same way and order. The only difference is that mutual exclusion covers a group of monitors. On the implementation side, handling multiple monitors does add a degree of complexity as the next few examples demonstrate.
1065
1066While deadlock issues can occur when nesting monitors, these issues are only a symptom of the fact that locks, and by extension monitors, are not perfectly composable. For monitors, a well-known deadlock problem is the Nested Monitor Problem~\cite{Lister77}, which occurs when a \code{wait} is made by a thread that holds more than one monitor. For example, the following pseudo-code runs into the nested-monitor problem:
1067\begin{multicols}{2}
1068\begin{pseudo}
1069acquire A
1070 acquire B
1071 wait B
1072 release B
1073release A
1074\end{pseudo}
1075
1076\columnbreak
1077
1078\begin{pseudo}
1079acquire A
1080 acquire B
1081 signal B
1082 release B
1083release A
1084\end{pseudo}
1085\end{multicols}
1086\noindent The \code{wait} only releases monitor \code{B} so the signalling thread cannot acquire monitor \code{A} to get to the \code{signal}. Attempting release of all acquired monitors at the \code{wait} introduces a different set of problems, such as releasing monitor \code{C}, which has nothing to do with the \code{signal}.
1087
1088However, for monitors as for locks, it is possible to write a program using nesting without encountering any problems if nesting is done correctly. For example, the next pseudo-code snippet acquires monitors {\sf A} then {\sf B} before waiting, while only acquiring {\sf B} when signalling, effectively avoiding the Nested Monitor Problem~\cite{Lister77}.
1089
1090\begin{multicols}{2}
1091\begin{pseudo}
1092acquire A
1093 acquire B
1094 wait B
1095 release B
1096release A
1097\end{pseudo}
1098
1099\columnbreak
1100
1101\begin{pseudo}
1102
1103acquire B
1104 signal B
1105release B
1106
1107\end{pseudo}
1108\end{multicols}
1109
1110\noindent However, this simple refactoring may not be possible, forcing more complex restructuring.
1111
1112% ======================================================================
1113% ======================================================================
1114\subsection{Internal Scheduling - In Depth}
1115% ======================================================================
1116% ======================================================================
1117
1118A larger example is presented to show complex issues for \textbf{bulk-acq} and its implementation options are analyzed. Listing \ref{lst:int-bulk-pseudo} shows an example where \textbf{bulk-acq} adds a significant layer of complexity to the internal signalling semantics, and listing \ref{lst:int-bulk-cfa} shows the corresponding \CFA code to implement the pseudo-code in listing \ref{lst:int-bulk-pseudo}. For the purpose of translating the given pseudo-code into \CFA-code, any method of introducing a monitor is acceptable, e.g., \code{mutex} parameters, global variables, pointer parameters, or using locals with the \code{mutex} statement.
1119
1120\begin{figure}[!t]
1121\begin{multicols}{2}
1123\begin{pseudo}[numbers=left]
1124acquire A
1125 //Code Section 1
1126 acquire A & B
1127 //Code Section 2
1128 wait A & B
1129 //Code Section 3
1130 release A & B
1131 //Code Section 4
1132release A
1133\end{pseudo}
1134\columnbreak
1136\begin{pseudo}[numbers=left, firstnumber=10,escapechar=|]
1137acquire A
1138 //Code Section 5
1139 acquire A & B
1140 //Code Section 6
1141 |\label{line:signal1}|signal A & B
1142 //Code Section 7
1143 |\label{line:releaseFirst}|release A & B
1144 //Code Section 8
1145|\label{line:lastRelease}|release A
1146\end{pseudo}
1147\end{multicols}
1148\begin{cfacode}[caption={Internal scheduling with \textbf{bulk-acq}},label={lst:int-bulk-pseudo}]
1149\end{cfacode}
1150\begin{center}
1151\begin{cfacode}[xleftmargin=.4\textwidth]
1152monitor A a;
1153monitor B b;
1154condition c;
1155\end{cfacode}
1156\end{center}
1157\begin{multicols}{2}
1159\begin{cfacode}
1160mutex(a) {
1161 //Code Section 1
1162 mutex(a, b) {
1163 //Code Section 2
1164 wait(c);
1165 //Code Section 3
1166 }
1167 //Code Section 4
1168}
1169\end{cfacode}
1170\columnbreak
1172\begin{cfacode}
1173mutex(a) {
1174 //Code Section 5
1175 mutex(a, b) {
1176 //Code Section 6
1177 signal(c);
1178 //Code Section 7
1179 }
1180 //Code Section 8
1181}
1182\end{cfacode}
1183\end{multicols}
1184\begin{cfacode}[caption={Equivalent \CFA code for listing \ref{lst:int-bulk-pseudo}},label={lst:int-bulk-cfa}]
1185\end{cfacode}
1186\begin{multicols}{2}
1187Waiter
1188\begin{pseudo}[numbers=left]
1189acquire A
1190 acquire A & B
1191 wait A & B
1192 release A & B
1193release A
1194\end{pseudo}
1195
1196\columnbreak
1197
1198Signaller
1199\begin{pseudo}[numbers=left, firstnumber=6,escapechar=|]
1200acquire A
1201 acquire A & B
1202 signal A & B
1203 release A & B
1204 |\label{line:secret}|//Secretly keep B here
1205release A
1206//Wakeup waiter and transfer A & B
1207\end{pseudo}
1208\end{multicols}
1209\begin{cfacode}[caption={Listing \ref{lst:int-bulk-pseudo}, with delayed signalling comments},label={lst:int-secret}]
1210\end{cfacode}
1211\end{figure}
1212
1213The complexity begins at code sections 4 and 8 in listing \ref{lst:int-bulk-pseudo}, which are where the existing semantics of internal scheduling needs to be extended for multiple monitors. The root of the problem is that \textbf{bulk-acq} is used in a context where one of the monitors is already acquired, which is why it is important to define the behaviour of the previous pseudo-code. When the signaller thread reaches the location where it should release \code{A & B}'' (listing \ref{lst:int-bulk-pseudo} line \ref{line:releaseFirst}), it must actually transfer ownership of monitor \code{B} to the waiting thread. This ownership transfer is required in order to prevent barging into \code{B} by another thread, since both the signalling and signalled threads still need monitor \code{A}. There are three options:
1214
1215\subsubsection{Delaying Signals}
1216The obvious solution to the problem of multi-monitor scheduling is to keep ownership of all locks until the last lock is ready to be transferred. It can be argued that that moment is when the last lock is no longer needed, because this semantics fits most closely to the behaviour of single-monitor scheduling. This solution has the main benefit of transferring ownership of groups of monitors, which simplifies the semantics from multiple objects to a single group of objects, effectively making the existing single-monitor semantic viable by simply changing monitors to monitor groups. This solution releases the monitors once every monitor in a group can be released. However, since some monitors are never released (e.g., the monitor of a thread), this interpretation means a group might never be released. A more interesting interpretation is to transfer the group until all its monitors are released, which means the group is not passed further and a thread can retain its locks.
1217
1218However, listing \ref{lst:int-secret} shows this solution can become much more complicated depending on what is executed while secretly holding B at line \ref{line:secret}, while avoiding the need to transfer ownership of a subset of the condition monitors. Listing \ref{lst:dependency} shows a slightly different example where a third thread is waiting on monitor \code{A}, using a different condition variable. Because the third thread is signalled when secretly holding \code{B}, the goal becomes unreachable. Depending on the order of signals (listing \ref{lst:dependency} line \ref{line:signal-ab} and \ref{line:signal-a}) two cases can happen:
1219
1220\paragraph{Case 1: thread $\alpha$ goes first.} In this case, the problem is that monitor \code{A} needs to be passed to thread $\beta$ when thread $\alpha$ is done with it.
1221\paragraph{Case 2: thread $\beta$ goes first.} In this case, the problem is that monitor \code{B} needs to be retained and passed to thread $\alpha$ along with monitor \code{A}, which can be done directly or possibly using thread $\beta$ as an intermediate.
1222\\
1223
1224Note that ordering is not determined by a race condition but by whether signalled threads are enqueued in FIFO or FILO order. However, regardless of the answer, users can move line \ref{line:signal-a} before line \ref{line:signal-ab} and get the reverse effect for listing \ref{lst:dependency}.
1225
1226In both cases, the threads need to be able to distinguish, on a per monitor basis, which ones need to be released and which ones need to be transferred, which means knowing when to release a group becomes complex and inefficient (see next section) and therefore effectively precludes this approach.
1227
1228\subsubsection{Dependency graphs}
1229
1230
1231\begin{figure}
1232\begin{multicols}{3}
1233Thread $\alpha$
1234\begin{pseudo}[numbers=left, firstnumber=1]
1235acquire A
1236 acquire A & B
1237 wait A & B
1238 release A & B
1239release A
1240\end{pseudo}
1241\columnbreak
1242Thread $\gamma$
1243\begin{pseudo}[numbers=left, firstnumber=6, escapechar=|]
1244acquire A
1245 acquire A & B
1246 |\label{line:signal-ab}|signal A & B
1247 |\label{line:release-ab}|release A & B
1248 |\label{line:signal-a}|signal A
1249|\label{line:release-a}|release A
1250\end{pseudo}
1251\columnbreak
1252Thread $\beta$
1253\begin{pseudo}[numbers=left, firstnumber=12, escapechar=|]
1254acquire A
1255 wait A
1256|\label{line:release-aa}|release A
1257\end{pseudo}
1258\end{multicols}
1259\begin{cfacode}[caption={Pseudo-code for the three thread example.},label={lst:dependency}]
1260\end{cfacode}
1261\begin{center}
1262\input{dependency}
1263\end{center}
1264\caption{Dependency graph of the statements in listing \ref{lst:dependency}}
1265\label{fig:dependency}
1266\end{figure}
1267
1268In listing \ref{lst:int-bulk-pseudo}, there is a solution that satisfies both barging prevention and mutual exclusion. If ownership of both monitors is transferred to the waiter when the signaller releases \code{A & B} and then the waiter transfers back ownership of \code{A} back to the signaller when it releases it, then the problem is solved (\code{B} is no longer in use at this point). Dynamically finding the correct order is therefore the second possible solution. The problem is effectively resolving a dependency graph of ownership requirements. Here even the simplest of code snippets requires two transfers and has a super-linear complexity. This complexity can be seen in listing \ref{lst:explosion}, which is just a direct extension to three monitors, requires at least three ownership transfer and has multiple solutions. Furthermore, the presence of multiple solutions for ownership transfer can cause deadlock problems if a specific solution is not consistently picked; In the same way that multiple lock acquiring order can cause deadlocks.
1269\begin{figure}
1270\begin{multicols}{2}
1271\begin{pseudo}
1272acquire A
1273 acquire B
1274 acquire C
1275 wait A & B & C
1276 release C
1277 release B
1278release A
1279\end{pseudo}
1280
1281\columnbreak
1282
1283\begin{pseudo}
1284acquire A
1285 acquire B
1286 acquire C
1287 signal A & B & C
1288 release C
1289 release B
1290release A
1291\end{pseudo}
1292\end{multicols}
1293\begin{cfacode}[caption={Extension to three monitors of listing \ref{lst:int-bulk-pseudo}},label={lst:explosion}]
1294\end{cfacode}
1295\end{figure}
1296
1297Given the three threads example in listing \ref{lst:dependency}, figure \ref{fig:dependency} shows the corresponding dependency graph that results, where every node is a statement of one of the three threads, and the arrows the dependency of that statement (e.g., $\alpha1$ must happen before $\alpha2$). The extra challenge is that this dependency graph is effectively post-mortem, but the runtime system needs to be able to build and solve these graphs as the dependencies unfold. Resolving dependency graphs being a complex and expensive endeavour, this solution is not the preferred one.
1298
1299\subsubsection{Partial Signalling} \label{partial-sig}
1300Finally, the solution that is chosen for \CFA is to use partial signalling. Again using listing \ref{lst:int-bulk-pseudo}, the partial signalling solution transfers ownership of monitor \code{B} at lines \ref{line:signal1} to the waiter but does not wake the waiting thread since it is still using monitor \code{A}. Only when it reaches line \ref{line:lastRelease} does it actually wake up the waiting thread. This solution has the benefit that complexity is encapsulated into only two actions: passing monitors to the next owner when they should be released and conditionally waking threads if all conditions are met. This solution has a much simpler implementation than a dependency graph solving algorithms, which is why it was chosen. Furthermore, after being fully implemented, this solution does not appear to have any significant downsides.
1301
1302Using partial signalling, listing \ref{lst:dependency} can be solved easily:
1303\begin{itemize}
1304 \item When thread $\gamma$ reaches line \ref{line:release-ab} it transfers monitor \code{B} to thread $\alpha$ and continues to hold monitor \code{A}.
1305 \item When thread $\gamma$ reaches line \ref{line:release-a} it transfers monitor \code{A} to thread $\beta$ and wakes it up.
1306 \item When thread $\beta$ reaches line \ref{line:release-aa} it transfers monitor \code{A} to thread $\alpha$ and wakes it up.
1307\end{itemize}
1308
1309% ======================================================================
1310% ======================================================================
1311\subsection{Signalling: Now or Later}
1312% ======================================================================
1313% ======================================================================
1314\begin{table}
1315\begin{tabular}{|c|c|}
1316\code{signal} & \code{signal_block} \\
1317\hline
1318\begin{cfacode}[tabsize=3]
1319monitor DatingService
1320{
1321 //compatibility codes
1322 enum{ CCodes = 20 };
1323
1324 int girlPhoneNo
1325 int boyPhoneNo;
1326};
1327
1328condition girls[CCodes];
1329condition boys [CCodes];
1330condition exchange;
1331
1332int girl(int phoneNo, int ccode)
1333{
1334 //no compatible boy ?
1335 if(empty(boys[ccode]))
1336 {
1337 //wait for boy
1338 wait(girls[ccode]);
1339
1340 //make phone number available
1341 girlPhoneNo = phoneNo;
1342
1343 //wake boy from chair
1344 signal(exchange);
1345 }
1346 else
1347 {
1348 //make phone number available
1349 girlPhoneNo = phoneNo;
1350
1351 //wake boy
1352 signal(boys[ccode]);
1353
1354 //sit in chair
1355 wait(exchange);
1356 }
1357 return boyPhoneNo;
1358}
1359
1360int boy(int phoneNo, int ccode)
1361{
1362 //same as above
1363 //with boy/girl interchanged
1364}
1365\end{cfacode}&\begin{cfacode}[tabsize=3]
1366monitor DatingService
1367{
1368 //compatibility codes
1369 enum{ CCodes = 20 };
1370
1371 int girlPhoneNo;
1372 int boyPhoneNo;
1373};
1374
1375condition girls[CCodes];
1376condition boys [CCodes];
1377//exchange is not needed
1378
1379int girl(int phoneNo, int ccode)
1380{
1381 //no compatible boy ?
1382 if(empty(boys[ccode]))
1383 {
1384 //wait for boy
1385 wait(girls[ccode]);
1386
1387 //make phone number available
1388 girlPhoneNo = phoneNo;
1389
1390 //wake boy from chair
1391 signal(exchange);
1392 }
1393 else
1394 {
1395 //make phone number available
1396 girlPhoneNo = phoneNo;
1397
1398 //wake boy
1399 signal_block(boys[ccode]);
1400
1401 //second handshake unnecessary
1402
1403 }
1404 return boyPhoneNo;
1405}
1406
1407int boy(int phoneNo, int ccode)
1408{
1409 //same as above
1410 //with boy/girl interchanged
1411}
1412\end{cfacode}
1413\end{tabular}
1414\caption{Dating service example using \code{signal} and \code{signal_block}. }
1415\label{tbl:datingservice}
1416\end{table}
1417An important note is that, until now, signalling a monitor was a delayed operation. The ownership of the monitor is transferred only when the monitor would have otherwise been released, not at the point of the \code{signal} statement. However, in some cases, it may be more convenient for users to immediately transfer ownership to the thread that is waiting for cooperation, which is achieved using the \code{signal_block} routine.
1418
1419The example in table \ref{tbl:datingservice} highlights the difference in behaviour. As mentioned, \code{signal} only transfers ownership once the current critical section exits; this behaviour requires additional synchronization when a two-way handshake is needed. To avoid this explicit synchronization, the \code{condition} type offers the \code{signal_block} routine, which handles the two-way handshake as shown in the example. This feature removes the need for a second condition variables and simplifies programming. Like every other monitor semantic, \code{signal_block} uses barging prevention, which means mutual-exclusion is baton-passed both on the front end and the back end of the call to \code{signal_block}, meaning no other thread can acquire the monitor either before or after the call.
1420
1421% ======================================================================
1422% ======================================================================
1423\section{External scheduling} \label{extsched}
1424% ======================================================================
1425% ======================================================================
1426An alternative to internal scheduling is external scheduling (see Table~\ref{tbl:sched}).
1427\begin{table}
1428\begin{tabular}{|c|c|c|}
1429Internal Scheduling & External Scheduling & Go\\
1430\hline
1431\begin{ucppcode}[tabsize=3]
1432_Monitor Semaphore {
1433 condition c;
1434 bool inUse;
1435public:
1436 void P() {
1437 if(inUse)
1438 wait(c);
1439 inUse = true;
1440 }
1441 void V() {
1442 inUse = false;
1443 signal(c);
1444 }
1445}
1446\end{ucppcode}&\begin{ucppcode}[tabsize=3]
1447_Monitor Semaphore {
1448
1449 bool inUse;
1450public:
1451 void P() {
1452 if(inUse)
1453 _Accept(V);
1454 inUse = true;
1455 }
1456 void V() {
1457 inUse = false;
1458
1459 }
1460}
1461\end{ucppcode}&\begin{gocode}[tabsize=3]
1462type MySem struct {
1463 inUse bool
1464 c chan bool
1465}
1466
1467// acquire
1468func (s MySem) P() {
1469 if s.inUse {
1470 select {
1471 case <-s.c:
1472 }
1473 }
1474 s.inUse = true
1475}
1476
1477// release
1478func (s MySem) V() {
1479 s.inUse = false
1480
1483 s.c <- false
1484}
1485\end{gocode}
1486\end{tabular}
1487\caption{Different forms of scheduling.}
1488\label{tbl:sched}
1489\end{table}
1490This method is more constrained and explicit, which helps users reduce the non-deterministic nature of concurrency. Indeed, as the following examples demonstrate, external scheduling allows users to wait for events from other threads without the concern of unrelated events occurring. External scheduling can generally be done either in terms of control flow (e.g., Ada with \code{accept}, \uC with \code{_Accept}) or in terms of data (e.g., Go with channels). Of course, both of these paradigms have their own strengths and weaknesses, but for this project, control-flow semantics was chosen to stay consistent with the rest of the languages semantics. Two challenges specific to \CFA arise when trying to add external scheduling with loose object definitions and multiple-monitor routines. The previous example shows a simple use \code{_Accept} versus \code{wait}/\code{signal} and its advantages. Note that while other languages often use \code{accept}/\code{select} as the core external scheduling keyword, \CFA uses \code{waitfor} to prevent name collisions with existing socket \textbf{api}s.
1491
1492For the \code{P} member above using internal scheduling, the call to \code{wait} only guarantees that \code{V} is the last routine to access the monitor, allowing a third routine, say \code{isInUse()}, acquire mutual exclusion several times while routine \code{P} is waiting. On the other hand, external scheduling guarantees that while routine \code{P} is waiting, no other routine than \code{V} can acquire the monitor.
1493
1494% ======================================================================
1495% ======================================================================
1496\subsection{Loose Object Definitions}
1497% ======================================================================
1498% ======================================================================
1499In \uC, a monitor class declaration includes an exhaustive list of monitor operations. Since \CFA is not object oriented, monitors become both more difficult to implement and less clear for a user:
1500
1501\begin{cfacode}
1502monitor A {};
1503
1504void f(A & mutex a);
1505void g(A & mutex a) {
1506 waitfor(f); //Obvious which f() to wait for
1507}
1508
1509void f(A & mutex a, int); //New different F added in scope
1510void h(A & mutex a) {
1511 waitfor(f); //Less obvious which f() to wait for
1512}
1513\end{cfacode}
1514
1515Furthermore, external scheduling is an example where implementation constraints become visible from the interface. Here is the pseudo-code for the entering phase of a monitor:
1516\begin{center}
1517\begin{tabular}{l}
1518\begin{pseudo}
1519 if monitor is free
1520 enter
1521 elif already own the monitor
1522 continue
1523 elif monitor accepts me
1524 enter
1525 else
1526 block
1527\end{pseudo}
1528\end{tabular}
1529\end{center}
1530For the first two conditions, it is easy to implement a check that can evaluate the condition in a few instructions. However, a fast check for \pscode{monitor accepts me} is much harder to implement depending on the constraints put on the monitors. Indeed, monitors are often expressed as an entry queue and some acceptor queue as in Figure~\ref{fig:ClassicalMonitor}.
1531
1532\begin{figure}
1533\centering
1534\subfloat[Classical Monitor] {
1535\label{fig:ClassicalMonitor}
1536{\resizebox{0.45\textwidth}{!}{\input{monitor}}}
1537}% subfloat
1539\subfloat[\textbf{bulk-acq} Monitor] {
1540\label{fig:BulkMonitor}
1541{\resizebox{0.45\textwidth}{!}{\input{ext_monitor}}}
1542}% subfloat
1543\caption{External Scheduling Monitor}
1544\end{figure}
1545
1546There are other alternatives to these pictures, but in the case of the left picture, implementing a fast accept check is relatively easy. Restricted to a fixed number of mutex members, N, the accept check reduces to updating a bitmask when the acceptor queue changes, a check that executes in a single instruction even with a fairly large number (e.g., 128) of mutex members. This approach requires a unique dense ordering of routines with an upper-bound and that ordering must be consistent across translation units. For OO languages these constraints are common, since objects only offer adding member routines consistently across translation units via inheritance. However, in \CFA users can extend objects with mutex routines that are only visible in certain translation unit. This means that establishing a program-wide dense-ordering among mutex routines can only be done in the program linking phase, and still could have issues when using dynamically shared objects.
1547
1548The alternative is to alter the implementation as in Figure~\ref{fig:BulkMonitor}.
1549Here, the mutex routine called is associated with a thread on the entry queue while a list of acceptable routines is kept separate. Generating a mask dynamically means that the storage for the mask information can vary between calls to \code{waitfor}, allowing for more flexibility and extensions. Storing an array of accepted function pointers replaces the single instruction bitmask comparison with dereferencing a pointer followed by a linear search. Furthermore, supporting nested external scheduling (e.g., listing \ref{lst:nest-ext}) may now require additional searches for the \code{waitfor} statement to check if a routine is already queued.
1550
1551\begin{figure}
1552\begin{cfacode}[caption={Example of nested external scheduling},label={lst:nest-ext}]
1553monitor M {};
1554void foo( M & mutex a ) {}
1555void bar( M & mutex b ) {
1556 //Nested in the waitfor(bar, c) call
1557 waitfor(foo, b);
1558}
1559void baz( M & mutex c ) {
1560 waitfor(bar, c);
1561}
1562
1563\end{cfacode}
1564\end{figure}
1565
1566Note that in the right picture, tasks need to always keep track of the monitors associated with mutex routines, and the routine mask needs to have both a function pointer and a set of monitors, as is discussed in the next section. These details are omitted from the picture for the sake of simplicity.
1567
1568At this point, a decision must be made between flexibility and performance. Many design decisions in \CFA achieve both flexibility and performance, for example polymorphic routines add significant flexibility but inlining them means the optimizer can easily remove any runtime cost. Here, however, the cost of flexibility cannot be trivially removed. In the end, the most flexible approach has been chosen since it allows users to write programs that would otherwise be hard to write. This decision is based on the assumption that writing fast but inflexible locks is closer to a solved problem than writing locks that are as flexible as external scheduling in \CFA.
1569
1570% ======================================================================
1571% ======================================================================
1572\subsection{Multi-Monitor Scheduling}
1573% ======================================================================
1574% ======================================================================
1575
1576External scheduling, like internal scheduling, becomes significantly more complex when introducing multi-monitor syntax. Even in the simplest possible case, some new semantics needs to be established:
1577\begin{cfacode}
1578monitor M {};
1579
1580void f(M & mutex a);
1581
1582void g(M & mutex b, M & mutex c) {
1583 waitfor(f); //two monitors M => unknown which to pass to f(M & mutex)
1584}
1585\end{cfacode}
1586The obvious solution is to specify the correct monitor as follows:
1587
1588\begin{cfacode}
1589monitor M {};
1590
1591void f(M & mutex a);
1592
1593void g(M & mutex a, M & mutex b) {
1594 //wait for call to f with argument b
1595 waitfor(f, b);
1596}
1597\end{cfacode}
1598This syntax is unambiguous. Both locks are acquired and kept by \code{g}. When routine \code{f} is called, the lock for monitor \code{b} is temporarily transferred from \code{g} to \code{f} (while \code{g} still holds lock \code{a}). This behaviour can be extended to the multi-monitor \code{waitfor} statement as follows.
1599
1600\begin{cfacode}
1601monitor M {};
1602
1603void f(M & mutex a, M & mutex b);
1604
1605void g(M & mutex a, M & mutex b) {
1606 //wait for call to f with arguments a and b
1607 waitfor(f, a, b);
1608}
1609\end{cfacode}
1610
1611Note that the set of monitors passed to the \code{waitfor} statement must be entirely contained in the set of monitors already acquired in the routine. \code{waitfor} used in any other context is undefined behaviour.
1612
1613An important behaviour to note is when a set of monitors only match partially:
1614
1615\begin{cfacode}
1616mutex struct A {};
1617
1618mutex struct B {};
1619
1620void g(A & mutex a, B & mutex b) {
1621 waitfor(f, a, b);
1622}
1623
1624A a1, a2;
1625B b;
1626
1627void foo() {
1628 g(a1, b); //block on accept
1629}
1630
1631void bar() {
1632 f(a2, b); //fulfill cooperation
1633}
1634\end{cfacode}
1635While the equivalent can happen when using internal scheduling, the fact that conditions are specific to a set of monitors means that users have to use two different condition variables. In both cases, partially matching monitor sets does not wakeup the waiting thread. It is also important to note that in the case of external scheduling the order of parameters is irrelevant; \code{waitfor(f,a,b)} and \code{waitfor(f,b,a)} are indistinguishable waiting condition.
1636
1637% ======================================================================
1638% ======================================================================
1639\subsection{\code{waitfor} Semantics}
1640% ======================================================================
1641% ======================================================================
1642
1643Syntactically, the \code{waitfor} statement takes a function identifier and a set of monitors. While the set of monitors can be any list of expressions, the function name is more restricted because the compiler validates at compile time the validity of the function type and the parameters used with the \code{waitfor} statement. It checks that the set of monitors passed in matches the requirements for a function call. Listing \ref{lst:waitfor} shows various usages of the waitfor statement and which are acceptable. The choice of the function type is made ignoring any non-\code{mutex} parameter. One limitation of the current implementation is that it does not handle overloading, but overloading is possible.
1644\begin{figure}
1645\begin{cfacode}[caption={Various correct and incorrect uses of the waitfor statement},label={lst:waitfor}]
1646monitor A{};
1647monitor B{};
1648
1649void f1( A & mutex );
1650void f2( A & mutex, B & mutex );
1651void f3( A & mutex, int );
1652void f4( A & mutex, int );
1653void f4( A & mutex, double );
1654
1655void foo( A & mutex a1, A & mutex a2, B & mutex b1, B & b2 ) {
1656 A * ap = & a1;
1657 void (*fp)( A & mutex ) = f1;
1658
1659 waitfor(f1, a1); //Correct : 1 monitor case
1660 waitfor(f2, a1, b1); //Correct : 2 monitor case
1661 waitfor(f3, a1); //Correct : non-mutex arguments are ignored
1662 waitfor(f1, *ap); //Correct : expression as argument
1663
1664 waitfor(f1, a1, b1); //Incorrect : Too many mutex arguments
1665 waitfor(f2, a1); //Incorrect : Too few mutex arguments
1666 waitfor(f2, a1, a2); //Incorrect : Mutex arguments don't match
1667 waitfor(f1, 1); //Incorrect : 1 not a mutex argument
1668 waitfor(f9, a1); //Incorrect : f9 function does not exist
1669 waitfor(*fp, a1 ); //Incorrect : fp not an identifier
1670 waitfor(f4, a1); //Incorrect : f4 ambiguous
1671
1672 waitfor(f2, a1, b2); //Undefined behaviour : b2 not mutex
1673}
1674\end{cfacode}
1675\end{figure}
1676
1677Finally, for added flexibility, \CFA supports constructing a complex \code{waitfor} statement using the \code{or}, \code{timeout} and \code{else}. Indeed, multiple \code{waitfor} clauses can be chained together using \code{or}; this chain forms a single statement that uses baton pass to any function that fits one of the function+monitor set passed in. To enable users to tell which accepted function executed, \code{waitfor}s are followed by a statement (including the null statement \code{;}) or a compound statement, which is executed after the clause is triggered. A \code{waitfor} chain can also be followed by a \code{timeout}, to signify an upper bound on the wait, or an \code{else}, to signify that the call should be non-blocking, which checks for a matching function call already arrived and otherwise continues. Any and all of these clauses can be preceded by a \code{when} condition to dynamically toggle the accept clauses on or off based on some current state. Listing \ref{lst:waitfor2} demonstrates several complex masks and some incorrect ones.
1678
1679\begin{figure}
1680\begin{cfacode}[caption={Various correct and incorrect uses of the or, else, and timeout clause around a waitfor statement},label={lst:waitfor2}]
1681monitor A{};
1682
1683void f1( A & mutex );
1684void f2( A & mutex );
1685
1686void foo( A & mutex a, bool b, int t ) {
1687 //Correct : blocking case
1688 waitfor(f1, a);
1689
1690 //Correct : block with statement
1691 waitfor(f1, a) {
1692 sout | "f1" | endl;
1693 }
1694
1695 //Correct : block waiting for f1 or f2
1696 waitfor(f1, a) {
1697 sout | "f1" | endl;
1698 } or waitfor(f2, a) {
1699 sout | "f2" | endl;
1700 }
1701
1702 //Correct : non-blocking case
1703 waitfor(f1, a); or else;
1704
1705 //Correct : non-blocking case
1706 waitfor(f1, a) {
1707 sout | "blocked" | endl;
1708 } or else {
1709 sout | "didn't block" | endl;
1710 }
1711
1712 //Correct : block at most 10 seconds
1713 waitfor(f1, a) {
1714 sout | "blocked" | endl;
1715 } or timeout( 10s) {
1716 sout | "didn't block" | endl;
1717 }
1718
1719 //Correct : block only if b == true
1720 //if b == false, don't even make the call
1721 when(b) waitfor(f1, a);
1722
1723 //Correct : block only if b == true
1724 //if b == false, make non-blocking call
1725 waitfor(f1, a); or when(!b) else;
1726
1727 //Correct : block only of t > 1
1728 waitfor(f1, a); or when(t > 1) timeout(t); or else;
1729
1730 //Incorrect : timeout clause is dead code
1731 waitfor(f1, a); or timeout(t); or else;
1732
1733 //Incorrect : order must be
1734 //waitfor [or waitfor... [or timeout] [or else]]
1735 timeout(t); or waitfor(f1, a); or else;
1736}
1737\end{cfacode}
1738\end{figure}
1739
1740% ======================================================================
1741% ======================================================================
1742\subsection{Waiting For The Destructor}
1743% ======================================================================
1744% ======================================================================
1745An interesting use for the \code{waitfor} statement is destructor semantics. Indeed, the \code{waitfor} statement can accept any \code{mutex} routine, which includes the destructor (see section \ref{data}). However, with the semantics discussed until now, waiting for the destructor does not make any sense, since using an object after its destructor is called is undefined behaviour. The simplest approach is to disallow \code{waitfor} on a destructor. However, a more expressive approach is to flip ordering of execution when waiting for the destructor, meaning that waiting for the destructor allows the destructor to run after the current \code{mutex} routine, similarly to how a condition is signalled.
1746\begin{figure}
1747\begin{cfacode}[caption={Example of an executor which executes action in series until the destructor is called.},label={lst:dtor-order}]
1748monitor Executer {};
1749struct Action;
1750
1751void ^?{} (Executer & mutex this);
1752void execute(Executer & mutex this, const Action & );
1753void run (Executer & mutex this) {
1754 while(true) {
1755 waitfor(execute, this);
1756 or waitfor(^?{} , this) {
1757 break;
1758 }
1759 }
1760}
1761\end{cfacode}
1762\end{figure}
1763For example, listing \ref{lst:dtor-order} shows an example of an executor with an infinite loop, which waits for the destructor to break out of this loop. Switching the semantic meaning introduces an idiomatic way to terminate a task and/or wait for its termination via destruction.
1764
1765
1766% ###### # ###### # # # ####### # ### ##### # #
1767% # # # # # # # # # # # # # # # ## ##
1768% # # # # # # # # # # # # # # # # # #
1769% ###### # # ###### # # # # ##### # # ##### # # #
1770% # ####### # # ####### # # # # # # # #
1771% # # # # # # # # # # # # # # # #
1772% # # # # # # # ####### ####### ####### ####### ### ##### # #
1773\section{Parallelism}
1774Historically, computer performance was about processor speeds and instruction counts. However, with heat dissipation being a direct consequence of speed increase, parallelism has become the new source for increased performance~\cite{Sutter05, Sutter05b}. In this decade, it is no longer reasonable to create a high-performance application without caring about parallelism. Indeed, parallelism is an important aspect of performance and more specifically throughput and hardware utilization. The lowest-level approach of parallelism is to use \textbf{kthread} in combination with semantics like \code{fork}, \code{join}, etc. However, since these have significant costs and limitations, \textbf{kthread} are now mostly used as an implementation tool rather than a user oriented one. There are several alternatives to solve these issues that all have strengths and weaknesses. While there are many variations of the presented paradigms, most of these variations do not actually change the guarantees or the semantics, they simply move costs in order to achieve better performance for certain workloads.
1775
1778A direct improvement on the \textbf{kthread} approach is to use \textbf{uthread}. These threads offer most of the same features that the operating system already provides but can be used on a much larger scale. This approach is the most powerful solution as it allows all the features of multithreading, while removing several of the more expensive costs of kernel threads. The downside is that almost none of the low-level threading problems are hidden; users still have to think about data races, deadlocks and synchronization issues. These issues can be somewhat alleviated by a concurrency toolkit with strong guarantees, but the parallelism toolkit offers very little to reduce complexity in itself.
1779
1780Examples of languages that support \textbf{uthread} are Erlang~\cite{Erlang} and \uC~\cite{uC++book}.
1781
1782\subsection{Fibers : User-Level Threads Without Preemption} \label{fibers}
1783A popular variant of \textbf{uthread} is what is often referred to as \textbf{fiber}. However, \textbf{fiber} do not present meaningful semantic differences with \textbf{uthread}. The significant difference between \textbf{uthread} and \textbf{fiber} is the lack of \textbf{preemption} in the latter. Advocates of \textbf{fiber} list their high performance and ease of implementation as major strengths, but the performance difference between \textbf{uthread} and \textbf{fiber} is controversial, and the ease of implementation, while true, is a weak argument in the context of language design. Therefore this proposal largely ignores fibers.
1784
1785An example of a language that uses fibers is Go~\cite{Go}
1786
1788An approach on the opposite end of the spectrum is to base parallelism on \textbf{pool}. Indeed, \textbf{pool} offer limited flexibility but at the benefit of a simpler user interface. In \textbf{pool} based systems, users express parallelism as units of work, called jobs, and a dependency graph (either explicit or implicit) that ties them together. This approach means users need not worry about concurrency but significantly limit the interaction that can occur among jobs. Indeed, any \textbf{job} that blocks also block the underlying worker, which effectively means the CPU utilization, and therefore throughput, suffers noticeably. It can be argued that a solution to this problem is to use more workers than available cores. However, unless the number of jobs and the number of workers are comparable, having a significant number of blocked jobs always results in idles cores.
1789
1790The gold standard of this implementation is Intel's TBB library~\cite{TBB}.
1791
1793While the choice between the three paradigms listed above may have significant performance implications, it is difficult to pin down the performance implications of choosing a model at the language level. Indeed, in many situations one of these paradigms may show better performance but it all strongly depends on the workload. Having a large amount of mostly independent units of work to execute almost guarantees equivalent performance across paradigms and that the \textbf{pool}-based system has the best efficiency thanks to the lower memory overhead (i.e., no thread stack per job). However, interactions among jobs can easily exacerbate contention. User-level threads allow fine-grain context switching, which results in better resource utilization, but a context switch is more expensive and the extra control means users need to tweak more variables to get the desired performance. Finally, if the units of uninterrupted work are large, enough the paradigm choice is largely amortized by the actual work done.
1794
1795\section{The \protect\CFA\ Kernel : Processors, Clusters and Threads}\label{kernel}
1796A \textbf{cfacluster} is a group of \textbf{kthread} executed in isolation. \textbf{uthread} are scheduled on the \textbf{kthread} of a given \textbf{cfacluster}, allowing organization between \textbf{uthread} and \textbf{kthread}. It is important that \textbf{kthread} belonging to a same \textbf{cfacluster} have homogeneous settings, otherwise migrating a \textbf{uthread} from one \textbf{kthread} to the other can cause issues. A \textbf{cfacluster} also offers a pluggable scheduler that can optimize the workload generated by the \textbf{uthread}.
1797
1798\textbf{cfacluster} have not been fully implemented in the context of this paper. Currently \CFA only supports one \textbf{cfacluster}, the initial one.
1799
1800\subsection{Future Work: Machine Setup}\label{machine}
1801While this was not done in the context of this paper, another important aspect of clusters is affinity. While many common desktop and laptop PCs have homogeneous CPUs, other devices often have more heterogeneous setups. For example, a system using \textbf{numa} configurations may benefit from users being able to tie clusters and/or kernel threads to certain CPU cores. OS support for CPU affinity is now common~\cite{affinityLinux, affinityWindows, affinityFreebsd, affinityNetbsd, affinityMacosx}, which means it is both possible and desirable for \CFA to offer an abstraction mechanism for portable CPU affinity.
1802
1804Given these building blocks, it is possible to reproduce all three of the popular paradigms. Indeed, \textbf{uthread} is the default paradigm in \CFA. However, disabling \textbf{preemption} on the \textbf{cfacluster} means \textbf{cfathread} effectively become \textbf{fiber}. Since several \textbf{cfacluster} with different scheduling policy can coexist in the same application, this allows \textbf{fiber} and \textbf{uthread} to coexist in the runtime of an application. Finally, it is possible to build executors for thread pools from \textbf{uthread} or \textbf{fiber}, which includes specialized jobs like actors~\cite{Actors}.
1805
1806
1807
1808\section{Behind the Scenes}
1809There are several challenges specific to \CFA when implementing concurrency. These challenges are a direct result of \textbf{bulk-acq} and loose object definitions. These two constraints are the root cause of most design decisions in the implementation. Furthermore, to avoid contention from dynamically allocating memory in a concurrent environment, the internal-scheduling design is (almost) entirely free of mallocs. This approach avoids the chicken and egg problem~\cite{Chicken} of having a memory allocator that relies on the threading system and a threading system that relies on the runtime. This extra goal means that memory management is a constant concern in the design of the system.
1810
1811The main memory concern for concurrency is queues. All blocking operations are made by parking threads onto queues and all queues are designed with intrusive nodes, where each node has pre-allocated link fields for chaining, to avoid the need for memory allocation. Since several concurrency operations can use an unbound amount of memory (depending on \textbf{bulk-acq}), statically defining information in the intrusive fields of threads is insufficient.The only way to use a variable amount of memory without requiring memory allocation is to pre-allocate large buffers of memory eagerly and store the information in these buffers. Conveniently, the call stack fits that description and is easy to use, which is why it is used heavily in the implementation of internal scheduling, particularly variable-length arrays. Since stack allocation is based on scopes, the first step of the implementation is to identify the scopes that are available to store the information, and which of these can have a variable-length array. The threads and the condition both have a fixed amount of memory, while \code{mutex} routines and blocking calls allow for an unbound amount, within the stack size.
1812
1813Note that since the major contributions of this paper are extending monitor semantics to \textbf{bulk-acq} and loose object definitions, any challenges that are not resulting of these characteristics of \CFA are considered as solved problems and therefore not discussed.
1814
1815% ======================================================================
1816% ======================================================================
1817\section{Mutex Routines}
1818% ======================================================================
1819% ======================================================================
1820
1821The first step towards the monitor implementation is simple \code{mutex} routines. In the single monitor case, mutual-exclusion is done using the entry/exit procedure in listing \ref{lst:entry1}. The entry/exit procedures do not have to be extended to support multiple monitors. Indeed it is sufficient to enter/leave monitors one-by-one as long as the order is correct to prevent deadlock~\cite{Havender68}. In \CFA, ordering of monitor acquisition relies on memory ordering. This approach is sufficient because all objects are guaranteed to have distinct non-overlapping memory layouts and mutual-exclusion for a monitor is only defined for its lifetime, meaning that destroying a monitor while it is acquired is undefined behaviour. When a mutex call is made, the concerned monitors are aggregated into a variable-length pointer array and sorted based on pointer values. This array persists for the entire duration of the mutual-exclusion and its ordering reused extensively.
1822\begin{figure}
1823\begin{multicols}{2}
1824Entry
1825\begin{pseudo}
1826if monitor is free
1827 enter
1829 continue
1830else
1831 block
1832increment recursions
1833\end{pseudo}
1834\columnbreak
1835Exit
1836\begin{pseudo}
1837decrement recursion
1838if recursion == 0
1839 if entry queue not empty
1841\end{pseudo}
1842\end{multicols}
1843\begin{pseudo}[caption={Initial entry and exit routine for monitors},label={lst:entry1}]
1844\end{pseudo}
1845\end{figure}
1846
1847\subsection{Details: Interaction with polymorphism}
1848Depending on the choice of semantics for when monitor locks are acquired, interaction between monitors and \CFA's concept of polymorphism can be more complex to support. However, it is shown that entry-point locking solves most of the issues.
1849
1850First of all, interaction between \code{otype} polymorphism (see Section~\ref{s:ParametricPolymorphism}) and monitors is impossible since monitors do not support copying. Therefore, the main question is how to support \code{dtype} polymorphism. It is important to present the difference between the two acquiring options: \textbf{callsite-locking} and entry-point locking, i.e., acquiring the monitors before making a mutex routine-call or as the first operation of the mutex routine-call. For example:
1851\begin{table}[H]
1852\begin{center}
1853\begin{tabular}{|c|c|c|}
1854Mutex & \textbf{callsite-locking} & \textbf{entry-point-locking} \\
1855call & pseudo-code & pseudo-code \\
1856\hline
1857\begin{cfacode}[tabsize=3]
1858void foo(monitor& mutex a){
1859
1860 //Do Work
1861 //...
1862
1863}
1864
1865void main() {
1866 monitor a;
1867
1868 foo(a);
1869
1870}
1871\end{cfacode} & \begin{pseudo}[tabsize=3]
1872foo(& a) {
1873
1874 //Do Work
1875 //...
1876
1877}
1878
1879main() {
1880 monitor a;
1881 acquire(a);
1882 foo(a);
1883 release(a);
1884}
1885\end{pseudo} & \begin{pseudo}[tabsize=3]
1886foo(& a) {
1887 acquire(a);
1888 //Do Work
1889 //...
1890 release(a);
1891}
1892
1893main() {
1894 monitor a;
1895
1896 foo(a);
1897
1898}
1899\end{pseudo}
1900\end{tabular}
1901\end{center}
1902\caption{Call-site vs entry-point locking for mutex calls}
1903\label{tbl:locking-site}
1904\end{table}
1905
1906Note the \code{mutex} keyword relies on the type system, which means that in cases where a generic monitor-routine is desired, writing the mutex routine is possible with the proper trait, e.g.:
1907\begin{cfacode}
1908//Incorrect: T may not be monitor
1909forall(dtype T)
1910void foo(T * mutex t);
1911
1912//Correct: this function only works on monitors (any monitor)
1913forall(dtype T | is_monitor(T))
1914void bar(T * mutex t));
1915\end{cfacode}
1916
1917Both entry point and \textbf{callsite-locking} are feasible implementations. The current \CFA implementation uses entry-point locking because it requires less work when using \textbf{raii}, effectively transferring the burden of implementation to object construction/destruction. It is harder to use \textbf{raii} for call-site locking, as it does not necessarily have an existing scope that matches exactly the scope of the mutual exclusion, i.e., the function body. For example, the monitor call can appear in the middle of an expression. Furthermore, entry-point locking requires less code generation since any useful routine is called multiple times but there is only one entry point for many call sites.
1918
1919% ======================================================================
1920% ======================================================================
1922% ======================================================================
1923% ======================================================================
1924
1925Figure \ref{fig:system1} shows a high-level picture if the \CFA runtime system in regards to concurrency. Each component of the picture is explained in detail in the flowing sections.
1926
1927\begin{figure}
1928\begin{center}
1929{\resizebox{\textwidth}{!}{\input{system.pstex_t}}}
1930\end{center}
1931\caption{Overview of the entire system}
1932\label{fig:system1}
1933\end{figure}
1934
1935\subsection{Processors}
1936Parallelism in \CFA is built around using processors to specify how much parallelism is desired. \CFA processors are object wrappers around kernel threads, specifically \texttt{pthread}s in the current implementation of \CFA. Indeed, any parallelism must go through operating-system libraries. However, \textbf{uthread} are still the main source of concurrency, processors are simply the underlying source of parallelism. Indeed, processor \textbf{kthread} simply fetch a \textbf{uthread} from the scheduler and run it; they are effectively executers for user-threads. The main benefit of this approach is that it offers a well-defined boundary between kernel code and user code, for example, kernel thread quiescing, scheduling and interrupt handling. Processors internally use coroutines to take advantage of the existing context-switching semantics.
1937
1938\subsection{Stack Management}
1939One of the challenges of this system is to reduce the footprint as much as possible. Specifically, all \texttt{pthread}s created also have a stack created with them, which should be used as much as possible. Normally, coroutines also create their own stack to run on, however, in the case of the coroutines used for processors, these coroutines run directly on the \textbf{kthread} stack, effectively stealing the processor stack. The exception to this rule is the Main Processor, i.e., the initial \textbf{kthread} that is given to any program. In order to respect C user expectations, the stack of the initial kernel thread, the main stack of the program, is used by the main user thread rather than the main processor, which can grow very large.
1940
1941\subsection{Context Switching}
1942As mentioned in section \ref{coroutine}, coroutines are a stepping stone for implementing threading, because they share the same mechanism for context-switching between different stacks. To improve performance and simplicity, context-switching is implemented using the following assumption: all context-switches happen inside a specific function call. This assumption means that the context-switch only has to copy the callee-saved registers onto the stack and then switch the stack registers with the ones of the target coroutine/thread. Note that the instruction pointer can be left untouched since the context-switch is always inside the same function. Threads, however, do not context-switch between each other directly. They context-switch to the scheduler. This method is called a 2-step context-switch and has the advantage of having a clear distinction between user code and the kernel where scheduling and other system operations happen. Obviously, this doubles the context-switch cost because threads must context-switch to an intermediate stack. The alternative 1-step context-switch uses the stack of the from'' thread to schedule and then context-switches directly to the to'' thread. However, the performance of the 2-step context-switch is still superior to a \code{pthread_yield} (see section \ref{results}). Additionally, for users in need for optimal performance, it is important to note that having a 2-step context-switch as the default does not prevent \CFA from offering a 1-step context-switch (akin to the Microsoft \code{SwitchToFiber}~\cite{switchToWindows} routine). This option is not currently present in \CFA, but the changes required to add it are strictly additive.
1943
1944\subsection{Preemption} \label{preemption}
1945Finally, an important aspect for any complete threading system is preemption. As mentioned in section \ref{basics}, preemption introduces an extra degree of uncertainty, which enables users to have multiple threads interleave transparently, rather than having to cooperate among threads for proper scheduling and CPU distribution. Indeed, preemption is desirable because it adds a degree of isolation among threads. In a fully cooperative system, any thread that runs a long loop can starve other threads, while in a preemptive system, starvation can still occur but it does not rely on every thread having to yield or block on a regular basis, which reduces significantly a programmer burden. Obviously, preemption is not optimal for every workload. However any preemptive system can become a cooperative system by making the time slices extremely large. Therefore, \CFA uses a preemptive threading system.
1946
1947Preemption in \CFA\footnote{Note that the implementation of preemption is strongly tied with the underlying threading system. For this reason, only the Linux implementation is cover, \CFA does not run on Windows at the time of writting} is based on kernel timers, which are used to run a discrete-event simulation. Every processor keeps track of the current time and registers an expiration time with the preemption system. When the preemption system receives a change in preemption, it inserts the time in a sorted order and sets a kernel timer for the closest one, effectively stepping through preemption events on each signal sent by the timer. These timers use the Linux signal {\tt SIGALRM}, which is delivered to the process rather than the kernel-thread. This results in an implementation problem, because when delivering signals to a process, the kernel can deliver the signal to any kernel thread for which the signal is not blocked, i.e.:
1948\begin{quote}
1949A process-directed signal may be delivered to any one of the threads that does not currently have the signal blocked. If more than one of the threads has the signal unblocked, then the kernel chooses an arbitrary thread to which to deliver the signal.
1950SIGNAL(7) - Linux Programmer's Manual
1951\end{quote}
1952For the sake of simplicity, and in order to prevent the case of having two threads receiving alarms simultaneously, \CFA programs block the {\tt SIGALRM} signal on every kernel thread except one.
1953
1954Now because of how involuntary context-switches are handled, the kernel thread handling {\tt SIGALRM} cannot also be a processor thread. Hence, involuntary context-switching is done by sending signal {\tt SIGUSR1} to the corresponding proces\-sor and having the thread yield from inside the signal handler. This approach effectively context-switches away from the signal handler back to the kernel and the signal handler frame is eventually unwound when the thread is scheduled again. As a result, a signal handler can start on one kernel thread and terminate on a second kernel thread (but the same user thread). It is important to note that signal handlers save and restore signal masks because user-thread migration can cause a signal mask to migrate from one kernel thread to another. This behaviour is only a problem if all kernel threads, among which a user thread can migrate, differ in terms of signal masks\footnote{Sadly, official POSIX documentation is silent on what distinguishes async-signal-safe'' functions from other functions.}. However, since the kernel thread handling preemption requires a different signal mask, executing user threads on the kernel-alarm thread can cause deadlocks. For this reason, the alarm thread is in a tight loop around a system call to \code{sigwaitinfo}, requiring very little CPU time for preemption. One final detail about the alarm thread is how to wake it when additional communication is required (e.g., on thread termination). This unblocking is also done using {\tt SIGALRM}, but sent through the \code{pthread_sigqueue}. Indeed, \code{sigwait} can differentiate signals sent from \code{pthread_sigqueue} from signals sent from alarms or the kernel.
1955
1956\subsection{Scheduler}
1957Finally, an aspect that was not mentioned yet is the scheduling algorithm. Currently, the \CFA scheduler uses a single ready queue for all processors, which is the simplest approach to scheduling. Further discussion on scheduling is present in section \ref{futur:sched}.
1958
1959% ======================================================================
1960% ======================================================================
1961\section{Internal Scheduling} \label{impl:intsched}
1962% ======================================================================
1963% ======================================================================
1964The following figure is the traditional illustration of a monitor (repeated from page~\pageref{fig:ClassicalMonitor} for convenience):
1965
1966\begin{figure}[H]
1967\begin{center}
1968{\resizebox{0.4\textwidth}{!}{\input{monitor}}}
1969\end{center}
1971\end{figure}
1972
1973This picture has several components, the two most important being the entry queue and the AS-stack. The entry queue is an (almost) FIFO list where threads waiting to enter are parked, while the acceptor/signaller (AS) stack is a FILO list used for threads that have been signalled or otherwise marked as running next.
1974
1975For \CFA, this picture does not have support for blocking multiple monitors on a single condition. To support \textbf{bulk-acq} two changes to this picture are required. First, it is no longer helpful to attach the condition to \emph{a single} monitor. Secondly, the thread waiting on the condition has to be separated across multiple monitors, seen in figure \ref{fig:monitor_cfa}.
1976
1977\begin{figure}[H]
1978\begin{center}
1979{\resizebox{0.8\textwidth}{!}{\input{int_monitor}}}
1980\end{center}
1981\caption{Illustration of \CFA Monitor}
1982\label{fig:monitor_cfa}
1983\end{figure}
1984
1985This picture and the proper entry and leave algorithms (see listing \ref{lst:entry2}) is the fundamental implementation of internal scheduling. Note that when a thread is moved from the condition to the AS-stack, it is conceptually split into N pieces, where N is the number of monitors specified in the parameter list. The thread is woken up when all the pieces have popped from the AS-stacks and made active. In this picture, the threads are split into halves but this is only because there are two monitors. For a specific signalling operation every monitor needs a piece of thread on its AS-stack.
1986
1987\begin{figure}[b]
1988\begin{multicols}{2}
1989Entry
1990\begin{pseudo}
1991if monitor is free
1992 enter
1994 continue
1995else
1996 block
1997increment recursion
1998
1999\end{pseudo}
2000\columnbreak
2001Exit
2002\begin{pseudo}
2003decrement recursion
2004if recursion == 0
2005 if signal_stack not empty
2009
2010 if entry queue not empty
2012\end{pseudo}
2013\end{multicols}
2014\begin{pseudo}[caption={Entry and exit routine for monitors with internal scheduling},label={lst:entry2}]
2015\end{pseudo}
2016\end{figure}
2017
2018The solution discussed in \ref{intsched} can be seen in the exit routine of listing \ref{lst:entry2}. Basically, the solution boils down to having a separate data structure for the condition queue and the AS-stack, and unconditionally transferring ownership of the monitors but only unblocking the thread when the last monitor has transferred ownership. This solution is deadlock safe as well as preventing any potential barging. The data structures used for the AS-stack are reused extensively for external scheduling, but in the case of internal scheduling, the data is allocated using variable-length arrays on the call stack of the \code{wait} and \code{signal_block} routines.
2019
2020\begin{figure}[H]
2021\begin{center}
2022{\resizebox{0.8\textwidth}{!}{\input{monitor_structs.pstex_t}}}
2023\end{center}
2024\caption{Data structures involved in internal/external scheduling}
2025\label{fig:structs}
2026\end{figure}
2027
2028Figure \ref{fig:structs} shows a high-level representation of these data structures. The main idea behind them is that, a thread cannot contain an arbitrary number of intrusive next'' pointers for linking onto monitors. The \code{condition node} is the data structure that is queued onto a condition variable and, when signalled, the condition queue is popped and each \code{condition criterion} is moved to the AS-stack. Once all the criteria have been popped from their respective AS-stacks, the thread is woken up, which is what is shown in listing \ref{lst:entry2}.
2029
2030% ======================================================================
2031% ======================================================================
2032\section{External Scheduling}
2033% ======================================================================
2034% ======================================================================
2035Similarly to internal scheduling, external scheduling for multiple monitors relies on the idea that waiting-thread queues are no longer specific to a single monitor, as mentioned in section \ref{extsched}. For internal scheduling, these queues are part of condition variables, which are still unique for a given scheduling operation (i.e., no signal statement uses multiple conditions). However, in the case of external scheduling, there is no equivalent object which is associated with \code{waitfor} statements. This absence means the queues holding the waiting threads must be stored inside at least one of the monitors that is acquired. These monitors being the only objects that have sufficient lifetime and are available on both sides of the \code{waitfor} statement. This requires an algorithm to choose which monitor holds the relevant queue. It is also important that said algorithm be independent of the order in which users list parameters. The proposed algorithm is to fall back on monitor lock ordering (sorting by address) and specify that the monitor that is acquired first is the one with the relevant waiting queue. This assumes that the lock acquiring order is static for the lifetime of all concerned objects but that is a reasonable constraint.
2036
2037This algorithm choice has two consequences:
2038\begin{itemize}
2039 \item The queue of the monitor with the lowest address is no longer a true FIFO queue because threads can be moved to the front of the queue. These queues need to contain a set of monitors for each of the waiting threads. Therefore, another thread whose set contains the same lowest address monitor but different lower priority monitors may arrive first but enter the critical section after a thread with the correct pairing.
2040 \item The queue of the lowest priority monitor is both required and potentially unused. Indeed, since it is not known at compile time which monitor is the monitor which has the lowest address, every monitor needs to have the correct queues even though it is possible that some queues go unused for the entire duration of the program, for example if a monitor is only used in a specific pair.
2041\end{itemize}
2042Therefore, the following modifications need to be made to support external scheduling:
2043\begin{itemize}
2044 \item The threads waiting on the entry queue need to keep track of which routine they are trying to enter, and using which set of monitors. The \code{mutex} routine already has all the required information on its stack, so the thread only needs to keep a pointer to that information.
2045 \item The monitors need to keep a mask of acceptable routines. This mask contains for each acceptable routine, a routine pointer and an array of monitors to go with it. It also needs storage to keep track of which routine was accepted. Since this information is not specific to any monitor, the monitors actually contain a pointer to an integer on the stack of the waiting thread. Note that if a thread has acquired two monitors but executes a \code{waitfor} with only one monitor as a parameter, setting the mask of acceptable routines to both monitors will not cause any problems since the extra monitor will not change ownership regardless. This becomes relevant when \code{when} clauses affect the number of monitors passed to a \code{waitfor} statement.
2046 \item The entry/exit routines need to be updated as shown in listing \ref{lst:entry3}.
2047\end{itemize}
2048
2049\subsection{External Scheduling - Destructors}
2050Finally, to support the ordering inversion of destructors, the code generation needs to be modified to use a special entry routine. This routine is needed because of the storage requirements of the call order inversion. Indeed, when waiting for the destructors, storage is needed for the waiting context and the lifetime of said storage needs to outlive the waiting operation it is needed for. For regular \code{waitfor} statements, the call stack of the routine itself matches this requirement but it is no longer the case when waiting for the destructor since it is pushed on to the AS-stack for later. The \code{waitfor} semantics can then be adjusted correspondingly, as seen in listing \ref{lst:entry-dtor}
2051
2052\begin{figure}
2053\begin{multicols}{2}
2054Entry
2055\begin{pseudo}
2056if monitor is free
2057 enter
2059 continue
2061 push criteria to AS-stack
2062 continue
2063else
2064 block
2065increment recursion
2066\end{pseudo}
2067\columnbreak
2068Exit
2069\begin{pseudo}
2070decrement recursion
2071if recursion == 0
2072 if signal_stack not empty
2076 endif
2077 endif
2078
2079 if entry queue not empty
2081 endif
2082\end{pseudo}
2083\end{multicols}
2084\begin{pseudo}[caption={Entry and exit routine for monitors with internal scheduling and external scheduling},label={lst:entry3}]
2085\end{pseudo}
2086\end{figure}
2087
2088\begin{figure}
2089\begin{multicols}{2}
2090Destructor Entry
2091\begin{pseudo}
2092if monitor is free
2093 enter
2095 increment recursion
2096 return
2097create wait context
2100 push self to AS-stack
2101 baton pass
2102else
2103 wait
2104increment recursion
2105\end{pseudo}
2106\columnbreak
2107Waitfor
2108\begin{pseudo}
2110 if found destructor
2111 push destructor to AS-stack
2112 unlock all monitors
2113 else
2114 push self to AS-stack
2115 baton pass
2116 endif
2117 return
2118endif
2119if non-blocking
2120 Unlock all monitors
2121 Return
2122endif
2123
2124push self to AS-stack
2126block
2127return
2128\end{pseudo}
2129\end{multicols}
2130\begin{pseudo}[caption={Pseudo code for the \code{waitfor} routine and the \code{mutex} entry routine for destructors},label={lst:entry-dtor}]
2131\end{pseudo}
2132\end{figure}
2133
2134
2135% ======================================================================
2136% ======================================================================
2137\section{Putting It All Together}
2138% ======================================================================
2139% ======================================================================
2140
2141
2143As it was subtly alluded in section \ref{threads}, \code{thread}s in \CFA are in fact monitors, which means that all monitor features are available when using threads. For example, here is a very simple two thread pipeline that could be used for a simulator of a game engine:
2144\begin{figure}[H]
2145\begin{cfacode}[caption={Toy simulator using \code{thread}s and \code{monitor}s.},label={lst:engine-v1}]
2146// Visualization declaration
2148Frame * simulate( Simulator & this );
2149
2150// Simulation declaration
2152void render( Renderer & this );
2153
2154// Blocking call used as communication
2155void draw( Renderer & mutex this, Frame * frame );
2156
2157// Simulation loop
2158void main( Simulator & this ) {
2159 while( true ) {
2160 Frame * frame = simulate( this );
2161 draw( renderer, frame );
2162 }
2163}
2164
2165// Rendering loop
2166void main( Renderer & this ) {
2167 while( true ) {
2168 waitfor( draw, this );
2169 render( this );
2170 }
2171}
2172\end{cfacode}
2173\end{figure}
2174One of the obvious complaints of the previous code snippet (other than its toy-like simplicity) is that it does not handle exit conditions and just goes on forever. Luckily, the monitor semantics can also be used to clearly enforce a shutdown order in a concise manner:
2175\begin{figure}[H]
2176\begin{cfacode}[caption={Same toy simulator with proper termination condition.},label={lst:engine-v2}]
2177// Visualization declaration
2179Frame * simulate( Simulator & this );
2180
2181// Simulation declaration
2183void render( Renderer & this );
2184
2185// Blocking call used as communication
2186void draw( Renderer & mutex this, Frame * frame );
2187
2188// Simulation loop
2189void main( Simulator & this ) {
2190 while( true ) {
2191 Frame * frame = simulate( this );
2192 draw( renderer, frame );
2193
2194 // Exit main loop after the last frame
2195 if( frame->is_last ) break;
2196 }
2197}
2198
2199// Rendering loop
2200void main( Renderer & this ) {
2201 while( true ) {
2202 waitfor( draw, this );
2203 or waitfor( ^?{}, this ) {
2204 // Add an exit condition
2205 break;
2206 }
2207
2208 render( this );
2209 }
2210}
2211
2212// Call destructor for simulator once simulator finishes
2213// Call destructor for renderer to signify shutdown
2214\end{cfacode}
2215\end{figure}
2216
2218As mentioned in section \ref{preemption}, \CFA uses preemptive threads by default but can use fibers on demand. Currently, using fibers is done by adding the following line of code to the program~:
2219\begin{cfacode}
2220unsigned int default_preemption() {
2221 return 0;
2222}
2223\end{cfacode}
2224This function is called by the kernel to fetch the default preemption rate, where 0 signifies an infinite time-slice, i.e., no preemption. However, once clusters are fully implemented, it will be possible to create fibers and \textbf{uthread} in the same system, as in listing \ref{lst:fiber-uthread}
2225\begin{figure}
2227//Cluster forward declaration
2228struct cluster;
2229
2230//Processor forward declaration
2231struct processor;
2232
2233//Construct clusters with a preemption rate
2234void ?{}(cluster& this, unsigned int rate);
2235//Construct processor and add it to cluster
2236void ?{}(processor& this, cluster& cluster);
2237//Construct thread and schedule it on cluster
2239
2240//Declare two clusters
2241cluster thread_cluster = { 10ms }; //Preempt every 10 ms
2242cluster fibers_cluster = { 0 }; //Never preempt
2243
2244//Construct 4 processors
2245processor processors[4] = {
2246 //2 for the thread cluster
2249 //2 for the fibers cluster
2250 fibers_cluster;
2251 fibers_cluster;
2252};
2253
2257 //Construct underlying thread to automatically
2258 //be scheduled on the thread cluster
2260}
2261
2263
2264//Declares fibers
2266void ?{}(Fiber& this) {
2267 //Construct underlying thread to automatically
2268 //be scheduled on the fiber cluster
2270}
2271
2272void main(Fiber & this);
2273\end{cfacode}
2274\end{figure}
2275
2276
2277% ======================================================================
2278% ======================================================================
2279\section{Performance Results} \label{results}
2280% ======================================================================
2281% ======================================================================
2282\section{Machine Setup}
2283Table \ref{tab:machine} shows the characteristics of the machine used to run the benchmarks. All tests were made on this machine.
2284\begin{table}[H]
2285\begin{center}
2286\begin{tabular}{| l | r | l | r |}
2287\hline
2288Architecture & x86\_64 & NUMA node(s) & 8 \\
2289\hline
2290CPU op-mode(s) & 32-bit, 64-bit & Model name & AMD Opteron\texttrademark Processor 6380 \\
2291\hline
2292Byte Order & Little Endian & CPU Freq & 2.5\si{\giga\hertz} \\
2293\hline
2294CPU(s) & 64 & L1d cache & \SI{16}{\kibi\byte} \\
2295\hline
2296Thread(s) per core & 2 & L1i cache & \SI{64}{\kibi\byte} \\
2297\hline
2298Core(s) per socket & 8 & L2 cache & \SI{2048}{\kibi\byte} \\
2299\hline
2300Socket(s) & 4 & L3 cache & \SI{6144}{\kibi\byte} \\
2301\hline
2302\hline
2303Operating system & Ubuntu 16.04.3 LTS & Kernel & Linux 4.4-97-generic \\
2304\hline
2305Compiler & GCC 6.3 & Translator & CFA 1 \\
2306\hline
2307Java version & OpenJDK-9 & Go version & 1.9.2 \\
2308\hline
2309\end{tabular}
2310\end{center}
2311\caption{Machine setup used for the tests}
2312\label{tab:machine}
2313\end{table}
2314
2315\section{Micro Benchmarks}
2316All benchmarks are run using the same harness to produce the results, seen as the \code{BENCH()} macro in the following examples. This macro uses the following logic to benchmark the code:
2317\begin{pseudo}
2318#define BENCH(run, result) \
2319 before = gettime(); \
2320 run; \
2321 after = gettime(); \
2322 result = (after - before) / N;
2323\end{pseudo}
2324The method used to get time is \code{clock_gettime(CLOCK_THREAD_CPUTIME_ID);}. Each benchmark is using many iterations of a simple call to measure the cost of the call. The specific number of iterations depends on the specific benchmark.
2325
2326\subsection{Context-Switching}
2327The first interesting benchmark is to measure how long context-switches take. The simplest approach to do this is to yield on a thread, which executes a 2-step context switch. Yielding causes the thread to context-switch to the scheduler and back, more precisely: from the \textbf{uthread} to the \textbf{kthread} then from the \textbf{kthread} back to the same \textbf{uthread} (or a different one in the general case). In order to make the comparison fair, coroutines also execute a 2-step context-switch by resuming another coroutine which does nothing but suspending in a tight loop, which is a resume/suspend cycle instead of a yield. Listing \ref{lst:ctx-switch} shows the code for coroutines and threads with the results in table \ref{tab:ctx-switch}. All omitted tests are functionally identical to one of these tests. The difference between coroutines and threads can be attributed to the cost of scheduling.
2328\begin{figure}
2329\begin{multicols}{2}
2330\CFA Coroutines
2331\begin{cfacode}
2332coroutine GreatSuspender {};
2333void main(GreatSuspender& this) {
2334 while(true) { suspend(); }
2335}
2336int main() {
2337 GreatSuspender s;
2338 resume(s);
2339 BENCH(
2340 for(size_t i=0; i<n; i++) {
2341 resume(s);
2342 },
2343 result
2344 )
2345 printf("%llu\n", result);
2346}
2347\end{cfacode}
2348\columnbreak
2350\begin{cfacode}
2351
2352
2353
2354
2355int main() {
2356
2357
2358 BENCH(
2359 for(size_t i=0; i<n; i++) {
2360 yield();
2361 },
2362 result
2363 )
2364 printf("%llu\n", result);
2365}
2366\end{cfacode}
2367\end{multicols}
2368\begin{cfacode}[caption={\CFA benchmark code used to measure context-switches for coroutines and threads.},label={lst:ctx-switch}]
2369\end{cfacode}
2370\end{figure}
2371
2372\begin{table}
2373\begin{center}
2374\begin{tabular}{| l | S[table-format=5.2,table-number-alignment=right] | S[table-format=5.2,table-number-alignment=right] | S[table-format=5.2,table-number-alignment=right] |}
2375\cline{2-4}
2376\multicolumn{1}{c |}{} & \multicolumn{1}{c |}{ Median } &\multicolumn{1}{c |}{ Average } & \multicolumn{1}{c |}{ Standard Deviation} \\
2377\hline
2378Kernel Thread & 241.5 & 243.86 & 5.08 \\
2379\CFA Coroutine & 38 & 38 & 0 \\
2380\CFA Thread & 103 & 102.96 & 2.96 \\
2381\uC Coroutine & 46 & 45.86 & 0.35 \\
2382\uC Thread & 98 & 99.11 & 1.42 \\
2383Goroutine & 150 & 149.96 & 3.16 \\
2384Java Thread & 289 & 290.68 & 8.72 \\
2385\hline
2386\end{tabular}
2387\end{center}
2388\caption{Context Switch comparison. All numbers are in nanoseconds(\si{\nano\second})}
2389\label{tab:ctx-switch}
2390\end{table}
2391
2392\subsection{Mutual-Exclusion}
2393The next interesting benchmark is to measure the overhead to enter/leave a critical-section. For monitors, the simplest approach is to measure how long it takes to enter and leave a monitor routine. Listing \ref{lst:mutex} shows the code for \CFA. To put the results in context, the cost of entering a non-inline function and the cost of acquiring and releasing a \code{pthread_mutex} lock is also measured. The results can be shown in table \ref{tab:mutex}.
2394
2395\begin{figure}
2396\begin{cfacode}[caption={\CFA benchmark code used to measure mutex routines.},label={lst:mutex}]
2397monitor M {};
2398void __attribute__((noinline)) call( M & mutex m /*, m2, m3, m4*/ ) {}
2399
2400int main() {
2401 M m/*, m2, m3, m4*/;
2402 BENCH(
2403 for(size_t i=0; i<n; i++) {
2404 call(m/*, m2, m3, m4*/);
2405 },
2406 result
2407 )
2408 printf("%llu\n", result);
2409}
2410\end{cfacode}
2411\end{figure}
2412
2413\begin{table}
2414\begin{center}
2415\begin{tabular}{| l | S[table-format=5.2,table-number-alignment=right] | S[table-format=5.2,table-number-alignment=right] | S[table-format=5.2,table-number-alignment=right] |}
2416\cline{2-4}
2417\multicolumn{1}{c |}{} & \multicolumn{1}{c |}{ Median } &\multicolumn{1}{c |}{ Average } & \multicolumn{1}{c |}{ Standard Deviation} \\
2418\hline
2419C routine & 2 & 2 & 0 \\
2420FetchAdd + FetchSub & 26 & 26 & 0 \\
2421Pthreads Mutex Lock & 31 & 31.86 & 0.99 \\
2422\uC \code{monitor} member routine & 30 & 30 & 0 \\
2423\CFA \code{mutex} routine, 1 argument & 41 & 41.57 & 0.9 \\
2424\CFA \code{mutex} routine, 2 argument & 76 & 76.96 & 1.57 \\
2425\CFA \code{mutex} routine, 4 argument & 145 & 146.68 & 3.85 \\
2426Java synchronized routine & 27 & 28.57 & 2.6 \\
2427\hline
2428\end{tabular}
2429\end{center}
2430\caption{Mutex routine comparison. All numbers are in nanoseconds(\si{\nano\second})}
2431\label{tab:mutex}
2432\end{table}
2433
2434\subsection{Internal Scheduling}
2435The internal-scheduling benchmark measures the cost of waiting on and signalling a condition variable. Listing \ref{lst:int-sched} shows the code for \CFA, with results table \ref{tab:int-sched}. As with all other benchmarks, all omitted tests are functionally identical to one of these tests.
2436
2437\begin{figure}
2438\begin{cfacode}[caption={Benchmark code for internal scheduling},label={lst:int-sched}]
2439volatile int go = 0;
2440condition c;
2441monitor M {};
2442M m1;
2443
2444void __attribute__((noinline)) do_call( M & mutex a1 ) { signal(c); }
2445
2447void ^?{}( T & mutex this ) {}
2448void main( T & this ) {
2449 while(go == 0) { yield(); }
2450 while(go == 1) { do_call(m1); }
2451}
2452int __attribute__((noinline)) do_wait( M & mutex a1 ) {
2453 go = 1;
2454 BENCH(
2455 for(size_t i=0; i<n; i++) {
2456 wait(c);
2457 },
2458 result
2459 )
2460 printf("%llu\n", result);
2461 go = 0;
2462 return 0;
2463}
2464int main() {
2465 T t;
2466 return do_wait(m1);
2467}
2468\end{cfacode}
2469\end{figure}
2470
2471\begin{table}
2472\begin{center}
2473\begin{tabular}{| l | S[table-format=5.2,table-number-alignment=right] | S[table-format=5.2,table-number-alignment=right] | S[table-format=5.2,table-number-alignment=right] |}
2474\cline{2-4}
2475\multicolumn{1}{c |}{} & \multicolumn{1}{c |}{ Median } &\multicolumn{1}{c |}{ Average } & \multicolumn{1}{c |}{ Standard Deviation} \\
2476\hline
2477Pthreads Condition Variable & 5902.5 & 6093.29 & 714.78 \\
2478\uC \code{signal} & 322 & 323 & 3.36 \\
2479\CFA \code{signal}, 1 \code{monitor} & 352.5 & 353.11 & 3.66 \\
2480\CFA \code{signal}, 2 \code{monitor} & 430 & 430.29 & 8.97 \\
2481\CFA \code{signal}, 4 \code{monitor} & 594.5 & 606.57 & 18.33 \\
2482Java \code{notify} & 13831.5 & 15698.21 & 4782.3 \\
2483\hline
2484\end{tabular}
2485\end{center}
2486\caption{Internal scheduling comparison. All numbers are in nanoseconds(\si{\nano\second})}
2487\label{tab:int-sched}
2488\end{table}
2489
2490\subsection{External Scheduling}
2491The Internal scheduling benchmark measures the cost of the \code{waitfor} statement (\code{_Accept} in \uC). Listing \ref{lst:ext-sched} shows the code for \CFA, with results in table \ref{tab:ext-sched}. As with all other benchmarks, all omitted tests are functionally identical to one of these tests.
2492
2493\begin{figure}
2494\begin{cfacode}[caption={Benchmark code for external scheduling},label={lst:ext-sched}]
2495volatile int go = 0;
2496monitor M {};
2497M m1;
2499
2500void __attribute__((noinline)) do_call( M & mutex a1 ) {}
2501
2502void ^?{}( T & mutex this ) {}
2503void main( T & this ) {
2504 while(go == 0) { yield(); }
2505 while(go == 1) { do_call(m1); }
2506}
2507int __attribute__((noinline)) do_wait( M & mutex a1 ) {
2508 go = 1;
2509 BENCH(
2510 for(size_t i=0; i<n; i++) {
2511 waitfor(call, a1);
2512 },
2513 result
2514 )
2515 printf("%llu\n", result);
2516 go = 0;
2517 return 0;
2518}
2519int main() {
2520 T t;
2521 return do_wait(m1);
2522}
2523\end{cfacode}
2524\end{figure}
2525
2526\begin{table}
2527\begin{center}
2528\begin{tabular}{| l | S[table-format=5.2,table-number-alignment=right] | S[table-format=5.2,table-number-alignment=right] | S[table-format=5.2,table-number-alignment=right] |}
2529\cline{2-4}
2530\multicolumn{1}{c |}{} & \multicolumn{1}{c |}{ Median } &\multicolumn{1}{c |}{ Average } & \multicolumn{1}{c |}{ Standard Deviation} \\
2531\hline
2532\uC \code{Accept} & 350 & 350.61 & 3.11 \\
2533\CFA \code{waitfor}, 1 \code{monitor} & 358.5 & 358.36 & 3.82 \\
2534\CFA \code{waitfor}, 2 \code{monitor} & 422 & 426.79 & 7.95 \\
2535\CFA \code{waitfor}, 4 \code{monitor} & 579.5 & 585.46 & 11.25 \\
2536\hline
2537\end{tabular}
2538\end{center}
2539\caption{External scheduling comparison. All numbers are in nanoseconds(\si{\nano\second})}
2540\label{tab:ext-sched}
2541\end{table}
2542
2543\subsection{Object Creation}
2544Finally, the last benchmark measures the cost of creation for concurrent objects. Listing \ref{lst:creation} shows the code for \texttt{pthread}s and \CFA threads, with results shown in table \ref{tab:creation}. As with all other benchmarks, all omitted tests are functionally identical to one of these tests. The only note here is that the call stacks of \CFA coroutines are lazily created, therefore without priming the coroutine, the creation cost is very low.
2545
2546\begin{figure}
2547\begin{center}
2549\begin{ccode}
2550int main() {
2551 BENCH(
2552 for(size_t i=0; i<n; i++) {
2555 perror( "failure" );
2556 return 1;
2557 }
2558
2560 perror( "failure" );
2561 return 1;
2562 }
2563 },
2564 result
2565 )
2566 printf("%llu\n", result);
2567}
2568\end{ccode}
2569
2570
2571
2573\begin{cfacode}
2574int main() {
2575 BENCH(
2576 for(size_t i=0; i<n; i++) {
2578 },
2579 result
2580 )
2581 printf("%llu\n", result);
2582}
2583\end{cfacode}
2584\end{center}
2585\begin{cfacode}[caption={Benchmark code for \texttt{pthread}s and \CFA to measure object creation},label={lst:creation}]
2586\end{cfacode}
2587\end{figure}
2588
2589\begin{table}
2590\begin{center}
2591\begin{tabular}{| l | S[table-format=5.2,table-number-alignment=right] | S[table-format=5.2,table-number-alignment=right] | S[table-format=5.2,table-number-alignment=right] |}
2592\cline{2-4}
2593\multicolumn{1}{c |}{} & \multicolumn{1}{c |}{ Median } &\multicolumn{1}{c |}{ Average } & \multicolumn{1}{c |}{ Standard Deviation} \\
2594\hline
2595Pthreads & 26996 & 26984.71 & 156.6 \\
2596\CFA Coroutine Lazy & 6 & 5.71 & 0.45 \\
2597\CFA Coroutine Eager & 708 & 706.68 & 4.82 \\
2598\CFA Thread & 1173.5 & 1176.18 & 15.18 \\
2599\uC Coroutine & 109 & 107.46 & 1.74 \\
2600\uC Thread & 526 & 530.89 & 9.73 \\
2601Goroutine & 2520.5 & 2530.93 & 61,56 \\
2602Java Thread & 91114.5 & 92272.79 & 961.58 \\
2603\hline
2604\end{tabular}
2605\end{center}
2606\caption{Creation comparison. All numbers are in nanoseconds(\si{\nano\second}).}
2607\label{tab:creation}
2608\end{table}
2609
2610
2611
2612\section{Conclusion}
2613This paper has achieved a minimal concurrency \textbf{api} that is simple, efficient and usable as the basis for higher-level features. The approach presented is based on a lightweight thread-system for parallelism, which sits on top of clusters of processors. This M:N model is judged to be both more efficient and allow more flexibility for users. Furthermore, this document introduces monitors as the main concurrency tool for users. This paper also offers a novel approach allowing multiple monitors to be accessed simultaneously without running into the Nested Monitor Problem~\cite{Lister77}. It also offers a full implementation of the concurrency runtime written entirely in \CFA, effectively the largest \CFA code base to date.
2614
2615
2616% ======================================================================
2617% ======================================================================
2618\section{Future Work}
2619% ======================================================================
2620% ======================================================================
2621
2622\subsection{Performance} \label{futur:perf}
2623This paper presents a first implementation of the \CFA concurrency runtime. Therefore, there is still significant work to improve performance. Many of the data structures and algorithms may change in the future to more efficient versions. For example, the number of monitors in a single \textbf{bulk-acq} is only bound by the stack size, this is probably unnecessarily generous. It may be possible that limiting the number helps increase performance. However, it is not obvious that the benefit would be significant.
2624
2625\subsection{Flexible Scheduling} \label{futur:sched}
2626An important part of concurrency is scheduling. Different scheduling algorithms can affect performance (both in terms of average and variation). However, no single scheduler is optimal for all workloads and therefore there is value in being able to change the scheduler for given programs. One solution is to offer various tweaking options to users, allowing the scheduler to be adjusted to the requirements of the workload. However, in order to be truly flexible, it would be interesting to allow users to add arbitrary data and arbitrary scheduling algorithms. For example, a web server could attach Type-of-Service information to threads and have a `ToS aware'' scheduling algorithm tailored to this specific web server. This path of flexible schedulers will be explored for \CFA.
2627
2628\subsection{Non-Blocking I/O} \label{futur:nbio}
2629While most of the parallelism tools are aimed at data parallelism and control-flow parallelism, many modern workloads are not bound on computation but on IO operations, a common case being web servers and XaaS (anything as a service). These types of workloads often require significant engineering around amortizing costs of blocking IO operations. At its core, non-blocking I/O is an operating system level feature that allows queuing IO operations (e.g., network operations) and registering for notifications instead of waiting for requests to complete. In this context, the role of the language makes Non-Blocking IO easily available and with low overhead. The current trend is to use asynchronous programming using tools like callbacks and/or futures and promises, which can be seen in frameworks like Node.js~\cite{NodeJs} for JavaScript, Spring MVC~\cite{SpringMVC} for Java and Django~\cite{Django} for Python. However, while these are valid solutions, they lead to code that is harder to read and maintain because it is much less linear.
2630
2631\subsection{Other Concurrency Tools} \label{futur:tools}
2632While monitors offer a flexible and powerful concurrent core for \CFA, other concurrency tools are also necessary for a complete multi-paradigm concurrency package. Examples of such tools can include simple locks and condition variables, futures and promises~\cite{promises}, executors and actors. These additional features are useful when monitors offer a level of abstraction that is inadequate for certain tasks.
2633
2635Simpler applications can benefit greatly from having implicit parallelism. That is, parallelism that does not rely on the user to write concurrency. This type of parallelism can be achieved both at the language level and at the library level. The canonical example of implicit parallelism is parallel for loops, which are the simplest example of a divide and conquer algorithms~\cite{uC++book}. Table \ref{lst:parfor} shows three different code examples that accomplish point-wise sums of large arrays. Note that none of these examples explicitly declare any concurrency or parallelism objects.
2636
2637\begin{table}
2638\begin{center}
2639\begin{tabular}[t]{|c|c|c|}
2640Sequential & Library Parallel & Language Parallel \\
2641\begin{cfacode}[tabsize=3]
2642void big_sum(
2643 int* a, int* b,
2644 int* o,
2645 size_t len)
2646{
2647 for(
2648 int i = 0;
2649 i < len;
2650 ++i )
2651 {
2652 o[i]=a[i]+b[i];
2653 }
2654}
2655
2656
2657
2658
2659
2660int* a[10000];
2661int* b[10000];
2662int* c[10000];
2663//... fill in a & b
2664big_sum(a,b,c,10000);
2665\end{cfacode} &\begin{cfacode}[tabsize=3]
2666void big_sum(
2667 int* a, int* b,
2668 int* o,
2669 size_t len)
2670{
2671 range ar(a, a+len);
2672 range br(b, b+len);
2673 range or(o, o+len);
2674 parfor( ai, bi, oi,
2675 []( int* ai,
2676 int* bi,
2677 int* oi)
2678 {
2679 oi=ai+bi;
2680 });
2681}
2682
2683
2684int* a[10000];
2685int* b[10000];
2686int* c[10000];
2687//... fill in a & b
2688big_sum(a,b,c,10000);
2689\end{cfacode}&\begin{cfacode}[tabsize=3]
2690void big_sum(
2691 int* a, int* b,
2692 int* o,
2693 size_t len)
2694{
2695 parfor (ai,bi,oi)
2696 in (a, b, o )
2697 {
2698 oi = ai + bi;
2699 }
2700}
2701
2702
2703
2704
2705
2706
2707
2708int* a[10000];
2709int* b[10000];
2710int* c[10000];
2711//... fill in a & b
2712big_sum(a,b,c,10000);
2713\end{cfacode}
2714\end{tabular}
2715\end{center}
2716\caption{For loop to sum numbers: Sequential, using library parallelism and language parallelism.}
2717\label{lst:parfor}
2718\end{table}
2719
2720Implicit parallelism is a restrictive solution and therefore has its limitations. However, it is a quick and simple approach to parallelism, which may very well be sufficient for smaller applications and reduces the amount of boilerplate needed to start benefiting from parallelism in modern CPUs.
2721
2722
2723% A C K N O W L E D G E M E N T S
2724% -------------------------------
2725\section{Acknowledgements}
2726
2727Thanks to Aaron Moss, Rob Schluntz and Andrew Beach for their work on the \CFA project as well as all the discussions which helped concretize the ideas in this paper.
2728Partial funding was supplied by the Natural Sciences and Engineering Research Council of Canada and a corporate partnership with Huawei Ltd.
2729
2730
2731% B I B L I O G R A P H Y
2732% -----------------------------
2733\bibliographystyle{plain}
2734\bibliography{pl,local}
2735
2736\end{document}
2737
2738% Local Variables: %
2739% tab-width: 4 %
2740% fill-column: 120 %
2741% compile-command: "make" %
2742% End: %
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Uniform motion (travel time) Calculator
Calculates the travel time from the velocity and travel distance.
Velocity v m/s km/h fps mph knot Distance d m km ft mi nmi 6digit10digit14digit18digit22digit26digit30digit34digit38digit42digit46digit50digit Travel time t
example. 4~6km/h on foot, 15km/h by bicycle$Uniform\ motion\\\hspace{30}v={\Large\frac{d}{t}},\hspace{30}d=vt,\hspace{30}t={\Large\frac{d}{v}}\\$
Uniform motion (travel time)
[1-3] /3 Disp-Num5103050100200
[1] 2018/09/07 14:09 - / Under 20 years old / High-school/ University/ Grad student / A little /
Purpose of use
For Mathematics homework
Comment/Request
Was confused on how they are getting the answers, when I do the work on paper I get a difference answer than what is on the website.
[2] 2017/01/26 11:15 Female / Under 20 years old / Elementary school/ Junior high-school student / Very /
Purpose of use
For science homework and to check teachers math. We had to drop a ruler and figure out how far it fell before we caught it. The only reference she gave us was for even numbers.
Comment/Request
This was really helpful because there is nothing else online like it that I could put in my own numbers. Had to use this for my Science homework. I may even start using their other calculators for my Math homework.
[3] 2009/10/23 23:40 Male / 40 level / A university student / Very /
Purpose of use
Testing it for now
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Great! Congratulations and thank you very much.
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Net Charge Of Amino Acids At Ph 7
Which group exhibits the greatest structural variability? Biochemistry. Certain \beta-amino acids are found in nature. In human bodies, pH is close to 7. 36 mA/cm2), using Ag/AgCl electrodes, was conducted across freshly excised hairless mouse skin. pI of an amino acid (8. At a pH greater than 10, the amine exists as a neutral base and the carboxyl as its conjugate base, so the alanine molecule has a net negative charge. At physiological pH we see here that the R group ionizes, and this is because that the R group for these two amino acids has a pKa value of about 4, meaning that at a physiological pH of 7, the proton on the carboxyl group is lost, leaving the molecule with a net negative charge on the R group. Nitrogen containing bases (e. The acid group will be deprotonated - not protonated, making it's charge -1. When performing a paper electrophoresis at these pHs, in which direction will the dipeptide move (anode or cathode) ? Solution. Basic amino acids have really high pKa values (>>7) so they are protonated at pH of 7 or 7. we have to calculate the net charge over the amino acid at the pH 7. And finally, once the pH climbs all the way up to a value of 12, we can expect the two basic functional groups to be deprotonated. When the pH = pKa the solution is a buffer, and therefore half of the amino groups are protonated and half are deprotonated. • They have a central α-carbon and α-amino and α-carboxyl groups • 20 different amino acids • Same core structure, but different side group (R) •The α-C is chiral (except glycine); proteins contain only L-isoforms. • Amino acids are a vital. For amino acids with neutral R-groups, at any pH below the pI of the amino acid, the population of amino acids in solution will: A. DSC for potato starch in the solutions of Asp, Glu, Lys, Gln and Asn with various concentrations was carried out at pH 7. Protonation can change the charge of these R groups. The side chains (R groups) of the amino acids can be divided into two major classes, those with non-polar side chains (shown here) and those with polar side chains. The Isoelectric Point (pI) is the pH at which any given protein has an equal number of positive and negative charges, in other word the protein has no charge or neutral. The acid group will be deprotonated - not protonated, making it's charge -1. P k values for α-amino groups of free amino acids is about 9. Amino acids are organic compounds that contain amine (-NH 2) and carboxyl (-COOH) functional groups, along with a side chain (R group) specific to each amino acid. Non polar side chains consist mainly of hydrocarbon. These differences in charge permit the electro-phoretic separation of acidic, neutral, and basic amino acids at pH 6, as illustrated in Figure 7. when the PH reaches its Pka (The net charge is +1 due to the protonated) - If the PH reaches the isoelectric point, so 100% of the amino acid is now in the Zwitterion form (The net charge is zero as the carboxyl group is 100% deprotonated. In a pH below the p K of the ionizable groups, these groups are protonated. What is the overall charge of the tripeptide at pH 12? What type of intermolecular attractions could the side chains of the amino acids make with the solvent (water) at pH 7. -log 10 (pK i + pK j). pptx), PDF File (. guanidino, imidazole or amino groups) with a net positive charge at neutral pH. have positive and negative charges in equal concentration. The atom with the lowest #pK_a# will be deprotonated. 1 Introduction. This negative charge masks the intrinsic charges on various types of R-groups of the amino acids of the protein. As we titrate with more hydroxide ions, we reach the pK a2, At this pH, all of the protons have been removed from the alpha-carboxyl group and half of the half of the protons have been removed from the R-group. Given the following amino acids, explain where they would be found in the tertiary structure of a protein and WHY (lack of explanation/incorrect explanations will NOT receive full credit; assume living system of pH = 7. The amino acids have a name, as well as a three letter or single letter mnemonic code:. Each amino acid has a unique “isoelectric point” [pI], or specific pH where it has NO net charge. However at certain pH an amino acid can exist as a ZWITTERION carrying no net charge. The net charge on the aspartate molecules is 0. The amino acid molecule appears to have a charge which changes with pH. A positive charge is observed on arginine and on lysine and a negative charge is observed on glutamic acid as well as aspartic acid. Serotonin lacks the carboxyl group of tryptophan. The list of pKas for all 20 amino acids can be found at the end of the “Problems” section of this problem set. For example when and has a net positive charge of 1. 4,thesidechainisalmost99% uncharged. 13 PH : about 7. Amino acids all contain the same backbone, which has both an acidic and a basic group. So between pH3 and 9 roughly, carboxyl will be deprotonated (-ve charge) and amino will be protonated (+ve charge). This is why biologists stick to D and L for amino acids. On the basis of this fact the "Alanine World" hypothesis was proposed. 0 arginine would be charged predominantly as follows: alpha-carboxylate -1, alpha-amino +1, guanidino +1, net charge +1 The disulfide bond between two cysteine molecules. AA is least soluble in water. In a pH above the p K of these groups, they lose. Every free amino acid has at least two ionizable functional groups, the amino group and the carboxylic acid group. The Isoelectric Point (pI) is the pH at which any given protein has an equal number of positive and negative charges, in other word the protein has no charge or neutral. It was found that T7 channel could not discriminate RK2 (Fig. At a specific pH called the isoelectric pH of the molecule, each such molecule exists as dipolar zwitterions bearing both anionic acid and cationic groups and minimum net charge. In the case of proteins, which are built up of many different amino acids containing weak acidic and basic groups, their net surface charge will change gradually as the pH of the environment changes i. Not all amino acids are neutral at physiologic pH, as lysine and arginine are positively charged and apartate and glutamate are negatively charged due to their respective side chains. When the pH is higher than the isoelectric point, the protein has negative net charge, and. PH influences the ionization of ionizable polar groups of amino acids, proteins, nucleic acids, Phospholipids, and mucopolysaccharides. Any functional groups they contain are uncharged at physiological pH and are incapable of participating in hydrogen bonding. Identify the common protein amino acid(s) that fits each description provided. Glycine electrical charge according to pH shift: pH<5: ; net charge =+1; 5≤pH<10: ; net charge =0. A zwitterion by definition is a molecule with 2 (zwitter) ions, one positive and one negative for a net zero charge. Mass, pI, composition and mol% acidic, basic, aromatic, polar etc. Predicting Intra- and Intermolecular Interactions of a Peptide 7. At $\mathrm{pH} = 2$, everything will be protonated because their $\mathrm{p}K_\mathrm{a} > \mathrm{pH}$. A positive charge is observed on arginine and on lysine and a negative charge is observed on glutamic acid as well as aspartic acid. amino acids can carry both positive and negative charges in solution. Learn vocabulary, terms, and more with flashcards, games, and other study tools. To complicate this a little bit more, proteins are made up of many amino acids each of which contributes to the proteins overall charge. Glutamine d. Amino acids contain both a carboxyl group which usually carries a negative charge and an amino group which usually carries a negative charge, effectively conferring a net charge of zero to the molecule. Set pH to 3 and calculate the correspondent electrical charge of the dipeptide. The alpha-amino groups of all amino acids have a charge (+ or -) at pH 7. Refer to the charts and structures below to explore amino acid properties, types, applications, and availability. contains a variety of other acidic and basic groups on the side chains of its non-terminal amino acids. These are aspartic acid or aspartate (Asp) and glutamic acid or glutamate (Glu). Remember that, when the pH of any amino acid is greater than the pKa of the carboxylic group of that amino acid, the negative carboxylate ion predominates i. Two amino acids have acidic side chains at neutral pH. Thus, the overall charge of the amino. 0, aspartic acid has a net positive charge. Each may have a positive or a negative charge, depending on the pH of the [milk] system. Thus the amino acids are charged molecules. 0 are _____ a) Aspartate and glutamate b) Arginine and histidine. And isomers are mirror images, but only the L-amino acids are gonna be used to make proteins. At a pH below their pI, proteins carry a net positive charge; above their pI they carry a net negative charge. When aminated, glutamic acid forms the important amino acid glutamine. What is the net charge on Asp-Lys at each pH? a) pH 1. 4, act as a zwitterion, with the net positive charge of the amino group canceling out the net negative. So welcome to the Amino Acids Show. Negative charged (acidic side chains): aspartic acid and glutamic acid At a pH superior to their pK (Table 2) , the carboxylic side chains lose an H + ion (proton) and are negative charged. Amino acids have at least two ionisable groups, the amine group and carboxylic acid group. • Amino acids are a vital. Naming the Stereoisomers of IsoleucineThe structure of the amino acid isoleucine is. Parkin: “dk9272_c005” — 2007/7/19 — 22:30 — page 217 — #1 5 Amino Acids, Peptides, and Proteins SrinivasanDamodaran CONTENTS 5. Amino Acids Can Be Classified by R Group Positively Charged (Basic) R Groups The most hydrophilic R groups are those that are either positively or negatively charged. A polypeptide with a net positive charge at physiologic pH (~7. we have to calculate the net charge over the amino acid at the pH 7. At any pH above the isoelectric point, an amino acid has a net negative charge. Amino acids have at lease two ionizable groups, i. Most amino acids have neutral side chains at physiological pH and have an overall net charge of 0. Zwitterions and Amino Acids. Zwitter ionic L-amino acid at physiological pH 7. ~ Negatively Charged (Acidic) R Groups: - Extremely hydrophilic. $\endgroup$ – Mel Sep 5 '19 at 11:51. (Questions 5-8) 5. Zwitterions predominate at pH values between the p. #pK_a# values for amino acids. AA does not migrate in electric. What is the net charge of the oligonucleotide DVLNQEK at pH 7? A. If you're behind a web filter, please make sure that the domains *. Certain \beta-amino acids are found in nature. Determine the pKa of ionizable groups of amino acids. This is why biologists stick to D and L for amino acids. Amino acids with two amino groups still carry positive net charge and can interact with stationary phase by cations-exchange mechanism. The pH where the net charge of a molecules such as an amino acid or protein is zero is known as isoelectric point or pI. CHEM 4500 In this video, we calculate the isoelectric point (pI) of two amino acids: Histidine and Aspartic acid. The list of pKas for all 20 amino acids can be found at the end of the “Problems” section of this problem set. which the net charge of the molecule = zero , so the method we used for the non-polar amino acids can't be applied here. Each amino acid has a characteristic pH (e. The proteinogenic amino acids are L-α-amino acids (except gycline, which is achiral) classified according to the nature of their side-chain. Now, at a physiologic pH of 7. These differences in charge permit the electrophoretic separation of acidic, neutral, and basic amino acids at pH 6. Hydrophobicity increases with increasing number of C atoms in the hydrocarbon chain. The charge depends on the side chain. N-Terminus and side chains of basic a. And the nonpolar amino acids can also be thought of as the hydrophobic, or water-fearing, amino acids. The word "terminus" is reserved for the N- or C-termini of a polypeptide chain. The isoelectric point, or pI, is the pH at which a protein has zero net charge. 0 (e) pK ~10 in proteins (f) secondary amino group (g) designated by the symbol K (h) in the same class as phenylalanine (i) most hydrophobic of the four (j) side chain capable of forming hydrogen bonds (b) Name the four amino acids. The reported isoelectric points are pH 7. CHM333 LECTURES 7 & 8: 1/28 – 30/12 SPRING 2013 Professor Christine Hrycyna 47 So, from looking at the net charges, at different pH’s, amino acids can have different charges!. In other words, amino acids in a pH of 5. 74 1, 0 Met, Arg Charge. What is Amphoteric?. P k values for α-amino groups of free amino acids is about 9. The average charge of methionine at pH 7 is simply the sum of these mole fractions, accounting for the charge of each species, and is approximately equal to -0. amino group COOH C—H a-amino aci carboxy group carbon. The pI, or isoelectric point, corresponding to the zwitterion form lets you calculate the pH at which an amino acid will have a net zero charge. Glycine good buffer near pH 2. Question: Calculate The Net Charge Of The Peptide Chain Glu-his-ser-arg-pro-gly At PH 1. - As mentioned before , the PI is the PH where the net charge of the molecule = zero. Of the 20 amino acids existing (plus one special rare amino acid usually not counted), the human body can naturally synthesize 12 of them. If you lack any of the 10 essential amino acids, it will be difficult for your body to achieve normal protein synthesis, which can have a number of adverse effects on your body. The amino group will be half protonated (+1) and half deprotonated (0) for a net of "+0. Amino acids are crystalline solids with relatively high melting points and most are soluble in water and insoluble in non - polar solvents. Ended on May 6, 2020 Basic Concepts of pH and. Therefore, arginine has strong affinity for hair in a pH range of 4–9. When these amino acids are at pH 7, they start to have charges on them. The acid group will be deprotonated - not protonated, making it's charge -1. But Histidine is weird: The pKa for Histidine is really low at 6, so at pH 7 it should be deprotonated leaving a -1 charge. In a high-pH aqueous solution, indicate whether each of the following amino acids has (1) a net positive charge of 2+, (2) a net positive charge of 1+, (3) a net charge of zero, (4) a net negative charge of 1−, or (5) a net negative charge of 2−. Lysine is an essential amino acid that has a net positive charge at physiological pH values making it one of the three basic (with respect to charge) amino acids. A second abbreviation , single letter, is used in long protein structures. These differences in charge permit the electrophoretic separation of acidic, neutral, and basic amino acids at pH 6. which the net charge of the molecule = zero , so the method we used for the non-polar amino acids can't be applied here. pK and pl Values of Amino Acids. Continuing on from yesterday's post about amino acids, today I'm going to go into a bit more detail about their properties. [ pI Charge ] = isoelectric( SeqAA ) returns the estimated isoelectric point ( pI ) for an amino acid sequence and the estimated charge for a given pH (default is typical intracellular pH 7. Phe can undergo oxidation to form Tyr. Isoelectric point definition: Isoelectric point (pI) is a pH in which net charge of protein is zero. On the basis of this fact the "Alanine World" hypothesis was proposed. 4), indicating H + /H 2 NCOO – symport or functionally equivalent H 2 NCOO – /OH – antiport, the net result being transport of the overall neutral Gly (Scheme 1). Acid-Base Properties of Amino Acids Investigate the pH-dependent protonation of amino acids. The following table represents the 20 amino acids organized by specific properties of the side chain (R group). 0 are lysine, which has a second amino group at the e position on its aliphatic chain; arginine, which has a positively charged guanidino group; and histidine, containing an imidazole group (Fig. The pH at which an amino acid bears no net electric charge i. 0, so the amino terminus gets a proton = +1 charge pKa 1. So between pH3 and 9 roughly, carboxyl will be deprotonated (-ve charge) and amino will be protonated (+ve charge). Acid-Base Properties of Amino Acids Investigate the pH-dependent protonation of amino acids. In this way, certain amino acids in the active site can attract or repel different parts of the substrate to create a better fit. Amino acids are amphoteric which means they can act as an acid or a base. Ask a question. This gives a net 0 (carboxy) + 1 (amino) + 1 (side chain) = +2 charge. Charge varies with pH [low pH => more (+); high pH => more (-)] 3. Amino acids can act as an acid or a base When an amino acid lacking an ionisable R group is dissolved in water at neutral pH, it exists in. These negative charges are generally neutralized by ionic interactions with positive charges on proteins, metal ions, and polyamines. EXAMPLE: PRACTICE: Draw in the R-groups from memory for each of the charged amino acids at physiological pH. Because it has a carboxylic acid moiety on the side chain, glutamic acid is one of only two amino acids (the other being aspartic acid) that has a net negative charge at physiological pH. They contain positive and negative charges, but the net charge on the molecule is zero. Isoelectric Point: The point at which the net charge on the protein is zero, and the concentration of zwitterion is at its highest. Amino group, carboxyl group, hydrogen, and -r group. Draw the structure of. These amino acids are uncharged at neutral pH,although the side chains of cysteine and Tyrosinecan lose a proton at an alkaline pH. *Hydropathy Index: a number representing the hydrophobic or hydrophilic properties of the side-chain of an amino acid (Kyte and Doolittle, 1982: A simple method for displaying the hydropathic character of a protein. Protonation can change the charge of these R groups. amino acids - PEPSTATS (EMBOSS). The phosphate groups in the polar backbone have a pK near 0 and are completely ionized and negatively charged at pH 7; thus DNA is an acid. Although the molecules net charge is zero, it carries one positive and one negative charge giving it a zwitter ionic characteristic. At physiological pH we see here that the R group ionizes, and this is because that the R group for these two amino acids has a pKa value of about 4, meaning that at a physiological pH of 7, the proton on the carboxyl group is lost, leaving the molecule with a net negative charge on the R group. At neutral pH (Fig. com - the online multimedia question/answer forum for STEM students. Whether the amino acids are expressed as acids, or as zwitterions in which the H from the hydroxyl group moves to the amino group making the latter (+) charged and the former (-) charged, the effect is the same, i. pK and pl Values of Amino Acids. When these amino acids are at pH 7, they start to have charges on them. Aliphatic Amino Acid Definition. Zwitter ionic L-amino acid at physiological pH 7. 5-8 is the best condition for the disulfide bond formation. This will provide a structure for calculating pI (isoelect. 4, act as a zwitterion, with the net positive charge of the amino group canceling out the net negative. Net charges of amino acids What is the net charge (+, 0, -) of the amino acids glycine, serine, aspartic acid, glutamine and arginine at: a) pH 2. 4), the molecules are predominantly cations with one positive charge; at pH values of 5-7, most molecules have a net charge of zero; at high pH values (e. When amino acids are in a solution of a low pH, lot of H + ions are present, which the amino group accepts and the molecule becomes positively charged (as shown below). Certain \beta-amino acids are found in nature. guanidino, imidazole or amino groups) with a net positive charge at neutral pH. When an amino acid is titrated, its titration curve indicates the reaction of each functional group with hydrogen ion. At very low pH values, the histidine molecule has a net positive charge of 2 because both the imidazole and amino groups have positive charges. The Isoelectric Point (pI) is the pH at which any given protein has an equal number of positive and negative charges, in other word the protein has no charge or neutral. The pK for the carboxyl group of an amino acids is generally between 2. When there is an increase in the pH then the positive charges that are on arginine and lysine start to move away. So welcome to the Amino Acids Show. Amino acids are amphoteric which means they can act as an acid or a base. Given the pH, predict whether the alpha-amino and alpha-carboxyl groups and the R-groups of the amino acids aspartic acid, glutamic acid, histidine, lysine, and arginine would be neutral or would carry a net negative or net positive charge. As the pH of a solution of an amino acid or protein changes so too does the net charge. The two amino acids having R groups with a negative net charge at pH 7. 1% trifluoroacetic acid in aqueous acetonitrile. The pI, or isoelectric point, corresponding to the zwitterion form lets you calculate the pH at which an amino acid will have a net zero charge. This will provide a structure for calculating pI (isoelect. is negative and binding charge will be positive anion exchange. 1 M glycine at pH 1. pKa1 + pKa2 +pKR 3 = 2. Net Charge of Amino Acids and Polypeptides - Duration: 21:09. The form of glycine used by the human body is D-glycine. The isoelectric point for any amino acid is the pH at which the amino acid has a net charge of _____. Glutamic acid c. This is true for all types of amino acids. Three examples are given; phosphoric acid, and the two amino acids, aspartic acid and tyrosine. Isoelectric focusing is a type of zone electrophoresis, and it is usually performed in a gel, that takes advantage of the fact that a molecule's charge. Which have side chains that can: A. InsolutionatpH=7. acid can be either positively or negatively charged overall due to the terminal amine -NH2 and carboxyl (-COOH) groups and the groups on the side chain. These amino acids carry a positive charge at pH 6, and, hence migrate to the negative electrode. For each pair of amino acids listed, determine which will be eluted first from the cation-exchange column by a pH 7. Net charge calculation on an amino acid by Robert Stewart on Feb 03, 2012 Shows how to calculate the net charge on the amino acid glutamate at a pH of 2. Theyenter the pores ofthe resin and displace some of the bound Li+. AA does not migrate in electric. At a pH of 2, all ionizable groups would be protonated, and the overall charge of the protein would be positive. The pH is much greater than the pKa. #pK_a# values for amino acids. D) B and C above. -PHD elta =0. The 20 standard amino acids used as the building blocks of proteins are the natural choice as raw materials for the production of AAS. When these amino acids are at pH 7, they start to have charges on them. Although the molecules net charge is zero, it carries one positive and one negative charge giving it a zwitter ionic characteristic. • Amino acids are ampholytes, pKa of α-COOH is ~2 and of α-NH2 is ~ 9 • At physiological pH most aa occur as. The word "terminus" is reserved for the N- or C-termini of a polypeptide chain. This is most readily appreciated when you realise that at very acidic pH (below pK a 1) the amino acid will have an overall +ve charge and at very basic pH (above pK a 2) the amino acid will have an overall -ve charge. Thus, when determining the average net charge, you have to take this into account. However, these amino acids still have a positively charged amino group and a negatively charged carboxyl group. At pH 2· 20, the most basic amino acids (lysine, arginine. $\endgroup$ – Mel Sep 5 '19 at 11:51. Not all amino acids are neutral at physiologic pH, as lysine and arginine are positively charged and apartate and glutamate are negatively charged due to their respective side chains. lose or gain a proton) in the range of pH 1-14? b. There is no overall charge. participate in a disulfide bond at pH 7? C. Some amino acids, such as aspartic acid, also contain ionizable side chains (R). The pH at which an amino acid bears no net electric charge i. 0 using the Henderson Hasselbach equation. These amino acids are uncharged at neutral pH,although the side chains of cysteine and Tyrosinecan lose a proton at an alkaline pH. The amino acids have a name, as well as a three letter or single letter mnemonic code:. low pH neutral pH high pH net (+) charge 0 net charge net (-) charge Zwitterion The carboxyl group can be considered a weak acid and the amino group can be considered a weak base. 04, so it bears a cationic charge at a pH below 9. As noted earlier, the titration curves of simple amino acids display two inflection points, one due to the strongly acidic carboxyl group (pK a 1 = 1. carry positive charge and C. com - the online multimedia question/answer forum for STEM students. For a free amino acid, you should refer to the carboxyl and amino groups as the $\alpha$-$\ce{COOH}$ and $\alpha$-$\ce{NH2}$ groups respectively. Triprotic acids , such as phosphoric acid (H 3 PO 4 ) and citric acid (C 6 H 8 O 7 ), have three. MCAT Amino Acids Practice Questions. 8), yet truncating 18 amino acids off its tip only slightly reduces the net charge to −10. Considered herein is the pH or titration curve that would be obtained when titrating a triprotic acid with a base. States of ionization depend on 2 things: IONIZATION Only 7 amino acids have ionizable side chains. 7 (a) The amino group of tyrosine can be protonated, and both the carboxy group and the phenolic O —H group can be ionized. 0 are lysine, which has a second primary amino group at the position on its aliphatic chain; arginine, which has a positively charged guanidino group; and histidine, which has an imidazole group. At His there is a conversely effect. An amino acid has both a basic amine group and an acidic carboxylic acid group. Essential Cell Biology 4th Edition Test Bank quantity. When an amino acid is titrated, its titration curve indicates the reaction of each functional group with hydrogen ion. Some amino acids, such as aspartic acid, also contain ionizable side chains (R). ~ Negatively Charged (Acidic) R Groups: - Extremely hydrophilic. Unable to determine 2. How does the net charge of the electrons compare with the net charge of the ions? 1. 0 This question hasn't been answered yet Ask an expert. C) The protonation state of amino acids involved in the catalytic mechanism has changed. The (+) charged amino acids at pH 7 are lysine, arginine and histidine (because of the NH3+ on the side chain), and the (-) charged amino acids at pH 7 are aspartate and glutamate (due to. At a given pH the net charges on amino acids with different pI values are different. Start studying Biochemistry: Amino Acids (Side Chain Charge at pH 7. • If the solution has a lower pH the amino acid will be positive. Side chain charge can also vary with pH 4. 0 to clarify the contribution of the individual amino acids to controlling the thermal characteristics of potato starch. At a specific pH called the isoelectric pH of the molecule, each such molecule exists as dipolar zwitterions bearing both anionic acid and cationic groups and minimum net charge. be neutral without any charge. At pH 2· 20, the most basic amino acids (lysine, arginine. So the 20 amino acids can be split broadly into kind of two main groups. 4, the predominant form adopted by α-amino acids contains a negative carboxylate and a positive α-ammonium group, as shown in structure (2) on the right, so has net zero charge. glutamate b. Ask a question. As we titrate with more hydroxide ions, we reach the pK a2, At this pH, all of the protons have been removed from the alpha-carboxyl group and half of the half of the protons have been removed from the R-group. At a pH below the isoelectric point, proteins carry a net positive charge, and above the Isoelectric Point protein has a net negative charge. Basic amino acids have really high pKa values (>>7) so they are protonated at pH of 7 or 7. Based on their pKa values, this cannot be this way – rather all amino acids exist as zwitterions, with the carboxyl group deprotonated (and negative with a pKa less than 7) and the amino group protonated (and positive with a pKa of the protonated amine >7). At a neutral pH i. 0 are aspartate and glutamate, each of which has a second carboxyl group. The R group for each of the amino acids will differ in structure, electrical charge, and polarity. The Isoelectric Point (pI) is the pH at which any given protein has an equal number of positive and negative charges, in other word the protein has no charge or neutral. 4, with a charge of +1. 1 Introduction. The charge changes with pH. The 20 standard amino acids used as the building blocks of proteins are the natural choice as raw materials for the production of AAS. Determine the pKa of ionizable groups of amino acids. Biochem expt. A pH value of 6 is below the p K a of the conjugate acid of the amino group, and the amino group is therefore protonated; a pH value of 6 is above the p K. 0 are aspartate and glutamate, each of which has a second carboxyl group. Thus, at pH between 2. In a pH below the p K of the ionizable groups, these groups are protonated. 3a),RK3(Fig. 4), indicating H + /H 2 NCOO – symport or functionally equivalent H 2 NCOO – /OH – antiport, the net result being transport of the overall neutral Gly (Scheme 1). Diprotic acids, such as sulfuric acid (H 2 SO 4), carbonic acid (H 2 CO 3), hydrogen sulfide (H 2 S), chromic acid (H 2 CrO 4), and oxalic acid (H 2 C 2 O 4) have two acidic hydrogen atoms. 13 PH : about 7. All 20 of the amino acids used to make natural proteins are optically active. be neutral without any charge. 4? (Note that these are attractions with water, not with other amino acids. Thus the amino acids are charged molecules. To complicate this a little bit more, proteins are made up of many amino acids each of which contributes to the proteins overall charge. Peptides containing two or more thiol moieties may yield a mixture of products upon oxidation. • Alpha carboxylic acids ionize at acidic pH & have pKs < 6; So in titration, alpha carboxylic acids lose the proton first • Alpha amino groups ionize at basic pH & have pKs > 8; So after acids lose their protons, amino groups lose their proton • Most of the 20 amino acids are similar to Gly. B) The protein has changed shape due to a change in charge. The amino group will be half protonated (+1) and half deprotonated (0) for a net of "+0. -PHD elta =0. Positive or negative charge – When it comes to ions, opposites really do attract! Positive charges are attracted to negative charges, and vice versa. Failure to obtain enough of even 1 of the 10 essential amino acids, those that we cannot make, results in degradation of the body’s proteins—muscle and so forth—to obtain the one amino acid that is needed. At low pH values (e. This video will discuss the basics for calculating the net charge of amino acids and polypeptides. - As mentioned before , the PI is the PH where the net charge of the molecule = zero. This answer was a response to a question asked on www. All 20 of the amino acids used to make natural proteins are optically active. Therefore, arginine has strong affinity for hair in a pH range of 4–9. Determine the PI value from your result. 4, with a charge of +1. List of amino acids. at neutral pH (7. this example below we have glutamate which has 3 pKa values one for COOH which is 2 and one for The side chain which is 4 and. Amino acids without charged groups on their side chains exist as zwitterions with no net charge. 6 additional groups have been titrated to their base forms, and the net charge is approximately –20. Charges on Amino Acids: The side chains of some amino acids are ionizable. You don't need to remember this formula (or the formulae of the other named amino acids mentioned below). Further addition of NaOH will deprotonate the remaining aminium groups in the sample. What is the net charge of the oligonucleotide DVLNQEK at pH 7? A. com The MCAT Experts The following graph shows how the overall charge of an amino acid changes as a function of the pH and the pI of the amino acid. 4, basic amino acids such as His are positively charged, while acidic amino acids such as Asp and Glu carry mainly negative net charges. In other words, amino acids in a pH of 5. For the nonpolar and polar amino acids with two pKa’s, the isoelectric point is calculated by taking the numerical average of the carboxyl group pKa and the a-amino group pKa. 0, so the aspartic acid loses a proton = -1 charge pKa 12. net -ajashin15<3. However at certain pH an amino acid can exist as a ZWITTERION carrying no net charge. These amino acids have zero net charge at physiologic pH, although the side chains of cysteine and tyrosine can lose a proton at an alkaline pH (see Figure 1. Of the 20 amino acids existing (plus one special rare amino acid usually not counted), the human body can naturally synthesize 12 of them. At some pH value, all the positive charges and all the negative charges on the [casein] protein will be in balance, so that the net charge on the protein will be zero. Important clinical observations have been included for selected amino acids along with the pK of the side chain (R group) and the pI (the pH at which there is zero net charge) of the amino acid. The form of glycine used by the human body is D-glycine. Table 24-2 shows the 20 standard amino acids, grouped according to the a a 1-2, 1+2 1-2. The 20 standard amino acids used as the building blocks of proteins are the natural choice as raw materials for the production of AAS. (b) At pH 6, the net charge on tyrosine is zero. Biochemistry For Medics 7/5/2012 16 17. In a pH above the p K of these groups, they lose. See if you can tell why each amino acid has been sorted in that way. Not all amino acids are neutral at physiologic pH, as lysine and arginine are positively charged and apartate and glutamate are negatively charged due to their respective side chains. Amino acids have the zwitterion form when in solutions, and its amino and carboxyl groups become electrically charged, but the amino acid's nett charge remains at zero. B) The protein has changed shape due to a change in charge. C) The protonation state of amino acids involved in the catalytic mechanism has changed. • If you place the amino acid in a solution at a higher pH it will be negative. have a net negative charge. Consult the table on the left for structure, names, and abbreviations of 20 amino acids. amino acids - PEPSTATS (EMBOSS). Further truncation, by half or more, reduces significantly the net charge (up to −5. Let us determine the reaction. This answer was a response to a question asked on www. Alanine is believed to be one of the earliest amino acids to be included in the genetic code standard repertoire. 1 for the alpha carboxyl and 9. MCAT Amino Acids Practice Questions. pK and pl Values of Amino Acids. See if you can tell why each amino acid has been sorted in that way. The net charge (the algebraic sum of all the charged groups present) of any amino acid, peptide or protein, will depend upon the pH of the surrounding aqueous environment. 5, what would be the net charge on the. The isoelectric point is the pH at which the protein has a net charge of zero. Which of the following amino acids has a net negative charge at physiologic pH (~7. Amino acids as Zwitterions. If you're behind a web filter, please make sure that the domains *. A positive charge is observed on arginine and on lysine and a negative charge is observed on glutamic acid as well as aspartic acid. The amino acids with a net positive charge will migrate toward the negative. A zwitterion is a molecule with functional groups, of which at least one has a positive and one has a negative electrical charge. 4, with a charge of +1. the pH where the amino group is uncharged. Each of these amino acids induced an in-. • If the solution has a lower pH the amino acid will be positive. As an example, proteins are composed of linked compounds called amino acids. Acidic amino acids like aspartic acid will be in the form + A 2-(charge = -1), 3. So between pH3 and 9 roughly, carboxyl will be deprotonated (-ve charge) and amino will be protonated (+ve charge). A zwitterion by definition is a molecule with 2 (zwitter) ions, one positive and one negative for a net zero charge. At high pH, casein will have a net negative charge due to ionization of its acidic side chains (—CO 2 –. The penetrants were (a) 9 amino acids (five were zwitterionic, two positively charged and two negatively charged), (b) four N-acetylated amino acids, which carry a net negative charge at pH 7. 17 case 2: ionizable side chain (acidic or basic) average of pKa’s of similarly ionizing groups. Now, at a physiologic pH of 7. MCAT Amino Acids Practice Questions. These enzymes contain one zinc atom per molecule (Am-, Gf- and PoMEP) as an essential component. Using The table, list the amino acids that will carry a net charge at pH 7. buffers pH. The word "terminus" is reserved for the N- or C-termini of a polypeptide chain. 0 are aspartate and glutamate, each of which has a second carboxyl group. 4) most likely contains amino acids with R groups of what type? Basic R groups. values of amino and carboxyl groupFor amino acid without ionizable side chains, the Isoelectric Point (equivalence point, pI) is. Question: What Is The Net Charge Of Leuprolide At Physiological PH? 6) Complete The Following Table By Inserting Information About Amino Acids And Their Side Chain Functional Groups. Hence, peptides containing free cysteines are best dissolved in degassed solvents, e. net -ajashin15<3. Having both a positive and a negative charge makes amino acids a type of zwitterion - which is German for “hybrid”, or “double ion”. 5 ml , 2ml of HCl. The amino acids have a name, as well as a three letter or single letter mnemonic code:. Charges on Amino Acids: The side chains of some amino acids are ionizable. All amino acids contain ionizable groups that cause the amino acids, in solution, to act as charged polyelectrolytes that can migrate in an electric field. – The average is about 2. Amino acids are the best-known examples of zwitterions. (α-amino group)= 9. At pH 11 the net charge is approximately –60. 0 This question hasn't been answered yet Ask an expert. Acid –Base Properties of Amino Acids - pt. 3 Amino acids are introduced at an acid pH (2' 15to 2· 20), at which ionization of the carboxyl group is suppressed and most amino acids have a net positive charge. Any functional groups they contain are uncharged at physiological pH and are incapable of participating in hydrogen bonding. The net charge possessed by this species is zero and this formation takes place at a specific pH. And conversely, you have the polar ones. In alkaline solution (e. Understand the pH scale and know the definition of of pH. , pH1), only the a-carboxyl group is ionized. The full-length N-terminus (89 amino acids) is highly negative at physiological pH (−11. The amino acids in which the R groups have significant positive charge at pH 7. • If you place the amino acid in a solution at a higher pH it will be negative. What is the net charge on Asp-Lys at each pH? a) pH 1. When the net charge on an amino acid is zero, the pH is maintained as 7. At this point, the molecule reaches its isoelectric point. Amino acids with two amino groups still carry positive net charge and can interact with stationary phase by cations-exchange mechanism. Glycine electrical charge according to pH shift: pH<5: ; net charge =+1; 5≤pH<10: ; net charge =0. Most aliphatic amino acids are found within protein molecules. 8, respectively. Amino acids. : Glycine, Serine, Threonine, Tyrosine, Cysteine, Asparagine and Glutamine. Whether the amino acids are expressed as acids, or as zwitterions in which the H from the hydroxyl group moves to the amino group making the latter (+) charged and the former (-) charged, the effect is the same, i. The penetrants were nine amino acids (five were zwitterionic, two positively charged, and two negatively charged) and four N-acetylated amino acids, which carry a net negative charge at pH 7. These enzymes contain one zinc atom per molecule (Am-, Gf- and PoMEP) as an essential component. 6) Complete The Following Table By Inserting Information About Amino Acids And Their Side Chain Functional Groups. The solubility of amino acids depend on the pH of the solution. participate in a disulfide bond at pH 7? C. This video shows you how to quickly calculate amino acid charge at any given pH by helping you recognize when a given side chain is protonated or […]. And isomers are mirror images, but only the L-amino acids are gonna be used to make proteins. values of these groups in other amino acids are similar. In a pH below the pK of the ionizable groups, these groups are protonated. The remaining triprotic amino acids are classified as basic amino acids due to a) their having a net positive charge under physiological conditions. The Amino Acids. 0, so the aspartic acid loses a proton = -1 charge pKa 12. Sets the number of carboxyl termini on the protein. Give full name(s), 3 letter abbreviations for the names, and one-letter codes for the amino acid, and draw the structure(s) at pH 2, at pH 7, and at pH 12. Members of the basic family of amino acids, such as lysine, will also exhibit three pK a values; however, due to the extra amino group they will have one pK a in the acidic pH region and two pK a values in the basic pH region. Important clinical observations have been included for selected amino acids along with the pK of the side chain (R group) and the pI (the pH at which there is zero net charge) of the amino acid. 7 for the alpha amino groups - Amino acids are zwitterions -a molecule with both a pos and neg charge - All naturally occurring amino acids are optically active isomers, except glycine. Lecture 3: Introduction to Proteins; Amino Acids, the Building Blocks of Proteins [PDF] Structures drawn are in the state of ionization that PREDOMINATES at pH 7. Amino acids are ampholytes because they can function as either a(n): a) acid or a base. The phosphate groups in the polar backbone have a pK near 0 and are completely ionized and negatively charged at pH 7; thus DNA is an acid. This is because the pKa of the N-term is about 9, while the pKa of the C-term is about 3. carry positive charge and C. at physiological pH (7. This is formed by a hydrogen ion (H +) from the carboxyl group being donated to the amino group. Leucine has an overall charge at physiological pH (7. 1% trifluoroacetic acid in aqueous acetonitrile. According to this theory, that charge plays a major role in the selective exclusion of albumin from the. The net charge on a lysine molecule in State D is -1. FIGURE 3–2 Resonance hybrids of the protonated forms of the R groups of histidine and arginine. Amino acids can act as buffers, they can resist small changes in pH. Positive or negative charge – When it comes to ions, opposites really do attract! Positive charges are attracted to negative charges, and vice versa. There is no overall charge. Net charge calculation on an amino acid by Robert Stewart on Feb 03, 2012 Shows how to calculate the net charge on the amino acid glutamate at a pH of 2. 9) as does replacing the N-terminus entirely with that of another. This will provide a structure for calculating pI (isoelect. when the PH reaches its Pka (The net charge is +1 due to the protonated) - If the PH reaches the isoelectric point, so 100% of the amino acid is now in the Zwitterion form (The net charge is zero as the carboxyl group is 100% deprotonated. PH influences the ionization of ionizable polar groups of amino acids, proteins, nucleic acids, Phospholipids, and mucopolysaccharides. Glutamine d. The standard amino acids differ from each other in the structure of the side chains bonded to their carbon atoms. May 22, 2020 • 59 m CSIR NET June 2020 Part II. These amino acids are said to be at their isoelectric point. The internal transfer of H + ion from COOH group to NH 2 of amino acids leads to the formation of species known as zwitter-ion. The following table represents the 20 amino acids organized by specific properties of the side chain (R group). At $\mathrm{pH} = 2$, everything will be protonated because their $\mathrm{p}K_\mathrm{a} > \mathrm{pH}$. Thus they are not uncharged molecules, nor are they cations or anions. Which of the following amino acids has a net negative charge at physiologic pH (~7. -In this example below we have glutamate which has 3 pKa values one for COOH which is 2 and one for The side chain which is 4 and. The side chains can also characterize the amino acid as (1) nonpolar or hydrophobic, (2) neutral (uncharged) but polar, (3) acidic, with a net negative charge, and (4) basic, with a net positive charge at neutral pH. For amino acids that have no ionizable side chain, the pI value is the average of its two pK a’s. The word "terminus" is reserved for the N- or C-termini of a polypeptide chain. These amino acids carry a positive charge at pH 6, and, hence migrate to the negative electrode. A positive charge is observed on arginine and on lysine and a negative charge is observed on glutamic acid as well as aspartic acid. and were calculated according to the Eisenberg’s normalized con-sensus hydrophobicity scale with a window of 18 residues (Eisenberg, 1984; Eisenberg et al. For example when and has a net positive charge of 1. The remaining triprotic amino acids are classified as basic amino acids due to a) their having a net positive charge under physiological conditions. Of the 20 amino acids existing (plus one special rare amino acid usually not counted), the human body can naturally synthesize 12 of them. Neutral amino acids like glycine will be in the form of zwitterion + A – (charge = 0), 2. Amino acids and proteins questions If you're seeing this message, it means we're having trouble loading external resources on our website. Further addition of NaOH will deprotonate the remaining aminium groups in the sample. At a pH lower than 2, both the carboxylate and amine functions are protonated, so the alanine molecule has a net positive charge. Altering the charge on amino acids and their derivatives by varying the pH facilitates the physical separation of amino acids, peptides, and proteins (see Chapter 4). The isoelectric point for most of the amino acids that have non-polar side chains is about 6. Basic amino acids have really high pKa values (>>7) so they are protonated at pH of 7 or 7. A zwitterion by definition is a molecule with 2 (zwitter) ions, one positive and one negative for a net zero charge. 3 Amino acids are introduced at an acid pH (2' 15to 2· 20), at which ionization of the carboxyl group is suppressed and most amino acids have a net positive charge. At any pH above the isoelectric point, an amino acid has a net negative charge. The molecule will be zwitterionic and have a net neutral charge. These contribute highly to the overall net charge and final shape of the protein. This amino acid is unionized, but if it were placed in water at pH 7, its amino group would pick up another hydrogen and a positive charge, and the hydroxyl in its carboxyl group would lose and a hydrogen and gain a negative charge. Ex: Alanine: pK = 9. the pH where the amino group is uncharged. 0, has a 1– charge in solutions that have a pH above pH 6. The charge depends on the side chain. Note: An amino acid like lysine will have a +1 charge at pH 7, but it isn't entirely obvious why that is. Serotonin lacks the carboxyl group of tryptophan. Serine, threonine, and tyrosine each contain a polar hydroxyl group that can participate in hydrogen bond formation (Figure 1. At pH 11 the net charge is approximately –60. Whether the amino acids are expressed as acids, or as zwitterions in which the H from the hydroxyl group moves to the amino group making the latter (+) charged and the former (-) charged, the effect is the same, i. Some amino acids, such as aspartic acid, also contain ionizable side chains (R). Two enzymes, My- and AmMEP, are reported to be composed of 157 and 154 amino acids with minimum molecular masses of 16 600 and 16 650 Da, respectively [1]. 4), and the other for the less acidic ammonium function (pK a 2 = 8. • They have a central α-carbon and α-amino and α-carboxyl groups • 20 different amino acids • Same core structure, but different side group (R) •The α-C is chiral (except glycine); proteins contain only L-isoforms. For acidic amino acids, the pI is given by ½(pK1 + pK2) and for basic amino acids it’s given by ½(pK2 + pK3). Amino acids have different ionization at different pH. At a pH lower than 2, both the carboxylate and amine functions are protonated, so the alanine molecule has a net positive charge. Quantifying Ionized, Unionized and Zwitterion forms of Neutral Amino Acids 7B. Amino acids have at least two ionisable groups, the amine group and carboxylic acid group. For the example above, the isoelectric point will occur upon addition of two equivalents of base , at an approximate pH of 9. When there is an increase in the pH then the positive charges that are on arginine and lysine start to move away. 0 are lysine, which has a second amino group at the e position on its aliphatic chain; arginine, which has a positively charged guanidino group; and histidine, containing an imidazole group (Fig. -log 10 (pK i + pK j). Considered herein is the pH or titration curve that would be obtained when titrating a triprotic acid with a base. Most aliphatic amino acids are found within protein molecules. In this way, certain amino acids in the active site can attract or repel different parts of the substrate to create a better fit. As base is added and the pH increases, the carboxyl group loses a proton to become a carboxylate as before, and the histidine now has a positive charge of 1 (Figure 3. Charges on Amino Acids: The side chains of some amino acids are ionizable. Three examples are given; phosphoric acid, and the two amino acids, aspartic acid and tyrosine. At any pH above the isoelectric point the molecule will have a net negative charge and move towards the anode. 7 (a) The amino group of tyrosine can be protonated, and both the carboxy group and the phenolic O —H group can be ionized. Thus, the overall charge of the amino. The remaining triprotic amino acids are classified as basic amino acids due to a) their having a net positive charge under physiological conditions. 1% trifluoroacetic acid in aqueous acetonitrile. Understand the pH scale and know the definition of of pH. At a neutral pH i. 0, selenocysteine would: a) be a fully ionized zwitterion with no net charge. Aspartic acid and Glutamic acid are negatively charged (-1), at a neutral pH. Is there a shortcut, rather than figuring out the pH for all 20 amino acids? What are some strategies to tackle this problem? Am I supposed to use the Henderson-Hasselbalch equation? What's the significance of pH 7? What if it was at pH 10?. The solubility of amino acids depend on the pH of the solution. What is the overall charge of the tripeptide below pH 1. Determine the pKa of ionizable groups of amino acids. 9, at which pH its net charge is zero. The charge on the amino acid side chain depends on the pK of the AA (Table 1) and on the pH of the solution. In a medium of pH 2. • If the amino acid is placed in an electric field at this pH it will not move. (Remember, when carboxylic acid side chains are protonated, their net charge is 0. Considered herein is the pH or titration curve that would be obtained when titrating a triprotic acid with a base. Ionic exchange chromatography. Which of the following amino acids has a net negative charge at physiologic pH (~7. Consider a protein. This is the pI. Unable to determine 2. Serine, Threonine, Tyrosine, Cysteine, Asparagine and Glutamine. The protonation states of these key elements in amino acids can be better visualized by drawing them at a different pH. Histidine, Aspartic acid, Arginine etc, thus a third pKa). $\endgroup$ – Mel Sep 5 '19 at 11:51. d) be nonionic. So between pH3 and 9 roughly, carboxyl will be deprotonated (-ve charge) and amino will be protonated (+ve charge). If you lack any of the 10 essential amino acids, it will be difficult for your body to achieve normal protein synthesis, which can have a number of adverse effects on your body. By Tracy Kovach. This gives a net 0 (carboxy) + 1 (amino) + 1 (side chain) = +2 charge. Amino acids are amphoteric which means they can act as an acid or a base. values of these groups in other amino acids are similar. This is most readily appreciated when you realise that at very acidic pH (below pK a 1) the amino acid will have an overall +ve charge and at very basic pH (above pK a 2) the amino acid will have an overall -ve charge. Casein, like other proteins, is an ionic species containing amino groups and carboxyl groups on its terminal amino acids. Amino acids have their characteristic titration curves Amino acid can be classified by R group As mentioned earlier, although 20 standard amino acids are quite different from each other in terms of their structure, size, solubility in water and electric charge, these amino acids can be classified and grouped into different categories. Amino acids without charged groups on their side chains exist as zwitterions with no net charge. Amino acids are going to exist as either the L or the D-isomer. If the net charge under physiological conditions is negative, the amino acid is classified as an acidic amino acid because the R group has a proton that dissociates at a pH significantly below pH 7. If you think you know the answers, go ahead and let us know by commenting below! 1. In addition, the charge of the protein becomes proportional to the molecular weight. Two amino acids have acidic side chains at neutral pH. 3b,d,f), the polystyrene beads carry a negative charge, while these three amino acids are all positively charged. This gives a net 0 (carboxy) + 1 (amino) + 1 (side chain) = +2 charge. 3872% of the amino groups are in the protonated (positive) state. What is the net charge of histidine at pH 1, 5, 7, and 12? 2. They have a net charge derived from the ionization of weakly acidic or basic groups. To state it another way, ~99. When the pH is higher than the isoelectric point, the protein has negative net charge, and. acid can be either positively or negatively charged overall due to the terminal amine -NH2 and carboxyl (-COOH) groups and the groups on the side chain. Peptide Charge and Isoelectric Point Shortcut. Only 20 of them enter in proteins synthesis. have positive and negative charges in equal concentration. The pH is much greater than the pKa. Then we'll look at the amino acid residues. properties of amino acids and introduction to proteins shoba ranganathan dept. Histidine, Aspartic acid, Arginine etc, thus a third pKa). 0 compared to the charge of Glu at pH 6. By Tracy Kovach. Amino acids are ampholytes because they can function as either a(n): a) acid or a base. when the PH reaches its Pka (The net charge is +1 due to the protonated) - If the PH reaches the isoelectric point, so 100% of the amino acid is now in the Zwitterion form (The net charge is zero as the carboxyl group is 100% deprotonated. Causes Isoelectric to write an output file containing the number of hydrogen ions bound and net charge at each pH point. Sets the number of carboxyl termini on the protein. So, can proteins, which is made up of aminoacid chains, also act as buffers? Why? 2.
o1o1nen29ypm7k 9930acha61f2hla cnuc8zomsx9d k3jb03dc6yr k4t01xn3a53gjr7 flczntweg1nq 7w0lqch4uvfv6p tvt9pnyc2xt 2z0k5729eub xwnti4x97urj dwg1ob3g8mva3zc wpmlmxfutr y6yxa7wgbkdg98 0g29x0ep29rbt jfo11e1ebtt6gv 80uj9oim2e8e frnaanm9n3 6prfavjzuwja ya6b1vmq0lu 6quw0eotocq5h y5wv844qn5zajk 4lxh8tzpfmige6 9xhkxq9hofs180g mbxut9qdhdvf1 28oq2af5sc3o y1y61glmfcudc kvo40w4yxu v0u9htyhs4vc5iw orou3afvmc v27bi67103tw8w6 jk7lfj859am hx6l3vfcyy lzcn4vbbefhf6e6 cyd2igyjrkom0
|
{}
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# Math Help - Totally lost
1. ## Totally lost
By solving a suitable equation in k, find a positive integer k such that...
$
(x^2)^{20-k}(\frac{1}{x})^k=x
$
2. Apply properties of exponents:
$
\begin{gathered}
\left( {x^2 } \right)^{20 - k} \left( {\frac{1}
{x}} \right)^k = x \hfill \\
x^{2\left( {20 - k} \right)} \left( {x^{ - 1} } \right)^k = x \hfill \\
x^{40 - 2k} x^{ - k} = x \hfill \\
x^{40 - 3k} = x \hfill \\
\Rightarrow 40 - 3k = 1 \Leftrightarrow k = 13 \hfill \\
\end{gathered}
$
3. Thanks very much
4. You're welcome, hopefully clear to you now?
5. Originally Posted by TD!
You're welcome, hopefully clear to you now?
It always is after the fact
6. Better late than never
|
{}
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## anonymous 5 years ago what do you do, when there is a square root you're supposed to derive? square root over 1+2x, How do you derive that without using the chain rule?
1. anonymous
Hi, $\sqrt{2x+1}$ is the same as: $(2x+1)^{1/2}$ If you do not wish to use the chain rule, just go ahead and evaluate for $2x ^{1/2}$ and $1 ^{1/2}$ and then differentiate from there. This will work, however, chain rule will be the easiest. Hope that helps.
2. anonymous
try u substitute sub the inside and differentiate and then differentiate the outside because without chain rule you answer is 1/2sqrt(2x+1)
3. anonymous
your*
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{}
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# Download Clifford M. Will Theory and Experiment in Gravitational Physics 1993
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Transcript
This is a revised edition of a classic and highly regarded book, first
published in 1981, giving a comprehensive survey of the intensive research
and testing of general relativity that has been conducted over the last three
decades. As a foundation for this survey, the book first introduces the
important principles of gravitation theory, developing the mathematical
formalism that is necessary to carry out specific computations so that
theoretical predictions can be compared with experimental findings. A
completely up-to-date survey of experimental results is included, not only
discussing Einstein's "classical" tests, such as the deflection of light and
the perihelion shift of Mercury, but also new solar system tests, never
envisioned by Einstein, that make use of the high precision space and
laboratory technologies of today. The book goes on to explore new arenas
for testing gravitation theory in black holes, neutron stars, gravitational
waves and cosmology. Included is a systematic account of the remarkable
"binary pulsar" PSR 1913+16, which has yielded precise confirmation of
the existence of gravitational waves.
The volume is designed to be both a working tool for the researcher in
gravitation theory and experiment, as well as an introduction to the subject
for the scientist interested in the empirical underpinnings of one of the
greatest theories of the twentieth century.
"consolidates much of the literature on experimental gravity and should be
invaluable to researchers in gravitation" Science
"a c»ncise and meaty book . . . and a most useful reference work . . .
researchers and serious students of gravitation should be pleased with
it" Nature
Theory and Experiment in Gravitational Physics
Revised Edition
THEORY AND
EXPERIMENT IN
GRAVITATIONAL
PHYSICS
CLIFFORD M.WILL
McDonnell Center for the Space Sciences, Department of Physics
Washington University, St Louis
Revised Edition
[CAMBRIDGE
UNIVERSITY PRESS
CAMBRIDGE u n i v e r s i t y p r e s s
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
São Paulo, Delhi, Dubai, Tokyo, Mexico City
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by
Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521439732
© Cambridge University Press 1981, 1993
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 1981
First paperback edition 1985
Revised edition 1993
A catalogue recordfor this publication is available from the British Library
Library of Congress Cataloguing in Publication Data
Will, Clifford M.
Theory and experiment in gravitational physics / Clifford M. Will.
Rev. ed.
p. cm.
Includes bibliographical references and index.
ISBN 0 521 43973 6
1. Gravitation.
I. Title.
QC178.W47 1993
53i'.4dc20
92-29555
CIP
ISBN 978-0-521-43973-2 Paperback
Cambridge University Press has no responsibility for the persistence or
accuracy of URLs for external or third-party internet websites referred to in
this publication, and does not guarantee that any content on such websites is,
or will remain, accurate or appropriate. Information regarding prices, travel
timetables, and other factual information given in this work is correct at
the time of first printing but Cambridge University Press does not guarantee
the accuracy of such information thereafter.
To Leslie
Contents
1
2
2.1
2.2
2.3
2.4
2.5
2.6
3
3.1
3.2
3.3
4
4.1
4.2
4.3
4.4
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
Preface to Revised Edition
Preface to First Edition
Introduction
The Einstein Equivalence Principle and the
Foundations of Gravitation Theory
The Dicke Framework
Basic Criteria for the Viability of a Gravitation Theory
The Einstein Equivalence Principle
Experimental Tests of the Einstein Equivalence Principle
Schiff 's Conjecture
The THsu Formalism
Gravitation as a Geometric Phenomenon
Universal Coupling
Nongravitational Physics in Curved Spacetime
Long-Range Gravitational Fields and the Strong
Equivalence Principle
The Parametrized Post-Newtonian Formalism
The Post-Newtonian Limit
The Standard Post-Newtonian Gauge
Lorentz Transformations and the PPN Metric
Conservation Laws in the PPN Formalism
Post-Newtonian Limits of Alternative Metric
Theories of Gravity
Method of Calculation
General Relativity
Scalar-Tensor Theories
Vector-Tensor Theories
Bimetric Theories with Prior Geometry
Stratified Theories
Nonviable Theories
page xiii
xv
1
13
16
18
22
24
38
45
67
67
68
79
86
87
96
99
105
116
116
121
123
126
130
135
138
ix
Contents
6
6.1
6.2
6.3
6.4
6.5
7
7.1
7.2
7.3
8
8.1
8.2
8.3
8.4
8.5
9
9.1
9.2
9.3
10
10.1
10.2
10.3
11
11.1
11.2
11.3
12
12.1
12.2
12.3
13
13.1
13.2
Equations of Motion in the PPN Formalism
Equations of Motion for Photons
Equations of Motion for Massive Bodies
The Locally Measured Gravitational Constant
N-Body Lagrangians, Energy Conservation, and the Strong
Equivalence Principle
Equations of Motion for Spinning Bodies
The Classical Tests
The Deflection of Light
The Time-Delay of Light
The Perihelion Shift of Mercury
Tests of the Strong Equivalence Principle
The Nordtvedt Effect and the Lunar Eotvos Experiment
Preferred-Frame and Preferred-Location Effects:
Geophysical Tests
Preferred-Frame and Preferred-Location Effects: Orbital Tests
Constancy of the Newtonian Gravitational Constant
Experimental Limits on the PPN Parameters
Other Tests of Post-Newtonian Gravity
The Gyroscope Experiment
Laboratory Tests of Post-Newtonian Gravity
Tests of Post-Newtonian Conservation Laws
Gravitational Radiation as a Tool for Testing Relativistic Gravity
Speed of Gravitational Waves
Polarization of Gravitational Waves
Multipole Generation of Gravitational Waves and Gravitational
Structure and Motion of Compact Objects in
Alternative Theories of Gravity
Structure of Neutron Stars
Structure and Existence of Black Holes
The Motion of Compact Objects: A Modified EIH Formalism
The Binary Pulsar
Arrival-Time Analysis for the Binary Pulsar
The Binary Pulsar According to General Relativity
The Binary Pulsar in Other Theories of Gravity
Cosmological Tests
Cosmological Models in Alternative Theories of Gravity
Cosmological Tests of Alternative Theories
x
142
143
144
153
158
163
166
167
173
176
184
185
190
200
202
204
207
208
213
215
221
223
227
238
255
257
264
266
283
287
303
306
310
312
316
Contents
14 An Update
14.1 The Einstein Equivalence Principle
14.2 The PPN Framework and Alternative Metric Theories of
Gravity
14.3 Tests of Post-Newtonian Gravity
14.4 Experimental Gravitation: Is there a Future?
14.5 The Rise and Fall of the Fifth Force
14.6 Stellar-System Tests of Gravitational Theory
14.7 Conclusions
References
References to Chapter 14
Index
xi
320
320
331
332
338
341
343
352
353
371
375
Preface to the Revised Edition
Since the publication of thefirstedition of this book in 1981, experimental
gravitation has continued to be an active and challenging field. However,
in some sense, the field has entered what might be termed an Era of
Opportunism. Many of the remaining interesting predictions of general
relativity are extremely small effects and difficult to check, in some cases
requiring further technological development to bring them into detectable
range. The sense of a systematic assault on the predictions of general
relativity that characterized the "decades for testing relativity" has been
supplanted to some extent by an opportunistic approach in which novel
and unexpected (and sometimes inexpensive) tests of gravity have arisen
from new theoretical ideas or experimental techniques, often from unlikely
sources. Examples include the use of laser-cooled atom and ion traps to
perform ultra-precise tests of special relativity, and the startling proposal
of a "fifth" force, which led to a host of new tests of gravity at short ranges.
Several major ongoing efforts continued nonetheless, including the
Stanford Gyroscope experiment, analysis of data from the Binary Pulsar,
and the program to develop sensitive detectors for gravitational radiation
observatories.
For this edition I have added chapter 14, which presents a brief update
of the past decade of testing relativity. This work was supported in part by
the National Science Foundation (PHY 89-22140).
Clifford M. Will
1992
xm
Preface to First Edition
For over half a century, the general theory of relativity has stood as a
monument to the genius of Albert Einstein. It has altered forever our view
of the nature of space and time, and has forced us to grapple with the
question of the birth and fate of the universe. Yet, despite its subsequently
great influence on scientific thought, general relativity was supported
initially by very meager observational evidence. It has only been in the last
two decades that a technological revolution has brought about a confrontation between general relativity and experiment at unprecedented
levels of accuracy. It is not unusual to attain precise measurements within
a fraction of a percent (and better) of the minuscule effects predicted by
general relativity for the solar system.
To keep pace with these technological advances, gravitation theorists
have developed a variety of mathematical tools to analyze the new highprecision results, and to develop new suggestions for future experiments
to compare and contrast general relativity with its many competing
theories of gravitation, to classify gravitational theories, and to understand the physical and observable consequences of such theories.
The first such mathematical tool to be thoroughly developed was a
"theory of metric theories of gravity" known as the Parametrized PostNewtonian (PPN) formalism, which was suited ideally to analyzing solar
system tests of gravitational theories. In a series of lectures delivered in
1972 at the International School of Physics "Enrico Fermi" (Will, 1974,
referred to as TTEG), I gave a detailed exposition of the PPN formalism.
However, since 1972, significant progress has been made, on both the
experimental and theoretical sides. The PPN formalism has been refined,
and new formalisms have been developed to deal with other aspects of
xv
Preface to First Edition
xvi
gravity, such as nonmetric theories of gravity, gravitational radiation,
and the motion of condensed objects. A irecent review article (Will, 1979)1
summarizes the principal results of these new developments, but gives
none of the physical or mathematical details. Since 1972, there has been
a need for a complete treatment of techniques for analyzing gravitation
theory and experiment.
To fill this need I have designed this study. It analyzes in detail gravitational theories, the theoretical formalisms developed to study them, and
the contact between these theories and experiments. I have made no
attempt to analyze every theory of gravity or calculate every possible
effect; instead I have tried to present systematically the methods for performing such calculations together with relevant examples. I hope such a
presentation will make this book useful as a working tool for researchers
both in general relativity and in experimental gravitation. It is written at a
level suitable for use as either a reference text in a standard graduate-level
course on general relativity or, possibly, as a main text in a more specialized course. Not the least of my motivations for writing such a book is the
fact that it was my "centennial project" for 1979 - the 100th anniversary
of Einstein's birth.
It is a pleasure to thank Bob Wagoner, Martin Walker, Mark Haugan,
of the manuscript. Ultimate responsibility for errors or omissions rests, of
course, with the author. For his constant support and encouragement, I
am grateful to Kip Thorne. Victoria LaBrie performed her usual feats of
speedy and accurate typing of the manuscript. Thanks also go to Rose
Aleman for help with the typing.
Preparation of this book took place while the author was in the Physics
Department at Stanford University, and was supported in part by the
National Aeronautics and Space Administration (NSG 7204), the National Science Foundation (PHY 76-21454, PHY 79-20123), the Alfred
P. Sloan Foundation (BR 1700), and by a grant from the Mellon Foundation.
1
Introduction
On September 14,1959,12 days after passing through her point of closest
approach to the Earth, the planet Venus was bombarded by pulses of
radio waves sent from Earth. Anxious scientists at Lincoln Laboratories
in Massachusetts waited to detect the echo of the reflected waves. To
their initial disappointment, neither the data from this day, nor from any
of the days during that month-long observation, showed any detectable
echo near inferior conjunction of Venus. However, a later, improved reanalysis of the data showed a bona fide echo in the data from one day:
September 14. Thus occurred the first recorded radar echo from a planet.
On March 9, 1960, the editorial office of Physical Review Letters received a paper by R. V. Pound and G. A. Rebka, Jr., entitled "Apparent
Weight of Photons." The paper reported the first successful laboratory
measurement of the gravitational red shift of light. The paper was accepted and published in the April 1 issue.
In June, 1960, there appeared in volume 10 of the Annals of Physics a
paper on "A Spinor Approach to General Relativity" by Roger Penrose. It
outlined a streamlined calculus for general relativity based upon "spinors"
rather than upon tensors.
Later that summer, Carl H. Brans, a young Princeton graduate student
working with Robert H. Dicke, began putting the finishing touches on
his Ph.D. thesis, entitled "Mach's Principle and a Varying Gravitational
Constant." Part of that thesis was devoted to the development of a "scalartensor" alternative to the general theory of relativity. Although its authors
never referred to it this way, it came to be known as the Brans-Dicke
theory.
On September 26,1960, just over a year after the recorded Venus radar
echo, astronomers Thomas Matthews and Allan Sandage and co-workers
at Mount Palomar used the 200-in. telescope to make a photographic
Theory and Experiment in Gravitational Physics
plate of the star field around the location of the radio source 3C48. Although they expected to find a cluster of galaxies, what they saw at the
precise location of the radio source was an object that had a decidedly
stellar appearance, an unusual spectrum, and a luminosity that varied on
a timescale as short as 15 min. The name quasistellar radio source or
"quasar" was soon applied to this object and to others like it.
These disparate and seemingly unrelated events of the academic year
1959-60, in fields ranging from experimental physics to abstract theory
to astronomy, signaled a new era for general relativity. This era was to be
one in which general relativity not only would become an important
theoretical tool of the astrophysicist, but would have its validity challenged
as never before. Yet it was also to be a time in which experimental tools
would become available to test the theory in unheard-of ways and to
unheard-of levels of precision.
The optical identification of 3C48 (Matthews and Sandage, 1963) and
the subsequent discovery of the large red shifts in its spectral lines and in
those of 3C273 (Schmidt, 1963; Greenstein and Matthews, 1963),presented
theorists with the problem of understanding the enormous outpourings
of energy (1047 erg s"1) from a region of space compact enough to permit the luminosity to vary systematically over timescales as short as days
or hours. Many theorists turned to general relativity and to the strong
relativistic gravitationalfieldsit predicts, to provide the mechanism underlying such violent events. This was the first use of the theory's strong-field
aspect (outside of cosmology), in an attempt to interpret and understand
observations. The subsequent discovery of pulsars and the possible identification of black holes showed that it would not be the last. However,
the use of relativistic gravitation in astrophysical model building forced
theorists and experimentalists to address the question: Is general relativity the correct relativistic theory of gravitation? It would be difficult
to place much confidence in models for such phenomena as quasars and
pulsars if there were serious doubt about one of the basic underlying
physical theories. Thus, the growth of "relativistic astrophysics" intensified the need to strengthen the empirical evidence for or against general
relativity.
The publication of Penrose's spinor approach to general relativity
(Penrose, 1960) was one of the products of a new school of relativity
theorists that came to the fore in the late 1950s. These relativists applied
the elegant, abstract techniques of pure mathematics to physical problems
in general relativity, and demonstrated that these techniques could also
aid in the work of their more astrophysically oriented colleagues. The
2
Introduction
3
bridging of the gaps between mathematics and physics and mathematics
and astrophysics by such workers as Bondi, Dicke, Sciama, Pirani, Penrose, Sachs, Ehlers, Misner, and others changed the way that research
(and teaching) in relativity was carried out, and helped make it an active
and exciting field of physics. Yet again the question had to be addressed:
Is general relativity the correct basis for this research?
The other three events of 1959-60 contributed to the rebirth of a program to answer that question, a program of experimental gravitation that
had been semidormant for 40 years.
The Pound-Rebka (1960) experiment, besides verifying the principle of
equivalence and the gravitational red shift, demonstrated the powerful
use of quantum technology in gravitational experiments of high precision. The next two decades would see further uses of quantum technology
in such high-precision tools as atomic clocks, laser ranging, superconducting gravimeters, and gravitational-wave detectors, to name only a few.
Recording radar echos from Venus (Smith, 1963) opened up the solar
system as a laboratory for testing relativistic gravity. The rapid development during the early 1960s of the interplanetary space program made
radar ranging to both planets and artificial satellites a vital new tool for
probing relativistic gravitational effects. Coupled with the theoretical discovery in 1964 of the relativistic time-delay effect (Shapiro, 1964), it provided new and accurate tests of general relativity. For the next decade
and a half, until the summer of 1974, the solar system would be the sole
arena for high-precision tests of general relativity.
Finally, the development of the Brans-Dicke (1961) theory provided a
viable alternative to general relativity. Its very existence and agreement
with experimental results demonstrated that general relativity was not a
unique theory of gravity. Many even preferred it over general relativity on
aesthetic and" theoretical grounds. At the very least, it showed that discussions of experimental tests of relativistic gravitational effects should
be carried on using a broader theoretical framework than that provided
by general relativity alone. It also heightened the need for high-precision
experiments because it showed that the mere detection of a small general
relativistic effect was not enough. What was now required was measurements of these effects to accuracy within 10%, 1%, or fractions of a percent and better, to distinguish between competing theories of gravitation.
To appreciate more fully the regenerative effect that these events had
on gravitational theory and its experimental tests, it is useful to review
briefly the history of experimental gravitation in the 45 years following
the publication of the general theory of relativity.
Theory and Experiment in Gravitational Physics
In deriving general relativity, Einstein was not particularly motivated
by a desire to account for unexplained experimental or observational
results. Instead, he was driven by theoretical criteria of elegance and simplicity. His primary goal was to produce a gravitation theory that incorporated the principle of equivalence and special relativity in a natural
way. In the end, however, he had to confront the theory with experiment.
This confrontation was based on what came to be known as the "three
classical tests."
One of these tests was an immediate success - the ability of the theory
to account for the anomalous perihelion shift of Mercury. This had been
an unsolved problem in celestial mechanics for over half a century, since
the discovery by Leverrier in 1845 that, after the perturbing effects of the
planets on Mercury's orbit had been accounted for, and after the effect
of the precession of the equinoxes on the astronomical coordinate system
in the perihelion of Mercury. The modern value for this discrepancy is
43 arc seconds per century (Table 1.1). A number of ad hoc proposals
were made in an attempt to account for this excess, including, among
others, the existence of a new planet, Vulcan, near the Sun; a ring of
planetoids; a solar quadrupole moment; and a deviation from the inversesquare law of gravitation (for a review, see Chazy, 1928). Although these
proposals could account for the perihelion advance of Mercury, they either
involved objects that were detectable by direct optical observation, or
predicted perturbations on the other planets (for example, regressions of
nodes, changes in orbital inclinations) that were inconsistent with observations. Thus, they were doomed to failure. General relativity accounted
Table 1.1. Perihelion advance of Mercury
Rate (arc s/century)
General precession (epoch 1900)
Venus
Earth
Mars
Jupiter
Saturn
Others
5025'.'6
211".%
9070
275
15376
773
072
Sum
Discrepancy
555770
559977
4277
4
Introduction
5
for the anomalous shift in a natural way without disturbing the agreement
with other planetary observations. This result would go unchallenged
until 1967.
The next classical test, the deflection of light by the Sun, was not only
a success, it was a sensation. Shortly after the end of World War I, two
expeditions set out from England: one for Sobral, in Brazil; and one for
the island of Principe off the coast of Africa. Their goal was to measure
the deflection of light as predicted by general relativity -1.75 arc seconds
for a ray that grazes the Sun. The observations had to be made in the
path of totality of a solar eclipse, during which the Moon would block
the light from the Sun and reveal thefieldof stars behind it. Photographic
plates taken of the star field during the eclipse were compared with plates
of the same field taken when the Sun was not present, and the angular
displacement of each star was determined. The results were 1.13 + 0.07
times the Einstein prediction for the Sobral expedition, and 0.92 ±0.17
for the Principe expedition (Dyson et al., 1920). The announcement of
these results confirming the theory caught the attention of a war-weary
public and helped make Einstein a celebrity. But Einstein was so convinced of the "correctness" of the theory because of its elegance and internal consistency that he is said to have remarked that he would have felt
sorry for the Almighty if the results had disagreed with the theory (see
Bernstein, 1973). Nevertheless, the experiments were plagued by possible
systematic errors, and subsequent independent analyses of the Sobral
plates yielded values ranging from 1.0 to 1.3 times the general relativity
value. Later eclipse expeditions made very little improvement (Table 1.2).
The main sources of error in such optical deflection experiments are unknown scale changes between eclipse and comparison photographic
plates, and the precarious conditions, primarily associated with bad
weather and exotic locales, under which such expeditions are carried out.
By 1960, the best that could be said about the deflection of light was
that it was definitely more than 0'.'83, or half the Einstein value. This
was the amount predicted from a simple Newtonian argument, by Soldner
in 1801 (Lenard, 1921),1 or from an extension of the principle of equivalence, by Einstein (1911). Beyond that, "the subject [was] still a live
one" (Bertotti et al., 1962).
The third classical test was actually thefirstproposed by Einstein (1907):
the gravitational red shift of light. But by contrast with the other two
1
In 1921, the physicist Philipp Lenard, an avowed Nazi, reprinted Soldner's
paper in the Annalen der Physik in an effort to discredit Einstein's "Jewish" science
by showing the precedence of Soldner's "Aryan" work.
Theory and Experiment in Gravitational Physics
6
tests, there was no reliable confirmation of it until the 1960 Pound-Rebka
experiment. One possible test was a measurement of the red shift of spectral lines from the Sun. However, 30 years of such measurements revealed
that the observed shifts in solar spectral lines are affected strongly by
Doppler shifts due to radial mass motions in the solar photosphere. For
example, the frequency shift was observed to vary between the center of
the Sun and the limb, and to depend on the line strength. For the gravitational red shift the results were inconclusive, and it would be 1962 before
a reliable solar red-shift measurement would be made. Similarly inconclusive were attempts to measure the gravitational red shift of spectral
lines from white dwarfs, primarily from Sirius B and 40 Eridani B, both
members of binary systems. Because of uncertainties in the determination
of the masses and radii of these stars, and because of possible complications in their spectra due to scattered light from their companions, reliable, precise measurements were not possible [see Bertotti et al. (1962)
for a review].
Furthermore, by the late 1950s, it was being suggested that the gravitational red shift was not a true test of general relativity after all. According
to Leonard I. Schiff and Robert H. Dicke, the gravitational red shift was
a consequence purely of the principle of equivalence, and did not test the
field equations of gravitational theory. Schiff took the argument one step
Table 1.2. Optical measurements of light deflection by the Suri*
Eclipse
Approximate
number
of stars
Minimum distance from
center of Sun
Result in
units of
Einstein
prediction
1919
1919
1922
1922
1922
1922
1929
1936
1936
1947
1952
1973"
7
5
92
145
14
18
17
25
8
51
10
39
2
2
2.1
2.1
2
2
1.5
2
4
3.3
2.1
2
1.13 + 0.07
0.92 + 0.17
0.98 ± 0.06
1.04 + 0.09
0.7 to 1.3
0.8 to 1.2
1.28 ±0.06
1.55 + 0.15
0.7 to 1.2
1.15 ±0.15
0.97 + 0.06
0.95 + 0.11
a
b
See Bertotti et al. (1962) for details.
Texas Mauritanian Eclipse Team (1976), Jones (1976).
Results from
different
analyses
1.0 to 1.3
1.3 to 0.9
1.2
0.9 to 1.2
1.55 ± 0.2
1.0 to 1.4
0.82 ± 0.09
Introduction
7
further and suggested that the gravitational red-shift experiment was
superseded in importance by the more accurate Eotvos experiment, which
verified that bodies of different composition fall with the same acceleration (Schiff, 1960a; Dicke, 1960).
Other potential tests of general relativity were proposed, such as the
Lense-Thirring effect, an orbital perturbation due to the rotation of a
body, and the de Sitter effect, a secular motion of the perigee and node
of the lunar orbit (Lense and Thirring, 1918; de Sitter, 1916), but the
prospects for ever detecting them were dim.
Cosmology was the other area where general relativity could be confronted with observation. Initially the theory met with success in its
ability to account for the observed expansion of the universe, yet by the
1940s there was considerable doubt about its applicability. According to
pure general relativity, the expansion of the universe originated in a dense
primordial explosion called the "big bang." The age of the universe since
the big bang could be determined by extrapolating the expansion of the
universe backward in time using the field equations of general relativity.
However, the observed values of the present expansion rate were so high
that the inferred age of the universe was shorter than that of the Earth.
One result of this doubt was the rise in popularity during the 1950s of the
steady-state cosmology of Herman Bondi, Thomas Gold, and Fred Hoyle.
This model avoided the big bang altogether, and allowed for the expansion of the universe by the continuous creation of matter. By this
means, the universe would present the same appearance to all observers
for all time.
But by the late 1950s, revisions in the cosmic distance scale had reduced
the expansion rate by a factor of five, and had thereby increased the age
of the universe in the big bang model to a more acceptable level. Nevertheless, cosmological observations were still in no position to distinguish
among different theories of gravitation or of cosmology [for a detailed
technical and historical review, see Weinberg (1972), Chapter 14].
Meanwhile, a small "cottage industry" had sprung up, devoted to the
construction of alternative theories of gravitation. Some of these theories
were produced by such luminaries as Poincare, Whitehead, Milne, Birkhoff, and Belinfante. Many of these authors expressed an uneasiness with
the notions of general covariance and curved spacetime, which were built
into general relativity, and responded by producing "special relativistic"
theories of gravitation. These theories considered spacetime to be "special
relativistic" at least at a background level, and treated gravitation as a
Lorentz-invariant field on that background. As of 1960, it was possible
Theory and Experiment in Gravitational Physics
8
to enumerate at least 25 such alternative theories, as found in the primary
research literature between 1905 and 1960 [for a partial list, see Whitrow
and Morduch (1965)].
Thus, by 1960, it could be argued that the validity of general relativity
rested on the following empirical foundation: one test of moderate precision (the perihelion shift, approximately 1%), one test of low precision
(the deflection of light, approximately 50%), one inconclusive test that
was not a real test anyway (the gravitational red shift), and cosmological
observations that could not distinguish between general relativity and
the steady-state theory. Furthermore, a variety of alternative theories
laid claim to viability.
In addition, the attitude toward the theory seemed to be that, whereas
it was undoubtedly of importance as a fundamental theory of nature, its
observational contacts were limited to the classical tests and cosmology.
This view was present for example in the standard textbooks on general
relativity of this period, such as those by Mcller (1952), Synge (1960), and
Landau and Lifshitz (1962). As a consequence, general relativity was cut
off from the mainstream of physics. It was during this period that one
because general relativity "had so little connection with the rest of physics
and astronomy" (his name: Kip S. Thorne).
However, the events of 1959-60 changed all that. The pace of research
in general relativity and relativistic astrophysics began to quicken and,
associated with this renewed effort, the systematic high-precision testing
of gravitational theory became an active and challengingfield,with many
new experimental and theoretical possibilities. These included new versions of old tests, such as the gravitational red shift and deflection of light,
with accuracies that were unthinkable before 1960. They also included
brand new tests of gravitational theory, such as the gyroscope precession,
the time delay of light, and the "Nordtvedt effect" in lunar motion, that
were discovered theoretically after 1959. Table 1.3 presents a chronology
of some of the significant theoretical and experimental events that occurred in the two decades following 1959. In many ways, the years 19601980 were the decades for testing relativity.
Because many of the experiments involved the resources of programs
for interplanetary space exploration and observational astronomy, their
cost in terms of money and manpower was high and their dependence
upon increasingly constrained government funding agencies was strong.
Thus, it became crucial to have as good a theoretical framework as possible
for comparing the relative merits of various experiments, and for pro-
Introduction
Table 1.3. A chronology: 1960-80
Time
Experimental or observational events
Theoretical events
1960
Hughes-Drever mass-anisotropy
experiments
Pound-Rebka gravitational red-shift
experiment
Discovery of nonsolar x-ray sources
Discovery of quasar red shifts
Princeton Eotvos experiment
Penrose paper on spinors
Gyroscope precession (Schiff)
1962
1964
Brans-Dicke theory
Bondi mass-loss formula
Kerr metric discovery
Time-delay of light (Shapiro)
Pound-Snider red-shift experiment
Discovery of 3K microwave
background
Singularity theorems in
general relativity
1966
1968
Reported detection of solar
oblateness
Discovery of pulsars
delay
Launch of Mariners 6 and 7
Acquisition of lunar laser echo
Element production in the
big bang
Nordtvedt effect and early
PPN framework
1970
CygXl: a black hole candidate
Mariners 6 and 7 time-delay
measurements
1972
1974
Moscow Eotvos experiment
Discovery of binary pulsar
1976
1978
1980
Rocket gravitational red-shift
experiment
Lunar test of Nordtvedt effect
Time delay results from Mariner 9
and Viking
Measurement of orbit period
decrease in binary pulsar
SS433
Discovery of gravitational lens
Preferred-frame effects
Refined PPN framework
Area increase of black holes in
general relativity
Quantum evaporation of
black holes
in alternative theories
Theory and Experiment in Gravitational Physics
10
posing new ones that might have been overlooked. Another reason that
such a theoretical framework was necessary was to make some sense of
the large (and still growing) number of alternative theories of gravitation.
Such a framework could be used to classify theories, elucidate their similarities and differences, and compare their predictions with the results of
experiments in a systematic way. It would have to be powerful enough to
be used to design and assess experimental tests in detail, yet general
enough not to be biased in favor of general relativity.
A leading exponent of this viewpoint was Robert Dicke (1964a). It led
him and others to perform several high-precision null experiments which
greatly strengthened our faith in the foundations of gravitation theory.
gravity and devises experiments to test them. The most important dividend of the Dicke framework is the understanding that gravitational
experiments can be divided into two classes. The first consists of experiments that test the foundations of gravitation theory, one of these foundations being the principle of equivalence. These experiments (Eotvos
experiment, Hughes-Drever experiment, gravitational red-shift experiment, and others, many performed by Dicke and his students) accurately
verify that gravitation is a phenomenon of curved spacetime, that is, it
must be described by a "metric theory" of gravity. General relativity and
Brans-Dicke theory are examples of metric theories of gravity.
The second class of experiments consists of those that test metric theories of gravity. Here another theoretical framework was developed that
takes up where the Dicke framework leaves off. Known as the "Parametrized Post-Newtonian" or PPN formalism, it was pioneered by Kenneth Nordtvedt, Jr. (1968b), and later extended and improved by Will
(1971a), Will and Nordtvedt (1972), and Will (1973). The PPN framework
takes the slow motion, weak field, or post-Newtonian limit of metric
theories of gravity, and characterizes that limit by a set of 10 real-valued
parameters. Each metric theory of gravity has particular values for the
PPN parameters. The PPN framework was ideally suited to the analysis
of solar system gravitational experiments, whose task then became one
of measuring the values of the PPN parameters and thereby delineating
which theory of gravity is correct. A second powerful use of the PPN
framework was in the discovery and analysis of new tests of gravitation
theory, examples being the Nordtvedt effect (Nordtvedt 1968a), preferredframe effects (Will, 1971b) and preferred-location effects (Will, 1971b,
1973). The Nordtvedt effect, for instance, is a violation of the equality
of acceleration of massive bodies, such as the Earth and Moon, in an
Introduction
11
external field; the effect is absent in general relativity but present in many
alternative theories, including the Brans-Dicke theory. The third use of
the PPN formalism was in the analysis and classification of alternative
metric theories of gravitation. After 1960, the invention of alternative
gravitation theories did not abate, but changed character. The crude attempts to derive Lorentz-invariant field theories described previously
were mostly abandoned in favor of metric theories of gravity, whose
development and motivation were often patterned after that of the BransDicke theory. A "theory of gravitation theories" was developed around
the PPN formalism to aid in their systematic study.
The PPN formalism thus became the standard theoretical tool for
analyzing solar system experiments, looking for new tests, and studying
alternative metric theories of gravity. One of the central conclusions of
the two decades of testing relativistic gravity in the solar system is that
general relativity passes every experimental test with flying colors.
But by the middle 1970s it became apparent that the solar system
could no longer be the sole testing ground for gravitation theories. One
reason was that many alternative theories of gravity agreed with general
relativity in their post-Newtonian limits, and thereby also agreed with all
solar system experiments. But they did not necessarily agree in other predictions, such as cosmology, gravitational radiation, neutron stars, or
black holes. The second reason was the possibility that experimental tools,
such as gravitational radiation detectors, would ultimately be available
to perform such extra-solar system tests.
This suspicion was confirmed in the summer of 1974 with the discovery
by Joseph Taylor and Russell Hulse of the binary pulsar (Hulse and Taylor, 1975). Here was a system that combined large post-Newtonian gravitational effects, highly relativistic gravitational fields associated with the
pulsar, and the possibility of the emission of gravitational radiation by
the binary system, with ultrahigh precision data obtained by radiotelescope monitoring of the extremely stable pulsar clock. It was also a
system where relativistic gravity and astrophysics became even more intertwined than in the case, say, of quasars. In the binary pulsar, relativistic
gravitational effects provided a means for accurate measurement of astrophysical parameters, such as the mass of a neutron star. The role of the
binary pulsar as a new arena for testing relativistic gravity was cemented
in the winter of 1978 with the announcement (Taylor et al., 1979) that
the rate of change of the orbital period of the system had been measured.
The result agreed with the prediction of general relativity for the rate of
orbital energy loss due to the emission of gravitational radiation. But it
Theory and Experiment in Gravitational Physics
12
disagreed violently with the predictions of most alternative theories, even
those with post-Newtonian limits identical to general relativity.
As a young student of 17 at the Poly technical Institute of Zurich,
Einstein studied closely the work of Helmholtz, Maxwell, and Hertz, and
ultimately used his deep understanding of electromagnetic theory as a
foundation for special and general relativity. He appears to have been
especially impressed by Hertz's confirmation that light and electromagnetic waves are one and the same (Schilpp, 1949). The electromagnetic
waves that Hertz studied were in the radio part of the spectrum, at 30
MHz. It is amusing to note that, 60 years later, the decades for testing
relativistic gravity began with radio waves, the 440 MHz waves reflected
from Venus, and ended with radio waves, the signals from the binary
pulsar, observed at 430 MHz.
During these two decades, that closed on the centenary of Einstein's
birth, the empirical foundations of general relativity were strengthened as
never before. But this does not end the story. The confrontation between
general relativity and experiment will proceed, using new tools, in new
arenas. Whether or not general relativity will continue to survive is a
matter of speculation for some, pious hope for another group, and supreme
confidence for others. Regardless of one's theoretical prejudices, it can
certainly be agreed that gravitation, the oldest known, and in many ways
most fundamental interaction, deserves an empirical foundation second
to none.
Throughout this book, we shall adopt the units and conventions of
Misner, Thorne, and Wheeler, 1973 (hereafter referred to as MTW).
Although we have attempted to produce a reasonably self-contained account of gravitation theory and gravitational experiments, the reader's
path will be greatly smoothed by a familiarity with at least the equivalent
of "track 1" of MTW. A portion of the present book (Chapters 4-9) is
patterned after the author's 1972 Varenna lectures "The Theoretical Tools
of Experimental Gravitation" (Will, 1974a, hereafter referred to as TTEG),
with suitable modification and updating. An overview of this book without
the mathematical details is provided by the author's "The Confrontation
between Gravitation Theory and Experiment" (Will, 1979). Other useful
reviews of this subject are of three types: (i) semipopular: Nordtvedt (1972),
Will (1972, 1974b); (ii) technical: Richard (1975), Brill (1973), Rudenko
(1978); (iii) "early": Dicke (1964a,b), Bertotti et al. (1962). The reader is
referred to these works for background or for different points of view.
The Einstein Equivalence Principle and the
Foundations of Gravitation Theory
The Principle of Equivalence has played an important role in the development of gravitation theory. Newton regarded this principle as such a
cornerstone of mechanics that he devoted the opening paragraphs of the
Principia to a detailed discussion of it (Figure 2.1). He also reported there
the results of pendulum experiments he performed to verify the principle.
To Newton, the Principle of Equivalence demanded that the "mass" of
any body, namely that property of a body (inertia) that regulates its
response to an applied force, be equal to its "weight," that property that
regulates its response to gravitation. Bondi (1957) coined the terms "inertial mass" mb and "passive gravitational mass" mP, to refer to these quantities, so that Newton's second law and the law of gravitation take the forms
F = m,a,
F = mPg
where g is the gravitational field. The Principle of Equivalence can then
be stated succinctly: for any body
mP = m1
An alternative statement of this principle is that all bodies fall in a gravitational field with the same acceleration regardless of their mass or internal structure. Newton's equivalence principle is now generally referred
to as the "Weak Equivalence Principle" (WEP).
It was Einstein who added the key element to WEP that revealed the
path to general relativity. If all bodies fall with the same acceleration
in an external gravitational field, then to an observer in a freely falling
elevator in the same gravitational field, the bodies should be unaccelerated
(except for possible tidal effects due to inhomogeneities in the gravitational field, which can be made as small as one pleases by working in a
sufficiently small elevator). Thus insofar as their mechanical motions are
Figure 2.1. Title page and first page of Newton's Principia.
PHILOSOPHISE
NATURALIS
PRINCIPIA
MATHEMATICA
Autore JS. UEfFTON, Trin. CM. Cantab. Soc. Mathefeos
Profeflbre Lucafuoto, & Sodetatis Regalis Sodali.
IMPRIMATUR
S. P E P Y S, Reg. Soc. P R R S E S.
Jutii 5. 1686.
L 0 N D I N /,
Juflii Societatis Regia ac Typis Jofepbi Streater. Proftat apud
plures Bibliopolas. Anno MDCLXXXVIl.
14
Figure 2.1 (continued)
MATHEMATICAL PRINCIPLES
OF
NATURAL PHILOSOPHY1
D eft
nitions
DEFINITION I
The quantity of matter is the measure of the same, arising from its density
and bulk, conjointly.2
T
HUS AIR of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be
understood of snow, and fine dust or powders, that are condensed
by compression or liquefaction, and of all bodies that are by any causes
whatever differently condensed. I have no regard in this place to a medium,
if any such there is, that freely pervades the interstices between the parts
of bodies. It is this quantity that I mean hereafter everywhere under the
name of body or mass. And the same is known by the weight of each body,
for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shown hereafter.
DEFINITION
IIs
The quantity of motion is the measure of the same, arising from the
velocity and quantity of matter conjointly.
The motion of the whole is the sum of the motions of all the parts; and
therefore in a body double in quantity, with equal velocity, the motion is
double; with twice the velocity, it is quadruple.
t l Appendix, Note 10.] [ 2 Appendix, Note 11.]
[ 3 Appendix, Note 12.]
CO
15
Theory and Experiment in Gravitational Physics
16
concerned, the bodies will behave as if gravity were absent. Einstein went
one step further. He proposed that not only should mechanical laws
behave in such an elevator as if gravity were absent but so should all the
laws of physics, including, for example, the laws of electrodynamics. This
new principle led Einstein to general relativity. It is now called the
"Einstein Equivalence Principle" (EEP).
Yet, it is only relatively recently that we have gained a deeper understanding of the significance of these principles of equivalence for gravitation and experiment. Largely through the work of Robert H. Dicke,
we have come to view principles of equivalence, along with experiments
such as the Eotvos experiment, the gravitational red-shift experiment, and
so on, as probes more of the foundations of gravitation theory, than of
general relativity itself. This viewpoint is part of what has come to be
known as the Dicke Framework described in Section 2.1, allowing one to
discuss at a very fundamental level the nature of space-time and gravity.
Within it one asks questions such as: Do all bodies respond to gravity
with the same acceleration? Does energy conservation imply anything
about gravitational effects? What types of fields, if any, are associated
with gravitation-scalar fields, vector fields, tensor fields... ? As one
product of this viewpoint, we present in Section 2.2 a set of fundamental
criteria that any potentially viable theory should satisfy, and as another,
we show in Section 2.3 that the Einstein Equivalence Principle is the
foundation for all gravitation theories that describe gravity as a manifestation of curved spacetime, the so-called metric theories of gravity. In
Section 2.4 we describe the empirical support for EEP from a variety
of experiments.
Einstein's generalization of the Weak Equivalence Principle may not
have been a generalization at all, according to a conjecture based on the
work of Leonard Schiff. In Section 2.5, we discuss Schiif 's conjecture,
which states that any complete and self-consistent theory of gravity that
satisfies WEP necessarily satisfies EEP. Schiff's conjecture and the Dicke
Framework have spawned a number of concrete theoretical formalisms,
one of which is known as the THsu formalism, presented in Section 2.6,
for comparing and contrasting metric theories of gravity with nonmetric
theories, analyzing experiments that test EEP and WEP, and proving
Schiff's conjecture.
2.1
The Dicke Framework
The Dicke Framework for analyzing experimental tests of gravitation was spelled out in Appendix 4 of Dicke's Les Houches lectures
Einstein Equivalence Principle and Gravitation Theory
(1964a). It makes two main assumptions about the type of mathematical
formalism to be used in discussing gravity:
(i) Spacetime is a four-dimensional differentiable manifold, with each
point in the manifold corresponding to a physical event. The manifold
need not a priori have either a metric or an affine connection. The hope
is that experiment will force us to conclude that it has both.
(ii) The equations of gravity and the mathematical entities in them are
to be expressed in a form that is independent of the particular coordinates
used, i.e., in covariant form.
Notice that even if there is some physically preferred coordinate system
in spacetime, the theory can still be put into covariant form. For example,
if a theory has a preferred cosmic time coordinate, one can introduce
a scalar field T{0>) whose numerical values are equal to the values of the
preferred time t:
T(0>) = t{0>),
0> a point in spacetime
If spacetime is endowed with a metric, one might also demand that VT
be a timelike vector field and be consistently oriented toward the future
(or the past) throughout spacetime by imposing the covariant constraints
VT-VT<0,
V<g>VT = 0
where V is a covariant derivative with respect to the metric. Other types
of theories have "flat background metrics" IJ; these can also be written
covariantly by defining i; to be a second-rank tensor field whose Riemann
tensor vanishes everywhere, i.e.,
Riem(>r) = 0
and by defining covariant derivatives and contractions with respect to i\.
In most cases, this covariance is achieved at the price of the introduction
into the theory of "absolute" or "prior geometric" elements (T, i/), that
are not determined by the dynamical equations of the theory. Some
authors regard the introduction of absolute elements as a failure of general
covariance (Einstein would be one example), however we shall adopt the
weaker assumption of coordinate invariance alone. (For further discussion
of prior geometry, see Section 3.3.)
Having laid down this mathematical viewpoint [statements (i) and (ii)
above] Dicke then imposes two constraints on all acceptable theories of
gravity. They are:
(1) Gravity must be associated with one or more fields of tensorial
character (scalars, vectors, and tensors of various ranks).
17
Theory and Experiment in Gravitational Physics
18
(2) The dynamical equations that govern gravity must be derivable
from an invariant action principle.
These constraints strongly confine acceptable theories. For this reason
we should accept them only if they are fundamental to our subsequent
arguments. For most applications of the Dicke Framework only the first
constraint is often needed. It is a fact, however, that the most successful
gravitation theories are those that satisfy both constraints.
The Dicke Framework is particularly useful for designing and interpreting experiments that ask what types of fields are associated with
gravity. For example, there is strong evidence from elementary particle
physics for at least one symmetric second-rank tensorfieldthat is approximated by the Minkowski metric i\ when gravitational effects can be
ignored. The Hughes-Drever experiment rules out the existence of more
than one second-rank tensor field, each coupling directly to matter, and
various ether-drift experiments rule out a long-range vectorfieldcoupling
directly to matter. No experiment has been able to rule out or reveal the
existence of a scalar field, although several experiments have placed
limits on specific scalar-tensor theories (Chapters 7 and 8). However, this
is not the only powerful use of the Dicke Framework.
2.2
Basic Criteria for the
Viability of a Gravitation Theory
The general unbiased viewpoint embodied in the Dicke Framework has allowed theorists to formulate a set of fundamental criteria that
any gravitation theory should satisfy if it is to be viable [we do not impose
constraints (1) and (2) above]. Two of these criteria are purely theoretical,
whereas two are based on experimental evidence.
(i) It must be complete, i.e., it must be capable of analyzing from
"first principles" the outcome of any experiment of interest. It is not
enough for the theory to postulate that bodies made of different material
fall with the same acceleration. The theory must incorporate a complete
set of electrodynamic and quantum mechanical laws, which can be used
to calculate the detailed behavior of bodies in gravitational fields. This
demand should not be extended too far, however. In areas such as weak
and strong interaction theory, quantum gravity, unified field theories,
spacetime singularities, and cosmic initial conditions, even special and
general relativity are not regarded as being complete or fully developed.
We also do not regard the presence of "absolute elements" and arbitrary
parameters in gravitational theories as a sign of incompleteness, even
though they are generally not derivable from "first principles," rather we
Einstein Equivalence Principle and Gravitation Theory
19
view them as part of the class of cosmic boundary conditions. Fortunately,
so simple a demand as one that the theory contain a set of gravitationally
modified Maxwell equations is sufficiently telling that many theories fail
this test. Examples are given in Table 2.1.
(ii) It must be self-consistent, i.e., its prediction for the outcome of
every experiment must be unique, i.e., when one calculates the predictions
by two different, though equivalent methods, one always gets the same
Table 2.1. Basically nonviable theories of gravitation - a partial list
Theory and references
Newtonian gravitation theory
Milne's kinematical relativity
(Milne, 1948)
Is not relativistic
Was devised originally to handle certain
cosmological problems. Is incomplete: makes
no gravitational red-shift prediction
Contain a vector gravitational field in flat
spacetime. Are incomplete: do not mesh with
the other nongravitational laws of physics
(viz. Maxwell's equations) except by imposing
them on the flat background spacetime. Are
then inconsistent: give different results for light
propagation for light viewed as particles and
light viewed as waves.
Action-at-a-distance theory in flat spacetime.
Is incomplete or inconsistent in the same
manner as Kustaanheimo's theories
Contains a vector gravitational field in flat
spacetime. Is incomplete or inconsistent in the
same manner as Kustaanheimo's theories.
Contains a tensor gravitational field used to
construct a metric. Violates the Newtonian
limit by demanding that p = pc2, i.e.
Kustaanheimo's various vector
theories (Kustaanheimo and
Nuotio, 1967; Whitrow and
Morduch, 1965)
Poincare's theory (as
generalized by Whitrow and
Morduch, 1965)
Whitrow-Morduch (1965)
vector theory
Birkhoff's (1943) theory
''sound
Yilmaz's (1971,1973) theory
=
"light-
Contains a tensor gravitational field used to
construct a metric. Is mathematically
inconsistent: functional dependence of metric
on tensor field is not well defined.
° These theories are nonviable in their present form. Future modifications or specializations might make some of them viable. If I have misinterpreted any theory here
I apologize to its proponents, and urge them to demonstrate explicitly its completeness, self-consistency, and compatibility with special relativity and Newtonian
gravitation theory.
Theory and Experiment in Gravitational Physics
20
results. An example is the bending of light computed either in the geometrical optics limit of Maxwell's equations or in the zero-rest-mass limit
of the motion of test particles. Furthermore, the system of mathematical
equations it proposes should be well posed and self-consistent. Table 2.1
shows some theories that fail this criterion.
(iii) It must be relativistic, i.e., in the limit as gravity is "turned off"
compared to other physical interactions, the nongravitational laws of
physics must reduce to the laws of special relativity. The evidence for this
comes largely from high-energy physics and from a variety of optical
ether-drift experiments. Since these experiments are performed at high
energies and velocities and over very small regions of space and time, the
effects of gravity on their outcome are negligible. Thus we may treat such
experiments as if they were being performed far from all gravitating
matter. The evidence provided by these experiments is of two types. First
are experiments that measure space and time intervals directly, e.g.,
measurements of the time dilation of systems ranging from atomic clocks
to unstable elementary particles, experiments that verify the velocity of
light is independent of the velocity of the source for sources ranging from
pions at 99.98% of the speed of light to pulsating binary x-ray sources at
10" 3 of the speed of light [for a thorough review and reference list, see
Newman et al. (1978)] and Michelson-Morley-type experiments [for recent high-precision results, see Trimmer et al. (1973) and Brillet and Hall
Second are experiments which reveal the fundamental role played by
the Lorentz group in particle physics, including verifications of fourmomentum conservation and of the relativistic laws of kinematics, electron and muon "g-2" experiments, and tests of esoteric predictions of
Lorentz-in variant quantumfieldtheories [Lichtenberg (1965), Blokhintsev
(1966), Newman et al. (1978), Combley et al. (1979), and Cooper et al.
(1979)].
The fundamental theoretical object that enters these laws is the Minkowski metric i\, with a signature of + 2, which has orthonormal tetrads
related by Lorentz transformations, and which determines the ticking
rates of atomic clocks and the lengths of laboratory rods. If we view q as
a field [Dicke statement (ii)], then we conclude that there must exist at
least one second-rank tensor field in the Universe, a symmetric tensor ^,
which reduces to r\ when gravitational effects can be ignored.
Let us examine what particle physics experiments do and do not tell
us about the tensor field V- First, they do not guarantee the existence of
global Lorentz frames, i.e., coordinate systems extending throughout
Einstein Equivalence Principle and Gravitation Theory
21
spacetime in which
(-1,1,1,1)
Nor do they demand that at each event 2P, there exist local frames related
by Lorentz transformations, in which the laws of elementary-particle
physics take on their special form. They only demand that, in the limit as
gravity is "turned off," the nongravitational laws of physics reduce to
the laws of special relativity.
Second, elementary-particle experiments do tell us that the times measured by atomic clocks in the limit as gravity is turned off depend
only on velocity, not upon acceleration. The measured squared interval,
ds2 = i^dx^dx", is independent of acceleration. Equivalently, but more
physically, the time interval measured by a clock moving with velocity vJ
relative to a coordinate system in the absence of gravity is
ds = (-q^tordx*)112
= dt(l - |v|2)1/2
independent of the clock's acceleration d2xi/dt2. (For a review of experimental tests, see Newman et al., 1978.)
We shall henceforth assume the existence of the tensor field $. (iv) It must have the correct Newtonian limit, i.e., in the limit of weak gravitational fields and slow motions, it must reproduce Newton's laws. Massive amounts of empirical data support the validity of Newtonian gravitation theory (NGT), at least as an approximation to the "true" relativistic theory of gravity. Observations of the motions of planets and spacecraft agree with NGT down to the level (parts in 108) at which post-Newtonian effects can be observed. Observations of planetary, solar, and stellar structure support NGT as applied to bulk matter. Laboratory Cavendish experiments provide support for NGT for small separations between gravitating bodies. One feature of NGT that has recently come under experimental scrutiny is the inverse-square force law. Despite one claim to the contrary (Long, 1976), there seems to be no hard evidence for a deviation from this law (other than those produced by post-Newtonian effects) over distances ranging from a few centimeters to several astronomical units (see Mikkelson and Newman, 1977; Spero et al., 1979; Paik, 1979; Yu et al., 1979; Panov and Frontov, 1979; and, Hirakawa et al., 1980). Thus, to at least be viable, a gravitation theory must be complete, self-consistent, relativistic, and compatible with NGT. Table 2.1 shows examples of theories that violate one or more of these criteria. Theory and Experiment in Gravitational Physics 2.3 22 The Einstein Equivalence Principle The Einstein Equivalence Principle is the foundation of all curved spacetime or "metric" theories of gravity, including general relativity. It is a powerful tool for dividing gravitational theories into two distinct classes: metric theories, those that embody EEP, and nonmetric theories, those that do not embody EEP. For this reason, we shall discuss it in some detail and devote the next section (Section 2.4) to the supporting experimental evidence. We begin by stating the Weak Equivalence Principle in more precise terms than those used before. WEP states that if an uncharged test body is placed at an initial event in spacetime and given an initial velocity there, then its subsequent trajectory will be independent of its internal structure and composition. By "uncharged test body" we mean an electrically neutral body that has negligible self-gravitational energy (as estimated using Newtonian theory) and that is small enough in size so that its coupling to inhomogeneities in external fields can be ignored. In the same spirit, it is also useful to define "local nongravitational test experiment" to be any experiment: (i) performed in a freely falling laboratory that is shielded and is sufficiently small that inhomogeneities in the external fields can be ignored throughout its volume, and (ii) in which self-gravitational effects are negligible. For example, a measurement of the fine structure constant is a local nongravitational test experiment; a Cavendish experiment is not. The Einstein Equivalence Principle then states: (i) WEP is valid, (ii) the outcome of any local nongravitational test experiment is independent of the velocity of the (freely falling) apparatus, and (iii) the outcome of any local nongravitational test experiment is independent of where and when in the universe it is performed. This principle is at the heart of gravitation theory, for it is possible to argue convincingly that if EEP is valid, then gravitation must be a curvedspacetime phenomenon, i.e., must satisfy the postulates of Metric Theories of Gravity. These postulates state: (i) spacetime is endowed with a metric g, (ii) the world lines of test bodies are geodesies of that metric, and (iii) in local freely falling frames, called local Lorentz frames, the nongravitational laws of physics are those of special relativity. General relativity, BransDicke theory, and the Rosen bimetric theory are metric theories of gravity (Chapter 5); the Belinfante-Swihart theory (Section 2.6) is not. The argument proceeds as follows. The validity of WEP endows spacetime with a family of preferred trajectories, the world lines of freely falling test bodies. In a local frame that follows one of these trajectories, Einstein Equivalence Principle and Gravitation Theory 23 test bodies have unaccelerated motions. Furthermore, the results of local nongravitational test experiments are independent of the velocity of the frame. In two such frames located at the same event, 9, in spacetime but moving relative to each other, all the nongravitational laws of physics must make the same predictions for identical experiments, that is, they must be Lorentz invariant. We call this aspect of EEP Local Lorentz Invariance (LLI). Therefore, there must exist in the universe one or more second-rank tensor fields i/t(1), ij/(2\ . . . , that reduce in a local freely falling frame to fields that are proportional to the Minkowski metric, (j)(1\^)tl, 0 (2) (^)«J,..., where 4>(A\0>) are scalar fields that can vary from event to event. Different members of this set of fields may couple to different nongravitationalfields,such as bosonfields,fermionfields,electromagnetic fields, etc. However, the results of local nongravitational test experiments must also be independent of the spacetime location of the frame. We call this Local Position Invariance (LPI). There are then two possibilities, (i) The local versions of ijf{A) must have constant coefficients, that is, the scalarfields4>(A\^) must be constants. It is therefore possible by a simple universal rescaling of coordinates and coupling constants (such as the unit of electric charge) to set each scalar field equal to unity in every local frame, (ii) The scalarfields<f>iA)(<P) must be constant multiples of a single scalar field${&), i.e., 4>{A\0>) = cA4>(0>). If this is
true, then physically measurable quantities, being dimensionless ratios,
will be location independent (essentially, the scalar field will cancel out).
One example is a measurement of the fine structure constant; another
is a measurement of the length of a rigid rod in centimeters, since such a
measurement is a ratio between the length of the rod and that of a standard
rod whose length is defined to be one centimeter. Thus, a combination
of a rescaling of coupling constants to set the cA's equal to unity (redefinition of units), together with a "conformal" transformation to a new
field ij/ = cj>~ V. guarantees that the local version of if/ will be ij.
In either case, we conclude that there exist fields that reduce to r\ in
every local freely falling frame. Elementary differential geometry then
shows that thesefieldsare one and the same: a unique, symmetric secondrank tensor field that we now denote g. This g has the property that it
possesses a family of preferred worldlines called geodesies, and that at
each event $* there exist local frames, called local Lorentz frames, that follow these geodesies, in which <W^) = 1** + 0(Y |X« - x\0>)\\ dgjdx* = 0, at 0> Theory and Experiment in Gravitational Physics 24 However, geodesies are straight lines in local Lorentz frames, as are the trajectories of test bodies in local freely falling frames, hence the test bodies move on geodesies of g and the Local Lorentz frames coincide with the freely falling frames. We shall discuss the implications of the postulates of metric theories of gravity in more detail in Chapter 3. Because EEP is so crucial to this conclusion about the nature of gravity, we turn now to the supporting experimental evidence. 2.4 Experimental Tests of the Einstein Equivalence Principle (a) Tests of the Weak Equivalence Principle A direct test of WEP is the Eotvos experiment, the comparison of the acceleration from rest of two laboratory-sized bodies of different composition in an external gravitational field. If WEP were invalid, then the accelerations of different bodies would differ. The simplest way to quantify such possible violations of WEP in a form suitable for comparison with experiment is to suppose that for a body of inertial mass m,, the passive mass mP is no longer equal to mv Now the inertial mass of a typical laboratory body is made up of several types of mass energy: rest energy, electromagnetic energy, weak-interaction energy, and so on. If one of these forms of energy contributes to mP differently than it does to m,, a violation of WEP would result. One could then write p = m, + I r]AEA/c2 (2.1) A where EA is the internal energy of the body generated by interaction A, and nA is a dimensionless parameter that measures the strength of the violation of WEP induced by that interaction, and c is the speed of light.1 For two bodies, the acceleration is then given by ^ ( + S r,AEA/m2Ag (2.2) where we have dropped the subscript I on mj and m2. 1 Throughout this chapter we shall avoid units in which c = 1. The reason for this is that if EEP is not valid then the speed of light may depend on the nature of the devices used to measure it. Thus, to be precise we should denote c as the speed of light as measured by some standard experiment. Once we accept the validity of EEP in Chapter 3 and beyond, then c has the same value in every local Lorentz frame, independently of the method used to measure it, and thus can be set equal to unity by appropriate choice of units. Einstein Equivalence Principle and Gravitation Theory 25 A measurement or limit on the relative difference in acceleration then yields a quantity called the "Eotvos ratio" given by K + a\ t \mC2 m2c2j v ' Thus, experimental limits on r\ place limits on the WEP-violation parameters rjA. Many high-precision Eotvos-type experiments have been performed, from the pendulum experiments of Newton, Bessel, and Potter to the classic torsion-balance measurements of Eotvos, Dicke, and Braginsky and their collaborators. The latter experiments can be described heuristically. Two objects of different composition are connected by a rod of length r, and suspended in a horizontal orientation by afinewire ("torsion balance"). If the gravitational acceleration of the bodies differs, there will be a torque N induced on the suspension wire, given by N = tjr(g x ew) er where g is the gravitational acceleration, and ew and er are unit vectors along the wire and rod, respectively (see Figure 2.2). If the entire apparatus is rotated about a direction <o with angular velocity |co|, the torque will be modulated with period 2JT/CO. In the experiments of Baron Roland von Eotvos, g was the acceleration of the Earth (note g and ew were not quite parallel because of the centripetal acceleration on the apparatus due to the Earth's rotation), and the apparatus was rotated about the direction of the wire. In the Princeton (Roll, et al., 1964) and Moscow (Braginski and Panov, 1972) experiments, g was that of the Sun, and the rotation of the Earth provided the modulation of N at a period of 24 hr. The modulated torque was determined either by measuring the torsional motion of the rod (Moscow) or by measuring the force required to counteract the torque and keep the rod in place (Princeton). The resulting upper limits on measurable torques |N| yielded limits on r\ given by \\ x 10" J 1 [Princeton] II x 10" 12 [Moscow] ( ' where the limits are \a formal standard deviations. For further discussion of the experiments, see Dicke (1964a) and Braginsky (1974). The primary sources of error in these experiments are seismic noise and coupling of the torsion balance to gradients in the external gravitational field (produced, for example, by the experimenters). Attempts to improve these results have centered on different forms of suspension of the masses, Theory and Experiment in Gravitational Physics 26 Figure 2.2. Schematic arrangement of a torsion-balance Eotvos experiment; g is the external gravitational acceleration, and to is the angular velocity vector about which the apparatus is rotated. The unit vectors e w and er are parallel to the wire and rod, respectively. In the Eotvos experiments, g was the acceleration toward the Earth, and to was parallel to e w ; in the Princeton and Moscow experiments g was that of the Sun, and co was parallel to the Earth's rotation axis. \\W\\\\\ including magnetic levitation (Worden and Everitt, 1974), flotation on liquids (Keiser and Faller, 1979), and free fall in orbit. Experiments to test WEP for individual atoms and elementary particles have been inconclusive or inaccurate, with the exception of neutrons (Fairbank et al, 1974 and Koester, 1976). Table 2.2 discusses various experiments and quotes the limits they set on Y) for different pairs of materials. Future improved tests of WEP must reduce noise due to thermal, seismic, and gravity-gradient effects, and may have to be performed in space using cryogenic techniques. Anticipated limits on r\ in such experiments range between 10""15 and 10" 18 (Worden, 1978). To determine the limits placed on individual parameters r\K by, say, the best of the torsion-balance experiments, we must estimate the co- Table 2.2. Tests of the weak equivalence principle Experiment Reference Method Substances tested Newton Bessel Eotvos Potter Renner Princeton Moscow Munich Stanford Boulder Orbital- Newton (1686) Bessel (1832) Eotvos, Pekar, and Fekete (1922) Potter (1923) Renner (1935) Roll, Krotkov, and Dicke (1964) Braginsky and Panov (1972) Koester (1976) Worden (1978) Keiser and Faller (1979) Worden (1978) Pendula Pendula Torsion balance Pendula Torsion balance Torsion balance Torsion balance Free fall Magnetic suspension Flotation on water Free fall in orbit Various Various Various Various Various Aluminum and gold Aluminum and platinum Neutrons Niobium, Earth Copper, tungsten Various " Experiments yet to be performed. Limit on \tj\ 2 5 2 2 x x x x 10" 3 10" 5 10" 9 10" 5 lO" 9 io-» lO" 1 2 3 x 10" 4 io- 4u 4 x 10" 10~15 - lO" 18 " Theory and Experiment in Gravitational Physics 28 efficients EA/m for the different interactions and for different materials. For laboratory-sized bodies, the dominant contribution to £A comes from the atomic nucleus. We begin with the strong interactions. The semiempirical mass formula (see, for example, Leighton, 1959) gives Es = -15.74 + 17.SA213 + 23.604 - 2Z)2A~l + 132/1"lS MeV (2.5) where Z and A are the atomic number and mass number, respectively, of the nucleus, and where S = 1 if {Z,A} = {odd, even}, 5 1 if {Z,A} = {even, even} and 8 = 0 if A is odd. Then, Es/mc2 = -1.7 x 1(T2 + 1.9 x 10" 2 ,4" 1/3 + 2.5 x 10~2(l - 2Z/A)2 + 1.41 x 10"M-2<5 (2.6) For platinum (Z = 78, A = 195), and aluminum (13,27) the difference in Es/mc2 is approximately 2 x 10"3, so from the limit \n\ < 10" 12 , we obtain the limit \ns\ < 5 x 10" 10 . In the case of electromagnetic interactions, we can distinguish among a number of different internal energy contributions, each potentially having its own n* parameter. For the electrostatic nuclear energy, the semiempirical mass formula yields the estimate £ ES = 0.71Z(Z - X)A ~1/3 MeV (2.7) Thus, EES/mc2 = 7.6 x 10"4Z(Z - \)A"4/3 (2.8) 3 with the difference for platinum and aluminum being 2.5 x 10" . The resulting limit on nES is |T/ES| < 4 X 10" 10 . Another form of electromagnetic energy is magnetostatic, resulting from the nuclear magnetic fields generated by the proton currents. To estimate the nuclear magnetostatic energy requires a detailed shell model computation. For example, the net proton current in any closed angular momentum shell vanishes, hence there is no energy associated with the magnetostatic interaction between such a closed shell and any particle outside the shell. For aluminum and platinum, Haugan and Will (1977) have shown (£MS/mc2)A1 = 4.1 x 10~7, us 6 (£M7mc2)pt = 2.4 x 10"7 (2.9) thus \r\ \ < 6 x 10 " . A third form of electromagnetic energy that has been studied is hyperfine, the energy of interaction between the spins of the nucleons and the magnetic fields generated by the proton and neutron magnetic moments. Computations by Haugan (1978) have yielded the Einstein Equivalence Principle and Gravitation Theory 29 estimate £ H F = (2n/V)ntig2pZ2 + g2(A - Z) 2 ] (2.10) where V is the nuclear volume, (iN is the nuclear magneton, and gp = 2.79 and ga = 1.91 are gyromagnetic ratios for the proton and neutron, respectively. Then, Em/mc2 = 2.1 x 10"5[>2Z2 + g2(A - Zf^A'2 (2.11) with the difference between aluminum and platinum being 4 x 10~ 6 ; thus|>7 HF |<2 x 10" 7. For some time, it was believed that the contribution of the weak interactions to nuclear energy was of the order of a part in 1012, and that the Eotvos experiment was not yet sufficiently accurate to test WEP for weak interactions (see for example, Chiu and Hoffman, 1964; Dicke, 1964a). However, these estimates took into account only the parity nonconserving parts of the weak interactions, which make no contribution to the energy of a nucleus in its ground state, to first order in the weak-interaction coupling constant G w . On the other hand, the parityconserving parts of the weak interactions do contribute at first order in Gw and yield a value E/me2 ~ 10" 8 (Haugan and Will, 1976). Specifically, in the Weinberg-Salam model for weak and electromagnetic interactions, the result is E^/mc2 = 2.2 x 10"8(iVZM2)[l + g(N,Z)], g(N,Z) = 0.295[i(iV - Z)2/ATZ + 4 sin2 0W + (Z/N) sin2 0W(2 sin2 0W - 1)] (2.12) where N = (A Z) is the neutron number, and where 0W ~ 20° is the "Weinberg" angle. For aluminum and platinum, the difference is 2 x 10~ 10 , yielding |>?w| < 10~2. Gravitational interactions are specifically excluded from WEP and EEP. In Chapter 3, we shall extend these two principles to incorporate local gravitational effects, thereby defining the Gravitational Weak Equivalence Principle (GWEP) and the Strong Equivalence Principle (SEP). These two principles will be useful in classifying alternative metric theories of gravity. In any case, for laboratory Eotvos experiments, gravitational interactions are totally irrelevant, since for an atomic nucleus Ea/mc2 ~ Gmp/Raucleusc2 ~ 10" 3 9 To test for gravitational effects in GWEP, it will be necessary to employ planetary objects and planetary Eotvos experiments (Section 8.1). Theory and Experiment in Gravitational Physics 30 (b) Tests of Local Lorentz Invariance Any experiment that purports to test special relativity (Section 2.2) also tests some aspect of Local Lorentz Invariance, since every Earthbound laboratory resides in a gravitational field (although it is only partially in free fall). However, very few of these experiments have been used to make quantitative tests of LLI in the same way that Eotvos experiments have been used to test WEP. For example, although elementary-particle experimental results are consistent with the validity of Lorentz invariance in the description of high-energy phenomena, they are not "clean" tests because in many cases it is unlikely that a violation of Lorentz invariance could be distinguished from effects due to the complicated strong and weak interactions. For instance, the observed violation of conservation of four momentum in beta decay was found to be due not to a violation of LLI, but to the emission of a hitherto unknown particle, the neutrino. However, there is one experiment that can be interpreted as a "clean" test of Local Lorentz invariance, and an ultrahigh precision one at that. This is the Hughes-Drever experiment, performed in 1959-60 independently by Hughes and collaborators at Yale University and by Drever at Glasgow University (Hughes et al., 1960; Drever, 1961). In the Glasgow version, the experiment examined the J = § ground state of the 7Li nucleus in an external magnetic field. The state is split into four levels by the magneticfield,with equal spacing in the absence of external perturbations, so the transition line is a singlet. Any external perturbation associated with a preferred direction in space (the velocity of the Earth relative to the mean rest frame of the universe, for example) that has a quadrupole (/ = 2) component will destroy the equality of the energy spacing and split the transition lines. Using NMR techniques, the experiment set a limit of 0.04 Hz (1.7 x 10" 16 eV) on the separation in frequency (energy) of the lines. One interpretation of this result is that it sets a limit on a possible anisotropy 3m\j in the inertial mass of the 7Li nucleus: |5mjJc2| ;$
1.7 x 10" 16 eV. If any of the forms of internal energy of the 7Li nucleus
suffered a breakdown of Local Lorentz Invariance, one would expect a
contribution to 5m{J of the form
dmij ~ X 5AEx/c2
(2.13)
A
where <5A is a dimensionless parameter that measures the strength of
anisotropy induced by interaction A. Using formulae from Section 2.4(a),
we can then make estimates of EA for 7Li and obtain the following limits
Einstein Equivalence Principle and Gravitation Theory
31
on<5\
|<5S| < 1(T 23 ,
|£ES| < 1 0 - 2 2 )
|<5HF| < 5 x 1(T 2 2 ,
|gW| < 5 x 1 Q -18
(2 .14)
Notice that the magnetostatic energy for 7Li is zero, since the proton
shell structure is ls1/2lp3/2 and there is no magnetostatic interaction
either within the closed s-shell (/ = 0) or between that shell and the valence
proton. Because of the remarkably small size of these limits, the HughesDrever experiment has been called the most precise null experiment ever
performed.
If Local Lorentz Invariance is violated, then there must be a preferred
rest frame, presumably that of the mean rest frame of the universe, or,
equivalently of the cosmic microwave background, in which the local
laws of physics take on their special form. Deviations from this form
would then depend on the velocity of the laboratory relative to the preferred frame. Since the anisotropy is a quadrupole effect, one would expect
it to be proportional to the square of the velocity w of the laboratory.
If <>o is a parameter that measures the "bare" strength of LLI violation,
then one would expect
For the motion of the Earth relative to the universe rest frame, w2 ~ 10 ~6.
Limits on the <5Q can then be inferred from Equation (2.14). As a special
case of this general argument, the Hughes-Drever experiment has also
been interpreted as a test of the existence of additional long-range tensor
fields that couple directly to matter (Peebles and Dicke, 1962; Peebles,
1962).
Other experiments that can be interpreted as tests of LLI include various ether-drift experiments, such as the Turner-Hill experiment (Dicke,
1964a; Haugan, 1979).
(c) Tests of Local Position Invariance
The two principal tests of Local Position Invariance are gravitational red-shift experiments that test the existence of spatial dependence
on the outcomes of local experiments, and measurements of the constancy
of the fundamental nongravitational constants that test for temporal
dependence.
Theory and Experiment in Gravitational Physics
32
Gravitational Red-Shift Experiments A typical gravitational red-shift
experiment measures the frequency or wavelength shift Z = Av/v =
AA/A between two identical frequency standards (clocks) placed at
rest at different heights in a static gravitational field. To illustrate how
such an experiment tests LPI, we shall assume that the remaining parts
of EEP, namely WEP and Local Lorentz Invariance, are valid. (In Sections 2.5 and 2.6, we shall discuss this question under somewhat different
assumptions.) WEP guarantees that there exist local freely falling frames
whose acceleration g relative to the static gravitational field is the same
as that of test bodies. Local Lorentz Invariance guarantees that in these
frames, the proper time measured by an atomic clock is related to the
Minkowski metric by
c2dx2 oc - r\^dx% dx\ oc c2dt\ - dx\ - dy\ - dz%
(2.15)
where x% are coordinates attached to the freely falling frame. However, in
a local freely falling frame that is momentarily at rest with respect to the
atomic clock, we permit its rate to depend on its location (violation of
Local Position Invariance), that is, relative to an arbitrarily chosen atomic
time standard based on a clock whose fundamental structure is different
than the one being analyzed, the proper time between ticks is given by
T = T(O)
(2.16)
where O is a gravitational potential whose gradient is related to the testbody acceleration by g = £V<I>.
Now the emitter, receiver, and gravitational field are assumed to be
static, therefore in a static coordinate system (ts,xs), the trajectories of
successive wave crests of emitted signal are identical except for a time
translation Ats from one crest to the next. Thus, the interval of time Afs
between ticks (passage of wave crests) of the emitter and of the receiver
must be equal (otherwise there would be a build up or depletion of wave
crests between the two clocks, in violation of our assumption that the
situation is static). The static coordinates are not freely falling coordinates, but are accelerated upward (in the +z direction) relative to the
freely falling frame, with acceleration g. Thus, for \gts/c\ ~ |#zs/c2| « 1
(i.e., for g uniform over the distance between the clocks), a sequence of
Lorentz transformations yields (MTW, Section 6.6)
ctF = (zs + c2/g)sinh{gts/c),
zF = (zs + c2/g)cosh(gts/c),
xF=xs,
yF = y s
(217)
Einstein Equivalence Principle and Gravitation Theory
33
Thus, the time measured by the atomic clocks (relative to the standard
clock) is given by
c2dx2 = T2(<D)(C2 dtl - dxl - dyl -
dz2)
= T2(<J>)[(1 + gzs/c2)2c2 dti - dx2 - dy2 - dz2}
(2.18)
Since the emission and reception rates are the same (1/Afj) when measured in static coordinate time, and since dxs = dys = dzs = 0 for both
clocks, the measured rates (v = AT" ') are related by
7
Vrec
Vem_,
~^T~
[T(g>rec)(l +
2
gZ)~\
rJc
rJc)\
Wc 22))JJ
U<"U(i +flWc
(
}
For small separations, Az = zrec z em , we can expand T(<D) in the form
T(4>rec) = T0 + ^ r ^ A z
(2.20)
where T 0 = T(<Dem), x'o = Bx/d<b\tm. Then
Z = (1 + a)At//c 2
(2.21)
where a =C 2 C~ 1 TJ ) /T 0 and where AC/ = g Az = g(zrec zem). If there
is no location dependence in the clock rate, then a = 0, and the red shift
is the standard prediction, i.e.,
Z = AU/c2
(2.22)
An alternative version of this argument assumes the validity of both
LLI and LPI and shows that, if the red shift is given by Equation (2.22),
then the acceleration of the local frames in which Lorentz and Position
Invariance hold is the same as that of test bodies, i.e., the local frames
are freely falling frames (Thorne and Will, 1971).
Although there were several attempts following the publication of the
general theory of relativity to measure the gravitational red shift of spectral lines from white dwarf stars, the results were inconclusive (see Bertotti
et al., 1962 for a review). The first successful, high-precision red-shift measurement was the series of Pound-Rebka-Snider experiments of 1960-65,
which measured the frequency shift of y-ray photons from Fe 57 as they
ascended or descended the Jefferson Physical Laboratory tower at Harvard University. The high accuracy achieved (1%) was obtained by
making use of the Mossbauer effect to produce a narrow resonance line
whose shift could be accurately determined. Other experiments since 1960
measured the shift of spectral lines in the Sun's gravitational field and
the change in rate of atomic clocks transported aloft on aircraft, rockets,
Table 2.3. Gravitational red-shift experiments
Experiment
Reference
Method
Pound-Rebka-Snider
Brault
Jenkins
Pound and Rebka (1960)
Pound and Snider (1965)
Brault (1962)
Jenkins (1969)
Snider
Jet-Lagged Clocks (A)
Snider (1972,1974)
Hafele and Keating (1972a,b)
Jet-Lagged Clocks (B)
Alley (1979)
Vessot-Levine Rocket
Red-shift Experiment
Null Red-shift Experiment
Close Solar Probe"
Vessot and Levine (1979)
Vessot et al. (1980)
Turneaure et al. (1983)
Nordtvedt (1977)
Fall of photons from
Mossbauer emitters
Solar spectral lines
Crystal oscillator clocks
on GEOS-1 satellite
Solar spectral lines
Cesium beam clocks on
jet aircraft
Rubidium clocks on
jet aircraft
Hydrogen maser on rocket
' Experiments yet to be performed.
Hydrogen maser vs. SCSO
Hydrogen maser or SCSO
on satellite
Limit on |a|
io- 2
5 x 10"2
9 x 10"2
6 x 10"2
10"1
2 x 10"2
2 x 10"4
io- 2
10 -6«
Einstein Equivalence Principle and Gravitation Theory
35
and satellites. Table 2.3 summarizes the important red-shift experiments
that have been performed since 1960.
Recently, however, a new era in red-shift experiments has been ushered
in with the development of frequency standards of ultrahigh stability parts in 1015 to 1016 over averaging times of 10 to 100 s and longer. Examples are hydrogen-maser clocks (Vessot, 1974), superconducting-cavity
stabilized oscillator (SCSO) clocks (Stein, 1974; Stein and Turneaure,
1975), and cryogenically cooled monocrystals of dielectric materials such
as silicon and sapphire (McGuigan et al., 1978). The first such experiment was the Vessot-Levine Rocket Red-shift Experiment that took
place in June, 1976. A hydrogen-maser clock was flown on a rocket to
an altitude of about 10,000 km and its frequency compared to a similar
clock on the ground. The experiment took advantage of the high frequency
stability of hydrogen-maser clocks (parts in 1015 over 100 s averaging
times) by monitoring the frequency shift as a function of altitude. A sophisticated data acquisition scheme accurately eliminated all effects of
the first-order Doppler shift due to the rocket's motion, while tracking
data were used to determine the payload's location and velocity (to evaluate the potential difference AU, and the second-order Doppler shift).
Analysis of the data yielded a limit (Vessot and Levine, 1979; Vessot et al.,
1980)
|a| < 2 x 10" 4
(2.23)
Coincidentally, the Scout rocket that carried the maser aloft stood 22.6 m
in its gantry, almost exactly the height of the Harvard Tower. In an interplanetary version of this experiment, a stable clock (H-maser or SCSO
clock) would be flown on a spacecraft in a very eccentric solar orbit
(closest approach ~ 4 solar radii); such an experiment could test a to a
part in 106 (Nordtvedt, 1977) and could conceivably look for "secondorder" red-shift effects of O(AC/)2 (Jaffe and Vessot, 1976).
Advances in stable clocks have also made possible a new type of redshift experiment that is a direct test of Local Position Invariance (LPI):
a "null" gravitational red-shift experiment that compares two different
types of clocks, side by side, in the same laboratory. If LPI is violated,
then not only can the proper ticking rate of an atomic clock vary with
position, but the variation must depend on the structure and composition
of the clock, otherwise all clocks would vary with position in a universal
way and there would be no operational way to detect the effect (since one
clock must be selected as a standard and ratios taken relative to that clock).
Theory and Experiment in Gravitational Physics
36
Thus, we must write for a given clock type A,
T
= t\U) = TA(1 - aA AU/c2)
(2.24)
Then a comparison of two different clocks at the same location would
measure
TA/TB =
( T A/ t B )o [ 1 _ (aA _ aB) A [// C 2]
£.25)
where (TA/TB)0 i s t n e constant ratio between the two clock times observed
at a chosen initial location.
A null red-shift experiment of this type was performed in April, 1978
at Stanford University. The rates of two hydrogen maser clocks and of
an ensemble of three SCSO clocks were compared over a 10 day period
(Turneaure et al., 1983). During this time, the solar potential U changed
sinusoidally with a 24 hour period by 3 x 10~13 because of the Earth's
rotation, and changed linearly at 3 x 10""12 per day because the Earth
is 90° from perihelion in April. However, analysis of the data set an upper
limit on both effects, leading to a limit on the LPI violation parameter
|aH_ascso|< 10-2
(2.26)
The art of atomic timekeeping has advanced to such a state that it may
soon be necessary to take red-shift and Doppler-shift corrections into
account in making comparisons between timekeeping installations at
different altitudes and latitudes.
Constancy of the constants The other key test of Local Position
Invariance is the constancy of the nongravitational constants over cosmological timescales (we delay discussion of the gravitational "constant"
until Section 8.5). We shall not review here the various theories and
proposals, originating with Dirac, that permit variable fundamental constants [for detailed review and references, see Dyson (1972)], rather we
shall cite the most recent observational evidence (Table 2.4). The observations range from comparisons of spectral lines in distant galaxies and
quasars, to measurements of isotopic abundances of elements in the solar
system, to laboratory comparisons of atomic clocks. Recently, Shlyakhter
(1976a,b) has made significant improvements in the limits on variations
in the electromagnetic, weak, and strong coupling constants by studying
isotopic abundances in the "Oklo Natural Reactor," a sustained U 235
fission reactor that evidently occurred in Gabon, Africa nearly two billion
years ago (Maurette, 1976). Measurements of ore samples yielded an
abnormally low value for the ratio of two isotopes of samarium (Sm149/
Table 2.4. Limits on cosmological variation of nongravitational constants
Constant k
Limit on kjk
per Hubble time
2 x 1010 yr
(H 0 = 5 5 k m s " 1 M p c " 1 )
Fine structure Constant:
a = e2/hc
4 x 10" 4
8 x 10" 2
8 x 10" 2
Weak Interaction Constant:
Method
Reference
Re 187 ft decay rate over
geological time
Mgll fine structure and
source at Z = 0.5
SCSO clock vs. cesium
beam clock
Dyson (1972)
Wolfe, Brown, and Roberts
(1976)
2
Re 187 , K 4 0 decay rates
Dyson (1972)
Electron-Proton Mass Ratio:
mjmp
1
Mass shift in quasar
spectral lines (Z ~ 2)
Pagel (1977)
10" 1
Mgll, 21 cm line
Wolfe, Brown, and Roberts
(1976)
8 x 10" 2
Nuclear stability
Davies (1972)
g me/mp
10" 7
Turneaure and Stein (1976)
P = g(mlc/h3
Proton Gyromagnetic Factor:
Limit from
Oklo reactor
(Shlyakhter, 1976a,b)
2 x 10~ 2
Strong Interactions:
gl
8 x 10" 9
Theory and Experiment in Gravitational Physics
38
Sm147). Neither of these isotopes is a fission product, but Sm149 can be
depleted by a dose of neutrons. Estimates of the neutron fluence (integrated dose) during the reactor's "on" phase, combined with the measured
abundance anomaly yielded a value for the neutron capture cross section
for Sm149 two billion years ago which agrees with the modern value.
However, the capture cross section is extremely sensitive to the energy
of a low-lying level (E ~ 0.1 eV) of Sm149, so that a variation of only
20 x 10 ~3 eV in this energy over 109 years would change the capture
cross section from its present value by more than the observed amount.
By estimating the contributions of strong, electromagnetic, and weak
interactions to this energy, Shlyakhter obtained the limits on the rate of
variation of the corresponding coupling constants shown in Table 2.4,
2.5
Schiff 's Conjecture
Because the three parts of the Einstein Equivalence Principle discussed above are so very different in their empirical consequences, it is
tempting to regard them as independent theoretical principles. However,
any complete and self-consistent gravitation theory must possess sufficient mathematical machinery to make predictions for the outcomes of
experiments that test each principle, and because there are limits to the
number of ways that gravitation can be meshed with the special relativistic
laws of physics, one might not be surprised if there were theoretical connections between the three subprinciples. For instance, the same mathematical formalism that produces equations describing the free fall of a
hydrogen atom must also produce equations that determine the energy
levels of hydrogen in a gravitational field, and thereby determine the
ticking rate of a hydrogen maser clock. Hence a violation of EEP in the
fundamental machinery of a theory that manifests itself as a violation of
WEP might also be expected to emerge as a violation of Local Position
Invariance. Around 1960, Leonard I. Schiff conjectured that this kind of
connection was a necessary feature of any self-consistent theory of gravity.
More precisely, Schiff's conjecture states that any complete, self-consistent
theory of gravity that embodies WEP necessarily embodies EEP. In other
words, the validity of WEP alone guarantees the validity of Local Lorentz
and Position Invariance, and thereby of EEP. This form of Schiff's conjecture is an embellished classical version of his original 1960 quantum
mechanical conjecture (Schiff, 1960a). His interest in this conjecture was
rekindled in November, 1970 by a vigorous argument with Kip S. Thorne
at a conference on experimental gravitation held at the California Insti-
Einstein Equivalence Principle and Gravitation Theory
39
tute of Technology. Unfortunately, his untimely death in January, 1971
cut short his renewed effort.
If Schiff's conjecture is correct, then the Eotvos experiments may be
seen as the direct empirical foundation for EEP, and for the interpretation
of gravity as a curved-spacetime phenomenon. Some authors, notably
Schiff, have gone further to argue that if the conjecture is correct, then
gravitational red-shift experiments are weak tests of gravitation theory,
compared to the more accurate Eotvos experiment. For these reasons,
much effort has gone into "proving" Schiff's conjecture. Of course, a
rigorous proof of such a conjecture is impossible, yet a number of
powerful "plausibility" arguments using a variety of assumptions can be
formulated.
The most general and elegant of these arguments is based upon the
assumption of energy conservation. This assumption allows one to perform very simple cyclic gedanken experiments in which the energy at the
end of the cycle must equal that at the beginning of the cycle. This approach was pioneered by Dicke (1964a), and subsequently generalized
by Nordtvedt (1975) and Haugan (1979).
Specifically, we restrict attention to theories of gravity in which there
is a conservation law of energy for nongravitating "test" systems that
reside in given static and external gravitational fields. To guarantee the
existence of such a law, it is sufficient for the theory to be based on an
invariant action principle [cf. Dicke's constraint (2)], but it is not necessary.
We consider an idealized composite body made up of structureless test
particles that interact by some nongravitational force to form a bound
system. For a system that moves sufficiently slowly in a weak, static
gravitationalfield,the laws governing its motion can be put into a quasiNewtonian form (we assume the theory has a Newtonian limit); in particular, the conserved energy function Ec associated with the conservation
law can be assumed to have the general form
Ec = MKc2 - MRU(X) + | M R F 2 + O(MRU2, MRV\MRUV2)
(2.27)
where X and V are quasi-Newtonian coordinates of the center of mass of
the body, MR is the "rest" energy of the body, U is the external gravitational potential, and c is a fundamental speed used to convert units of
mass into units of energy. If EEP is violated, we must allow for the
possibility that the speed of light and the limiting speed of material
particles may differ in the presence of gravity; to maintain this possibility
we do not set c = 1 automatically in Equation (2.27) (see also footnote,
p. 24). Note that V is the velocity relative to some preferred frame. In
Theory and Experiment in Gravitational Physics
40
problems involving external, static gravitational potentials, the preferred
frame is generally the rest frame of the external potential, while in problems
involving cosmological gravitational effects where localized potentials
can be ignored, the preferred frame is that of the universe rest frame.
[In problems involving both kinds of effects, the simple form of Equation (2.27) no longer holds.] The possible occurrence of EEP violations
arises when we write the rest energy MRc2 in the form
MRc2 = Moc2 - £B(X, V)
(2.28)
where M o is the sum of the rest masses of the structureless constituent
particles, and £ B is the binding energy of the body. It is the position and
velocity dependence of £ B , a dependence that in general is a function
of the structure of the system, which signals the breakdown of EEP.
Roughly speaking, an observer in a freely falling frame can monitor the
binding energy of the system, thereby detecting the effects of his location
and velocity in local nongravitational experiments. Haugan (1979) has
made this more precise by showing that in fact it is the possible functional
difference in £B(X, V) between the system under study and a "standard"
system arbitrarily chosen as the basis for the units of measurement that
leads to measurable effects. Because the location and velocity dependence
in £ B is a result of the external gravitational environment, it is useful
to expand it in powers of U and V2. To an order consistent with the
quasi-Newtonian approximation in Equation (2.27), we write
£B(X,V) = El + 8myUiJ(X) - tfntfV'V1 + O(E%U2,...)
(2.29)
iJ
where U is the external gravitational potential tensor [cf. Equation (4.28)]; it is of the same order as U and satisfies U" = U. The quantities <5w# and bm\' are called the anomalous passive gravitational and
inertial mass tensors, respectively. They are expected to be of order t\E%,
where r\ is a dimensionless parameter that characterizes the strength of
EEP-violating effects; they depend upon the detailed internal structure
of the composite body. Summation over repeated spatial indices i, j is
assumed. The conserved energy can thus be written, to quasi-Newtonian
order,
Ec = (M0c2 - £g) - [(Mo -
E°c-2)S»
^ ' J + O(M0U2,...)
(2.30)
We first give examples of violations of Local Position and Lorentz
Invariance generated by £B(X, V). Consider two different systems at rest
in the gravitational potential. Each system makes a transition from one
Einstein Equivalence Principle and Gravitation Theory
41
quantum energy level to another, and emits a quantum of frequency
v = AEc/h. The ratio of the two frequencies is given from Equation (2.30) by
L (AEE)x
(AEg) 2 Jc 2
In the case where <5mj/ oc diJ, the quantity in square brackets can be
identified as the coefficient a. 1 a 2 in Equation (2.25). Thus the anomalous
passive gravitational mass tensor dmtf produces preferred-location effects
in a null gravitational red-shift experiment. Consider the same two systems
far from gravitating matter, but moving relative to the universe rest frame
with velocity V. Then the ratio of the two frequencies is given by
(A£g)21
1
jAQSmiV
\ (A£°)x
J
A(<5m{V2l V'V
2
(A£°)2 J c
Thus, the anomalous inertial mass tensor produces preferred-frame effects
in an experiment such as the Hughes-Drever experiment, where the two
systems in question are two excited states of 7 Li nuclei of different azimuthal quantum numbers in an external magnetic field. In this case,
because the Zeeman splitting is the same for all levels in the absence of
a preferred-frame effect, (A£B)X = (A£ B ) 2 , however because of the possible
anisotropy in dm?, one would expect A(dm\J) to differ for transitions
between different pairs of levels [for further details, see Section 2.6]. Thus
dmy is responsible for violations of Local Position Invariance and <5m{J
is responsible for violations of Local Lorentz Invariance.
In order to verify Schiff s conjecture, it remains only to show that 5mj,J
and dm\3 also produce violations of WEP. To do this we make use of a
cyclic gedanken experiment first used by Dicke (1964a). We begin with a
set of n free particles of mass m0 at rest at X = h. From Equation (2.27),
the conserved energy is simply nm 0 c 2 [l t/(h)/c 2 ]. We then form a
composite body and release the binding energy £B(h,0), in the form of
free particles of rest mass m 0 , stored in a massless reservoir. The conserved
energy of the composite body is [nmoc2 £ B (h,0)][l - [7(h)/c2] and
that of the reservoir is £ B (h,0)[l t/(h)/c 2 ]. The composite body falls
freely to X = 0 with an acceleration assumed to be A = g 4- <5A while
the stored test particles fall with acceleration g = Vt/ (by definition). At
X = 0 we bring both systems to rest, and place the energies thus gained,
- [nm0 - £B(0, V)/c 2 ]A h - dniJgihi,
and
- EB(h, 0)g h/c 2
into the reservoir (we have assumed g, h, and V are parallel). Dropping
terms of order (g h) 2 , we see that the reservoir now contains conserved
Theory and Experiment in Gravitational Physics
42
energy
£B(h,O)[l - C/(0)/c2] - £jjg h/c 2 - (nmo - E°/c2)A h - &nitfh>
From this we extract enough energy £ B (0,0)[l t/(0)/c 2 ] to disassemble
the composite system into its n constituents, and enough energy nmog h
to give the particles sufficient kinetic energy to return to their initial
state of rest at X = h. The cycle is now closed, and if energy is to be conserved, the reservoir must be empty. To quasi-Newtonian order, this
implies
£B(h, 0) - £B(0,0) - (nm0 - E$/c2)SA h - <5m{W = 0 (2.33) £B(h, 0) - £B(0,0) = dmy\UlJ h (2.34) Since we obtain A' = g> + (5mik/MR)U{f - (<5mp/MRV (2.35) where M R s nm0 £ B /c 2 . The first term is the universal gravitational acceleration that would be expected in a theory satisfying WEP. The remaining terms depend upon the body's structure through the anomalous mass tensors in £B(X, V). Hence a violation of Local Lorentz or Position Invariance implies a violation of WEP. Equivalently, WEP {dm^ = dm{k = 0) implies Local Lorentz and Position Invariance. Equivalently, WEP implies EEP. The gravitational red-shift experiment can also be studied within this framework, using a cyclic gedanken experiment suggested by Nordtvedt (1975). The cycle begins as before with a set of n free particles of mass m0 at rest at X = h. We form a composite body and release the binding energy £ B (h,0)[l L/(h)/c2] in the form of a massless quantum which propagates to X = 0. Its energy there, compared to the energy £ B (0,0)[l l/(0)/c 2 ] of a quantum emitted from an identical system at X = 0, is assumed to be given by (1 - Z)£ B (0,0)[l - t/(0)/c 2 ]. This energy is stored in a reservoir. Our goal is to evaluate the red shift Z. The body is then allowed to fall freely to X = 0, where it is brought to rest, with the kinetic energy of motion, -{nm0 - £ B (0,V)/c 2 ]A h - <5mJW added to the reservoir. If we substitute for A from Equation (2.35), we see that the reservoir now contains energy (to quasi-Newtonian order) (1 - Z)£ B (0,0)[l - t/(0)/c 2 ] - [rnno - £ B (0,0)/c 2 ]g h - Einstein Equivalence Principle and Gravitation Theory 43 We extract from the reservoir enough energy £B(0,0)[l - t/(O)/c2] to disassemble the system, and enough energy nmog h to return the n free particles to the starting point. Again, conservation of energy requires that the reservoir be empty and therefore that Z must satisfy (to first order in g h) - ZE% + Elg h/c 2 - dmyVUiJ h = 0 (2.36) or Z = [At/ - (8r4Jc2/E$) AUir\/c2
(2.37)
where A C / s g h and AUiJ = Vt/° h. By a similar analysis one can
show that the second-order Doppler shift between an emitter moving at
velocity V and a receiver at rest, relative to a preferred universe rest
frame, is given by (Haugan, 1979)
Z D = - i V2/c2 +
ttSnf/E&VV'
(2.38)
Thus, the simple assumption of energy conservation has allowed us to
"prove" Schiff's conjecture, as well as elucidate the empirical consequences
of possible violations of the three aspects of the Einstein Equivalence
Principle.
Thorne, Lee, and Lightman (1973) have proposed a more qualitative
"proof" of Schiff's conjecture for that class of gravitation theories that
are based on an invariant action principle, so-called Lagrangian-based
theories of gravity. They begin by defining the concept of "universal
coupling": a generally covariant Lagrangian-based theory is universally
coupled if it can be put into a mathematical form (representation) in
which the action for matter and nongravitational fields / NG contains
precisely one gravitational field: a symmetric, second-rank tensor # with
signature + 2 that reduces to J/ when gravity is turned off; and when ^ is
replaced by if, 7NG becomes the action of special relativity. Clearly, among
all Lagrangian-based theories, one is universally coupled if and only if it
is a metric theory (for details see Thorne, Lee, and Lightman, 1973).
Let us illustrate this point with a simple example. Consider a
Lagrangian-based theory of gravity that possesses a globally flat background metric i\ and a symmetric, second-rank tensor gravitational field
h. The nongravitational action for charged point particles of rest mass
m0 and charge e, and for electromagnetic fields has the form
/NG = h + /in, + hm
(2-39)
Theory and Experiment in Gravitational Physics
44
where
Io= -m0 jdr,
dx2 = -(rj^ + h^)dx"dx\
^ ^
d4x
(2.40)
where F^ = AVfll A ^ , and where
IMNIWI 1
'
(2.41)
We work in a coordinate system in which if = diag( 1,1,1,1). To see
whether this theory is universally coupled, the obvious step is to assume
that the single gravitational field i/^v is given by
"/V = V + V
(2.42)
This would make J o and Jinl appear universally coupled. However, in the
electromagnetic Lagrangian, we obtain, for example,
tf* _ W = yj,** _ }f% + O(h3)
(2.43)
1
where ||^""|| = | | ^ | | " . Thus, there is no way to combine */ and h^ into
a single gravitational field in ING, hence the theory is not universally
coupled. To see that the theory is also not a metric theory, we transform
to a frame in which at an event 2P,
Note that in this frame, h^
into the form
^ 0 in general, thus the action can be put
/NG = ^SRT + A/
(2.44)
where
JSRT
= -m 0 Jdx + e JAndx* - (len)'1
j^F^F^-fj)1!2d*x,
AI =
+ O(h3F2)
flC
9
(2.45)
wherefc = h* ]? ¥= 0. So in a local Lorentz frame, the laws of physics
are not those of special relativity, so the theory is not a metric theory.
Notice that in this particular case, for weak gravitationalfields(|fy,v| « 1),
the theory is metric to first order in h^, while the deviations from metric
form occur at second order in h^. In the next section, we shall present a
Einstein Equivalence Principle and Gravitation Theory
45
mathematical framework for examining a class of theories with nonuniversal coupling and for making quantitative computations of its
empirical consequences.
Consider now all Lagrangian-based theories of gravity, and assume
that WEP is valid. WEP forces 7NG to involve one and only one gravitational field (which must be a second-rank tensor ty which reduces to r\
far from gravitating matter). If / NG were to involve some other gravitational fields <j>, Kp, h^,... they would all have to conspire to produce
exactly the same acceleration for a body made largely of electromagnetic
energy as for one made largely of nuclear energy, etc. This is unlikely
unless i/^v and the other fields appear everywhere in JNG in the same form,
for example, /((p)^^ if a scalar field is present, i//^ + ah^ if a tensor field
is present, and so on. In this case, one can absorb these fields into a new
field g^ and end up with only one gravitational field in JNG. This means
that the theory must be universally coupled, and therefore a metric
theory, and must satisfy EEP.
One possible counterexample to Schiff 's conjecture has been proposed
by Ni (1977): a pseudoscalar field <j> that couples to electromagnetism in
a Lagrangian term of the form ^""F^F^, where s*11 is the completely
antisymmetric Levi-Civita symbol. Ni has argued that such a term,
while violating EEP, does not violate WEP, although it does have the
observable effect of producing an anomalous torque on systems of electromagnetically bound charged particles. Whether this torque then can
lead to observable WEP violations is an open question at present.
2.6
The THs/i Formalism
The discussion of Schiff's conjecture presented in the previous
section was very general, and perhaps gives compelling evidence for the
validity of the conjecture. However, because of the generality of those
arguments, there was little quantitative information. For example, no
means was presented to compute explicitly the anomalous mass tensors
(5mj/ and 8m\J for various systems. In order to make these ideas more
concrete, we need a model theory of the nongravitational laws of physics
in the presence of gravity that incorporates the possibility of both nonmetric (nonuniversal) and metric coupling. This theory should be simple,
yet capable of making quantitative predictions for the outcomes of experiments. One such "model" theory is the THe/x formalism, devised by
Lightman and Lee (1973a). It restricts attention to the motions and electromagnetic interactions of charged structureless test particles in an external, static, and spherically symmetric (SSS) gravitational field. It
Theory and Experiment in Gravitational Physics
46
assumes that the nongravitational laws of physics can be derived from
an action / N G given by
f NG
=
Jo + hat + Iem>
\{T -
E2 - li-^d'x
(2.46)
(we use units here in which x and t both have units of length) where mOa,
ea, and x£(t) are the rest mass, charge, and world line of particle a, x° = t,
v»a = dxljdt, E = \A0 - A o, B = (V x A), and where scalar products
between 3-vectors are taken with respect to the Cartesian metric 8ij.
The functions T, H, e, and n are assumed to be functions of a single external gravitational potential 4>, but are otherwise arbitrary. For an SSS
field in a given theory, T, H, e, and /x will be particular functions of O. It
turns out that, for SSS fields, equations (2.46) are general enough to encompass all metric theories of gravitation and a wide class of nonmetric
theories, such as the Belinfante-Swihart (1957) theory and the nonmetric
theory discussed in Section 2.5. In many cases, the form of / N G in equation
(2.46) is valid only in special coordinate systems ("isotropic" coordinates
in the case of metric theories of gravity). An example of a theory that
does not fit the THsfi form of / N G is the Naida-Capella nonmetric theory
(see Lightman and Lee, 1973a for discussion). Cases such as this must
then be analyzed on an individual basis. For an "en" formalism, see
Dicke (1962).
(a) Einstein Equivalence Principle in the THe/x formalism
We begin by exploring in some detail the properties of the formalism as presented in equations (2.46). Later, we shall discuss the physical
restrictions built into it, and shall apply it to the interpretation of experiment.
In order to examine the Einstein Equivalence Principle in this formalism
we must work in a local freely falling frame. But we do not yet know
whether WEP is satisfied by the THsn theory (and suspect that it is not,
in general), so we do not know to which freely falling trajectories local
frames should be attached. We must therefore arbitrarily choose a set
of trajectories: the most convenient choice is the set of trajectories of
neutral test particles, i.e., particles governed only by the action l0, since
Einstein Equivalence Principle and Gravitation Theory
47
their trajectories are universal and independent of the mass mOa. We make
a transformation to a coordinate system x" = (?, x) chosen according to
the following criteria: (i) the origins of both coordinate systems coincide,
that is, for a selected event 3P, xx{@) - x\0>) = 0, (ii) at 0>, a neutral test
body has zero acceleration in the new coordinates, i.e., d2xJ'/dt2^ = 0,
and in the neighborhood of 9 the deviations from zero acceleration are
quadratic in the quantities Ax* = x* x%0>), and (iii) the motion of the
neutral-test body is derivable from an action Jo. The required transformation, correct to first order in the quantities g0? and gj, x, assumed small, is
x = Hy\x
+ |tf 0-»Togof2 + ±Ho ' H^2xg 0 x - gox2)]
(2.47)
where the subscript (0) and superscript (') on the functions T, H, E, and
fi denote
To = T(x* = xs = 0),
r 0 = ^r/a<D|x.=xa=0
(2.48)
and where
go = V*
(2.49)
The action Io in the new coordinates then has the form
'o = - I > o a fd - v2a)il2 dt{\ + O[(xa)2]}
(2.50)
where va = dxjdf. Note that our choice of the multiplicative factors Tj / 2
and HQ12 resulted in unit coefficients in / 0 , making it look exactly like
that of special relativity. Similarly the actions /int and Iem can be rewritten
in the new coordinates, with the result
/in, = 2 X UfV? dt{\ + O[(XS)2]},
Im =
(2.51)
(Snr1eoTlo'2Ho1
- To
'HOEE V O
+ H0TE
1/2
'^(l -
JT'OTO
l
HE
1/2
A o go x)
r 0 f g 0 -(E x S)(l - To 'HoEE V O 1 ) } ^
+ [corrections of order (x*)2]
(2.52)
where
A% =
(dx*/dxii)Aa
E = *At - A o,
6 = Vx A
(2.53)
Theory and Experiment in Gravitational Physics
48
and
Ao = (2r o /T' o )(^o * + iT'oTo ' - iH'oHo *)
(2.54)
Let us now examine the consequences for EEP of physics governed by
I NG . Focus on the form of JNG at the event 0>(xj = t = 0), since local test
experiments are assumed to take place in vanishingly small regions surrounding 3". Because such experiments are designed to be electrically
neutral overall, we can assume that the E and B fields do not extend
outside this region. Then at 9
/NG= - X > o a f ( l -v2a)
+ (8TT)- hoT^Ho
1/2
J [ £ 2 - (To 'Hoeo Vo X)B2] d*x
(2.55)
We first see that, in general, /NG violates Local Lorentz Invariance. A
simple Lorentz transformation of particle coordinates and fields in 7NG
shows that JNG is a Lorentz invariant if and only if
To 1 /f o £oVo 1 = l
or eofio=TolHo
(2.56)
Since we have not specified the event 0>, this condition must hold throughout the SSS spacetime. Notice that the quantity (TQ lHtfo Vo x ) 1/2 plays
the role of the speed of light in the local frame, or more precisely, of the
ratio of the speed of light clight to the limiting speed c0 of neutral test
particles, i.e.,
To 'Hoeo Vo l = (clight/c0)2
(2.57)
Our units were chosen in such a way that, in the local freely falling frame,
c0 = 1; equivalently, in the original THsfi coordinate system [cf. Equation (2.46)]
c0 = (To/Ho)1'2,
clight = (EoAio)-1/2
(2.58)
These speeds will be the same only if Equation (2.56) is satisfied. If not,
then the rest frame of the SSS field is a preferred frame in which / NG takes
its THe/j. form, and one can expect observable effects in experiments that
move relative to this frame. Thus, the quantity 1 ToHo lfioMo plays the
role of a preferred-frame parameter: if it is zero everywhere, the formalism
is locally Lorentz invariant; if it is nonzero anywhere, there will be preferred-frame effects there. As we shall see, the Hughes-Drever experiment
provides the most stringent limits on this preferred-frame parameter.
Einstein Equivalence Principle and Gravitation Theory
49
Next, we observe that / NG is locally position invariant if and only if
o 1/2 = [constant, independent of 9\
o 1/2 = [constant, independent of ^>]
(2.59)
Even if the theory is locally Lorentz invariant (TQ 1H0EQ VO * = 1>
independent of &) there may still be location-dependent effects if the
quantities in Equation (2.59) are not constant. This would correspond,
for example, to the situation discussed in Section 2.3, in which different
parts of the local physical laws in a freely falling frame couple to different
multiples of the Minkowski metric; in this case, free particle motion
coupling to 7 itself, electrodynamics coupling to the position-dependent
tensor i\* =. eTi/2H'i/2ti in the manner given by the field Lagrangian
ri*'"'rivfFllvFltf. The nonuniversality of this coupling violates EEP and
leads to position-dependent effects, for example, in gravitational redshift experiments (also see Section 2.4). An alternative way to characterize
these effects in the case where Local Lorentz Invariance is satisfied is to
renormalize the unit of charge and the vector potential at each event
& according to
e*a = ettso 1/2To 1/4 H S/4,
Af = A^T^H^
(2.60)
then the action, (2.55), takes the form
+ (8TT)- * j(E*2 - B*2) d4x
(2.61)
This action has the special relativistic form, except that the physically
measured charge e* now depends on location via Equation (2.60), unless
E0TII2HQXI2
is independent of 9. In the latter case, the units of charge
can be effectively chosen so that everywhere in spacetime,
soTlol2Ho112
= 1
(2.62)
Note, however, that if LPI alone is satisfied, one can renormalize the
charge and vector potential to make either £0TQI2HQ 1/2 = 1 or
fioTll2Ho 1/2 = 1, but not both, thus in general LLI need not be satisfied.
Combining Equations (2.56), (2.59), and (2.62), we see that a necessary
and sufficient condition for both Local Lorentz and Position Invariance
to be valid is
e0 = n0 = (Ho/T0)112, for all events 9
(2.63)
Theory and Experiment in Gravitational Physics
50
Consider now the terms in / NG , in Equations (2.50)-(2.52) that depend
on the first-order displacements x, t from the event 9. These occur only
in / em , and presumably produce polarizations of the electromagnetic fields
of charged bodies proportional to the external "acceleration" g0 = V4>.
One would expect these polarizations to result in accelerations of composite bodies made up of charged test particles relative to the local freely
falling frame (i.e., relative to neutral test particles), in other words, to
result in violations of WEP. These terms are absent if F o = Ao = 0, and
U i.e.,
eoTlol2Ho
1/2
= const,
noTl'2Ho
EoHo^HoTo1
m
= const,
(2.64)
Again, the units of charge can be normalized so that
e0 = Ho = (H0/T0)1'2, for all 9
(2.65)
But this condition also guarantees Local Lorentz and Position Invariance.
Thus, within the THe/j. formalism, for SSS fields
[Equation (2.65)] => WEP,
[Equation (2.65)] => EEP
(2.66)
However, the above discussion suggests that WEP alone may guarantee
Equation (2.65) and thereby EEP. We can demonstrate this directly by
carrying out an explicit calculation of the acceleration of a composite
test body within the THsfi framework. The resulting restricted proof of
Schiff's conjecture was first formulated by Lightman and Lee (1973a).
(b) Proof of Schiff's conjecture
We work in the global THsfi coordinate system in which JNG
has the form Equation (2.46). Variation of / NG gives a complete set of
particle equations of motion and "gravitationally modified" Maxwell
(GMM) equations, given by
(d/dt)(HW~ \ ) + {W- XV(T - Hvl) = aL(xa),
aL(x0) = (ea/mOa){VAo(xa) + V[va A(xfl)] - dA{xa)/dt},
V (EE) = 4np,
V x ( ^ » B ) = 4TTJ + d{eE)/dt
where W = {T-
(2.67)
Hv2a)112,p = Y.aea<53(x - xfl), J = £ a e a y a 5 3 (x - xa), and
aL is the Lorentz acceleration of particle a. These equations are used to
Einstein Equivalence Principle and Gravitation Theory
51
calculate the acceleration from rest of a bound test body consisting of
charged point particles. A number of approximations are necessary to
make the computation tractable. First, the functions T, H, e, and fi,
considered to be functions of <D are expanded about the instantaneous
center of mass location X = 0, in the form
T(O) = To + T'ogo x + O(g0 x)2
(2.68)
where To = T(x = 0), T'o = dT/d<S>\x=0. As long as the body is small
compared to the scale over which d> varies, we can assume that g0 x « 1,
and work to first order in g0 x. Second, we assume that the internal
particle velocities and electromagnetic fields are sufficiently small so we
can expand the equations of motion and GMM equations in terms of
the small quantities
v1 ~ e2/mor « 1
where r is a typical interparticle distance. By analogy with the postNewtonian expansion to be described in Chapter 4, we call this a postCoulombian expansion; for the purpose of the present discussion we
shall work to first post-Coulombian order. We expect the single particle
acceleration to contain terms that are O(g0) (bare gravitational acceleration), O(v2) (Coulomb interparticle acceleration), O(gf0t;2) (post-Coulombian gravitational acceleration), O(v4) (post-Coulombian interparticle
acceleration), O(g0v*) (post-post-Coulombian gravitational acceleration),
and so on. To O(g0v2), we obtain
o + itf'oHo'goi;2
+ (T'oTo 1 - H'0H0- l)g0 vavfl + Ty2H» X W
(2-69)
To write the Lorentz acceleration aL(xa) directly in terms of particle
coordinates, we must obtain the vector potential A^ in this form to an
appropriate order. In a gauge in which
£Mo,o - V A = 0
(2.70)
the GMM equations take the form
V 2 4 0 - ej^o.oo = 4ns~1p-e~1\e2
V A - e/xA.oo = ~^nJ
(\A0 - A,o),
x
+ (sp)- V(sp)V A - ^ V / z x (V x A) (2.71)
These equations can be solved iteratively by writing
Ao = A^ + A%\
A = A(0) + A(1)
(2.72)
Theory and Experiment in Gravitational Physics
52
where A(v/A(°y ~ O(g0), and solving for each term to an appropriate
order in v2. The result is
A o = -4>
A = A(0) + O(0O)
(2.73)
where
a
The resulting single-particle acceleration is inserted into a definition of
center of mass. It turns out that to post-Coulombian order, it suffices
to use the simple center-of-mass definition
Y-
-iv
-v
2 J wOaXa>
A = m
a
m
(2-75)
m
= ZJ
0a
a
We then compute d2X/dt2, substituting the single-particle equations of
motion to the necessary order, and using the fact that, at t = 0, X = 0,
dX/dt = 0. The resulting expression is simplified by the use of virial
theorems that relate internal structure-dependent quantities to each
other via total time derivatives of other internal quantities. As long as
we restrict attention to bodies in equilibrium, these time derivatives can
be assumed to vanish when averaged over intervals of time long compared
to internal timescales. Errors generated by our choice of center-of-mass
definitions similarly vanish. To post-Coulombian order, the required
virial relation is
=o
(2.76)
where angular brackets denote a time average, and where
(2.77)
ab
a
where xab = xa xb, rab = |xa(,|, and the double sum over a and b excludes
the case a = b. The final result is
d2Xl/dt2 = g{-
^[TE1'%1rojea/m)
+ 0 J '[ro 1/2 eo 1 (l-T o Ho 1 eo/Xo)]
x (JEjf/m + <5yE E »
(2.78)
where
Vo
(2-79)
Einstein Equivalence Principle and Gravitation Theory
53
where F o is given by Equation (2.54), and where
Ef? = <Q">,
EES = ( £
Q
")
(2-8°)
The first term gl is the universal acceleration of a neutral test body (governed by Io alone); the other two terms depend on the body's electromagnetic self-energy and self-energy tensor. These terms vanish for all
bodies (i.e., WEP is satisfied) if and only if
(2.81)
at any event 0>, which is equivalent to Equation (2.63). Hence
WEP => EEP
and Schiff's conjecture is verified, at least within the confines of the
THEH formalism.
It is useful to define the gravitational potential U whose gradient
yields the test-body acceleration g; modulo a constant
U(x)= ~^T'0Ho
x
g0-x
(2.82)
If the functions T, H, e, and fi are now considered as functions of U
instead of <D, then because of Equations (2.48), (2.54), (2.58), and (2.82),
T o = -c20(
]|x=0
(2.83)
(c) Energy conservation and anomalous mass tensors
Because the THefi formalism is based on an action principle, it
possesses conservation laws, in particular a conservation law for energy,
and so is amenable to analysis using the conserved-energy framework
described in Section 2.5. The main products of that framework are the
anomalous inertial mass tensor Sm[J and passive gravitational mass tensor
Sm'j obtained from the conserved energy. These two quantities then yield
expressions for violations of WEP, Local Lorentz Invariance, and Local
Position Invariance.
As a concrete example (Haugan, 1979), we consider a classical bound
system of two charged particles. As in the above "proof" of Schiff's conjecture we work to post-Coulombian order and to first order in g 0 x. We
first formulate the equations of motion in terms of a truncated action
?NG = h + An,
(2-84)
Theory and Experiment in Gravitational Physics
54
where 7im is rewritten entirely in terms of particle coordinates by substituting the post-Coulombian solutions for A^, Equation (2.73), into I int .
Variation of 7NG with respect to particle coordinates then yields the complete particle equations of motion. We identify a Lagrangian L using the
definition
7NG = $Ldt (2.85) We next make a change of variables in L from xu x2, vl5 and v2, to the center of mass and relative variables X = ( m ^ + m2x2)/m, x = xt x2, V = dX/dt, v = dx/dt (2.86) where m = mx + m2, n = mlm2/m. A Hamiltonian H is constructed from L using the standard technique Pj s dL/dV{ pi = BL/dv3, H s PJVj + pV - L (2.87) The result is H = Tj'2m(l + iTiTo »g0 X) + n'2HE lP2/2m eie2/r)go X - Tj/2Ho 2[(p P)2 o \e,e2lr)\P2 + (n P)2]/2m2} + hT0li0Ho \m2 - m1)(e1e2/r)(p P + ii pfi + O(p4) + O(P4) (2.88) 4 4 where n = x/r. The post-Coulombian terms O(p ) and O(P) neglected in Equation (2.88) do not couple the internal motion and the center-of-mass motion and thus do not lead to violations of EEP. We now average H over several timescales for the internal motions of the bound two-body system, assumed short compared to the timescale for the center-of-mass motion. The average is simplified using virial theorems obtained from Hamilton's equations for the internal variables derived from H. The relevant expressions are + post-Coulombian terms), H j n (n p)]> (2.89) (2.90) Einstein Equivalence Principle and Gravitation Theory 55 Notice that although the post-Coulombian corrections in Equation (2.89) may depend on the center of mass variables P or X, this dependence does not affect the form of if; it is only the explicit dependence on P and X in Equation (2.88) that generates the center-of-mass motion. The resulting average Hamiltonian is then rewritten in terms of V using VJ = d<H>/dPJ. The conserved energy function Ec used in Section 2.5 is then defined to be Ec = Tll2Ho\H}, so that at lowest order, Ec = mc\ = m(ToHo *). The result is Ec = M(T0Ho ') + i x (£«% )] |[ ^^o} xT'oHo'go-X (2.91) where M = m + <tfo V / t y + To-1/2£0- ^ W ) (2-92) By defining the "binding" energy and energy tensor by Ef = -c o 2 {i/ o -y/2/i + To 1 ES E & 1/2 eo'e^/r) + [post-Coulombian corrections], l Ef? (2.93) and using Equation (2.82), we cast Equation (2.91) into the form £ c = MRC2, - MRt/(X) + \MKV2 (2.94) MRC2. = mcl - Ef + tymiWVi - dm\!Ui} (2.95) dm? = 2(1 - Toffo'eoMoX^w + £^)/cS. j/ i s i)<5 ij ' (2.96) where with Substitution of these formulae into Equation (2.35) for the center-of-mass acceleration of the system yields precisely Equation (2.78). One advantage of the Hamiltonian approach is that it can also be applied to quantum systems (Will, 1974c). This is especially useful in discussing gravitational red-shift experiments since it is transitions between quantized energy levels that produce the photons whose red shifts are measured. For the idealized gravitational red shift experiments discussed in Section 2.5, only the anomalous passive mass tensor 5m^ is needed. The simplest quantum system of interest is that of a charged Theory and Experiment in Gravitational Physics 56 particle (electron) moving in a given external electromagnetic potential of a charged particle (proton) at rest in the SSS field, i.e., a hydrogen atom. For such a system the truncated Lagrangian [Equation (2.85)] has the form L = - me(T0 - Hov2)112 - eA^if (2.97) where m0 = me and e + \e\ for the electron. We shall ignore the spatial variation of T, H, e, and \i across the atom, hence we evaluate each at x = 0. The Hamiltonian obtained from L is given by H = n / 2 [m e 2 + Ho > + eA| 2 ] 1/2 + eA0 (2.98) where Pj = dL/dvK Introducing the Dirac matrices where / is the two-dimensional unit matrix and <rt are the constant Pauli spin matrices, we perform the "square root" in H and obtain + Ho 1/2 a (p + <?A)] + eAol H = Ti^lmJ (2.100) The gravitationally modified Dirac (GMD) equation is then H\\l>y = ih(8/dt)\il/y (2.101) For most applications it is more convenient to use the semirelativistic approximation to if obtained by means of a Foldy-Wouthuysen transformation, yielding H = Ty\me + Ho J |p + eA\2/2me - Ho 2p*/Sm2 + HQ leho B/2m,] + eA0-HQ l {eh/4m2)a - ( E x p - % i h \ xE) (2.102) where we have made the usual identification p -» ih\ and have ignored the effects of spatial variations in T0,H0,s0, or fi0 on the atomic structure. For a charged particle with magnetic moment M p at rest at the origin, the vector potential as obtained from the GMM equations is given (to the necessary accuracy) by Ao = -e/sor, A = iu 0 M p x x/r 3 (2.103) The Hamiltonian then takes the form H = Hr + Hs + H( + HM + O(p6) (2.104) Einstein Equivalence Principle and Gravitation Theory 57 where Hr = Tl'2me, H(=-Tlo'2Ho2p4/^m2 - Hvh^{e2hl4m2er3)o Hu = T^ 2 [Ho l(ehl2me)o B] L, (2.105) where L = r x p is the angular momentum of the electron. The four pieces of H are the usual rest mass (Hr), Schrodinger (Hs), fine-structure (H{) and hyperfine-structure (Hhf) contributions. We have ignored the Darwin term (oc V E). The magnetic field produced by the proton is given by B = V x A = - i ^ o { [ M p - 3n(fi M p )]/r 3 - (87t/3)Mp<53(x)} (2.106) We must first identify the proton magnetic moment. From the hyperfine term Hhl, it is clear that the magnetic moment of the electron is given by M e = T£ /2 #o H - eh/2me)o (2.107) It is then reasonable to assume that the magnetic moment of the proton has the same dependence on T o and Ho, Mp = T^Ho l (gpeh/2mjap (2.108) where gp is the gyromagnetic ratio of the proton and mp is its mass. Then Hbf = -yoToHo2(gpe2h2/4memp) x ae {|>p - 3fi(il ffp)]/r3 - (8rt/3)«Tp^3(x)} (2.109) Solving for the eigenstates of the Hamiltonian using perturbation theory yields £ = Ty\me + £p(HoTolEo2) + *AH0TZ h^ 2 ) 2 l (2.H0) where ip, Su and SM are the usual expressions for the principal, finestructure, and hyperfine-structure energy levels in terms of atomic constants me, e, mp, gp, h, and quantum numbers. In order to calculate the anomalous mass tensors <5mj/, we must determine the manner in which E varies as the location of the atom is changed. Expanding E to first order in g X, substituting Equation (2.82), and converting to the conserved energy function Ee - E(Tk'2/H0), we obtain Equation (2.30) (with V = 0), with E% = Ef + El-¥EW (2.111) Theory and Experiment in Gravitational Physics 58 where £ B e 0 <5p, F £ B ttoio 3 £g =-Wo 'hf e0 0(, (2-112) and H* 2r o (£| s /cg)^ ii (2.113) = 4ro{El/cl)Sij (2.114) 2 = (3r 0 - A0)(E^/c )8^ (2.115) Compare Equation (2.113) with Equation (2.96). A useful fact that emerges from the solution for the energy eigenstates is that the Bohr radius is given by a = (e0Ti'2/H0){h2/mee2) (2.116) This will be important in analyzing the gravitational red shift of microwave cavities. (d) Limitations of the THefi formalism The THefi formalism is a very strong - perhaps overly strong idealization of the coupling of electromagnetism to gravity. The question naturally arises, can the formalism be applied to realistic physical situations where there are no SSSfieldsand where strong and weak interactions may be present? We shall discuss each of these points in turn. (i) SSS Fields In practical experimental situations, say in an Earthbound laboratory, there are, strictly speaking, no SSS fields: orbital and rotational motions of the planets cause the gravitational potentials to change with time, and the superposition offieldsfrom the Sun and planets leads to asphericalfields.However, the evolution of the gravitational fields occurs on a much longer timescale than the internal (atomic) timescales of typical laboratory experiments, and so the fields can be treated quasistatically. Furthermore, most experiments of interest single out one static, nearly spherical gravitational field by exploiting a symmetry, by modulation, or by some other technique. (For example, singling out of the solar field by searching for a torque with a 24 h period in the DickeBraginsky versions of the Eotvds experiment.) A potentially more serious criticism of the SSS restriction is the possibility of relativistic, nonisotropic effects due for example to the orbital motion of the planets, or to the motion of the solar system relative to the mean rest frame of the universe. These ef- Einstein Equivalence Principle and Gravitation Theory 59 fects would produce off-diagonal terms in the action / NG , such as F v in Jo or G (E x B) in / em , where F and G are vector gravitational functions. In the case of the overall motion of the solar system, one can see that the frame in which the solar potential is spherical is in motion relative to the frame in which the cosmic background field is spherical (isotropic), therefore there must be two limiting actions of the THe/i form, one applicable to each situation. These limiting cases can be handled by a single action of the THefi form only if the theory is Lorentz invariant, i.e., only if TH~1e/i = 1. Nevertheless, if either of these off-diagonal effects occurs, they will be smaller than the dominant SSS effects by factors of order |v| ~ [orbital velocity of planets] ~ 10 ~4 or |w| ~ [solar system velocity] ~ 10"3. The simplest way to summarize is as follows: the restriction to SSSfieldsis an approximation that may overlook observable effects, however, the experimental consequences that emerge from the pure SSS version are sufficiently interesting and, we believe, sufficiently generic to a broad class of gravitational theories, that powerful conclusions about the nature of gravity can be made within the standard THs/i framework. With this caveat in mind, for most of the remainder of this chapter we will assume that every experiment discussed takes place in a SSS field. (ii) Weak and strong interactions The coupling of classical electromagnetic fields to gravitation is well understood within metric theories of gravity (see Section 3.2) and has been formulated in many nonmetric theories. By contrast, the laws of weak and strong interactions have only recently been given an adequate mathematical representation even in the absence of gravity, and the problem of their coupling to gravity is made even more complicated by the fact that the theories of these interactions fundamentally involve quantum field theory. Thus, at present, electromagnetism is the only interaction amenable to a detailed analysis of EEP using something like the THe/j. formalism. Nevertheless, a violation of EEP by electrodynamics alone can lead to many observable effects, barring fortuitous cancellations, and to several important experimental tests. Consequently, for the remainder of this discussion we shall simply ignore the strong and weak interactions, or if necessary assume that they obey EEP. (e) Application to tests of EEP We now turn to the experiments that test EEP and study the constraints they place on the coupling of electromagnetism to gravity in SSS gravitational fields. Theory and Experiment in Gravitational Physics 60 Tests of WEP Equation (2.78) gives the acceleration of a composite body through post-Coulombian order in an external SSS field. However, for the purpose of comparing the predicted acceleration with the results of Eotvos experiments, that expression is not accurate enough. The WEP-violating terms in Equation (2.78) are of order EBS/m ~ 10" 3 for atomic nuclei; therefore, WEP-violating terms of order (EES/m)2 ~ {EES/m)v2 ~ 10 ~6 would also be strongly tested by Eotvos experiments accurate to a part in 1012. To obtain these terms, Haugan and Will (1977) extended the Lightman-Lee computation to post-post-Coulombian order (the Hamiltonian method could also have been used). When specialized to composite bodies that are spherical on average (a good approximation for experimental situations), the resulting acceleration is given by d2X/dt2 = g{l + (Efs/Mc2)[2r0 - f(l - ToH (2.117) where [cf. Equations (2.77), (2.80), and (2.93)] Ef = ab o Vo ( l eaebr;b\va yb + (vfl ab 2 V Vo ( abI W a V [ v a yb- (ya *ab)(yb x j r i ] ) (2.118) \ I Because we shall shortly obtain a very tight upper limit on the coefficient 1 T0HQ 1e0^i0 from the Hughes-Drever experiment, we shall simply set it equal to zero in Equation (2.117). Then the results for the Eotvos ratios defined in Equation (2.2) are ^ES = |2T0|, r, = |2A0| (2.119) The quantities E| S an< l £ B S given by Equation (2.118) were estimated for various substances in Section 2.4 [Equations (2.8) and (2.9)] and provided experimental limits on nES and nm that are equivalent to |r o | < 2 x 10- 10 , |A0| < 3 x 10- 6 Recall that if EEP is satisfied, r 0 = Ao = 0. (2.120) Einstein Equivalence Principle and Gravitation Theory 61 Tests of LLI The Hughes-Drever experiment can now be analyzed in detail using the TH&n formalism (Haugan, 1978). Equations (2.95) and (2.96) demonstrate the possibility of an inertial mass anisotropy 3m\j that leads to a contribution to the binding energy given by SEB = -$8m\'ViVj
(2.121)
where V is the velocity of the body relative to the THefi coordinate
system. This term could lead to energy shifts of states having different
values of 5m\j and thus to observable effects in a quantum mechanical
transition between these states. In the case of the Hughes-Drever experiment, the system, a 7Li nucleus, can be approximated as a two-body
system consisting of a J = 0 core (two protons and four neutrons) of charge
+ 2, and a valence proton in a ground state with angular momentum of
1. The spin of the proton couples to its angular momentum to yield a
total angular momentum J = f. In an applied magnetic field, the four
magnetic substates Af, = ±j, + § are split equally in energy, giving a
singlet emission line for transitions between the three pairs of states. How
does SEB alter the energies of these four states? The isotropic part of
dm\J oc EBsd'J simply shifts all four levels equally, since < JMi\e1e2r~ l\ JM3}
is independent of Mj. However, the other contribution to 8m\J oc £{[? does
shift the levels unequally. We first decompose V into a component V^
parallel to the applied magnetic field and a component V± perpendicular
to it. Then
0]
(2.122)
where 0, (j> are polar coordinates appropriate to the orbital wave function
Aim, =/('")5inii(0>0)- By combining the orbital wave function and spin
states into states of total J, Ms, we then calculate the expectation value
of (x'xJ/r3)ViVJ in states of different M,. Inserting these results into the
formula, (2.121), for 8EB, and taking the difference in the energy shifts
between adjacent Af, states, we find that the singlet line splits into a triplet with relative energies
Mi - -i) = o,
M-i--!)=-<5
(2-123)
S = &(£f/cg)(l - ToHo lHH0){Vl - 2VD
(2.124)
where
Theory and Experiment in Gravitational Physics
62
In the notation of Section 2.4, Equation (2.13), we have
<5ES = £ ( 1 - ToHo 'eolhKVl - 2F(j)
(2.125)
The limit set by the experiment was |<5ES| < 10~ 22 . If we treat the laboratory
as being in motion in the SSS field of the Sun, then V^ ~ VL ~ 10" 4 ;
hence, as evaluated at the Earth,
|1 - ToHo'Bo/ioU = I1 - (Co/clighl)2U < 10" 1 3
(2.126)
We can also assume the laboratory to be moving in the quasistatic, spherically symmetric background field of the universe, with velocity V^ ~
VL ~ 10" 3 , then for that portion of the THEH fields associated with the
asymptotic cosmological model (labeled by the subscript oo), we obtain
Il-TVO^^HT 15
(2.127)
Although there may be observable effects due to the possible nonmeshing
of these two SSS fields into a single THefi field, they are unlikely to cancel
the effects we have derived and negate the limits obtained above.
The central conclusion is that to within at least a part in 1013, Local
Lorentz Invariance is valid.
Tests of Local Position Invariance Consider gravitational
red-shift experiments. Suppose, for example, one measures the gravitational red shift of photons emitted from various transitions of hydrogen,
such as principal transitions, fine-structure transitions within a principal
level, or a hyperfine transition in, say, the ground state (21 cm line, basis
for hydrogen maser clocks). Then, substituting Equations (2.113)(2.115)
into Equation (2.37), we obtain (Will, 1974c)
Zp=[l-2r0]AC//c2,
Zf=[l-4r0]Al//c&
Z hf = [1 - (3r 0 - Ao)] AU/c2.
(2.128)
Notice that the three shifts are different in general. Thus the gravitational
red shift depends on the nature of the clock whose frequency shift is
being measured unless F o = Ao = 0, i.e., unless LPI is satisfied [Equation
(2.59)]. The red-shift parameters a discussed in Section 2.4(c) can thus be
read off from Equations (2.128). The Vessot-Levine Rocket red-shift experiment thus sets the limit
|(3r 0 - Ao)| < 2 x KT 4
(2.129)
Einstein Equivalence Principle and Gravitation Theory
63
To analyze the Stanford null gravitational red-shift experiment, we must
calculate the energy of a microwave cavity. The energy in question is that
of an electromagnetic mode whose wavelength is determined by the
length of the cavity. The vector potential for the mode can be written, in
second quantized notation,
A = N(a t eexp[i(k x - cot)] + h.c.)
(2.130)
+
where a is a creation operator, e a polarization vector, N a normalization constant, and h.c. denotes Hermitian conjugate. We have suppressed
the sum over k and e. The GMM equations (2.67) yield the dispersion
relation
|k| 2 -
E0H0CO2
=0
(2.131)
The energy of the electromagnetic field obtained from the canonical
Hamiltonian is
E = %(aa< + a^a)hco
(2.132)
However, for a stationary mode, the wave number k must satisfy
k L = nn
(2.133)
where |L| is the length of the cavity and n is an integer. But it is clear that
|L| is proportional to an integer (number of atoms in a line along the
length of the cavity) times the Bohr radius a (which determines the interatomic spacing). But from Equation (2.116) we find L cc(eoTo/2Ho l) x
(atomic constants, integers), hence, |k| cc H0TQ 1/2SQ 1. Combining Equations (2.131) and (2.132), we finally obtain
E = ^csoHoV/Vo^eo3/2
(2-134)
where <^SGSO depends only on atomic constants and integers. Expanding
in terms of g0 x, and calculating the conserved energy function Ec, we
obtain Equation (2.30) with
pSCSO _ _ v
C
B
0
-1/2.-3/2
scsoA t o
fc
o
>
6n#= i(3r 0 + A o )(EF°/co)* y
(2.135)
Thus for a superconducting-cavity stabilized oscillator clock
Zscso = [1 - l(3r 0 + Ao)] AL//c2
(2.136)
or, in the comparison between a cavity clock and a hydrogen maser
clock [see Equation (2.31)]
+ f (r 0 - A 0 )t//c 2 ]
(2.137)
Theory and Experiment in Gravitational Physics
64
The experimental limit is thus
|r o -A o |<l(T 2
(2.138)
(f) The Belinfante-Swihart nonmetric theory
As a specific example of the application of the THz\i formalism
to the analysis of gravitational theory and experiment we consider the
Belinfante-Swihart (1957a,b,c) theory. This theory treats gravity as a
symmetric second rank tensor field B on a Riemann-fiat background
metric (prior geometry), t\. We first define a "particle metric" g^ according
to
( 2 - 139 )
H ~ &,) = $where K is an arbitrary constant, and where indices on B^ and A^v are raised and lowered using n^. In a coordinate system in which t\ = diag( 1, 1,1,1), the nongravitational action can be put into the form (Lee and Lightman, 1973) r( dx11 dxv\1/2 /NO = - 1 «o. J ( - 0,v -£-ft) c dt + ^ea JA.ixl) dx" - (2.140) where, through second order in B, H^ is related to the Maxwell field H, v = FMV(1 + i B + i^ 2 ) + 2FAUBJ,(1 + B) - 2Fx(MBt}Bl - 2Fi.B£,B;, + O(Ffi3) (2.141) It turns out that, to first order in B, the electromagnetic part of the action can be put into metric form (see Section 3.2 for discussion of this form), but not to second and higher orders. The particle and interaction parts of / N G are already in metric form. The action for the gravitational field IG = -(167T)-1 §(aB»JB& + fB<aB'*)(-rj)ll2d*x (2.142) where a and / are arbitrary constants. In the weak field, post-Newtonian limit appropriate for application to solar system experiments (see Chapter 5), the theory can be made to agree with all experiments performed to date. Thus, the theory was thought Einstein Equivalence Principle and Gravitation Theory 65 to be a completely viable alternative to general relativity. However, because of the deviations from metric form in the electromagnetic action, the theory violates EEP. We therefore expect it to violate WEP, although at second order in B^. To demonstrate that this is indeed the case, we first compute B^ for a SSS field, then recast /NG into THefi form. The solution of the gravitational field equations (Section 5.5) yields the form B oo = b0, Bij & b&j (2.143) where b0 and b t are functions of a gravitational potential U. Then, from Equations (2.143) and (2.139), we find to O(b2), 0Oo = -(l-bo-2Kb + fb2 + 2Kbb0 + K2b2), 2Kb + Jfef - 2Kbb1 + K2b2) g..= ^.(l + bt- (2.144) where b = b0 + 3b ^ We have assumed for simplicity that far from the gravitating source, b0 and b^ vanish (see Section 5.5 for discussion). Substituting Equations (2.144) and (2.141) into Equation (2.140) puts / NG into THe/j. form to O(b2), with T = 1 - b0 - 2Kb + Ibl + 2Kbb0 + K2b2, + K2b2, / * = [ ! + i(*o + *»i)] (2-145) In the weak-field limit, it turns out that the SSS solutions for b0 and fcx have the form (see Section 5.5) bo = 2CoU, b1 = 2C1U (2.146) where U is the Newtonian gravitational potential and Co and Ct are arbitrary constants. Then T = 1 - 21/ + 2t/ 2 [i + C o ] + O(l/ 3 ), H = 1 + 2U[C0 + d - 1] + C/2[(C! + C0)(3C1 + Co) - 4CX - 2C 0 + 1] + O(t/ 3 ), £ = 1 + U(C0 + d ) + U2(C0 + Cx)2 + O(t/ 3 ), H = 1 + U(C0 + C t ) + O(t/ 3 ) where we have chosen the values of C o , d , Co + 2K(3Cl - Co) = 1 an< i^ (2.147) sucn tnat (2.148) in order to ensure that T = 1 1U + .... This will guarantee that the particle Lagrangian will yield the correct Newtonian limit. Notice that, Theory and Experiment in Gravitational Physics 66 to first order in U, these functions satisfy the EEP constraint in Equation (2.63), but to second order they do not in general. Now in solar system tests of post-Newtonian effects, where the consequences of electromagnetic violations of EEP are negligible, the coefficients (^ + Co) and (Co + C t 1) in T and H are simply the PPN parameters /? and y (see Chapter 4). Solar system measurements of light deflection, radar-time delay, and the perihelion shift of Mercury (see Chapter 7) constrain these parameters by |2C0 + 4C t - 7| < 0.1, \C0 + d - 2| < 0.002 (2.149) Equations (2.83) and (2.147) then yield r0 = -2c o (c o + cx)u + o(t/2), Ao = 2C1(C0 -I- d)U + O(U2) (2.150) Using the above constraints on Co and C t along with the value U = [/Q s 10"8, the relevant local potential for the Princeton-Moscow Eotvos experiments, we obtain |r o | ^ 1.7 x 10- 8 (2.151) which violates the experimental limit, Equation (2.120), by a factor 80. Thus, the Belinfante-Swihart theory is unviable. Gravitation as a Geometric Phenomenon The overwhelming empirical evidence supporting the Einstein Equivalence Principle, discussed in the previous chapter, has convinced many theorists that only metric theories of gravity have a hope of being completely viable. Even the most carefully formulated nonmetric theory - the Belinfante-Swihart theory - was found to be in conflict with the Moscow Eotvos experiment. Therefore, here, and for the remainder of this book, we shall turn our attention exclusively to metric theories of gravity. In Section 3.1, we review the concept of universal coupling, first defined in Section 2.5. Armed with EEP and universal coupling, we then develop, in Section 3.2, the mathematical equations that describe the behavior of matter and nongravitational fields in curved spacetime. Every metric theory of gravity possesses these equations. Metric theories of gravity differ from each other in the number and type of additional gravitational fields they introduce and in the field equations that determine their structure and evolution; nevertheless, the onlyfieldthat couples directly to matter is the metric itself. In Section 3.3, we discuss general features of metric theories of gravity, and present an additional principle, the Strong Equivalence Principle that is useful for classifying theories and for analyzing experiments. 3.1 Universal Coupling The validity of the Einstein Equivalence Principle requires that every nongravitational field or particle should couple to the same symmetric, second rank tensorfieldof signature 2. In Section 2.3, we denoted this field g, and saw that it was the central element in the postulates of metric theories of gravity: (i) there exists a metric g, (ii) test bodies follow geodesies of g, and (iii) in local Lorentz frames, the nongravitational laws of physics are those of special relativity. Theory and Experiment in Gravitational Physics 68 The property that all nongravitational fields should couple in the same manner to a single gravitational field is sometimes called "universal coupling" (see Section 2.5). Because of it, one can discuss the metric g as a property of spacetime itself rather than as a field over spacetime. This is because its properties may be measured and studied using a variety of different experimental devices, composed of different nongravitational fields and particles, and, because of universal coupling, the results will be independent of the device. Consider, as a simple example, the proper time between two events as measured by two different clocks. To be specific, imagine a Hydrogen maser clock and a SCSO clock at rest in a static spherically symmetric gravitational field. If each clock is governed by a Hamiltonian H, then the proper time (number of clock "ticks") between two events separated by coordinate time dt is given by where E is the eigenstate energy of the Hamiltonian (or energy difference, for a transition). The results of Section 2.6 show that if, for instance, the THefi formalism is applicable, and if EEP is satisfied, e0 = /x0 = (Ho/To)112 everywhere, thus using Equations (2.110) and (2.134) we obtain for each clock JVH oc dt(H0To oc dt (H0TZ m Mo 1/2 6o 3/2 ) = TV2 dt where the proportionality constants are fixed by calibrating each clock against a standard clock far from gravitating matter. Thus, each clock measures the same quantity T o (in metric theories of gravity, in SSS fields, To gOo) a n d the proper time between two events is a characteristic of spacetime and of the location of the events, not of the clocks used to measure them. Consequently, if EEP is valid, the nongravitational laws of physics may be formulated by taking their special relativistic forms in terms of the Minkowski metric r\ and simply "going over" to new forms in terms of the curved spacetime metric g, using the mathematics of differential geometry. The details of this "going over" are the subject of the next section. 3.2 Nongravitational Physics in Curved Spacetime In local Lorentz frames, the nongravitational laws of physics are those of special relativity. For point, charged, test particles coupled to electromagnetic fields, for example, these laws may be derived from the Gravitation as a Geometric Phenomenon 69 action a -(167T)- 1 U A V^F A v -F^(-f?) 1 / 2 d 4 x (3.1) where F .= A A fj = det||^,|| (3.2) Here, n^ is the Minkowski metric, which in Cartesian coordinates has the form In the local Lorentz frame, t}^ is assumed to have this form only up to corrections of order [x s x s (^)] 2 , where x s (^) is the coordinate of a chosen fiducial event in the local frame, in other words, r\^ is described more precisely as -1,1,1,1), According to the discussion in Section 3.1, the general form of these laws in any frame is obtained by a simple coordinate transformation from the freely falling frame to the chosen frame. This transformation is given by (3.4) Then, the vectors and tensors that appear in JNG transform according to n-^ = (dx«/dx»)(dxll/dxi)rlxlh dx* = {dx*/dx*)dx*, where J is the Jacobian of the transformation. Partial derivatives of fields, as for example in the formula for F^, transform according to a '" + dx* dx* * ( } However, in the local frame, n^^ = 0. Thus, d2x* _dx*_dif_dx>_ n - ^ * - ^ ? a ? &? *"- y+ 82xp dx» a* d & ex 1 n< + * dx* & dx* dx dx*n" { ' Theory and Experiment in Gravitational Physics 70 Using the fact that (dxydx^idx^/dx*) = <5f (3.8) we obtain n y ^' dx^dx* d2xs dx* dx° d2xd n ~ ~ fa* ~W dx^dx^ »~fa7~fa7 "dx^dx1* n<* { ' If we now define 9*e = 1ae, (3-10) M»IUl--gg-r. (in) then Equation (3.9) can be written or, using Equations (3.11), (3.12), and (3.13), the Christoffel symbols F^y (also known as connection coefficients) take the form % P,y y,D - gyfi,s) (3.14) Then Equation (3.6) becomes dx" dx0 ^^faJdx1^11'1^^ (115) We define the covariant derivative ";" by Ax;P = A^ - n,A, (3.16) and notice that it transforms as a tensor; it can be shown that Atf^A'f + VnA1 (3.17) where A* = gafiAp. Taking the determinant of Equation (3.5) yields n = [det(Sx7dx*)]2g (3.18) where g = det g^, then 1 / 2 y i 2 (3.19) Gravitation as a Geometric Phenomenon 71 Substituting these results into / NG gives JNG _ r( dx" dxv\112 . = ~ L mOa J I -g^ I$
(
3
r,
Ju
A + 2, eB Jf iA^dx"
.
2
0
)
where
We notice that the transformation to an arbitrary frame has resulted
simply in the replacements
%v by #
"comma" by "semicolon"
(-f7)1/2d4x by (-gY'Wx
(3.22)
This is the mathematical manifestation of EEP. We must point out that
the specific mathematical forms given above for the Christoffel symbols,
transformation laws, and so on are valid only in coordinate bases (see
MTW, Chapter 10 for further discussion). However, in this book we shall
work exclusively in coordinate bases.
Generally speaking, then, the procedure for implementing EEP is: put
the local special relativistic laws into a frame-invariant form using Lorentzinvariant scalars, vectors, tensors, etc., then make the above replacements.
It is simple to show that the same rules apply to the field equations and
equations of motion derived from the Lagrangian. In the local frame they
are
(3.23)
where
dx = ( - f,p. dx" dx")112,
if = dx*ld%,
rfr (3.24)
However, these are not in frame-invariant form. We must write
=
MvU/iip,
(3.25)
Theory and Experiment in Gravitational Physics
72
where we have used the fact that for the four-dimensional delta function
(fj)~ll2SA is invariant (since J<S4d4x = 1 or 0 regardless of the frame).
Then in the general frame the equations are
(3.26)
Ffvv
(3.27)
= 47r./"
where
fa = (-9^ dx" dxv)112,
DuJDx = u"umv,
/" = I ea(-g)-V25\x
u" = dx"/dt,
- xa)dx»/dt
(3.28)
a
However, here there is a potential ambiguity in the application of EEP
to electrodynamics if one writes Maxwell's equations, (3.27), in terms of
the vector potential A^. In the local Lorentz frame, Maxwell's equations
have the special relativistic form
Ay~A»-»,v=
-47tJ"
(3.29)
It is always possible to choose a gauge (Lorentz gauge) in which A" = 0,
thus, since AV'")V = A"^11, we have
= A»'\v = -47tJ*
(3.30)
It is tempting then to apply the rules of EEP to this equation to obtain
ngA" s A":v.v = <TM?vA = -4nJ»,
A% = 0
(3.31)
However, there is another alternative. The curved-spacetime Maxwell
equation, Equation (3.27), yields
(3.32)
But covariant derivatives of vectors and tensors do not commute in
curved spacetime, in fact in general
A?* = Ke + *UAV
(3-33)
where R%ap is the Riemann curvature tensor, given by
*U = rf^. - r^
+ r ^ r j , - rj^rj.
(3.34)
Then
Ar-».v = A " , * + R^A",
(3.35)
Gravitation as a Geometric Phenomenon
73
where R% is the Ricci tensor given by
Ri = g**Ry»
Ryf = R-U
(3.36)
This version of Maxwell's equations in Lorentz gauge becomes
n0A»-R$Al>= -4nJ", A], = 0 (3.37) It is generally agreed that this second version is correct (although there is no experimental evidence one way or the other). To resolve such ambiguities, the following rule of thumb should be applied: the simple replacements (i; -* g, comma -* semicolon) should be used without curvature terms in equations involving physically measurable quantities (F"v is physically measurable, A* is not); and coupling to curvature should occur only with good physical reason (as in tidal coupling). (For a fuller discussion, see MTW, box 16.1.) An uncharged test body follows a trajectory given by Equation (3.26) with e = 0, namely Du^/Dx = 0. This equation can be written using Equations (3.17) and (3.28) in the form d V / d r 2 + r^(dxx/dr)(dxp/dz) = 0 (3.38) This is the geodesic equation. The mathematics of measurements made by atomic clocks and rigid measuring rods follow the same rules since the structure of such measuring devices is governed by solutions of the nongravitational laws of physics. In special relativity, the proper time between two events separated by an infinitesimal coordinate displacement dx", as measured by any atomic clock moving on a trajectory that connects the events, is given by dT = (_^ v dx"dx v ) 1/2 (3.39) v if the separation is timelike, i.e., »/ dx" dx < 0. The proper distance between two events as measured by a rigid rod joining them is given by ds = (r,liydx»dxv)112 (3.40) if the separation is spacelike, i.e., n^ dx* dxv > 0. These results are independent of the coordinates used. Then in curved spacetime we have [timelike] «> g^dx"dxv < 0, [spacelike] «» Sllv dx" dx" > 0 (3.41) Theory and Experiment in Gravitational Physics 74 There is a third class of separation dx* between events, those for which rjllvdx"dxv = 0 (3.42) These are called null or lightlike separations, and pairs of events that satisfy this condition are connectible by light rays. It is a tenet of special relativity that light rays move along straight, null trajectories, i.e., if k" = dx"/da is a tangent vector to a light-ray trajectory, then dW/d<r = 0, iffc'ifc,= 0 (3.43) where a is a parameter labeling points along the trajectory. It should not be forgotten, however, that this is at bottom a consequence of Maxwell's equations, valid only in the "geometrical optics" limit, in which the characteristic wavelength X [a^fc 0 )" 1 ] is small compared to the scale £P over which the amplitude of the wave changes. (For example, if might be the radius of curvature of a spherical wavefront.) Since the first of equations (3.43) can be written, in flat spacetime dkf/do = {dx*/da)k*y = kvk% = 0 (3.44) then EEP yields the equations /cv/c?v = 0, fe"/cv^v = 0 (3.45) i.e., the trajectories of light rays in the geometrical optics limit are null geodesies. It is useful to derive this result directly from the curved-spacetime form of Maxwell's equations, in order to illustrate the role and the limits of validity of the geometrical-optics assumption. In curved spacetime, the geometrical-optics limit requires that X be small compared both to ££ and to ffl, the scale over which the background geometry changes {01 is related to the Riemann curvature tensor), i.e., A/(min{&, Si}) = 1/L « 1 (3.46) In this limit, the electromagnetic vector potential can be written in terms of a rapidly varying real phase and a slowly varying complex amplitude in the form (see MTW, Section 22.5 for details) K = (a, + <* + y / £ (3.47) where 6 is the real phase, a,,, b^,... are complex, and e is a formal expansion parameter that keeps track of the powers oiXjL. Ultimately, one takes only the real part of A^ in any physical calculations. We define the Gravitation as a Geometric Phenomenon 75 wave vector K = e,v> k " = /V0 v (3-48) Then Maxwell's equations in Lorentz gauge [Equations (3.37)], yield 0 = A% = [(i/s)kv(av + sbv) + a]v + 0(e)]ei9'E, 0 = DgA" - R$A'
= [ - s - 2Jfc,fcV +fib")+ 2(i/e)fc'aJ + (i/e)fef^a" + 0(e °j]emie
(3.49)
Setting the coefficients of each power of e equal to zero, we obtain for the
fa, = 0,
(3.50)
k% = 0,
(3.51)
in other words, the amplitude is orthogonal to the wave vector, and the
wave vector is null. Taking the gradient of Equation (3.51) and noting
that &. = kv;il since /cM itself is a gradient, we get
fcyc" = 0
(3.52)
which is the geodesic equation for k". The trajectory x^a) of the ray can
then be shown to be related to k" by the differential equation
dx"(a)/da =
fe"(xv)
(3.53)
where a is an affine parameter along the ray. For further discussion of the
higher-order terms in Equation (3.49), see MTW, Section 22.5.
Another useful and important form of the equations of motion for
matter and nongravitational fields can be derived in the case where the
equations are obtained from a covariant action principle. This will essentially always be the case, for the following reason: in special relativity,
all modern viable theories of nongravitationalfieldsand their interactions
take an action principle as their starting point, leading to an action / NG .
The use of EEP does not alter the fact that the equations of motion are
derivable from an action. Consequently, one is led in curved spacetime
to an action of the general form
=
J
( 3 - 54 )
where qA and qAifl are the nongravitational fields under consideration and
their first partial derivatives (e.g., M", A^, A^,...)
and g^ and g^^ are
Theory and Experiment in Gravitational Physics
76
the metric and its first derivative. (The extension to second and higher
derivatives is straightforward). The action principle <5/NG = 0 is covariant,
thus, under a coordinate transformation, i? N G must be unchanged in functional form, modulo a divergence [see Trautman (1962) for discussion].
Consider the infinitesimal coordinate transformation
x* -> x" + <5x",
<5x" = £"
(3.55)
Then the metric changes according to [cf. Equation (3.5)],
a
= - g^% - gvx^ - g^J"
(3.56)
Assume the matter and nongravitational field variables change according
to
. - «A.*£V
<5<7A = «
(3-57)
where d^v are functions of x". Under this transformation, JSfNG changes by
S 9
^
(3 58)
-
Substituting Equations (3.56) and (3.57), integrating by parts, dropping
divergence terms, and demanding that JS?NG be unchanged for arbitrary
functions £*, yields the "Bianchi identities"
w
J
l
=0
(3.59)
where S£CNO/3qx is the "variational" derivative of i ? N G defined, for any
variable \j/, by
dx" V # /
and T*" is the "stress-energy tensor," defined by
T^^2(-g)-^8^NG/dg,v
(3.61)
Using the fact that
(-ff).V2 = ( - 0 ) 1 / 2 n .
we can rewrite Equation (3.59) in the form
(3.62)
Gravitation as a Geometric Phenomenon
11
However, the nongravitational field equations and equations of motion
are obtained by setting the variational derivative of J£NG with respect to
each field variable qA equal to zero, i.e.,
.= 0
(3.64)
which by Equation (3.63) is equivalent to
Tl,y = 0
(3.65)
Thus, the vanishing of the divergence of the stress-energy tensor T"v is a
consequence of the nongravitational equations of motion. This result
could also have been derived, first working with i ? N G in flat spacetime,
to obtain the equation T)j>v = 0 by the above method, then using EEP
to obtain Equation (3.65). Notice that Equation (3.65) is a consequence
purely of universal coupling (EEP) and of the invariance of the nongravitational action, and is valid independent of the field equations for
the gravitational fields.
The stress-energy tensor T*"1 for charged particles and electromagnetic
fields may be obtained from the action 7 NG , Equation (3.20), by first
rewriting it in the form
.
_
^
,,
dx'^W*
-NG
(3.66)
Since only g^ (and not its derivatives) appears in 7 NG , we obtain
= £ mOau"u\uorl{-g)-ll28\x
- xa(x)]
a
+ (4*)- \F^F\
- kg"vF^Fa0)
(3.67)
where we have used the fact that
(3-68)
Throughout most of this book we shall use the perfect fluid as our
model for matter. This model is an average of the properties of matter
over scales that are large compared to atomic scales, but small compared
to the scales over which the bulk properties of the fluid vary. Thus, one
can speak of density, pressure, velocity of fluid elements at a point within
Theory and Experiment in Gravitational Physics
78
the fluid. A perfect fluid is one that has negligible viscosity, heat transport,
and shear stresses. It is then possible to show that the stress-energy tensor
for the fluid has the following property: in a local Lorentz frame, momentarily comoving with a chosen element of the fluid, the stress-energy
tensor for that element has the form
T"v = diag[p(l + n),p,p,p-]
(3.69)
where p is the rest-mass-energy density of atoms in the fluid element, II
is the specific density of internal kinetic and thermal energy in the fluid
element, and p is the isotropic pressure. This can also be written in the
covariant form
+ J/"V)
(3.70)
where u" = dx^jdi is the four-velocity of the fluid element (=<58 m the
comoving frame). Then in curved spacetime, T"v has the form
T"v = (p + PU + p)u"Mv + pgT
(3.71)
This can also be derived from Equation (3.67) using suitable techniques
in relativistic kinetic theory (see Ehlers, 1971).
To obtain a complete metric theory of gravity one must now specify
field equations for the metric and for the other possible gravitational
fields in the theory. There are two alternatives. The first is to assume that
these equations, like the nongravitational equations can be derived from
an invariant action / G which will be a function of the gravitational fields
(£A (which could include #):
IG = IG(<I>A,<1>AJ
(3.72)
The complete action is thus
1=
IG(4>A,<I>\J
+
/NGOZA^A.^V.^V./S)
(3.73)
Variation with respect to <f>K yields the gravitational field equations
or, using Equation (3.61),
dSeol64>K= -U-g)ll2T^dgJdcl>A
(3.75)
Theories of this type are called Lagrangian-based covariant metric theories of gravity. Many important general properties of such theories are
described by Lee, Lightman, and Ni (1974). The other alternative is to
specify gravitational field equations that are not derivable from an action.
Gravitation as a Geometric Phenomenon
79
These are called non-Lagrangian-based theories. Although many such
theories have been devised, they have not met with great success in
agreeing with experiment. All the metric theories to be described in
Chapter 5 that agree with solar system experiments are Lagrangian based.
3.3
Long-Range Gravitational Fields and the
Strong Equivalence Principle
In any metric theory of gravity, matter and nongravitational fields
respond only to the spacetime metric g. In principle, however, there could
exist other gravitational fields besides the metric, such as scalar fields,
vectorfields,and so on. If matter does not couple to thesefieldswhat can
their role in gravitation theory be? Their role must be that of mediating the
method by which matter and nongravitationalfieldsgenerate gravitational
fields and produce the metric. Once determined, however, the metric alone
interacts with the matter as prescribed by EEP.
What distinguishes one metric theory from another, therefore, is the
number and kind of gravitational fields it contains in addition to the
metric, and the equations that determine the structure and evolution of
these fields. From this viewpoint, one can divide all metric theories of
gravity into two fundamental classes: "purely dynamical" and "prior geometric." (This division is independent of whether or not the theory is
Lagrangian based.)
By "purely dynamical metric theory" we mean any metric theory whose
gravitational fields have their structure and evolution determined by coupled partial differential field equations. In other words, the behavior of
each field is influenced to some extent by a coupling to at least one of the
otherfieldsin the theory. By "prior geometric" theory, we mean any metric
theory that contains "absolute elements,"fieldsor equations whose structure and evolution are given a priori and are independent of the structure
and evolution of the other fields of the theory. These "absolute elements"
could include flat background metrics IJ, cosmic time coordinates T, and
algebraic relationships among otherwise dynamical fields, such as
where h^ and k^ may be dynamicalfields.Note that afieldmay be absolute
even if it is determined by partial differential equations, as long as the
equation does not involve any dynamicalfields.For instance, a flat background metric is specified by the field equation
Riemfa) = 0
(3.76)
Theory and Experiment in Gravitational Physics
80
or a cosmic time function is specified by the field equations
v w v v r = 0,
VT VT = - 1
where the gradient and inner product are taken with respect to a nondynamical background metric, such as i\.
General relativity is a purely dynamical theory since it contains only one
gravitational field, the metric itself, and its structure and evolution is
governed by a partial differential equation (Einstein's equations). BransDicke theory is a purely dynamical theory; thefieldequation for the metric
involves the scalar field (as well as the matter as source), and that for the
scalar field involves the metric. Rosen's bimetric theory is a priorgeometric theory: it has a flat background metric of a type described in
Equation (3.76), and thefieldequations for the physical metric g involve t\.
In Chapter 5, we will discuss these and other theories in more detail.
By discussing metric theories of gravity from this broad, "Dicke" point
of view, it is possible to draw some general conclusions about the nature of
gravity in different metric theories, conclusions that are reminiscent of the
Einstein Equivalence Principle, but that will be given a new name: the
Strong Equivalence Principle.
Consider a local, freely falling frame in any metric theory of gravity. Let
this frame be small enough that inhomogeneities in the external gravitational fields can be neglected throughout its volume. However, let the
frame be large enough to encompass a system of gravitating matter and its
associated gravitationalfields.The system could be a star, a black hole, the
solar system, or a Cavendish experiment. Call this frame a "quasilocal
Lorentz frame". To determine the behavior of the system we must calculate the metric. The computation proceeds in two stages. First, we determine the external behavior of the metric and gravitational fields, thereby
establishing boundary values for thefieldsgenerated by the local system,
at a boundary of the quasilocal frame "far" from the local system. Second,
we solve for thefieldsgenerated by the local system. But because the metric
is coupled directly or indirectly to the otherfieldsof the theory, its structure
and evolution will be influenced by thosefields,particularly by the boundary values taken on by thosefieldsfar from the local system. This will be
true even if we work in a coordinate system in which the asymptotic form
of g^ in the boundary region between the local system and the external
world is that of the Minkowski metric. Thus, the gravitational environment
in which the local gravitating system resides can influence the metric generated by the local system via the boundary values of the auxiliary fields.
Consequently, the results of local gravitational experiments may depend
Gravitation as a Geometric Phenomenon
81
on the location and velocity of the frame relative to the external environment. Of course, local nongravitational experiments are unaffected since
the gravitational fields they generate are assumed to be negligible, and
since those experiments couple only to the metric whose form can always
be made locally Minkowskian. Local gravitational experiments might
include Cavendish experiments, measurements of the acceleration of massive bodies, studies of the structure of stars and planets, and so on. We can
now make several statements about different kinds of metric theories (Will
and Nordtvedt, 1972).
(a) A theory that contains only the metric g yields local gravitational
physics that is independent of the location and velocity of the local
system. This follows from the fact that the only field coupling the local
system to the environment is g, and it is always possible to find a coordinate
system in which g takes the Minkowski form at the boundary between the
local system and the external environment. Thus, the asymptotic values of
g^ are constants independent of location, and are asymptotically Lorentz
invariant, thus independent of velocity. General relativity is an example of
such a theory.
(b) A theory that contains the metric g and dynamical scalar fields
</>A yields local gravitational physics that may depend on the location of
the frame but which is independent of the velocity of the frame. This follows
from the asymptotic Lorentz invariance of the Minkowski metric and of
the scalar fields, except now the asymptotic values of the scalar fields may
depend on the location of the frame. An example is Brans-Dicke theory,
where the asymptotic scalarfielddetermines the value of the gravitational
constant, which can thus vary as <j> varies.
(c) A theory that contains the metric g and additional dynamical
vector or tensor fields or prior-geometric fields yields local gravitational
physics that may have both location- and velocity-dependent effects. This
will be true, for example, even if the auxiliary field is a flat background
metric IJ. The background solutions for g and t\ will in general be different,
and therefore in a frame in which g^ takes the asymptotic form diag
( 1,1,1,1), r\^ will in general have a form that depends on location and
that is not Lorentz invariant (although it will still have vanishing curvature). The resulting location and velocity dependence in q will act back on
the local gravitational problem. (For a clear example of this, see Rosen's
theory in Chapter 5.) Be reminded that these effects are a consequence of
the coupling of auxiliary gravitational fields to the metric and to each
other, not to the matter and nongravitational fields. For metric theories
of gravity, only g^ couples to the latter.
Theory and Experiment in Gravitational Physics
82
These ideas can be summarized in the form of a principle called the
Strong Equivalence Principle that states that (i) WEP is valid for selfgravitating bodies as well as for test bodies (GWEP), (ii) the outcome of any
local test experiment is independent of the velocity of the (freely falling)
apparatus, and (iii) the outcome of any local test experiment is independent
of where and when in the universe it is performed. The distinction between
SEP and EEP is the inclusion of bodies with self-gravitational interactions
(planets, stars) and of experiments involving gravitational forces (Cavendish experiments, gravimeter measurements). Note that SEP contains EEP
as the special case in which gravitational forces are ignored.
It is tempting to ask whether the parallel between SEP and EEP extends
as far as a Schiff-type conjecture; e.g., "any theory that embodies GWEP
also embodies SEP." As in Section 2.5, we can give a plausibility argument
in support of this, for the special case of metric theories of gravity with a
conservation law for total energy (Haugan, 1979). Generally speaking, this
means Lagrangian-based theories. Consider a local gravitating system
moving slowly in a weak, static, and external gravitational field. We assume
that the laws governing its motion can be put into a quasi-Newtonian
form, with the conserved energy Ec given by
(3.77)
where
MR = M 0 - E B ( X , V ) ,
£B(X, V) = £g + 8my UiJ(X) - $Sm[j VlV> + O ( E g t / 2 , . . . ) (3.78) (see Section 2.5 for detailed definitions). Here, we use units in which the speed of light as measured far from the local system is unity. The position and velocity dependence in £ B can manifest itself, for example, as position and velocity dependence in the locally measured gravitational constant. For two bodies in a local Cavendish experiment, the gravitational constant is given by Gcavendish = r2Fr/mim2 = r\dE^dr)lmxm1 (3.79) and thus the anomalous mass tensors will contribute to GCavendUh (see Section 6.4). However, a cyclic gedanken experiment identical to that presented in Section 2.5 shows that the anomalous mass tensors bml4 and 8m\J also generate violations of GWEP A1 = g' + (Smf/MJU* - (<5mjJ/AW (3.80) Gravitation as a Geometric Phenomenon 83 where g = \U. Hence, GWEP (dm$ = Sm{k = 0) implies no preferredlocation or preferred-frame effects, thence SEP. In Chapters 4, 5, and 6
we shall see specific examples of GWEP and SEP in action in the postNewtonian limits of arbitrary metric theories of gravity, and in Chapter 8,
shall study experimental tests of SEP.
The above discussion of the coupling of auxiliary fields to local gravitating systems indicates that if SEP is valid, there must be one and only one
gravitational field in the universe, the metric g. Those arguments were only
suggestive however, and no rigorous proof of this statement is available
at present.
The assumption that there is only one gravitational field is the foundation of many so-called derivations of general relativity. One class of
derivations uses a quantum-field-theoretic approach. One begins with
the assumption that, in perturbation theory, the gravitational field is
associated with the exchange of a single massless particle of spin 2 (corresponding to a single second-rank tensor dynamicalfield),and by making
certain reasonable assumptions that the S-matrix be Lorentz invariant
or that the theory be derivable from an action, one can generate the full
classical Einstein field equations (Weinberg, 1965; Deser, 1970). Another
class of derivations attempts to build the most general field equation for
g out of tensors constructed only from g, subject to certain constraints
(no higher than second derivatives, for instance). By demanding that the
field equations should imply the matter equations of motion Tfvv = 0,
one is led (except for the possible cosmological term) to Einstein's equations. For a review of these and other derivations of general relativity the
reader is referred to MTW, box 17.2. However, the implicit use of SEP in
all these derivations cannot be emphasized enough. Empirically, it has
been found that every metric theory other than general relativity introduces auxiliary gravitational fields, either dynamical or prior geometric,
and thus predicts violations of SEP at some level. General relativity seems
to be the only metric theory that embodies SEP completely. Thus, the
wide variety of derivations of general relativity assuming SEP, plus evidence from alternative theories lends some credence to the conjecture
SEP => [General Relativity]
(3.81)
In Chapters 8 and 12, we shall discuss experimental evidence for the
validity of SEP.
This qualitative discussion of alternative metric theories of gravity has
neglected two subjects, each of which could generate a monograph of its
Theory and Experiment in Gravitational Physics
84
own. The first is "torsion." In applying EEP to the nongravitational laws
of physics we assumed the rule "comma goes to semicolon," where semicolon denoted covariant derivative with respect to the metric g [Equations
(3.14), (3.16), and (3.17)]. However, it is possible that the correct covariant
derivative is given by
A% = A'j, + §y}A>
(3.82)
{?,} = T% + S$y (3.83) where with Sjiy antisymmetric on ft and y, i.e., Sfryi-Si, (3.84) In general, S^y is a tensor called the "torsion" tensor, and thus does not vanish in the local Lorentz frame. Torsion has been introduced into gravitation theory either as a means to incorporate quantum mechanical spin in a consistent way, as a byproduct of attempts to construct gauge theories of gravitation, and as a possible route to a unified theory of gravity and electromagnetism. However, in almost all experiments discussed in this book, the observable effects of torsion are negligible [see, however, Ni (1979)]. Instead, torsion has an effect primarily in the realm of elementary particle physics or in the very early universe. Thus, we shall neglect torsion completely for the rest of this book, and shall refer the interested reader to the review by Hehl et al. (1976). The second topic to be neglected falls under the heading "general relativity with R2 terms." Although this is an old subject (Weyl, 1919; Eddington, 1922), it has recently attracted some interest. The standard gravitational action of classical general relativity (Section 5.2) has the form (3.85) where R is the Ricci scalar given by R = g^R,, (3.86) However, some attempts to make a renormalizable quantum theory of gravity based on general relativity lead to the introduction of "counter terms" into the action, to eliminate the nonrenormalizable infinities. These counter terms are quadratic and higher in the Riemann tensor, Ricci tensor, and Ricci scalar, leading to a gravitational action of the form 1G = (16TT)-l J(R + aR2 + bR^R*"* + cR^R^i-g)1'2dAx (3.87) Gravitation as a Geometric Phenomenon 85 Since the theory has only one gravitational field #, one suspects that it satisfies SEP, and so represents a possible counter example to our conjecture that SEP => general relativity. However, in most theories of this type, the constants a, b, and c [units of (length)2] have sizes ranging from the Planck length, 10~ 33 cm, to nuclear dimensions, 10" 1 3 cm, so the observable effects of these terms will be confined to elementary particle interactions or to the very early universe. Thus the issue of "R2 terms," too, will be ignored throughout this book (see Havas, 1977). The Parametrized Post-Newtonian Formalism We have seen that, despite the possible existence of long-range gravitational fields in addition to the metric in various metric theories of gravity, the postulates of metric theories demand that matter and nongravitational fields be completely oblivious to them. The only gravitational field that enters the equations of motion is the metric g. The role of the other fields that a theory may contain can only be that of helping to generate the spacetime curvature associated with the metric. Matter may create these fields, and they, plus the matter, may generate the metric, but they cannot interact directly with the matter. Matter responds only to the metric. Consequently, the metric and the equations of motion for matter become the primary theoretical entities, and all that distinguishes one metric theory from another is the particular way in which matter and possibly other gravitational fields generate the metric. The comparison of metric theories of gravity with each other and with experiment becomes particularly simple when one takes the slow-motion, weak-field limit. This approximation, known as the post-Newtonian limit, is sufficiently accurate to encompass all solar system tests that can be performed in the foreseeable future. The post-Newtonian limit is not adequate, however, to discuss gravitational radiation, where the slowmotion assumption no longer holds, or systems with compact objects such as the binary pulsar, where the weak-field assumption is not valid, or cosmology, where completely different assumptions must be made. These issues will be dealt with in later chapters. In Section 4.1, we discuss the post-Newtonian limit of metric theories of gravity, and devise a general form for the post-Newtonian metric for a system of perfectfluid.This form should be obeyed by most metric theories, with the differences from one theory to the next occurring only in the The Parametrized Post-Newtonian Formalism 87 numerical coefficients that appear in the metric. When the coordinate system is appropriately specialized (standard gauge), and arbitrary parameters used in place of the numerical coefficients, the result, described in Section 4.2, is known as the Parametrized post-Newtonian (PPN) formalism, and the parameters are called PPN parameters. In Section 4.3, we discuss the effect of Lorentz transformations on the PPN coordinate system, and show that some theories of gravity may predict gravitational effects that depend on the velocity of the gravitating system relative to the rest frame of the universe (perferred-frame effects). In Section 4.4, we analyze the existence of post-Newtonian integral conservation laws for energy, momentum, angular momentum, and center-of-mass motion within the PPN formalism and show that metric theories possess such laws only if their PPN parameters obey certain constraints. This formalism then provides the framework for a discussion of specific alternative metric theories of gravity (Chapter 5) and for the analysis of solar system tests of relativistic gravitational effects (Chapters 7-9). Most of this chapter is an updated version of Chapter 4 of TTEG (Will, 1974a). 4.1 The Post-Newtonian Limit (a) Newtonian gravitation theory and the Newtonian limit In the solar system, gravitation is weak enough for Newton's theory of gravity to adequately explain all but the most minute effects. To an accuracy of about one part in 105, light rays travel on straight lines at constant speed, and test bodies move according to a = \U (4.1) where a is the body's acceleration, and U is the Newtonian gravitational potential produced by rest-mass density p according to 1 = -4np, U(x, t) = [-~Ar d2x' (4.2) |x x | A perfect, nonviscous fluid obeys the usual Eulerian equations of hydrodynamics dp/dt + V (pv) = 0, pdv/dt^ pVU - Vp, d/dt = d/dt + v V (4.3) 1 We use "geometrized" units in which the speed of light is unity and in which the gravitational constant as measured far from the solar system is unity. Theory and Experiment in Gravitational Physics 88 where v is the velocity of an element of the fluid, p is the rest-mass density of matter in the element, p is the total pressure (matter plus radiation) on the element, and d/dt is the time derivative following the fluid. From the standpoint of a metric theory of gravity, Newtonian physics may be viewed as a first-order approximation. Consider a test body momentarily at rest in a static external gravitational field. From the geodesic Equation (3.38), the body's acceleration a* = d2xk/dt2 in a static (t,x) coordinate system is given by «*=-n<> = ie*W, (4-4) Far from the Newtonian system, we know that in an appropriately chosen coordinate system, the metric must reduce to the Minkowski metric (see subsection (c)) ff,,,-»if,, = diag(-1,1,1,1) (4.5) In the presence of a very weak gravitational field, Equation (4.4) can yield Newtonian gravitation, Equation (4.1) only if gfi-S*, goo^-l+2U (4.6) It can be straightforwardly shown that with this approximation and a stress-energy tensor for perfect fluids given by T00 = p, TOj = pvJ, TJk = pvV + p5Jk (4.7) the Eulerian equations of motion, (4.3), are equivalent to T?; ~ r?; + rgoT 00 = o (4.8) where we retain only terms of lowest order in v2 ~ U ~ p/p. But the Newtonian limit no longer suffices when we begin to demand accuracies greater than a part in 105. For example, it cannot account for Mercury's additional perihelion shift o f ~ 5 x 10 ~7 radians per orbit. Thus we need a more accurate approximation to the spacetime metric that goes beyond or "post" Newtonian theory, hence the name postNewtonian limit. (b) Post-Newtonian bookkeeping The key features of the post-Newtonian limit can be better understood if we first develop a "bookkeeping" system for keeping track of "small quantities." In the solar system, the Newtonian gravitational potential U is nowhere larger than 10" 5 (in geometrized units, U is dimensionless). Planetary velocities are related to U by virial relations The Parametrized Post-Newtonian Formalism 89 which yield v2 Z U (4.9) The matter making up the Sun and planets is under pressure p, but this pressure is generally smaller than the matter's gravitational energy density pU; in other words P/P £ U, (4.10) 5 10 {p/p is ~10~ in the Sun, ~10~ in the Earth). Other forms of energy in the solar system (compressional energy, radiation, thermal energy, etc.) are small: the specific energy density II (ratio of energy density to rest-mass density) is related to U by nzu (4.ii) (II is ~ 10" 5 in the Sun, ~ 10~9 in the Earth). These four small quantities are assigned a bookkeeping label that denotes their "order of smallness": U ~ v2 ~ p/p ~ n ~ O(2). (4.12) Then single powers of velocity i; are O(l), U2 is O(4), Uv is O(3), UH is O(4), and so on. Also, since the time evolution of the solar system is governed by the motions of its constituents, we have d/dt ~ v V and thus, \s/et\ \d/e> O(l) (4.13) We can now analyze the "post-Newtonian" metric using this bookkeeping system. The action, Equation (3.20), from which one can derive the geodesic Equation (3.38) for a single neutral particle, may be rewritten -i - Cf dx»dx*yi2 - ' o - -m0 Jl ~a^~Jf~^fJ dt (- 000 - 2gop' - gjkvV)112 dt (4.14) The integrand in Equation (4.14) may thus be viewed as a Lagrangian L for a single particle in a metric gravitational field. From Equation (4.6), we see that the Newtonian limit corresponds to L = (1 - 2(7 - v2)112 (4.15) Theory and Experiment in Gravitational Physics 90 as can be verified using the Euler-Lagrange equations. In other words, Newtonian physics is given by an approximation for L correct to O(2). Post-Newtonian physics must therefore involve those terms in L of next highest order, O(4). What happened to odd-order terms, O(l) or O(3)? Odd-order terms must contain an odd number of factors of velocity v or of time derivatives d/dt. Since these factors change sign under time reversal, odd-order terms must represent energy dissipation or absorption by the system. But conservation of rest mass prevents terms of O(l) from appearing in L, and conservation of energy in the Newtonian limit prevents terms of O(3). Beyond O(4), different theories may make different predictions. In general relativity, for example, the conservation of post-Newtonian energy prohibits terms of O(5). However, terms of O(7) can appear; they represent energy lost from the system by gravitational radiation. In order to express L to O(4), we must know the various metric components to an appropriate order: L = {1 - 2V - v2 - 0oo[O(4)] V<} 1/2 2gOJ[p0)y (4.16) Thus the post-Newtonian limit of any metric theory of gravity requires a knowledge of 0OO to O(4), g0J to O(3), gJk to O(2) (4.17) The post-Newtonian propagation of light rays may also be obtained using the above approximations to the metric. Since light moves along null trajectories (dx 0), the Lagrangian L must be formally identical to zero. In the first order Newtonian limit this implies that light must move on straight lines at speed 1, i.e., 0 = L = (1 - v2)112, v2 = 1 (4.18) In the next, post-Newtonian order, we must have 0 = L={l-2U -v2- gjk[O(2)~]vJvk}112 (4.19) Thus to obtain post-Newtonian corrections to the propagation of light rays, we need to know goo to O(2), gjk to O(2) (4.20) The Parametrized Post-Newtonian Formalism 91 In a similar manner, one can verify that if one takes the perfect-fluid stress-energy tensor T"v = (p + pU + p)u"u" + pg»v (4.21) expanded through the following orders of accuracy: T 00 T0J TJk to pO(2), to pO(3), to pO(4) (4.22) and combined with the post-Newtonian metric, then the equation of motion 7?vv = 0 will yield consistent "post-Eulerian" equations of hydrodynamics. (c) Post-Newtonian coordinate system To discuss the post-Newtonian limit properly, we must specify the coordinate system. We imagine a homogeneous isotropic universe in which an isolated post-Newtonian system resides. We choose a coordinate system whose outer regions far from the isolated system are in free fall with respect to the surrounding cosmological model, and are at rest with respect to a frame in which the universe appears isotropic (universe rest frame). In these outer regions, one expects the physical metric to vary according to ds2 = -dt\+ [a(t)/ao]2(l + kr2IAal)'2dijdxidxi + h^dx^dx1 (4.23) where the first two terms comprise the Robertson-Walker line element appropriate to a homogeneous isotropic cosmological model and the third term represents the perturbation due to the local system. Here, r is the distance from the local system to the field point, a = a{t)[aQ = a(toj] is the cosmological scale factor, and k is the curvature parameter (k = 0, + 1). At a given radius r0 and at a particular moment t0, we can transform to a coordinate system t' == t, xy = x\l - krl/4al)-1 (4.24) in which ds2 = (ifc, +fcJJdx»' dxv' (4.25) This must be done at a value of r0 large enough that we can then regard n^ as the asymptotic form of g^, i.e., that h^ ~ M/r0 « 1, where M is the mass of the isolated system, yet small enough that the deviation of the cosmological metric from n^ for r « r0 is small, in fact smaller than Theory and Experiment in Gravitational Physics 92 the post-Newtonian terms in h^ of order (M/r)2. The value of r0 that optimizes these constraints is given by (M/r0)2 > {ro/ao)2, or M « r0 <, (Mao)1/2. Since a0 ~ 1010 light yr, we have, for the solar system r0 <: 1011 km ^ 103 a.u., with maximum deviations from n^ of order (ro/ao)2 ~ 10"24. These are much smaller than the expected postNewtonian deviations (M/r)2 > 10" 16 that influence solar system experiments. Thus, to a precision of about one part in 1022, we can regard the space time metric of the solar system as being asymptotically Minkowskian in its outer regions, out to 103 a.u., with deviations of order M/r and {M/r)2 in its interior. The above discussion ignores the variation of the cosmological scale factor a(t) with time. However, because this variation takes place over a timescale (1010 yr) long compared to a dynamical timescale (1 yr) for the solar system, we can treat the effects of the variation adiabatically. The coordinate system thus constructed we shall call "local quasiCartesian coordinates." In this coordinate system it is useful to define the following conventions and quantities: (i) Unless otherwise noted, spatial vectors are treated as Cartesian vectors, with x* = xk. (ii) Repeated spatial indices or the symbol |x| denotes a Cartesian inner product, for example xkxk = Xkxk = xkxk = |x|2 s x2 + y2 + z2 (4.26) 3 (iii) The volume element d x = dxdydz. (d) Post-Newtonian potentials We assume throughout that the matter composing the solar system can be idealized as perfect fluid. For the purposes of most solar system experiments in the coming decades, this is an adequate assumption (see, however, Section 9.2). As we shall see in more detail in Chapter 5, the post-Newtonian limit for a system of perfect fluid in any metric theory of gravity is best calculated by solving the field equations formally, expressing the metric as a sequence of post-Newtonian functionals of the matter variables, with possible coefficients that may depend on the matching conditions between the local system and the surrounding cosmological model and on other constants of the theory. The evolution of the matter variables, and thence of the metric functionals, is determined by means of the equations of motion Tfvv = 0 using the matter stress-energy tensor and the post-Newtonian metric all evaluated to an The Parametrized Post-Newtonian Formalism 93 order consistent with the post-Newtonian approximation. The evolution of the cosmological matching coefficients is determined by a solution of the appropriate cosmological model. Thus, the most general postNewtonian metric can be found by simply writing down metric terms composed of all possible post-Newtonian functionals of matter variables, each multiplied by an arbitrary coefficient that may depend on the cosmological matching conditions and on other constants, and adding these terms to the Minkowski metric to obtain the physical metric. Unfortunately, there is an infinite number of such functionals, so that in order to obtain a formalism that is both useful and manageable, we must impose some restrictions on the possible terms to be considered, guided in part by a subjective notion of "reasonableness" and in part by evidence obtained from known gravitation theories. Some of these restrictions are obvious: (i) The metric terms should be of Newtonian or post-Newtonian order; no post-post-Newtonian or higher terms are included. (ii) The terms should tend to zero as the distance |x x'| between the field point x and a typical point x' inside the matter becomes large. This will guarantee that the metric becomes asymptotically Minkowskian in our quasi-Cartesian coordinate system. (iii) The coordinates are chosen so that the metric is dimensionless. (iv) In our chosen quasi-Cartesian coordinate system, the spatial origin and initial moment of time are completely arbitrary, so the metric should contain no explicit reference to these quantities. This is guaranteed by using functionals in which the field point x always occurs in the combination x x', where x' is a point associated with the matter distribution, and by making all time dependence in the metric terms implicit via the evolution of the matter variables and of the possible cosmological matching parameters. (v) The metric corrections h00, h0J, and hJk should transform under spatial rotations as a scalar, vector, and tensor, respectively, and thus should be constructed out of the appropriate quantities. For variables associated with the matter distribution, examples are: scalar, p, |x x'|, v'2, v' (x x') etc.; vector, v), (x x')f, and tensor, (x x')/* x')k, VjVk, etc. For variables associated with the structure of the field equations of the theory or with the cosmological matching conditions, there are only two available quantities in the rest frame of the universe: scalar cosmological matching parameters or numerical coefficients; and a tensor, Sjk. In the rest frame of an isotropic universe, no vectors or anisotropic Theory and Experiment in Gravitational Physics 94 tensors can be constructed. [If the universe is assumed to be slightly anisotropic, other terms may be possible (Nordtvedt, 1976).] (vi) The metric functionals should be generated by rest mass, energy, pressure, and velocity, not by their gradients. This restriction is purely subjective, and can be relaxed quite easily if there is ever any reason to do so. No reason has yet arisen. A final constraint is extremely subjective: (vii) The functionals should be "simple." With those restrictions in mind, we can now write down possible terms that may appear in the post-Newtonian metric. (1) gJk to O(2): From condition (v), gjk must behave as a threedimensional tensor under rotations, thus the only terms that can appear are gjk[O(2)-]:U3jk,Ujk (4.27) where Ujk is given by Ujk s f PV,t)(x-x')M-x')k ^ (4 2g) X X The term Ujk can be expressed more conveniently in terms of the "superpotential" %(x, t), given by 3 X(x,t)=-jp(x',t)\x-x'\d Xjk=-SjkU+UJk, x', V 2 x=-2t/ (4.29) Thus, the only terms that we shall consider are gjk[O(2)l. U5jk,x,Jk (4.30) (2) gOj to O(3): These metric components must transform as threevectors under rotations, and thus contain only the terms : VJtWj where y CPWMp J \x x'\ (4.31) The Parametrized Post-Newtonian Formalism 95 The functionals V} and Wj are also related to the superpotential % by X.oj =VJ-WJ (4.33) (3) goo t° O(4): This component should be a scalar under rotations. The only terms we shall consider are 0oo[O(4)]: U\<bw,<bu<b2,<bi,<bt,s/,a (4.34) where Y ,,, 2 2 x-x f,,,, J x-x' (4.35) | | |x-x'| dt Restriction (vii) has been used liberally to eliminate otherwise possible metric functionals, for example VJVJU-\ '\'Y d3x\... Should one of these terms ever appear in the post-Newtonian metric of a gravitational theory, the formalism could be modified accordingly. There are a number of simple and useful relationships satisfied by the functionals that we have included in the metric: ! = -4npv2, V2O2 = -AnpU, V2O4 = -4np, = rf + ® - <E>X (4.36) To derive many of these relationships one makes repeated use of the formula, obtained using the continuity Equation (4.3), 8 r r n(x' at J t)f(x x')d3x' = o(x' tW V'/Yx x'1ii3jc'ri + O<2)1 > (4 371 Theory and Experiment in Gravitational Physics 4.2 96 The Standard Post-Newtonian Gauge We can restrict the form of the post-Newtonian metric somewhat by making use of the arbitrariness of coordinates embodied in statement (ii) of the Dicke framework. An infinitesimal coordinate or "gauge" transformation [see Equations (3.5), (3.13), and (3.17)] xu = x" + £"(xv) (4.38) changes the metric to *«» - £*, U (439) We wish to retain the post-Newtonian character of g^ and the quasiCartesian character of the coordinate system, and to remain in the universe rest frame, thus the functions £ must satisfy: (i) £mv + £v;/, are post-Newtonian functions; (ii) £. + £v;A, -* 0, far from the system; and (iii) |^"|/|x"| -> 0, far from the system. The only "simple" functional that has this property is the gradient of the superpotential x,»- Thus, we choose and obtain, to post-Newtonian order 9~jk = 9jk ~ 2A2Xjk> 9oo = 9oo ~ 2AiX,oo + 2X2HoX,j (4.41) To the necessary order, the Christoffel symbol rJ00 is equal to Uj [see Equation (3.14)]. We must also transform the functional integrals over xk' that appear in g^ into integrals over x F . The only place where this changes anything is in g00 ^ 1 + 2U(x,l), where - f P(X'J Now the quantity p(x',T) is an invariant; it is the rest-mass density as measured in a comoving local Lorentz frame. Furthermore, the quantity (g)ll2u°d3x is an invariant proper volume element, where u° is the fourvelocity of the matter. Thus, d3x' = d3x'[(-g)ll2i/i/(-g)ll2u0'] (4.42) The Parametrized Post-Newtonian Formalism 97 Using Equation (4.41) plus the relation u° = dt/dt, we get to the required order, p'dV = p'd3x'[l + 212[/(x',l)] (4.43) We also have k |x x'l ~ |x-x'| 2 |x-xf Thus, l/(x,T) = U{x,7) + 212O2 - k2 J P ^ _ ~ _ ^ , | 3 ? T d3x (4.45) Using Equations (4.33), (4.36), and (4.41), we obtain, finally 9jk = 9jk ~ ^iXjk, 055 = 0oo - 2A2(C/2 +OW-$ 2 ) - 2X^
+<%-*>!)
(4.46)
By an appropriate choice of kt and X2 w e c a n eliminate certain terms
from the post-Newtonian metric. We will thus adopt a standard postNewtonian gauge - that gauge in which the spatial part of the metric is
diagonal and isotropic (i.e., x,jk eliminated) and in which g00 contains no
term Si. There is no physical significance in this gauge choice; it is purely
a matter of convenience.
We now have a very general form for the post-Newtonian perfect-fluid
metric in any metric theory of gravity, expressed in a local, quasi-Cartesian
coordinate system at rest with respect to the universe rest frame, and in a
standard gauge. The only way that the metric of any one theory can
differ from that of any other theory is in the coefficients that multiply
each term in the metric. By replacing each coefficient by an arbitrary
parameter we obtain a "super metric theory of gravity" whose special
cases (particular values of the parameters) are the post-Newtonian metrics
of particular theories of gravity. This "super metric" is called the parametrized post-Newtonian (PPN) metric, and the parameters are called
PPN parameters.
This use of parameters to describe the post-Newtonian limit of metric
theories of gravity is called the Parametrized Post-Newtonian (PPN) Formalism. A primitive version of such a formalism was devised and studied
Theory and Experiment in Gravitational Physics
98
by Eddington (1922), Robertson (1962), and Schiff (1967). This EddingtonRobertson-Schiff formalism treated the solar system metric as that of a
spherical nonrotating Sun, and idealized the planets as test bodies moving
on geodesies of this metric. The metric in this version of the formalism
9oj = 0.
9jk
= (1 + 2yM/r)6jk
(4.47)
where M is the mass of the Sun, and )5 and y are PPN parameters.
These two parameters may be given a physical interpretation in this
formalism. The parameter y measures the amount of curvature of space
produced by a body of mass M at radius r, in the sense that the spatial
components of the Riemann curvature tensor are given to post-Newtonian
order by [see Equations (3.14) and (3.34)]
Riju = (3yM/r3)(«j«^a + n^S^ - nfi^j, - n/i,<5tt - f 8jkSa + ^Sikdjt)
where
n = x/r
independent of the choice of post-Newtonian gauge. The parameter ft
is said tb measure the amount of nonlinearity (M/r)2 that a given theory
puts into the g00 component of the metric. However, this statement
is valid only in the standard post-Newtonian gauge. The coefficient of
U2 = (M/r)2 depends upon the choice of gauge, as can be seen from
Equation (4.46). In general relativity, for example (/? = y = 1), the (M/r)2
term can be completely eliminated from g00 by a gauge transformation
that is the post-Newtonian limit of the exact coordinate transformation from isotropic coordinates to Schwarzschild coordinates for the
Schwarzschild geometry. Thus, this identification of fi should be viewed
only as a heuristic one.
Schiff (1960b) generalized the metric [Equation (4.47)] to incorporate
rotation (Lense-Thirring effect, Section 9.1), and Baierlein (1967) developed a primitive perfect-fluid PPN metric. But the pioneering development of the full PPN formalism was initiated by Kenneth Nordtvedt, Jr.
(1968b), who studied the post-Newtonian metric of a system of gravitating
point masses. Will (1971a) generalized the formalism to incorporate matter
described by a perfect fluid. A unified version of the PPN formalism was
then presented by Will and Nordtvedt (1972) and summarized in TTEG.
The Whitehead term Ow was added by Will (1973). Henceforth, we shall
The Parametrized Post-Newtonian Formalism
99
term), altered to conform with MTW signature and index conventions, and
with minor notational modifications (see Table 4.1). As in the EddingtonRobertson-Schiff version of the PPN formalism, we introduce an arbitrary PPN parameter in front of each post-Newtonian term in the metric.
Ten parameters are needed; they are denoted y, j8, & alt a2, a3, d, d> (3.
and C4. In terms of them, the PPN metric reads
0Oo = - 1 + 217 - 2pU2 - 2&bw + (2y + 2 + a3 + Ci + 2(3y - 2/J + 1 + £2 + £)<D2 + 2(1 + {3)«>3
9oj = -i(4y + 3 + ax - a2 + d - 2 $^ - | ( 1 + a2 - Ci ^ = (1 + 2yU)5jk (4.48) Although we have used linear combinations of PPN parameters in Equation (4.48), it can be seen quite easily that a given set of numerical coefficients for the post-Newtonian terms will yield a unique set of values for the parameters. The linear combinations were chosen in such a way that the parameters a t , a2, a3, £l5 f2, £3, and £4 will have special physical significance. Other versions of the formalism have been developed to deal with point masses with charge (Section 9.2), fluid with anisotropic stresses (MTW Section 39), and isolated systems in an anisotropic universe (Nordtvedt, 1976). 4.3 Lorentz Transformations and the PPN Metric In Section 4.1, the PPN metric was devised in a coordinate system whose outer regions are at rest with respect to the universe rest frame. For some purposes - for example, the computation of the post-Newtonian metric in a given theory of gravity - this is a useful coordinate system. But for other purposes, such as the computation of observable postNewtonian effects in systems, such as the solar system, that are in motion relative to the universe rest frame, it is not a convenient coordinate system. In such cases, a better coordinate system might be one in which the center of mass of the system under study is approximately at rest. Again, this is a matter of convenience; the results of experiments cannot be affected by our choice of coordinate system. Because many of our computations will be carried out for such moving systems, it is useful to reexpress the PPN metric in a moving coordinate system. This will also yield some insight into the significance of the PPN parameters au oe2, and <x3 (Will, 1971c). Theory and Experiment in Gravitational Physics 100 To do this we make a Lorentz transformation from the original PPN frame to a new frame which moves at velocity w relative to the old frame. In order to preserve the post-Newtonian character of the metric, we assume that |w| is small, i.e., of O(l). This transformation from rest coordinates (t,\) to moving coordinates (T,£) can be expanded in powers of w to the required order: this approximate form of the Lorentz transformation is sometimes called a post-Galilean transformation (Chandrasekhar and Contopoulos, 1967), and has the form x = | + (1+ » + ±({ w)w + O(4) x {, T 4 t = T(1 + W + fw ) + (1 + W)S w + O(5) x T (4.49) where wr is assumed to be O(0). We use the standard transformation law, and express the functional that appear in gaf(x, t) in terms of the new coordinates. Since p, n , and p are all measured in comoving local Lorentz frames, they are unchanged by the transformation: for any given element of fluid, p(x,t) = p(Z,i), p(x,t) = p«,t) (4.51) If v(x, t) and v(£, T) are the matter velocities in the two frames, they are related by v = v + w + O(3) (4.52) The elements of volume d3x' and d3£' in the two frames are related by the transformation law [Equation (4.42)] - v' w - W + O(4)] (4.53) The quantity x(t) x'(t) that appears in the post-Newtonian potentials transforms according to - T') + K«T) - S'(T')] WW + O(4), 0 = (t - T')(1 + W) + [«t) - £'(*')] w + O(3) (4.54) The Parametrized Post-Newtonian Formalism 101 But in the (§,T) coordinates, the quantity £ §' must be evaluated at the same time x, hence we must use the fact that £'(T') = %{x) + v'(t' - t) + O ( T ' - T) 2 (4.55) Combining Equations (4.54) and (4.55), we obtain 1 , |X - 1 .,, {1 + i(w fi')2+ (w fi')(v' n') + O(4)} S II S ,, = .« .,, X| |S (4.56) where S'| (4.57) We then find, using Equations (4.51)-(4.53), and (4.56), along with the definitions of the metric functions, Equations (4.2), (4.32), and (4.35), that U(x,t) = (1 - WWlr) - wtVj&z) + T) + 2WiVj($, T) + W2 U(Z, T) + O(6),
.4)($,t) + O(6), + 2 ^ W J « , T ) + w V ^ ( { , t ) + O(6), Vj(x, t) = Vj{l t) + wy!/(«, T) + O(5), ) W5(x, t) = Wj(Z, x) + wkUjk($, T) + O(5)
(4.58)
Applying the transformation Equations (4.49) and (4.50) to the PPN
metric Equation (4.48) and making use of Equations (4.58) we obtain, for
the metric in the moving (£, T) system, to post-Newtonian order,
- 2)5 + 1 + C2 + )« 2 «,T) + 2(1
2(3y + 3
(a, - a2
(2a3 - a ^ F / t r ) - (1 - a2 3 + «t - a2
,T) - | ( 1 - a 2 - Ci
T)],5;t
J
(4.59)
Theory and Experiment in Gravitational Physics
102
Because we now have available an additional post-Newtonian variable,
w, we have an additional gauge freedom that can be employed without
altering the standard PPN gauge, which is valid in the frame in which
w = 0 (and which, incidently, was not affected by the post-Galilean transformation). By making the gauge transformation
T = T + J(l - <x2 - d + 2Z)w%j,
V =V
(4.60)
we can eliminate the terms
- ( 1 - a 2 - Ci + 2£)wJx>Oj from g00,
-Ul - « 2 ~ Ci + 2f)w*JU from g0J
This then becomes part of the standard PPN gauge in a coordinate system
moving at velocity w relative to the universe rest frame: that gauge in
which gjk is diagonal and isotropic, and in which the terms 36 and wJx,Oj
are absent from g00. It is then possible to show that a further post-Galilean
transformation (plus a possible gauge transformation to maintain the
standard gauge) does not alter the form of the PPN metric, it merely
changes the value of the coordinate system velocity w that appears there.
At first glance, one might be disturbed by the presence of metric terms
that depend on the coordinate system's velocity w relative to the universe
rest frame. These terms do not violate the principles of special relativity
since they are purely gravitational terms, while special relativity is valid
only when the effects of gravitation can be ignored; but they do suggest
that the gravitation generated by matter may be affected by motion relative to the universe (violation of the Strong Equivalence Principle). Nevertheless, the results of physical measurements must not depend on the
velocity w (this is a consequence of general covariance). For a system
such as the Sun and planets, the only physically measureable velocities are
the velocities of elements of matter relative to each other and to the center
of mass of the system, and the velocity, w0, of the center of mass relative
to the universe rest frame (as measured for example by studying Doppler
shifts in the cosmic microwave radiation). Thus, the PPN prediction for
any physical effect can depend only on these relative velocities and on
w0, never on w. Therefore, the terms in the PPN metric that depend on
w must signal the presence of effects that depend on w0. This can be seen
most simply by working in a coordinate system in which the system under
study is at rest, i.e., where w = w0. Then, if any one of the set of parameters
{ai,a 2 ,a 3 } is nonzero, there may be observable effects which depend on
w0; if a t = a2 = a3 = 0, there is no reference to w or w0 in the metric in
The Parametrized Post-Newtonian Formalism
103
any coordinate system, and no such effects pan occur. Thus, we see that the
parameters au a2, and <x3 measure the extent and manner in which motion
relative to the universe rest frame affects the post-Newtonian metric and
produces observable effects. These parameters are called "preferred-frame
parameters" since they measure the size of post-Newtonian effects produced by motion relative to the "preferred" rest frame of the universe. If
all three are zero, no such effects are present, and there is no preferred
frame (to post-Newtonian order).
Notice that even if one works in the universe rest frame, where w = 0,
physical effects will be unchanged, for even though the explicit preferredframe terms are absent, the velocities of elements of matter vJ that appear
in the PPN metric and in the equations of motion must be decomposed
according to
v = w0 + v"
where v is the velocity of each element relative to the center of mass, and,
unless alt <x2, and <x3 are all zero, the same effects dependent upon w0
will result.
At this point the PPN metric has taken on its standard form. Table 4.1
summarizes the basic definitions and formulae that enter the PPN formalism and compares the present version with previous versions.
Table 4.1. The parametrized post-Newtonian formalism
A. Coordinate system: the framework uses a nearly globally Lorentz coordinate
system [Section 4.1(c)] in which the coordinates are (t,xl,x2,x3). Three-dimensional, Euclidean vector notation is used throughout. All coordinate arbitrariness
("gauge freedom") has been removed by specialization of the coordinates to the
standard PPN gauge (Section 4.2).
B. Matter variables:
1. p = density of rest mass as measured in a local freely falling frame momentarily
comoving with the gravitating matter.
2. v' = (dx'/dt) = coordinate velocity of the matter.
3. w' = coordinate velocity of PPN coordinate system relative to the mean rest
frame of the universe.
4. p = pressure as measured in a local freely falling frame momentarily comoving
with the matter.
5. n = internal energy per unit rest mass. It includes all forms of nonrest mass,
nongravitational energy - e.g., energy of compression and thermal energy.
C. PPN parameters:
v, P, L «i, &2, «3, Ci, £2, C3> C4
Theory and Experiment in Gravitational Physics
104
Table 4.1. (continued)
D. Metric:
goo= -1 + 2U- 2PU1 - 2^w + (2y + 2 + a 3 + Ci - 2{)®t
+ 2(3y - 20 + 1 + C2 + £)* 2 + 2(1 + C3)O3 + 2(3y + 3
- (£, - 2§st - (a t - « 2 - a 3 )w 2 l/ - a 2 wWl7 y + (2a3
0oi = - i ( 4 y + 3 + a, - <x2 + Ct - 2{W - ftl + a2 - i ( B l - 2*2)wtU >
gtj = (1 + 2yU)Sij
E. Metric potentials:
| - xI
®W =
/ p'p"(X - X') ( X' - X"
-j
^ j -j
77s
J
|x - x | 3
J
J
Xn
|x x |
\ |x x I
'[v'-(x-xQP
|x - x'| 3
X
d
X
|x x I /
pV^
f
J |x - x I
|
J |x - x'|
X" \
7T ) d
J
I
|x - x I
|x x |
F. Stress-energy tensor (perfect fluid)
T00 = p(l + U + v2 + 217)
T0i = p(l + Tl + v2 + 2U + p/p)v'
TiJ = pt>V(l + n + v2 + 217 + p/p) + p5iJ(l - 2yU)
G. Equations of motion
1. Stressed Matter,
7?vv = 0
2. Test Bodies
d2x"/dX2 + TUdxVd).)(dxx/dX) = 0
3. Maxwell's Equations
H. Differences between this version and the TTEG version
1. Adoption of MTW signature ( 1,1,1,1) and index convention (Greek indices
run 0,1,2,3; Latin run 1,2,3)
2. New symbol for Whitehead parameter: ^ instead of Cn- as in Will (1973)
3. Modified conservation-law parameters incorporating effects of Whitehead
term (see Lee et al., 1974)
The Parametrized Post-Newtonian Formalism
105
4.4
Conservation Laws in the PPN Formalism
Conservation laws in Newtonian gravitation theory are familiar:
for isolated gravitating systems, mass is conserved, energy is conserved,
linear and angular momenta are conserved, and the center of mass of
the system moves uniformly. This does not apply to every metric theory
of gravity, however. Some theories violate some of these conservation
laws at the post-Newtonian level, and it is the purpose of this section to
explore such violations using the PPN formalism.
One can distinguish two kinds of conservation laws: local and global.
Local conservation laws are laws that are valid in any local Lorentz
frame, and are independent of the metric theory of gravity. They depend
rather, upon the structure of matter that one assumes. Global conservation laws, however, are statements about gravitating systems in asymptotically flat spacetime. Because they incorporate the structure of both
the matter and the gravitational fields, they depend on the metric theory
in question.
(a) Local conservation laws
Conservation of baryon number is one of the most fundamental
laws of physics, and should certainly be valid in the presence of gravity.
This law can be expressed as a continuity equation for the baryon number
density n: in a local Lorentz frame momentarily comoving with the matter,
the equation expressing conservation of baryon number 5A
0 = d(SA)/dt = dind V)/dt
is equivalent to
dn/dt + V (nv) = 0
(4.61)
(4.62)
where v is the baryon velocity in the comoving frame (v = 0 but V v =
SV~1d(SV)/dt # 0). The Lorentz-invariant version of this continuity
equation, valid in any local Lorentz frame is
0 = ^(n«°) + ^ ( n ^ ) = (nu")>,
(4.63)
where w" is the baryon four-velocity given by u" = dx^jdx. Equation (4.63)
can then be generalized to any frame in curved spacetime using the standard "comma-goes-to-semicolon" rule
0 = (nu").^
(4.64)
This is the law of baryon conservation in covariant form. If the matter
is assumed to have a chemical composition that is homogeneous and
Theory and Experiment in Gravitational Physics
106
static, then there is a direct proportionality between the baryon number
density (we assume negligible numbers of antibaryons) and the rest mass
density p of the atoms in the element of fluid, namely
p = \m
(4.65)
where \i is the mean rest mass per baryon in the element and a constant.
Proceeding by a similar argument to the one presented above, one obtains
the law of rest-mass conservation,
(pu% = 0.
(4.66)
By combining this equation with the equations of motion for stressed
matter Tfvv = 0 along with the assumption that matter is a perfect fluid,
we obtain a third local law, the law of local energy conservation or the
law of isentropic flow. The equation
ujt:
= 0
(4.67)
may be evaluated, using Equation (3.71). We work in a local Lorentz
frame, momentarily comoving with the element 8V of fluid. From Equation (4.67),
(d/dt)(p + pU) + V (p + pll + p)\ = 0
(4.68)
This can be rewritten
(d/dt){p + pU) + (p + pU) V v + p V v = 0
(4.69)
or,
{d/dt)[(p + pU)8V\ + pd(5V)/dt = 0
(4.70)
So, in a local comoving inertial frame, the change in the total energy (restmass plus internal) of an element of fluid is balanced by the work done
[pd(8V)2: this simply expresses Local Conservation of Energy or Isentropic
Flow, since from the First Law of Thermodynamics, and from Equation
(4.70)
d(energy) + pdV = 2f(heat) = TdS = 0
(4.71)
Actually, the absence of heat flow was built into the stress-energy tensor
from the start by assuming the perfect-fluid form. Had we permitted heat
1
v
heat
ZM
H
where q is a "heat-flux four-vector." For further discussion of nonperfect
fluids see MTW, Section 22.3 and Ehlers (1971).
The Parametrized Post-Newtonian Formalism
107
Because of the conservation of rest mass, pSV is constant, and Equation
(4.70) can be written in the form
pdTl/dt - (p/p)dp/dt = 0
(4.72)
Then in frame-invariant language, Equation (4.72) has the form
«*[n + p(l/p)J = 0
(4.73)
We can obtain a useful form of the law of conservation of rest mass
(or baryon number) by noticing that for any four-vector field, A11,
A^ = (-grll2i{-g)mA"lll
(4.74)
hence
(pu% = (-gr1/2l(-g)ll2pu"l,
=0
(4.75)
In a coordinate system (t, x), Equation (4.75) can thus be written
0 = ip(-g)ll2u0l0 + ip(-g)ll2u°v%
J
(4.76)
J
since u = u°v .
By defining the "conserved density" p*
P* = p(-g)ll2u°
(4.77)
we can cast Equation (4.75) in the form of an "Eulerian" continuity equation, valid in our (t,x) coordinate system:
dp*/dt + V p*v = 0
(4.78)
This "conserved" density is useful because for any function /(x,t)
defined in a volume V whose boundary is outside the matter
(d/dt) j y p*f d3x = J K p*(df/dt)d3x
(4.79)
Notice that Equation (4.79) implies
dm/dt^Q,
m=[ p * d 3 x
(4.80)
where m is the total rest mass of the particles in the volume V; from Equation (4.77), we get,
m \ [pu°(g)ll2~\d3x = Jpd(proper volume)
= total rest mass of particles
(4.81)
(b) Global conservation laws
The conservation laws discussed above are purely local conservation laws; they depend only on properties of matter as measured in local,
Theory and Experiment in Gravitational Physics
108
comoving Lorentz frames, where relativistic and gravitational effects are
negligible (hence they are theory independent). Equation (4.80) represents
our first "global" or "integral" conservation law; it is really nothing more
than conservation of baryons coupled with our specific model for matter.
However, when we attempt to devise more general integral conservation laws, such as for total energy (as opposed to exclusively rest mass),
total momentum, or total angular momentum, we run into difficulties. It
is well known that integral conservation laws cannot be obtained directly
from the equation of motion for stressed matter Tfvv = 0 because of the
presence of the Christoffel symbols in the covariant derivative. Rather,
one searches for a quantity 0" v which reduces to T"v in flat spacetime
and whose ordinary divergence in a coordinate basis vanishes, i.e.,
0?vv = 0
(4.82)
Then, provided 0*" is symmetric, one finds that the quantities
P" = £ ©"v p ^
J"v = 2 £ xl»®vU <*%
(4.83)
are conserved, i.e., the integrals in Equation (4.83) vanish when taken over
a closed three-dimensional hypersurface E. If one chooses a coordinate
system (t, x) in which £ is a constant-time hypersurface that extends infinitely far in all spatial directions, then, provided 0" v vanishes sufficiently rapidly with spatial distance from the matter, P" and J1" are independent of time and are given by
J"v = 2 J x ^ 0V>° d3x
(4.84)
An appropriate choice of 0" v allows one to interpret the components of
P" and J* in the usual way: as measured in the asymptotically flat spacetime far from the matter, P° is the total energy, PJ is the total momentum,
JiJ is the total angular momentum, and J0J determines the motion of the
center of mass of the matter. If 0" v exists but is not symmetric, then P"
is conserved but J"v varies according to
dJ"v/dt= -2 J © M d 3 x
(4.85)
The quantity 0" v , normally called the stress-energy complex, has been
found for the exact versions of general relativity (Landau and Lifshitz,
1962), Brans-Dicke theory (Nutku, 1969b), and others (Lee et al, 1974).
A wide variety of nonsymmetric stress-energy complexes have been devised and discussed within general relativity, but only the symmetric
version guarantees conservation of angular momentum.
The Parametrized Post-Newtonian Formalism
109
There is a close connection between integral conservation laws and
covariant Lagrangian formulations of metric theories. It has been shown
(Lee et al., 1974) that every Lagrangian based, generally covariant metric
theory of gravity that either (i) is purely dynamical (possesses no absolute
variables), or (ii) contains prior geometry, with a simple constraint on the
symmetry group of its absolute variables (a constraint satisfied by all
specific metric theories known), possesses conservation laws of the form
0?vv = 0
where 0" v is a function of certain variational derivatives of the Lagrangian
of the theory that reduces to T** in the absence of gravity. When there
are no absolute variables, the conservation laws are the result of invariance under coordinate transformations, and the stress-energy complexes
0" v are not tensors (or tensor densities); moreover, there may be infinitely
many of them. When absolute variables are present, their symmetry group
produces the conservation laws and 0" v typically are tensors (or tensor
densities). Although &lv is guaranteed to exist for any Lagrangian-based
metric theory, there is no guarantee that it will be symmetric, and no
general argument is known to determine the conditions under which it
will be symmetric.
In the post-Newtonian limit, the existence of conservation laws of the
form of Equation (4.82) can be translated into a condition on values of
some of the PPN parameters. The form of 0" v that we shall attempt to
construct is given by
0*v = (1 - aU)(T"v + t"v)
(4.86)
where a is a constant, and t"v is a quantity (which may be interpreted
under some circumstances as "gravitational stress energy") which vanishes
in flat spacetime, and which is a function of the fields U, UJk, ®w, Vp
Wp ..., their derivatives, and w (and may also contain the matter variables p, II, p, and v). We reject terms in 0" v of the form
w2T»"
since such terms do not vanish in general in regions of negligible gravitational field.
By combining Equations (3.65), (4.82), and (4.86), we find that, to postNewtonian order, t"v must satisfy
"v
(4.87)
Theory and Experiment in Gravitational Physics
110
In our attempt to integrate Equation (4.87) we will make use of Table 4.1
and Equations (4.36) along with the following identity, which is valid for
any function/:
+ 17,,V2/
(4.88)
where
riJ(f)=UAif,j)-iSijVU-\f
(4.89)
Another useful identity is
-2rij(^w
+ 3/4172 - VX Vl/)
^U^ - <5,/l/,0)2] + (2n)-l(d/dt)(U,iU,0)
+ U,£(4ny*V2^ + pv2 + 2p- (87t)-^VL/j2]
(4.90)
where \j/j is the solution of the equation
VV ; = -4npUj
(4.91)
Then, Equation (4.87) can be put into the form
4nt°; = 47i(t°0° + t°>)
2a- 5)|Vt/| 2 ]
+ a - 3 ) l / j ^ , n + (3y + a - 2)UtiU<0\
(4.92)
5/5t[(4y + 4 + OLXWJVM + i(4y + 2 + a, - 2a2 + 2C1)C/,iC/,0
- (5y + a - l)UV2Vi + ia1wI-[/V2t/ + a2Uti(<n
+ a/3x^{[l - (f2 + 4£ - a)C7 + i(a 3 - a i )w 2 ]
+
+
2ri7(0») + (2a3 (1 + a 2 - Ci
2(4y + 4 + «I)(T^,
{Ay + 4 + aiMl/
2 + a i - 2a2 + 2tl)5ij(U,0)2
Vl/) 2 - 17.ow Vl/]
(5y + a- l)U(pv'vJ + pdij) + ziJ} + 4nQ'
(4.93)
The Parametrized Post-Newtonian Formalism
111
where
<£ = i(2y + 2 + « 3 + Ci - 2{)®i + (3y - 2/3 + 1 + C2 +
+ (1 + £3)<J>3 + (3y + 3C4 - 2 0 * 4 ,
(4-94)
x'J = iaiWit/V 2 ^- + ajWj-t/^-t/.o - OL2WJU,,{W VC7),
Q' = t/j[i(«3 +
CI)P»2
2
+ (8n)-K2\vu\
(4.95)
+ c 3 pn
2
+ 3Up + (SnrKiV ** + «3pv w]
(4.96)
It has been found to be impossible to write Q, as a combination of gradients and time derivatives of gravitational fields and matter variables.
Thus, integrability of Equations (4.92) and (4.93) requires that each of the
terms in Q( vanish identically, i.e.,
« 3 = fl = Ci = t 3 = C4 3E 0
(4.97)
These constraints must be satisfied by any metric theory in order that
there be conservation laws of the form of Equation (4.82).
If these conditions hold, then expressions for the conserved energy and
momentum can be obtained using Equations (4.84), (4.86), (4.92), and
(4.93). The results are (after integrations by parts):
,
(4.98)
n + p/p] - | ( i + 0L2)Wi
- fawjUtj} d3x
(4.99)
where we have used the PPN version of the conserved density [Equation
(4.77)]
p* = p[i + %V2 + 3yU + O (4)]
(4.100)
In the expression for P°, the first term is the total conserved rest mass of
particles in the fluid. The other terms are the total kinetic, gravitational,
and internal energies in the fluid, whose sum is conserved according to
Newtonian theory (which can be used in any post-Newtonian terms).
Thus, P° is simply the total mass energy of the fluid, accurate to O(2)
beyond the rest mass, and is conserved irrespective of the validity of the
conditions in Equation (4.97). However, if those conditions were violated,
one would expect violations of the conservation of P° at O(4).
An alternative derivation of the conserved momentum uses Chandrasekhar's (1965) technique of integrating the hydrodynamic equations of
motion T = 0 over all space, and searching for a quantity P' whose time
derivative vanishes. This procedure is blocked by a term ^Qtd3x where Qt
is given by Equation (4.96). This integral can be written as a total time
Theory and Experiment in Gravitational Physics
112
derivative only if (?, can be written as a combination of time derivatives
and spatial divergences (which lead to surface integrals at infinity that
vanish). But according to the reasoning given above, this can be true only
if the five parameter constraints of Equation (4.97) are satisfied. Then
Qi 25 0 and the conserved P' derived by this method agrees with Equation
(4.99).
We now see the physical significance of the parameters <x3, £u £ 2 , C3,
and £4: they measure the extent and manner in which a given metric theory
of gravity predicts violations of conservation of total energy and momentum. If all five are zero in any given theory, then energy and momentum are
conserved; if some are nonzero, then energy and momentum may not be
conserved. According to the theorem of Lee, et al., 1974, every Lagrangianbased metric theory of gravity has all five conservation law parameters
zero. Notice that the parameter a 3 plays a dual role in the PPN formalism, both as a conservation-law parameter and as a preferred-frame
parameter.
In order to guarantee conservation of the angular momentum tensor
J^, t"v must be symmetric. Equations (4.92) and (4.93) show that there are
nonsymmetric terms, xiJ [Equation (4.95)], in tiJ, and that tOi # t'°. However, in integrating Equations (4.92) and (4.93), we have the freedom to
add to the nominal solutions for t"v any quantity S"v that satisfies
Sfvv = 0
(4.101)
iJ
However, we have been utterly unable to find an S that will eliminate or
symmetrize the offending terms tiJ in t'J. As for the toi and ti0 components,
the best we can do is to make use of the identity
d/dt(UV2U + |VC/|2) + d/dxJ(UW2Vj-
Ui0Uj - 2U_kVlkJ s 0
(4.102)
to eliminate or symmetrize one of the offending terms. A convenient
choice is to match the term involving f/V2 Vt in ti0 with an identical term
in toi. With this choice, all dependence on the constant a is eliminated
from ®"v. The result is
2a2)UtOUti 2
-&lWiUV U
2ti-n
=
- a2l/>;w VU,
(4.103)
2 T WI
= ai U Wli V 2 V n - 2a 2 l/ >o w [| .l/, jl + 2a2w[il/,J.jW Vl/
(4.104)
v
Symmetry of t" requires that each of the terms in Equations (4.103) and
(4.104) vanish identically, i.e.,
a!=«2E0
(4.105)
The Parametrized Post-Newtonian Formalism
113
We apply the name Fully Conservative Theory to any theory of gravity
that possesses a full complement of post-Newtonian conservation laws:
energy, momentum, angular momentum, and center-of-mass motion, i.e.,
whose PPN parameters satisfy
ttl
= a2 s a 3 = d = £2 = C3 = U = 0
(4.106)
A fully conservative theory cannot be a preferred frame theory to postNewtonian order since a t = a2 = a3 = 0. For such theories, only three
PPN parameters, y, /?, and f may vary from theory to theory, and &"v
and t"v have the form
0"v = [1 + (5y - 1)[/](T"V + t"v),
too= (8w)-1(4y + 3) |Vt/|2,
tot = t.o = (47t)-1[(2y + lyUjUjo + 4(y
tu = [i _ ( 5 y + 4 |
- 8( T
-i(2y + 1)<50([/,0)2,
O == i(2y + 2 2^)Oj + (3y 2/? + 1 + (3y - 2{)0>4
(4.107)
and the conserved quantities are
P° = J p *(l + ^ 2 - \\J + Jl)d3x
F = Jp*[V(l + iu2 - ^{7 + n + p/p) - i^ £ ] d3x,
J" = 2 Jp*x[I>J1[l + *»2 + (2y +1)C/ + n + p/p]
JOi = Jp*x'(l + if2 - i l / + n)<i3x - PH
(4.108)
1
By defining a center of mass X given by
fp*x'(l+ ?v2 - i[7 + U)d3x
X' = ^
(4.109)
Theory and Experiment in Gravitational Physics
114
we find from Equations (4.108) and the constancy of J 0 ' that
(4.110)
i.e., the center of mass moves uniformly with velocity P'/P°.
Some theories of gravity may possess only energy and momentum
conservation laws, i.e., their parameters may satisfy
a3 = d = C2 = C3 = U s 0,
one of { ai ,a 2 } # 0
(4.111)
We call such theories Semiconservative Theories. Their conserved P"
may be obtained from Equations (4.98) and (4.99); their nonconserved
J"v may be obtained from Equations (4.84), (4.92), and (4.93). A peculiar
feature of the semiconservative case is that in a coordinate system at rest
with respect to the universe, w = 0, and the spatial components t'J are
automatically symmetric, irrespective of the values of at and a2 (since
rij = 0 if w = 0). Thus, spatial angular momentum J'j is a conserved
quantity in this frame, whereas it is not in a moving frame. The center-ofmass component J°\ however, is not conserved in any frame, since
%oi _£ T>o for a n v w jjjj s discrepancy Can be understood by noting that
the distinction between JiJ and J0J is not a Lorentz-invariant distinction.
Because the PPN metric is post-Galilean invariant, the quantities P"
and J"v should transform as a vector and antisymmetric tensor respectively under post-Galilean transformations. This can be verified explicitly
by applying the transformation Equation (4.49) to the integrals that
comprise P" and J"v, with the result, valid to post-Newtonian order
P0'
P'
fr
Ji0'
= P°(l + |u 2 ) - u P,
= P - (1 + i«2)uP° + |u(u P),
= fJ _ j * W + 2(1 + %u2)JoliuJ\
= J'°(l + lu 2 ) - uJJiJ - yu}JJ0
(4.112)
J
where u = u e, is the velocity of the boost. Thus a boost from the universe
rest frame where (d/di)JiJ = 0 to a frame moving with velocity w yields
jtr
=
2J°(V1[1 + O(w2)]
(4.113)
thus, the violation of angular momentum conservation is intimately
connected with.the violation of uniform center-of-mass motion. This is
our reason for stating that semiconservative theories of gravity possess
only energy and momentum conservation laws. Equation (4.113) may be
verified explicitly using Equations (4.103), (4.104), and the fact that
JiJ = 2 J tmd3x,
joi = 2 J tli0]d3x
(4.114)
Every Lagrangian-based theory of gravity is at least semiconservative.
The Parametrized Post-Newtonian Formalism
115
Nonconservative Theories possess no conservation laws (other than
the trivial one for P°); their parameters satisfy
oneoffo.fc.Ca.CW^O
(4.115)
Table 4.2 summarizes these conservation law properties of metric theories
of gravity, and Table 4.3 summarizes the significance of the various PPN
parameters.
Table 4.2. Post-Newtonian integral conservation laws
PPN parameter values
{C1.C2.C3.C4,1*3}
{«i.«2}
Type of theory
all zero
all zero
may be nonzero
all zero
may be nonzero
any values
Fully conservative
Semiconservative
Nonconservative
Conserved
quantities
P", J"v
P"
pOa
" In nonconservative theories, P° is only conserved through lowest Newtonian
order, i.e., to O(2) beyond the conserved rest mass.
Table 4.3. The PPN parameters and their
Parameter
y
/}
£
What it measures, relative
to general relativity"
How much space-curvature
is produced by unit
rest mass?
How much "nonlinearity"
is there in the superposition
law for gravity?
Are there preferred-location
effects?
Are there preferred-frame
effects?
Is there violation of conservation
of total momentum?
significance
Value in Value in
general
semiconservative
relativity theories
Value in
fully conservative
theories
1
y
y
1
/?
j?
Of
0
0
0
0
0
0
0
(
at
a
02
0
0
0
0
0
0
0
0
0
0
0
" These descriptions are valid only in the standard PPN gauge, and should not be construed as
covariant statements. For examples of the misunderstandings that can arise if this caution is not
heeded, especially in the case of P, see Deser and Laurent (1973), and Duff (1974).
Post-Newtonian Limits of Alternative
Metric Theories of Gravity
We now breathe some life into the PPN formalism by presenting a chapter
full of metric theories of gravity and their post-Newtonian limits. This
chapter will illustrate an important application of the PPN formalism,
that of comparing and classifying theories of gravity. We begin in Section
5.1 with a discussion of the general method of calculating post-Newtonian
limits of metric theories of gravity. The theories to be discussed in this
chapter are divided into three classes. The first class is that of purely
dynamical theories (see Section 3.3). These include general relativity in
Section 5.2; scalar-tensor theories, of which the Brans-Dicke theory is a
special case in Section 5.3; and vector-tensor theories in Section 5.4.
The second class is that of theories with prior geometry. These include
bimetric theories in Section 5.5; and "stratified" theories in Section 5.6.
The theories described in detail in these five sections are those of which
we are aware that have a reasonable chance of agreeing with present
solar system experiments, to be described in Chapters 7, 8, and 9. Table
5.1 presents the PPN parameter values for the theories described in these
five sections. The third class of theories includes those that, while perhaps
thought once to have been viable, are in serious violation of one or more
solar system tests. These will be described briefly in Section 5.7.
5.1
Method of Calculation
Despite the large differences in structure between different metric
theories of gravity, the calculation of the post-Newtonian limit possesses
a number of universal features that are worth summarizing. It is just these
common features that cause the post-Newtonian limit to have a nearly
universal form, except for the values of the PPN parameters. Thus, the
computation of the post-Newtonian limits of various theories tends to
Table 5.1. Metric theories of gravity and their PPN parameter values
PPN parameters"
Cosmological
Theory and its
gravitational fields
Arbitrary functions
or constants
ma idling
parameters
y
P
(a) Purely dynamical theories
(i) General relativity (g)
(ii) Scalar-tensor (g, <j>)
none
none
l
1
0
0
0
0
00
1 +0)
2 + 0)
1+A
0
0
0
0
BWN
«1
a2
(«3,0
Bekenstein's VMT
o)(0),r,
th
1 +0)
2 + 0)
1+A
0
0
0
0
Brans-Dicke
CO
ih
1+0)
2 + o)
1
0
0
0
0
0)
K
K
K
y
a'2
i
1
0
0
0
0
0
0
i
i
1
1
(iii) Vector-tensor (g, K)
General
Hellings-Nordtvedt
Will-Nordtvedt
(b) Theories with prior geometry
(iv) Bimetric theories
Rosen (g, 9)
Rastall(g,ir,K)
BSLL(g, V) B)
(v) Stratified theories
none
none
none
a,f,k
Wo
<Po
K
l
co,cua,b,c,d
co,cua,b,c,d
0
0
0
P
0
0
0
bc0
0
%
(co/ci)-l
a'2
a'2
aco/cl
«2
0
0
0
0
0
" Prime over a PPN parameter (e.g., / ) denotes a complicated function of arbitrary constants and cosmological matching parameters. See text
for explicit formulae.
Theory and Experiment in Gravitational Physics
118
have a repetitive character, the major variable usually being the amount
of algebraic complexity involved. In order to streamline the presentation
of specific theories in the following sections, and to establish a uniform
notation, we present a "cookbook" for calculating post-Newtonian limits
of any metric theory of gravity.
Step 1: Identify the variables: (a) dynamical gravitational variables
such as the metric g^, scalar field <f>, vector field K", tensor field J5^v,
and so on; (b) prior-geometrical variables such as a flat background
metric n^, cosmic time function t, and so on; and (c) matter and nongravitational field variables.
Step 2: Set the cosmological boundary conditions. Assume a homogeneous isotropic cosmology, and at a chosen moment of time and
asymptotic coordinate system define the values of the variables far from
the post-Newtonian system. With isotropic coordinates in the rest frame
of the universe, a convenient choice that is compatible with the symmetry
of the situation is, for the dynamical variables,
9^ -> gfv =
diag{-co,ci,cuci),
<p -></> 0 ,
*->(*:, o,o,o),
B^ -* B$= diag(coo, <ou cou coj (5.1) and for the prior-geometric variables (these values are valid everywhere, since these variables are independent of the local system), t = t, with Vt = (l,0) (5.2) The relationships among and the evolution of these asymptotic values will be set by a solution of the cosmological problem. Because these asymptotic values may affect the values of the PPN parameters, a complete determination of the post-Newtonian limit may in fact require a complete cosmological solution. This can be very complicated in some theories. For the present, we shall avoid these complications by simply assuming that the cosmological matching constants are arbitrary constants (or more precisely, arbitrary slowly varying functions of time). In Chapter 13, we shall turn to the cosmological question and discuss the relationship between cosmological models and observations that may fix the asymptotic values of the fields and post-Newtonian gravity. Notice that if a flat background metric q is present, it is almost always most convenient to work in a coordinate system in which it has the Minkowski form, for in Post-Newtonian Limits 119 many theories the resulting field equations involve flat-spacetime wave equations, which are easy to solve. Then the asymptotic form of g shown is determined by the cosmological solution. If r\ is present it is not generally possible (unless in a special cosmology or at a special cosmological epoch) to make both it and g have the asymptotic Minkowski form simultaneously. Of course, once the post-Newtonian metric g has been determined, one can always choose a local quasi-Cartesian coordinate system [see Section 4.1(c)] in which it takes the asymptotic Minkowski form. The form that IJ now takes is irrelevant since, unlike g, it does not couple to matter. In theories without if, it is usually convenient to choose asymptotically Minkowski coordinates right away. Step 3: Expand in a post-Newtonian series about the asymptotic values: Guv Gfiv ' 'Vv) <f> = 4>0 + <p, K^iK + ko,fcl5fe2,/c3), B,v = &°> + & (5.3) Generally, the post-Newtonian orders of these perturbations are given by ~ O(2) Hh 0(4), 9 ~ O(2) H- 0(4), k0 ~ O(2) HH 0(4), "00 boo~ O(2) H- 0(4), » /<y ~ 0(3), htj ~ 0(2), /c,- ~ 0(3), Z»oi ~ 0(3), fty ~ O(2) (5.4) Step 4: Substitute these forms into the field equations, keeping only such terms as are necessary to obtain a final, consistent post-Newtonian solution for h^. Make use of all the bookkeeping tools of the postNewtonian limit (Section 4.1), including the relation (d/dt)/(d/dx) ~ O(l). For the matter sources, substitute the perfect-fluid stress-energy tensor T"v and associated fluid variables. Step 5: Solve for h00 to O(2). Only the lowest post-Newtonian order equations are needed. Assuming that h00 -»0 far from the system, one obtains the form /loo = 2aU (5.5) where U is the Newtonian gravitational potential [Equation (4.2)], and where a. may be a complicated function of cosmological matching parameters and of other coupling constants that may appear in the theory's field equations (such as a "gravitational constant"). To Newtonian order, Theory and Experiment in Gravitational Physics 120 the metric thus has the form 0 o o = -co + 2ccU, gOJ = 0, giJ = Sif1 (5.6) To put the metric into standard Newtonian and post-Newtonian form in local quasi-Cartesian coordinates, we must make the coordinate transformation x5 = (c o ) 1 / 2 x 0 , x1 = (Cl)ll2xJ 065 = Co ^ o o , 06J = ( c o c i ) " il2g0j, (5.7) then 9iJ = cf £7 = c x t7 1 gii, (5.8) and goo = - 1 + 2(cc/coCl)U, 0sj = 0, gij = dtj (5.9) Because we work in units in which the gravitational constant measured today far from gravitating matter is unity, we must set Gtoday = a/coC! = 1 (5.10) The constraint provided by this equation often simplifies other calculations, however there is no physical constraint implied; it is merely a definition of units. Step 6: Solve for hu to O(2) and h0J to O(3). These solutions can be obtained from the linearized versions of the field equations. The field equations of some theories have a gauge freedom, and a certain choice of gauge often simplifies solution of the equations. However, the gauge so chosen need not be the standard PPN gauge (Section 4.2), and a gauge (coordinate) transformation into the standard gauge [Equations (4.40) and (4.46)] may be necessary once the complete solution has been obtained. Step 7: Solve for h00 to O(4). This is the messiest step, involving all the nonlinearities in the field equations, and many of the lower-order solutions for the gravitational variables. The stress-energy tensor T"v must also be expanded to post-Newtonian order. Using Equations (3.71), (5.6), and (5.10), we obtain T00 = Co V [ l + n + 2cxU + Co'cy T' J = Co'pv'vJ + c^'pS1' + pO(4) + O(4)], (5.11) Step 8: Convert to local quasi-Cartesian coordinates [Equation (5.7)] and to the standard PPN gauge (Section 4.2). Post-Newtonian Limits 121 Step 9: By comparing the result for g^ with Equation (4.48), or with Table 4.1 (with w = 0), read off the PPN parameter values. In obtaining these post-Newtonian solutions, the following formulae are useful u = -iv 2 x , \\U\2 = V2(iU2 - O2) (5.12) along with Equations (4.29), (4.33), (4.36), and (4.37). 5.2 General Relativity (a) Principal references: Standard textbooks such as MTW and Weinberg (1972). (b) Gravitationalfieldspresent: the metric g. (c) Arbitrary parameters and functions: None (we shall ignore the cosmological constant, which is too small to be measured in the solar system). (d) Cosmological matching parameters: None. (e) Field equations: The field equations are derivable from an invariant action principle 51 = 0, where +W«A,0U (5-13) where R is the Ricci scalar [Equation (3.86)] and JNG is the universally coupled nongravitational action, and G is the gravitational coupling constant. By varying the action with respect to g^, we obtain the field equations (5.14) (f) Post-Newtonian limit: Because g is the only gravitational field present, we can choose it to be asymptotically Minkowskian without affecting any otherfields.Thus we have initially c0 = cl = 1. It is convenient to rewrite the field Equation (5.14) in the equivalent form R^ = 8TTG(T,V - i ^ T ) where T = T^^. the form (5.15) To the required order in the perturbation h^, R^v has j nk0,jk + nkk,Oj ~ fcy - fcoo.« + Kk.ii ~ hki,kj ~ hkJ,k,i) Theory and Experiment in Gravitational Physics 122 (i) h00 to O(2): To the required order, Roo = -iV 2 fc 0 0 , TOo = - T * p, 0oo = - 1 (5-17) thus V2h00 = -8nGp, h00 = 2GU (5.18) We now choose units in which G = 1, hence Ko = 21/ (5.19) (ii) hy to O(2): If we impose the three gauge conditions (i = 1,2,3) K, - \Ki = o, K = yf*hH (5.20) Equation (5.16) for Rtj becomes V2fcy=-8«piw, hiJ = 2UStJ (5.21) (iii) h0J to O(3): If we impose the further gauge condition ^ - R o = -ifcoco (5-22) Equation (5.15) becomes V2h0j + U.oj = 16npvj (5.23) or, using Equations (4.29), (4.32), and (4.33), h0J = -4Vj + ix.o; =-ty-iWj (5.24) It is useful to check that the solutions for h00, hOj, and htJ do satisfy the gauge conditions, Equations (5.20) and (5.22), to the necessary order. (iv) h00 to O(4): In the chosen gauge, Roo evaluated correctly to O(4) using the known lower-order solutions for h^v where possible, has the form Roo = -iV 2 (/j 0 0 + 2U2 - 8<D2) (5.25) To the necessary order, we also have Too - k o o T = M l + 2(v2 -U + ±I1 + fp/p)] (5.26) Then the solution to Equation (5.15) is h00 = 21/ - 2U2 + 4$ t + 4<D2 + 2<D3 + 6O4
(5.27)
(v) g^ and the PPN parameters: The final form for the metric is
<?oo = - 1 + 21/ - 2[/ 2 + 4 ^ + 4<D2 + 2«D3 + 64>4,
gtJ = (1 + 2l/)5y
(5.28)
Post-Newtonian Limits
123
Since the metric is already in the standard PPN gauge, the PPN parameters can be read off immediately
y = p = 1,
£ = 0,
ai
= a 2 = a 3 = Ci = C2 = C3 = C4 = 0
(5.29)
(g) Discussion: Notice that general relativity is a fully conservative
theory of gravity (af = £; = 0) and predicts no preferred-frame effects
(a* = 0).
5.3
Scalar-Tensor Theories
A variety of metric theories of gravity have been devised which
postulate in addition to the metric, a dynamical scalar gravitational field
<t>. The most general such theory was examined by Bergmann (1968) and
Wagoner (1970), and special cases have been studied by Jordan (1955),
Thiry (1948), Brans and Dicke (1961), Nordtvedt (1970b), and Bekenstein
(1977). We shall examine the Bergmann-Wagoner theory in detail, then
shall discuss the various special cases.
(a) Principle references: Bergmann (1968), Wagoner (1970).
(b) Gravitational fields present: the metric g, a dynamical scalarfield<j>.
(c) Arbitrary parameters and functions: Two arbitrary functions of (j),
the coupling function a>{4>) and the cosmological function A(#).
(d) Cosmological matching parameters: <f>0.
(e) Field equations: The field equations are derived from the action
^
(5.30)
The resulting field equations are
(5.31)
(532)
The field equation for (j> can be rewritten by substituting the contraction
of Equation (5.31) into Equation (5.32), with the result
dco
^ 4
The cosmological function l(<f>) causes two effects in this theory. First,
in the field equation for g, it plays the same role as the cosmological
Theory and Experiment in Gravitational Physics
124
constant in general relativity. Second, in the field equation for <j>, it gives
the scalar field <l> a range I related to X, co and their derivatives, in the
sense that the solutions for cf> for an isolated system contain Yukawa-like
terms exp(r/l). The result in g00 (Wagoner, 1970) is a "Newtonian
gravitational potential" U of the form
^x,0 = J ^ j ^ ^ ^ V
(5.34)
where the effective gravitational "constant" is given by
G(x - x') = a + bexp(- |x - x'\/l)
(5.35)
Experiments that test the inverse square law for gravitation (see Section 2.2) could thus set limits on the cosmological function X. However,
henceforth we shall assume X = 0.
(f) Post-Newtonian limit: We choose coordinates (local quasi-Cartesian)
in which g is asymptotically Minkowskian; <j) takes the asymptotic
value (f>0 (which presumably varies on a Hubble timescale as the universe
evolves). We define
co = co(<p0),
co =
A == <u'(3 + 2co)- 2(4 + 2co)-1
(5.36)
Following the method of Section 5.1, we obtain for the post-Newtonian
metric
goo= -1 +2U - 2 ( 1 +A)U2
+ 4 ( , . - A )<E>2 + 2«D3 + 6 ( ^ - |<D4
In going to geometrized units, we have set
1
<5J8)
Notice that if c/>0 changes as a result of the evolution of the universe,
then Gtoday may change from its present value of unity (see Section 8.4).
Post-Newtonian Limits
125
The PPN parameters may now be read off:
a t = a 2 = a 3 = d = C2 = £3 = U = 0
(5.39)
For details of the derivation see Nutku (1969a), Nordtvedt (1970b).
(g) Other theories and special cases: (i) Nordtvedt's (1970b) scalar-tensor
theory is equivalent to the Bergmann-Wagoner theory in the special
case of zero cosmological function X = 0. Its PPN parameters are the
same as in the Bergmann-Wagoner theory. We shall denote these general
versions the BWN scalar-tensor theories.
(ii) Brans-Dicke theory is the special case a> = constant, 1 = 0. Its
PPN parameters may be obtained from the BWN PPN parameters
by setting a>' = 0 s A. In the limit a> -* oo, the Brans-Dicke theory
reduces to general relativity.
(iii) Bekenstein's (1977) Variable Mass Theory (VMT) is a special case
of the BWN theory with a restricted form for the coupling function
oi((j)). Beginning with a theory in which the rest masses of elementary
particles are allowed to vary in spacetime via a scalar field <f>, the variation
being determined by a field equation with two arbitrary parameters r
and q, Bekenstein has shown that, when transformed to a metric representation, the theory is a BWN scalar-tensor theory with
l][r + (1 - r)qf(<f>)y2,
0 = [1 - qf(4>)~\f(<t>rr
(5-40)
Note that for chosen values for r and q, the present values of a> and A
are determined by the asymptotic value </>0, which in turn is found through
a cosmological solution using the theory. For further details, see Bekenstein and Meisels (1978,1980) and Bekenstein (1979).
(iv) Barker's Constant G Theory (1978) is the special case in which
0 * 0 = (4 - 30)/(20 - 2)
(5.41)
G,oday = 1 = [constant]
(5.42)
A = (1 - </>o)/2<Ao = - ( 8 + 4co)-x
(5.43)
thus
and
Theory and Experiment in Gravitational Physics
126
(h) Discussion: We note that scalar-tensor theories are all fully conservative theories (a, = £, = 0), with no preferred-frame effects (<x; = 0).
In the limit a -* oo, they reduce to general relativity, both in the postNewtonian limit and in the exact, strong-field theory, for all except a
set of measure zero of pathological coupling functions co(<t>). In particular,
this is true for Brans-Dicke theory, Bekenstein's VMT, and Barker's
theory. Generally speaking, for large values of a>($0), these theories make predictions at the current epoch for all gravitational situations - postNewtonian limit, neutron stars, black holes, gravitational radiation, cosmology - that differ from general relativity at most by corrections of O(l/co). However, in theories in which co is a function of <t>, there could be significant differences with general relativity in the early universe, even if the present value of co(0o) is large (Chapter 13). In Brans-Dicke theory (constant co) all predictions are within O(l/co) of those of general relativity [see Ni (1972) for an extensive list of references for Brans-Dicke theory]. 5.4 Vector-Tensor Theories Within the class of purely dynamical metric theories of gravity, one simple way to devise a theory that is different from the scalar-tensor theories is to postulate a dynamical four-vector gravitational field K" in addition to the metric, thus obtaining a vector-tensor theory of gravity. A broad class of such theories can be analyzed if we restrict attention to Lagrangian-based theories, and to theories whose differential equations for the vector field are linear and at most of second order. The most general gravitational action for such theories is given by i G = (167CG)-1 §[atR + a2KliK"R + aJPlTR^ 2 + a4Kp.vK":v (5.44) (we have ignored the possible term K^ICg^, since it presumably plays the same role as the cosmological function A in scalar-tensor theories). In fact this action is too general; it can be simplified by an integration by parts, dropping divergence terms which do not contribute to the variation of /. Thus the sixth term in JG can be eliminated. Furthermore, the constant a! can be absorbed into G, resulting in a four-parameter set of vector-tensor theories. (a) Principal references: Will and Nordtvedt (1972), Hellings and Nordtvedt (1973). Post-Newtonian Limits 127 (b) Gravitational fields present: The metric g, a dynamical vector field K (assumed timelike). (c) Arbitrary parameters and functions: Four arbitrary parameters co, rj, e, T. (d) Cosmological matching parameters: K. (e) Field equations: The field equations are derived from the action + xK^K^J - gf'2 d4x + ING(qA, gj (5.45) where F,v = *,. - KK, (5.46) The resulting field equations are = SnGT^, eFfvv v v + \tK% - ^coK'R - iriK R$ = 0
(5.47)
(5.48)
where
0£> - K^R
+K ^
- \g^K2R - (K%v
= 2K"K(I1RV)X - faj
+ (K*K(fl;V) K*(flKv) K ( / J K^). a
(5.49)
where K2 = K^K". Throughout, we assume that one of {e, T} is nonzero
in order to have a well-defined free dynamical vector field. An important
property of these equations is worth examining here. If one takes a co variant divergence of the left-hand side of Equation (5.47), one finds explicitly
that it vanishes, in agreement with the law Tfvv = 0, in other words, no
additional constraint on the fields is imposed by the vanishing divergence
of T"v. This is a result of the fact that the action / is generally covariant
and contains no prior-geometric variables [see Lee, Lightman, and Ni
(1974) for discussion]. However, a divergence of the left-hand side of
Equation (5.48) yields the constraint
ll
- (a)K»R + tilPR/i).,, = 0
(5.50)
Theory and Experiment in Gravitational Physics
128
This is a result of the fact that the action is not fully gauge invariant,
i.e., invariant under the transformation
K, -> K^ + A
(5.51)
where A is a scalar function. Only the term involving F p v is gauge invariant.
A Lagrangian that admits such a partial gauge group is called "singular,"
and can be shown to satisfy a "Bianchi identity," which, in the case of
the partial gauge group of a vector field, has the form
. .,. = 0
(5.52)
This is equivalent to Equation (5.50). This means that, in general, the
solution for K^ will be constrained. It is useful to examine the form that
this constraint takes in the linearized approximation, in which we write
g^ = n^ + Ky
K
? =
K3
° +K
( 5 - 53 )
If we adopt a coordinate system (coordinate "gauge" as opposed to vector
gauge) in which
>,*; - ±h-° = 0
(5.54)
where indices on h^ and kv are raised and lowered using if, and where
h == hi, Equation (5.50), to first order in h^ and fcp, takes the form
{3v{?k*v jK(a> + ^n \x)h 0 } = 0
(5.55)
Since this equation must be satisfied for arbitrary sources, then to first
order in h^ and k^ we must have
+ %n- %i)ht0 = 0
(5.56)
In the weak field limit, in the chosen gauge, h^v must have the form
h00 = 2(7,
hu = 2yUStJ - (y - l)x,,j
(5.57)
where y is the PPN parameter. Then Equation (5.56) becomes
T/cyv - 2K{co + \n - ±r)(2y - 1)17.0 = 0
(5.58)
In the case x =£ 0, this represents a constraint on the gauge of the vector
field k^ imposed by the lack of full gauge invariance of the action /. In
the case x = 0, no constraint is placed on the vector field; however, in
order to obtain consistent solutions of the equations, with a hope of
agreeing with experiment, we must have co + \r\ = 0, since K # 0 and
t / 0 ^ 0 in general, and since experiments (Chapter 7) place the value of
Post-Newtonian Limits
129
y close to unity, so that 2y 1 ^ 0. These constraints will be important
in our discussion of the post-Newtonian limit.
(f) Post-Newtonian limit: We choose local quasi-Cartesian coordinates
in the universe rest frame, with K^ taking the asymptotic form K8®, where
K may vary on a Hubble timescale. Following the method of Section 5.1,
we compute the post-Newtonian limit, and obtain for the PPN parameters
_ 1 + K2[co - 2co{2co + n~ -Q/(2e - T)]
7
~
1 - K2[co + 8« 2 /(2e - T )]
/*= i(3 + y) + M i
ai
+ y(v - 2)/G],
= 4(1 - y)[l - (2e - T)A] + 4coK2 Aa,
a2 = 3(1 - y)[l - |(28 - T)A] + 2coK2 Aa -
\bK2IG,
<*3 = Ci = Ci = Cs = U - 0
(5.59)
The quantities a, A, a, and b are given by
(1 - coK2)(2co -n + 2£) _ (1
- T) - 8o>2K2
A = {(2e - t)[l - K2(co + n - T)] + i ( ^ -
2
T)2K2}-\
a = (2B - r)(3y - 1) - 2(n - T)(2 7 - 1),
f(2o> + n - x)[(2y - l)(r + 1) + <r(y-2)]
= \ -(2y - l)2(2co + i/)[l - T - 1(2co + »,)],
0,
%# 0
T= 0
(5.60)
and G is related to the other parameters by our choice of geometrical
units, namely
Gtoday s G[i(y + 1) + f coK2(y - 1) - i(ij - T)K 2 (1 + <r)] "» = 1
(5.61)
(g) (Mer theories and special cases: (i) The Will-Nordtvedt (1972)
theory is the special case a> = n = s = 0, x = 1. Its PPN parameters are
given by
y = fi = 1,
ox = 0,
f = a 3 = d = C2 = C3 = C4 = 0,
a 2 = K 2 /(l + | K 2 )
(5.62)
with
2
)=l
(5.63)
Theory and Experiment in Gravitational Physics
130
(ii) The Hellings-Nordtvedt (1973) theory is the special case T = 0,
e = 1, r\ = 2ft). Its PPN parameters are given by
l + coK2
o
.
.
/ 1 + coy
= « 3 = d = C2 = Cs = C* = 0,
4coK\2(l + co)y + co(y - 1)]
1 + a)X 2 (l + co)
0(2
_2c»K[y + c o ( y l ) ]
~ 1 + coKHl + co)
(5>64)
with
G,oday = G[cDK\y + 1)] - 1 = 1
(5.65)
We point out that the original computations of Hellings and Nordtvedt
(1973) were in error, since their method failed to take into account the
constraint Equation (5.58).
(h) Discussion: These vector-tensor theories are semiconservative
(a3 = C, = 0) with possible post-Newtonian preferred-frame effects (one
of{a x ,a 2 } ¥" 0). In the limit {co, r\, E, T} -» 0, they reduce to general relativity
both in the post-Newtonian limit, and in the exact, strong field theory.
However, there are other possible limiting cases in which the theories
may agree with general relativity (and thus with experiment) in the postNewtonian limit. For instance, in the limit K -»0, the PPN parameters
coalesce with those of general relativity. However, the present value of
K depends upon a solution of the cosmological problem, and in the
early universe K could be sufficiently large to produce significant differences.
5.5
Bimetric Theories with Prior Geometry
Theories in this class contain dynamical scalar, vector, or tensor
gravitational fields, and a nondynamical metric ij of signature + 2. In
typical theories, t\ is chosen to be Riemann flat everywhere in spacetime,
that is
Rlem(i/) = 0
(5.66)
(in some versions, IJ is chosen to correspond to a spacetime of constant
curvature). Because of the above constraint, we can always choose global
coordinates in which t]^ = diag( 1,1,1,1); this is usually the most convenient choice for the computation of the post-Newtonian metric.
Post-Newtonian Limits
131
Rosen's bimetric theory
(a) Principal references: Rosen (1973,1974,1977,1978), Rosen and Rosen
(1975), Lee et al. (1976).
(b) Gravitationalfieldspresent: the metric g, a flat, nondynamical metric
flic) Arbitrary parameters and functions: None.
(d) Cosmological matching parameters: co,cy.
(e) Field equations: The field equations are derived from the action
= (647TG)x (-V)ll2d*x + ING(qA,g,v)
(5.67)
where the vertical line " |" denotes covariant derivative with respect to
The field equations may be written in the form
Riemfo) = 0
(5.68)
1
where , is the d'Alembertian with respect to q, and T s T^g .
(f) Post-Newtonian limit: We choose coordinates in which if has the
form diag( 1,1,1,1) everywhere. In the universe rest frame, g then has
the asymptotic form diag(c^c^c^c^) [see Equation (5.1)], where c0
and c t may vary on a Hubble timescale. Following the method of Section
5.1, we obtain for the PPN parameters (Lee et al., 1976)
y = p = 1,
Kl = 0,
£ = « 3 = Ci = C2 = C3 = U = 0,
a2 = (c o / Cl ) - 1
(5.69)
with
Gtoday = G{coCl)112
= 1
(5.70)
(g) Discussion: The PPN parameters are identical to those of general
relativity except for a 2 , which may be nonzero if c0 # cx. Notice that
the ratio cjco is equal to the square of the velocity of weak gravitational
waves, in units in which the speed of light is unity. This can be seen as
follows. In a quasi-Cartesian coordinate system, in which gffl = diag( 1,
1,1,1 ),»; must have the form
n^ = diag( - Co \ c^ \ erf *, c r ' )
and the vacuum, linearized field equations for g^v (wave equations for
weak gravitational waves) take the form
(co/cite^oo -
V
V =0
(5.71)
112
whose solution is a wave propagating with speed vg = (cjco) . Thus,
in Rosen's theory, the PPN parameter a2 measures the relative difference
Theory and Experiment in Gravitational Physics
132
in speed (as measured by an observer at rest in the universe rest frame)
between electromagnetic and gravitational waves. The values of c 0 and
Cj are determined by a solution of the cosmological problem. They can
also be related to the covariant expressions
CQ i + 3cf* = /7JIV0<O)"V
c 0 + 3cj = n^gtg!,
Rastall's theory
(a) Principal references: Rastall (1976, 1977a,b,c, 1979).
(b) Gravitational fields present: the metric g, a dynamical timelike vector
field K, a nondynamical flat metric r\.
(c) Arbitrary parameters and functions: None.
(d) Cosmological matching parameters: K.
(e) Field equations: The physical metric g is an algebraic function of
the fields if and K, given by
g = (1 + rfKJS.,)-^2^
+ K ® K)
(5.72)
where \\rfp\\ = H^H" 1 . The field equations are derivable from the action
:+ W«A»M
(5-73)
where indices on K^ are raised using g, and where
F(N) = - N(2 + N)~ \
N = g^K^
(5.74)
We also have Riem(if) = 0. The resulting field equations are
0)-
1/2
(0" v - k
= 87tG(l + n^KxK^yll2{T"v
- ^VT)KV
(5.75)
where F'(N) = dF/dN,
+ F'{N)KX'IIKO[.I)K'1KV
(5.76)
and & = ©''"^v, T= T" v ^ v . In varying the action /, with respect to
Kp, we have taken account of the fact that the dependence on K^ is both
explicit and implicit via gMV, thus for example, although the action for
matter and nongravitational fields 7NG contains only g^, we have
(5.77)
Post-Newtonian Limits
133
(f) Post-Newtonian limit: We choose coordinates in which n^ =
diag( 1,1.1.1), then from Equation (5.72) g^ takes the form, to postNewtonian order
0oo = - c o ( l - Kco2k0 - |CQ 4feg),
gOJ =
Kc^kj,
gij = ColSJk(l + Kco2k0)
(5.78)
where c 0 = (1 K2)112, \K\ < 1. Solving the field equations for k^ to the
required order, substituting into Equation (5.78), and transforming to
local quasi-Cartesian coordinates in the standard PPN gauge yields the
PPN parameters
y = /? = 1,
Z. = a 3 = d = C2 = C3 = U = 0,
In choosing geometrical units, we set
Gtoday = G = 1
(5.80)
(g) Discussion: RastalFs theory is semiconservative (a 3 = Ci 0), with
preferred-frame effects (a2 # 0). Its PPN parameters are identical to those
of general relativity, except for a2, which maybe nonzero. The value of
<x2 depends upon K, whose value is determined by a solution of the cosmological problem.
The BSLL bimetric theory
This theory is a variant of the Belinfante-Swihart nonmetric
theory of gravity, discussed in Section 2.6. Instead of the nongravitational
action / N G shown in Equations (2.140) and (2.141), one chooses a universally coupled action, thereby obtaining a metric theory of gravity
(Lightman and Lee, 1973b). Otherwise the equations of the theory are
the same as those presented in Section 2.6.
(a) Principal references: Belinfante and Swihart (1957a,b,c), Lightman
and Lee (1973b).
(b) Gravitational fields present: the metric g, a dynamical second rank
tensor field B, a nondynamical flat metric i\.
(c) Arbitrary parameters and functions: three arbitrary parameters a,
f,K.
(d) Cosmological matching parameters: a>0, (o^.
Theory and Experiment in Gravitational Physics
134
(e) Field equations: The metric is constructed algebraically from t\ and
B according to the equations
; - ± 3 D = <5v
(5-81)
where indices on Apv and B^ only are raised and lowered using n^;
indices on all other tensors are raised and lowered using g^; B = B^n*".
The field equation for i\ is Riem(//) = 0. The field equations for B are
derived from the action
/ = -(167c)" 1 j(aB"^B^x
+ /B^X-i/)1'2**** +
W«A,0,«)
(5-82)
where vertical line denotes a covariant derivative with respect to r\. The
resulting field equations are
, ,
(5.83)
which may be rewritten in the form
D ^
= -(4w/fl)to/f7)1/27^[0|5 - f(a + 4f)-ieil>r,»%d]
(5.84)
where
Kl = 8gxl,/dB,v
(5.85)
(f) Post-Newtonian limit: We work in the universe rest frame, choose
coordinates in which n^ = diag( 1,1,1,1), and assume that
o.co^co^co!). We further assume that |coo| « 1, \a>^ « 1, assumptions that turn out to be consistent with experimental limits. Then to the
necessary order, g^ has the form
fifoo = -Do + E0b00 - Fob - K2b2 - 2Kbb00 - |fcg 0 .
g0J = HbOj,
SiJ
= Ddtj + EbtJ + FdiP
(5.86)
where
Do = 1 - 2Kco -coo + K2co2 + 2Kcoco0 + |coo + O(co3),
Eo = 1 - 2Kco - f a»0 + O(co2),
Fo = -2K
+ 2K2co + 2Kco0 + O(co2),
H = 1 - 2Kco - |(co o - co^ + O(a>2),
D = 1 - 2Kw + w1+ K2co2 - 2Kcoco1 + |cof + O(co3),
E = 1 - 2Kco + Icoj + O(co2),
F= -2K
+ 2K2co - 2Kco1 + O(co2),
co = 3(0! coo
(5.87)
Post-Newtonian Limits
135
Solving the field equations for fcJlv, substituting into Equation (5.86), then
transforming to quasi-Cartesian coordinates and to the standard PPN
gauge yields the PPN parameters
p = i [ l + l a " 1 - ia-^Sa2
- 3a)1/2(a + 4/)- 1 / 2 ] + O(co),
i = a 3 = Ci = £2 = C3 = U = 0,
a t = (2a)~1[ojo + » ! - (8X - 2)w] + O(co2),
a 2 = (<o0 + coi) + O(co2)
(5.88)
In using geometrized units, we set
_a + 3/-4Xa-16*2a
"today
/) j
. jn
T- vj^ti); 1.
(J.O?,)
2a(a + 4/)
(g) Discussion: The BSLL Theory is semiconservative (a3 = Ct = 0),
with potential preferred-frame effects if a>0 or col are nonzero. However,
solar system experiments (Chapter 8) demand that la^ and |a 2 | be small,
in keeping with our original assumption that jcoo| « 1, leo^ « 1. Whether
a>0 and co1 in fact satisfy this constraint depends upon a solution of the
cosmological problem. Notice that if m^ ~ a>0 0, the PPN parameters
can be made identical to those of general relativity if
0 = (i -A,*}
(5-90)
5.6
Stratified Theories
These theories are characterized by the presence, in addition to a
flat background metric t\, of a nondynamical scalar field t whose gradient
is covariantly constant and timelike with respect to tj, i.e.,
This scalar field selects out preferred spatial sections or "strata" in the
universe that are orthogonal to \t. In a frame in which V / = <5°, the equations of a stratified theory take on some special form.
(a) Principal references: Lee, Lightman, and Ni (1974), Ni (1973).
(b) Gravitational fields present: the metric g; dynamical scalar, vector,
and symmetric tensor fields <p, K, B; nondynamical flat metric i\ and scalar
field t.
(c) Arbitrary parameters and functions: functions /1 (</>), fii^
parameters e, KU K2.
(d) Cosmological matching parameters: co,cud,a,b, c.
Theory and Experiment in Gravitational Physics
136
(e) Field Equations: The field equations for the prior-geometric variables are
Riem(i;) = 0,
*l,* = 0,
t/^=-l,
= 0,
= 0
(5.91)
The last two equations constrain the vector and tensor fields to have
components only in the strata orthogonal to \t. The metric g is constructed
algebraically from t/, <j>, t, B, and K according to
9 = / 2 ( # / - E/iW>) - / 2 (0)]dt ® d t + K ® d t + d t ® K + B
(5.92)
The field equations for the dynamical variables are derived from the
action
- < £ > * - \_Mcj>)
+ / N G (q A ,^ v )
(5.93)
where all indices on the variables <f>, t, B, and K are raised and lowered
using i\. The result is
\_M<t>)
(5.94)
The constraints on the prior geometric variables allow one to choose
a global coordinate system in which t]^ = diag( 1,1,1,1), ttll = d°,
Ko = B^o = 0, and in which the field equations simplify to
'1((/») -
Tilf'2(4>)l
(5.95)
In this preferred frame (presumably the universe rest frame), g^v has the
form
9oj =
Kj,
9u = 8M4)
+ Bij
(5-96)
Post-Newtonian Limits
137
(f) Post-Newtonian limit: In the preferred frame we expand <j>, Kit and
(j> = 4>0 + <P,
Kt = k,,
By =
fi)15y+fty
(5.97)
We then define the cosmological matching parameters c 0 , cu a, b, c, d
according to
M<t>) = c0 - 2c<p + 2bc2(p2 + O(<p3),
fi((t>) = (ci - cOi) + 2ac<p + O(q>2),
M4) = d + O(<p)
Then to post-Newtonian order, g^ has the form
(5.98)
0oo = - c 0 + 2ccp - 2bczcp2,
9oj = kj,
Qii = ci3u + 2acq>5tj + bu
(5.99)
Solving the field Equation (5.95) for $, kj, and bip and transforming to quasi-Cartesian coordinates and the standard PPN gauge, we obtain the PPN parameters y = aco/ci, P = bc0 + (K 2 /8K 1 C)(C 0 /C 1 ), £, = (K 2 /8K 1 C)(C 0 /C 1 ), 12 a t = 2e/(coc1) ' a 3 = Ci = t2 = C3 = U = 0, - 4a(co/Cl) - 4, a 2 = - 1 - ( c o / c J t a ^ c + (d + K22/4Kl)(l + K2/4Kl)-x] (5.100) In choosing geometrical units, we set Gtoday = c2c\'2co3/2(l + K1/4K!)-1 = 1 (5.101) (g) Other theories and special cases: (i) Ni's (1973) stratified theory is the special case K^ * = K2 = 0 (no tensor field). Its PPN parameters can be obtained from Equation (5.100) by setting K2 = 0, with the result y = acQ/cu P = bc0, i = a 3 = Ci = C2 = C3 = U = 0, ax = 2e/(coci)1'2 - 4a(co/Cl) - 4, a 2 =-l-d(c 0 / Cl ) (5.102) l (ii) Ni's (1972) stratified theory is the special case e = K^ = K2 = 0 (no vector or tensor field). However, as we shall see in the next section, this theory is not viable because its PPN parameter a1 satisfies ai = -(4y + 4) which is in serious violation of experiment. (5.103) Theory and Experiment in Gravitational Physics 138 5.7 Nonviable Theories All the metric theories of gravity previously discussed have the property that by making an appropriate choice of values for arbitrary constants and for cosmological matching parameters, one can produce PPN parameter values in agreement with present-day solar system experiments, to be described in Chapters 7,8, and 9. In some theories, a particular choice of these quantities can yield PPN parameters that are identical with general relativity at the current epoch. Therefore, in order to test and possibly rule out some of these competing theories, we will have to explore new arenas for testing relativistic gravity outside the solar system, such as gravitational radiation, the binary pulsar, and cosmology. However, there is a sizable set of metric theories that, while perhaps once thought to have been viable, are now known to be in serious violation of solar system experiments. Some of these theories agree with the "classical" tests: deflection of light, time delay, perihelion shift of Mercury (see Chapter 7 for discussion). But this is not enough. There are now many further solar system tests, discovered through the use of the PPN formalism, that place tight limits on the preferred-frame parameters a1? and a2, on conservation-law parameters such as a3, and on the parameter £,. Many theories violate these limits. The lesson to be learned is that it is no longer sufficient for the inventor of an alternative gravitation theory to compare the predictions of the theory with experiment by simply deriving the static spherically symmetric solution (analogue of the Schwarzschild solution in general relativity), obtaining the PPN parameters ft and y. He or she must determine the full post-Newtonian metric for a dynamical system of bodies or fluid, possibly moving relative to the universe rest frame, including cosmological matching parameters. Only with a complete set of values for the PPN parameters can the theory be compared with the results of solar system experiments. Many of the nonviable theories that we shall describe were discussed in more detail in TTEG. We shall touch upon them here only briefly, referring the interested reader to TTEG and the original references for details. (a) Quasilinear theories Quasilinear theories of gravity are theories whose postNewtonian metric, in a particular post-Newtonian gauge, contains only linear potentials, in particular lacks the potentials U2 and <SW. This is a property of many theories that attempt to describe gravity by means of a linearfieldtheory on a flat spacetime background. If the gauge in which Post-Newtonian Limits 139 this occurs is not the standard PPN gauge, then a gauge transformation, as in Equations (4.38) and (4.40) yields 06o = 0oo ~ 2X2(U2 + ®w- <D2) - 21^,00 (5.104) Since g00 did not contain U2 or <bw, we see immediately that {=P (5.105) We shall see that this is in severe violation of Earth-tide measurements (Chapters 8 and 9). The most famous example of a quasilinear theory is Whitehead's (1922) theory. The theory has a nondynamical flat background metric IJ, and a physical metric constructed algebraically from IJ and the matter variables according to = n,,v 2 ( / ) - = X" - (*)-, w2 = (/)-(u M )-, da = rifl,dx"dxy (w-)3iA »/ r> (yT(y»)~ = o, u" = dx^/da, (5.106) where the superscript () indicates quantities to be evaluated along the past i/-light cone of the field point x*. The post-Newtonian metric has y = P = £, = 1, {Ci,C2,C3,C*}^0 oti = <x2 = a3 = 0, (5.107) Although the theory was thought for a long time to have been viable, the value £ = 1 is now known to be in violation of Earth-tide measurements. Another group of theories in this class is known as Linear FixedGauge (LFG) theories. The standard field theoretic approach to the construction of a tensor gravitation theory on a flat spacetime background is to use the gauge invariant action for a spin-two tensor field h^, combined with the universally coupled nongravitational action to yield - h*v[ V l J ( - f ) 1 / 2 d 4 x + /NG(<2A,0/1V) (5.108) where g^ = */ + h^. However, the Lagrangian is singular: the gravitational part is invariant under the gauge transformation hpv -> h ^ £(!) (5.109) Theory and Experiment in Gravitational Physics 140 while / N G is not. The Bianchi identity associated with this partial gauge invariance is (5.110) = 0 which is in conflict with the equation of motion that results from the general coordinate invariance of / [Equation (3.63)], T?v=0 (5.111) LFG theories seek to remedy this by breaking the gauge invariance of the gravitational action through the introduction of auxiliary gravitational fields that couple to h in such a way as to fix the gauge of h. Nevertheless these theories, devised by Deser and Laurent (1968) and Bollini et al. (1970), turn out to be quasilinear in the sense defined above, and predict <* = P in violation of experiment (see Will, 1973). (b) Stratified theories with time-orthogonal space slices These theories are special cases of the stratified theories discussed in Section 5.6, in which there is no vector field K^, i.e., e = 0. Table 5.2. Nonviable metric theories of gravity Theory" (a) Quasilinear theories Whitehead Deser-Laurent Bollini-Giambiagi-Tiomno Description For some gauge, U2 Predict galaxy induced perihelion shifts and and Q>w are absent from 0oo; thus £ = fi Earth tides, in violation of observation (b) Stratified theories with time-orthogonal space slices Einstein (1912) Metric is given by Whitrow-Morduch g = /idt ® it + f2t\; Rosen thus a : = 4(y + 1) Papapetrou Ni (2 versions) Yilmaz Page-Tupper Coleman (c) Conformally flat theories Nordstrom Einstein-Fokker Ni (2 versions) Whitrow-Morduch Littlewood-Bergmann 1 Reasons for nonviability Predict preferred-frame effects on Earth's rotation rate and on perihelion shifts, in violation of observation Metric is given by Predict no deflection or g = fii; thus y = 1 time delay of light, in violation of observation For discussion and references, see TTEG, Ni (1972), and Will (1973). Post-Newtonian Limits 141 They therefore have the property that f "0T + W)g^ = - K" = 0 (5.112) independently of the nature of the source. In the preferred frame, this means goj = 0. However, under a possible coordinate transformation to put the post-Newtonian limit of the theory into the standard PPN gauge, g0J becomes goj = sx,oj = sVj-eWj (5.113) By comparing this with the PPN metric [Equation (4.48)], it is possible to obtain in a straightforward manner, independently of £, Bl = -(4y + 4) (5.114) This is a gross violation of geophysical experiments that demand loc^ « 1, while time-delay measurements demand y « 1 (see Chapters 7 and 8). Prior to the placing of the limit on ax, theories of this type were popular alternatives to general relativity, largely because of their mathematical simplicity. Table 5.2 lists nine theories of this type, all nonviable. (c) Conformallyflattheories These theories typically possess a flat background metric IJ and a scalar field <f>. The metric g is constructed from r\ and </> according to g = /(#/ (5.H5) where / is some function of <f>. However, in order to obtain the correct Newtonian limit, /($) must have the form (in a suitable coordinate system)
/ = 1-217 + 0(4)
(5.116)
Thus,
flfy = [1 - 21/+ O(4)]5 y
(5.117)
hence y = 1. We shall see in Chapter 7 that this implies zero bending
of light and zero time delay, in violation of experiment. This result can
also be deduced from the conformal invariance of Maxwell's equations
(i.e., invariance under the transformation g^ -» </>#): propagation of light
rays in the metric f(4>)ti is identical to propagation in the flat spacetime
metric i/, namely straight-line propagation at constant speed. Table 5.2
lists six conformally flat theories, all nonviable.
Equations of Motion in the PPN Formalism
One of the consequences of the fundamental postulates of metric theories
of gravity is that matter and nongravitational fields couple only to the
metric, in a manner dictated by EEP. The resulting equations of motion
include
Tfvv = 0, [stressed matter and nongravitational fields]
wvwfv = 0, [neutral test body: geodesies]
F?vv
v
= 4nJ", [Maxwell's equations]
/c /cfv = 0, [light rays: geodesies]
(6.1)
(6.2)
(6.3)
(6.4)
(see Section 3.2 for discussion). In Chapter 4, we developed the general
spacetime metric through post-Newtonian order as a functional of matter
variables and as a function of ten PPN parameters. If this metric is
substituted into these equations of motion, we obtain coupled sets of
equations of motion for matter and nongravitational field variables in
terms of other matter and nongravitational field variables. For specific
problems, these equations can be solved using standard techniques to
obtain predictions for the behavior of matter in terms of the PPN parameters. These predictions can then be compared with experiment. It is the
purpose of this chapter to cast the above equations of motion into a form
that can be simply applied to specific situations and experiments. That
application will be made in Chapters 7, 8, and 9. In Section 6.1, we carry
out this procedure for light rays. Section 6.2 deals with massive, selfgravitating bodies and presents appropriate n-body equations of motion.
In Section 6.3, we derive the relative acceleration between two bodies,
including the effects of nearby gravitating bodies and of motion with
respect to the universe rest frame, and put it into a form from which one
Equations of Motion in the PPN Formalism
143
can identify a "locally measured" Newtonian gravitational constant.
Section 6.4 specializes to semiconservative theories and presents an n-body
action from which the semiconservative n-body equations of motion can
be derived. We also develop in Section 6.4 a conserved-energy formalism
of the type discussed in Section 2.5, and discuss the Strong Equivalence
Principle from this viewpoint. In Section 6.5, we analyze equations of
motion for spinning bodies.
6.1
Equations of Motion for Photons
We begin with the geodesic equation obtained from Maxwell's
equations in the geometrical-optics limit [Equation (6.4)]:
fev/cfv = 0
(6.5)
11
where k is the wave vector tangent to the "photon" trajectory, with
*"* = 0
(6.6)
Substituting k" = dx*/da where a is an "affine" parameter measured along
the trajectory, we obtain
We can rewrite Equation (6.7) using PPN coordinate time t = x° rather
than a as affine parameter by noticing that
Then the spatial components of Equation (6.7) can be rewritten
~dW +
Equation (6.6) can be written
^v^!^L = 0
(6.10)
To post-Newtonian accuracy, Equations (6.9) and (6.10) take the form
(see Table 6.1 for expressions for the ChristorTel symbols T*k):
2 =[/,.
dt
1 + y
dx2S
dt
0 = 1 - 2C7 - \dx/dt\2(l + 2yU)
(6.11)
The Newtonian, or zeroth order solution of these equations is
x£ - n\t - t0),
\n\ = 1
(6.12)
Theory and Experiment in Gravitational Physics
144
Table 6.1. Christoffel symbols for the PPN metric
3 + a, - a2 + C, - 2 © ^ + i(l + a2 t/,J-) + a2
+ y)U2
3 + B l - a, + Ci - 2 « ^ +1(1 + a2 - d
(at - 2a2)w'U + ajvWl/y],
C/>0 - i(4y + 4 + a j ^ ^ - i
where
2 + <x3 + {, - 2^a>x + (3y - 2)3
(3y + 3C4
in other words, straight-line propagation at constant speed |dx N /di| = 1.
By writing
xj = n\t - to) + xJp
(6.13)
and substituting into Equation (6.11) we obtain post-Newtonian equations
for the deviation xJp of the photon's path from uniform, straight line
motion:
^ £ = (l + y)[yu
dx
i'-£=-(l+y)U
- 2n(n VC7)],
(6.14)
(6.15)
In Chapter 7 we shall use these equations to derive expressions for the
deflection and the time delay of photons passing near the Sun.
6.2
Equations of Motion for Massive Bodies
One method of obtaining equations of motion for massive bodies
is to assume that each body moves on a test-body geodesic in a spacetime whose PPN metric is produced by the other bodies in the system as
well as by the body itself (with proper care taken of infinite self-field
terms). However, the resulting equations of motion cannot be applied to
massive self-gravitating bodies, such as planets, stars, or the Sun (except
in general relativity, as it turns out), because such bodies do not necessarily
follow geodesies of any PPN metric. Rather, their motion may depend
Equations of Motion in the PPN Formalism
145
upon internal structure (a violation of GWEP). This was first demonstrated by Nordtvedt (1968b).
Therefore, one must treat each body realistically, as a finite, selfgravitating "clump" of matter and solve the stressed-matter equations of
motion [Equation (6.1)] to obtain equations of motion for a suitably
chosen center of mass of each body. For the purposes of solar system
experiments, it is adequate to treat the matter composing each body as
perfect fluid (see Will, 1971a for discussion).
In Newtonian gravitation theory, this program is straightforward. By
defining an inertial mass and a center of mass for each body according to
pd3x,
ma = I
Joth body
xa = m- 1 f pxd3x
(6.16)
Ja
one can show, using the Newtonian equation of continuity [Equation (4.3)]
that
dmjdt = 0,
va = dxjdt = m~1 ja p\d3x,
= dyjdt = m;l I p{dv/dt)d3x
K
(6.17)
By using the Newtonian perfect-fluid equations of motion [Equation (4.3)]
we obtain the following expression for aa
S \
\ Ql #
L
b*a \Jab
r
ab
(^)]
(6-18)
J
where mb is the inertial mass of the bth body, Q'J is its quadrupole moment
defined by
J
i
| |
2
)
3
(6.19)
and \ab and rah are given by
xai, = xa ~ xb,
r^ = IxJ
(6.20)
We now wish to generalize these equations to the post-Newtonian
approximation, using the PPN formalism. Because there are many different "mass densities" in the post-Newtonian limit - rest-mass of baryons
p, mass-energy density p{\ + U), "conserved" density p*, and so on there is a variety of possible definitions for inertial mass and center of
mass. The definition we shall adopt is chosen in order to yield the simplest
closed-form result for the equations of motion. It turns out that as long
Theory and Experiment in Gravitational Physics
146
as we average the equations of motion over several internal dynamical
timescales of each body (assumed short compared to the orbital dynamical
timescale), the final equation of motion is insensitive to the precise form
of the definition. We define the inertial mass of the ath body to be
ma = f p*(l + iF 2 - \V + n)d3x
(6.21)
Ja
where p* is the conserved density [Equation (4.77)], v = v va(0), where
vY
= f n*\d3x
(6 77\
o(0) I r
and
" " *
\v.£^.)
U = £ p(x',t)\x -x'l'1
d3x'
(6.23)
Note that, roughly speaking, ma is the total mass energy of the body rest mass of particles plus kinetic, gravitational, and internal energies - as
measured in a local, comoving, nearly inertial frame surrounding the
body. As long as we ignore tidal forces on the ath body, then according
to our discussion of conservation laws in the PPN formalism [see Equation (4.108)], ma is conserved to post-Newtonian accuracy, i.e.,
dmjdt = 0
(6.24)
This can also be shown by explicit calculation using Equations (6.21),
(6.22), and (6.23). We now define the center of inertial mass
xa s m~1 f p*(l + iv2 - \V + U)xd3x
(6.25)
Ja
By making use of the equation of continuity for p* [Equation (4.78)]
and by using Newtonian equations of motion in any post-Newtonian
terms, we obtain
vfl = dxjdt = m~1 f [p*(l + iu 2 - iU + IT)v + pv - |p*W] d3x
(6.26)
where
The acceleration aa is thus given by
ao = d\Jdt
= m'1 < Ja p*(l + it; 2 - if/ + U)(d\/dt)d3x
+ v{ £ pjf d3x + £ |>>ov - (p/p*)Vp] d3x
3
P*Wd
x + \g-a - \«r*a + g>>\
(6.28)
Equations of Motion in the PPN Formalism
147
where &~a, 3~*, and 0>a are determined purely by the internal structure
of the ath body. Formulae for these and other "internal" terms are given
in Table 6.2. Notice that the acceleration of our chosen center of mass is
more than just the weighted average of the accelerations of individual
fluid elements, as it is in Newtonian theory.
We now evaluate the first integral in Equation (6.28) using the PPN
perfect-fluid equations of motion. We substitute the post Newtonian
expressions for T"v (Table 4.1) and Y*x (Table 6.1) into the equation of
motion, (6.1), and rewrite it in terms of the conserved density p*. The
Table 6.2. Integrals for massive bodies in the PPN equations of motion.
Vector integrals
'
"II
'13
X X I [X X I
o*p*'p*"(x' - x") (x - x')(x - x'Y
r
H
y
u,
**v
~>d3Xd3X',
^
|x - x'| 3
- I P*P*'lr " (x ~ x')]2(* ~ x')3
y*i
X - X'
5
Tensor and scalar integrals:
P*v'vJd3x
n
« = -ij»prz^p r f
xd x
> "-= -*J.-prr7j-J xd
I'J = £ p*(x - xj'(x - x a )^ 3 x,
/. = ja p*\x - xa\2 d3x
Theory and Experiment in Gravitational Physics
148
result is
p*dvJ/dt = p*Utj - |>(1 + 3yU)lj + Pj&2
+ U + pip*)
v\p*Ui0 - p,0) - i(l + <x2 iP*[(4y + 4 + atf + (ax p*(d/dxJ)[® - £«V - i ( d -
- (2jS - 2)U + 3yp/p*]
(6.29)
where O is given in Table 6.1.
We now substitute this expression for p*d\/dt into Equation (6.28) and
perform the integration, using Newtonian equations where necessary to
simplify post-Newtonian terms. Considerable simplification of the equations results if we average over several internal dynamical timescales of
each body. Then we can set equal to zero any total time derivatives of
internal quantities. This is a reasonable approximation for the solar
system, since any secular changes in the structure of the sun or planets
that would prevent the vanishing of such averaged time derivatives occurs
over timescales much longer than an orbital timescale. This allows us to
use several Newtonian virial relations to simplify post-Newtonian expressions. These relations, easily derived using the Newtonian equations of
motion have the form for each massive body
H"= -<Q> = 0,
ST*> + 33T**J - Q*i - 0>J = (~
\dt J
=(j
[p*WWx)
/
Jp*VJ d3x)=0
= 0,
(6.30)
Equations of Motion in the PPN Formalism
149
The final form of the equation of motion is
K = (aAelf + (aa)Newt + (aJnbody
(6-31)
where
{ai)seK = -m^iu**
+ CM + Ci(ri - ir**J)
vttfHkJ,
(6.32)
(6.33)
(27 + 2f})
{(7
f) rr± (y
I ^r
b*a
ab
rr
(.
p
C2) ^
fa
r
ab
ab
+(2j8-l-2«-C 2 ) E ^+(2y + 2/?-2£) ^
c*a* r6c
c*ab rac
.,2
> + 2 + a2+<x3)»i
a 2 )(v 6 Kb)2 + | « 2 ( w fla!,)2 + 3a 2 (w hab)(vb nab)
|(7 +
^+
1
2
+ C1)E
I
*#a rab c*ab
#
r
bc
- t Z ^ ( ^ - 3 « 4 ^ ) E "C^
+ I 5 xa6 [(2y + 2)va - (2y
1
*w
~ 5z I ^x a 6 -[(47 + 4+a1)vfl
b*a "ab
~ * I
z
^ r x«fc' [«i v « - ( a i ~ 2«2)T6 + 2a2w]w^'
(6.34)
i>#n "aft
where nab = \ab/rab
The first six terms in (a a ) self , Equation (6.32), involving terms such as
t{, 3~'a, and so on, depend only on the internal structure of the ath massive
body, and thus represent "self-accelerations" of the body's center of mass.
Such self-accelerations are associated with breakdowns in conservation
of total momentum, since they depend on the PPN conservation-law
parameters a 3 , £i> ^2> C3, and £4- In any semiconservative theory of
Theory and Experiment in Gravitational Physics
150
gravity,
« 3 EE d = C2 = C3 s C4 = 0
(6.35)
and these self-accelerations are absent. Also note that spherically symmetric bodies suffer no acceleration regardless of the theory of gravity,
since for them the terms t{, PJa, ^~**j, Q.{, &{, and &[ are identically zero.
The same is true for a composite massive body made up of two bodies in
a nearly circular orbit, when the self-acceleration is averaged over an
orbital period. Thus, there is little hope of testing the existence of these
terms in the solar system. However, in the binary pulsar, for instance,
where the orbit eccentricity is large, there may be a potential test. We
shall discuss this possibility in Section 9.3.
The next term in Equation (6.32), m~1a.3(w + vafHk.j, is a selfacceleration which involves the massive body's motion relative to the
universe rest frame. It depends on the conservation-law/preferred-frame
parameter <x3, which is zero in any semiconservative theory of gravity.
For any static body, v = 0, thus HkJ is zero, but for a body that rotates
uniformly with angular velocity o>,
v = <o x (x - xa)
(6.36)
and
\X X I
llm
= e/ co\Sla)
Jm
(6.37)
For a nearly spherical body, the isotropic part of QJm makes the dominant
contribution to Equation (6.37), i.e.,
(Qaym * &jmna,
HkJ =s ±e*WQ.
(6.38)
Then the acceleration term in Equation (6.32) becomes
-!<x3(Qa/ma)(w + ya) x to
(6.39)
In Chapter 8, we shall see that this term may produce strikingly large
observable effects in the solar system, if a 3 is different from zero.
The next term, (ao)Newt in Equation (6.31) is the quasi-Newtonian acceleration of the massive body. Here (mP)a* is the "passive gravitational
mass tensor" given by
(mP)ik=ma{<5*[l + (4/J - y - 3. - 3{ - a, + a 2 - Ci W.M, - 3£nafcnam^i7ma]
C2)fiJ*MJ
(6.40)
Equations of Motion in the PPN Formalism
151
and U(xa) is the quasi-Newtonian potential, given by
U(xa) = £ & ^ ) i
r
ab
where [mA(habf\b is the "active gravitational mass" of the bth body, given by
(6-42)
Note that the active and passive gravitational mass tensors may be functions of direction n^ relative to the other bodies. It is useful to rewrite
the quasi-Newtonian acceleration in a form involving inertial, active and
passive mass tensors that are independent of position, and a gravitational
potential U'm, as follows
W'm= £
frJTK&r*
(6.43)
where
(ax - a2 + C i R M ] + (a2 - d
(4/J - y - 3 - 3fl«./<| - ZflT/
(4/? - y - 3 - 3{ - i« 3 - Ki + Ca^M, - (fa3 + Ci - X&M
- « - KiJOfM} (6-44)
In Newtonian theory, the active gravitational mass, the passive gravitational mass, and the inertial mass are the same, hence each massive body's
acceleration is independent of its mass or structure ("Equivalence principle"). However, according to Equation (6.44), passive gravitational mass
need not be equal to inertial mass in a given metric theory of gravity (and
in fact both may be anisotropic); their difference depends on several PPN
parameters, and on the gravitational self energy (Q and Qik) of the body.
This is a breakdown in the gravitational Weak Equivalence Principle
(GWEP) (see Section 3.3), also called the "Nordtvedt effect" after its discoverer (Nordtvedt, 1968a, b). The possibility of such an effect was first
noticed by Dicke [1964b; see also Dicke (1969), Will (1971a)]. The observable consequences of the Nordtvedt effect will be discussed in Chapter
8. Its existence does not violate EEP or the Eotvos experiment (Chapter 2),
because the laboratory-sized bodies considered in those situations have
negligible self gravity, i.e., (il/m)^^ bodies < 10~39. In Section 6.3, we shall
see that there is a close connection between violations of GWEP and
the existence of preferred-location and preferred-frame effects in postNewtonian gravitational experiments.
Theory and Experiment in Gravitational Physics
152
According to Equation (6.44), active gravitational mass for massive
bodies may also differ from inertial mass and from passive gravitational
mass. In Newtonian gravitation theory, the uniform center-of-mass motion of an isolated system is a result of the law "action equals reaction,"
i.e., of the law "active gravitational mass equals passive gravitational
mass." In the PPN formalism, one can still use such Newtonian language
to describe the quasi-Newtonian acceleration (ao)Newt. From Section 4.4,
we know that uniform center-of-mass motion is a property of fully conservative theories of gravity, whose parameters satisfy
a, = a2 = oc3 = Ci = C2 = C3 = U = 0
(6.45)
By substituting these values into Equation (6.44), we find that for fully
conservative theories, the inertial mass is equal to ma, and the active and
passive mass tensors are indeed equal, and are given by
(«ptf=(«A)^ = ^ { ^ [ l + ( 4 / » - 7 - 3 - 3 W . / m J - { n f / m . }
(6.46)
equivalently, (a£)Newt can be written to post-Newtonian order in the form
(ae^/TS*^
\ma mj\ r%b
sxjjj^Xj
r^ ))
{6A7)
The term in braces is manifestly antisymmetric under interchange of a
and b, hence action equals reaction, and £ , ma(a{)Newt = 0. Note that in
general relativity, the mass tensors of Equation (6.44) are isotropic and
equal to the inertial mass, i.e., (dropping the Kronecker deltas)
fhl = fhp = mA = ma [general relativity]
(6.48)
There is no Nordtvedt effect in general relativity. However, in scalartensor theories, there is in general a Nordtvedt effect, since
mP = mA = ma{\ + [(2 + co)"1 + 4A]fta/ma}
(6.49)
For most practical situations, we may assume that the bodies in question
are spherically symmetric, then using the equation ClJak m %SJkQa to simplify the mass tensors, we may write
« = I (MAV^
b*
(6.50)
Equations of Motion in the PPN Formalism
153
where (we combine (m{k)~1 and ml into one quantity mP)
K) fl /m a = 1 + (40 - y - 3 - Aft -
Kl
+ | a 2 - f£
K ) » M = 1 + (4/J - y - 3 - ^ - i« 3 - Ki
+ Cs^/m - (|a 3 + d - SCJPJm,
(6.51)
The remaining term (aj nbody in Equation (6.31) is called the n-body
term. It contains the post-Newtonian corrections to the Newtonian
equations of motion which would result from treating each body as a
"point mass" moving along a geodesic of the PPN metric produced by
all the other bodies, assumed to be point masses, taking account of
certain post-Newtonian terms generated by the gravitational field of the
body itself. It is the n-body acceleration which produces the "classical"
perihelion shift of the planets, as well as a host of other effects, to be
examined in Chapters 7 and 8. For the case of general relativity, the n-body
terms in Equation (6.34) are in agreement with the equations obtained by
de Sitter (1916) [once a crucial error in de Sitter's work has been corrected],
Einstein, Infeld, and Hoffmann (1938), Levi-Civita (1964), and Fock (1964).
6.3
The Locally Measured Gravitational Constant
Here, we derive an equation which is not really an equation of
motion, but is nevertheless a fundamental result in the PPN formalism.
In the previous section, we found that some metric theories of gravity
could predict a violation of GWEP (Nordtvedt effect). Such effects would
represent violations of the Strong Equivalence Principle (SEP). As discussed in Section 3.3, the existence of preferred-frame and preferredlocation effects in local gravitational experiments would also represent
violations of SEP. One such local gravitational experiment is the Cavendish experiment. In an idealized version of such a Cavendish experiment
one measures the relative acceleration of two bodies as a function of their
masses and of the distance between them. Distances and times are measured by means of physical rods and atomic clocks at rest in the laboratory. The gravitational constant G is then identified as that number with
dimensions cm3 g" 1 s" 2 which appears in Newton's law of gravitation
for the two bodies. This quantity is called the locally measured gravitational constant GL.
The analysis of this experiment proceeds as follows: a body of mass mt
("source") falls freely through spacetime. A test body with negligible mass
moves through spacetime, maintained at a constant proper distance rp
from the source by a four-acceleration A. The line joining the pair of
masses is nonrotating relative to asymptotically flat inertial space. An invariant "radial" unit vector Er points from the test mass toward the source.
Theory and Experiment in Gravitational Physics
154
Then according to Newton's law of gravitation the radial component of
the four acceleration of the test mass is given by
/KEr=-GlmJr2p
(6.52)
for rp small compared to the scale of inhomogeneities in the external gravitational fields. Since the quantity A Er is invariant, we can calculate it in
a suitably chosen PPN coordinate system, then use Equation (6.52) to
read off the locally measured GL.
Before carrying out the computation, however, it is instructive to ask
what might be expected for the form of A Er to post-Newtonian order.
We imagine that the source and the test body are moving with respect to
the universe with velocity w1 and are in the presence of some external
sources, idealized as point masses of mass ma at location xa. It is simplest
to do the calculation in a PPN coordinate system in which the source
is momentarily at rest. Then we would expect A Er to contain postNewtonian corrections to the equation A E, = m1frl
rl of the form
m1ml
A E
mt ma
m^ ma
r -2'
72>
-rzr>
mYY 22
72-K)
V
P ia
'P l"
P
;
Y
r
P
r
r
r
m
(6-53)
where rla = |xx xa|. In obtaining this form, we have neglected the variation of the external gravitational potentials across the separation rp. This
variation will produce the standard Newtonian tidal gravitational force,
which is of the form
(AE) - m a r
r
la
and post-Newtonian corrections to this force. The latter we shall neglect
throughout. The first term in Equation (6.53) represents post-Newtonian
modifications in the two-body motion of the test body about the source,
which can be understood and analyzed separately from a discussion of GL.
The third term represents effects due to the gradients of the external fields;
however, if we fit A Er to an r~2 curve in order to determine GL, these
terms will have no effect [in most practical situations, they are negligibly
small anyway (Will, 1971d)]. Both of these types of terms will be dropped
throughout the analysis. Thus, we retain only terms of the form (m,/
rl)(mjrla) or (m1/^)(wj).
The form of the PPN metric that we shall use is given by the expression
in Table 4.1, where now the velocity w is the source's velocity relative to
the mean rest frame of the universe, denoted w t . We label the test body by
a = 0, the source by a 1, and the remaining bodies by a = 2, 3 , . . . Initially, both the source and test body are at rest, i.e.,
Vl (t
= 0) = vo(t = 0) = 0
(6.54)
Equations of Motion in the PPN Formalism
155
We separate the Newtonian gravitational potential Ux due to the source
from that due to the other bodies in the system:
l/(x) = Ufa) + £ mjra
(6.55)
where rx = |x xx\, ra = |x xo|, and Ut is assumed for simplicity to be
spherically symmetric.
The proper distance between the test body and the source is given by
[see Equation (3.41)]
rp = £ [1 + yU(x(X)) + O(4)]|dx/dA|dk
(6.56)
where to sufficient accuracy we may choose a straight coordinate line to
join the two points:
x(l) = x o (l - X) + \tX,
0< X < 1
(6.57)
Then
Neglecting the variation of the external gravitational potential across the
rP = rj\ + y E - ) + ? \T U^da
(6.59)
The proper distance rp is to be kept constant by the four-acceleration A,
thus
drp/dt s d\/dt2 = 0
(6.60)
with the result, at t 0,
)
where we have used the fact that Vj = v0 = 0 at t = 0, and have neglected
time derivatives of the external potential. For the rest of this discussion,
it is sufficient to drop the final term in Equation (6.59) (it leads only to
terms that we previously decided to ignore) and to treat the coefficient
of r01 as a constant. Thus,
(
)
(6-62)
Theory and Experiment in Gravitational Physics
156
We now assume that the source follows a geodesic of spacetime, but
that the four-acceleration of the test body is A. Thus,
^source^source;v
*A
V
«, eS,«rest;v = A",
uJU,^ = 0
In PPN coordinates, Equation (6.53) may be written, at t = 0,
uuv o /
dt '
(6-63)
\dt
,4° = 0
(6.64)
where, for the test body,
j )
= 1 - 2l/ 1 (x 0 ) - 2 £ mjrla
+ O(4)
(6.65)
where we have again ignored the variation of the external potential in
evaluating it at xt instead of at x 0 . We make use of the PPN Christoffel
symbols (Table 6.1) evaluated for the external point masses [substitute
p = p*{l jv2 3yU), \ap* d3x = ma] and use the Newtonian equations
of motion to simplify any post-Newtonian terms. We retain only the
terms discussed above; for illustration we also keep the Newtonian tidal
force. Substituting Equations (6.64) into (6.61) yields, finally,
A - x 1 0 __
'10
^
a*l
marloeJek(3n{
n\.
-3a
~ $Jk) "la '10 a* I r l a L + Vt/f'(x0) |"1 a 2 wX - { X ^"'""'"l r Z 10 1_ #] ?"la (6-66) J where p*(x',t)d3x' 9 - XI -x)(xo-x)dx (6^7) Equations of Motion in the PPN Formalism 157 For a spherically symmetric source, it is possible to show straightforwardly that O(6), VjUf (x0) = (mJr\o){Moe e - 2x%5l)i) l - 2x%dl» - 4o<5*') + O(4) l k l (6.68) where mt and It are the rest mass and spherical moment of inertia of the source, given by mx = I p*d3x, / t = I p*r2d3x (6.69) We must now compute the invariant radial unit four-vector Er. Its components at x 0 are simply those of the tangent vector to the curve x(A) joining the two bodies, E} = adx\k)ldk = - oxJ01, £r° = 0 (6.70) ErvE; == 1 = a 2 |x 01 | 2 1 + 2V X ~ I (6-71) The normalization a is obtained from \ 0*1 ~\aj where we have retained only the necessary terms. Thus (6-72) Then the invariant radial component of the four-acceleration A is (6.73) The final result is (Will, 1971d, 1973; Nordtvedt and Will, 1972) A E, = X m a r 10 [3(n lo e) 2 - l>r. 3 T(WI a 2 ot3)w1 5 a 2 ( w i ' e ) + t z^ fl#l ( n io ' e ' '"la Theory and Experiment in Gravitational Physics 158 The first term in Equation (6.74) is simply the Newtonian tidal acceleration. From the second term we may read off the locally measured gravitational constant, (675) H ^ y i ^ ) ^ where U& = X man{an\Jrla, UeU = [/£, (6.76) Here, we see a direct example of the possibility of violations of the Strong Equivalence Principle, via preferred-frame or preferred-location effects in local Cavendish experiments. The preferred-frame effects depend upon the velocity Wj of the source relative to the universe rest frame, and are present unless the PPN preferred-frame parameters a1; a2, and <x3 all vanish. The preferred-location effects depend upon the gravitational potentials Unt and (/£*, of nearby bodies, and are present in general unless the PPN parameters satisfy £, = (4)3 y 3 £2) = 0- ^n next section we shall develop a conserved energy formalism for the special case of semiconservative theories of gravity that will reveal a direct connection between violations of local Lorentz and position invariance in Cavendish experiments, and the violations of GWEP described in Section 6.2. We note here that general relativity predicts GL = 1 6.4 (6.77) N-Body Lagrangians, Energy Conservation, and the Strong Equivalence Principle In the previous two sections we showed that some metric theories of gravity may predict violations of GWEP and of LLI and LPI for gravitating bodies and for gravitational experiments. In the special case of theories of gravity that possess conservation laws for energy and tne Equations of Motion in the PPN Formalism 159 momentum, namely semiconservative theories, it is possible to derive a direct relationship between these violations. The method is the same as that developed in Section 2.5: derive a conserved energy expression for a composite system in a quasi-Newtonian form, from which one can read off the anomalous inertial and passive gravitational mass tensors Sm[J and 5m'J, respectively. The use of cyclic gedanken experiments, parallel to those used in Section 2.5, then reveals that violations of GWEP as well as of LLI and LPI depend upon these anomalous mass tensors. The derivation of these results proceeds as follows (Haugan, 1979): We first restrict attention to semiconservative theories of gravity, thus <x3 = d = £2 = £3 = £4 == 0, and to systems in which the basic particles are point masses. We then build composite bodies out of point masses moving in their mutual gravitational fields. We work in a PPN coordinate frame at rest with respect to the universe rest frame. The equations of motion for the particles then consist of the standard Newtonian acceleration plus the post-Newtonian n-body acceleration anbody, Equation (6.34) with w = 0 and with semiconservative PPN parameters, b*a 'ab L 'ab 'ab c*ab rbc c*ab rac a2) r c*ab ac - i(4? + 4 + ax)va \b + i(2y + 2 + a 2 ) ^ - f (1 + a2)(> 'ab c±ab 'be -11b*a 5ab 0* - Wto I * fir - T r r \' 'a r b - (2y 'ab 4 + a i )v o - (4y + 2 + ax - 2a 2 )v fc ]^ (6.78) Theory and Experiment in Gravitational Physics 160 It is then possible to show straightforwardly that these equations of motion can be derived from the Euler-Lagrange equations obtained by varying the trajectory xq(t), vq(t) of the qth particle in the action ^ (6.79) where 3+ ai - a2)va vfc - i(l + a2)(vo nab)(v6 fij Consider a system consisting of a body of mass mQ and a composite body made up of bodies of mass ma. We assume that m0 » Xom«> anc* that the massive body is situated at rest at the origin a distance |X| from the composite body, where |X| is large compared to the size of the composite body. Because it is more massive, the distant body may be assumed to remain at rest, thereby providing an external potential in which the composite body resides and moves. (We ignore coupling of the body to inhomogeneities in the external potential.) We now make a change of variables in L from xa to center-of-mass and relative variables X and x a , respectively, where X = m~1 £ maxa, m = Ym<" a a xa=xa-X (6.81) We also have \a = dxjdt, V = dX/dt (6.82) A Hamiltonian H is then constructed from L using the standard technique PJ = 8L/dVJ, pJa = dL/dvJa, H = PJVJ + X rfpl - L (6.83) Equations of Motion in the PPN Formalism 161 The result is P2 H = m + 2m v-i Pa mm0 ab *** ** 1 > 2ma R + ~ 2 % rab K r ab 'ab ab ab r ab ab m 'ab + Jd + «2) - I ? (».,' P)(«.. P.) + O(p') + O(P*) m (6.84) aft 'ab where /? = |X|, n = X/R, and nofc = xab/fab. We have neglected postNewtonian terms O(p 4 ) and O(P*) in H that do not couple the internal motion and the center-of-mass motion of the composite system. We now average H over several timescales of the internal motions of the composite system, and make use of virial theorems for the internal variables, + mO(4)\ (6.85) As in Section 2.6, we argue that although the post-Newtonian terms in Equation (6.85) may depend on P or X, this dependence does not affect the form of H. The resulting average Hamiltonian is then rewritten in terms of V using the equation V = 5<H>/5P. The result for the conserved energy function is £ c = M + \{M5ij + [(a t - a2)Q<5y + a 2 Q 0 ]} VW - {M8'J + [(4j8 - y - 3 - 3£)Q<5ij' - ^QiJ2}mon'nj/R (6.86) where p « 2m a 1 V m-m>\ l ab T ab I (6.87) ab Theory and Experiment in Gravitational Physics 162 By comparing Equation (6.86) with Equations (3.77) and (3.78) we may read off the anomalous mass tensors dm\J = (a! - tx2)Q8iJ + <x2QiJ, 3my = (4/J - y - 3 - 3{)IM<> - &J (6.88) Substituting these results into Equation (3.80) yields 3ak = M-l\jAP - y - 3 - 3£)Q<5U - £&J~\(d/dXk)(mon'nJ/R) + A T '[(o^ - a 2 )iW w + oc2Qk}]m0xJ/R3 (6.89) This is in complete agreement with the GWEP-violating terms in ajSjewt. Equations (6.40), (6.43), and (6.44) if we substitute the semiconservative values of the PPN parameters, and take into account that the potential U'm is that due to a single distant point mass, i.e., U im = monlnm/R (6.90) To determine the influence of the internal structure of the composite body on its center-of-mass motion, we fixed its structure and focussed on the explicit P and X (or V and X) dependence of H. Now, to study the effect of a body's motion on its internal structure, and thereby obtain an expression for GL, we must fix the center-of-mass motion (P,X), and focus on the explicit p and x dependence of H. Using the Newtonian virial theorem [Equation (6.85)] to simplify the post-Newtonian terms in H, we obtain the conserved energy function Ec = M + i{M8iJ - [(<*! - oc2)Qdij + a 2 Q y ] }PiPj/M2 - {M5ij + [(4)3 - y - 3 - 3£)Q<5>V - ^Q iJ ]}m o n i « J 7^ (6-91) where M and Q'v are given by Equation (6.87). Notice that the quantities in square brackets are precisely 5m[J and 5m^, of Equation (6.88), but that the sign in front of dm[J is opposite to that in Equation (6.86) (a result of expressing Ec in terms of P rather than V). Let us suppose for simplicity that the composite body is composed of two point masses in a local Cavendish experiment. Then with Ec written in the above form, it is possible to show straightforwardly that the effective force between the two particles is given by F=-(V£c)p,XfUed (6.92) Then the effective local gravitational constant is - [(4/3 - y - 3 - 3£)8iJ - ^e^monW/R (6.93) Equations of Motion in the PPN Formalism 163 where e = x 12 /r 12 , and Q'J = mlm2e'eJ/r12. This is precisely Equation (6.75), with P/M = w1; monlnJ/R == U'Jxt, and with lx = 0. Again, we see the explicit connection between violations of GWEP and violations of LLI and LPI, for the case of semiconservative theories of gravity. 6.5 Equations of Motion for Spinning Bodies The motion of spinning bodies (gyroscopes, planets, elementary particles) in curved spacetime has been a subject of considerable research for many years. This research has been aimed at discovering (i) how a body's intrinsic angular momentum (spin) alters its trajectory (deviations from geodesic motion), and (ii) how a body's motion in curved spacetime alters its spin. No really satisfactory solution is available for the first problem, outside of approximate solutions, or solutions in special spacetimes, because of the difficulties in defining rigorously a center of mass of a spinning body in curved spacetime. The most successful attempts at a solution have been made by Mathisson (1937), Papapetrou (1951), Corinaldesi and Papapetrou (1951), Tulczyjew and Tulczyjew (1962) and Dixon (1979). The central conclusion of these calculations has been that the intrinsic spin S1" (i.e., J"v evaluated in the body's "center-of-mass" frame) of a body should produce deviations from geodesic motion of the form mSa* ~ Sv V K ? a (6.94) where W is the body's four velocity, and R*lX is the Riemann curvature tensor. However, these calculations differ greatly in details and interpretation. For a spinning body moving with velocity v in a Newtonian gravitational potential U ~ M/r, these deviations are, in order of magnitude: 5a ~ (|S*|/m)|v|(M/r3) ~ (i'A/rHM/r) 1 ' 2 ^ (6.95) where b is the radius of the body, and k its rotational angular velocity. For a planet rotating near break-up velocity (X2 ~ m/b3), we have Sa £ (m/b)1'2(M/r)1'2(b/r)aIhwl % 10 ~J 2aNewt (6.96) and for a 4 cm-radius gyroscope orbiting the Earth (frequency 200 rps), 5a £ 10~20 aNewt (6.97) Thus, for the most part, spin-induced deviations from geodesic motion can be ignored in the solar system. In our derivation of massive-body Theory and Experiment in Gravitational Physics 164 equations of motion (Section 6.2), we ignored the effects of tidal gravitational forces (Riemann curvature tensor); and thus our equation of motion, (6.31), does not include the effects of spin. Even for a rapidly rotating neutron star such as the binary pulsar (b ~ 10 km, A ~ 102 Hz, m ~ lmG, r ~ 106 km), (5a;glO-10aNewt (6.98) and can be ignored (see Chapter 12). It is problem (ii), the effects of a body's motion on its spin, which is better understood. All calculations to date have shown that, as long as the direct effects of tidal gravitational forces (Riemann curvature tensor) on the spinning body can be neglected, the spin S is Fermi-Walker transported along the body's world line. Here the four-vector S has the components S"s^Vi,> u"S,, = 0 (6.99) The equation of Fermi-Walker transport is then uvS?v = ul'id'S,,) (6.100) where a" is the body's four-acceleration, given by a" = uvufv (6.101) The reader is referred to MTW, Section 40.7 for further discussion of Fermi-Walker Transport. The following derivation is patterned after that section. It is convenient to analyze Equation (6.100) in a local Lorentz frame which is momentarily comoving with the body. The basis vectors of this frame are related to those of the PPN coordinate system by a Lorentz transformation plus a normalization, and are given by e% = W, e°j =vj + O(3), 4 = (1 - yU)8) + %Vjvk + O(4) (6.102) where all quantities in Equation (6.102) are assumed to be evaluated along the world line of the body. Thus, because of Equation (6.99), the spin is a purely spatial vector in this frame, i.e., S6 s egS, = ! « = 0 (6.103) Equations of Motion in the PPN Formalism 165 We now calculate the precession of the spatial components of the spin Sj. Since efu^ = 0, we have, from Equation (6.100), 0 = efifS^ = i?Sj.v - SMuv4v (6-104) and since Sj is a scalar (scalar product of two vectors), we have uvS/;v = wvS;>v = dS}/dT (6.105) The second term in Equation (6.104) is most easily evaluated in the PPN coordinate frame. Using Equation (6.102), we first obtain relations between SM and Sf S0=-VJSJ+O(3)SJ, Sj=Sj+O(2)S} (6.106) Then after some simplification, we get, to post-Newtonian order, dSj/dx = SlVuak] + g0lKSi - (2y + l)vuU,k{\ (6.107) This can be written in three-dimensional vector notation dS/dx = ft x S, ft = -%\ x a - ^V x g + (y + i)v x Vt/, g= In Equation (6.108) it does not matter whether the vectors entering into ft are evaluated in the PPN coordinate frame or in the comoving frame, since their spatial basis vectors differ only by terms of O(2). We have calculated the precession of the spin relative to a comoving frame which is rotationally tied to the PPN coordinate frame, and whose axes are fixed relative to the distant galaxies. Thus, we have calculated the spin's precession angular velocity ft relative to a frame fixed with respect to the distant galaxies. We shall discuss the observable consequences of this precession in Chapter 9. The Classical Tests With the PPN formalism and its associated equations of motion in hand, we are now ready to confront the gravitation theories discussed in Chapter 5 with the results of solar system experiments. In this chapter, we focus on the three "classical" tests of relativistic gravity, consisting of (i) the deflection of light, (ii) the time delay of light, and (iii) the perihelion shift of Mercury. This usage of the term "classical" tests is a break with tradition. Traditionally, the term "classical tests" has referred to the gravitational redshift experiment, the deflection of light, and the perihelion shift of Mercury. The reason is largely historical. These were among the first observable effects of general relativity to be computed by Einstein. However, in Chapter 2 we saw that the gravitational red-shift experiment is really not a test of general relativity, rather it is a test of the Einstein Equivalence Principle, upon which general relativity and every other metric theory of gravity are founded. Put differently, every metric theory of gravity automatically predicts the same red-shift. For this reason, we have dropped the red-shift experiment as a "classical" test (that is not to deny its importance, of course, as our discussion in Chapter 2 points out). However, we can immediately replace it with an experiment that is as important as the other two, the time delay of light. This effect is closely related to the deflection of light, as one might expect, since any physical mechanism in Maxwell's equations (refraction, dispersion, gravity) that bends light can also be expected to delay it. In fact, it is a mystery why Einstein did not discover this effect. It was not discovered until 1964, by Irwin I. Shapiro. The simplest explanation seems to be that Shapiro had the benefit of knowing that the space technology of the 1960s and 1970s would make feasible a measurement of a delay of the expected size (200 us for a round Classical Tests 167 trip signal to Mars). No such technology was known to Einstein. He was aware only of the known problem of Mercury's excess perihelion shift of 43 arcseconds per century, and of the potential ability to measure the deflection of starlight. But the lack of available technology may not be the whole story. After all, Einstein derived the gravitational red-shift at a time when the hopes of measuring it were marginal at best (a reliable measurement was not performed until 1960), and other workers such as Lense and Thirring, and de Sitter derived effects of general relativity, with little or no hope of seeing them measured using the technology of the day. Why then, did no one at the time take the step from deflection to time delay, if only as a matter of principle? Nevertheless, despite its late arrival, the time delay deserves a place in the triumvirate of "classical" tests, not the least because it has given one of the most precise tests of general relativity to date! We begin this chapter with the deflection of light (Section 7.1), turn to the time delay (Section 7.2), and finally to the perihelion shift of Mercury (Section 7.3). 7.1 The Deflection of Light An expression for the deflection of light can be obtained in a straightforward way using the PPN photon equations of motion, (6.14) and (6.15). Consider a light signal emitted at PPN coordinate time te at a point xe in an initial direction described by the unit vector ft, where n n = l. Including the post-Newtonian correction xp, the resulting trajectory of the photon has the form x°(t) = t, x(t) = x£ + %t - O + xp(t) (7.1) where we have imposed the boundary condition xp(te) = 0. We decompose xp into components parallel and perpendicular to the unperturbed trajectory: Xp(f)|| = ft Xp(t), x p « i = xp(t) - n[n xp(t)] (7.2) Equations (6.14) and (6.15) then yield ^ (7.3) = (1 + y){Uj - n^n Vt/)] (7.4) Theory and Experiment in Gravitational Physics 168 For simplicity, we assume that the Newtonian gravitational potential U is produced by a static spherical body of mass m at the origin (Sun), i.e., (7.5) Along the unperturbed path of the photon, U then has the form m fi(t - te)\ r(t) (7.6) To post-Newtonian order, then, Equation (7.4) can be integrated along the unperturbed photon path using Equation (7.6) with the result d r ( dt' M } mA lx(t) a i xe ft d> \ r(t) (7.7) where d =n X (X, xft) (7.8) Note that d is the vector joining the center of the body and the point of closest approach of the unperturbed ray (see Figure 7.1). Equation (7.7) represents a change in the direction of the photon's trajectory, toward the sun (in the direction -d). We then have Consider an observer at rest on the Earth (©) who receives the photon from the source and a photon from a reference source located at a different Figure 7.1. Geometry of light-deflection measurements. Reference Source Source Earth Classical Tests 169 position on the sky, x r . The angle 9 between the directions of the two incoming photons is a physically measurable quantity, and can be given an invariant mathematical expression. The tangent four-vectorsfeM= dx"/dt andfefr)= dx$r)/dt of the paths x"(t) and x("r)(t) of the two incoming
photons are projected onto the hypersurface orthogonal to the observer's
four-velocity t/ using the projection operator
( 71 °)
PI = K + "X
The inner product between the resulting vectors is related to the cosine
offlby
If we ignore the velocity of the Earth, which only produces aberration,
then Equation (7.11) simplifies to
coSe=l-(g00)-1gllvk%)
(7.12)
By substituting Equations (7.1) and (7.9) into Equation (7.12) we get, to
post-Newtonian accuracy,
'*
where
M M
(x, x 8,)
(7.14)
It is useful to note that, to sufficient post-Newtonian accuracy in Equation
(7.13),
d = n x (x e x fl), dr = nr x (x e x
fir)
(7.15)
We now define the angle 60 to be the angle between the unperturbed paths
of the photons from the source and from the reference source, i.e.,
cos0 o sfl-ii r
(7.16)
and we define the "deflection" of the measured angle from the unperturbed
angle to be
d6 = 6-d0
(7.17)
There are two interesting cases to consider. This first is an idealized situation that leads to a simple formula. We suppose that the Sun itself is the
Theory and Experiment in Gravitational Physics
170
reference source, then, dr = 0, the second term inside the braces in Equation (7.13) vanishes,
(7.18)
and
d \ re
re
For a photon emitted from a distant star or galaxy,
re»r®,
xe-n/re^-l
(7.20)
Also, to sufficient accuracy,
x e fi/r®~ nr n = cos 90
(7.21)
thus,
(! + ,) * ( . + - * )
For general relativity (y = 1) this is in agreement with results obtained
by Shapiro (1967) and Ward (1970).
It is interesting to note that the classic derivations of the deflection of
light that used only the principle of equivalence or the corpuscular theory
of light (Einstein, 1911, Soldner, 1801) yield only the "1/2" part of the
coefficient in front of (Am/d)(\ + cos0o)/2 in Equation (7.22). That does
not invalidate these calculations however; they are correct as far as they
go. But the result of these calculations is the deflection of light relative
to local straight lines, as denned for example by rigid rods; however,
because of space curvature around the Sun, determined by the PPN
parameter y, local straight lines are bent relative to asymptotic straight
lines far from the Sun by just enough to yield the remaining factor "y/2".
The first factor "1/2" holds in any metric theory, the second "y/2" varies
from theory to theory. Thus, calculations that purport to derive the full
deflection using the equivalence principle alone are incorrect (see Schiff,
1960a, and the critique by Rindler, 1968).
The deflection is a maximum for a ray which just grazes the Sun, i.e.,
for 60~0,d^Ro^
6.96 x 105 km, m = mQ = 1.476 km. In this case,
<50max = I d + 7)1"75
(7.23)
The second case to consider is more closely related to the actual method
of measuring the light deflection using the techniques of radio interfero-
Classical Tests
171
metry. There one chooses a reference source near the observed source and
monitors changes 80 in their angular separation. If we define $and <S>r to be the angular separation between the Earth-Sun direction and the unperturbed direction of photons from the two sources, as in Figure 7.1, then cos O = x e ft/r9, cos <J>r = x e nr/rm (7.24) Assuming again that the two sources are very distant, we obtain = / I + y\r4m /cos*, cos<J>cos0o\ / I + cos$\
)\ T \
) )
sin*sin9 0
Am /cos <Dr cos 0O cos 4>\ / I + cos<J>A~]
~T \
sin*rsin0o
)V
2
) \
K1J5)
If the observed source direction passes very near the Sun, while the reference source remains a decent angular distance away, we can approximate
$« <br, and thus, 60 =s <Dr - 4> cos x + O(* 2 /^r) (7.26) where / is the angle between the Sun-source and Sun-reference directions projected on the plane of the sky (Figure 7.1). The resulting deflection is This result shows quite clearly how the relative angular separation between two distant sources may vary as the lines of sight of one of them passes near the Sun (d ~ RQ, dr » d, % varying). The prediction of the bending of light by the Sun was one of the great successes of Einstein's general relativity. Eddington's confirmation of the bending of optical starlight observed during a total solar eclipse in the first days following World War I helped make Einstein famous. However, the experiments of Eddington and his co-workers had only 30% accuracy, and succeeding experiments weren't much better: the results were scattered between one half and twice the Einstein value, and the accuracies were low (for reviews, see Richard, 1975; Merat et al., 1974; Bertotti et al., 1962). The most recent optical measurement, during the solar eclipse of 30 June 1973 illustrates the difficulty of these experiments. It yielded a value |(1 + y) = 0.95 ± 0.11 [lo- error] (7.28) (Texas Mauritanian Eclipse Team, 1976 and Jones, 1976). The accuracy was limited by poor seeing (caused by a dust storm just prior to the Theory and Experiment in Gravitational Physics 172 eclipse, and by clouds and rain during the follow-up expedition in November, 1973) that drastically reduced the number of measurable star images. There were also variable scale changes between eclipse- and comparison-field exposures. Recent advances in photoelectric and astrometric techniques may make possible optical deflection measurements without the need for solar eclipses (Hill, H. 1971). The development of long-baseline radio interferometry has altered this situation. Long-baseline and very-long-baseline (VLBI) interferometric techniques have the capability in principle of measuring angular separations and changes in angles as small as 3 x 10 ~4 seconds of arc. Coupled with this technological advance is a series of heavenly coincidences: each year, groups of strong quasistellar radio sources pass very close to the Sun (as seen from the Earth), including the group 3C273, 3C279, and 3C48, and the group 0111 + 02, 0119 + 11 and 0116 + 08. The angular position of each quasar determines a phase in the radio signal at the output of the radio interferometer that depends on the wavelength of the radiation and on the baseline between the radio telescopes. The angular Figure 7.2. Results of radio-wave deflection measurements 1969-75. Value of i (1 + 7) 0.88 i i 0.92 I i 0.96 i i 1.00 r Radio Deflection Experiments 1969 1.04 i 1.08 r I Muhleman et al. (1970) Seielstad et al. (1970) | Hill (1971) I 1970 Shapiro (quoted in Weinberg, 1972) I Sramek (1971) | a, 1971 Sramek (1974) x w Riley(1973) 1972 Weileretal. (1974) I t- Counselman et al. (1974) 1973 Weileretal. (1974) 1974 Fomalont and Sramek (1975) 1975 Fomalont and Sramek (1976) 5 10 2040=o Value of ScalarTensor GO i Classical Tests 173 separation between a pair of quasars is determined by a difference in phases. As the Earth moves in orbit, changing the lines of sight of the quasars relative to the Sun, the angular separation 89 varies [Equation (7.25)], resulting in a variation in the phase difference. The time variation in the quantities d, dr, d>, and 3>r in Equation (7.25) is determined using an accurate ephemeris for the Earth and initial directions for the quasars, and the resulting prediction for the phase difference as a function of time is used as a basis for a least-squares fit of the measured phase differences, with one of the fitted parameters being the coefficient ^(1 + y). A number of measurements of this kind over the past decade have yielded an accurate determination of ^(1 + y), which has the value unity in general relativity. Those results are shown in Figure 7.2. One of the major sources of error in these experiments is the solar corona which bends radio waves much more strongly than it bent the visible light rays that Eddington observed. Advancements in dual frequency techniques have improved accuracies by allowing the coronal bending, which depends on the frequency of the wave, to be measured separately from the gravitational bending, which does not. Fomalont and Sramek (1977) provide a thorough review of these experiments, and discuss the prospects for improvement. 7.2 The Time Delay of Light Because of the presence of the gravitational field of a massive body, a light signal will take a longer time to traverse a given distance than it would if Newtonian theory were valid. An expression for this "time delay" can be obtained simply from Equation (7.3). Integrating the equation using Equation (7.6), we obtain ] (729) Then from Equation (7.1), the coordinate time taken to propagate from the point of emission to x is given by ^ l l ^ ] (7.30) For a signal emitted from the Earth, reflected off a planet or spacecraft at xp, and received back at Earth, the roundtrip travel time At is given by At - 2|xe - xp| + 2(1 + y>»to[(r« + * ' - y ' - X ' - * ) ] (7.31) Theory and Experiment in Gravitational Physics 174 where ft is the direction of the photon on its return flight. Here we have ignored the motion of the Earth and planets during the round trip of the signal. To be completely correct, the round trip travel time should be expressed in terms of the proper time elapsed during the round trip, as measured by an atomic clock on Earth; but this introduces no new effects, so we will not do so here. The additional "time delay" 8t produced by the second term in Equation (7.31) is a maximum when the planet is on the far side of the Sun from the Earth (superior conjunction), i.e., when xffi n ~ r$,
x p n ~ rp,
(7.32)
then
5t = 2(1 + y)mln(4r9rp/d2)
= i(l + y) [240 ps - 20 JIS In ( ^ - Y (f\\
(7.33)
where R o is the radius of the Sun, and a is an astronomical unit. For
further discussion of the time delay see Shapiro (1964,1966a,b), Muhleman
and Reichley (1964), and Ross and Schiff (1966).
In the decade and a half since Shapiro's discovery of this effect, a
number of measurements of it have been made using radar ranging to
targets passing through superior conjunction. Since one does not have
access to a "Newtonian" signal against which to compare the round trip
travel time of the observed signal, it is necessary to do a differential
measurement of the variations in round trip travel times as the target
passes through superior conjunction, and to look for the logarithmic
behavior. To achieve this accurately however, one must take into account
the variations in round trip travel time due to the orbital motion of the
target relative to the Earth [variations in |x e x p | in Equation (7.31)].
This is done by using radar-ranging (and possibly other) data on the
target taken when it is far from superior conjunction (i.e., when the timedelay term is negligible) to determine an accurate ephemeris for the target,
using the ephemeris to predict the PPN coordinate trajectory xp(t) near
superior conjunction, then combining that trajectory with the trajectory
of the Earth xffi to determine the quantity |x$ x p | and the logarithmic term in Equation (7.31). The resulting predicted round trip travel times in terms of the unknown coefficient 5(1 + y) are then fit to the measured travel times using the method of least squares, and an estimate obtained for |(1 + y). [This is an oversimplification, of course. The reader is referred to Anderson (1974) for further discussion.] Classical Tests 175 Three types of targets have been used. The first type is a planet, such as Mercury or Venus, used as a passive reflector of the radar signals ("passive radar"). One of the major difficulties with this method is that the largely unknown planetary topography can introduce errors in round trip travel times as much as 5 /zs (i.e., the subradar point could be a mountaintop or a valley), which introduce errors in both the planetary ephemeris and, more importantly, in the round trip travel times at superior conjunction. Several sophisticated attempts have been made to overcome this problem. The second type of target is an artificial satellite, such as Mariners 6 and 7, used as active retransmitters of the radar signals ("active radar"). Here topography is not an issue, and the on-board transponders permit accurate determination of the true range to the spacecraft. Unfortunately, spacecraft can suffer random perturbing accelerations from a variety of sources, including random fluctuations in the solar wind and solar radiation pressure, and random forces from on-board attitude-control devices. These random accelerations c^in cause the trajectory of the spacecraft near superior conjunction to differ by as much as 50 m or 0.1 us from the predicted trajectory in an essentially unknown way. Special methods of analyzing the ranging data ("sequentialfiltering")have been devised to alleviate this problem (Anderson, 1974). The third target is the result of an attempt to combine the transponding capabilities of spacecraft with the imperturbable motions of planets by anchoring satellites to planets. Examples are the Mariner 9 Mars orbiter and the Viking Mars landers and orbiters. In all of these cases, as in the radio-wave deflection measurements, the solar corona causes uncertainties because of its slowing down of the radar signal. Again, dual frequency ranging helps reduce these errors, in fact, it is the corona problem that provides the limiting accuracy for the most recent time-delay measurements. The results for the coefficient |(1 + y) of all radar time-delay measurements performed to date are shown in Figure 7.3. Recent analyses of Viking data have resulted in a 0.1% measurement (Reasenberg et al. 1979). From the results of light-deflection and time-delay experiments, we can conclude that the coefficient ^(1 + y) must be within at most 0.2% of unity. Most of the theories shown in Table 5.1 can select their adjustable parameters or cosmological boundary conditions with sufficient freedom to meet this constraint. Scalar-tensor theories must have co > 500 to be within 0.1% or w > 250 to be within 0.2% of unity. Theory and Experiment in Gravitational Physics 176 Value of i (1+7) 0.88 I I 0.92 I T 0.96 T I 1.00 1.04 1.08 i TimeDelay Measurements Passive Radar to Mercury and Venus Shapiro (1968) Shapiro etal. (1971) - 1 Active Radar Mariner 6 and 7 Anderson et al. (1975) Anchored Spacecraft Mariner 9 Anderson et al. (1978), Reasenberg and Shapiro (1977) Viking Shapiro et al. (1977) Cain etal. (1978) Reasenberg etal. (1979) (±0.001) i 5 10 2040°° Value of Scalar-Tensor co Figure 7.3. Results of radar time-delay measurements 1968-79. 7.3 The Perihelion Shift of Mercury The explanation of the anomalous perihelion shift of Mercury's orbit was another of the triumphs of general relativity. However, between 1967 and 1974, there was considerable controversy over whether the perihelion shift was a confirmation or a refutation of general relativity because of the apparent existence of a solar quadrupole moment that could contribute a portion of the observed perihelion shift. Although this controversy has abated somewhat, the question of the size of the solar quadrupole moment has yet to be conclusively answered. The PPN prediction for the perihelion shift can be obtained from the PPN equation of motion [Equation (6.31)]. We consider a system of two bodies of inertial masses n^ and m2, and self-gravitational energies Q^ and Q2 The first body has a small quadrupole moment Q'{. We assume that the entire system is at rest with respect to the universe rest frame (w = 0) and that there are no other gravitating bodies near the system. In Chapter 8, we shall return to the effects of motion and of distant bodies (preferred-frame and preferred-location effects) on the perihelion Classical Tests 111 shift. For the moment we ignore them. We work in a PPN coordinate system in which the center of mass of the system is at rest at the origin. Making use of the fact that each body is nearly spherical, Qf m we obtain from Equation (6.31) the acceleration of each body ! - - ^ F(2y + 20 ^ a, = 4 + ai )v! v2 - i(2y + 2 + a2 + a3)t>l + f(1 + «2)(v2 n) 2 l - ^ £(2y + 2*, - (2y + l)v 2 j v, + Y7^' (4y + 4 + ai)Vl - (4r + 2 + a i - 2a >2 k , a2 = {l<-*2;x-^ - x } (7.34) where x = x 21 , n = x/r. Including the Newtonian contribution of the quadrupole moment in the quasi-Newtonian potential produced by body 1, we have xJ (UJi = (mA)2 p-, (Uj)2 = -(mA)x ^-\^r(^nknlW - 25H1) (7.35) where (mA)j and (mA)2 are the active gravitational masses, given by Equation (6.51). For a body which is axially symmetric about an axis with direction e, Qf can be shown to have the form Qf = mxR\J2W^k - 3^2*) (7.36) where J2 is a dimensionless measure of the quadrupole moment, given by J2={C- A)/mR2 (7.37) where C = [moment of inertia about symmetry axis], A = [moment of inertia about equatorial axis], R = [radius] (7.38) Theory and Experiment in Gravitational Physics 178 (The subscript 2 on J2 denotes that it is associated with the quadrupole, or / = 2 moment of the body.) Since the center of mass of the system is at rest, we may, to sufficient accuracy in the post-Newtonian terms in Equation (7.34), replace vx and v2 by Vi = -(m2/m)v, v2 = (mi/m)v (7.39) where v = v2-v1, m = m1+m2 (7.40) We also define the reduced mass fi = mlm2/m (7.41) Then the relative acceleration a 2 - a , 5 a takes the form a = - ^ + * "1*^2(1) ocl+a2+ ^ [ [ 1 5 ( g oc3)^v2 m ^ . ft)2fi _ 6 ( e . m + | ( 1 + a2)-^-(v n) m ] (7.42) where m* = (mP/w)2(mA)1 + (mp/m = m(l + [self-energy terms for bodies 1 and 2]). (7.43) The self-energy terms from Equation (6.51) that appear in the above expression are at most ~ 1 0 ~ 5 for the Sun, and are constant. Thus the difference between m* and m is unmeasurable, so we simply drop the (*) in Equation (7.42). We consider a planetary orbit with the following instantaneous orbit elements (see Smart, 1953, for detailed discussion of the definitions): inclination i relative to a chosen reference plane, the angle Q from a chosen reference direction in the reference plane to the ascending node, the angle co of perihelion from the ascending node measured in the orbital plane, the eccentricity e and semi-major axis a. The sixth element T, the time of periastron passage, is an initial'condition and is irrelevant for our purposes. Classical Tests 179 For the solar system, the reference plane is chosen to be the plane of the Earth's orbit (ecliptic) and the reference direction is the Earth-Sun direction at spring equinox. Following the standard procedure for computing perturbations of orbital elements [Smart (1953), Robertson and Noonan (1968)], we resolve the acceleration a [Equation (7.42)] into a radial qomponent M, a component "W, normal to the orbital plane, and a component £f normal to Si and iV, and calculate the rates of change of the orbital elements using the formulae [in the notation of Robertson and Noonan (1968)]: da, -r-= at p® ^ip + r) . , iTr . / - -r cos 4> + -, sin 4> cot i sin(<w + <p), he he h di Wr ~ = cos(co + <t>), (7.44) (7.47) r sm{w + <f>) i ; : ~ (7.48) dt h sm i where h is the angular momentum per unit mass of the orbit, <j> is the angle of the planet measured from perihelion, and p is the semi-latus rectum given by p = a(l - e2) (7.49) The variables r and 0 are related to the instantaneous orbit elements by the definitions r == p(l + ecos 4>)~l, r2 d4>/dt = h = (mp)1'2 (7.50) Now, because observations of the planets are made with reference to geocentric coordinates, the perihelion measured is the perihelion relative to the equinox, <o, given by (o = co + Qcos i (7.51) Then the rate of change "of c5 is given by deb pM Sr(p + r) = -y cos 4> H T sm$
dt
he
he
(7.52)
Theory and Experiment in Gravitational Physics
180
where we have used the fact that, for all the planets, i is small, so that
sin i« 1. For the perturbing acceleration in Equation (7.42) (we drop the
subscript " 1 " on m, R and J 2 )
-11 (2y + 20) - - yv2 + (2y + 2)(v ii)2
(2 + «! - 2f2) £ - i(6 + «! + a 2 + a3)
y = - 3(m/?2J2/r4)(e n)(e 2)
+ j j (v n)(v X)|~(2y + 2) - £ (2 - a t + «2) 1
<7-53)
where 2 is a unit vector in the plane of the orbit in the direction of the
orbital motion, normal to ii. For Mercury's orbit, the solar symmetry axis
is essentially normal to the orbital plane, hence e n ^ 0. Then substituting
Equations (7.53) into (7.52) and integrating over one orbit using Equation
(7.50) yields
AS = (67tm/p)[i(2 + 2y - 0)
- a 2 + a 3 + 2£2)fi/m + J2(R2/2mp)-]
(7.54)
This is the only secular perturbation of an orbital element produced by the
post-Newtonian terms in Equation (7.42); however the quadrupole terms
can be shown to produce secular changes in i and Q proportional to sin 0
and sin 0/sin i respectively, where 6 is the tilt of the Sun's symmetry or
rotation axis relative to the ecliptic (9 « 7°). The elements a and e suffer
no secular changes under either of these perturbations.
The first term in Equation (7.54) is the classical perihelion shift, which
depends upon the PPN parameters y and p. The second term depends
upon the ratio of the masses of the two bodies (Will, 1975); it is zero in
any fully conservative theory of gravity (al = a2 = a3 = £2 = 0); it is also
negligible for Mercury, since /x/m ~ m^/mo ~ 2 x 10" 7 . We shall drop
this term henceforth. The third term depends upon the solar quadrupole
moment J2. For a Sun that rotates uniformly with its observed surface
angular velocity, so that the quadrupole moment is produced by centri-
Classical Tests
181
fugal flattening, one may estimate J2 to be ~1 x 10~7. Normalizing J2
by this value and substituting standard orbital elements and physical
constants for Mercury and the Sun (Allen, 1976), we obtain the rate of
perihelion shift c5, in seconds of arc per century,
5 = 42795 Ape"1
Xv = [i(2 + 2y - P) + 3 x IO-V2/IO- 7 )]
(7.55)
The measured perihelion shift is accurately known: after the effects of
the general precession of the equinoxes (~5000" c""1) and the perturbing
effects of the other planets (280" c~1 from Venus, 150" c'1 from Jupiter,
100" c~1 from the rest) have been accounted for, the remaining perihelion
shift is known (a) to a precision of about one percent from optical observations of Mercury during the past three centuries (Morrison and Ward,
1975), and (b) to about 0.5% from radar observations during the past
decade (Shapiro et al., 1976). Unfortunately, measurements of the orbit
of Mercury alone are incapable at present of separating the effects of
relativistic gravity and of solar quadrupole moment in the determination
of Xp. Thus, in two recent analyses of radar distance measurements to
Mercury, J2 was assumed to have a value corresponding to uniform rotation (effect on Xp negligible), and the PPN parameter combination was
estimated. The results were
1
s( + 7
fl.005 ± 0.020(1966-1971 data: Shapiro et al., 1972)
P) ~ | 1 0 0 3 ± 0.005(1966-1976 data: Shapiro et al, 1976)
(7.56)
where the quoted errors are 1CT estimates of the realistic error (taking into
account possible systematic errors).
The origin of the uncertainty that has clouded the interpretation of
perihelion-shift measurements is a series of experiments performed in 1966
by Dicke and Goldenberg (see Dicke and Goldenberg, 1974, for a detailed
review). Those experiments measured the visual oblateness or flattening
of the Sun's disk and found a difference in the apparent polar and equatorial angular radii of AR = (43'.'3 ± 3'.'3) x 10"3. By taking into account
the oblateness of the surface layers of the Sun caused by centrifugal
flattening, this oblateness signal can be related to J2 by (Dicke, 1974)
J2 = §(A*/KG) - 5.3 x 10" 6
(7.57)
which gives (i?G = 959")
J2 = (2.47 ± 0.23) x 10"5 (Dicke and Goldenberg, 1974) (7.58)
Theory and Experiment in Gravitational Physics
182
A value of J2 this large would have contributed about 4" c~1 to Mercury's
perihelion shift, and thus would have put general relativity in serious
disagreement with the observations, while on the other hand supporting
Brans-Dicke theory with a value co ^ 5, whose post-Newtonian contribution to the perihelion shift would thus have been 39" per century.
These results generated considerable controversy within the relativity
and solar physics communities, and a mammoth number of papers was
produced, both supporting and opposing solar oblateness. One recurring
line of argument in opposition to the Dicke-Goldenberg result was that
their method of measuring the difference in brightness between the solar
pole and the solar equator of an annulus of the solar limb produced around
an occulting disk placed in front of the Sun, could equally well be interpreted by assuming a standard solar model (with a small J2 ~ 10 ~7
produced by centrifugal flattening) with a temperature difference on the
solar surface between the equator and the pole, leading to a brightness
difference indistinguishable from that due to a geometrical oblateness.
Such a brightness difference, it was suggested, could also be produced by
an equatorial excess in the number of solar faculae. Refutations of these
arguments by Dicke and his supporters, and counter-refutations abounded
in the literature.
The controversy abated somewhat in 1973, when Hill and his collaborators performed a similar visual oblateness measurement that yielded
AR = (9'.'2 ± 6'.'2) x 1(T3 or
J2 = 0.10 ± 0.43 x 10" 5
(Hill et al., 1974)
(7.59)
an upper limit five times smaller than Dicke's value. (See also Hill and
Stebbins, 1975). The disagreement between these two observational results
remains unresolved.
One of the major difficulties in relating visual solar oblateness results
to J2 is that a considerable amount of complex solar physics theory must
be employed. There is, however, a way of determining J2 unambiguously,
namely by probing the solar gravity field at different distances from the
Sun, thereby separating the effects of J2 from those of relativistic gravitation through their different radial dependences [see Equation (7.42)].
One method would compare the perihelion shifts of different planets. But
the perihelion shifts of Venus, Earth, and Mars are not known to sufficient
accuracy, although Shapiro et al. (1972) pointed out that several more
years of radar observations of the inner planets may permit such a comparison. Another method would take advantage of Mercury's orbital
eccentricity (e ~ 0.2) and search for the different periodic orbital pertur-
Classical Tests
183
bations induced by J2 and by relativistic gravity. The accuracy required
for such measurements would necessitate tracking of a spacecraft in orbit
around Mercury, but preliminary studies have shown that J2 could be
determined to within a few parts in 107 (Anderson et al., 1977, Wahr and
Bender, 1976). Finally, and most promisingly, a mission currently under
study by NASA for the 1980s known as the Solar Probe, a spacecraft in
a high-eccentricity solar orbit with perihelion distance of four solar radii
("Arrow to the Sun"), could yield a measurement of J2 with a precision of
a part in 108 (Nordtvedt, 1977, Anderson et al., 1977). Such missions would
also lead to improved determinations of y and /?. The possibility of determining y and j8 from measurements of the precessions of the pericenters
of the inner satellites of the gas giant planets has recently been considered
by Hiscock and Lindblom (1979).
8
Tests of the Strong Equivalence Principle
The next class of solar system experiments that test relativistic gravitational effects may be called tests of the Strong Equivalence Principle
(SEP). That principle states that (i) WEP is valid for self-gravitating
bodies as well as for test bodies (GWEP), (ii) the outcome of any local
test experiment, gravitational or nongravitational, is independent of the
velocity of the freely falling apparatus, and (iii) the outcome of any local
test experiment is independent of where and when in the universe it is
performed. In Section 3.3, we pointed out that many metric theories of
gravity (perhaps all except general relativity) can be expected to violate
one or more aspects of SEP. In Chapter 6, working within the PPN framework, we saw explicit evidence of some of these violations: violations of
GWEP in the equations of motion for massive self-gravitating bodies
[Equations (6.33) and (6.40)]; preferred-frame and preferred-location
effects in the locally measured gravitational constant GL [Equation (6.75)];
and nonzero values for the anomalous inertial and passive gravitational
mass tensors in the semiconservative case [Equation (6.88)].
This chapter is devoted to the study of some of the observable consequences of such violations of SEP, and to the experiments that test for
them. In Section 8.1, we consider violations of GWEP (the Nordtvedt effect), and its primary experimental test, the Lunar Laser-Ranging"E6tvos"
experiment. Section 8.2 focuses on the preferred-frame and preferredlocation effects in GL. The most precise tests of these effects are obtained
from geophysical measurements. In Section 8.3, we consider preferredframe and preferred-location effects in the orbital motions of planets.
Perihelion-shift measurements are important tests of such effects. Another
violation of SEP would be the variation with time of the gravitational
constant as a result of cosmic evolution. Tests of such variation are de-
Tests of the Strong Equivalence Principle
185
scribed in Section 8.4. In Section 8.5, we summarize the limits on the
values of the PPN parameters y, /?, £, al5 a2, and <x3 that are set by the
classical tests and by tests of SEP, and discuss the consequences for
the metric theories of gravity described in Chapter 5.
8.1
The Nordtvedt Effect and the Lunar Eorvos Experiment
The breakdown in the Weak Equivalence Principle for massive,
self-gravitating bodies (GWEP), which many metric theories predict, has
a variety of observable consequences. In Chapter 6, we saw that this violation could be expressed in quasi-Newtonian language by attributing to
each massive body inertial and passive gravitational mass tensors m{k and
m£* which may differ from each other. The quasi-Newtonian part of the
body's acceleration may be written [see Equation (6.43)]
(mi)i*Kftew« = (mP)lrUlJ
(8.1)
lra
where U is a quasi-Newtonian gravitational potential, and {fh^f and
(fhp)'" are given by
(«i)? = ma{o-}k[l + (ax - a2 + C ^ / m J + (a2 - Ci + C2)OfM,},
(mP)'am = ma{5"»[l + (4/8 - y - 3 - 3£)Qa/ma] - {Qf/m.}
(8.2)
where Qa and Q£* are the body's internal gravitational energy and gravitational energy tensor (see Table 6.2), and ma is the total mass energy of the
body.
Now, most bodies in the solar system are very nearly spherically symmetric, so we may approximate
a* * &a5Jk
(8.3)
J
Any "Nordtvedt" effects that arise from the anisotropies in Q * in Equation (8.2) are expected to be too small to be measurable in the foreseeable
future (see Will, 1971b, for an example). With the above approximation
we write the quasi-Newtonian Equation (8.1) in the form
Wkw. = K M . U , .
(8.4)
where
(mp/ifi). = 1 + (40 - y - 3 - 4ft - ax + §<x2 - Ki - K ^ M . ,
U = Z (mAV\,V
(8-5)
The most important consequence of the Nordtvedt effect is a polarization of the Moon's orbit about the Earth [Nordtvedt (1968c)]. Because
Theory and Experiment in Gravitational Physics
186
the Moon's self-gravitational energy is smaller than the Earth's, the Nordtvedt effect causes the Earth and Moon to fall toward the Sun with slightly
different accelerations. Including their mutual attraction, we have [from
Equations (8.4) and (8.5), and neglecting quadrupole moments],
(8.6)
where X and Xo are vectors from the Sun to the Earth and Moon, respectively, and x is a vector from the Earth to the Moon (Figure 8.1). The
relative Earth-Moon acceleration a, denned by
®
(8.7)
Figure 8.1. (a) Geometry of the Earth-Moon-Sun system.
(b) The Nordtvedt effect - a polarization of the Moon's orbit with the
apogee always directed along the Earth-Sun line.
Sun
Moon
Earth
(a)
(b)
Tests of the Strong Equivalence Principle
187
is then given by
a = ~m*x/r3 + i/[(Q//n)© - (Q/m)J/n o X/K 3
+ {m^m^m^X/R3 - XJR3)
(8.8)
where
G
^+1^-1^-^2
(8-9)
The first term in Equation (8.8) is the Newtonian acceleration between
the Earth and Moon and the second term is the difference between the
Earth's and Moon's acceleration toward the Sun (Nordtvedt effect). The
third term is the classical tidal perturbation on the Moon's orbit; since
it is a purely nonrelativistic perturbation, we will not consider it for the
moment. Hence, the equation of motion of the Moon relative to the Earth,
including the perturbation arising from the Nordtvedt effect, is
a = -m*x/r3
+ ff[(O/m)e - (n/m\~}mQX/R3
(8.10)
We assume that the Moon's unperturbed orbit is circular with angular
velocity co0 and in the x-y plane, and also that the orbit of the Earth
around the Sun is circular with angular velocity a>s in the same plane.
We work in an inertial PPN coordinate system centered at the Sun. Then
the acceleration a and the angular momentum per unit mass of the EarthMoon orbit are given by
a = d2x/dt2,
h = x x (dx/dt)
(8.11)
and the following relations hold
d2r/dt2 = x a/r + h2/r\
dh/dt= (x x a)
(8.12)
where r = \x\. Thus, by making use of Equation (8.10) and by defining
da s ij[(|n|/m)e - (p\/m\-]mQ/R2
(8.13)
we obtain
d2r/dt2 = -m*/r2
+ h2fr3 + SacosAt,
dh/dt= -rSa sin At
(8.14)
where
cosAt s n x/r,
sin At = (n x x/r) z ,
A = co0 ws
(8.15)
Theory and Experiment in Gravitational Physics
188
where n = X/R. Note that At is the angle between the Earth-Sun and
Earth-Moon directions. We next linearize about a circular orbit:
r = ro + 3r,
h = ho + 5h
(8.16)
2
and use m*jr% = hl/r% = co ,. Integration of the resulting equations yields
5h = (r0/A)5acosAt,
(8.17)
(8.18)
Equation (8.18) represents a polarization of the Earth-Moon system
by the external field of the Sun. This polarization of the orbit is always
directed toward the Sun if r\ > 0 (away from the Sun if r\ < 0) as it rotates
around the Earth (see Figure 8.1).
Using Equations (8.13) and (8.18) and the values mQ/R2 st 5.9 x 10" 6
km s~2, w° ^ 13.4OJS * 2.7 x 1(T 6 s" 1 , (Q/m)e * - 4.6 x 1(T 10 , and
(Q/m)a =* - 0 . 2 x 10~ 10 (Allen, 1976), we obtain
dr =s 8.0»/ cos(<o0 - cos)t m
(8.19)
Actually, a more accurate calculation would take into account the effect
of the Nordtvedt perturbation on the tidal acceleration term in Equation
(8.8) and that of the tidal perturbation on the Nordtvedt term; this modifies the coefficient of 8r by a factor of approximately 1 + 2cos/co0 ^ 1.15,
giving
8r ^ 9.2/7 cos(a>0 - ojs)t m
(8.20)
Since August, 1969, when the first laser signal was reflected from the
Apollo 11 retroreflector on the Moon, the Lunar Laser-Ranging Experiment (LURE) has made regular measurements of the round trip travel
times of laser pulses between McDonald Observatory in Texas and the
lunar retroreflectors, with accuracies of 1 ns (30 cm) (see Bender et al,
1973, Mulholland, 1977). These measurements were fit using the method
of least squares to a theoretical model for the lunar motion that took
into account perturbations due to the other planets, tidal interactions,
and post-Newtonian gravitational effects. The predicted round trip travel
times between retroreflector and telescope also took into account the
librations of the Moon, the orientation of the Earth, the location of the
observatory, and atmospheric effects on the signal propagation. The
"Nordtvedt" parameter, //, along with several other important parameters
of the model were then estimated in the least-squares method.
Tests of the Strong Equivalence Principle
189
An important issue in this analysis is whether other perturbations of
the Earth-Moon orbit could mask the Nordtvedt effect. Most perturbations produce effects in 5r, which, when decomposed into sinusoidal components, occur at frequencies different from that of the Nordtvedt term
(e.g., at angular frequencies co0, 2A), and thus can be separated cleanly
from it using a multi-year span of data. However, there is one perturbation, due to the tidal term that we neglected in Equation (8.8), that does
have a component at the frequency A. To see this, we expand X and Xo
about Xc, the center of mass of the Earth-Moon system, using
Xo = Xc + (roe/m*)x,
X = Xc - (mjm*)x
(8.21)
where we now ignore all post-Newtonian self-energy corrections to masses.
Then the tidal acceleration in Equation (8.8) becomes
a I l X f e i y e ]
I - 2mc/m*) ~ [nc - 5(nc e)% + 2(nc S)e]
(8.22)
Kc
where fic = Xc/Rc, e = x/r. It is the second term, of order {r^/Rf), in the
above expression that leads to a perturbation in Sr of frequency A. Applying Equations (8.12), (8.15), (8.16), and integrating, it is possible to show
straightforwardly that
2mA
8 [col-A
+ [terms proportional to cos2At, cos 3At...]
(8.23)
Using the expressions
col = rn*/r30,
cof = mQ/R?
(8.24)
we may rewrite Equation (8.23) in the useful form
where Q = toj(o0 x 0.075. Again, a more accurate computation, taking
into account the mutual effect of the two terms in Equation (8.22), modifies
Equation (8.25) by corrections that depend upon Q, with the result (Brown,
1960)
Theory and Experiment in Gravitational Physics
190
where
F(Q) = 1 + (81/15)Q + ^ 1.64
(8.27)
Substituting numerical values (Allen, 1976) yields
<5>"tidai ^ 110 cos At km
(8.28)
Although this term is ten thousand times larger than the nominal amplitude of the Nordtvedt effect, it turns out, fortunately, that the parameters that appear in Equation (8.26) are known with sufficient accuracy
that the tidal term can be accounted for to a precision of about 2 cm. The
values of Rc, mQ/m*, and Q. are known to sufficient precision from other
data, while the values of mjm* and r0 are estimated using the laserranging data via their effects on the lunar orbit at frequencies other than A.
Two independent analyses of the data taken between 1969 and 1975
were carried out, both finding no evidence, within experimental uncertainty, for the Nordtvedt effect. Their results for n were
n
_ fO.OO ± 0.03 [Williams et al. (1976)],
~ (0.001 + 0.015 [Shapiro et al. (1976)]
(8.29)
where the quoted errors are la, obtained by estimating the sensitivity of
n to possible systematic errors in the data or in the theoretical model.
The formal statistical errors that emerged from the data analysis were
typically much smaller, of order <x(>7)fOrmai ~ + 0.004.
This represents a limit on a possible violation of GWEP for massive
bodies of 7 parts in 1012 (compare Table 2.2). For Brans-Dicke theory,
these results force a lower limit on the coupling constant a> of 29 (2a,
Shapiro result).
Improvements in the measurement accuracy and in the theoretical
analysis of the lunar motion may tighten this limit by an order of magnitude (Williams et al., 1976), while a comparable test of the Nordtvedt
effect may be possible using the Sun-Mars-Jupiter system (Shapiro et al.,
1976). Other potentially observable consequences of the Nordtvedt effect
include shifts in the stable Lagrange points of Jupiter (measurable by
ranging to the Trojan asteroids), and modification of Kepler's third law
(Nordtvedt, 1968a, 1970a, 1971a,b).
8.2
Preferred-Frame and Preferred-Location Effects:
Geophysical Tests
In Section 6.3, we found that some metric theories of gravity predict preferred-frame and preferred-location effects in the locally measured
gravitational constant GL, measured by means of Cavendish experiments.
Tests of the Strong Equivalence Principle
191
These effects represent violations of SEP. Unfortunately, present-day
Cavendish experiments are only accurate to about one part in 105 in absolute measurements of GL (Rose et al, 1969), and so cannot discern the
post-Newtonian corrections to GL in Equation (6.75).
However, there is a "Cavendish" experiment that can detect corrections
in GL, one in which the source is the Earth and the test body is a gravimeter on the surface of the Earth. A gravimeter is a device that measures
the force required to keep a small "proof" mass stationary with respect
to the center of the Earth. This is exactly the physical situation assumed
in our derivation of GL in Section 6.3. Because of uncertainties in our
knowledge of the internal structure and composition of the Earth, it is
impossible to determine the absolute value of GL by this method with
sufficient precision to detect post-Newtonian effects. Instead, gravimeters
are powerful tools for measuring variations in the gravitational force. In
Newtonian geophysics, these variations are known as solid-Earth tides;
in post-Newtonian geophysics, measurements of these variations can test
for variations in GL, with high precision.
We therefore shall apply Equation (6.75) for GL to a gravimeter "Cavendish" experiment, and shall focus on the post-Newtonian terms that
vary with time. A detailed justification of the application of Equation
(6.75) to this situation is given by Will (1971d). Recall that
GL = 1 - [4)8 - y - 3 - C2 - «3 + //mr2)] Uext
- Aw.n
^/ m r 22Vw.
z\22 + «1 - 3//mr 2 )t/f xt e^
- 3//mr
)(we e)
2
U
(8.30)
where /, m, and r are the spherical moment of inertia, mass, and radius of
the Earth, e is a unit vector directed from the gravimeter to the center of
the Earth, and
U{kn = Z manJ9an%JrBa,
Uext = UHt
(8.31)
Consider the first post-Newtonian term in Equation (8.30). Because of
the Earth's eccentric orbital motion, the external potential produced by
the Sun varies yearly on Earth by only a part in 1010, too small to be detected with confidence by Earth-bound gravimeters or Cavendish experiments. The time-varying effects of other bodies (planets, the galaxy) are
even smaller.
Next, consider the preferred-frame terms. The Earth's velocity w e is
made up of two parts, a uniform velocity w of the solar system relative to
Theory and Experiment in Gravitational Physics
192
the preferred frame, and the Earth's orbital velocity v around the Sun, thus
w | = w2 + 2w v + v2,
(w e 6)2 = (w e)2 + 2(w e)(v e) + (v e)2
(8.32)
So because of the Earth's rotation (changing e) and orbital motion (changing v), there will be variations in the gravimeter measurements of GL,
given by (we retain only terms which vary with amplitude larger than
a 3 - ax)w v
+ i<x2[(w g)2 + 2(w e)(v e) + (v e) 2 ]
(8.33)
where we have used the fact that, for the Earth,
I C* mr2/2
(8.34)
Finally, we consider the preferred-location term. According to our discussion of the PPN formalism (see Section 4.1) the potential I/£, must
include all local gravitating matter that is not part of the cosmological
background used to establish the asymptotically Lorentz PPN coordinate
system. Therefore it must include the Sun, planets, stars, the galaxy, and
possibly the local cluster of galaxies. In this case, [/£, is dominated by our
galaxy (Ua ~ 5 x 10~7), followed by the Sun (UQ ~ 1 x 10~8), thus,
AGJGh = - K t / G ( e eG)2 - ^Uo(e
e0)2
(8.35)
In order to compare this variation in G with gravimeter data, we must
perform a harmonic analysis of the terms in Equations (8.33) and (8.35).
The frequencies involved will be the sidereal rotation rate of the Earth Q,
due to the changing direction of e relative to the fixed direction of w and
e G , and its orbital sidereal frequency co due to the changing direction of v
relative to w, along with harmonics and linear combinations of these frequencies. We work in geocentric ecliptic coordinates, and assume a circular Earth orbit, with the Earth at vernal equinox at t = 0. Then,
e 0 = cos cotex + sin a>tey,
v = i;(sin cotex cos atey),
w s w[cos /^(cos Xwex + sin lwey) + sin /fwez],
eG = cos /SG(cos AGex + sin lGey) + sin /?Ge2
(8.36)
The latter two equations define the ecliptic coordinates (lw,/?w) and
For a
(^G,PG)gravimeter stationed at Earth latitude L,
e = cosLcos(Qt e)ex + [cosLsin(lQt e)cos0 + sinLsin0]e y
- [cos L sin(Qt e) sin 6 sin L cos 0]ez
(8.37)
Tests of the Strong Equivalence Principle
193
where £ is related to the longitude of the gravimeter on the Earth, and 0 is
the "tilt" (23^°) of the Earth relative to the Earth's orbit (ecliptic). Equations (8.36) and (8.37) give
w v = wv cos fiw sin(a>t Xw),
(8.38)
(w e)2 = w 2 [i + | ( i - sin2 <5J(i - sin2 L)
+ \ sin 2<5W sin 2L cos(fit £ ocw)
+ icos 2 <5 H ,cos 2 Lcos2(Qf - £ - <xj],
(8.39)
(w e)(v e) = wi;{yCOS)S)1,sin(a»t - AJ
+ (i sin 2 L)[^cos jSw sin(cot X^ + § sin <5W sin 0 cos cot]
+ j sin dw(l cos 6) sin 2L sin[(Q + co)t e]
5Cos<5 w sin0sin2Lcos[(Q + (o)t & a w ]
\ sin ^ w (l + cos 6) sin 2L sin[(Q a>)t s]
jcos 6W sin 9 sin 2Lcos[(Q a>)t £ a w ]
\ - cos^)cos 2 Lsin[(2fi + a»)f - 2E - a w ]
l + cos0)cos 2 Lsin[(2Q - co)t - 2E - a w ]},
(8.40)
2
2
2
2
(v e) = i> {± + | ( i - sin L)(i - ^sin 0)
| ( i sin 2 L)sin 2 0cos2cof + |sin20sin2Lsin(Qf - e)
i sin 0(1 - cos 9) sin 2L sin[(fi + 2co)t - e]
+ jsin 9(1 + cos 0) sin 2L sin[(Q - 2co)t - s]
+ \ sin2 9 cos 2 L cos 2(Qf - s)
i ( l - cos0)2 cos2 Lcos[2(Q + co)t - 2e]
| ( 1 + cos 9)2 cos 2 Lcos[2(Q - co)t - 2E]},
2
2
(8.41)
2
(SG e) = i + | ( i - sin «5G)(i - sin L)
+ -j sin 2<5G sin 2Lcos(Q( s aG)
+ \ cos 2 ^ G cos 2 L cos 2(Q{ - £ - aG ),
(8.42)
(e o e)2 = H | ( i - sin2 L)(i - ^sin2 9)
+ i(j - sin2L)sin29cos2cot + i sin 29 sin 2L sin(Qt - E)
+ | sin 0(1 - cos 0) sin 2L sin[(Q + 2co)t - e]
5 sin 0(1 + cos 0) sin 2L sin[(O - 2co)t e\
+ i sin2 0 cos 2 L cos 2(Q( - E)
+ | ( 1 - cos0) 2 cos 2 Lcos[2(Q + co)t - 2e]
+ i ( l + cos0) 2 cos 2 Lcos[2OQ - co)t - 2e]
(8.43)
Theory and Experiment in Gravitational Physics
194
where we have used both the ecliptic coordinates (Xw, /?w), (XG, fic) and the
equatorial coordinates (OLW,5W), (<XG,<SG) (Smart, 1960) corresponding to
the directions of w and eG in order to simplify the various expressions.
These coordinate systems are related by
sin 8 = sin /? cos 9 + cos /? sin 9 sin X,
cos 8 cos a = cosficos X,
cos 8 sin a = sin f$sin 9 + cos /? cos 9 sin A (8.44) Equations (8.38)-(8.43) reveal four different types of variations in GL. (i) Semidiurnal variations: These are the terms that vary with frequency around 2Q: 2fi, 2Q, + co,2Q- a>, 2(Q. + co), 2(Q - co); i.e., that have periods around twelve hours (co « Q) and vary with latitude according to cos 2 L. These variations are completely analogous to the twelve hour solid-Earth tides produced by the Sun and Moon, called "semidiurnal sectorial waves" [Melchior (1966)]. The true gravimeter measurements for these tides are affected not only by the variation in G, but also by the displacement of the Earth's surface relative to the center of the Earth, and by the redistribution of mass inside the Earth. This variation in gravimeter readings is related to the variation in G by (AfifMemidiurnal = 1.16(AG/G)semidiurnal (8.45) where the factor 1.16 is a combination of "Love numbers," which depend on the detailed structure of the Earth (Melchior, 1966). A more accurate calculation of Ag/g would take into account the fact that in the Earth's interior the perturbing force generated by the variations in GL is proportional to pV U, whereas the tidal perturbing force is proportional to the distance from the center of the Earth. If the Earth's density were uniform, then pWU would be proportional to r and the Love numbers would be the same as in the Newtonian tidal case. However, in Newtonian tidal theory, the Love number for gravimeter measurements, (1.16), is not very sensitive (+ 5%) to variations in. the model for the Earth, thus we do not expect it to be sensitive to a different disturbing force law. (ii) Diurnal variations: These are the terms that vary with a frequency around Q: Q, SI + co, fi co, Q + 2co, Q 2co; i.e., have periods around 24 hours, and vary with latitude according to sin 2L. These variations are analogous to the 24 hour "diurnal tesseral waves" of the solid Earth, and give gravimeter readings related to the variation in G by the same factor: (A<7M,iurnal = 1.16(AG/G)diurnaI (8.46) Tests of the Strong Equivalence Principle 195 (iii) Long-period zonal variations: These are the variations with frequencies co and 2co, and with latitude dependence (5 sin2 L), that are analogous to the long-period tides produced by the Sun and Moon, called "long-period zonal waves." These long-period zonal waves produce variations in the Earth's moment of inertia, which in turn cause variations in the rotation rate of the Earth. These rotation-rate variations are related to the amplitude of the zonal variations by (Mintz and Munk, 1953; Melchior, 1966) (AQ/QL^, = 0.41^zonal (8.47) where Azonai is related to the zonal variations in G in Equations (8.40), (8.41), and (8.43) by (AG/G)zonal = AnJk - sin2 L) (8.48) (iv) Long-period spherical variations: These are the variations [Equations (8.38) and (8.40)] which have frequency <x>, but no latitude dependence; they represent a yearly variation in the strength of G, and have no counterpart in Newtonian tidal theory. These variations produce a purely spherical deformation of the Earth, as opposed to the sectorial, tesseral, and zonal waves which produce purely quadrupole deformations. This yearly spherical "breathing" of the Earth as G varies causes a variation in the Earth's moment of inertia, which in turn causes a variation in the rotation frequency, given by (AQ/Q)spherioal= -(AJ//) spherical (8.49) However, because this effect has no counterpart in Newtonian tidal theory, there is no Love number factor to relate A/// to AG/G. Instead we must do an explicit calculation to determine the factor. We assume the Earth is spherically symmetric and momentarily at rest with respect to the PPN coordinate frame. Since we are focusing on long-period variations of GL (1 yr), we can assume that the Earth is in hydrostatic equilibrium at each moment of time, and changes only quasistatically. Then, from Equations (6.52) and (6.75), or from the PPN perfect-fluid equation of motion, Equation (6.29), keeping only the terms leading to significant long-period spherical perturbations, we find that the equation of hydrostatic equilibrium may be written -T- = P -jr; [1 + i(<*2 + «3 - «i)w©] - j<x2w}@w%p ^ (8.50) Theory and Experiment in Gravitational Physics 196 For a spherically symmetric body, it is straightforward to show that dU__ m(r) ~3r~~ ~~P~' ^ (851) where m(r) = 4TT P pr 2 dr, I(r) = 4n f' pr4 dr (8.52) %J0 JO Substituting Equations (8.32), (8.38), (8.40), and (8.51) into Equation (8.50), and keeping only the spherical terms yields GL(t) = 1 + (a 3 + | a 2 - a t)wt; cos j8w sin(co£ - ^ J (8.54) Using m(r) instead of r as independent variable, we may integrate Equation (8.53), to obtain f =? (8.55) where m e is the mass of the Earth. By definition, p must vanish at the surface of the Earth, i.e., pirn®) = 0. As GL(t) changes, the pressure distribution changes, causing a change £ = ^e in the position of each element of matter. For a given shell of matter, the mass inside that shell is constant, by conservation of mass. Then if GL changes by AGL, we get from Equations (8.52) and (8.55), Am = 0, Ap = p(AGL/GL) + O(f) (8.56) But the volume of each element of matter changes, and this change can be related to the pressure change using the bulk modulus K (we ignore temperature changes) Ap = - K(AV/V) = - KV S, = - (K/r2)(r20,r (8.57) Integrating Equation (8.57) and combining with Equation (8.56), we obtain £ ('/'V2 d' O(£2) (8.58) Tests of the Strong Equivalence Principle 197 The spherical moment of inertia is given by (8.59) and the change caused by the displacement of each shell of matter is A / = 2 [M r^dm (8.60) Combining Equations (8.58) and (8.60) yields A7 = - 8TT ^ J * prdr £ (p'/Ky2 dr' (8.61) Numerical integration of this expression for a reasonable Earth model yields A / / / = -0.17AG L /G L (8.62) (see Lyttleton and Fitch, 1978; Nordtvedt and Will, 1972). We now substitute numerical values for the quantities that appear in Equations (8.35)-(8.43). For the galaxy, Ua * 5 x 10- 7 , <xG = 265°, AG = 266°, fio 5G = -29° = -6°, (8.63) For the velocity w of the solar system relative to the preferred frame, we use the results of the most recent measurements of the anisotropy of the 3 K microwave background. Our motion through this radiation causes the measured effective temperature to be Doppler shifted differently in the front and back directions. From measurements taken using a 33 GHz Dicke radiometer flown on a U-2 aircraft (to get above a substantial amount of the Earth's atmosphere), Smoot et al. (1977) obtained a value w = 390 ± 60 km s~1 in the direction <xw = 165° ± 9°, 5W = 6° ± 10°. We shall adopt the values <xw^165°, K-&, ^164°, j6w^0° (8.64) We also have i>^30kms-\ 0 = 23.5° (8.65) Using these values, we first compute the amplitudes of the dominant components of the Earth tides, as listed in Table 8.1 (unconnected for Love numbers). For comparison, Table 8.1 also gives the amplitudes of the tidal potential for the dominant Newtonian tides in the frequency bands of interest. Theory and Experiment in Gravitational Physics 198 Table 8.1. Amplitudes of earth tides Angular frequency0 Doodson label PPN tidal amplitude (108 Ag/g)" (a) Semidiurnal tides (latitude dependence cos2 L) 2fi-3co 0 T2 _ 2fl-2co s 2 2fi - c o 2.9 a2 f 17 a2 K 2fi (9.6 { 2Q + m 2Q + 2a> - (b) Diurnal tides (latitude dependence sin 2L) Q-2co Pi 0.7 a2 ii-w s, ("3.5 a 2 a n+w Q + 2co 0.6 a 2 Predicted Newtonian amplitude (108 Ag/g)* 0.14 2.4 0.02 0.67 0 0 1.3 0.03 4.1 0.03 0.06 " The angular frequencies of the Earth's rotation and the Earth's orbit are Q, and a>, respectively. b Amplitudes are uncorrected for Love numbers. An entry of zero denotes precise absence of a tide at that frequency, while an entry of a dash denotes that the nominal amplitude is smaller than 10 ~9 g. Recent advances in superconducting techniques in the design and construction of gravimeters have resulted in highly stable devices capable of measuring periodic changes in the local gravitational acceleration g as small as 10" n g. Using such superconducting gravimeters, Warburton and Goodkind (1976) have analyzed an 18 month record of gravimeter data taken at Pinon Flat, California (33°59 N, 116?46W) in search of anomalous PPN tidal amplitudes. From a harmonic analysis of the record, they obtained amplitudes and phases of the tides at the frequencies shown in Table 8.1. They then subtracted (vectorially) from these measured tides the predicted Newtonian tides (corrected by an accurately known Love number factor of 1.160). The remaining amplitudes and phases, known as "load vectors," are thought to be due primarily to the complex effect of ocean tides, which can influence gravimeter readings even at the centers of continents. To take this "ocean loading" into account, they assumed that the anomalous load vectors at the diurnal Pt harmonic and at the semidiurnal T2 (or S2) harmonic, where the PPN effect is negligible or absent, were entirely due to ocean loading. Since the effect of ocean loading is not believed to be strongly frequency dependent over Tests of the Strong Equivalence Principle 199 the narrow (few cycles per year) frequency bands under consideration, the P x and T2 load vectors were simply subtracted from the Kt and from the R2 and K2 load vectors, respectively. Small corrections for barometric effects were also made. The remaining load vectors had amplitudes smaller than 3 x 10" 1 0 g for Ku 1 x 1(T 10 g for K2, and 1 x 1 0 " u g for R2. (Compare with the PPN amplitudes in Table 8.1.) Furthermore, the phases of the remaining load vectors did not agree with the relationships among the phases predicted by Equations (8.39)-(8.43). The result was upper limits on the PPN parameters <x2 and t, given by |a2|<4xl(T4, |£|<HT3 (8.66) The other important post-Newtonian geophysical effect is the possibility of periodic (co, 2co) variations in the Earth's rotation rate produced by the zonal and spherical variations in GL. The zonal variations have amplitudes [see Equations (8.40), (8.41), and (8.43)] {AGJGh)mnil ~ 3 x 10~8a2[frequency co], ~ 3 x 10" 10 a 2 [frequency 2a>], ~ 3 x 10-10£[frequency 2co] (8.67) However, because of the tight limits on a2 and £ set by gravimeter data, we shall ignore these variations. The spherical variations [Equation (8.54)] have amplitude (AGL/GL)spherical * 1.2 x 10" 7 [a 3 + | a 2 - a t ] [frequency co] (8.68) resulting in annual variations of the Earth's moment of inertia [Equation (8.62)] with amplitude |A///| =* 2.0 x 10- 8 [a 3 + | a 2 - a,] (8.69) Now, the observed annual variations in the Earth's rotation rate, of amplitude |Afl/fi| 2^4 xl0~ 9 can be accounted for as an effect of seasonal variations in the angular momentum J wind of atmospheric winds, to a level of 4 parts in 1010 (Rochester and Smylie, 1974). Then, from conservation of angular momentum, we have A7 T Q i/Q <4xlO~10 (8.70) Thus, comparing Equations (8.69) and (8.70) we obtain (8.71) Theory and Experiment in Gravitational Physcis 8.3 200 Preferred-Frame and Preferred-Location Effects: Orbital Tests There are a number of observable effects of a preferred-frame and preferred-location type in the orbital motions of bodies governed by the H-body equation of motion, (6.31). The most important of these effects are perihelion shifts of planets in addition to the "classical" shift discussed in Section 7. To determine these effects, we consider a two-body system whose barycenter moves relative to the universe rest frame with velocity w, and that resides in the gravitational potential UG of a distant body (the galaxy is the dominant such body). In the n-body equations of motion, (6.31), we shall ignore all the self-acceleration terms except the term (6.39) that depends on a3 and w. We shall also ignore the Newtonian acceleration, the Nordtvedt terms, and all the post-Newtonian terms that were included in the classical perihelion-shift calculation. Thus, from Equations (6.32), (6.33), and (6.39) we have the additional accelerations +I(a 1 -a 2 -a 3 )w 2 +ia 1 w v 1 +|(a 1 -2a 2 -2a 3 )w v2 +fa 2 (wn) 2 .^ r (w n)(v2 n) J!° [2(fiG x )n G - 3x(nG n) 2 ] + a 2 - | (x w)v2 rG r i ^ x3 . r aL i V _(( aai,-2a,)i lVl 2 2 r " > (8.72) where x = x 21 , n = x/r, ra = |x1G|, nG = x 1G /r G . In obtaining Equation (8.72) we have ignored terms of order mGr/rQ, mGr2/rG, and so on. The first two terms inside the braces in Equation (8.72) are constant, therefore they can simply be absorbed into the Newtonian acceleration by redefining the gravitational constant [they are related to the constant corrections to GL in Equation (6.75)]. Since our two-body system will consist of the Sun and a planet, we can ignore Q/m for the planet. If body 1 is chosen to be the Sun, then the relative acceleration <5a = 5a2 d*i Tests of the Strong Equivalence Principle 201 is given by <5a = \^a.l(dm/m)yi v + |a 2 (w ft)2] + ZULU* [2(flG x)nG - 3x(nG n) 2 ] r r o j - [^a^m/mjv + a2w]w + %tx3(Q/m)QY/ x <o (8.73) where we have made use of Equations (7.39) and (7.40), and where Sm = my m 2 . Following the method described in Section 7.3, we calculate the secular change in the perihelion position. We assume that m2«m1, that e«l, and that co is perpendicular to the orbital plane, then to zeroth order in e, we obtain for the secular change in a> over one orbit, A<3= - 2 4 rG w 2 \ m JQ\ me (8.74) where vvP, wQ, «P, and nQ are the respective components of w and flG m the direction of the planet's perihelion (wP, nP) and in the direction at right angles to this (wQ,«Q) in the plane of the orbit. The perturbations in Equation (8.73) can also be shown to produce secular changes in e, i, and Q. We now evaluate this additional perihelion shift for Mercury and Earth, using standard values for the orbital elements (Allen, 1976), numerical values for the Sun's gravitational energy and rotational angular velocity ^ 4 x 10~6, |t»|Q =* 3 x 10~6 s" 1 (8.75) the direction of the galactic center, and our adopted value for w (see Section 8.2.). Including the "classical" contributions (Section 7.3), the result, in seconds of arc per century, is = 43.0[i(2y + 2 - 123a! + 92a2 + 1.4 x 105a3 = 3.8[i(2y + 2 - 0)] - 198at + 12a2 + 2.4 x 106a3 + 14£ c" l (8.76) Theory and Experiment in Gravitational Physics 202 Note that the effect of J2 on the Earth's perihelion shift is below the experimental uncertainty. The measured perihelion shifts are (^®)meas^3'.'8 + 0'.'4C-1 (8.77) By combining Equations (8.76) and (8.77), eliminating the term involving y and /?, and treating J 2 as an experimental uncertainty with maximum value given by Hill's observations, \J2\ < 5 x 10~6 (Section 7.3), we obtain the following limit on the parameters a t , a2, a3, and £ |49at - a2 - 6.3 x 105a3 - 2.2£| < 0.1 (8.78) It is clear that <x3 must be extremely small, |a3|<2xKT7 (8.79) otherwise there would be major violations of perihelion-shift data. Nonzero values of OLU a2, a3, or t, can also lead to periodic perturbations in orbits, most notably in the lunar orbit, with nominal amplitudes ranging from 70 km, for terms dependent upon oc3, to several meters, for terms dependent upon a1; a2, or £. For a partial catalogue of these effects, see Nordtvedt and Will (1972) and Nordtvedt (1973). In Section 9.3, we shall obtain an even tighter limit on a3 than that shown in Equation (8.79) by considering the effect of the acceleration term equation, (6.39), on the motion of pulsars. 8.4 Constancy of the Newtonian Gravitational Constant Most theories of gravity that violate SEP predict that the locally measured Newtonian gravitational constant may vary with time as the universe evolves. For the theories listed in Table 5.1, the predictions for G/G can be written in terms of time derivatives of the asymptotic dynamical fields or of the asymptotic matching parameters. Other, more heuristic proposals for a changing gravitational constant, such as those due to Dirac cannot be written this way. Dyson (1972) gives a detailed discussion of these proposals. Where G does change with cosmic evolution, its rate of variation should be of the order of the expansion rate of the universe, i.e., G/G = oH0 (8.80) where Ho is the Hubble expansion parameter whose value is Ho cz 55 km s" 1 Mpc" 1 s ( 2 x 1010 yr)~\ and a is a dimensionless parameter whose value depends upon the theory of gravity under study and upon the detailed cosmological model. Tests of the Strong Equivalence Principle 203 For very few theories has a systematic study of values of a been carried out. For general relativity, of course, G is precisely constant {a = 0). For Brans-Dicke theory a ranges from a 2= 3qo(a> + 2)" 1 for q0 « 1 to a =s -(co + 2)" 1 for q0 = i (flat Friedman cosmology) to a ^ 3.34<jJ/2(ct) + 2)" 1 for <j0 » 1, where q0 is the deceleration parameter of the cosmology [see Section 16.4 of Weinberg (1972) for review and references]. In Bekenstein's variable-mass theory, generic cosmological models with chosen values of r and q (see Section 5.3) evolve to states at the current epoch in which a < 5 x 10""3 (Bekenstein and Meisels, 1980). But for most other theories, detailed computations of this sort have not been performed (see Chapter 13). However, several observational constraints can be placed on G/G, using methods that include studies of the evolution of the Sun, observations of lunar occultations (including analyses of ancient eclipse data), planetary radar-ranging measurements, lunar laser-ranging measurements, and yet-to-be-performed laboratory experiments. The present status of these experiments is summarized in Table 8.2 [for a review of some of these methods see Halpern (1978)]. Some authors, chiefly Van Flandern (1975,1978), have claimed that the nonzero results for o shown in Table 8.2 are significant and support the hypothesis of a varying gravitational constant, while others, notably Reasenberg and Shapiro (1978) have argued that unavoidable errors in the models used in the numerical estimation Table 8.2. Tests of the constancy of the gravitational constant Method a = (G/G) x (2 x 1O10 yr) Reference Solar evolution Lunar occultations and eclipses H<2 Planetary and spacecraft radar H<8 W<3 Viking radar Lunar laser ranging Laboratory experiments \a\ < 0.6 Chin and Stothers (1976) Morrison (1973) Van Flandern (1975, 1976, 1978) Muller(1978) Newton (1979) Shapiro et al. (1971) Reasenberg and Shapiro (1976, 1978), Anderson et al. (1978) Anderson (1979) Williams et al. (1978) Braginsky and Ginzberg (.1974), Braginsky et al. (1977), Ritterand Beams (1978) |CT| < 0.8 a = -(0.6 ±0.3) <r= -(0.5 + 0.3) cr= -(2.5 ±0.7) ' Experiments yet to be performed. Theory and Experiment in Gravitational Physcis 204 of parameters such as G/G may seriously degrade such estimates. The laser-ranging and radar-ranging results are regarded as being consistent with G/G = 0. Reasenberg and Shapiro (1976) have pointed out that, because the errors in the radar observations of G/G decrease as T~5/2 where T is the time span of the observations, one can expect from that method an accuracy of A|G/G| < 10" l l yr" 1 by 1985. Anderson et al. (1978) and Wahr and Bender (1976) have shown that radar observations of Viking or of a Mercury orbiter over two-year missions could yield 8.5 Experimental Limits on the PPN Parameters We now summarize the results of the solar system experiments described in Chapters 7 and 8, in the form of a set of limits on the PPN parameters. For the purposes of this summary, we shall consider only semiconservative theories of gravity, i.e., theories for which <x3 = £i = £2 = (3 = £4 = 0. Our reasons are the following: (i) we wish to keep things simple; (ii) all currently interesting metric theories of gravity are Lagrangian based, and are thus automatically semiconservative; (iii) we have already seen that |a3| < 2 x 10~7; and (iv) decent experimental limits on the parameters Ci, (i> Cs, and £4 are hard to obtain, the only known exceptions being a limit |£3| < 0.06 from the Kreuzer experiment, and a possible limit on |£2| from the binary pulsar (see Chapter 9 for discussion of these tests). We thus have the la experimental limits y = 1.000 ± 0.002 [Viking time delay], \{2y + 2 - j8) = 1.00 ± 0.02 [perihelion shift, Hill's value for J 2 ], \40 - y - 3 - ^ |£|<10" - <*! + fa 2 | < 0.015 [lunar laser ranging], 3 |a2| < 4 x 10" 4 |fa2 - ax| < 0.02 |49ax a2 2.2^| < 0.1 (8.81) (8.82) (8.83) [Earth tides], (8.84) [Earth tides] (8.85) [Earth rotation rate], (8.86) [anomalous perihelion shifts] (8.87) One useful way to represent these results pictorially is to construct "PPN theory space," a five-dimensional space whose axes are the five semiconservative PPN parameters. A given theory, with chosen values for its adjustable constants and matching parameters, occupies a point in Tests of the Strong Equivalence Principle 205 this space. If we choose as variables y 1, /? 1, £, al9 and a2, then general relativity occupies the origin, scalar-tensor theories with co > 0 occupy the left hand (y - l)-(j8 - 1) plane, Rosen's bimetric theory occupies the a2 axis, and so on (see Figures 8.2 and 8.3). The results of solar system experiments can be viewed as "squeezing" the available theory space into smaller and smaller portions. For example, Figure 8.2 shows the y-fi-% subspace of PPN theory space, and indicates the constraints imposed by time delay, lunar-laser ranging, perihelion shift, and Earth tide measurements. The resulting available theory space is the "pill box" around the origin (general relativity) shown. Figure 8.3 shows the a t - a 2 plane, and indicates the constraints placed by Earth tide, Earth rotation rate, and perihelion-shift measurements. Figure 8.2. The (y 1)-(P l)-£ space. Brans-Dicke theory occupies the negative (y l)-axis(/? = 1), while the generalized scalar tensor theories of Bergmann, Wagoner, Nordtvedt, and Bekenstein occupy the half-plane (y 1) < 0. The numbers on the negative (y 1) axis are the corresponding values of co. General relativity resides at the origin. Shown are limits on the PPN parameters placed by the Viking time delay (dotted lines), lunarlaser ranging (dashed lines), and perihelion shift (dot-dashed lines) measurements. The remaining available PPN theory space is the box shown, of thickness 2 x 10" 3 in the direction. 0- / 0.03; Scalar-Tensor (BWN, Bekenstein) / 0.02: / 25 :; BransDicke Theory and Experiment in Gravitational Physics 206 For specific theories discussed in Chapter 5, these constraints can be translated into constraints on adjustable constants or matching parameters if the theory is to hope to remain viable. From the la constraints listed above and from the formulae given in Chapter 5, we obtain the limits (i) Scalar-tensor theories: co > 500, A < 10" 3 (ii) Will-Nordtvedt theory: K2 < 4 x 10" 4 (iii) Hellings-Nordtvedt theory: |coX2| < 2 x 10~4, co2K2 < 5 x 10~4 (iv) Rosen's bimetric theory: \co/c1 1| < 4 x 10" 4 (v) Rastall's theory: K2 < 3 x 10" 2 (8.88) Because many theories can be made to agree within experimental error with all solar system tests performed to date, we shall ultimately be forced, beginning in Chapter 10, to turn to new arenas for testing relativistic gravitation. a/ Ii I j Hellings-Nordtvedt : I 7 - IK0.002 Figure 8.3. The a!-a 2 plane. The Rosen, Rastall, and Will-Nordtvedt theories occupy parts of the a2-axis shown. The Hellings-Nordtvedt theory, constrained by Viking time-delay measurements of y, occupies the shaded region. General relativity and scalar-tensor theories (ST) reside at the origin. Shown are limits placed by Earth tide (dotted lines), perihelion shift (dashed line), and Earth rotation rate (dot-dashed line) measurements. Other Tests of Post-Newtonian Gravity There remains a number of tests of post-Newtonian gravitational effects that do not fit into either of the two categories, classical tests or tests of SEP. These include the gyroscope experiment (Section 9.1), laboratory experiments (Section 9.2), and tests of post-Newtonian conservation laws (Section 9.3). Some of these experiments provide limits on PPN parameters, in particular the conservation-law parameters Ci, d> £3* £4. that were not constrained (or that were constrained only indirectly) by the classical tests and by tests of SEP. Such experiments provide new information about the nature of post-Newtonian gravity. Others, however, such as the gyroscope experiment and some laboratory experiments, all yet to be performed, determine values for PPN parameters already constrained by the experiments discussed in Chapters 7 and 8. In some cases, the prior constraints on the parameters are tighter than the best limit these experiments could hope to achieve. Nevertheless, it is important to carry out such experiments, for the following reasons: (i) They provide independent, though potentially weaker, checks of the values of the PPN parameters, and thereby independent tests of gravitation theory. They are independent in the sense that the physical mechanism responsible for the effect being measured may be completely different than the mechanism that led to the prior limit on the PPN parameters. An example is the gyroscope test of the Lense-Thirring effect, the dragging of inertial frames produced purely by the rotation of the Earth. It is not a preferred-frame effect, yet it depends upon the parameter <xx. (ii) The structure of the PPN formalism is an assumption about the nature of gravity, one that, while seemingly compelling, could be incorrect. This viewpoint has been expounded by Irwin Shapiro (1971) and others. They argue that one should not prejudice the design, performance, and Theory and Experiment in Gravitational Physics 208 interpretation of an experiment by viewing it within any single theoretical framework. Thus, the parameters measured by light-deflection and timedelay experiments could in principle be different according to this viewpoint, while according to the PPN formalism they must be identical [i(l + y)]- We agree with this viewpoint because although theoretical frameworks such as the PPN formalism have proved to be very powerful tools for analyzing both theory and experiment, they should not be used in a prejudicial way to reduce the importance of experiments that have independent, compelling justifications for their performance. (iii) Any result in disagreement with general relativity would be of extreme interest. 9.1 The Gyroscope Experiment Since 1960, when Leonard Schiff proposed it as a new test of general relativity, much effort has been directed toward the gyroscope experiment (Schiff, 1960b,c; Everitt, 1974; Lipa et al, 1974). The object of the experiment is to measure the precession of a gyroscope's spin axis S relative to the distant stars as the gyroscope orbits the Earth. According to the PPN formalism, this precession is given by (see Section 6.5) dS/dt = ft x S, ft = _ i v x a - |V x g + (y + |)v x VC/, g = QofiJ (9.1) where a is the spatial part of the gyroscope's four-acceleration, which is zero for a body in free-fall orbit. In a chosen PPN coordinate system, Equation (9.1) along with the expression for gOj in Table 4.1 yields ft = i(4y + 4 + at)V x V - |a t w x\U + (y + frr x \U (9.2) where w is the velocity of the coordinate system relative to the universe rest frame, and where V = Vjej (9.3) For a system of nearly spherical bodies of masses ma, angular momenta J a , and velocities va, we have V = £ mavjra - | £ x a x Ja/r3a + O(r8~ 3) (9.4) Other Tests of Post-Newtonian Gravity 209 where xa is the vector from the ath body to the gyroscope. Then " = (7 + i) £ (v - va) x \{mjra) - i(y + i + i«i) I [J. - 3fia(fifl Jfl)]/rfl3 - i*i I (w + vj x V(m>a) - ± £ va x V(ma/ra) a (9.5) a where na = xa/ra. The first term in Equation (9.5) is called the geodetic precession, a consequence of the curvature of space near gravitating bodies. For a circular orbit around the Earth, the Earth's potential (a = ©) leads to a secular change in the direction of the gyroscope spin given, over one orbit, by as = -2n{y + i)(me/a)(S x h) (9.6) Figure 9.1. Precession of gyroscopes in a polar Earth orbit. The gyroscope with its axis in the plane of the orbit undergoes a geodetic precession, while the gyroscope with its axis normal to the orbital plane suffers a precession due to the dragging of inertial frames. 8"/year / Theory and Experiment in Gravitational Physics 210 where a is the orbital radius, and h is a unit vector normal to the orbital plane. For a gyroscope whose initial direction lies in the orbital plane, the angular precession 39 ( = |5S|/|S|) per year is given by «) 5/2 yr~' (9-7) There is also a correction of ~0'.'01 yr~' due to the Earth's oblateness. Another secular contribution comes from the Sun's potential (a = Q), given by (^geodetic)© =* O'.'Q2ft(2y + 1)] yr~ x (9.8) where we have assumed a circular orbit for the Earth around the Sun. The second term in Equation (9.5) is known as the Lense-Thirring precession or the "dragging of inertial frames" (for further discussion of this effect, see MTW Sections 19.2 and 33.4). For a circular orbit around the Earth, it leads to a secular precession per orbit given by <SS = i(y + 1 + i a i ) ( P / a 3 ) [ J e ~ 3h(B J e ) ] x S (9.9) where P is the orbital period of the satellite. For a gyroscope in a polar orbit (fi J @ = 0) or an equatorial orbit (fi Jffi = |J®|), the precession is given by <5SPOL = i(y + 1 + ia 1 )(P/a 3 )J® x S, SSm = - i ( y + 1 + £ a i )(P/a 3 )J e x S (9.10) with angular precessions, in arcseconds per year 50poL * 0'.'05[±(y + 1 + i a i )](/le/fl) 3 sin0 yr" 1 , d0EQ ~ O'.'ll[i(y + 1 + W K / V a ) 3 sin «£ yr" 1 (9.11) where <j> is the angle between the spin vectors of the Earth and gyroscope. The third term in Equation (9.5) is a preferred-frame effect, dependent upon the velocity of the ath body relative to the universe rest frame. For an Earth-orbiting satellite, the dominant effect comes from the solar term (a = O), leading to periodic precession of the form (5S = - i a ^ W o x v e ) x S (9.12) where vffi is the Earth's orbital velocity around the Sun and wG = w + v 0 . This leads to a periodic angular precession with a one year period, with amplitude <50p.F. £ 5 x l O " 3 ' ^ (9.13) Other Tests of Post-Newtonian Gravity 211 Since the ultimate goal of the experiment is to measure precessions to 10" 3 arcseconds per year, this latter effect is probably too small to be of interest. The last term in Equation (9.5) would appear to be anomalous, since it depends upon the velocity of each body va with respect to our arbitrarily chosen PPN coordinate frame. However, this is simply a result of the fact that the spin precession dSj/dx that we have calculated is not a truly measurable quantity, since the basis vectors e s were not tied to physical rods and clocks. A correct physical choice, and one that is closely related to the actual experimental method, is to use the directions of distant stars as basis directions (Wilkins, 1970). From Equations (7.1) and (7.9), the tangent vector to the trajectory of an incoming photon in the PPN coordinate frame is given by - (1+ y)l/] + (1 + y)93 (9.14) where |n|2 = 1 is a unit spatial vector in the direction of the unperturbed trajectory from the chosen star, and where 2> is equal to the right-hand side of Equation (7.7), summed appropriately over all the gravitating bodies in the system, and gives the gravitational deflection of the incoming signals. We now project A onto the inertial basis of Equation (6.102), and normalize the spatial components, so that X/(^j)2 = 1, to obtain Aje;= n - n x (v x n)(l + v n) - ^v x (n x v) + (1 + y)@ (9.15) We now wish to show that the precession of the components of J on this basis is independent of the velocity of the PPN coordinate frame. In Equation (9.5), only the final term has this dependence, so we write it in the form - i I (v. - vB) x V(mjra) - hs * di/dt (9.16) a where vB is the velocity of the solar system barycenter relative to the PPN coordinate frame, and where we have used the fact that, for a freely falling gyroscope, dx/dt = £ V mjra (9.17) a The first term is now independent of the coordinate frame and so may be dropped. The second term may be integrated immediately to obtain <5SB = -i(v B x «5v) x S o (9.18) Theory and Experiment in Gravitational Physics 212 where the subscript B denotes that we retain only the terms that depend on vB. Then the change in the components of S with respect to A is given by 5(S;A;)B = <5SB A + S <5AB = - [i(v B x <5v) x S o ] A + i S 0 [vB(<5v A) - c5vvB A] = 0 (9.19) Thus, as expected, there is no physically measurable dependence on the coordinate-system velocity. In any case, the final term in Equation (9.5) produces only periodic precessions of negligible amplitude. A variety of technical problems has caused the gyroscope experiment to be almost a quarter of a century in the making, from its inception in 1960 to projected launch, in the middle 1980s. Among the more difficult technological hurdles that have had to be overcome in order to produce a spaceworthy experiment that can measure gyroscope precessions accurate to 10" 3 arcseconds per year, or equivalently to 10" 1 6 rad/s, include: (i) Fabrication of a gyroscope that is spherical and homogeneous to a part in a million. For this purpose, a 2 cm radius quartz sphere is used. This constraint is necessary to reduce torques on the gyroscope. Even if this constraint is satisfied, there must be no residual gravitational forces on the gyroscope larger than 10~ 9 g. This necessitates a drag-free satellite. (ii) Readout of the direction of the spin axis. Conventional methods of determining the spin direction of the gyroscope require violations of its sphericity and homogeneity, and thus introduce unacceptable torques. Thus a "London moment" readout method has been adopted. The gyroscope is coated uniformly with a superconducting film. When spinning, the superconductor develops a magnetic dipole moment M parallel to its spin axis. Any change in the direction of M can be determined by measuring the current induced in a superconducting loop surrounding the gyroscope. For this method to be viable, however, it was necessary to develop a magnetic shield that could reduce the ambient magnetic field below 10 ~ 7 G, otherwise the gyroscope could contain trapped magnetic flux of sufficient size to produce anomalous readout signals. By comparison, the ambient magnetic field of the Earth is about 0.5 G. (iii) Determination of basis directions. The precession of the gyroscope's spin axis is measured relative to the direction of a chosen reference star, as observed by a telescope mounted on the gyroscope housing. This direction must be monitored to better than 10" 3 arcseconds per year, so the design of a suitable optical system has been a major problem. Other Tests of Post-Newtonian Gravity 213 Further details of the experimental problems and progress are found in Lipa and Everitt (1978) and Cabrera and Van Kann (1978). A variant of the gyroscope experiment has recently been proposed by Van Patten and Everitt (1976) in which the "gyroscope" is itself the orbit of a satellite around the Earth. The dragging of inertial frames causes the plane of the orbit to rotate about an axis parallel to the Earth's rotation axis. Assume the Earth is at rest, and rotates with angular momentum J. The substitution of Equation (9.4) for F, into the equations of motion (Section 4.2) yields the additional acceleration on a body near the Earth da = -i(4y + 4 + a,) \ [2v x J - 3(v n)(ii x J) + 3nv (n x J)] (9.20) where v is the body's velocity, and fi = x/r. For an orbit characterized by inclination i relative to the plane normal to J, angle of the ascending node Q and orbit elements p, e, and co, the use of the orbit perturbation Equations (7.47) and (7.48) yields, over one orbit 5i = 0, 8Q = 2n{y + 1 + i^pl/imp3)1'2 (9.21) Thus the "spin" vector S orthogonal to the orbital plane precesses about the direction of J according to dS/dt = ft x S (9.22) ft = (y + 1 + iut)Ja- 3(1 - e2)- 3/2 (9.23) where For a body in a nearly circular orbit, this yields an annual angular precession 5Q = O'.'22rj(y + 1 + U^jRJa)3 yr" l (9.24) In order to eliminate the effects of other sources of precession (such as the quadrupole moment of the Earth) two satellites counterrotating in nearly identical orbits are necessary. With the use of drag-free satellites and with two to three years of orbit data, an experiment with results within 3% accuracy may be possible. 9.2 Laboratory Tests of Post-Newtonian Gravity Because the gravitational force is so weak, most tests of postNewtonian effects in the solar system require the use of the Sun and planets as sources of gravitation. One disadvantage of such experiments Theory and Experiment in Gravitational Physics 214 is that the experimenter has no control over the sources, and so is unable to manipulate the experimental configuration to test or improve the sensitivity of the apparatus, or at the very least, to repeat the experiment. Despite this disadvantage of solar system-sized experiments, the weakness of post-Newtonian gravity has effectively prohibited laboratory experiments, with one exception. That exception is the Kreuzer experiment (Kreuzer, 1968) that compared the active and passive gravitational masses of fluorine and bromine. Kreuzer's experiment used a Cavendish balance to compare the Newtonian gravitational force generated by a cylinder of Teflon (76% fluorine by weight) with the force generated by that amount of a liquid mixture of trichloroethylene and dibromomethane (74% bromine by weight) that had the same passive gravitational mass as the cylinder, namely the amount of liquid displaced by the cylinder at neutral buoyancy. In the actual experiment, the Teflon cylinder was moved back and forth in a container of the liquid, with the Cavendish balance placed near the container. Had the active masses of Teflon and displaced liquid differed at neutral buoyancy, a periodic torque would have been experienced by the balance. The absence of such a torque led to the conclusion that the ratios of active to passive mass for fluorine and bromine are the same to 5 parts in 105, that is (mA/mP)F{ - (mA/wP)Br < 5 x 10-5 (mA/mP)Br (9.25) [For further discussion of Kreuzer's experiment, see Gilvarry and Muller (1972) and Morrison and Hill (1973)]. If the active mass were to differ from the passive mass for these substances, the major contribution to the difference would come from the nuclear electrostatic energy (as it does, say in the Eotvos experiment). Since Ee/m ~ 10" 3 , one could regard such effects as post-Newtonian corrections. However, the perfect-fluid P P N formalism of Chapter 4 is poorly suited to a discussion of nuclear matter. A better approximation is one in which the P P N metric is generated by charged point masses, with gravitational potentials generated by masses, microscopic velocities, charges, and so on. Using this metric, one can calculate the active to passive mass ratio of a bound system (nucleus) of point charges, with the result, for a spherically symmetric body (Will, 1976a), mjmv = 1 + T£(£e/mP) (9.26) Other Tests of Post-Newtonian Gravity 215 where Ee is the electrostatic energy of the system of charges and £ is a combination of PPN parameters derived from the charged-point-mass metric. However, it can be shown that if the perfect-fluid PPN metric of Table 4.1 is simply a macroscopic average of the point-mass metric (as one would expect in most reasonable theories of gravity), then the combination of charged-point-mass parameters that makes up e is precisely the same as the fluid PPN parameter £3. Thus, in any such theory of gravity, mJrn? = 1 + K a ^ / H O (9.27) (For further details, see Will, 1976a). The semiempirical mass formula (see Equation 2.8) yields mjmp = 1 + 3.8 x 10~4£3Z(Z - l ) ^ " 4 ' 3 (9.28) For fluorine Z = 9, A = 19, and bromine Z = 35, A = 80, Equations (9.25) and (9.28) yield |C3| < 6 x 1(T2 (9.29) This generalizes and corrects a previous result of Thorne et al. (1971). Advancing technology may make several laboratory post-Newtonian experiments possible in the coming decades (Braginsky et al., 1977). The progress that makes such experiments feasible is the development of sensing systems with very low levels of dissipation, such as torque-balance systems made from fused quartz or sapphire fibers at temperatures <;0.1 K, massive dielectric monocrystals cooled to millidegree temperatures, and microwave cavities with superconducting walls. Among some of the experimental possibilities are a measurement of the gravitational spin-spin coupling of two rotating bodies; searches for time variations of the gravitational constant, preferred-frame, and preferred-location effects; and a measurement of the dragging of inertial frames by a rotating body. The reader is referred to Braginsky et al. (1977) for detailed discussion and references. 9.3 Tests of Post-Newtonian Conservation Laws Of the alternative metric theories of gravity discussed in detail in Chapter 5, all are Lagrangian based, that is, all possess integral conservation laws for energy and momentum. In the post-Newtonian limit, their PPN parameters satisfy the semiconservative constraints a3 = Ci = £2 = £3 s U = 0 What is the experimental evidence for these constraints? (9.30) Theory and Experiment in Gravitational Physics 216 In Chapter 8, we obtained the upper limit |a3|<2xl(T7 (9.31) from perihelion-shift data. The effect there was a combined preferredframe effect and self-acceleration of a massive body, in particular of the Sun. However, this limit can probably be tightened considerably, although with somewhat less rigor, by applying the self-acceleration term, Equation (6.39) to pulsars. For these bodies, assumed to be rotating neutron stars, |Q/m| ~ 0.1, and 2 s" 1 < \co\ < 200 s" 1 , thus their self-acceleration has the form Keifl <* 6 x 103|a where v is the pulsar frequency, and 9 is the angle between the pulsar spin axis and its velocity relative to the universe rest frame. Although strictly speaking, the post-Newtonian limit does not apply to pulsars, we feel this is a reasonable estimate of the size of the effect in any theory with a3 # 0. This acceleration will cause a change in the pulse period P p given by = a self -n (9.33) where n is a unit vector along the line of sight to the pulsar. Thus, -2 x independently of Pp, where O is the angle between aseIf and the line of sight ii. For the 90 pulsars reported by Manchester and Taylor (1977) whose values of dPJdt have been measured, those values range between 4 x 10" 13 (Crab Pulsar) and 1 x 10" 18 (PSR 1952 + 29), with half of them lying between 10" 14 and 10" 15 . In all cases, dPJdt > 0, i.e., all pulsars are slowing down. Now for the 40 or so pulsars with 10" 14 > dPp/dt > 10" 15 , it is extremely unlikely that either sin6 = 0 or cos® = 0 for all of them, furthermore if a 3 # 0, we would expect as many pulsars with dPp/dt < 0 as with dPp/dt > 0, assuming their spin directions were oriented randomly. Thus, a conservative limit on <x3 can be obtained by setting sin 0 = cos <I> = 5 in Equation (9.34), and imposing the 10" 14 upper limit on an anomalous dPp/dt, giving |a3l < 2 x 10" 10 (9.35) Other Tests of Post-Newtonian Gravity 217 There may be one promising way to set a limit on the parameter £2 involving an effect first pointed out, incorrectly, by Levi-Civita (1937). The effect is the secular acceleration of the center of mass of a binary system. Levi-Civita pointed out that general relativity predicted a secular acceleration in the direction of the periastron of the orbit, and found a binary system candidate in which he felt the effect might one day be observable. Eddington and Clark (1938) repeated the calculation using de Sitter's (1916) n-body equations of motion. After first finding a secular acceleration of opposite sign to that of Levi-Civita, they then discovered an error in de Sitter's equations of motion, and concluded finally that the secular acceleration was zero. Robertson (1938) independently reached the same conclusion using the Einstein-Infeld-Hoffmann equations of motion, and Levi-Civita later verified that result. In fact, the secular acceleration does exist, but only in nonconservative theories of gravity; that is, it depends on the PPN parameters a 3 and £2 (Will, 1976b). The simplest way to derive this result is to treat the two-body system as a single composite "body" in otherwise empty space, and to focus on the self acceleration in the equation of motion, (6.32). For two point masses, Equation (6.32) and the formulae in Table 6.2 give i(a 3 + Ci){mim2x/r3)(vl - v2) + C1(w1rn2/r3)[v2(v2 x) - v ^ x) - fx(v 2 ft)2+ f x ^ ft)2] i ~ tn2)x/r4' + a3m1m2(w + V) vx/r3 (9.36) where x = x 2 - \ u r = |x|,ft= x/r, v = v2 - v ls V is the center-of-mass velocity with respect to the PPN coordinate system, va = va V, and ms £ mJil+ffi-fa/r) [b # a] (9.37) a=l, 2 Substituting vx s -(m 2 /m)v, v2 s {rnjnifs along with the expressions appropriate for a Keplerian orbit x = p(l + e cos 4>)~l{ex cos</> + ej,sin<£), v = (m/p)1/2[ e x sin0 + ey(e + cos<£)], r2 dWdt = (mp)112 and averaging (a)self over one orbit, we obtain (9.38) (9.39) <(«)self > = («3 + C2) - i a 3 ( l - e2y\Q/m)(Y/ + V) x 0 (9.40) Theory and Experiment in Gravitational Physics 218 where eP = ex = [unit vector in the direction of the periastron of m x ], (o = (2n/P)ez = (mean angular velocity vector of orbit], Q = < m1m2/r) = m1m2/a (9.41) The second term in Equation (9.40) is the same as the term in Equation (6.39) except for the numerical factor (3 compared to j ( l e 2 )" 1 ], which arises from the difference in averaging for a stationary, nearly spherical body, and for a binary system. However, because of the limit we have already obtained for a 3 , its effects on the self acceleration of a binary system will be negligible. Thus, we shall set a3 = 0, leaving <(a)self > = C 2 In the solar system, this has effects that are utterly unmeasurable. For example, the self acceleration of the Earth-Moon binary system produces a perihelion shift for the Earth of the order dcom ~ 10~ 5 per century. A more promising testing ground for this effect would be a close binary system, such as iBoo, with m t = 1.35mo, m2 = 0.68mQ, P = 0.268 day. The resulting change in the periods (inverse frequencies), say, of the spectral lines of the stars in iBoo would be P~l(dPJdt) = 8.8 x 10- 7 £ 2 e(l - e 2 )" 3 / 2 sin co sin i yr" 1 (9.43) where i is the inclination of the orbit relative to the plane of the sky, and w is the angle of periastron. Unfortunately, because of Doppler broadening, the frequencies of spectral lines are not known to sufficient accuracy to make such a change observable. However, the discovery of the binary pulsar (Chapter 12) has changed the situation. The characteristics of the orbit are very similar to that of iBoo, however the pulsar provides a much more precise and stable time standard than do spectral lines. This enables one not only to measure changes in the pulse period with high accuracy, but also to determine the parameters of the orbit and thereby the change of the oribit period Pb with high accuracy. The results are (Table 12.1) P^dPJdt l = (4.617 ± 0.005) x 10~ 9 yr" 1 P b - dPJdt = - (2.4 ± 0.4) x 10 - 9 yr - J (9.44) However, because the binary pulsar is a "single-line spectroscopic binary," the individual masses are not known from the velocity curve data (we Other Tests of Post-Newtonian Gravity 219 shall see that they can be determined if one assumes a particular metric theory of gravity), rather, the known quantities are (see also Table 12.1) e =* 0.62, P b =* 27907 s 3 ft = (m2sini) /m2 c^ 0.13mo co si 179° + 4.23°(t - t o )/(l yr) (9.45) where / t is known as the mass function, and where t0 [September, 1974]. Then the predicted period change for both the pulsar and the orbit is given by where X = mjm2 = w pu , sar /m companion , and where we have used the fact that, from Equation (9.45), sin co st - 7 x 10" 2 (t - to)/(l yr), t - t0 < 10 yr (9.47) Note too, that the second derivatives of the periods are given by p ; 1 d2Pp/dt2 = p ^ 1 d2Pb/dt2 Now, from data covering a time span of several years, the error on Pp 1 dPJdt was found to be 10" x 1 y r " l . In other words, P;1 dPJdt did not change by more than 10" 1X yr" 1 in a year, that is, \p-^d2Pvjdt2\ < lO-^yr" 2 (9.49) Assuming that the secular acceleration is responsible for no more than this amount, in other words, that there is no fortuitous cancellation between this effect and other sources of period change (Section 12.1), we obtain from Equations (9.48) and (9.49) the limit |C2| < 2 x 10" 4 (m o /m) 2/3 |(l + X) 2 /4X(1 - X)\ (9.50) Now, without assuming a particular metric theory of gravity, we do not know the values of m and X, so the limit on £2 is uncertain (if the masses are equal, for example, X 1, and there is no secular acceleration, by virtue of symmetry). If, for example, we assume that general relativity is valid except for the sole possibility of a violation of momentum conservation manifested Theory and Experiment in Gravitational Physics 220 by £2 # 0, then we can use the values of m and X obtained from periastronshift data and from the gravitational red shift-second-order Doppler shift data (see Chapter 12 for details), m ^ 2.85mo, X ~ 1.007 ± 0.1 (9.51) Although the data are not yet sufficiently accurate to exclude X = 1, it is of interest to substitute the nominal value of X into Equation (9.50) to obtain |£2| < 10"2. As long as \X - 1| > 10" 3, we will still have |£2| < 0.1. Of the remaining three conservation-law parameters, only £3 has been tested experimentally, as we saw in the previous section where we obtained the limit |£3| < 0.06 from the Kreuzer experiment. No feasible experiment or observation has ever been proposed that would set direct limits on the parameters £1 or £4. Note, however, that these parameters do appear in combination with other PPN parameters in observable effects, for example in the Nordtvedt effect (see Section 8.1). 10 Gravitational Radiation as a Tool for Testing Relativistic Gravity Our discussion of experimental tests of post-Newtonian gravity in Chapters 7, 8, and 9 led to the conclusion that, within margins of error ranging from 1% to parts in 10" 7 (and in one case even smaller), the post-Newtonian limit of any metric theory of gravity must agree with that of general relativity. However, in Chapter 5, we also saw that most currently viable theories of gravity could accommodate these constraints by appropriate adjustments of arbitrary parameters and functions and of cosmological matching parameters. General relativity, of course, agrees with all solar system experiments without such adjustments. Nevertheless, in spite of their great success in ruling out many metric theories of gravity (see Sections 5.7, 8.5), it is obvious that tests of post-Newtonian gravity, whether in the solar system or elsewhere, cannot provide the final answer. Such tests probe only a limited portion, the weak-field slow-motion, or post-Newtonian limit, of the whole space of predictions of gravitational theories. This is underscored by the fact that the theories listed in Chapter 5 whose post-Newtonian limits can be close to, or even coincident with, that of general relativity, are completely different in their formulations, One exception is the Brans-Dicke theory, which for large co, differs from general relativity only by modifications of O(l/a>) both in the postNewtonian limit and in the full, exact theory. The problem of testing such theories thus forces us to turn from the post-Newtonian approximation toward new areas of "prediction space," new possible testing grounds where the differences among competing theories may appear in observable ways. The remaining four chapters will be devoted to these new arenas for testing relativistic gravity. One new testing ground is gravitational radiation. Almost from the outset, general relativity was known to admit wavelike solutions analogous Theory and Experiment in Gravitational Physics 222 to those of electromagnetic theory (Einstein, 1916). However, unlike the case with electromagnetic waves, there was considerable doubt as to the physical reality of such waves. Eddington (1922) suggested that they might represent merely ripples of the coordinates of spacetime and as such would not be observable. This lingering doubt was dispelled conclusively in the late 1950s by the work of Hermann Bondi and his collaborators, who demonstrated in invariant, coordinate-free terms that gravitational radiation was physically observable, that it carried energy and momentum away from systems, and that the mass of systems that radiate gravitational waves must decrease (Bondi et al., 1962). The pioneering work of Joseph Weber initiated the experimental search for gravitational radiation. Although no conclusive evidence for the direct detection of gravitational waves exists at present [see Douglass and Braginsky (1979) for a review], gravitational-wave astronomy may ultimately open a new window on the universe. Virtually any metric theory of gravity that embodies Lorentz in variance, on at least some crude level, in its gravitational field equations, predicts gravitational radiation. Thus, the existence of gravitational radiation does not represent a particularly strong test of gravitation theory. It is the detailed properties of such radiation that will concern us here. While the post-Newtonian approximation may be described as the weak-field, slow motion "near-zone" limit, our discussion of gravitational radiation will center on the weak-field, slow motion, "far-zone" limit. In this limit, one finds that metric theories of gravity may differ from each other and from general relativity in at least three important ways: (i) they may predict a difference between the speed of weak gravitational waves and the speed of light (see Section 10.1); (ii) they may predict different polarization states for generic gravitational waves (see Section 10.2); and (iii) they may predict different multipolarities (monopole, dipole, quadrupole, etc.), of gravitational radiation emitted by given sources (see Section 10.3). The use of gravitational-wave speed and polarization as tests of gravitation theory requires the regular detection of gravitational radiation, a prospect that may be far off (see Douglass and Braginsky, 1979). However, the multipolarity of gravitational waves can be studied by analyzing the back influence of the emission of radiation on the source (radiation reaction) for different multipoles. One example is the change in the period of a two-body orbit caused by the change in the energy of the system as a result of the emission of gravitational radiation. Such a test is now possible in the binary pulsar (Chapter 12). Gravitational Radiation: Testing Relativistic Gravity 223 10.1 Speed of Gravitational Waves The Einstein Equivalence Principle demands that in every local, freely falling frame, the speed of light must be the same - unity, if one works in geometrized units. The speed of propagation of all zero rest-mass nongravitational fields (neutrinos, for example) must also be the same as that of light. However, EEP demands nothing about the speed of gravitational waves. That speed is determined by the detailed structure of the field equations of each metric theory of gravity. Some theories of gravity predict that weak, short-wavelength gravitational waves propagate with exactly the same speed as light. By weak, we mean that the dimensionless amplitude /zMV that characterizes the waves is in some sense small compared to the metric of the background spacetime through which the wave propagates, i.e., IIU/IICII«i and by short wavelength, we mean that the wavelength X is small compared to the typical radius of curvature 0t of the background spacetime, i.e., |A/£| « 1 This is equivalent to the geometrical optics limit, discussed in Chapter 3 for electromagnetic radiation. In the case of general relativity, for example, one can show (see MTW, Exercise 35.15) that the gravitational wave vector /" is tangent to a null geodesic with respect to the "background" spacetime, i.e., i*r$» = o,
i% = o
where "slash" denotes covariant derivative with respect to the background
metric. In a local, freely falling frame, where gfj = rj^, the speed of the
radiation is thus the same as that of light. Gravitational radiation propagates along the "light cones" of electromagnetic radiation.
General relativity
A simple method to derive this result in general relativity, which
can then be applied to other metric theories, is to solve the vacuum field
equations, linearized (weak fields) about a background metric chosen
locally to be the Minkowski metric. Physically, this is tantamount to
solving the propagation equations for the radiation in a local Lorentz
frame. As long as the wavelength is short compared to the radius of
curvature of the background spacetime, this method will yield the same
Theory and Experiment in Gravitational Physics
224
results as a full geometrical-optics computation. We thus write
G^ = Vw + V
(10.1)
Then the linearized vacuum field equations (5.15) take the form
( 10 - 2 )
Av + K* - <** - K, = °
where indices are raised and lowered using r\. We choose a gauge (Lorentz
gauge) in which
Then
D A* = °
whose plane-wave solutions are
V = O'"**'
'"'X* = 0
(10.5)
Thus, the electromagnetic and gravitational light cones coincide, i.e.,
the gravitational waves are null.
Scalar-tensor theories
The linearized vacuum field equations are (see Section 5.1 for
discussion of notation)
DV + h
-K
= 0,
t
(io.6)
>0 V,^v
Choosing a gauge in which
1 4 , - ^ , - ^ =0
(10.7)
we obtain
U,9 = D A , = 0
(10.8)
|ix
whose plane-wave solutions are proportional to e" " where
/"'V = 0
(10.9)
So in scalar-tensor theories, gravitational waves are null.
Vector-tensor theories
In this case the linearizedfieldequations are much more complex
than in the scalar-tensor theories, with the propagation of linearized
metric disturbances (h^) being strongly influenced by the background
225
cosmological value K of the vector field. In general there are ten different
solutions, each with its own characteristic speed and polarization. For
one of these solutions, for example (for derivation see Section 10.2) the
speed is
v2g = (1 - «K 2 )/[1 - (o> - i, - t)K 2 ]
(10.10)
Rosen's bimetric theory
We have already discussed weak gravitational waves in this
theory, in Section 5.5(g). The resulting speed was given by v\ = Cx/c0
where c1 and c 0 are cosmological matching parameters (see Section 5.5
for discussion). If we take into account not only the cosmological boundary
conditions but also a gravitational potential t/ ext due to an external
gravitating body (galaxy, sun), with the wavelength of the radiation
being short compared to the scale over which l/ext varies, then c 0 and ct
may be replaced by c o (l 2l/ cxt ) and c t (l +2£/ ext ), where c 0 and cx denote
the purely cosmological values, and thus
v2g = (c!/c o )(l + 4[/ ext )
(10.11)
Therefore, the velocity of gravitational radiation may depend both upon
cosmological parameters and on the local distribution of matter. Notice
that solar system limits on a 2 constrain v2 to be within ~ 4 x 10~ 4 ofunity.
RastalFs theory
The (extremely complicated) linearized vacuum field equations
for the vector field K^ in the rest frame of the universe, where
K, = K8° + k^
(10.12)
yield three independent polarizations for k^, one having a different
velocity than the other two. However, to first order in the cosmological
matching parameter K, which is constrained to be small by Earth-tide
measurements (see Sections 5.5, 8.5), the velocities are the same,
Vg
= 1 + %K2
and the polarizations for a wave traveling in the z-direction are given by
k(1)oce0-ez,
k(2)azex,
k (3) oce y
(10.14)
(These results are valid only in the universe rest frame.)
Table 10.1 summarizes the velocities of gravitational waves in these
and other theories of grayity discussed in Chapter 5. Generally speaking,
there are two ways in which the speed of gravitational waves may differ
Theory and Experiment in Gravitational Physics
226
Table 10.1. Properties of gravitational radiation in alternative metric theories of gravity.
Gravitational
wave speed
Theory
General relativity
Scalar-tensor theory
Vector-tensor theory
Rosen's bimetric theory
Rastall's theory
BSLL theory
Stratified theories
1
1
various
(cx/c0)1/:2
1 + iK2 + O(K3)
1 + K^o + °>i) + O(co2)
a
E(2) class
N2
N3
n'
ni 5
n6
n«
" Speed is a complicated function of parameters.
from that of light. The first is through the cosmological matching parameters, i.e.,
vg =* vgc
(10.15)
where vgc denotes the cosmologically determined speed. The second is
through the local distribution of matter. If we take into account a nearly
constant, but noncosmological gravitational potential t/ ext («1), the
matching parameters may be modified by terms of O(l/ ext ), resulting in
a speed
vg =* 1^(1 + a[/ ext )
(10.16)
Solar system experiments limit some of the parameters that appear in
the expressions for vgc, but only to accuracies of order 10~ 3 . A crucial
test of such theories would be provided by high-precision measurements
of the relative speed of gravitational and electromagnetic waves (Eardley
et al., 1973). By comparing the arrival times for gravitational waves and
for light that come from a discrete event such as a supernova, one could
set a limit on the relative speeds that, for a source in the Virgo cluster
(11 Mpc from Earth) for example, would yield
precision in measuring
. . .
.
time lag, in weeks
(10.17)
Another possible way to test whether vg = 1 has been described by
Caves (1980) within the context of Rosen's bimetric theory. If vg < 1,
then high-energy particles are prevented from being accelerated to speeds
greater than vg by gravitational-radiation damping forces that accompany
the nearly divergent gravitational radiation flux emitted by a particle at
velocities near vg. The indirect observation of cosmic rays with energies
227
exceeding 1019 eV places a very tight upper limit, if this analysis is correct,
on 1 - vg in Rosen's theory. Similar conclusions would be expected to
follow in any theory in which vg < 1.
10.2
Polarization of Gravitational Waves
(a) The E{2) classification scheme
General relativity predicts that weak gravitational radiation has
two independent states of polarization, the " + " and " x " modes, to use
the language of MTW, (Section 35.6), or the + 2 and 2 helicity states,
to use the language of quantum field theory. However, general relativity
is probably unique in that prediction; every other known, viable metric
theory of gravity predicts more than two polarizations for the generic
gravitational wave. In fact, the most general weak gravitational wave
that a theory may predict is composed of six modes of polarization, expressible in terms of the six "electric" components of the Riemann tensor
ROiOj that govern the driving forces in a detector (Eardley et al., 1973;
Eardley, Lee, and Lightman 1973).
Consider an observer in a local freely falling frame. In the neighborhood of a chosen fiducial world line ^(t), construct a locally Lorentz
orthonormal coordinate system {t,xj) with t as proper time along the
world line and^(f) as spatial origin ("Riemann normal coordinates").
The metric has the form (MTW, Section 13.6)
9 m = "m + hjn
(10.18)
where
+ O(|x|3),
* + O(|x|3),
% = - i / l a y ^ x * + O(|x|3)
*
(10.19)
where R^a, are components of the Riemann tensor. For a test particle
with spatial coordinates x\ momentarily at rest in the frame, the acceleration relative to the origin is
at = 1*66,? = ~ Kof6j* ;
are tne
(10.20)
where Roio}
"electric" components of Riem due to gravitational
waves or other external gravitational influences. Note that despite the
possible presence of auxiliary gravitational fields in a given metric theory
of gravity, the acceleration is sensitive only to Riem. [This is not necessarily true if the body has self-gravitational energy, as has been emphasized
by Lee (1974).]
Theory and Experiment in Gravitational Physics
228
Thus, a gravitational wave may be completely described in terms of
the Riemann tensor it produces. We define a weak, plane, nearly null
gravitational wave in any metric theory [Eardley, Lee, and Lightman
(1973)] to be a weak, propagating vacuum gravitational field characterized, in some local Lorentz frame, by a linearized Riem with components
that depend only on a retarded time u, i.e.,
R*yi = KpyM
(10.21)
(henceforth we shall drop the caret on indices) where the "wave vector"
! which is normal to surfaces of constant u, defined by
is almost null with respect to the local Lorentz metric, i.e.,
rfvlX = e,
|e| « 1
(10.23)
where e is related to the difference in speed, as measured in a local Lorentz
frame at rest in the universe rest frame, between light and the propagating
gravitational wave, i.e.,
e = (c/vg)2 - 1
(10.24)
We now wish to analyze the general properties of Riem for a weak,
plane, nearly null gravitational wave. To do this, it is useful to introduce,
instead of the locally Lorentz orthonormal basis (t, xJ), a locally null basis.
Consider a null plane wave (light, for instance) propagating in the + z
direction in the local Lorentz frame. The wave is described by functions
of retarded time u, where
u =t - z
(10.25)
(we use units in which the locally measured speed of light is unity). A
similar wave propagating in the z direction would be described by functions of advanced time v, where
v= t +z
(10.26)
We now define the vector fields I and n to be I = Fe^, n = n^e,,, where
These vectors are tangent to the propagation directions of the two null
plane waves. In the (t, xJ) basis they have the form
/" = (1,0,0,1),
n" = i ( l , 0,0,-1)
(10.28)
229
and are null with respect to 17, i.e.,
I'l'V
=
rt\(
= 0
(10.29)
We also introduce the complex null vectors m and m, where the bar denotes complex conjugation, denned by m = m"e,,, where
m" = (2)- ^(O,1, i,0),
m" = (2)" 1/2 (0,1, - i,0)
(10.30)
and where
m"mvjjpv = m"mvf/^v = 0
(10.31)
These null vectors obey the orthogonality relations
>f v = -2J ( "n v) + 2m("mv)
(10.32)
In a Cartesian basis, they are constant. For the remainder of this section,
we shall use roman subscripts (excluding i, j , k) to denote components
of tensors with respect to the null tetrad basis I, n, m, in, i.e.,
Zarb_ = Za$1..<f1fiV... (10.33) where a, b, c,... run over I, n, m, and m, while p, q, r,... run over only I, m, and m. Because the null tetrad I, n, m, and m is a complete set of basis vectors, we may expand the gravitational wave vector Tin terms of them; however, since the gravitational wave is not exactly null, this expansion will depend in general upon the velocity of the observer's local frame relative to the universe rest frame. Choose a "preferred" observer, whose frame is at rest in the universe, and let /" in this frame have the form I" = f ( l + e,) + enn" + emm" + ejn" (10.34) where {£,,£, £,,£} ~ O(E). However, this observer is free (i) to orient his spatial basis so that the gravitational wave and his null wave are parallel, i.e., so that V oc V, (10.35) and (ii) to choose the frequency of his positively propagating null wave to be equal to that of the gravitational wave, i.e., 7° = 1° (10.36) Hence, em = em = 0, £, = -|fi n , and ? = / " - sj^l" - n") (10.37) Theory and Experiment in Gravitational Physics 230 Now, because the Riemann tensor is a function of retarded time u alone, Thus, using the orthogonality relations among the null tetrad vectors, ( 10 - 39 ) Rw = ° The linearized Bianchi identities Rab[cdie] = 0 then yield RatPq,n = O(BnR) (10.40) which, except for a trivial nonwavelike constant, implies Rabpq = ^ W = O(£nR) (10.41) Thus the only components of Riem that are not O(en) are of the form Rnpnq. There are only six such components and all other components of Riem can be expressed in terms of them. They can be related to particular tetrad components of the irreducible parts of Riem; the Weyl tensor, the traceless Ricci tensor, and the Ricci scalar (see MTW Section 13.5 for definitions). These components are called Newman-Penrose quantities, denoted T, <J>, and A, respectively, (Newman and Penrose, 1962). For our nearly null plane wave in the preferred tetrad, they have the form (i) Weyl tensor: ¥ 0 ~ O(s2nR), V2=~i;RnM V1 ~ O(enR), + O(£nR), ^ 3 = -iRnlnm + O(EnR), *4 = -Rnmnm (10.42) (ii) traceless Ricci tensor:$ 0 0 ~ O(en2R),
<D01 ~ O 1 0 =* O 0 2 ^ 3) 20 =s O(enR),
®i2 = *2i = ¥ 3 + O(enR)
(10.43)
(iii) Ricci scalar:
A = - i « F 2 + O(enR)
(10.44)
To describe the six independent components of Riem we shall choose
the set W2, *P3, *P4, and <D22 (¥3 and *P4 are complex). The above results
231
y
y
o
o
(/
/)
(a)
(b)
Im * 4
31
o
J)
(c)
(d)
i
\
(e)
*
\
;
\
/
)
(0
Figure 10.1. The six polarization modes of a weak, plane gravitational
wave permitted in any metric theory of gravity. Shown is the displacement
that each mode induces on a sphere of test particles. The wave propagates
in the +z direction and has time dependence cos cot. The solid lirie is a
snapshot at cot = 0, the broken line one at cot = n. There is no displacement perpendicular to the plane of the figure. In (a), (b), and (c) the wave
propagates out of the plane; in (d), (e), and (f), the wave propagates in the
plane.
are valid for a gravitational wave as detected by the preferred observer.
Now in order to discuss the polarization properties of the waves, we must
consider the behavior of these components as observed in local Lorentz
frames related to the preferred frame by boosts and rotations. However,
we must restrict attention to observers who agree with the preferred observer on the frequency of the gravitational wave and on its direction;
such "standard" observers can then most readily analyze the intrinsic
Theory and Experiment in Gravitational Physics
232
polarization properties of the waves. The Lorentz frames of these standard
observers are related by a subgroup of the group of Lorentz transformations that leave \ unchanged. The most general such transformation of
the null tetrad that leaves T [cf. Equation (10.37)] fixed is given by
I' =
n' =
m' =
m' =
(1 - aa£n)\ - en(am + am) + O(£2),
(1 aa£B)(n + aal + am + am) + O(e2),
(1 - aaeje'^m + al) - e^e'^n + am) + O(E 2 ),
(1 - aaejg-'^m + al) - £nae"'>(n + oan) + O(e2)
(10.45)
where a is a complex number that produces null rotations (combinations
of boosts and rotations) asd cp is an arbitrary real phase (0 < q> < 2n)
that produces a rotation about ez. The parameter a is arbitrary except for
the restriction
aa«e n " 1
(10.46)
This expresses the fact that our results are valid as long as the velocity
of the frame, w, is not too close either to the speed of light or of the gravitational wave, whichever is less; note that for nearly null waves e~ * » 1
and almost any velocity that is not infinitesimally close to unity is permitted, since
aa ~ w2/(l - w2)
(10.47)
For exactly null waves en = 0, and arbitrary velocities w < 1 are permitted.
Under the above set of transformations, the amplitudes of the gravitational wave change according to
W2 = V2 + O(snR),
+ 6a2*F2) + O{snR),
2af 3 + 6aa»P2 + O(snR)
(10.48)
Consider a set of observers related to each other by pure rotations
about the direction of propagation of the wave (a = 0). A quantity that
transforms under rotations by a multiplicative factor e's<? is said to have
helicity s as seen by these observers. Thus, ignoring the correction terms
of O(£nR), we see that the amplitudes {^ 2 ,¥ 3 ,'F 4 ,4) 22 } have helicities
T 2 : s = 0,
O 22 :s = 0,
¥ 3 :s=-l,
¥ 4 :s=-2,
¥ 3 :s=+l,
?4:5=+2
(10.49)
233
However, these amplitudes are not observer-independent quantities, as
can be seen from Equation (10.48). For example, if in one frame *F2 # 0,
*P4 ^ 0, then there exists a frame in which ¥4 = 0. Thus, the presence or
absence of the components of various helicities depends upon the frame.
amplitudes, within the small corrections of O(snR). These statements
comprise a set of quasi-Lorentz invariant classes of gravitational waves.
Each class is labeled by the Petrov type of its nonvanishing Weyl tensor
and the maximum number of nonvanishing amplitudes as seen by any
observer. These labels are independent of observer.
For exactly null waves, the classes are:
Class II6; *F2 # 0. All standard observers measure the same value for
*F2, but disagree on the presence or absence of all other modes.
Class III5: *F2 = 0, ¥3 ^ 0. All standard observers agree on the
absence of *F2 and on the presence of ¥3, but disagree on the presence
or absence of *P4 and <I>22.
Class N3: f 2 s ^ 5 0> * 4 # 0. $22 # 0. Presence or absence of all modes is observer-independent. Class N2: *¥2 = ¥3 = 4>22 = 0, ¥ 4 =£ 0. Independent of observer. Class O1: x¥2 = x¥3 = *¥4 = 0, <D22 # 0. Independent of observer. Class Oo: *¥2 = x¥3 = x¥4 = 0>22 = 0. Independent of observer: No wave. For nearly null waves, simply replace the vanishing of modes (=0) with the nearly vanishing of modes [~O(enR)]. This scheme, developed by Eardley et al. (1973), is known as the E(2) classification for gravitational waves, since in the case of exactly null plane waves (en = 0), the transformation equations, (10.45), are the "little group" E(2) of transformations, a subgroup of the Lorentz group. The E(2) class of a particular metric theory is defined to be the class of its most general wave. Although we have confined our attention to plane gravitational waves, one can show straightforwardly (Eardley, Lee, and Lightman, 1973) that these results also apply to spherical waves far from an isolated source provided one considers the dominant l/R part of the outgoing waves, where R is the distance from the source. (b) E(2) classes of metric theories of gravity To determine the E(2) class of a particular theory, it is sufficient to examine the linearized vacuum field equations of the theory in the limit of plane waves (observer far from source of waves). The resulting Theory and Experiment in Gravitational Physics 234 classes for the theories discussed in Chapter 5 are shown in Table 10.1. Here, we present some examples. Some useful identities that can be obtained from Equations (10.32) and (10.41) are Rnl = RnM + O(snR), Rm = 2Rnmnih + O(enR), Rnm = Kinm + O(enR), R = - 2Rnl + O(snR) (10.50) If Riem is computed from a linearized metric perturbation h^{u), then and W2 = A + O(snR), ^ 4 = ifc*» + O{anR), W3 = O22 O(snR), ^ + O(enR) (10.52) General relativity The vacuum field equations are (10.53) R,v = 0 The waves are null (en = 0). Thus, KM = Rnn,nm = Rnlnn, = 0 (10.54) or V2 = «P3 = O 2 2 = 0 (10.55) The only unconstrained mode is *F4 ^ 0, so general relativity is of E(2) class N 2 . Scalar-tensor theories In a local freely falling frame, the linearized vacuum field equations are ,9 = o, R = O(<p2) (10.56) (see Section 5.3 for details), where <f>0 is the cosmological boundary value of the scalar field (f>. The solution to the first of Equation (10.56) for a plane wave is cp = (10.57) Gravitational Radiation: Testing Relativistic Gravity 235 where tj^lT 0. Then, from Equation (10.56), «(1,= -^oV/'-V» (10.58) Thus, Rnn * 0, Rnl s Rnm = 0, thus, V2 = V3 = 0, <D 2 2 #0, ¥4#0 (10.59) and scalar-tensor theories are of class N 3 . Vector-tensor theories In a local freely falling frame in the universe rest frame, the linearized vacuum field equations take the form - coK2h00iltv - 2coKk0tllv - (co + \r\ - - i(co - (if - ?) - ifa ^ » + (i, + TJK/C^O = 0, / (10.60) (6 - |T)D A - efcf,, - i « X ^ ( n ^ - ^ ) + i(»? - t ) X D ^ , - ftg,^) = 0 (10.61) By substituting plane wave forms h^ = h^u) and fcM = k^(u), we can turn the field equations into a set of algebraic equations for the amplitudes hMV and k^. We now project these equations onto the null tetrad I, n, m, and m, and obtain ten homogeneous algebraic equations for the ten unknowns hab, ka, with coefficients that depend upon the parameters a>, n, x, e, and K, and upon en [see Equation (10.37)]. These equations are of the form [(1 - coK2)en(l - K ) - ±fo - t)K2]hmm = 0, «AihM + pBiLm + Pnhu + PB3K XAIKM + *A3km = 0, + PB4.h'nn + pBt'k, + pB6kn = 0 (10.62) (10.63) (10.64) where A = 1,2,3, and B = 1 , . . . , 6. One mode is given by hmm * 0, en(l - i O = kin ~ r)K\l - coK2yl (10.65) Since -2e n (l - hn) = £ = (vgy2 - 1 (10.66) Theory and Experiment in Gravitational Physics 236 the speed of this mode is given by In this case it can be shown that (except for special values of the parameters) the remaining amplitudes satisfy Km = fej = K = km = 0, eJL> - (1 - Ktfi. s 0, (1 - hn)hnl + (1 - a ^ = 0 = (1 - !£)£ + 2£n/I"nn (10.68) whose consequence is *P2 = *F3 =$22 = 0. Hence, this mode is N2. There
are in principle nine other modes with timm = 0, each with its own speed
and characteristic polarization, some as general as II 6 .
Rosen's bimetric theory
The linearized field equations are of the form D,^MV = 0 with no
restrictions on the h^, hence all modes are nonzero in general, hence the
theory is of class II 6 .
Rastall's theory
Since the linearized physical metric g in the universe rest frame
has the form [Equation (5.78)]
g0J =
KCQ
l
kp
l
gik = Co 8jk + Kco*k05ik
(10.69)
where k^ = k^u), then we have, after transforming to local Lorentz coordinates,
h'tt <x k0 + kz,
^
0,
tilih
oc kg,,
hmih<xk0
(10.70)
However, from the solution of the linearized field equations discussed in
the previous section [Equation (10.14)], it is clear that for all solutions,
n
'u = O(snR), hence,
¥ 2 = O(enR)
(10.71)
Notice that for thek ( 1 ) mode, only <J>22 °c timjh # 0, so this mode is O t ; for
thek (2) andk (3) modes, *P3 oc hm # 0, so these modes are III 5 . The vanishing of *P4 a hmih is valid only in the universe rest frame, a result of the
237
special form of g^ there. The most general wave therefore is III 5 , hence
Rastall's theory is of class III 5 .
It is possible to show that the other theories discussed in Chapter 5 are
of class II6 (see Table 10.1).
(c) Experimental determination of the E(2) class
Consider an idealized gravitational-wave polarization experiment. An observer uses an array of gravitational-wave detectors to determine via Equation (10.20) the six electric components ROiOj of Riem for
an incident wave (for discussion of possible devices and arrays see Eardley,
Lee, and Lightman, 1973; Paik, 1977; and Wagoner and Paik, 1977). Let
us suppose that the waves come from a single localized source with spatial
wave vector k (which the observer may or may not know a priori). If the
observer expresses his data as a 3 x 3 symmetric "driving force matrix"
StJ(t) = Roioj
(10.72)
then, for a wave with k = e2, Equations (10.28), (10.30), (10.42), and (10.43)
give the following form for S,7 in terms of the wave amplitudes
-2/2Rex¥3
(10.73)
2/2,
where the standard xyz orientation of the matrix elements is assumed.
Now, if the observer knows the direction k a priori, either by associating
the wave with an independently observed event such as a supernova, or
by correlating signals detected at two widely spaced antennas (gravitational-wave interferometry), then by choosing a z-axis parallel tok, one
can determine uniquely the amplitudes as given in Equation (10.73), and
thereby the class of the incident wave. Because a specific source need not
emit the most general wave possible, the E(2) class determined by this
method would be the least general class permitted by any metric theory
of gravity.
However, if the observer does not know the direction a priori, it is not
possible to determine the E(2) class uniquely, since there are eight unknowns (six amplitudes and two direction angles) and only six observables
(Sy). In particular, any observed StJ can be fit by an appropriate wave of
class II6 and an appropriate direction. However, for certain observed
Sjj, the E(2) class may be limited in such a way as to provide a test of gravitational theory. For example, if the driving forces remain in a fixed plane
Theory and Experiment in Gravitational Physics
238
and are pure quadrupole, i.e., if there is a fixed coordinate system in which
n
-X
\0
o\
n
0
(10.74)
0 0/
then the wave may be either II 6 (unknown direction), or N 2 (direction
parallel to z axis of new coordinate system). If this condition is not fulfilled, the class cannot be N 2 . Such a result would exemplify evidence
against general relativity. Eardley, Lee, and Lightman (1973) provide a
detailed enumeration of other possible outcomes of such polarization
measurements.
10.3
Multipole Generation of Gravitational Waves and
It is common knowledge that general relativity predicts the lowest multipole emitted in gravitational radiation is quadrupole, in the sense
that, if a multipole analysis of the gravitational field in the radiation zone
far from an isolated system is performed in terms of tensor spherical harmonics, then only the harmonics with / ^ 2 are present (see Thorne, 1980
for a thorough discussion of multipole-moment formalisms). For material
sources, this statement can be reworded in terms of appropriately denned
multipole moments of the matter and gravitational-field distribution
within the near-zone surrounding the source: the lowest source multipole
sources, such as binary star systems, quadrupole radiation is in fact the
dominant multipole. (Some have argued that this is true for any slowmotion source, whether weak field or not. One exponent of this viewpoint
is Thorne, 1980.) The result is a gravitational waveform in the radiation
zone given by
hmm = (2/R)Imm
(10.75)
where R is the distance from the source, 7 y is the moment of inertia of the
source, and dots denote derivatives with respect to retarded time. The
waveform h^m is related to the measured electric components of Riem by
Equation (10.52),
«-*=-*4=-i«**
(10.76)
The flux of energy at infinity that results from this waveform is given by
dE/dt = - J < W
(10.77)
239
where J y is the trace-free moment of inertia tensor of the system, given to
lowest order in a post-Newtonian expansion by
/ = J p ( x , t){xtXj - !<50x2) d3x
(10.78)
and where angular brackets denote an average over several periods of
oscillation of the source (for a recent discussion, see Walker and Will,
1980a).
These comments apply to the asymptotic properties of the outgoing
radiation field. However, we are interested not in the properties of the outgoing radiation field (those were relevant for Sections 10.1 and 10.2), but
in the back reaction of the source to the emission of the radiation. A variety of computations have led to the conclusion that the energy flux at infinity given by Equation (10.77) is balanced by an equal loss of mechanical
or orbital energy by the system, and that this energy loss can be derived
from a local radiation-reaction force (MTW Section 36.11)
F(react)=
_ (2/5)mi\?Xj
(10.79)
where the superscript (5) denotes five time derivatives (Walker and Will,
1980b). However, one school of thought maintains that these conclusions
have not been satisfactorily derived from a fully self-consistent, approximate solution of Einstein's equations (Ehlers et al., 1976). It is not the purpose of this section to enter into this controversy. Instead, we shall simply
make the assumption that in any semiconservative metric theory of gravity, there is an energy balance between the flux of gravitational-wave
energy at infinity and the loss of mechanical energy of the source, provided
one averages over several periods of oscillation, and that the energy flux
can be determined using a slow-motion, weak-field approximation
scheme of a kind suggested by Epstein and Wagoner (1975).
If we now focus on binary systems with total mass m = mx + m2, reduced mass ii = m^m^m, orbital separation r, and relative velocity v,
to a loss of orbital energy at a rate (Peters and Mathews, 1963)
^=_/^A(12^-ll^)\
(10.80)
dt
\ r4
/
where r = dr/dt, and where angular brackets denote an average over an
orbit. This loss of energy results in a decrease in the orbital period P given
by Kepler's third law,
(10.81)
Theory and Experiment in Gravitational Physics
240
of the system, and to a corresponding decrease in the eccentricity of the
orbit (see Wagoner, 1975, for references and a summary of the formulae).
Faulkner (1971) has pointed out that these effects of quadrupole gravitational radiation may play an important role in the evolution of ultrashortperiod binary systems (see also Ritter, 1979). But probably the most
by observations of period changes P in the binary pulsar (Chapter 12).
Unlike general relativity, however, nearly every alternative metric
theory of gravity predicts the presence in gravitational radiation of all
multipoles-monopole and dipole, as well as quadrupole and higher
multipoles (Eardley, 1975; Will and Eardley, 1977; and Will, 1977). For
binary star systems, the presence of these additional multipole contributions has two effects on the energy-loss-rate formula, (10.80): (a) modification of the numerical coefficients in (10.80) and (b) generation of an
additional term (produced by dipole moments) that depends on the selfgravitational binding energy of the stars. The resulting formula for dE/dt
may be written in a form that contains dimensionless parameters whose
values depend upon the theory under study. Two parameters, KV and K2,
are denoted "PM parameters" because they refer to that part of dE/dt that
corresponds to the Peters-Mathews (1963) result for general relativity. A
parameter KD refers to the dipole self-gravitational contribution, where, at
least schematically, we may write
{dE/dt)dipole
= - i K f l < D D>
(10.82)
where D is the dipole moment of the self-gravitational binding energy Gla
of the bodies
D = £ Qaxo
(10.83)
a
(Within each specific theory of gravity the details are more complicated
than this, however.) For a binary system, the result is
f = - (~£- iUw2 - K2f2) + i*fl®2])
(10-84)
where <3 is the difference in the self-gravitational binding energy per unit
mass between the two bodies.
In this section, we shall derive these results using a post-Newtonian
gravitational radiation formalism developed by Epstein and Wagoner
(1975) and Wagoner and Will (1976). However, because of the complexity
of many alternative theories of gravitation beyond the post-Newtonian
241
approximation, it has proven impossible to devise a general formalism
analogous to the PPN framework, beyond writing Equation (10.84) with
arbitrary parameters. But, we can provide a general description of the
method used to arrive at Equation (10.84) within a chosen theory of gravity, emphasizing those features that are common to many currently viable
theories. Later, we shall describe the specific computations within selected
theories. The method proceeds as follows:
Step 1: Select a theory. Restrict the adjustable constants and cosmological matching parameters to give close agreement with solar system
tests (Chapters 7, 8, and 9).
Step 2: Derive the "reduced field equations." Working in the universe
rest frame, expand the gravitational fields about their asymptotic values,
and, using any gauge freedom available, express thefieldequations in the
"reduced" form
(v~2d2/dt2 + V2) [terms linear in perturbations of fields]
= - ten [source] (10.85)
where vg, the gravitational-wave speed, is a function of adjustible constants and matching parameters, and where the "source" consists of matter
and nongravitational field stress energies, and of "gravitational" stress
energies consisting of terms quadratic and higher in gravitational-field
perturbations. If we denote the linear term by xjt (it can be a tensor of any
rank) and the source by x, then the solution of Equation (10.85) that has
outward propagating disturbances at infinity is
ij/(x,t) = 4 J\(r - v;x\x - x'|,x')|x - x'l" 1 d3x'
(10.86)
For field points far from the source (R = |x| » r = "size" of source, \i//\ «
1), we have
\j/(x,t)=4R~l jxit-v^R
+ v^t x',x')d3x' + O(r/R)2 (10.87)
where n = x/R. If we assume that the motions of the source are sufficiently
slow (source within wave zone, r < k/2n = wavelength/27t « R), then
Equation (10.87) may be expanded in the form
\x(t-v-lR,x'){n-x')md2x'
(10.88)
m=O
For further use, we note that
f; = - V9 %^,o + O(r/R2)
(10.89)
Theory and Experiment in Gravitational Physics
242
Step 3: Determine the energy loss rate in terms of \jt. Let us restrict
attention to Lagrangian-based theories of gravity (such as the currently
viable theories described in Chapter 5). Such theories possess conservation
laws of the form (see Section 4.4)
(10.90)
where ©*" reduces in flat spacetime to the stress-energy tensor for matter
T"v. Hence, we can define quantities P" that are conserved for a localized
source, except for a possible flux 0 " j of energy-momentum far from the
source: when integration is performed over a constant-time hypersurface,
we have
P" = J0«° d3x,
dP"/dt = J0f o ° d3x = - J s 0 W ' d2Sj (10.91)
where S is a closed two-dimensional surface surrounding the region of
integration. For each theory, it turns out that 0'"' may be written
0" v = f(il/)T"v + t"v
(10.92)
where /(if/) -> 1 as \j/ -* 0, and t"v is an expression at least quadratic in the
first-order perturbations (i/0 of the gravitational fields. If we choose for
S a sphere of radius R in the wave zone far from the source, we have for
dP°/dt = -R2j>
tOinj dQ
(10.93)
Substituting Equation (10.88) into the expression for t0J provided by the
equations of the theory yields an expression for dP°/dt in terms of time
derivatives i^>0 of the gravitational fields, evaluated in the far zone.
Step 4: Make a post-Newtonian expansion of the "source" x (see
Chapter 4 for discussion). For this purpose, use the near-zone, postNewtonian forms for matter variables and gravitational fields obtained
in Chapter 5 in the solution for the post-Newtonian metric, appropriately
transformed to the gauge adopted in Step 2. Depending on the nature of
^i, the sources x are of even ("electric") order in the post-Newtonian sense
[O(0),O(2),...] or odd ("magnetic") order [O(l),O(3),.. .] For electric
sources, x typically contains terms of the form
l
electric
~ P,pn,pv2,pU,p
(10.94)
modulo total divergences whose moments [monopole, dipole, etc.; see
Equation (10.88)] can be shown to be negligible upon integration by
243
parts [see Epstein and Wagoner (1975) for discussion]. For magnetic
sources, x typically has the form (modulo divergences)
^magnetic ~ M P&U P^U, pvh2, PVJ, PV\ PWj
(10.95)
Step 5: Simplify i// using integral conservation laws. Because \jj,
Equation (10.88), involves time derivatives of integrated moments of the
source T, and since time derivatives of ij/ will ultimately be used, it is convenient to employ the integral conservation laws obtained from Equation (10.90) to extract from the integrated moments terms that are constant
in time, linear in time, etc. Some of these terms reflect the imprints of the
mass, momentum, angular momentum, and center of mass of the source
on the far-zone field, and do not contribute to gravitational radiation.
Since these integral conservation laws are to be applied only to near-zone
integrals, we neglect surface integrals such as the one in Equation (10.91)
(retaining them would only yield higher-order corrections to the energy
loss-rate formula). These integral conservation laws give the following
useful results (valid in the near zone)
(d/dt)
(d2/dt2) §®°°xJxkd3x=2
(d/dt) J0' o (fi x)d3x = j®Jknkd3x
(10.96)
Notice that, because we are dealing with semiconservative theories of
gravity, 0" v is not necessarily symmetric, so we have retained the contributions of the antisymmetric parts of 0" v where necessary. However,
as we saw in Section 4.4, these terms depend upon the PPN parameters
<*! and a2 and so they will be small if we impose the experimental constraints on aj and a2 discussed in Chapter 8, or will be zero if we adopt
a version of the theory with a t = a2 = 0 (i.e., a fully conservative version).
Step 6: Apply to binary systems. We consider a system made up of
two bodies that are small compared to their separations (d « r); that is,
we ignore all tidal interactions between them. We may thus treat each
body's structure as static and spherical in its own rest frame. We then
follow the procedure of Section 6.2: for a given element of matter in body
a, we write
v = vfl [static structure],
x = Xfl + x
(10.97)
Theory and Experiment in Gravitational Physics
244
where
Xa = m-1 £ p*(l + II - if/)xd 3 x,
ma = P°a= f p*(l +
ya = dXJdt,
n-^O)d3x,
0 = Jo p*(x', t)|x - x'|"' d3x'
(10.98)
We note that ma is conserved to post-Newtonian order, as long as tidal
forces are neglected. The full Newtonian potential U for spherical bodies
is given by
U(x, t)= Ua+ £ mb\x - Xb\ ~J
[x inside body a]
b*a
= X mb\x Xj.1"1
[x outside body a]
(10.99)
b
Then the total energy of the system P° [cf. Equation (4.108)] is given by
(KU00)
where rafc = |Xa Xb|. For a binary system we may evaluate the orbital
terms in Equation (10.100) to the required order using Keplerian equations (see Section 7.3 for definitions of orbital elements);
r = rab = a(l - e 2 )(l + e cos <^)"r
(10.101)
The result is
°
(10.102)
where m = ma + mb, n = mamb/m, and where the semi-major axis a is
related to the orbital period P to the necessary order by (P/2n)2 a3/m.
In the emission of gravitational radiation whose source is the orbital
motion, the quantities ma and mb will be unchanged because of our neglect
of tidal forces and internal motions. Invoking energy balance, we thus have
= dP°/dt
(10.103)
We now use the above procedure to split the moments oft that determine
\ji into orbital parts (v2 ~ m/r) and "self" parts associated with each body
(£7 ~ II ~ p/p ~ m/d » m/r). In terms of the quantities m/r and m/d, we
find that electric ij/ fields have the schematic form [Equations (10.88),
245
(10.94), (10.96)]
^eiectnc ~ 4(m/R) | [constant]
[ml r / m V l [mm]r
, J
J
mono
~ + I ~ I + ~~ 3 L
[dipole]
-r) r b i v J
(10.104)
and magnetic \jt fields have the form [Equations (10.88), (10.95), (10.96)]
^magnetic ~ 4(m/R) \ [constant]
+ j
(10.105)
Because the energy flux tOi [Equation (10.93)] typically depends on (i/'.o)2*
the constant terms in Equations (10.104) and (10.105) do not contribute
to the radiation. In Equations (10.104) and (10.105) it is the (m/r) term
that yields the PM contribution, since t/^0 ~ (m/R)2(m/r2)2v2. The terms
of O(m/r)2 and O(m/r)3'2 in Equations (10.104) and (10.105) are postNewtonian corrections of a kind discussed by Wagoner and Will (1976)
for general relativity. The terms of O[(w/r)(m/d)] effectively renormalize
the masses that appear in the PM result by corrections of O{m/d).
The terms of O[(m/d)(m/r) 1/2 ] produce the dipole radiation of interest:
(fo) 2 ~ (m/R)2(m/d)2{mll2/r3l2)2v2
~ (m/R)2(m/r2)2{rn/d)2. Cross terms
produce effects that are down from these by powers of (d/r) or that vanish
on integration over solid angle [Equation (10.93)]. Hence, we retain only
terms in \jj of order (m/r) and (m/d)(m/r)1/2.
In evaluating the "self" terms, we employ the standard virial theorem
for static spherically symmetric bodies:
3 japd3x
+ na = 0
(10.106)
Theory and Experiment in Gravitational Physics
246
where to the necessary order
Qfl= -^ap0d3x=
-^ap{x)p{x')\x-x'\-1d3xd3x'
(10.107)
Step 7: Calculate the average energy loss over one orbit, using Newtonian equations of motion to simplify the Newtonian P M contribution
and the post-Newtonian dipole contribution to the radiation.
To illustrate this method, we shall now focus on three metric theories:
general relativity, scalar-tensor theories, and Rosen's bimetric theory.
For other theories, such as the BSLL theory and Ni's theory, see Will
(1977).
General relativity
By defining
0"v = /i"v - \rf"h
(10.108)
and choosing a gauge ("Lorentz" gauge) in which
0?vv = 0
(10.109)
where indices are raised and lowered using the Minkowski metric, one
can show that Einstein's equations are equivalent to the reduced field
equations
I|0"v = -
16TCT"V
(10.110)
where
T"V = T"v + t"v
(10.111)
v
v
with t" a function of quadratic and higher order in 0^ and its derivatives.
Because of the gauge condition, Equation (10.109), T"V satisfies
T^VV = 0
(10.112)
Then
e»» = 4R-1
f; (l/ml)(d/dt)m {^(t-R,
m=0
x')(n x')md3x'
(10.113)
J
Because of the gauge condition, Equation (10.109), and the retarded
nature of 0"v, we need to determine only the 01J components, since
247
Now, because the source T"V for 6"v satisfies its own conservation law
tfvv = 0, and is symmetric, we may make use of Equations (10.96), with
TMV in place of 0*v to show that
6iJ = 2R-\d2ldt2)\
xoo(t - R, x)xixid3x
+ [quadrupole moments of xOj, TJ*] + !
(10.115)
Notice that the monopole and dipole moments of zij have been reexpressed as second time derivatives of quadrupole moments. Since each
time derivative (8/dt)x ~ v ~ (m/r)112, there can be no "dipole" contribution to 0** of the form (m/d)(m/r)112. Thus, the only contribution to the
field to the required order comes from the lowest order, "Newtonian"
part of T 00 , namely
t 0 0 = p [ i + O(2)]
(10.116)
For a binary system, Equation (10.115) becomes
9iJ = 2R ~ \d2ldt2) X max\xi + O(m/r)3'2
(10.117)
a
The conservation laws for T"V also imply that the center of mass of the
system is unaccelerated, so, decomposing xa into center-of-mass and
relative coordinates to Newtonian order, using
X = m~i(mtx1 + m2x2),
x = x2 x t
(10.118)
we obtain, modulo a constant,
6iJ = (2n/R)(d2/dt2){xixj) + O(m/r)312
(10.119)
Now, to determine the energy-loss rate, it is most convenient to use
for 0'"' the conserved quantity
= ( - g)(T»v + til),
0fL,v = 0
(10.120)
where ££L is the Landau-Lifshitz pseudotensor, given for example by
MTW, Equation (20.22). (Actually, we could equally well have used T*"1
for this purpose, since one can show that both quantities yield identical
equations of motion for matter and identical integral conservation-law
results as, for example, in Equation (10.91). The Landau-Lifshitz version
is simpler because t£L contains only first derivatives of O1"1.) Evaluating
t££ for use in Equation (10.93), using Equation (10.114), and defining the
Theory and Experiment in Gravitational Physics
248
transverse traceless (TT) part of 6lJ by
%PiJpklekl,
p j = <s< - n%
(10.121)
we obtain the energy-loss rate
dP°/dt = -(R2/32n) (Jjfl¥r,o0Tr,oda
(10.122)
Substituting Equation (10.119) into Equation (10.122), performing the
angular integrations, and averaging over several oscillations of the source
yields
dp°/dt = -*<*;/«>.
hj = I*(X,XJ - i v 2 )
( 10 - 123 )
Using the Newtonian equations of motion, d\/dt = mx/r3, to evaluate
the time derivatives in Equation (10.123) to the required accuracy yields
the Peters-Mathews formula, Equation (10.80). Thus, for general relativity Kt = 12, K2 = 11, KD = 0.
Scalar-tensor theories
By defining
f
(10.124)
(see Section 5.1) and choosing a gauge in which
0?vv = 0
(10.125)
we can write the field equations for scalar-tensor theories in the form
, 0 " v = - 16TE t"v,
<? = - 16TI S
(10.126)
where
S = -(6 + 4w)- 1 r[l - $0 - W^o - 2c»'(3 1 [<p^v6>"v + ^ " V y - a»'(3 + 2co)->?"] W,<?)3) (10.127) where co = a>(</>0), cu' = dco/d^l^, T = g^T"*, and indices on 0"v and <pf|1 are raised and lowered using i\. The quantity t"v is a function of quadratic and higher order in 0"v and q>. Now, because of the conservation law satisfied by z'"', it is clear that 6Jk can be reexpressed as second time derivatives of quadrupole moments of x 00 , as in general relativity, and thus will not contribute any dipole terms. However, the source, S, does Gravitational Radiation: Testing Relativistic Gravity 249 lead to dipole terms, as follows. We first evaluate the post-Newtonian forms of 0"v and q> in the near zone. From the post-Newtonian limit as calculated in Section 5.3, for instance, we obtain 600 = 2(1 + y)U + O(4), 60J = 2(1 + y)VJ + O(5), 0° = O(4), q> = (1 - y)4>0U + O(4) (10.128) where y is the PPN parameter, given by y = (1 + o>)/(2 + co), (10.129) and where we have used Equation (5.38) to convert to geometrized units. Equations (10.127) and (10.128) then yield, to the necessary order (10.130) To simplify the source, S, and its moments, we use the post-Newtonian forms for conserved quantities in the near zone as given in Equation (4.107). Then, for a system containing compact objects, the general procedure described above yields, for the far zone to the required accuracy, 6iJ = (1 + y)R-\d2ldt2) £ rnXxi + O(m/r)3'2, n P + 2[1 + 2ca'(3 - [1 + 4co'(3 <ab 2o,'(3 + 2a,) - 2 + O(m/r)312 (10.131) For a binary orbit we obtain (modulo constants), diJ = 2(1 + y)(fj,/R)(v'vj cp=-(l- mx'xj/r3) y)4>0{nlR){v2 - (n v)2 + [1 + 4a.'(3 + 2©)"2~\m/r + m(a x) 2 /r 3 + 2[1 + 2co'(3 + 2a>)"2]G(n v)} (10.132) where S is given by S = Q i M - Q 2 /m 2 (10.133) Theory and Experiment in Gravitational Physics 250 The most useful conserved quantity appropriate for determining the energy flux is given by 0"* = ( - g # 0o \T>" + t£D (10.134) where tfx is the scalar-tensor theory analogue of the Landau-Lifshitz pseudotensor, as given by Nutku (1969b) [for alternative conserved quantities, see Lee (1974)]. Evaluating t°{ a n d using Equations (10.114) and (10.121), we obtain dP°/dt = -(R2/32n)(t>o <j> [^'T.O^TT.O + (4© + 6#o V.oP.o] <« (10.135) Substituting Equation (10.132) into Equation (10.135) and integrating over solid angle yields Equation (10.84) with = 12 - 5/(2 + co), K2 = 11 - 45(1 + fa + ia 2 )/(8 + 4co), KD = 2(1 + a) 2 /(2 + co) (10.136) Kl where 2 co) (10.137) Rosen's bimetric theory For simplicity, we choose the version of the bimetric theory whose post-Newtonian limit is identical to that of general relativity, that is, we choose c0 = c t (see Section 5.5). This is equivalent to assuming that, far from the local system, both g and i; have the asymptotic form diag( 1,1,1,1). Our final results will then be valid up to corrections of O(l c o /ci), which, according to Earth-tide measurements (limits on a 2 ), must be small. We then define (10.138) and write the field equations, (5.68), in the form (10.139) v The post-Newtonian forms for 6" in the near zone are (see Section 5.5) 00° = 41/ + Q(4), 6Oi = 4VJ, 0'-> = O(4) (10.140) Gravitational Radiation: Testing Relativistic Gravity 251 To the necessary order, Equation (10.139) then yields T 0 0 = p(l + n + v2 + 2U), T'J' = pvlv> + p5iJ, tOj = p{v\\ + n + v2 + AU + pip) - 2V]~] (10.141) The conserved quantity associated with the bimetric theory is The near-zone conserved quantities 0 0 0 and 0 O j can be determined from Equations (10.140) and (10.142) or taken directly from Equation (4.107), since we are using the fully conservative version of the theory. For a system of compact objects, we then obtain 000 = 4R-1 \P° + n P + X a + X mJ(nva)2 O(m/r)3'2, + E m.»i»i - | X Qa(n v j a 3 a 312 + O{m/r) (10.143) For a binary orbit, we obtain (ignoring constant terms) 600 = 4(n/R){[ih v)2 - m/r - m(n x) 2 /r 3 ] - <Sn v}, 601 = 4(|i/J?){[uJ'(fi v) - mxJ(ii x)/r 3 ] - | S u J } , 0iJ' = 4(/i/i?)[i;^-i + i®(B v)5 y ] (10.144) We now evaluate the energy flux tOj in the far zone using Equations (10.89), (10.138), and (10.142) and obtain dp°/dt = - (R 2 /32TI) <j) {efte^o - i e oeiO) rfn (10.145) Theory and Experiment in Gravitational Physics 252 Substituting Equation (10.144) into (10.145) and integrating over solid angle yields Equation (10.84), with *i = - ¥ , *2= - ¥ , *D=- ¥ (10.146) Other theories Calculation of the PM and dipole parameters within this formalism has been carried through for the BSLL theory and for Ni's stratified theory (Chapter 5), restricting attention to those versions whose post-Newtonian limits are identical to that of general relativity (see Will, 1977, for details). The results are shown in Table 10.2. We note the surprising result that, for all the theories listed in Table 10.2, except scalar-tensor theories and general relativity, the dipole radiation carries negative energy, i.e., increases the energy of the system (KD < 0), and that the PM radiation may carry either positive or negative energy, depending on the theory and on the nature of the orbit. It could be argued (and presumably will be argued by some) that this prediction alone should be sufficient grounds to judge each such theory unviable. However, this is a theorist's constraint that has little experimental foundation in the case of gravitation, and so we will restrict attention to observational evidence for or against such an effect. Such evidence will be provided by the binary pulsar (Chapter 12). The only theory shown in Table 10.2 that automatically predicts no dipole radiation is general relativity. Scalar-tensor theories can also avoid dipole radiation for particular choices of the function co(</>). For example, if «(<£) = (4 - 3^)/(2<£ - 2) (Barker's constant G Theory), then 1 + 2o>'(3 + 2a>)~2 = 0 = KD. In this case, to post-Newtonian order, the theory satisfies the strong equivalence principle (Section 3.3); the locally measured gravitational constant GL is truly constant, and the theory predicts no Nordtvedt effect (4/? 7 3 = 0). The other theories in Table 10.2 violate SEP. This suggests the general conjecture that a theory of gravity predicts no dipole gravitational radiation if and only if it satisfies SEP to the appropriate order of approximation. In Section 11.3, we shall see more directly how the violation of SEP can manifest itself in dipole gravitational radiation. It is also interesting to note the strong correlation between the sign of the energy carried by gravitational radiation and the E(2) class of the theory, as summarized in Table 10.1. General relativity and scalar-tensor theories predict waves of the least general E(2) classes (N2 and N3), of definite helicity (±2; ±2, 0), and of positive energy. The other theories Table 10.2. Multipole gravitational radiation parameters in metric theories of gravity PM parameters Dipole parameter Theory General relativity Scalar-tensor: 12 BWN, Bekenstein 12 Brans-Dicke 12 5 2 + co 11 - 45 No Yes No Yes No No -125/3 No No No No No No -400/3 No No 2 2 + co a Vector tensor Definite helicity? Yes 45 11 Sign of energy Yes 11 2 + ca Satisfies SEP? 6 Bimetric : Rosen Rastall BSLL Stratified": Lee-Lightman-Ni -21/2 -23/2 a -21/2 18 73/8 19 -20/3 a " Calculations have not been performed to determine these values. 6 We adopt that version of each theory whose PPN parameters are identical to those of general relativity. Theory and Experiment in Gravitational Physics 254 in Table 10.2 predict waves of more general classes, of indefinite helicity, and of negative or positive energy. It is perhaps not surprising that such theories predict indefinite sign for the emitted energy, since - according to quantum field theory - definite helicity, quantizibility, and positive definiteness of energy go hand in hand. Whether or not a general conjecture along these lines can be proved is an open question. One of the drawbacks of the post-Newtonian method for deriving formulae for energy loss is that it assumes that gravitational fields are weak everywhere. This assumption is no longer valid in systems containing compact objects (neutron stars or black holes), such as the binary pulsar. In the next chapter we shall describe a formalism that retains the essential post-Newtonian features of the orbital motion of such systems but that permits one to take into account the highly relativistic nature of any compact objects in the system. Nevertheless, the basic conclusions summarized in Table 10.2 will be unchanged. 11 Structure and Motion of Compact Objects in Alternative Theories of Gravity Within general relativity, the structure and motion of relativistic, condensed objects-neutron stars and black holes-are subjects that have attracted enormous interest in the past two decades. The discovery of pulsars in 1967, and of the x-ray source Cygnus XI in 1971, have turned these "theoretical fantasies" into potentially viable denizens of the astrophysical zoo. However, relatively little attention has been paid to the study of these objects within alternative metric theories of gravity. There are two reasons for this. First, as potential testing grounds for theories of gravitation, the observations of neutron stars and black holes are generally thought to be weak, because of the large uncertainties in the nongravitational physics that is inextricably intertwined with the gravitational effects in the structure and interactions of such bodies. Examples are uncertainties in the equation of state for matter at neutronstar densities, and uncertainties in the detailed mechanisms for x-ray emission from the neighborhood of black holes. Second, compared with the simplicity of the post-Newtonian limits of alternative theories and the consequent availability of a PPN formalism, the equations for neutronstar structure and black hole structure are so complex in many theories, and so different from theory to theory, that no systematic study has been possible. Neutron stars were first suggested as theoretical possibilities within general relativity in the 1930s (Baade and Zwicky, 1934). They are highly condensed stars where gravitational forces are sufficiently strong to crush atomic electrons together with the nuclear protons to form neutrons, raise the density of matter above nuclear density (p ~ 3 x 1014 g cm"3), and cause the neutrons to be quantum-mechanically degenerate. A typical neutron-star model has m ^ lm 0 , R =* 10 km. Theory and Experiment in Gravitational Physics 256 However, they remained just theoretical possibilities until the discovery of pulsars in 1967 and their subsequent interpretation as rotating neutron stars. Since that time much effort has been directed toward calculating detailed neutron-star models within general relativity, with particular interest in masses, moments of inertia, and internal structure. These quantities are important in understanding both steady changes and discontinuous jumps ("glitches") in the observed periods of pulses from pulsars. The principle uncertainty in these computations is the equation of state of matter above nuclear density (for a review see Baym and Pethick, 1979). In a certain sense, black hole theory has a longer history than neutronstar theory, as it dates back to a 1798 suggestion by Laplace that such objects might exist in Newtonian gravitation theory (see Hawking and Ellis, 1973, Appendix A). Within general relativity, two key events in the history of black holes were the discovery of the Schwarzschild metric (Schwarzschild, 1916) and the analysis of gravitational collapse across the Schwarzschild horizon (Oppenheimer and Snyder, 1939). However, theoretical black hole physics really came into its own with the discovery in 1963 of the Kerr metric (Kerr, 1963), now known to be the unique solution for a stationary, vacuum, and rotating black hole (with the Schwarzschild metric being the special case corresponding to no rotation). It was the discovery in 1971 of the rapid variations of the x-ray source Cygnus XI by telescopes aboard the UHURU satellite that took black holes out of the realm of pure theory. The source of x-rays was observed to be in a binary system with the companion star HDE 226868; analysis of the nature of the companion and of its orbit around the x-ray source, and detailed study of the x-rays, led to the conclusion that the unseen body was a compact object (white dwarf, neutron star, or black hole) with a mass exceeding 9m© (Bahcall, 1978). Since the maximum masses of white dwarfs and neutron stars are believed to be approximately 1.4m© and 4m©, respectively, the simplest conclusion was that the object was a black hole. The source of the x rays was believed to be the hot, inner regions of an accretion disk around the black hole, formed by gas stripped from the atmosphere of the companion star. Since 1971, other potential black hole candidates in x-ray binary systems have been found, and studies of the central regions of some galaxies and globular clusters have indicated the possible existence of supermassive black holes [see Blandford and Thorne (1979) for a review]. However, a crucial link in the chain of argument that leads to the black hole conclusion for Cygnus XI is that the maximum mass of a neutron star is less than 4m©. There are three Structure and Motion of Compact Objects 257 possible sources of uncertainty in this limit (the maximum mass of white dwarfs is much more certain). The first is the equation of state. However, it has been possible to obtain bounds on the maximum mass of between 3 and 5mQ using arguments that are independent of the details of the high-density equation of state (Hartle, 1978). The second is rotation. However, most analyses indicate that rotation cannot increase the maximum mass beyond about 20%. The third is the theory of gravitation. Although alternative theories of gravitation may have post-Newtonian limits close to that of general relativity, their predictions for the highly nonlinear, strong-field regime of neutron-star structure may differ markedly from those of general relativity. Indeed, some theories predict no maximum mass for neutron stars. Since the only present evidence for black holes crucially depends upon the maximum-mass argument, these results within alternative theories are used by many authors as reasons for caution in making the black hole interpretation, rather than as tests of competing theories. As we shall see, some alternative theories do not even predict black holes. However, the discovery of the binary pulsar (Chapter 12) has made the study of neutron-star structure and motion an important tool for testing gravitation theory. The precise orbital data obtained for that system permits for the first time the direct measurement of the mass of a neutron star and the study of relativistic orbital effects (such as periastron shifts) in systems containing condensed objects. In alternative theories of gravity, the nonlinear gravitational effects involved in the neutron star can make significant differences in many relativistic effects, even though in the postNewtonian limit, these effects would have been the same as in general relativity. Crucial tests of competing theories may then be possible. Discussion of these tests will be presented in Chapter 12; this chapter sets the framework for that discussion. In Section 11.1, we analyze the equations of neutron-star structure and, in Section 11.2, the equations of black hole structure in alternative theories of gravitation. In Section 11.3 we present a framework for discussing the motion of compact objects, such as neutron stars, in competing theories. As we noted above, very little systematic study of these issues has ever been carried out, so we shall merely present a few relevant examples. 11.1 Structure of Neutron Stars In Newtonian gravitation theory, the equations of stellar structure for a static, spherically symmetric star composed of matter at zero temperature (T 0 is an adequate approximation for neutron-star matter) Theory and Experiment in Gravitational Physics 258 are given by dp/dr = p dU/dr [Hydrostatic equilibrium], P = P(P) [Equation of state], 2 2 (d/dr)(r dU/dr)= -4nr p [Field equation] (11.1) where p(r) is the pressure, p{r) the density, and U(r) the Newtonian gravitational potential. In any metric theory of gravity, it is simple to write down the equations corresponding to the first two of these three equations, because they follow from the Einstein Equivalence Principle (Chapter 2), which states that in local freely falling frames the nongravitational laws of physics are those of special relativity, Tfvv = 0, and p = p(p). Thus, we have in any basis, Tfvv = 0, p = p(p) (11.2) For a perfect fluid, T"v = {p + p)«"uv + pg"" (11.3) where we have lumped the internal energy pTl into p [compare Equation (3.71)]. It is useful to rewrite these equations in a form that parallels the first two parts of Equation (11.1). For a static, spherically symmetric spacetime, there exists a coordinate system in which the metric has the form ds2 = -e20ir)dt2 - TV{r)drdt + e2Mr)dr2 + e2mr\dQ2 + sin2 0 d(f>2) (11.4) For theories of gravity with a preferred frame, this coordinate system must be at rest in that frame. There still exists the freedom to change the t coordinate by the transformation t = t'-f(r) (11.5) where f(r) can be chosen to eliminate the off-diagonal term in the metric, namely f(r) = J ' m(r)e ~ 2o<r) dr (11.6) There is the further freedom to change the coordinate r by r = 9(r') (11.7) Structure and Motion of Compact Objects 259 If the radial coordinate is chosen so that fi(r) = 0, the coordinates are called "curvature coordinates;" in such coordinates, 2nr measures the proper circumference of circles of constant r. In general relativity, they are known as Schwarzschild coordinates. If the radial coordinate is chosen so that n(r) = A(r), the coordinates are called "isotropic coordinates." However, in two-tensor theories of gravity, such as those with a background flat metric q, these transformations are usually best carried out after the solution to the field equations has been obtained. The reason is that the above transformations will in general make the second tensor a complicated nondiagonal function of r, which may result in worse complications in the field equations than those introduced by starting with the general nondiagonal physical metric, Equation (11.4). For example, if the second tensor field is t\, the field equations may take their simplest form in coordinates in which ij = diag(-l,l,r 2 ,r 2 sin 2 0) (11.8) In such a coordinate system there is no freedom to alter <J>, A, T, or fx a priori. Now, for hydrostatic equilibrium, the equations of motion Tfvv = 0 may be written in the form where j runs over r, 6, (p. For spherical symmetry only the j = r component is nontrivial, and, using the fact that u = (e~0(r), 0,0,0), we obtain dp/dr = (p + p)d<b/dr, P = p{p) (11.10) Notice that in the Newtonian limit, p « p,$ =s U and we recover the
first two parts of Equation (11.1). Equation (11.10) is valid independently
of the theory of gravity. The field equations for <J>, A, *F, and fi will depend
upon the theory. In constructing a stellar model, boundary conditions
must be imposed. These are
dp/dr\r=0 = 0,
p{R) = 0,
e2A(r)
R = [stellar surface],
e2M
,.
Ci
(1LU)
Theory and Experiment in Gravitational Physics
260
The first of these conditions is a continuity condition for the matter, the
second defines the stellar surface and its radius R. The remaining four
are asymptotic boundary conditions on the metric functions [see Equation (5.6)]. They guarantee that in asymptotically Lorentz coordinates,
and in geometrized units (Gtoda}, = 1),
0oo -» - 1 + 2m/r,
gtJ - r\i}
(11.12)
and thus that the Kepler-measured mass of the star will be m.
Unfortunately, this exhausts the common features of the equations of
relativistic stellar structure, so we must now turn to specific theories.
General relativity
In curvature coordinates [*F(r) = /x(r) = 0], the field Equation
(5.14) takes the form
d/dr[r(l - e~ 2A )] = 8nr2p
(11-13)
with the solution
e 2A = ( l - 2 r n ( r ) / r r 1
(11.14)
where
, .
. rr 2
m(r) = 4JI
t par,
Jo
or
dm
= 4nr2 p
dr
......
(11.15)
and
dO
dr
m + 4nr3p
r(r 2m)
(11.16)
This equation together with Equations (11.10), (11.14), and (11.15), and
the boundary conditions, Equation (11.11) (with c 0 = c1 = 1), are sufficient
to calculate neutron-star models, given an equation ofstate. These equations are called the TOV (Tolman, Oppenheimer, and Volkoff) equations
for hydrostatic equilibrium. They form the foundation for the study
of relativistic stellar structure within general relativity. For reviews of
neutron-star structure, see Baym and Pethick (1979), Arnett and Bowers
(1977), and Hartle (1978).
Scalar-tensor theories
Using curvature coordinates [*P(r) = ft(r) = 0] and defining
e 2A = [1 - 2 m ( r ) / r ] - 1
(11.17)
Structure and Motion of Compact Objects
261
we can put the field equations for scalar-tensor theories, Equations (5.31)
and (5.33), into the form
2m
drr(r-
2m)(l
1.18)
[Note that the equations quoted in Rees, Ruffini, and Wheeler (1975,
p. 13) are in error.] The present value of G is related to the asymptotic
value of <j>, by [see Equation (5.38)]
l
(11.19)
For the special cases of Brans-Dicke theory (co = constant) and the
Variable-Mass Theory [co(</>) satisfies Equation (5.40)], it has been shown
that for values of co consistent with solar system experiments (i.e., a> ;> 500),
all features of neutron-star structure differ from those predicted by general
relativity only by relative corrections of O(l/co) (see Salmona, 1967;
Hillebrandt and Heintzmann, 1974; Bekenstein and Meisels, 1978).
Rosen's bimetric theory
In coordinates in which the flat background metric has the form
t, = diag( - Co \ el \ cl V 2 , ci V 2 sin2 9)
(11.20)
the field equations take the form
V 2 $+ |D~1e""2<I>"2A|VxP - 2»PV<I>|2 1/2 <I>+A+2 = 47tG(c0c1) e 2 l 2 2A y V A + \D- e- *- \\ ¥ 1/2 \ 1/2 ''D (p + - 24*VA|2 *+A 2 1 / 2 = -4nG(c0c1)ii2e">+A+2>lD1i2(p - p + 2(p - p), Y - 2*PVA) De 2 A ~ 2 " = 0 Theory and Experiment in Gravitational Physics 262 where D = 1 + *p2 e - 2 < I - 2 A , V = tr(d/dr), V2 = r ~ 2{d/dr)r2(d/dr) (11.22) Here, we see an example of the loss of freedom to vary the metric functions *P and pi a priori. Thus, for example, the substitutions *F = 0, ju s 0 (Schwarzschild form of the metric) do not lead to a solution of the equations for general matter distributions. However, the metric function *P alone is freely specifiable, for the following reason. If, instead of Equation (11.20) for IJ, we had chosen the equally valid flat metric whose line element is dsLt = - c o ' [ A + f(r)drf + c^[dr2 + r2(dd2 + sin 2 0# 2 )] where /(r) is an arbitrary function of r, then had changed coordinates to put this metric into the form of Equation (11.20), the result would be to change the function ¥ in g^ by an arbitrary amount. Thus, for example, we are free to choose *P = 0. This is consistent with the fact that *P = 0 is a solution of the fourth field equation, (11.21). The free choice of *F is part of the absolute, prior-geometric character of i\, and represents the freedom to "tip" the null cones of i; relative to those of g. Different choices of *F lead to physically different spacetimes (this point has been overlooked by most authors). The simple choice T s O leads to the field equations V20> = 47tG(coc1)1/2e*+3A(p + 3p), V2A = -47tG(c o c 1 ) 1/2 e* +3A (p - p), H=A (11.23) Henceforth, we shall adopt the choice T s O . The boundary conditions on <I> and A are given by$
0,
A
0
(11.24)
Notice that the matching of the tensors i\ and g to the external world
influences the structure of the star (violation of SEP) via the effective
gravitational constant G^QCJ) 1 ' 2 . We now recast the field equations into
the form
dfb/dr = Gom*(r)/r2,
dA/dr=-GomA(r)/r2
(11.25)
Structure and Motion of Compact Objects
263
where
G o = (c0Ci)1/2G = 1 [geometrized units],
= An £ e*+3A(p - p)r2 dr
(11.26)
Outside the star, r > R,
<D = _ MJr,
A = MJr
(11.27)
where M^ = m^R), MA = mA(R). In quasi-Cartesian coordinates, the
exterior metric then has the form
ds2 = - exp( - 2MJr) dt2 + exp(2MJr)(dx2 + dy2 + dz2)
(11.28)
A variety of numerical integrations of the field equations, (11.25), and
the hydrostatic equilibrium equations, (11.10), have been carried out using
various equations of state (see Rosen and Rosen, 1975; Caporaso and
Brecher, 1977; and Will and Eardley, 1977). Generally, neutron stars with
Kepler-measured masses M& much larger than those permitted by general
relativity are possible, with maximum masses ranging from ~8m o for
soft equations of state, to ~80m o for equations of state of the form
p = p - p0 for p > p0 ~ 1014 g cm" 3 .
NVs stratified theory
In coordinates in which
i/ = diag(-l,l,r 2 ,r 2 sin 2 0),
(11.29)
we have [see Section 5.6(g)-(i)]
e 2 * = /i(*X
V=-Kr,
e2A = e2" = f2(<t>),
X9 = K 0 = O
(11.30)
The field equation for Kr is
V2Kr - r-2Kr = -47t e (/ 2 // 1 D) 1 / 2 X r p
(11.31)
where D = 1 + K2(f1f2)~1. One immediate solution of this equation is
Kr = 0 (no tipping of null cones). Then, thefieldequation for (f> is given by
V20 = 27t(/ 1 / 3 ) 1 ' 2 [p(/' 1 // 1 ) - 3p(/'2//2)]
(11.32)
where f\ = dfjd<f>, f2 = df2/d<j>. The boundary condition on <p is
4> Tzz^t 0. Outside the star, <p is given by
<p(r)= -MJr
(11.33)
Theory and Experiment in Gravitational Physics
264
where
M, = 2n J* (fJiy'WJA)
- 3p(/'2//2)]r2 dr
(11.34)
Asymptotically, the functions fx and / 2 are assumed to have the forms
M4)
= c 0 - 2c<\> + O(4>2),
/ 2 (0) = c, + O(0)
(11.35)
In coordinates in which g^ is asymptotically Minkowskian, g00 then
has the form
goo=-l+2GoMJr
(11.36)
where
Go = C2C\I2CQ 3/2 = 1
[geometrized units],
Mikkelson (1977) has numerically integrated these equations using reasonable equations of state, after first assuming a specific form for the functions fi(<j>) and f2(4>), designed to yield agreement with general relativity
in the post-Newtonian limit. These forms were
with a being an adjustable parameter (note co = ct = c = 1). For a = 1,
the maximum Kepler-measured mass was ~1.4m o ; for a = 64, it was
~840m o ; and in the limit a-* oo, the maximum mass was unbounded.
The stiffer the equation of state used, the larger the maximum masses.
Thus, neutron-star models in alternative theories of gravity can be
very different from their counterparts in general relativity, the known
exceptions to this rule being scalar-tensor theories. In particular, the
maximum mass of a neutron star may be orders of magnitude larger than
that in general relativity.
11.2
The Structure and Existence of Black Holes
General relativity predicts the existence of black holes. Black
holes are the end products of catastrophic gravitational collapse in which
the collapsing matter crosses an event horizon, a surface whose radius
depends upon the mass of matter that has fallen across it, and which is a
one-way membrane for timelike or null world lines. Such world lines can
Structure and Motion of Compact Objects
265
cross the horizon moving inward but not outward. The interior of the
black hole is causally disconnected from the exterior spacetime. There is
now considerable evidence to support the claim that any gravitational
collapse situation, whether spherically symmetric or not, with zero net
charge and zero net angular momentum, results in a black hole, whose
metric (at late times after the black hole has become stationary) is the
Schwarzschild metric, given in Schwarzschild coordinates by
ds2 = - ( 1 - 2M/r)dt2 + (1 - 2M/r)-ldr2 + r2(d62 + sin 2 0# 2 ) (11.39)
(If the collapsing body has net rotation, the black hole is described by
the Kerr metric.) Much is now known about the theoretical properties
of black holes within general relativity, and there are strong candidates
for observed black holes in Cygnus XI and elsewhere. For reviews of
this subject see Giacconi and Ruffini (1978) and Hawking and Israel
(1979).
However, the existence of black holes is not an automatic byproduct
of curved spacetime. To be sure, curved spacetime is essential to the
existence of horizons as one-way membranes for the physical interactions,
but whether or not a horizon occurs depends crucially on the field equations that determine the curvature of spacetime. In the following examples,
we shall illustrate this point. Throughout this section, we restrict ourselves to nonrotating, spherically symmetric systems.
Scalar-tensor theories
As one might have expected, scalar-tensor theories, being in some
sense the least violent modification of general relativity, predict black
holes. However, what is unexpected is that they predict black holes whose
geometry is identical to the Schwarzschild geometry. The reason is that
the scalar field <>/ is a constant throughout the exterior of the horizon,
given by its asymptotic cosmological value (f>0. Thus, the vacuum field
equation, (5.31), for the metric is Einstein's vacuum field equation, and
the solution is the Schwarzschild solution. The scalar field has no effect
other than to determine the value of the gravitational constant. (This
result also holds for rotating and charged black holes.) In Brans-Dicke
theory, for instance, the most direct way to verify this is to use the vacuum
field equation for cp = cj) (j)0, \3g<P = 0, and to integrate the quantity
<pOg(p over the exterior of the horizon between two spacelike hypersurfaces at different values of coordinate time. After an integration by
parts, we obtain
J<P,*9'*(-9) ll2 d*x - I j(<p2y*dZx = 0
(11.40)
Theory and Experiment in Gravitational Physics
266
Now the surface integrals over the spacelike hypersurfaces cancel because
the situation is stationary; that over the hypersurface at infinity vanishes
because <p~r~1 asymptotically; and that over the horizon vanishes
because dE a is parallel to the generators of the horizon and is thus in a
direction generated by the symmetry transformations of the black hole
(Killing direction) whereas Op2)'" is orthogonal to that direction, since
the derivatives of q> along symmetry directions must vanish. Thus,
= 0
(11.41)
But cptX is spacelike, since cp is stationary, so (pAqy* > 0 everywhere, and
Equation (11.41) thus implies (px = 0. Further details and other arguments
can be found in Thorne and Dykla (1971), Hawking (1972), Bekenstein
(1972), and Bekenstein and Meisels (1978).
Rosen's bimetric theory
For the case *P = 0, the static spherically symmetric vacuum field
equations are [see Equation (11.23)]
fi = A,
V20> = V2A = 0
(11.42)
with solutions
<D = -MJr,
n = A = MJr
(11.43)
and
ds2 = - exp( - 2MJr) dt2 + exp(2MJr)(dx2 + dy2 + dz2)
(11.44)
There is no horizon in this spacetime, only a naked singularity at r = 0.
Thus, at least within the subset of vacuum spacetimes specified by *F = 0,
there are no black holes in Rosen's theory.
11.3
The Motion of Compact Objects:
A Modified EIH Formalism
In Chapter 6, we derived the n-body equations of motion for
massive, self-gravitating bodies within the parametrized post-Newtonian
(PPN) framework [see Equations (6.31)-(6.34)]. A key assumption that
went into that analysis was that the weak-field, slow-motion limit of gravitational theory applied everywhere, in the interiors of the bodies as well
as between them. This assumption restricted the applicability of the equations of motion to systems such as the solar system.
However, when dealing with a system such as the binary pulsar in which
there is a neutron star with a highly relativistic interior, one can no longer
Structure and Motion of Compact Objects
267
apply the assumptions of the post-Newtonian limit everywhere, except
possibly in the interbody region between the relativistic bodies. Instead,
one must employ a method for deriving equations of motion for compact
objects that, within a chosen theory of gravity, involves (a) solving the full,
relativistic equations for the regions inside and near each body, (b) solving
the post-Newtonian equations in the interbody region, and (c) matching
these solutions in an appropriate way in a "matching region" surrounding
each body. This matching presumably leads to constraints on the motions
of the bodies (as characterized by suitably denned centers of mass); these
constraints would be the sought after equations of motion. Such a procedure would constitute a generalization of the Einstein-Infeld-Hoffmann
(EIH) approach (see Einstein, Infeld, and Hoffmann, 1938).
Let us first ask what would be expected from such an approach within
general relativity. In the full post-Newtonian limit, we found that the
motion of post-Newtonian bodies is independent of their internal structure, i.e., there is no Nordtvedt effect. Each body moves on a geodesic of
the post-Newtonian interbody metric generated by the other bodies, with
proper allowance for post-Newtonian terms contributed by its own interbody field. This is the EIH result. It turns out however, that this conclusion is valid even when the bodies are highly compact (neutron stars or
black holes). The only restriction is that they be quasistatic, nearly spherical, and sufficiently small compared to their separations that tidal interactions may be neglected. The effects of rotation (Lense-Thirring effects)
are also neglected. This would be a bad approximation for a neutron star
about to spiral into a black hole, for example, but is a good approximation
for the binary pulsar (rpulsar/rorbit ^ 10"5).
Although this conclusion has not been proven rigorously, a strong
argument for its plausibility can be presented by considering in more detail the matching procedure discussed above. We first note that the solution for the relativistic structure and gravitational field of each body is
independent of the interbody gravitational field, since we can always
choose a coordinate system for each body that is freely falling and approximately Minkowskian in the matching region and in which the body is
at rest. Thus, there is no way for the external fields to influence the body
or its field, provided we can neglect tidal effects due to inhomogeneities
of the interbody field across the interior of the matching region. Only
the velocity and acceleration of the body are affected. Now, provided the
body is static and spherically symmetric to sufficient accuracy, its external
gravitational field is characterized only by its Kepler-measured mass m,
and is independent of its internal structure. Thus, the matching procedure
Theory and Experiment in Gravitational Physics
268
described above must yield the same result, whether the body is a black
hole of mass m or a post-Newtonian body of mass m. In the latter case,
the result is the EIH equations of motion (see Section 6.2), so it must be
valid in all cases. A slightly different way to see this is to note that because
the local field of the body in the freely falling frame is spherically symmetric, depends only on the constant mass m, and is unaffected by the
external geometry, the acceleration of the body in the freely falling frame
must vanish, so its trajectory must be a geodesic of some metric. The
metric to be used is a post-Newtonian interbody metric that includes
post-Newtonian terms contributed by the body itself, but that excludes
self-fields. This conclusion has been verified for nonrotating black holes
by D'Eath (1975), and for the Newtonian acceleration of post-postNewtonian bodies by Rudolph and Borner (1978). D'Eath (1975) gives a
detailed presentation of the matching procedure described above.
A key element of this derivation is the validity of the Strong Equivalence
Principle within general relativity (see Chapter 3 for discussion), which
guarantees that the structure of each body is independent of the surrounding gravitational environment. By contrast, most alternative theories of gravity possess additional gravitational fields, whose values in the
matching region can influence the structure of each body, and thereby
affect its motion. Consider as a simple example a theory with an additional scalar field (scalar-tensor theory). In the local freely falling coordinates, although the interbody metric is Minkowskian up to tidal terms,
the scalar field has a value <Ao(0- I n a solution for the structure of the
body, this boundary value of <j>0{i) will influence the resulting mass according to m = m((j)0). Thus, the asymptotic metric of the body in the
matching region may depend upon its internal structure via the dependence of m on <j)0 (essentially, the matching conditions will depend upon
m, dm/d(j),...). Furthermore, the acceleration of the body in the freely
falling frame need not be zero, as we saw in Sections 2.5 and 3.3. If the
energy of a body varies as a result of a variation in an external parameter,
we found, using cyclic gedanken experiments that assumed only conservation of energy, that [see Equation (3.80)]
a - a8eodesic ~ \EB(X, V) ~ (dm/dWt
(11.45)
Thus, the bodies need not follow geodesies of any metric, rather their
motion may depend strongly on their internal structure.
In practice, the matching procedure described above is very cumbersome (D'Eath, 1975). A simpler method, within general relativity, for
Structure and Motion of Compact Objects
269
obtaining the EIH equations of motion, is to treat each body as a "point"
mass of inertial mass ma and to solve Einstein's equations using a pointmass Lagrangian or stress-energy tensor, with proper care taken to neglect
"infinite" selffields.In the action for general relativity we thus write
JGR = (lenG)-1 JR(-g)l/2d*x
- ^ m f l jdxa,
(11.46)
where xa is proper time along the trajectory of the ath body. By solving
the field equations to post-Newtonian order, it is then possible to derive
straightforwardly from the matter action an n-body EIH Lagrangian in
the form
J EIH =
u
...xn,\u...
yn)dt
(11.47)
written purely in terms of the variables (xfl, vo) of the bodies. The result is
Equation (6.80) with the PPN parameters corresponding to general relativity. The n-body EIH equations of motion are then given by
-ft-a
dt dv'a
«-l,...,n
(11.48)
dx'a
In alternative theories of gravity, the only difference is the possible
dependence of the mass on the boundary values of the auxiliaryfields.In
the quasi-Newtonian limit (Sections 2.5 and 3.3) this was sufficient to
yield the complete quasi-Newtonian acceleration of composite bodies including modifications (Nordtvedt effect) due to their internal structure.
Thus, following the suggestion of Eardley (1975)1 we merely replace the
constant inertial mass ma in the matter action with the variable inertial
mass ma{\j/A), where \j/K represents the values of the external auxiliary
fields, evaluated at the center of the body (we neglect their variation across
the interior of the matching region), with infinite self-field contributions
excluded. The functional dependence of ma upon the variable i//A will
depend on the nature and structure of the body. Thus, we write
1 = JG ~ £ Jm-{^A[x.(Tj]} dxa
(11.49)
a
In varying the action with respect to thefieldsg^v and i^A the variation of
ma must then be taken into account. In the post-Newtonian limit, where
the fields i//A are expanded about asymptotic values i//1^ according to
1
Parts of this section are developed from unpublished notes by Douglas
Eardley.
Theory and Experiment in Gravitational Physics
<AA
270
= <AA * + <5<AA> it is generally sufficient to expand ma(i/jA) in the form
+ \ Z (5 W # k O ) # i ? V l M * B +
(11-50)
A,B
Thus, the final form of the metric and of the n-body Lagrangian will
depend on ma and on the parameters dmjdxj/^, and so on. We shall use
the term "sensitivity" to describe these parameters, since they measure the
sensitivity of the inertial mass to changes in the fields \j/ A. Thus, we shall
denote
s<A) = - 3(ln mJ/a^A31
["first sensitivity"]
AB
2
>)
3)
s<, >' = -d (lnm a )/#£ #B
["second sensitivity"]
(11.51)
and so on. The final result is a "modified EIH formalism."
By analogy with the PPN formalism, a general EIH formalism can be
constructed using arbitrary parameters whose values depend on the theory under study, and, in this case, on the nature of the bodies in the system.
However, to keep the resulting formalism simple, we shall make some
restrictions. First, we restrict attention to fully conservative theories of
gravity. Technically, this means any theory whose EIH Lagrangian is
post-Galilean invariant. Now, every Lagrangian-based metric theory of
gravity will possess an EIH Lagrangian (thus all the theories discussed
in Chapter 5 fall into this class), however not every theory is fully conservative. Only general relativity and scalar-tensor theories are automatically
fully conservative. Other theories can be fully conservative, in their postNewtonian limits, at least, only for special choices of adjustable constants
and cosmological matching parameters that make the PPN parameters
<*! and oe2 equal to zero (Rosen's bimetric theory with co = cu for example).
It is not known whether these choices are sufficient to guarantee that the
EIH Lagrangian also be post-Galilean invariant. Nonetheless, the experimental upper limits on the PPN parameters OL1 and <x2 (Chapter 8) obtained
from searches for post-Newtonian preferred-frame effects make it unlikely
that the analogous effects in the EIH formalism will be of much interest.
Therefore we shall adopt a fully conservative EIH formalism.
We shall also restrict attention to theories of gravity that have no
Whitehead term in the post-Newtonian limit (i.e., £ = 0). The experimental
constraints on £ (Chapter 8), from searches for galaxy-induced effects in
the solar system, likewise make the analogous effects in the EIH formalism
of little interest.
Structure and Motion of Compact Objects
271
Each body is characterized by an inertial mass ma, defined to be the
quantity that appears in the conservation laws for energy and momentum
that emerge from the EIH Lagrangian. We then write, for the metric,
valid in the interbody region and far from the system,
000 = - 1 + 2 £ «fl*ma|x - xa| ~ * + O(4),
a
9OJ= 0(3),
9u = ( l + 2 1 7>a\* ~ xa| -*) Su
(11.52)
where a* and y* are functions of the parameters of the theory and of the
structure of the ath body. For test-body geodesies in this metric, the quantities x*ma and ^a*wJ a are the Kepler-measured active gravitational
masses of the individual bodies and of the system as a whole. In general
relativity, a* = y* = 1.
To obtain the EIH Lagrangian, we first generalize the post-Newtonian
semiconservative n-body Lagrangian [Equation (6.80)] in a natural way,
to obtain
= - I ma{l - \^vl
- i^<2><] + \a,b
r
ab
a*b
X
h*J
flj(
flj]
(i i-53)
where nab = xjrab. The quantities s/«\ st, 9^,, ®ah, 9abc, <£ttb, and Sah
are functions of the parameters of the theory and of the structure of each
body, and satisfy
^ab
*(<.»)>
Wab (abY
«ab ~ *(ab),
In general relativity, all these parameters are unity. In the true postNewtonian limit of semiconservative theories (with t, 0), for structureless
masses (no self gravity), the parameters have the values [compare Equation (6.80)]
^ab = 7(4? + 3 + at - a2),
Sab= 1 + a2
(11.55)
In the fully conservative case, including contributions of the self-gravitational binding energies of the bodies, one can show to post-Newtonian
Theory and Experiment in Gravitational Physics
272
order, that
1,
9A = 1 + (4/? - y - 3)(QJma + Qb/mb),
3),
<fa6 = l
(11.56)
where Qa is the self-gravitational energy of the ath body.
We now impose post-Galilean invariance on the Lagrangian in Equation (11.53). We make a low-velocity Lorentz-transformation from (t,x)
to (T, £) coordinates, given by
x = { + (1 + |W 2 )WT + | ( £ w)w + O(4)
t = T(1 + W
4
+ fw ) + (1 + W)S
x &
w + O(5) x T
(11.57)
We required that L be invariant, modulo a divergence, i.e.,
L(l T) = L(x, f) dt/dt + # / d t
(11.58)
for some function ^. From the transformation Equation (11.57), we have
v = v + w jw2v v(v w) jw(v w),
r*1 = C, 1 [1 + i(w fi^)2 + w fi> n^,]
(11.59)
where v = d£/dz, and n^b = %abl£,ab. Substituting these results into Equations (11.53) and (11.58), and dropping constants and total time derivatives,
yields
L«,T)= - I ma{\ - \
a
2
+ va2(va w) + (v. w) 2 ]}
+i($ab + ^ab - T#ab)(w + 2va W) } (11.60) Thus the action is invariant if and only if ab = Furthermore, the scale of L and the constant term £ a ma are irrelevant, thus, we can always scale the values of ma so that .a?*,1' = 1, and choose Structure and Motion of Compact Objects 273 the constant to be £ a ma in terms of the rescaled mass. This merely guarantees that the inertial mass obtained from the Hamiltonian constructed from L agrees with that obtained from the equations of motion. Thus, the final form of the modified EIH Lagrangian is ad U)] yb - &Jya HJfo nj I (11.62) Since our ultimate goal is to apply this formalism to binary systems containing compact objects, such as the binary pulsar, let us now restrict attention to two-body systems. Denning & = <g12 and Si = @12, we obtain from L the two-body equations of motion [compare Equation (7.34)] r3 [ r r - 2vt v 2 ) - ^ fl)2 (v2 - v j x [(SF a2 = { 1 ^ 2 , x - > - x } (11.63) where a s dsjdt, x = x 2 x l5 r = |x|, n = x/r. It is possible to show straightforwardly from these equations that if we define ma=ma + \mav2a - \^mjnhjrah, a# b X = (mjXj + m2x2)/("'i + "»2) (11.64) then the "center of mass" X of the system is unaccelerated, i.e., 2 =0 (11.65) This agrees with the fully conservative nature of the EIH Lagrangian and justifies our identification of ma as the inertial rest mass of each body. If we now choose the center of mass to be at rest at the origin, X = X 2 0, then, to sufficient post-Newtonian accuracy we may write x 2 = [nti/m + O(2)]x (11.66) Theory and Experiment in Gravitational Physics 274 As in Section 7.3, we define y=v2-\u a = a 2 -a 1; m = ml + m 2 , n = m1m2/fn, dm = m2 mt (11.67) then the equations of motion for the relative orbit take the form dm r m f ( ) ^ J m (11.68) r In the Newtonian limit of the orbital motion, we have a=-m^x/r3 (11.69) with Keplerian orbit solutions x = p(l + ecos^>)~1(exCOS0 + e,,sin</>), r2 d4>ldt = hs CSmp)112, p = a{\ - e2), v = (&m/p)il2[-exsin<l> + ey(cos<p + e)\ 2 (Pb/2n) = a3/&m (11.70) where a, p, and e are the semi-major axis, semi-latus rectum, and eccentricity, respectively; h is the angular momentum per unit mass; and P b is the orbital period. In this solution we have chosen the x direction to be in the direction of the periastron. The post-Newtonian terms in Equation (11.68) can then be viewed as perturbations of the Keplerian orbit. Using the method of perturbations of osculating orbital elements outlined in Section 7.3, we find that the periastron advance is given by 1 (11.71) where 0> q)(% .). ^<§2 _ ^(/nj:^ 2 1 1 + w 2 ® 122 )/m (11.72) This is the only secular perturbation produced by the post-Newtonian terms in Equation (11.68). In the PPN limit, this result agrees with Equation (7.54), for fully conservative theories (with £ = 0). In obtaining the modified EIH equations of motion, we assumed that the field equations obtained from Equation (11.49) were solved for the interbody gravitational fields through post-Newtonian order. However, Structure and Motion of Compact Objects 275 those equations can also be solved for the gravitational-radiation fields in the far zone, and for the rate of energy loss via gravitational radiation. The method parallels that presented in Chapter 10, except that now, the self-gravitational corrections in the sources of the fields ip (ifr may include the metric itself) are automatically taken into account to all orders via the sensitivities s [see Equation (11.51)]. The only terms that we need to retain in order to determine the lowest order quadrupole and dipole contributions to the energy loss rate are [compare Equations (10.104) and (10.105)] Aeiectric ~ 4(m/R) < - [1 + (s) + ] [monopole and quadrupole] }, [dipole] ] [dipole] + [1 + (s) + ] 1 [quadrupole] (11.73) The only other differences from the post-Newtonian method described in Section 10.3 are the use of the conservation laws and Newtonian equations of motion obtained from the modified EIH Lagrangian. We shall ultimately be interested primarily in the energy loss due to dipole gravitational radiation, so it is useful to rewrite the dipole portion of Equation (10.84) using terms more suited to the modified EIH formalism, namely where S is related to the difference in sensitivities between the two bodies. As an illustration of these methods, we shall again focus on specific theories: general relativity, Brans-Dicke theory, and Rosen's bimetric theory. General relativity As we have already seen, the EIH equations of motion for compact objects within general relativity are identical to those of the full postNewtonian limit. In other words, &ab = 0&ab = 3)abc = 1, independently of the nature of the bodies. Furthermore, the gravitational radiation produced by the orbital motion is dominated by quadrupole radiation (no Theory and Experiment in Gravitational Physics 276 dipole radiation), and the energy loss rate is the same as in the pure postNewtonian case, obeying the Peters-Mathews formula, Equation (10.80) with Kj = 12, K2 = 11, KD = 0. (The same caveats regarding the rigor with which this result has been established apply here as in Section 10.3.) Brans-Dicke theory The modified EIH formalism was first developed by Eardley (1975) for application to Brans-Dicke theory. It makes use of the fact that only the scalar field (f> produces an external influence on the structure of each compact body via its boundary values in the matching region. In fact the boundary value of <t> is related to the local value of the gravitational constant as felt by the compact body by (j) = G'1(4 + 2o})/(3 + 2co) (11.75) Hence, we shall regard the inertial mass ma of each body as being a function of G, or more specifically, of In G. Then, if post-Newtonian, interbody gravitational fields lead to variations in 4> away from its asymptotic (cosmological) value <p0 according to 4> = 4>o + <?>tnen w e m a v write (1L76) Defining the sensitivities sa and s'a of the inertial mass of body a to changes in the local value of G, following Equation (11.51), by s'a=-[d2(\nma)/d(\nG)2]0 (11.77) and dropping the 0 subscript, we obtain ma(4) = mil + sa{cpl4>Q) - Wa - st + sa)(cp/ct>0)2 + O(W0o) 3 ] (11.78) The action for Brans-Dicke theory is then written l z a (11.79) where the integrals in the matter action are to be taken along the trajectories of each particle, and where infinite "self-fields" are to be ignored. Structure and Motion of Compact Objects 277 The resulting field equations are (compare Section 5.3) Dg<l> = Y ^ [T - 24>dT/8(j>-] (11.80) where T*v = (-gy112 I m s ( ^ « ' ( u 0 ) - ^ 3 ( x - xfl), dT/d<l> = - ( - 9 ) - 1 / 2 2 (5ma(</.)/^)(u°)-^3(x - x j (11.81) The equations of motion take the form T?vv - (dT/dtfrW = 0 (11.82) Performing a post-Newtonian expansion following the method outlined in Chapter 5, we obtain to lowest order <p/4>0 = (2 + a ) - 1 X m a (l - lsa)/ra, a g00 = - 1 + 2 X (ma/ra)[l - s./(2 + <»)] + O(4), a 9iJ = dJl + 2y X (ma/ra)[l + s./(l + ai)]J (11.83) where rfl = |x xo|, y = (1 + a;)/(2 + co), and we have chosen units in which G s l . Notice that the active gravitational mass as measured by test-body Keplerian orbits far from each body is given by W , = oi*ma = ma[l - sj(2 + <»)] (11.84) In the full post-Newtonian limit, where sa =* QJma, this agrees with Equation (6.49). If we define a "scalar mass" (ms)a by K ) a = i ( 2 + o J )- 1 m a (l-2s a ) (11.85) so that ( ) (11.86) the metric can be written S'oo = ~ 1 + 2 X a gtj = dv {l + 2 I [(mA)a - 2(ms)J/r.J (11.87) Theory and Experiment in Gravitational Physics 278 From the active mass and the scalar mass, it is useful to define a "tensor mass" (mT)o, (mT)a = (mA)a - (ms)a = ( | ^ ) *. (11.88) It can then be shown (Lee, 1974) that the tensor mass (mT)a is associated with a conservation law of the form U-gMT**+ ni* = 0 (11.89) where V is a symmetric stress-energy "pseudotensor" given by Lee (1974). This result is consistent with the identification of ma as the inertial mass of the ath body. The full post-Newtonian solution for g^ and cp may now be obtained, and the results substituted into the matter action, which, for the ath body takes the form I.= - jma(cj>)dt(-g00 - 2g0Jvi - g,/^1'2 (11.90) To obtain an n-body action in the form of Equation (11.47), we first make the gravitational terms in / manifestly symmetric under interchange of all pairs of particles, then take one of each such term generated in la, and sum over a. The resulting n-body Lagrangian then has the form of Equation (11.62) with <§ab = 1 - (2 + coy ^s. + sb- 2Sasb), 1 ®ab = 4(2y + 1) + i ( 2 + oi)- (so + sb- 2sasb), 9abc = 1 - (2 + Co)"x(2sfl + sb + se) + (2 + co)- 2 [(l - 4sa)sksc + (5 + 2(o)sB{sb + sc) - (s'a - s2a)(l - 2sb)(l - 2s c )] (11.91) The quasi-Newtonian equations of motion obtained from the EIH Lagrangian are K = ~ I K x X ) [ l - (2 + coy l(sa + sb- 2Sasby] (11.92) b*a In the full post-Newtonian limit, the product term s^ may be neglected, and the acceleration may be written a. a -[(mp)./mj £ {m^Jrl (11.91) b*a where (mA)b is given by Equation (11.84) and where W . = m.[l - sj(2 + co)-] (11.92) Structure and Motion of Compact Objects 279 in agreement with our results of Section 6.2, Equation (6.49). However, if the bodies are sufficiently compact that sa ~ sb ~ 1, then because of the product term s^,,, it is impossible to describe the quasi-Newtonian equations simply in terms of active and passive masses of individual bodies. Roughly speaking, the sensitivity s ~ [self-gravitational binding energy]/[mass], so s e ~ 10~10, s o ~ 1(T6, swhitedwarf ~ 10~3. For neutron stars, whose equation of state is of the form p = p(p), a model is uniquely determined (for a given value of <w) by the local value of G and by the central density pc or the total baryon number JV. Now, the sensitivity s is to be computed holding N fixed; it can then be shown that fdlnm\ (dlnm\ dlnN\ /SlnnA fdlnN\ For fixed equation of state and fixed central density, a simple scaling argument reveals that m and N scale as G~3/2, so Note that (d In m/d In N)G is the injection energy per baryon. Then it can be shown that S'NS = ( ! - sNS)(dsNS/d In m) G (11.95) Equations (11.94) and (11.95) actually hold in any theory of gravity in which the local structure depends upon a single external parameter whose role is that of a gravitational "constant." For a variety of neutron star models, Eardley (1975) has shown that s ranges from s ^ 0.01 for m = 0.13mo to s ^ 0.39 for m = 1.41mo. For black holes, we have seen (in Section 10.2) that the scalar field is constant in the exterior of the hole, thus from Equation (11.83) sBH = | ; equivalently mBH scales as G~1/2. Note that the quasi-Newtonian equation of motion for a test black hole and a companion (mBH « mc) is thus given, from Equation (11.92), by = - [(3 + 2cu)/(4 + 2o))]mcx/r3 (11.96) therefore the Kepler-measured mass experienced by a test black hole is the tensor mass mT of the companion (Hawking, 1972). The energy loss rate due to dipole gravitational radiation can be computed simply in this formalism (the PM radiation can also be calculated, but the result is not particularly illuminating). The wave-zone form of cp Theory and Experiment in Gravitational Physics 280 is given, from Equations (11.78), (11.80), and (11.81) by <P = y ^ ^ | E ma(l - 2s a )[l + ii va + O(m/r)]| (11.97) For a binary system, modulo constants, we obtain 9 = - [ 4 / ( 3 + 2co)]K~ V<»(n v) (11.98) where ©ss 2 - S l (11.99) Then, following the method of Section 10.3, we find that the rate of energy loss is given by Equation (11.74) with KD = 2/(2 + co). Rosen's bimetric theory In Rosen's theory, the flat background metric ij can influence the structure of a compact object, in spite of the fact that it is a nondynamical field. In a coordinate system in which the physical metric g is asymptotically Minkowski, and thus in which i\ has the form i; = diag(-Co 1 ,cr 1 ,cr 1 r 2 ,c 1 -V 2 sin 2 0) (11.100) we found in Section 11.1 that the equations of structure for static stars depend only on the quantity (CQCJ) 1 ' 2 . This quantity, as we discovered from the post-Newtonian limit of Rosen's theory (Equation 5.70), plays the role of the local gravitational constant G. Let us now adopt the fully conservative version of the theory, i.e., the version in which the cosmological values of the matching parameters are c 0 = ct = 1. Consider a body moving with velocity v through some given postNewtonian interbody field. In asymptotically Minkowski coordinates, the metric has the form ds2 = -e2*'dt2 + ix'jdxJdt + e2A'(dx2 + dy2 + dz2) (11.101) where <t7 ~ O(2), A7 ~ O(2), x'j ~ O(3), with the superscript / denoting interbody values. Now to determine the structure of the body, we must transform to coordinates x 4 in which g has the Minkowski form in the matching region. But in this coordinate system, the components of i; are essentially the inverse of those of g (we ignore variations of the components across the interior of the matching region); to post-Newtonian order we have Structure and Motion of Compact Objects 281 We must also transform to coordinates xf in which the body is at rest. For |v| ~ O(l), we have, using Equation (4.49), ri-oo = - e - 2 * ' [ l - 2v2(& - A') + 2X' v], i? 8 j=-z} + 2i;/<&/-Af), mi= e-^'dtj + 2viVj(^ - A') - Iv^'n (11.103) Now the nondiagonal components of nm are of O(3) or O(4), and by the nature of the local field equations, (11.21), for static spherically symmetric bodies, they contribute only quadratically, i.e., at O(6) or higher. Thus, to O(4) we can determine the local values of c 0 and cx by (cokcai = -tooo)' 1 = e2<D*[l + 2» 2 (* / - A1) - 2 Z J v], (11.104) (cxkoa. = (iriu)-1 = e 2A '[l - iv2(<D' - A') + h' v] Thus, the local structure to O(4) is determined by the "local value of G," given by G = (coCi)1'2 = e* +A [l + f» 2 (* - A) - fv z ] (11.105) where we have dropped the superscript /. If we view the inertial mass ma as a function of the interbody values of the metric coefficients (and of the velocity), then we may write ma{gj = m.{\- sfl[(D + A + §i>2(«> - A) - | v Z ] - Wa ~ s2a)(® + A)2 + } (H-106) where sa and s'a are given by Equation (11.77). The action for the theory can then be written (see Section 5.5) .igjdr. (H-107) where we again ignore infinite "self-field" contributions to ma. In coordinates in which 17 = diag(-1,1,1,1), the field equations are D ^ v - g'^g^yg^s = - IGnig/r,)1'2^ - k, v T) (11.108) where 1 Tfov, * (0) + 1 (1). 112 = (-g)' I M . t e ^ u 0 ) - 1 ^ - xfl), a Tft = -2{-g)-112 E (dmjdg^iu0)-^^ - xa) (11.109) Theory and Experiment in Gravitational Physics 282 A post-Newtonian expansion using the methods of Chapter 5 yields g00 = - 1 + 2 £ mjra + O(4), gu = 6U [l + 2 I m.(l - fsj/r.j (11.110) hence a? = l, 7fl* = l - | s f l (11.111) We note that, unlike the case in Brans-Dicke theory, the active mass as measured by test-body Keplerian orbits is equal to the inertial mass ma. The full post-Newtonian metric can now be obtained (the function A must also be determined to O(4) for use in ma), and the results for it and for mjig^) substituted into the matter action a(gjdxa (11.112) The resulting n-body Lagrangian is of the EIH form, with Values of sa and s^ for neutron stars range from sa m 0.05, s'a ss 0.07 for ma =i 0.4 m o to sa ^ 0.6, s'a ^ 0.2 for m =: 12 m o (Will and Eardley, 1977). Calculation of the dipole gravitational radiation energy loss rate proceeds as in Section 10.3, but using Equations (11.106), (11.108), and (11.109), with the result given by Equation (11.74), with KD = -20/3 as before, and ® = s2 s1. 12 The Binary Pulsar The summer of 1974 was an eventful one for Joseph Taylor and Russell Hulse. Using the Arecibo radio telescope in Puerto Rico, they had spent the time engaged in a systematic survey for new pulsars. During that survey, they detected 50 pulsars, of which 40 were not previously known, and made a variety of observations, including measurements of their pulse periods to an accuracy of one microsecond. But one of these pulsars, denoted PSR 1913 + 16, was peculiar: besides having a pulsation period of 59 ms - shorter than that of any known pulsar except the one in the Crab Nebula - it also defied any attempts to measure its period to ± 1 us, by making "apparent period changes of up to 80 fis from day to day, and sometimes by as much as 8 us over 5 minutes" (Hulse and Taylor, 1975). Such behavior is uncharacteristic of pulsars, and Hulse and Taylor rapidly concluded that the observed period changes were the result of Doppler shifts due to orbital motion of the pulsar about a companion. By the end of September, 1974, Hulse and Taylor had obtained an accurate "velocity curve" of this "single line spectroscopic binary." The velocity curve was a plot of apparent period of the pulsar as a function of time. By a detailed fit of this curve to a Keplerian two-body orbit, they obtained the following elements of the orbit of the system: Ku the semiamplitude of the variation of the radial velocity of the pulsar; Pb, the period of the binary orbit; e, the eccentricity of the orbit; <a the longitude of periastron at a chosen epoch (September, 1974); ax sin i, the projected semi-major axis of the pulsar orbit, where i is the inclination of the orbit relative to the plane of the sky; and /j = (m2 sin i)3l{ml + m2)2, the mass function, where mx and m2 are the mass of the pulsar and companion. In addition, they obtained the "rest" period Pp of the pulsar, corrected for orbital Doppler shifts at a chosen Theory and Experiment in Gravitational Physics 284 epoch. The results are shown in the first column of Table 12.1 (Hulse and Taylor, 1975). However, at the end of September 1974, the observers switched to an observation technique that yielded vastly improved accuracy (Taylor et al., 1976). That technique measures the absolute arrival times of pulses (as opposed to the period, or the difference between adjacent pulses) and compares those times to arrival times predicted using the best available pulsar and orbit parameters. The parameters are then improved by means of a least-squares analysis of the arrival-time residuals. With this method, it proved possible to keep track of the precise phase of the pulsar over intervals as long as six months between observations. This was partially responsible for the improvement in accuracy. The results of this analysis using data up to August 1980 are shown in column 2 of Table 12.1 (Taylor, 1980). The discovery of PSR 1913 + 16 caused considerable excitement in the relativity community (to say nothing of the editorial offices of the Astrophysical Journal Letters), because it was realized that the system could provide a new laboratory for studying relativistic gravity. Post-Newtonian orbital effects would have magnitudes of order v2 ~ K\ ~ 5 x 10"7, m/r ~ fxlax sin i ~ 3 x 10~7, a factor often larger than the corresponding quantities for Mercury, and the shortness of the orbital period (~ 8 hours) would amplify any secular effect such as the periastron shift. This expectation was confirmed by the announcement in December, 1974 (Taylor, 1975) that the periastron shift had been measured to be 4.0° ± 1.5° yr" 1 (compare with Mercury!). Moreover, the system appeared to be a "clean" laboratory, unaffected by complex astrophysical processes such as mass transfer. The pulsar radio signal was never eclipsed by the companion, placing limits on the geometrical size of the companion, and the dispersion of the pulsed radio signal showed little change over an orbit, indicating an absence of dense plasma in the system, as would occur if there were mass transfer from the companion onto the pulsar. These data effectively ruled out a main-sequence star as a companion: although such a star could conceivably fit the geometrical constraints placed by the eclipse and dispersion measurements, it would produce an enormous periastron shift (>5000° yr" 1 ) generated by tidal deformations due to the pulsar's gravitational field (Masters and Roberts, 1975 and Webbink, 1975). Another suggested companion was a helium main-sequence star, which could accommodate the geometrical and periastron-shift constraints. Estimates of the distance to the pulsar (5 kpc: Hulse and Taylor, 1975) and extinction along the line of sight (~ 3.3 mag: Davidsen, et al., 1975) indicated that such Table 12.1. Measured parameters of the binary pulsar" Parameter Symbol (units) Right ascension (1950.0) Declination (1950.0) Pulse period Derivative of period Velocity curve half-amplitude Projected semi-major axis Orbital eccentricity Orbital period Mass function Longitude of periastron (9/74) Periastron advance rate Red-shift-Doppler parameter Sine of inclination angle Derivative of orbital period Reference a 5 PP(s) P P (ss- x ) K^kms" 1 ) ay sin i (cm) e Pb(s) fi(fnQ) CO cb (deg y r " l ) nsinS)i P^ss" 1 ) Value from period data (summer, 1974) Value from arrival-time data (9/74-8/80) 19h13m13s ± 4s +16°00'24" + 60" 0.059030 + 1 <KT12 199 ± 5 (6.96 + 0.14) x 1010 0.615 + 0.010 27908 + 7 0.13 + 0.01 179° ± 1° Hulse and Taylor (1975) 19h13m12'469 + K014 16°O1'O8'.'15 + O'.'2O 0.0590299952695 + 8 (8.636 ± 0.010) x 10" 1 8 *** (7.0208 + 0.0012) x 1010 0.617138 + 8 27906.98157 ± 6 *** 178.867° + 0.002° 4.226 + 0.001 0.0044 + 0.0003 <0.96 (-2.1+0.4) x HT 1 2 Taylor (1980) " The entry of a dash (-) denotes that a determination of the parameter was beyond the accuracy of the data; the entry (***) denotes that an accurate value of the parameter is not needed. Theory and Experiment in Gravitational Physics 286 a helium star would have an apparent magnitude mR ~ 21. A number of searches have failed to detect any such object within an error circle of radius 0'.'5 around the radio pulsar position derived from the analysis of the arrival-time data (Kristian et al., 1976; Shao and Liller, 1978; Crane et al., 1979; and Elliott et al., 1980). Other possible companions to the pulsar are the condensed stellar objects: white dwarf, neutron star, or black hole. None of these is expected to be observable optically, and there is no evidence for a second (companion) pulsar in the system. Attempts to delineate further the nature of the companion involved constructing scenarios for the formation and evolution of the system. The favored scenario appears to be evolution from an x-ray binary phase whose end product is two neutron stars (Flannery and van den Heuvel 1975, Webbink 1975, and Smarr and Blandford 1976). However alternative scenarios have been constructed that lead to white dwarf companions (Van Horn et al. 1975, Smarr and Blandford 1976), black hole companions (Webbink 1975, Bisnovatyi-Kogan and Komberg 1976, Smarr and Blandford 1976) and helium star companions (Webbink 1975, Smarr and Blandford 1976). As we shall see, the nature of the companion is crucial for discussion of various relativistic and astrophysical effects in the system. One of the most important of these effects is the emission of gravitational radiation by the system, and the consequent damping of the orbit (Wagoner, 1975). The observable effect of this damping is a secular change in the period of the orbit. However, the timescale for this change, according to general relativity, is so long (~ 109 yr) that it was thought that 10 to 15 years of arrival-time data would be needed to detect it. However, with improved data acquisition equipment and continued ability to "keep in phase" with the pulsar, Taylor and his collaborators surpassed all expectations, and announced in December 1978 a measurement of the rate of change of the orbital period in an amount consistent with the prediction of gravitational radiation damping in general relativity (Taylor et al. 1979, Taylor and McCulloch 1980, Taylor 1980). This chapter presents a detailed discussion of the confrontation between gravitation theory and the binary pulsar. In Section 12.1, we develop an arrival-time formalism analogous to that used by the observers to analyze the data from the binary pulsar, and we discuss the important relativistic gravitational effects (and some competing nonrelativistic effects) in the system. We encounter a new and unexpected role for relativistic gravitational theory: that of a practical, quantitative tool for measuring astrophysical parameters (such as the mass of a pulsar). In Section 12.2, we use general relativity to interpret the data from the system. In Section 12.3, we Binary Pulsar 287 interpret the data using alternative theories of gravitation, and discover that one theory, Rosen's bimetric theory, faces a killing test. In fact, we conjecture that for a wide class of metric theories of gravity, the binary pulsar provides the ultimate test of relativistic gravity. 12.1 Arrival-Time Analysis for the Binary Pulsar Because the pulsar is the only object seen to date in the system, the analysis of its radio signal is equivalent to that of optical stellar systems in which spectral lines from only one of the members are observed. Such systems are known as "single-line spectroscopic binaries," and standard methods exist for analyzing them. However, there are important differences in the binary pulsar, including the possibility of large relativistic effects, and the ability to measure directly the arrival times of individual pulses, instead of the pulse period. For this reason it is worthwhile to develop a "single-line spectroscopic binary" arrival-time analysis tailored to systems like the binary pulsar. Such an analysis was first carried out in detail by Blandford and Teukolsky (1976) and extended by Epstein (1977) (see also Wheeler, 1975). We begin by setting up a suitable coordinate system. We choose quasiCartesian coordinates (t, x) in which the physical metric is of post-Newtonian order everywhere except possibly in the neighborhood of the pulsar and its companion, and is asymptotically flat. The origin of the coordinate system coincides with a suitably chosen "center of mass" of the binary system. The "reference plane" (Figure 12.1) is denned to be a plane perpendicular to the line of sight from the Earth to the pulsar (plane of the sky), passing through the origin. The "reference direction" is the direction in the reference plane from the origin to the north celestial pole. At any instant, the orbit of each member of the binary system is tangent to a Keplerian ellipse ["osculating" orbit; see Smart (1953) for discussion of these concepts and definitions]. This ellipse lies in a plane that intersects the reference plane along a line (line of ascending nodes) at an angle Q (angle of nodes) from the reference direction. The orbital plane is inclined at an angle i from the reference plane. The periastron of the osculating orbit of the pulsar occurs at an angle a> from the line of nodes, measured in the orbital plane. The other elements of the osculating relative orbit are the semi-major axis a, the eccentricity e, and the time of periastron passage T o . Then the instantaneous relative coordinate position x = x2 x t (1 = [pulsar], 2 = [companion]) is given by x = -a[(cos£ - e)eP + (1 - e2)1'2sin£§Q] (12.1) Theory and Experiment in Gravitational Physics 288 To Observer Figure 12.1. Geometry and orbit elements for the binary pulsar. where eP is a unit vector in the direction of the periastron of the pulsar, and eQ is a unit vector at right angles to this in the orbital plane (measured in the direction of motion of the pulsar). The quantity E is the eccentric anomaly, related to coordinate time t by E - e sin E - {2n/Pb){t - To) (12.2) where Pb is the binary orbit period. (For the purposes of this arrival-time analysis, it is more convenient to use E than the true anomaly 4>.) The relative separation is given by r = |x| = a(\ ecos£) (12.3) By solving the quasi-Newtonian limit of the modified EIH equations of motion (See Section 11.3), taking into account the modifications due to the self-gravity of the pulsar and its companion, we find [from Equation (11.70)] PJ2n = (12.4) where m = mt + m2 is the sum of the inertial masses of the bodies. To quasi-Newtonian order, the center of mass [Equation (11.64)] may be Binary Pulsar 289 chosen to be at rest at the origin, i.e., X = m~ 1 (m 1 x 1 +m 2 x 2 )s0 (12.5) then Xj = (m2/m)x, x2 = (m1/m)x (12.6) Now, any perturbation of the orbit, whether relativistic or not, is to be viewed as causing changes in the orbit elements Q, i, a>, a, and e of the osculating Keplerian orbit; given a set of values of these elements at any instant, Equations (12.1), (12.2), and (12.6) define the coordinate locations of the two bodies. The changes in the osculating elements produced by perturbations can be either periodic or secular. We next consider the emission of the radio signals by the pulsar. Let x be proper time as measured by a hypothetical clock in an inertial frame on the surface of the pulsar. The time of emission of the Nth pulse is given in terms of the rotation frequency v of the pulsar by N = No + vr + ivt 2 + £vt3 + (12.7) where N o is an arbitrary integer constant, and v = dv/di\z=0, v = d2v/dt2\t=0. We shall ignore the possibility of discontinuous jumps ("glitches") in the frequency of the pulsar. We ultimately wish to determine the arrival time of the Nth pulse on Earth. Outside the pulsar and its companion, the metric in our chosen coordinate system is given by Equation (11.52), 0OO = - 1 + 2 X «;»./|x - x a(0| + 0(4), <J=1,2 goj = O(3), 2 X y*mj\x - xa{t)\ + O(4)) (12.8) where ma is the inertial mass of the ath body, and a* and y* are factors that take into account the possibility of self-gravitational corrections to the "gravitational" masses if any of the bodies are compact (see Section 11.3 for discussion). Because we are interested in the propagation of the pulsar signal away from the system, we shall ignore the possibility of large, beyond-post-Newtonian corrections to the metric in the close neighborhood of the pulsar and the companion. The main result of such corrections will be either constant additive terms in the arrival-time formula equation, (12.7), that can be absorbed into the arbitrary value of No, or constant multiplicative factors (such as the red shift at the surface of the neutron Theory and Experiment in Gravitational Physics 290 star) that can be absorbed into the unknown intrinsic value of v. Modulo such factors, proper time x at the pulsar's point of emission can be related to coordinate time t by dx = dt[\ - <x%m2jr - \v\ + O(4)] (12.9) where we have dropped the constant contribution of the pulsar's gravitational potential a}'m1/|xem x t |, and where we have ignored the difference in the potential and the velocity between the emission point and the center x t of the pulsar. The two correction terms in Equation (12.9) are the gravitational red shift and the second-order Doppler shift. We can rewrite Equation (12.9) using Equations (12.1) and (12.2), which yield v\ = <g(ml/m)(2/r - I/a) (12.10) with the result (modulo constants) dx/dt = 1 - afmjr - ^m\jmr (12.11) Using Equations (12.2) and (12.3) we may integrate this equation to obtain T = t - (m2/a)(<x£ + gm2/m)(PJ2n)e sin E (12.12) Although the constants that have been dropped in integrating Equation (12.11) may actually undergo secular or periodic variations in time due to orbital perturbations and other effects (such as in a), the correction term in Equation (12.12) is already sufficiently small that such variations will have negligible effect. Now, the pulsar signal travels along a null geodesic. We can therefore use the method in Sections 6.1, 7.1, and 7.2 to calculate the coordinate time taken for the signal to travel from the pulsar to the solar system barycenter x 0 , with the result tarr ~ * = |*o('arr) ~ * l W | + (a? + yf)m2 ln{2ro(tm)/[r(t) + x(t) i]} (12.13) where r0 = |xo|, n = xo/ro, and where we have used the fact that r0 » r. The second term in Equation (12.13) is the time delay of the pulsar signal in the gravitational field of the companion; the time delay due to the pulsar's field is constant to the required accuracy, and has been dropped. Our ultimate goal is to express the timing formula equation, (12.7), in terms of the arrival time farr. In practice, one must take into account the fact that the measured arrival time is that at the Earth and not at the barycenter of the solar system, and will therefore be affected by the Earth's position in its orbit and by its own gravitational red shift and Binary Pulsar 291 Doppler-shift corrections. In fact, it is the effect of the Earth's orbital position on the arrival times that permits accurate determinations of the pulsar position on the sky. It is also necessary to take into account the effects of interstellar dispersion on the radio signal. These effects can be handled in a standard manner [see Blandford and Teukolsky (1976), for example], and will not be treated here. Now, because r 0 » r, we may write \*o(tm) - XiW| = ro{tm) - x,(t) n + O(r/r 0 ) (12.14) Combining Equations (12.13) and (12.14), and using the resulting formula to express xx(r) in terms of x 1 (t arr ) to the required post-Newtonian order, we obtain t = *arr ~ ^0 + *l(ttn ~ r0) fi + (*i(t m - r0) fi)(Xl(tarr - r0) ft) + [O(3)t arr ] (12.15) where the time-delay term is [O(3)t arr ]. We now choose the constant in Equation (12.2) so that E - e sin E = (2n/Ph){tMt - r0) (12.16) then, combining Equations (12.12), (12.15), and (12.16), substituting Equations (12.1) and (12.6), and noting from Figure 12.1 that eP n = sinisintu, eQ n = sinicosco (12.17) we find, modulo constants, T = tatI - s?(cosE - e) - (@ + ^ ) s i n £ - (2n/Pb)(l - e cos E)~ l{d sin E-36 cos £)[j/(cos E - e)+& sin £ ] + [O(3)t arr ] (12.18) where s4 = at sin i sin co, 0$ = (1 e 2 ) 1/2 a t sin i cos co,
V = (w|/ma 1 )(a? + &m2/m)(PJ2n)e
(12.19)
a, = (m2fm)a
(12.20)
with
The timing formula then takes the form
N = N0 + vtarr - va/(cos£ - e) - \{08 + ^sinE
-v(2n/Pb){l-ecosE)-i(s/[email protected])[#?(cosE-e)[email protected]'}
- e) + & sin E] + |vfa3rr +
(12.21)
Theory and Experiment in Gravitational Physics
292
The quantity N is to be regarded as a function of the time tarr and of
the parameters No, v, v, v, alsin/, a>, e, P b , tQ, c£. From an initial guess
for the values of these parameters a prediction for the arrival time of a
given N is made. The difference between the predicted arrival time and
the observed arrival time is used to correct the parameters using the
method of least squares. Possible variations with time of the parameters
resulting from perturbations of the system can also be determined, for
example, by substituting
CO-KO + <bt + ,
e -* e + et + ,
P b - + P b + $P b t + --- (12.22) and so on, into Equation (12.21). (The factor \ in the formula for P b comes from the formal definition of P b in terms of osculating elements.) We now turn to a discussion of the important measurable parameters and their interpretation. (a) The pulsar period The terms linear, quadratic, and cubic in tarr in the timing formula, Equation (12.21), determine the effective pulse period (at a chosen epoch) and its derivatives. The results of least-squares fits using data up to August 1980 were (Table 12.1) p p = v -» = 0.0590299952695 ± 8 s, P p = -vv~2 = (8.636 ± 0.010) x 1(T 18 s s" 1 (12.23) where the epoch was September, 1974. No determination of P p has been possible to date except for the crude limit set by the fact that P p has not changed by more than the experimental error over timescales of one year, thus, |Pp|<6 x 10"28ss-2 (12.24) Despite the fact that the pulsar's period is the second shortest known, its "spin-down rate," P p , is anomalously small, i.e., Pp/Pp = (4.617 ± 0.005) x 10" 9 yr" 1 . The most popular explanation for this is that the pulsar has a weak magnetic field, leading to small braking torques caused by magnetic Lorentz forces, thence a small P p . However, its short period is a remnant of an earlier phase in the evolution of the system, during which accretion of matter onto the pulsar caused it to be "spun up," essentially to its present period [for discussion see Smarr and Blandford (1976), BisnovatyiKogan and Komberg (1976)]. Binary Pulsar 293 (b) The Keplerian velocity curve The terms vs/(cosE e) v^tsinE in Equation (12.21) will be referred to as the "Keplerian velocity curve." The time derivative of these terms yields simply the first-order Doppler shift of the pulsar frequency, given by Av/v oc vt ii. This variation in frequency is the quantity usually measured in spectroscopic binaries, and was the quantity measured in the binary pulsar until the method of arrival-time measurements was adopted in late 1974. By fitting the measurements of arrival times to cos£ and sin is curves using Equation (12.21), a determination of the parameters P b , e, stf, and 88 can be made. From these parameters it is conventional to determine (i) the periastron direction at a given epoch: tan w = (1 - e2)- 1/2 J%s/ (12.25) (ii) the projected semi-major axis of the pulsar: ax sin i = [ y 2 + m\\ - e2yxY'2 (12.26) (iii) the mass function of the pulsar: A M * ! sin 0 3 (iV2*r 2 (12.27) The observed values for these quantities are shown in Table 12.1. Using Equations (12.4) and (12.20), we may also express fx in the form / 1 = Sr(m2 sin i)3/m2 (12.28) These interpretations assume afixedKeplerian orbit with constant values of the orbit elements. In reality, variations with time of the elements as given for example by Equation (12.22), make it necessary to treat the above values as being valid at a chosen epoch, and to perform further least-squares fits to determine rates of change such astit,Pb, e, and so on. We shall discuss some of these quantities below. (c) The periastron shift By substituting <x>-*a> + cat into the expressions for si and 88 in the Keplerian velocity curve, one can make an accurate determination of cb. The best value to date is d> = 4?226 ± 0!001 yr~ 1 . There are several possible sources of periastron shift in the binary pulsar. The first is relativistic: in Section 11.3, we found that in a binary system with compact objects, the periastron shift rate in a fully conservative theory of gravity in the modified EIH formalism took the form Theory and Experiment in Gravitational Physics 294 [from Equation (11.71)] + m23>122)/m (12.29) \M here should not be confused with that defined by Equation (12.19).] Substituting the known values of Pb and e and using Equation (12.4), we obtain d»rel = 2!lO(m/m 0 ) 2/3 ^«r 4/3 yr~* (12.30) In general relativity 9 s ^ == 1. The second possible source is a noninverse square gravitational potential produced by tidal deformation of the companion by the pulsar. The resulting rate is given by d)tidal =* 30nk2Ph-\$m1/m2)(R2/a)5f(e),
/( e ) = ( l - e 2 ) - 5 ( l + | e 2 + i e 4 )
(12.31)
where R2 is the radius of the companion (Cowling, 1938). The quantity
k2 is a dimensionless factor that depends on the mass distribution of the
companion and is of order 10 ~2 for white dwarfs or helium stars. In
obtaining Equation (12.31), we have assumed that the companion is not
a neutron star or a black hole in order to avoid the additional complications of self-gravitational effects on the tidal effects. Because of the
{R2/a)5 dependence, the tidal contribution of such objects would be
negligible in any case. Substituting numerical values we obtain
where X = mxfm2. For a white dwarf companion (R2 5> 104km), the
tidal effect is also negligible. Only for a "helium-star" companion, for
which (fc2/10~2)(i?2/105 km)5 ~ 6(m2/mQf-6, can the tidal periastron advance be significant (Roberts et al., 1976).
The third possible source of periastron shift is the noninverse square
potential produced by rotational deformation of the companion. For a
body that rotates with angular velocity Q about an axis that is inclined
by an angle 8 relative to the plane of the orbit, the result is
d)rol * 27t/c2Pb-1(3Wm2)(i?2/a)5(l + X-1)(n/n)2g(e)P2(cos9),
g(e) 3 (1 - e2r2,
n = 2n/Pb
(12.33)
This result is valid provided one assumes that the angular momentum
of the companion is small compared to that of the orbit, an assumption
Binary Pulsar
295
that is valid for most reasonable companions [see Smarr and Blandford
(1976) for discussion]. With numerical values, we find
,5
\mo)
x
K
^ 'V 1 0 ~ 2 A 1 0 5 km
a O?83a(^m/mo)"2/3P2(cos0) yr" 1
(12.34)
where
a = %k2(Cl2Rl/m2)(R2/l03 km)2
(12.35)
For stable, uniformly or differentially rotating white dwarf models, for
example, a may range from zero to ~ 15 (Smarr and Blandford, 1976).
Notice that the rotationally induced periastron motion can be either an
advance [P2(cos 6) > 0], for example, when the spin axis is normal to the
orbital plane, or a regression [P2(cos 9) < 0], as when the spin axis lies
in the orbital plane.
If the companion is a neutron star, a black hole, or a nonrotating white
dwarf, then only the relativistic periastron precession is present. The
observed advance shown in Table 12.1 then yields via Equation (12.30) a
relation between the masses of the bodies:
m = 2.85m G ^- 3/2 # 2
(12.36)
where & and ^ are functions of mu m2, and the structure of the pulsar
and possibly of the companion.
If the companion is a rotating white dwarf, only the relativistic and
rotational contributions are significant, thus we may write
(b = 2°10(/n/m G ) 2/3 ^«r 4/3 + O°83a(m/mGr2/3«r2/3P2(cos0) yr" 1
(12.37)
If the companion is a helium star with rotation axis perpendicular to
the orbital plane, all three sources of periastron precession may be
present, with
/ m \2
\mQJ
2/3
mQ
(12.38)
Theory and Experiment in Gravitational Physics
296
(d) The gravitational red shift and second-order Doppler shift:
the way to weigh the pulsar
The term <^sin£ in the timing formula, Equation (12.21), represents the combined effects of the gravitational red shift of the pulsar
frequency produced by the gravitational field of the companion and of
the second-order Doppler shift produced by the pulsar's motion. In some
theories of gravity, there is another effect that contributes to the timing
formula at the same order as <if sin E and should be included here (Eardley,
1975). That effect is the following: in theories of gravity that violate SEP,
the local gravitational constant at the location of the pulsar may depend
on the gravitational potential of the companion, i.e.,
GL = G 0 (l - »*m2/r)
(12.39)
If the companion is a white dwarf or a helium star, for example, the
parameter n* is simply the combination of PPN parameters n* = 4p
y 3 (fully conservative theories with £ = 0), however, if the companion
is a neutron star or a black hole, rj* could be more complicated and could
depend upon the internal structure of the companion. As GL then varies
during the orbital motion, the structure of the pulsar, its moment of
inertia, and thence its intrinsic rotation frequency will vary, according to
v
/
GL
_
K
,*^
rr
(12 .40)
where K determines the response of the moment of inertia to the changing
G. The contribution of this variation to the timing formula is given by
J Av dt = - vKn*(m2/a)(Pb/2n)e sin E
(12.41)
modulo constants. Thus the parameter W is actually given by
V = (mi/ffMiHaJ + « W m + Kn*){PJ2n)e
(12.42)
With numerical values it takes the form
« a* 2.93 x Kr 3 (m 2 /m)(mAn 0 ) 2/3 «r
1/3
(<x| + <Zm2/m + Kn*)s
(12.43)
However, were it not for the presence of periastron precession in the
system, this parameter would be entirely unmeasurable, since for constant values of sd and 8$, the term %> sin E is degenerate with the two Keplerian velocity curve terms, i.e., it cannot be separated from them in a least-squares fit (Brumberg et al. 1975, Blandford and Teukolsky, 1975). However, the variation of co at 4° per year causes$4 and J 1 themselves to vary with approximately a 90-year period. Thus, over a suf-
Binary Pulsar
297
ficiently long time span (though much shorter than 90 years, fortunately),
a separate determination of si, 88, and < can be made. Using data through
August, 1980, Taylor (1980) in fact reports
<«?~4.4±0.3 x H T 3 s
(12.44)
Equations (12.43) and (12.44) yield a further relation between the masses
of the bodies. When combined with the mass relations provided by the
mass function [Equation (12.28)], and by the periastron shift [Equations
(12.36), (12.37), or (12.38)], and with assumptions about the nature of
the companion and about the theory of gravitation, they permit a unique
(within experimental errors) determination of the masses m^ and m2 and
of sin i. This is that unique new role of relativistic gravity alluded to in
the introduction to Chapter 12. Not only does a relativistic effect, the
periastron shift, yield a constraint on the masses of the bodies, it also
enables the determination of a second relativistic effect, the red-shift
Doppler coefficient (. Nowhere in astrophysics has relativistic gravity
played such a direct, quantitive role in the measurement of astrophysical
parameters.
(e) Post-Newtonian effects and sin i
In Equation (12.15), we dropped the explicit term arising from
the time delay, and denoted it [O(3)farr]. There are additional terms in
the timing formula that are also of [O(3)£arr], produced by post-Newtonian
deviations of the orbital motion from a pure Keplerian ellipse. Within
general relativity, these terms have been analyzed in detail by Epstein
(1977), and included in the data analysis by Taylor, et al. (1979). They
provide an independent means to determine the parameters of the system,
especially the inclination angle i. This is a valuable consistency check
for any interpretation of the data. In fact the data are just accurate enough
to be sensitive to these effects, and the limit on sin i quoted in Table 12.1
was obtained from these terms. Note that this particular result is valid
only in general relativity; the corresponding analysis of the [O(3)farr]
terms using the modified EIH formalism has not been carried out.
(f) Decay of the orbit: a test for the existence of
A variety of effects may cause the orbital period P b of the system
to undergo a secular change with time, but the most important is the
effect of the emission of gravitational radiation. According to the quadrupole formula of general relativity (see Section 10.3), a binary system
Theory and Experiment in Gravitational Physics
298
should lose energy to gravitational radiation at a rate given by Equation
(10.80),
dE_ _
/n2m2
dt ~
where /i is the reduced mass of the system, and
F{e) = (1 + He 2 + Me4)(l - e2)'1'2
(12.46)
The resulting rate of change of P b is given, from Kepler's third law, by
Pb-1 dPJdt = -IE"1 dE/dt = -¥{nm2/aA)F{e)
(12.47)
where E = \\an/a. For the known parameters of the binary pulsar we
find
jn\5'*
X_
= -(1.91xlO-9)(^l
T T - ^ y r - 1 (12.48)
(1
KmQ)
As we pointed out in Section 10.3, most theories of gravitation alternative to general relativity predict the existence of dipole gravitational
radiation. Since the magnitude of the effect in binary systems depends
upon the self-gravitational binding energies of the two bodies, the binary
pulsar provides an ideal testing ground. In general relativity, neutronstar binding energies can be as large as half their rest masses, and in other
theories even larger, so the dipole effect, if present, could produce more
rapid period changes than the general relativistic quadrupole effect. The
predicted energy loss rate is given by
ah/at = 2KD\ & n m vs> /r }
where KD is a parameter whose value depends upon the theory in question, and S is related to the difference in "sensitivities" (s2 Si) between
the two bodies, where sa is a measure of the self-gravitational binding
energy per unit mass of the ath body. In general relativity, KD = 0. The
rate of change of period is thus given by
l + | e 2 )(l - e 2 r 5 / 2
(12.50)
where now E = \ 'S^mja. For the parameters of the binary pulsar, we
obtain
- "(3.09 x w - ^ J ^ j ^ y , - '
(,2.M,
Binary Pulsar
299
For a theory of gravity with |/cD| ~ 1, this can be several orders of magnitude larger than the general relativistic quadrupole prediction, unless,
for instance, the two bodies are identical, in which case there is no dipole
radiation, by virtue of the symmetry of the situation.
However, before these effects can be used as a reliable test for the
existence of gravitational radiation or as a test of alternative gravitation
theories, other possible sources of period change must be accounted for.
Since we have previously discussed tests of gravitational theory involving
detecting changes in the pulsar period as well as in the orbital period
(see Section 9.3), we shall review possible sources of both.
(i) Tidal dissipation. Tides raised on the companion by the gravitational field of the pulsar will change both the energy of the orbit and the
rotational energy of the companion via viscous heating. The corresponding tides raised on the pulsar are negligible because of its small size
and by the same token, if the companion is a neutron star or a black hole,
tidal dissipation is negligible. For a companion with rotation axis normal
to the plane of the orbit, the rate of change of the orbital period is given by
(Alexander, 1973)
Ph
672TC
3 W
) - 6 ^ ^i
fete2
(12.52)
2
Pb
25
' \m 2)\a ) \ m2 ) \ ' nj
where n = 2n/Pb is the orbital mean anomaly, </*> is an "average" coefficient of viscosity of the companion given by
(1-e
2
v
fir8dr
(12.53)
where n is the local coefficient of viscosity in units of gem" 1 s" 1 , and
h(e2,Q/n) is a complicated function of e2 and Q/n of the following general
form
He2,0/n) = ht(e2) - (i/n)h2(e2)
(12.54)
For circular orbits h1 = h2 = 1, however for the binary pulsar (e m 0.6)
they could be an order of magnitude larger (but hx ^ h2). We note that if
Q < «(companion counter rotates relative to the orbit), tidal dissipation
always decreases the orbit energy and thus the period, whereas if Q/n >
hx(e2)/h2(e2) (companion rotates faster than the orbit by some factor of
order unity), dissipation increases the orbit energy (at the expense of
rotational energy) and causes the period to increase. Notice than even if
the companion is in synchronous (tidally locked) rotation, Q = n, there
can still be tidal dissipation due to the time-changing deformation of the
Theory and Experiment in Gravitational Physics
300
companion resulting from the eccentric motion of the pulsar. Substituting
the observed parameters of the system, we obtain
= -2x
mo
R2 \ 9
T73T </*>i3*(e2,n/»)yr~1
(12.55)
105 km/
where </x\ 3 s 10"13<(^>. For standard molecular viscosity, (n~) ~ 1,
i.e., <ju>13 ~ 10" 13 , and tidal dissipation is completely negligible. However, if the source of viscosity is tidally driven turbulence (Press et al.,
1975; Balbus and Brecher, 1976), </i>13 could be as large as unity. For a
helium star companion, PJPb could then be comparable to the general
relativistic quadrupole radiation damping rate. For a white dwarf companion {R2 < 104 km), tidal dissipation is negligible unless the white
dwarf is very rapidly rotating (|Q| » n), and a very strong source of viscosity, such as magnetic viscosity «M>i3 ~ 103), is present (Smarr and
Blandford, 1976).
(ii) Mass loss from the system. The emission of energy of various forms
(particles, electromagnetic radiation, etc.) from the pulsar results in a
decrease in its rotational kinetic energy, and thus in an increase in its
pulse period, given by
Erol = -l{2nlPpfPJPp
(12.56)
where / is the moment of inertia of the pulsar. The loss of mass energy
from the pulsar leads to a change in the orbital period at a rate
PJPb = -\rhjm
(12.57)
Now, if the emission of energy is dominated by relativistic particles
(photons, for example) then most of the mass loss will occur at the
expense of rotational kinetic energy, i.e.,
£rot < m,
(12.58)
PJPb ~ 1 x 10" 6 {ml2.MmQ)-HA5(PpIPp)
(12.59)
Thus,
where 745 = 7/1045 g cm2. Since the observed value for Pp/Pp is ~ 4 x
10~9 yr~x (Table 12.1), then PJPb due to energy loss must be ~10~ 1 4
yr" 1 .
(iii) Acceleration of the binary system. If the center of mass of the binary
system suffers an acceleration relative to that of the solar system, then
Binary Pulsar
301
both the orbital and pulsar periods will change at a rate given by
PJPb = Pp/Pp = r 0 = a n + r0-J [V - (v n)2]
(12.60)
where v and a are, respectively, the relative velocity and acceleration
between the binary system and the solar system. The first term is the
projection of the acceleration along the line of sight, while the second
represents the effect of variation of the line of sight. Accelerations may
also lead to observed second-time derivatives of periods, given by
PJPb = Pp/Pp = r0 + 2(P/P) 0 r 0 - Tr%
(12.61)
where (P/P)o is the observed relative rate of change for the corresponding
period.
One possible source of acceleration was discussed in Section 9.3, namely
a violation of conservation of total momentum in some theories of
gravity. There, we used the observed limits on Pp/Pp to set a potential
limit on the PPN conservation law parameter £2 [ m t n a t c a s e > t n e second
and third terms in Equation (12.61) are negligible compared to the first].
Another source is the differential rotation of the galaxy. If we assume
that the binary system (b) and the solar system (©) are in circular orbits
around the galaxy with angular velocities Qb and Q 0 , distances from the
galactic center rb and rQ, and longitudes relative to the galactic center
cf>b and <t>Q, then Equation (12.60) takes the form
Pp/Pp = PJPb = («o - 0^2r0Vo
'
x [cos(^ b - <t>0) - rQrbro2sin2(^b
- <£0)]
(12.62)
Estimates of the location and distance of the binary pulsar (Hulse and
Taylor, 1975) yield
ro~5kpc,
rQ~10kpc,
rb~8kpc,
<£ b -<£ 0 ~3O°
(12.63)
Using the standard galactic rotation law, Q(r) ~ 250 (km s"1)/?-, we find
PJPb = K/Pp ~ 2 x lO" 1 3 yr" 1
(12.64)
This is too small to be of importance (Will 1976b, Shapiro and Terzian
1976).
Another possible source of acceleration is a third massive body in the
vicinity of the binary system. For a body of mass m3, and for a circular
orbit with orbit elements a3, co3, and i3, we have
PJPb = PJPp~ - f e Y ( - ^ ~ ) a 3 s i n i 3 c o s ( c o 3 + 4>) (12.65)
Theory and Experiment in Gravitational Physics
302
where <>
/ is the orbital true anomaly, and
Pb/Pb = Pp/Pp * ( ^ J ( - ^ L _ ) a3 sin i, sin(a>3 + </»)
(12.66)
It is then simple to show that if t] represents the observed upper limit on
\PP/Pp\, then the contribution of a third body to period changes is limited
by
\PJPp\ = |P b /P b | < (7 x lO-^if/meY'^/lO-11
yr- 2 ) 4 ' 7
x |cot(0 + a>3)sm(4> + o)3)3/7| yr" 1
(12.67)
where / is the mass function of the binary system relative to the third
body, given by
/ = (m3 sin i3)3(m + m 3 )" 2
(12.68)
Since the observed valued of Pp/Pp has not changed by more than its
experimental error in a year (Table 12.1), we may conclude that rj <
10"11 yr~2. An explicit determination of r\ from the timing data that
improves this limit would help to determine the likelihood that a third
body is responsible for part of the observed orbit period change.
(g) Precession of the pulsar's spin axis
If the pulsar is a rapidly rotating neutron star, it should experience
the same relativistic precession effects on its spin axis as does a gyroscope
in orbit around the Earth (see Section 9.1). The dominant effects are the
geodetic precession due to the companion's gravitational field, and a
Lense-Thirring-type precession due to the companion's "magnetic" gravitational field generated by its orbital motion [see Equations (9.2) and
(9.4)]. The Lense-Thirring precession due to the possible rotation of the
companion is negligible. By substituting Equations (9.4) and (9.2) with
J = 0 into Equation (9.1), inserting the orbital elements for the binary
pulsar, and averaging over an orbit, one finds
fdt = ftxS
il = (3n/Ph)[m22/ma(l -
2
e
)][i(2y + l) + f(y+ 1 +ia1)(mi/m2)]fi
(12.69)
where y and a t are PPN parameters and h is a unit vector normal to the
orbital plane (Barker and O'Connell, 1975; Hari Dass and Radakrishnan,
1975; and Rudolph, 1979). In obtaining this result we have ignored the
possibility of modified-EIH-formalism corrections to effective masses in
alternative theories of gravity. The magnitude of ft is about one degree
per year (compare with an Earth-orbiting gyroscope in Section 9.1); note,
Binary Pulsar
303
however, that no precession occurs if the pulsar's spin axis is normal to
the plane of the orbit.
If precession does occur, it could be viewed as a means to test gravitational theory. However, it may be more fruitful to use the relativistic
precession as a means to probe the nature of the pulsar's emission mechanism. As the pulsar precesses, the observer's line of sight intersects the
surface of the neutron star at different latitudes, thus it may be possible
to obtain two-dimensional information on the shape of the emitted beam,
as well as to study the variation of spectrum and polarization with latitude. Unfortunately, in most pulsar models, the radio pulses are emitted
in a pencil beam, so the pulsar might one day disappear altogether.
We now turn to the confrontation between the binary pulsar and gravitation theory. It is here that the philosophy of testing gravitation theory
must depart somewhat from that adopted in Chapters 2 through 9. There,
we regarded experiments as "clean" tests of gravitational theory. Because
the underlying nongravitational physics associated with solar system and
laboratory experiments was reasonably well understood, the experimental
results could be viewed as limiting the possible alternative theories of
gravity, in a theory-independent way. The use of the PPN formalism was
a clear example of this approach. The result was to "squeeze theory space"
in a manner suggested by Figures 8.2 and 8.3.
However, when complex astrophysical systems such as the binary pulsar
are used as gravitational testing grounds, one can no longer be so certain
about the underlying physics. In such cases, a gravitation-theory-independent approach is not useful. Instead, a more appropriate approach would
be to assume, one by one, that individual theories are correct, then
use the observations to make statements about the possible compatible
physics underlying the system. The viability of a theory would then
be called into question if the resulting "available physics space" were
squeezed into untenable, unreasonable, or ad hoc positions. Such a method
would be most powerful for theories that make qualitatively different predictions in such systems. We shall illustrate this philosophy of "squeezing
physics space" (using relativistic gravity to determine astrophysical parameters) with general relativity, Brans-Dicke theory, and Rosen's bimetric
theory.
12.2
The Binary Pulsar According to General Relativity
The confrontation between relativistic gravity and binary pulsar
data takes its simplest and most natural form within general relativity.
In general relativity, there are no EIH self-gravitational mass corrections
Theory and Experiment in Gravitational Physics
304
due to violations of SEP (see Section 11.3), and there is no dipole gravitational radiation. Thus, ^ E g = aj H yf = 1 and KD = n* 0. The relevant measured parameters of the system are then given by the following
expressions
Mass Function:
fi = (m2 sin ifjm2
(12.70)
Orbital Period:
PJ2n = ( a » 1 / 2
Periastron Shift:
(12.71)
2?10(m/mG)2/3 yr""1, [black hole, neutron star,
nonrotating white dwarf companion]
(12.72)
2?10(m/mo)2/3 + O?83a(m/m o r 2/3 P 2 (cos0)yr-\
[rotating white dwarf companion]
(12.73)
a; =
105km/
[aligned rotating helium star companion]
(12.74)
Red-shift-Doppler Parameter:
^ = 2.93 x 10"31
m\213
m
(12.75)
VAJgr.quad = -(1.91 x 10-9)(m/mG)5/3X(l + X)'2 yr" 1
(12.76)
Together with the measured values shown in Table 12.1, these equations determine constraints on the possible masses of the pulsar and
companion, and on the inclination i. The most convenient way to display
these constraints is to plot m1 vs. m2. The results are shown in Figure 12.2.
One constraint is provided by the mass function fx and by the fact that
sin i < 1. The periastron shift constrains the system to lie along the
straight line BH-NS-WD if the companion is a black hole, neutron star,
or nonrotating white dwarf [Equation (12.72)]. This line represents a
total mass m = 2.85 mQ. It is useful to remark that the maximum mass
of a nonrotating white dwarf is ~ 1.4 solar masses. If the companion is a
rapidly rotating white dwarf (with "U" denoting uniform rotation and
305
Binary Pulsar
"D" denoting differential rotation), the system could lie in the regions
denoted U and D [Equation (12.73)]. The regions to the left of the BHNS-WD line correspond to white dwarfs with spin axes aligned perpendicular to the orbital plane (6 = 0). In this case, the rotational contribution
to d> is positive so the inferred system mass m must be less than 2.85m0.
The regions to the right of the BH-NS-WD line correspond to white
dwarfs with spin axes in the orbital plane (0 = n/2). Here the rotational
periastron shift is retrograde and thus m > 2.85mo. Values of the parameter a that depends upon the structure and rotation rate of the white
dwarf were given by Smarr and Blandford (1976). Figure 12.2 also shows
the configuration if the companion is a helium star, tidally locked into
the orbital rotation rate (Q = n). Values for k2 and R2 for helium stars
were given by Roberts, Masters, and Arnett (1976). The red-shift-Doppler
parameter then constrains the system to lie between the lines marked #.
Finally, if we attribute all the observed orbit-period decay to gravitationalradiation damping, then the system must lie between the lines marked
Figure 12.2. The mx-m2 plane in general relativity. The shaded region
fits all the formal observational constraints. The point marked "a" is the
most likely configuration.
3.0
\
\
\
^H-NS »',
- N.
2.0
i
\
\
\
y
,
i
\
\
\
fo,
s
D-WD"
I- D
""
^ - - '
gr.§C
\HE
.-
1.0
-^^^BH-NS-WD
sin i> 1
i
i
1.0
i
2.0
Mass of Pulsar m,/m0
I
1
3.0
1
Theory and Experiment in Gravitational Physics
306
P b . This leaves the shaded region available. The most natural physical
interpretation therefore seems to be that the companion is a black hole,
neutron star, or nonrotating white dwarf (point "a" in Figure 12.2) of mass
m2 = 1.42 ± 0.07m©. The mass of the pulsar is then mx = 1.43 ± 0.07m©
and the sine of the inclination angle (from the mass function) is sin i =
0.72 + 0.04. This interpretation is also consistent with the constraint
sin i < 0.96 obtained by taking into account in the timing formula postNewtonian effects such as the time delay [Equation (12.13)] and periodic
perturbations of the Keplerian orbit (Taylor, 1980). Before this interpretation can be accepted with confidence, however, some account must be
taken of the possible nonrelativistic sources of orbit period change discussed in the previous section, in particular tidal dissipation and a third
body.
Thus, barring these remote possibilities, general relativity leads to a
natural physical configuration for the system, and the results support the
conclusion that the measurement of Ph represents the first observation of
the effects of gravitational radiation. They also lend support to the validity
least as a good approximation, and rule out the possibility that gravitational waves are composed of half-retarded plus half-advanced fields and
therefore carry no energy at all (Rosen, 1979).
12.3
The Binary Pulsar in Other Theories of Gravity
(a) Brans-Dicke theory
Because solar-system experiments constrain the coupling constant a to be large (co > 500) we expect the predictions of scalar-tensor
theories to be within corrections of order (1/co) of their general relativistic
counterparts for the binary pulsar. The self-gravitational mass renormalizations merely introduce corrections of order (l/co) (see Section 11.3).
Thus, the mt m2 plane in scalar-tensor theories is largely indistinguishable from that in general relativity. Even the added possibility of dipoie
gravitational radiation does not seriously constrain either "physics" space
or the coupling constant co. Substituting the value KD = 2/(2 + co) into
Equation (12.51) we find
(A/PbWie = - ( 1 x 10-9)(500/a>)(S/0.1)2(AVm©) yr" l
(12.77)
For neutron star models with masses around 1.4m©, s ~ 0.39 (Eardley,
1975). Thus, (PJPb) dipoie could be significant if the companion is a white
dwarf or a neutron star whose mass differs from that of the pulsar by
greater than ~ 10%. In such an event, it might be possible to push the
Binary Pulsar
307
coupling constant even higher than 500. However, the data can equally
well be fit (for <x> ~ 500) by a system with two nearly equal-mass neutron
stars, or by one of the above possibilities with a small contribution to
Pb from some nonrelativistic source.
This is a case in which the theoretical predictions are sufficiently close
to those of general relativity, and the uncertainties in the physics still
sufficiently large that the viability of the theory cannot be judged reliably.
We would expect roughly the same conclusions to be valid in general
scalar-tensor theories such as Bekenstein's VMT (see Section 5.3).
(b) Rosen's bimetric theory
In the bimetric theory, however, the situation is very different.
The EIH self-gravitational mass corrections (of Section 11.3) lead to
qualitative differences for two reasons. First, the correction terms in
^, 0*, etc. are ~s, and second, s can be much larger for bimetric neutronstar models than for their general relativistic counterparts. Table 12.2
Table 12.2. EIH sensitivities, s, s', in
Rosen's bimetric theory."
Inertial mass
m(mQ)
normal star
white dwarf
neutron stars
0.097
0.165
0.409
0.635
0.865
1.158
1.868
2.371
3.217
4.553
7.177
10.93
12.34
14.45C
s
s'
~10~6
glO"3
~10~6
0.006
0.018
0.048
0.071
0.096
0.128
0.206
0.258
0.331
0.410
0.494
0.561
0.582
0.628
b
£io- 3
b
0.065
0.099
0.132
0.174
0.265
0.294
0.271
0.223
0.175
0.152
0.222
b
" For equations of state from Canuto (1975).
Accuracy + 3 in last place.
* Accurate value not computed.
c
Maximum mass.
308
Theory and Experiment in Gravitational Physics
shows values of s and s' for normal stars and white dwarfs, and for neutronstar models with inertial masses up to 14.5wo (Will and Eardley, 1977).
From these values, we compute values for'S and 0, given by [see Equations
(11.72) and (11.113)]
(12.78)
and plot the corresponding m1 m2 plane for the bimetric theory, shown
in Figure 12.3. [For simplicity, we have ignored the effect of changes in
GL on the parameter <, Equation (12.43). It is only significant if the companion is a neutron star {t\* %s2), and is expected to modify <<? by only
Figure 12.3. The mx-m2 plane in Rosen's bimetric theory. Note the scale
of masses is almost double that of Figure 12.2. The numbers shown are
the predicted values of P b /P b due to gravitational radiation, including
6.0 -
5.0 -
N.
4.0 -
/+10-*
\ +10"'
V-+10- 6
a,
<S 3.0
s
s
^
2.0 -
1.0 -
^
\
D-WD
^
JT>VC
' * ' - ' ' ' + 1(
^ - ' ^ + 1 0 ^ - ^
sin i> 1
/"^U-WD
I
1.0
i
I
i
2.0
3.0
4.0
Mass of Pulsar nij/nio
i
5.0
Binary Pulsar
309
In Figure 12.3, we notice that the companion cannot be a nonrotating
white dwarf, since such a configuration would violate the condition
sini < 1. If the companion is a neutron star, the system must lie along
the curve "NS," with total inertial mass ~ 7 m o . When the red-shiftDoppler constraint (curves 'V') is folded in, the theory is left with a major
problem. Dipole gravitational radiation causes the system to gain energy
and the period to increase at a rate
( i V n W i e =* +(2 x l(T 5 )(6/O.3) 2 (Mn 0 ) yr" 1
(12.79)
with specific values for various companions shown in several locations
in Figure 12.3. In order to agree with the observed value of PJPb ^
(2.4 + 0.4) x 10~9 y r ~ \ the theory must produce a mechanism (tidal
dissipation, third body) to cancel this predicted increase and account for
the observed period decrease. The contrived and ad hoc nature of such
mechanisms deals a convincing blow to the viability of this theory.
(c) The ultimate test of gravitation theory?
This result may, in fact, apply to many other theories, particularly
those with "prior geometry." In such theories, SEP is violated, and the
differences between the theories and general relativity become larger the
stronger the gravitational fields. Thus, one can expect qualitative EIH
mass renormalizations similar to those in the bimetric theory. Furthermore, all such theories are expected to predict dipole gravitational
radiation of magnitude comparable to that in the bimetric theory. So it
is very likely that the binary pulsar data will be able to rule out a broad
class of alternative gravitation theories.
However, the class of "purely dynamical" theories has the property
that the effects of the additional gravitational fields can usually be made
as small as one chooses, both in weak-field and in strong-field or gravitational-radiation situations, by choosing sufficiently weak coupling constants (co'1 -> 0 in Brans-Dicke, for instance). Thus, Brans-Dicke theory,
with to ^ 500, is consistent with the present binary pulsar data, even
though it, too, predicts dipole gravitational radiation. Such theories, that
merge smoothly and continuously with general relativity, can never be
truly distinguished from it (as long as experiments continue to be consistent with general relativity). Except for such cases, the binary pulsar may
provide the "ultimate" test of gravitation theory.
13
Cosmological Tests
Since the discovery by Hubble and Slipher in the 1920s of the recession
of distant galaxies and the inferred expansion of the universe, cosmology
has been a testing ground for gravitational theory. That discovery was
thought at the time to be a great confirmation of general relativity for
two reasons. First, general relativity, in its original form, predicted a
dynamical universe that necessarily either expands or contracts. Of course,
Einstein had later modified the theory by introducing the "cosmological
constant" into the field equations in order to obtain static cosmological
solutions in accord with the current, pre-Hubble observations. To his
great joy, following Hubble's discovery, Einstein was allowed to drop the
cosmological constant.
Second, was simply the fact that general relativity was capable of dealing
with the structure and evolution of the universe as a whole, a capability
not shared by Newtonian theory (unless special assumptions are made).
However, this capability is more a consequence of the Einstein Equivalence Principle (alternatively of the metric-theory postulates) than a
property of general relativity. Because of EEP, spacetime is endowed
with a metric g which determines the results of observations made using
nongravitational equipment (light rays, telescopes, spectrometers, etc.)
and the motion of test bodies (galaxies). Via the field equations provided
by each metric theory of gravity, the distribution of matter then determines
the metric g, and thereby the entire physical spacetime in which observations are made. By contrast, in Newtonian cosmology, space and time
are fixed a priori, and one is faced either with the problem of specifying
and interpreting the boundary of afiniteuniverse or with the mathematical
problems associated with an infinite universe in Newtonian theory [see
Sciama (1975) for a discussion of Newtonian cosmology].
Cosmological Tests
311
Despite the success of general relativity in treating the expansion of
the universe, there remained doubts. Chief among these was the "timescale
problem." The early values for the Hubble constant Ho, the ratio between
recession velocity and distance, implied an age of the universe since the
beginning of the expansion ("big bang") that was shorter than the estimated
ages of the stars (from stellar evolution theory) and of the radioactive
elements on the Earth. However, by the late 1950s, revisions in the extragalactic distance scale (increase by a factor of five) and the consequent
reduction of the Hubble constant increased the age of the universe to a
value greater than that of our galaxy, thus resolving the timescale problem.
But the crucial confirmation of the big bang model came in 1965 with
the discovery of the 3K cosmic microwave background radiation (Penzias
and Wilson, 1965). This discovery implied that the universe was once
much hotter and much denser than it is today [see Weinberg (1977) for a
detailed account of the discovery and of its interpretation]. In particular,
it made the steady-state theory of Bondi, Gold, and Hoyle untenable. It
also made it possible to resolve the discrepancy between the observed
cosmic abundance of helium (20-30% by weight) and estimates of the
production of helium in stars (a few percent at most). Calculations by
Peebles (1966) and by Wagoner, Fowler, and Hoyle (1967) of nucleosynthesis in a hot (109 K) big bang yielded helium abundances precisely
within the observed range [for a review, see Schramm and Wagoner
(1977)]. The hot big-bang model within general relativity is today the
standard working model for cosmology [for reviews of general relativistic
cosmology, see Peebles (1971), Weinberg (1972), MTW, Sciama (1975)
and Zel'dovich and Novikov (1983)].
However, cosmological models within alternative theories of gravity
have not undergone a systematic study with a view toward testing them
in a cosmological arena. One reason is that in their exact, strong-field
formulations, alternative theories are sufficiently different that it has not
been possible to date to devise a general scheme, analogous to the PPN
formalism, for classification, comparison, and confrontation with observations. Also, cosmological observations are not "clean" tests of gravitation since much "dirty" astrophysics often goes into their interpretation.
But many alternative theories of gravity, even those whose postNewtonian limits are identical to or close to that of general relativity, are
different enough in their full formulations that they may predict qualitatively different cosmological histories. These may be sufficiently different
that observational data such as the mere existence of the microwave
background or the observed abundance of helium, however imprecise,
Theory and Experiment in Gravitational Physics
312
may suffice to rule out some theories in spite of the astrophysical and
observational uncertainties.
Section 13.1 outlines the general approach to be used in building
cosmological models in alternative metric theories of gravity. In Section
13.2, we present a brief and qualitative survey of what little is known at
present about cosmology in such theories.
13.1
Cosmological Models in Alternative Theories of Gravity
We begin by making two important assumptions about the nature
of the universe that should hold in any metric theory of gravity:
Assumption 1: The Einstein Equivalence Principle (EEP) is valid.
Assumption 2: The Cosmological Principle is valid.
As we saw in Chapter 2, the validity of EEP is equivalent to the adoption
of a metric theory of gravity. The cosmological principle states that the
universe presents the same aspect to all observers at any fixed epoch of
cosmic time, or equivalently, that the universe is homogeneous and
isotropic, at least on large scales (~ 100 Mpc). The cosmological principle
may be justified by noting the observations of isotropy of the universe,
especially of the microwave background, and by assuming that we occupy
a typical, not special place in the universe (Copernican principle). Neither
of these two assumptions is open to much question (see, however, Ellis
et al., 1978) although there has been considerable study of cosmological
models within general relativity that, while approximately isotropic today,
were highly anisotropic in the past (for a review, see MacCallum, 1979).
Because of EEP, spacetime is endowed with a metric g in whose local
Lorentz frames the nongravitational laws of physics take their special
relativistic forms. The cosmological principle then demands that the line
element of g must take the Robertson-Walker form (MTW, Section 27.6)
ds2 =^gflvdxlidxv
= -dt2 + a(t)2[(l - kr2rldr2
+ r2(d62 + sin2 0 # 2 ) ]
(13.1)
where r, 9, and 4> are dimensionless coordinates, t is proper time as
measured by an atomic clock at rest, a(t) is the expansion factor (units of
distance), and k e {+1,0} is a constant. Each element of cosmic matter
(galaxy) is assumed to be at rest in these coordinates. If k = 1, the universe
is closed (i.e., has closed spatial sections), if k = 1, the universe is open,
and if k = 0, the universe is open, with Euclidean spatial sections. The
alternative form of the metric that was used in Section 4.1 to establish
the asymptotically flat PPN metric can be obtained from this by making
Cosmological Tests
313
the transformation to the new radial coordinate r' given by
r = {r'lao){l + kr'2IAaly"
(13.2)
where a0 is the value of a(t) at the present epoch.
Although the present value of the scale factor a is difficult to measure,
its rate of variation with time is subject to observation. In particular one
defines the Hubble constant H o and the deceleration parameter q0 by
H o = (d/a)0,
qo=-
H« 2(a/a)0
(13.3)
where a dot denotes d/dt and the subscript "0" denotes present values.
These parameters may be measured by a variety of techniques, such as
the magnitude-red-shift relation or the angular-size-red-shift relation for
distant galaxies. The present "best" values for these parameters are
H o ^ 60 ± 2 0 km s" 1 M p c ~ \
-l<qo^2
(13.4)
The large uncertainty in q0 is a result of the uncertain effects of galactic
evolution on the intrinsic luminosities of distant galaxies used as "standard
candles" in magnitude-red-shift measurements.
The validity of EEP also allows one to determine the behavior of the
matter in the universe, independently of the theory of gravity. If we
idealize that matter as a homogeneous perfect fluid, then the equations
of motion Tfvv = 0 can be shown (MTW, Section 27.7) to yield the following
equations for the evolution of the mass-energy density p(t) and the
pressure p(t):
P(t) = p mO |>oMt)] 3 + Pro[ao/a(t)T,
Pit) = yrolao/a(t)Y
(13.5)
where pm0 and pr0 denote the present mass-energy densities of matter
and radiation, respectively. These equations will be valid for temperatures
less than about 1010 K, when the electrons and positrons annihilated.
We now turn to an outline of the recommended method for obtaining
cosmological models in any metric theory of gravity.
Step 1: Use the cosmological principle to determine the mathematical forms in Robertson-Walker coordinates to be taken by all the
dynamical and nondynamical fields of the theory. For the dynamical
fields listed in Section 5.1, these forms are
Metric:
ds2 = -dt2 + a(t)2da2,
Scalar:
0(r),
Vector:
K^dx* = K{t)dt,
Tensor:
B^dx"dxv
= co0(t)dt2 + (ox{t)do2
(13.6)
Theory and Experiment in Gravitational Physics
314
where
da2 = (1 - /cr2)"1 dr2 + r2{d62 + s i n 2 0 # 2 )
(13.7)
For a nondynamical flat background metric r\ governed by the equation
Riem(i/) = 0, the general form for its line element dy2 = tj^dx" dx" in
Robertson-Walker coordinates is
dSf2 = - i(t)2 dt2 + %{i)2 da2
2
dSf = -i(t)
2
2
dt + da
2
[k = - 1 ] ,
[k = 0]
(13.8)
where r(t) is a function of t, with i = dt/dr (as in Section 11.3, we shall
ignore the possibility of "tipping" of the t] cones relative to the g cones).
Note that there is no solution for the case k = 1. Thus, it is very unlikely
that any theory of gravity with a flat background metric can have a
closed (k = 1) cosmological model for the physical metric g. For a nondynamical cosmic time coordinate T, it is sufficient to assume that T =
T(t). The matter variables have the form
p = p(t),
p = p(t),
u" = (1,0,0,0)
(13.9)
Step 2: Substitute these forms into the field equations of the theory.
Step 3: Set boundary conditions on the fields, in particular on their
present values <j>0,K0,x0,a0, etc. These values are related in general to such
measurable quantities as H0,q0, and k, as well as to the PPN parameters
and the present rate of variation of G, or (G/G)o. Use the present experimental values or limits on these parameters to limit the class of cosmological models to be considered.
Step 4: Integrate the field equations and the equations of motion
backward in time (using numerical methods as a rule), taking into account
possible changes in the equation of state for the matter variables as the
universe becomes hotter and denser (see MTW, Section 28 for discussion).
Step 5: The tests. Although cosmological data is sketchy and imprecise, there are two pieces of evidence about the early universe about
which there seems to be little disagreement, the 3K cosmic microwave
background radiation and the cosmic abundance of helium.
(i) The microwave background: There is now general consensus that
the microwave background is the relic of a hotter, denser phase of the
universe, where the temperature exceeded 4 x 103 K. No reasonable
mechanism has yet been devised to produce the background during later
epochs (T < 4 x 103 K) that agrees both with the observed high degree
of isotropy of the radiation [after the effects of the Earth's motion (see
Section 8.2) have been subtracted] and with the close agreement of the
Cosmological Tests
315
spectrum with that of a black body. Prior to the epoch T = 4 x 103 K, a
variety of physical processes are consistent with the observed background,
ranging from recombination of electrons and protons to form hydrogen
to the quantum evaporation of primordial "mini"-black holes (m <
1015 g). Thus, in order to predict cosmological models with the microwave
background, the theory must guarantee that the universe evolved from a
state with T>4 x 103 K (p/p0 > 109, a/a0 < 10~3). An example of an
unviable cosmological model would be one that contracts from some
earlier dispersed state to a maximum density and temperature below the
above limits, then bounces and reexpands to the present observed state.
Such a model would contain no reasonable explanation for the microwave
background. A class of models in Rosen's bimetric theory has this property (see Section 13.2).
(ii) The helium abundance: It is also generally believed that stellar
nucleosynthesis can account for only a small fraction of the observed
20-30% abundance by weight of helium, and thus, most of the helium
was produced in the early universe. Similar claims have been made for
the deuterium abundance (observed to be ~2 parts in 105 by weight),
but in this case the contributions of galactic production (and destruction)
and of chemical fractionation are more uncertain, so we shall focus on
helium [see Schramm and Wagoner (1977)]. Primordial nucleosynthesis
requires temperatures in excess of 109 K and baryon number densities
> 10~6 cm" 3 , and therefore a viable cosmological model must predict a
state at least this hot and dense. Furthermore, the fraction of helium
produced is sensitive to the rate of expansion of the universe at the epoch
of nucleosynthesis. The reason is as follows: when nucleosynthesis occurs,
essentially all the neutrons go into helium nuclei, so the abundance of
helium depends only on the neutron-proton abundance ratio at the time
tN of nucleosynthesis, i.e.,
X(He4) = 2(n/p)(l + n/p)-%N
(13.10)
where X denotes the mass fraction and n/p is the neutron-proton density
ratio. This ratio n/p is determined by two factors. First is the (n/p) ratio
at the moment ("freeze out") when weak interactions are no longer fast
enough to maintain the neutrons and protons in chemical equilibrium;
at freeze out their ratio is thus given by (n/p)F exp[(mn mp)//c7V],
where mn and mp are the proton and neutron rest masses and TF is the
temperature at freeze out. Second is the interval of time between freeze
out and nucleosynthesis, during which the neutrons undergo free decay.
The faster the expansion rate at a given temperature, the earlier the
Theory and Experiment in Gravitational Physics
316
weak interactions freeze out, thus TF is higher and (n/p) is closer to unity.
In addition, the time between freeze out and nucleosynthesis is shorter
and fewer neutrons decay. The result is a higher abundance of helium.
The opposite occurs for a lower expansion rate. In some cosmological
models, the expansion rate during nucleosynthesis can be expressed
phenomenologically as
a^da/dt^ZQnp)1'2
(13.11)
where ^ is a parameter whose value is 1 in the standard model of general
relativity, and p is the total mass-energy density. The resulting helium
abundance is given approximately by
X(He4) ~ 0.26 + 0.38 log £
(13.12)
for a present density p o ~ 1 0 ~ 3 O g m c m ~ 3 (see Schramm and Wagoner,
1977, for discussion). Thus, a value of £ greater than about 3 or less than
about 5 would do serious violation to observed helium abundances. Since
the scale a{t) of the universe was 109 times smaller at this epoch than at
present, this is a very restrictive result for a generic theory of gravity.
Other possible tests of cosmological models, such as the question of
galaxy formation or the problem of the observed ratio of the number of
photons to the number of baryons (nY/nb ~ 108) are so poorly understood
within general relativity that they are unlikely to be useful tools for
testing alternative theories in the foreseeable future.
13.2
Cosmological Tests of Alternative Metric Theories of Gravity
For specific theories of gravity, results for the confrontation
between theory and cosmology are sparse. No systematic study of cosmological models in alternative theories has been carried out, and of those
analyses that have been performed within specific theories, few have
addressed such questions as the microwave background and the helium
abundance. Thus, we shall confine ourselves to a brief list, without details
and largely without comment, of those few results that are known.
General relativity
The "standard big bang model" (MTW, Section 28) agrees at
least qualitatively with all observations, although there remain problems
when one pushes for more precision or more detailed comparison with
observation such as galaxy formation, the photon-to-baryon ratio puzzle,
the initial singularity, the value of k, the abundances of deuterium and
the other light elements, the mean density of the universe, and so on.
Cosmological Tests
317
Brans-Dicke theory
Several computations (Greenstein 1968, Weinberg 1972) have
shown that a wide class of cosmological models in Brans-Dicke theory
are in qualitative agreement with all observations, including the helium
abundance. The models begin from a singular big bang as in general
relativity, one difference being the uncertainty in the boundary condition
to be placed on the scalar field (f> at t = 0. However, choices can be made
for this boundary condition that yield results similar to those of general
relativity for similar values of the present uncertain matter density p0.
Moreover, the larger the value of a>, the closer the agreement with general
relativity. In all cases, the present value of G/G is below the experimental
uncertainty (see Chapter 8).
Bekenstein's variable-mass theory (VMT)
By contrast with Brans-Dicke theory, the VMT can have cosmological models that begin the expansion from a nonsingular "bounce"
(which presumably was preceded by a contraction phase). Bekenstein
and Meisels (1980) have studied a variety of such models that satisfy the
following constraints: at the initial moment of expansion, /($) is small (Equation 5.40), i.e., co{4>) ~ § (required for the model to start from a minimum radius), and a c± 1016-1017 cm (appropriate for initial temperatures of ~ 101 * K). After numerical integration of the field equations for a variety of values of the curvature parameter k and the arbitrary constants r and q (see Chapter 5), they reached the following conclusions: (i) Although the initial value of a> was quite small, its present value in many models exceeded 500, thus yielding close agreement with all experimental tests, and with the predictions of general relativity for neutron stars, black holes, gravitational waves, the binary pulsar, etc. (ii) The gravitational constant G decreased by between 36 and 40 orders of magnitude between the initial moment and the present, thereby accounting for the "large number" puzzle that Gm^/hc^ 10" 38 , where mp is the proton mass. Because of the large variation in G, this ratio was initially near unity, (iii) Despite the large variation in G, the present value of G/G, in most cases, was well below the experimental upper limits. Because the universe in these models began from a hot, dense (though nonsingular) state, it permits origins for the cosmic microwave radiation as naturally as does general relativity. However, the helium abundance remains an open question at this writing. At the time of nucleosynthesis (T ~ 109 K) the expansion rate would have been very different from that of general relativity, since ca was very small then (perhaps of order f). Only a Theory and Experiment in Gravitational Physics 318 detailed computation can determine whether there are VMT cosmological models that are consistent with the helium abundance. Rosen's bimetric theory Because the theory has a flat background metric i/, there are no closed (k = 1) cosmological models. The Euclidean (k = 0) models have been studied by Babala (1975) and by Caves (1977). There are only two classes of models that have a physically reasonable expansion phase. One class expands from a singular state at a finite proper time in the past. These models make the definite prediction {G/G)o ;> 0.51H0[l + 3Om0(l + «2o)~'] (13.13) where Qm0 = 4npmO/3Hl, and a2o is the present value of the PPN parameter a2. Experiments (Chapter 8) place the limit |a20| « 1, and observations indicate Qm0 < 0.1 for Ho ^ 55 km s" 1 Mpc" 1 . This prediction could thus be tested by future measurements or limits on (G/G)o. The other class of models have a bounce at a minimum radius given by (Caves, 1977) amjao £ [1 + (1 + a 2 0 )/3Q m 0 r 2 £ <To (13.14) too large to permit a natural origin of the microwave background. The open (k = 1) models have, among other possibilities, expansion from a singular state at finite proper time in the past, and a similar expansion from a singular state at an infinite proper time in the past (Goldman and Rosen, 1976). These models have not been meshed with the present values of the PPN parameters, Ho, or {G/G)o. Rosen (1978) has also studied models in which the background metric t\ is notflat,but rather corresponds to a spacetime of constant curvature. The helium abundance has not been studied in any models in the bimetric theory. Rastall's theory As in Rosen's theory, the presence of a flat background metric rules out closed (k = 1) cosmological models. Rastall (1978) has shown that the k = 0 models predict a contraction phase, a nonsingular bounce, then an expansion phase. However, the bounce occurs at a radius fl min/«o TS> t o ° large to provide an explanation of either the microwave background or the helium abundance. Although the results presented here are very sketchy, they illustrate an important lesson. For some theories of gravitation, cosmology may provide do-or-die tests. This applies particularly to theories whose Cosmological Tests 319 predictions for present-day gravitational phenomena (post-Newtonian limit, neutron stars, gravitational waves, and present cosmological observations) are indistinguishable from those of general relativity, viz. the VMT. For such theories, gravitational effects in the early universe may be sufficiently different from those predicted by general relativity that the cosmic microwave background and the abundances of the light elements may help to determine the most viable theory of gravitation. 14 An Update In this chapter, we present a brief update of the past decade of testing relativity. Earlier updates to which the reader might refer include "The Confrontation between General Relativity and Experiment: An Update " (Will, 1984), "Experimental Gravitation from Newton's Principia to Einstein's General Relativity" (Will, 1987), "General Relativity at 75: How Right Was Einstein?" (Will, 1990a), and "The Confrontation Between General Relativity and Experiment: a 1992 Update" (Will, 1992a). For a popular review of testing general relativity, see "Was Einstein Right?" (Will, 1986). 14.1 The Einstein Equivalence Principle (a) Tests of EEP Several recent experiments that constitute tests of the Weak Equivalence Principle (WEP) were carried out primarily to search for a "fifth-force" (Section 14.5). In the "free-fall Galileo experiment" performed at the University of Colorado (Niebauer, McHugh and Faller, 1987), the relative free-fall acceleration of two bodies made of uranium and copper was measured using a laser interferometric technique. The "Eot-Wash" experiment (Heckel et al., 1989; Adelberger, Stubbs et al., 1990) carried out at the University of Washington used a sophisticated torsion balance tray to compare the accelerations of beryllium and copper. The resulting upper limits on q [Equation (2.3)] from these and earlier tests of WEP are summarized in Figure 14.1 Dramatically improved " mass isotropy " tests of Local Lorentz Invariance (LLI) (Section 2.4(b)) have been carried out recently using lasercooled trapped atom techniques (Prestage et al., 1985; Lamoreaux et al., 1986; Chupp et al., 1989). By exploiting the narrow resonance lines made Theory and Experiment in Gravitational Physics 1 1 1 1 1 1 1 1 1 1 :T I - 321 i Renner 10r-9 " 10r io - 1free-Fall - Boulder Princeton 1 10" 10.-12 1 +«2) i 1900 i EoMVash | LURE Moscow 1 «i i I- 1 1920 1 t 1 1940 l i i 1960 1970 1 1 1980 1990 Year of experiment Figure 14.1. Selected tests of the Weak Equivalence Principle, showing bounds on r/, which measures fractional difference in acceleration of different materials or bodies. Free-fall and Eot-Wash experiments originally performed to search for the fifth force. Hatched and dashed line show current bounds on t] for gravitating bodies (test of the Strong Equivalence Principle) from lunar laser ranging (LURE). possible by the suppression of atomic collisions in the traps, these experiments have all yielded extremely accurate results, quoted as limits on the parameter 3 [Equation (2.13)] in Figure 14.2. In the THeju framework 2 (Section 2.6), S = !], where c0 and ce are re1 = spectively the limiting speed of test particles and the speed of light. Also included for comparison is the corresponding limit on 5 obtained from Michelson-Morley type experiments. Recent advances in atomic spectroscopy and atomic timekeeping have made it possible to test LLI by checking the isotropy of the one-way propagation of light (as opposed to the round-trip speed of light, as tested in the Michelson-Morley experiment). In one experiment, for example (" JPL" in Figure 14.2), the relative phases of two hydrogen maser clocks at two stations of NASA's Deep Space Tracking Network were compared over five rotations of the Earth by propagating a light signal one-way along an ultrastable fiberoptic link connecting them (Krisher, Maleki et al., An Update 322 1 1 1 1 1 1 1 i i i i Michelson-Morley « JPL Joos TPAl T 10"8 X, - Brillet-Hall I I 10" Hughes-Drever 10" 10,-20 T t 8= - e l l 1900 NIST l J l 1920 lHarvard_ U. Washington 1 1 1940 1 1 1 1960 1970 1 1980 1 * 1 1990 Year of experiment Figure 14.2. Selected tests of local Lorentz invariance showing bounds on parameter S, which measures degree of violation of Lorentz invariance in electromagnetism. Michelson-Morley, Joos, and Brillet-Hall experiments test isotropy of the round-trip speed of light, the later experiment using laser technology. Two-photon absorption (TPA) and JPL experiments test isotropy of the one-way speed of light. The remaining four experiments test isotropy of nuclear energy levels. Limits assume the speed of Earth is 300 km/s relative to the mean rest frame of the universe. 1990). In another ("TPA"), the isotropy of the Doppler shift was studied as a function of direction using two-photon absorption in an atomic beam (Riis et al., 1988). Although the bounds from these experiments are not as tight as those from mass-isotropy experiments, they probe directly the fundamental postulates of special relativity, and thereby of LLI. A number of novel tests of the gravitational redshift (Local Position Invariance) were carried out. The varying gravitational redshift of Earthbound clocks relative to the highly stable millisecond pulsar PSR 1937 + 21, caused by the Earth's monthly motion in the solar gravitational field around the Earth-Moon center of mass (amplitude 4000 km), has been measured to about 10 % (Taylor, 1987), and the redshift of stable oscillator clocks on the Voyager spacecraft caused by Saturn's gravitational field yielded a one percent test (Krisher, Anderson and Campbell, 1990). The Theory and Experiment in Gravitational Physics Pound-Rebka 10" IT 10,-2 I ,-3 10" ' -,- Null T T Y 1 323 Millisecond Pulsar Solar I Redshift t Redshift Y Saturn * H-Masser Y I960 1970 1980 Year of experiment 1990 Figure 14.3. Selected tests of local position invariance via gravitational redshift experiments, showing bounds on a, which measures degree of deviation of redshift from the formula Av/v = AU/c2. solar gravitational redshift has been tested to about 2 % using infrared oxygen triplet lines at the limb of the Sun (LoPresto, Schrader and Pierce, 1991). Figure 14.3 summarizes the bounds on a [Equation (2.21)] that result from these and earlier experiments. It is now routine to take redshift and time-dilation corrections into account in making comparisons between timekeeping installations at different altitudes and latitudes, and in navigation systems, such as the NAVSTAR Global Positioning System, which use Earth-orbiting atomic clocks. (b) The c2 formalism The THefi formalism (Section 2.6) can be applied to tests of local Lorentz invariance, but in this context it can be simplified (Haugan and Will, 1987; Gabriel and Haugan 1990). Since most such tests do not concern themselves with the spatial variation of the functions T, H, e, and fi, but rather with observations made in moving frames, we can treat them as spatial constants. Then by rescaling the time and space coordinates, the charges and the electromagnetic fields, we can put the THefi action in Equation (2.46) into the form f (1 - vy2dt +£ea > -c2B2)d4x, (14.1) An Update 324 where c2 = // 0 /r o e 0i u 0 = cl/cl. This amounts to using units in which the limiting speed c0 of massive test particles is unity, and the speed of light is c. If c # 1, LLI is violated; furthermore, the form of the action above must be assumed to be valid only in some preferred universal rest frame. The natural candidate for such a frame is the rest frame of the cosmic microwave background. The electrodynamical equations which follow from Equation (14.1) yield the behavior of rods and clocks, just as in the full THsfi formalism. For example, the length of a rod moving through the rest frame with velocity V in a direction parallel to its length will be observed by a rest observer to be contracted relative to an identical rod perpendicular to the motion by a factor 1 V2/2 + O(VA). Notice that c does not appear in this expression. The energy and momentum of an electromagnetically bound body which moves with velocity V relative to the rest frame are given by E = M R + ^M R F 2 + ^ M f F'F^, (14.2a) p> = M R V + 8M\V\ s (14.2b) s where M R = MoE% , Mo is the sum of the particle rest masses, E# is the electrostatic binding energy of the system, and SMI is the anomalous inertial mass tensor, given by SMf = - 2 [ ~ llgfif^+JS 8 *!, (14.3) where ^ E (14.4a) (14.4b) ab Note that (c~2 1) here corresponds to the parameter S plotted in Figure 14.2. The electromagnetic field dynamics given by Equation (14.1) can also be quantized, so that we may treat the interaction of photons with atoms via perturbation theory. The energy of a photon is ft times its frequency co, while its momentum is fico/c. Using this approach, one finds that the difference in round-trip travel times of light along the two arms of the interferometer in the Michelson-Morley experiment is given by L0(v2/c)(c~2 1). The experimental null result then leads to the bound on Theory and Experiment in Gravitational Physics 325 (c~2 1) shown on Figure 14.2. Similarly the anisotropy in energy levels is clearly illustrated by the tensorial term in Equation (14.2a); by evaluating £|Sl> for each nucleus in the various Hughes-Drever-type experiments and comparing with the experimental limits on energy differences, one obtains the extremely tight bounds also shown on Figure 14.2. The behavior of moving atomic clocks can also be analysed in detail (Gabriel and Haugan, 1990), and bounds on (c~2 1) can be placed using results from tests of time dilation and of the propagation of light (Riis et al., 1988; Krisher, Maleki et al., 1990; Will, 1992b). The bound obtained from the " J P L " test of the isotropy of the one-way speed of light (see below) was based on the prediction for the time dilation of hydrogen maser clocks (Gabriel and Haugan, 1990) namely , (14.5) where a = -f(l-c2). (14.6) (c) Kinematical frameworks for studying LLI There are a number of frameworks for studying tests of special relativity (or of LLI) that are kinematical in nature, dating back to H. P. Robertson [see Haugan and Will (1987) and Mac Arthur (1986) for recent reviews]. One particularly useful version was developed by Mansouri and Sexl (1977a,b,c) (see also Abolghasem, Khajehpour and Mansouri, 1988, 1989). It assumes that there exists a preferred universal reference frame E:(7", X) in which the speed of light is isotropic (with unit speed in the appropriate units). The transformation between £ and a moving inertial frame S:{t, x) is given by T=a-\t-e-x), l i i (14.7a) X = d- x-(d~ -b- )yvxY//w 2 + <wT, (14.7b) where w is the velocity of the moving frame, a, b, and dare functions of w2, and E is a vector determined by the procedure adopted for the global synchronization of clocks in S. In special relativity, the functions a, b, and a? have the special forms a'1 = b = y = (1 w2)'1'2, and d = 1, but E can be arbitrary, depending upon the procedure for synchronization; with either Einstein (round-trip light signals) or clock-transport synchronization, £ = w. In the low-velocity limit, it will be useful to expand the functions a, b, d, An Update 326 and E in powers of w2 using arbitrary parameters. Adopting a slightly different convention from Mansouri and Sexl, we write a(w)x, l+(a-4)w 2 + ( a 2 - 4 K + . . . , b(w)x l+0?+i)w 2 + 0?2+f)w4 + ..., (14.8a) (14.8b) d(w) xl+Sw2 (14.8c) + S2w4 + ..., .. (14.8d) In SRT, a, a2, fi /?2, 8 and <S2 all vanish, and with standard synchronization, so do s and e2. The physics that results from experiments should not depend on the synchronization procedure, except measurements which depend on a direct, one-time comparison of separated clocks. Thus a measurement of the absolute value of the speed of light in S by a time-of-flight technique between two points will depend on the synchronization of the two clocks (a particularly perverse choice of synchronization can make the apparent speed between those points infinite, for example). However, a study of the isotropy of the speed between the same two clocks as the orientation of the line connecting them varies relative to £ should not depend on how they were synchronized, as long as they were synchronized by some procedure initially. Similarly, a measurement of the Doppler shift of an atomic spectral line using a single "clock" as receiver of the signal should not depend on synchronization, provided that the velocity of the atom is expressed in terms of observables measured by a single clock. This point has been misunderstood by numerous authors who have argued against the efficacy of tests of the one-way speed of light. An advantage of the Mansouri-Sexl framework is that it allows one to understand explicitly the role of synchronization in a given experiment. A disadvantage of this and similar kinematical frameworks is that they do not allow for the dynamical effects revealed by the c2 framework. Thus, the transformation of Equation (14.7) must be understood as being based on measurements made by a standard rod and a standard atomic clock. Measurements made using different rods or clocks would not yield the same relationships between the two frames. Nevertheless, for some experiments, such as the JPL experiment or the two-photon-absorption (TPA) experiment which involved only a single type of atom or atomic clock and the propagation of light, the Mansouri-Sexl formalism can be put to good use (Will, 1992b). In the JPL experiment, for example, the phases of two hydrogen maser oscillators of frequency v separated by a baseline of L = 21 kilometers were Theory and Experiment in Gravitational Physics 327 compared by propagating a laser carrier signal along a fiberoptic link connecting them. The phase comparisons could be performed simultaneously at each end using signals propagated in both directions along the fiber. The phase differences were monitored over afive-dayperiod as the baseline rotated relative to the Earth's velocity w through the cosmic microwave background. The predicted phase differences as a function of direction are, to first order in w A^«2a(wn-wn0), (14.9) where$ = 2nvL, and where n and n,, are unit vectors along the direction of
propagation of the light, at a given time, and at the initial time, respectively.
The initial phase difference has been set arbitrarily to zero; this is
tantamount to choosing a convention for synchronization. The observed
limit on a diurnal variation in the relative phase resulted in the bound
|a| < 1.8 x 1(T4; this gives a limit on (c~ 2 -l) using Equation (14.6). The
bound from the TPA experiment was |«| < 1.4 x 10~6. The best bound from
such isotropy experiments comes from "Mossbauer-rotor" experiments
(Champeney, Isaak and Khan, 1963; Isaak, 1970), which test the isotropy
of time dilation between a gamma ray emitter on the rim of a rotating disk
and an absorber placed at the center; the result is |a| < 9 x 10~8.
(d) Other frameworks for analysing EEP
A number of alternative formalisms have been developed to
analyse EEP and Schiff's conjecture in detail. Ni (1977) devised an
extension of the THe/i formalism in which the action for test particles and
electromagneticfieldscouples minimally to a metric gm, but in which there
is an additional electromagnetic coupling to a scalar field of the form
(167T)-1 J \/(-g)</>£"v'"'FllvFpacl4x. EEP is satisfied if and only if <f> s 0. On the
other hand, electromagnetically bound test bodies satisfy WEP, but
experience anomalous torques if <p is non-zero. This model thus represents
a counterexample to the simple version of Schiff's conjecture. A bound
d(j>/dt < 0.1 Ho, where Ho is the Hubble parameter was set by showing that
this electromagnetic coupling would cause rotations in the plane of
observed (Carroll and Field, 1992). Ni (1987) has extended this formalism
to incorporate non-abelian gauge fields.
Bekenstein (1982) focussed on a particular model for violation of EEP:
a coupling of electromagnetism to a dynamical, dimensionless scalar field
that manifests itself as a spacetime variation of thefinestructure constant.
The dynamics of the scalar field is determined more or less uniquely by a
An Update
328
set of reasonable postulates together with the requirement that the
fundamental scale that determines its dynamics be of the order of but no
smaller than the Planck scale (Gh/c3)1'2 x 10~33 cm. He found, however,
that the spatial variation of the field is so severely constrained by the
Eotvos experiment that the length scale must be smaller than 10~3 Planck
lengths. This, he argued, rules out any variability of the fine structure
constant.
Coley (1982, 1983a,b,c) studied an extension of the THe/i formalism to
non-metric theories that possess both a metric and an independent affine
connection, retaining the restriction to static, spherically symmetric (SSS)
fields. The model contains seven independent functions, whose forms can
be constrained by various experimental tests of EEP.
Horvath et al. (1988) extended the THe/x formalism to include weak
interactions in a "gravitationally modified" standard model. Such a
formalism could be used to calculate explicitly the possible WEP-violating
effects of weak interactions, which were only estimated by Haugan and
(e) Is spacetime symmetric ?
Our statement of the metric theory postulates included the
assumption that the metric is symmetric, corresponding to a standard
pseudo-Riemannian spacetime. It turns out that a nonsymmetric metric,
even if coupled to matter fields in a universal way, does not satisfy the
postulates of EEP (Will, 1989; Mann and Moffat, 1981). Consider a class
of theories in which the action for charged test particles and electromagnetic fields coupled to gravity is given by the "minimally coupled"
form of Equation (3.20) where now gm =£ gyft and g1" is the inverse of gm
such that gM*gm = g^g^ = S"v. (Mann, Palmer and Moffat (1989) and
Gabriel et al. (1991a) consider a broader class of electromagnetic actions,
but the minimally coupled version illustrates the essential features.) We
consider nonsymmetric theories having the property that, in an SSS
gravitational field, a Cartesian coordinate system can be found in which
the nonsymmetric g^ takes the form
9W = -T(r),
g^H^Sij,
gm = -ga = L(r)n,
(14.10)
where T, H, and L are functions of r = |x|, «,. = xjr. The inverse of gm is
given by
gm = g~l
(14.11)
Theory and Experiment in Gravitational Physics
329
where (-g) = H3T{\~L2/HT).
Substituting into Equation (3.20), and
identifying Fm = Et and Ftj = eijkBk, we obtain
/ = - ! % f(f-W2«fc + £ea UtfA
a
J
a
J
+ ^- [{eE2-n-\B2-co{n-Bf]}dix,
ore J
(14.12)
where
e = (H/T)ll2(\ -L2/HTyw,
co = L2/HT.
ft = (/f/T)1/2(l
-L2/HT)i/2,
(14.13)
Apart from the term co(n B)2, this action is that of the THefi formalism.
The condition for validity of EEP, s = fx = (H/T)v2 for all r is violated by
the action (14.12), if L ^ 0 (nonsymmetric metric).
For example, in the THs/i formalism, the acceleration of an electrically
neutral, composite body of charged particles with total mass M and
internal electrostatic energy £ ES is given by Equation (2.117), dropping the
magnetostatic terms. In order to apply this directly to nonsymmetric
theories, it suffices to show that the to(n-B)2 term in Equation (14.12)
makes no contribution to a, to electrostatic order. This can be shown by
direct calculation, extending the Lightman-Lee procedure appropriately;
it can be seen heuristically by noting that, to the required order,
0[g(EES/M)], the only part of the vector potential A that results in a
contribution to the acceleration of the composite body is that part
produced by the acceleration of each charged particle in the external
gravitational field. This part of A is therefore parallel to g, and thus to n,
and hence the relevant part of n B vanishes; as a consequence, the a>(n B)2
term will have no effect, to the electrostatic order considered. Higher-order
magnetostatic effects will result from that term, but, as we saw in Section
2.4(a), these are significantly smaller than electrostatic effects. For systems
that move through the SSS field with velocity V, the co(n B)2 terms will also
produce effects of order V2EEB/M (Gabriel et al., 1991b).
The given forms of s and fi imply that T^H^E^
= 1. Assuming that
7"« 1 + 0(m/r), Hx\+ O(m/r), and L2 <4 TH, where m is the mass of
the external source, we obtain from Equations (14.13) and (2.83),
To as (r2/2m)dL2/dr.
(14.14)
Thus nonsymmetric theories in this class violate WEP, and consequently,
Eotvos experiments can test their validity. The significance of the resulting
An Update
330
constraints on the nonsymmetric part of the metric will depend on the
specific form of L(r).
In one nonsymmetric theory, Moffat's NGT (Moffat 1979a,b, 1987,
1989; Moffat and Woolgar, 1988; for a recent review see Moffat, 1991),
L(r) = / 2 /> 2 , where / 2 is a parameter (which can be negative) defined by
e1 = f^/(-g)S°d3x,
where S" is a conserved current (W(-g)S")i/1 = 0)
of hitherto unspecified microscopic origin, and the integral is over the
gravitating source. Thus, in this theory, with the minimal coupling of
Equation (3.20), r o = 2^/mr3. However, because of an additional matter
coupling in the Lagrangian of NGT, there is an extra WEP-violating term
in the gravitational acceleration of a body that depends on the value of its
^-parameter, namely, <5a = g(2/ 2 /r 3 )(/ b 2 /M), where £ refers to the source
and <?b refers to the body. Combining the two terms, we obtain for the
parameter tj in minimally-coupled NGT,
Thus the constraint placed on NGT will depend on the model adopted for
the f2 parameter. For bulk, electrically neutral, stable matter consisting of
neutrons, protons, and electrons, it is straightforward to show that the
most general form of / 2 is f2 =fB2B+fL2L, where B and L are the total
baryon and lepton numbers of the body, and / B 2 and fL2 are arbitrary
coupling parameters (which can be negative) having units of (length)2.
Thus tests of WEP will constrain the fB2 /L2 plane. Because of the r~3
dependence in Equation (14.15), the most sensitive tests use the Earth as
the gravitating source, and for this purpose, the Eot-Wash III experiment
(Adelberger, Stubbs et al., 1990) is the most stringent. We determine EJM,
B/M and L/M for each of the test masses in this experiment, and we note
that, for the Earth, L9 « B^/2.05. The experimental limits from E6t-Wash
III then provide the constraints on the coupling parameters shown in
Figure 14.4. With the coupling parameters constrained by the rough bound
2 x 10"44 cm2, we obtain the limit \£92\ < (100 m)2.
It should be noted that this result applies only to the minimally coupled
electromagnetic action. Mann, Palmer and Moffat (1989) have presented
an alternative class of couplings of F^ to the nonsymmetric metric, one of
which satisfies WEP to electrostatic order, and thus evades the bound given
above. In this model, e = ju = (H/T)"2, and so the only EEP-violating
effects come from the o>(n-B)2 term in Equation (14.12).
In fact, this (n B)2 term is generic to all nonsymmetric theories, and has
important observable consequences. It will produce perturbations in the
Theory and Experiment in Gravitational Physics
4
331
-
-A -
Figure 14.4. Constraints on j \ and J\ of minimally coupled Moffat
NGT from the Eot-Wash III experiment. The hatched region is excluded.
energy levels of an atomic system that depend on the orientation of the
system's wave function relative to the direction n (anisotropies in inertial
mass). Such perturbations can be constrained by energy-isotropy experiments of the type used to test local Lorentz invariance (Gabriel et al.,
1991b). The violation of EEP by this term also produces observable effects
in the propagation of light, such as polarization dependence in the
propagation of light near the Sun (Gabriel et al., 1991c). One consequence
of this is a depolarization of the Zeeman components of spectral lines
emitted by extended, magnetically active regions near the limb of the Sun;
observations of the residual polarization of such lines place the stringent
bound Ko2| < (535 km)2, substantially smaller than the values preferred by
Moffat (1991).
14.2
The PPN Framework and Alternative Metric Theories of Gravity
The PPN framework of Chapter 4 is the standard tool for studying
experiments and gravitational theories in the weak-field slow motion limit
appropriate to the solar system. Other versions of the PPN formalism have
been developed to deal with bodies with strong internal gravity (Nordtvedt,
1985), and post-post-Newtonian effects (Epstein and Shapiro, 1980;
Fischbach and Freeman, 1980; Richter and Matzner, 1982a,b; Nordtvedt,
An Update
332
1987; Benacquista and Nordtvedt, 1988; Benacquista, 1992). A version of
the formalism with potentials substantially more complicated than the
canonical version has also been proposed (Ciufolini, 1991).
Despite the experimental bound of co > 500 on the coupling constant of
Brans-Dicke theory, variants of the theory became popular again during
the 1980s, as a result of developments in cosmology and elementaryparticle physics. Inflationary models of cosmology involving Brans-Dickelike scalar fields coupled to gravity have been developed and studied in
detail. Scalar fields coupled to gravity or matter are also ubiquitous in
particle-physics-inspired models of unification, such as string theory. In
many models, the coupling to matter leads to violations of WEP, which can
be tested by Eotvos-type experiments. In many models the scalar field is
massive; if the Compton wavelength is of macroscopic scale, its effects are
those of a "fifth force" (see Section 14.5). Only if the theory can be cast as
a metric theory with a scalar field of infinite range or of range long
compared to the scale of the system in question (solar system) can the PPN
framework be applied. If the mass of the scalarfieldis sufficiently large that
its range is microscopic, then, on solar-system scales, the scalar field is
suppressed, and the theory is typically equivalent to general relativity. In
any event, the bounds from solar system experiments can provide
constraints on such speculations. The post-Newtonian limit of a class of
massive scalar-tensor theories, including the Yukawa potentials that result
from the massive scalar field, was derived by Helbig (1991) and Zaglauer
(1990).
14.3
Tests of Post-Newtonian Gravity
(a) The classical tests
Improvements in the accuracy of very long baseline interferometry
(VLBI) to the level of hundreds of microarcseconds made new tests of the
deflection of light possible. For example, a series of transcontinental and
primarily to monitor the Earth's rotation ("VLBI" in Figure 14.5) was
sensitive to the deflection of light over almost the entire celestial sphere (at
90° from the Sun, the deflection is still 4 milliarcseconds). The data yielded
a value 5(1+7)= 1.000 + 0.001, comparable to the Viking test of the
Shapiro time delay (Robertson and Carter, 1984; Robertson, Carter and
Dillinger, 1991; Shapiro, 1990). A measurement of the deflection of light
by Jupiter using VLBI was recently reported (Truehaft and Lowe, 1991);
50% accuracy.
Theory and Experiment in Gravitational Physics
'ft'
1.2
T
1919
Expedition
333
Deflection of light
optical
1.1
VLBI
VLBI
1.0
0.9
Expedition
1.1
PSR 1937 + 21
Viking
IVoyager
JVo
1.0
Shapiro time delay
4 two-way
D one way
0.9
0.8
-
J L
_L
J_
1920 1930 1940 1950 1960 1970 1980
Year of experiment
1990
Figure 14.5. Measurements of the coefficient (l + y)/2 from light
deflection and time delay measurements. The general relativity value is
unity. Arrows denote anomalously large values from 1929 and 1936
expeditions. Shapiro time-delay measurements using Viking spacecraft
and VLBI light deflection measurements yielded agreement with general
relativity to 0.1 per cent.
Recent" opportunistic " measurements of the Shapiro time delay include
a measurement of the one-way time delay of signals from the millisecond
pulsar PSR 1937 + 21 (Taylor, 1987), and measurements of the two-way
delay from the Voyager 2 spacecraft (Krisher, Anderson and Taylor,
1991). The results for the coefficient 5(1 + y) of all light deflection and timedelay measurements performed to date are shown in Figure 14.5.
Continued radar ranging to Mercury and the other planets has resulted
in further improvements in the measured perihelion shift of Mercury. After
the perturbing effects of the other planets have been accounted for, the
excess shift is now known to about 0.1 % (Shapiro 1990) with the result
that & = 42"98 (1.000 + 0.001)^' [see Equation (7.55)]. [For an amusing
history of how the theoretical value of 42"98 has been misquoted in
An Update
334
numerous books, including the first edition of this book, see Nobili and
moment may be approaching a resolution. Beginning around 1980, the
observation and classification of modes of oscillation of the Sun have made
it possible to obtain information about its internal rotation rate, thereby
constraining the possible centrifugal flattening that leads to an oblateness;
current results favor a value J2 as 1.7 x 10~7 (Brown et al., 1989), making
the correction to Xp [Equation (7.55)] from the solar quadrupole moment
smaller than the experimental error. If further studies of solar oscillations
continue to support this interpretation, the perihelion shift of Mercury will
once again be a triumph for general relativity.
(b) Parametrized post-Newtonian ephemerides
Improvements in the accuracy of planetary and spacecraft
tracking and in the ability of theorists to model their motions has made it
useful to adopt a slightly different attitude toward tests such as the time
delay and the perihelion shift. As we remarked in Section 7.2, the
measurement of the time delay of light involves a multiparameter leastsquares fit of tracking data to a model for the trajectory of the planet or
spacecraft and for the propagation of the radar signal. The " time delay "
as a distinct phenomenon is never measured directly. Similarly the
"perihelion shift" of Mercury is not observed, rather the least-squares
method estimates various parameters (ft, y, J2, etc.) that determine part of
the shift. Although this point of view takes some of the glamour out of the
subject, it is the standard approach in the analysis of relativistic solarsystem dynamics.
The goal is to determine the parameters in a model for the relativistic
motion of bodies in the solar system. One might call this model a
"parametrized post-Newtonian ephemeris". The current model (Hellings,
1984; Reasenberg, 1983) includes such parameters as: (i) the initial
positions and velocities of the nine planets and the Moon; (ii) the masses
of the planets, and of the three asteroids Ceres, Pallas and Vesta; (iii) the
mean densities of 200 of the largest asteroids whose radii are known; (iv)
the Earth-Moon mass ratio; (v) the value of the astronomical unit; (vi)
PPN parameters, y, fi, a,,...; (vii) J2 of the Sun; (viii) other parameters
relevant to specific data sets, such as station locations, rotation and
libration of bodies, known systematic errors or corrections, etc. The model
also includes PPN equations of motion for the bodies, and PPN equations
for the propagation of the tracking signal. In some applications, the model
also includes equations that tie the coordinate system associated with the
Theory and Experiment in Gravitational Physics
335
ephemerides to a system tied to distant stars via VLBI. The output of the
model might, for example, be a predicted " range " (round-trip travel time)
from a particular station to a planet or spacecraft at a particular epoch, as
a function of the parameters. The parameters are then adjusted in the leastsquares sense to minimize the difference between the predicted and
observed ranges.
One circumstance that has made it possible to obtain improved
determinations of the parameters is the ability to combine different data
sets in an unambiguous way. In the orbit of Mercury, the effects of /?, y and
J2 are large, but not separable using Mercury radar data alone. In the orbit
of Mars, their effects are much smaller (and that of J2 smaller still than that
of fi and y), but the accuracy of Viking lander ranges is so much better that
the effects can be seen more clearly than with Mercury data. Lunar laserranging data has also been incorporated into the data set. In the coming
years, analysis of PPN ephemerides will further improve our knowledge of
PPN parameters, J2, and the dynamics of the solar system (for reviews, see
Kovalevsky and Brumberg, 1986; and Soffel, 1989).
(c) Tests of the strong equivalence principle
Recent analyses of lunar laser-ranging data continue to find no
evidence, within experimental uncertainty, for the Nordtvedt effect
(Section 8.1). Their results for n [Equation (8.9)] are
n = 0.003+0.004, (Dickey et al., 1989)
n = 0.000 ± 0.005, (Shapiro, 1990)
n = 0.0001 ±0.0015, (Muller et al., 1991)
(14.16)
where the quoted errors are \a, obtained by estimating the sensitivity of n
to possible systematic errors in the data or in the theoretical model.
The third of these results represents a limit on a possible violation of
WEP for massive bodies of 7 parts in 1013 (compare Figure 14.1). For
Brans-Dicke theory, these results force a lower limit on the coupling
constant co of 600. Nordtvedt (1988a) has pointed out that, at this level of
precision, one cannot regard the results of lunar laser ranging as a clean
test of SEP because the precision exceeds that of laboratory tests of WEP.
Because the chemical compositions of the Earth and Moon differ in the
relative fractions of iron and silicates, an extrapolation from laboratory
Eotvos-type experiments to the Earth-Moon system using various nonmetric couplings to matter (Adelberger, Heckel et al., 1990) yields bounds
on violations of WEP only of the order of 2 x 10"!2. Thus if lunar laser
An Update
336
Table 14.1. Constancy of the gravitational constant
Method
Lunar Laser Ranging
Binary Pulsar"
Pulsar PSR 0655 + 64"
G/G (10-12 yr"1)
0+11
2+4
-2±10
11±11
<55
Reference
Miiller et al. (1991)
Hellings et al. (1983)
Shapiro (1990)
Damour and Taylor (1991)
Goldman (1990)
" Bounds dependent upon theory of gravity in strong-field regime and on
neutron star equation of state.
ranging is to test SEP at higher accuracy, tests of WEP must keep pace; to
this end, a proposed satellite test of the equivalence principle (Section 14.4)
In general relativity, the Nordtvedt effect vanishes; at the level of several
centimeters and below, a number of non-null general relativistic effects
should be present (Mashhoon and Theiss, 1991; Gill et al., 1989;
Nordtvedt, 1991).
An improved limit on the "preferred frame" PPN parameter a, of
4x 10~4 was reported by Hellings (1984), from analyses of Mercury and
Mars ranging data. Nordtvedt (1987) has placed an improved bound on
the parameter <x2 of 4 x 10~7 by showing that the failure of conservation
of angular momentum in a frame moving relative to the universe when
a2 / 0 [Equations (4.104) and (4.114)] would lead to anomalous torques
on the Sun that would cause the angle between its spin axis and the ecliptic
to be arbitrarily far from its observed value.
Improved observational constraints have recently been placed on G/G,
using ranging measurements to Viking (Hellings et al., 1983; Shapiro,
1990), lunar laser-ranging measurements (Miiller et al., 1991), and pulsar
timing data (Damour, Gibbons and Taylor, 1988; Goldman, 1990;
Damour and Taylor, 1991). Recent results are shown in Table 14.1. The
best limits on G/G come from ranging measurements to Viking. The
combination of three factors: (i) extremely accurate range measurements
made possible by anchoring of the landers and orbiters, (ii) the unexpectedly long lifetime of the spacecraft (Lander 2 survived for 6 years), and (iii)
the ability to combine Viking data consistently with other data sets such as
data, made it possible to look for G/G at levels below 10~u yr""1. The major
factors limiting the accuracy of these estimates (and responsible in part for
Theory and Experiment in Gravitational Physics
337
the difference between the two Viking estimates in Table 14.1, despite being
based upon similar data sets) are the uncertainty in the masses and
distributions of the asteroids, and the level of correlations among the many
parameters to be estimated in the model. It has been suggested that radar
observations of a Mercury orbiter over a two-year mission (30 cm accuracy
in range) could yield A(G/G) ~ l O ^ y r 1 (Bender et al., 1989).
Although bounds on G/G using solar-system measurements can be
obtained in a phenomenological manner through the simple expedient of
replacing G by G0 + G0(t10) in Newton's equations of motion, the same
does not hold true for pulsar and binary pulsar timing measurements
(Nordtvedt 1990). The reason is that, in theories of gravity that violate
SEP, the "mass" and moment of inertia of a gravitationally bound body
may vary with variation in G. Because neutron stars are highly relativistic,
the fractional variation in the mass can be comparable to AG/G, the
precise variation depending both on the equation of state of neutron star
matter and on the theory of gravity in the strong-field regime. The
variation in the moment of inertia affects the spin rate of the pulsar, while
the variation in the mass can affect the orbital period in a manner that
can add to or subtract from the direct effect of a variation in G, given by
PJPb = -jG/G. Thus, the bounds quoted in Table 14.1 for the binary
pulsar PSR 1913 + 16 and the pulsar PSR 0655 + 64 are theory dependent
and must be treated as merely suggestive.
(d) Tests of post-Newtonian conservation laws
Of the five "conservation law" PPN parameters £ f2, £3, £4, and
<x3, only three, C2> C3 and <x3, have been constrained directly with any
precision. The bound |<x3| < 2 x 10~10 was obtained in Section 9.3 using
pulsar timing measurements.
A remarkable planetary test of Newton's third law was reported by
Bartlett and van Buren (1986), leading to an improved constraint on £3
(Section 9.2). They noted that current understanding of the structure of the
Moon involves an iron-rich, aluminum-poor mantle whose center of mass
is offset about 10 km from the center of mass of an aluminum-rich, ironpoor crust. The direction of offset is toward the Earth, about 14° to the east
of the Earth-Moon line. Such a model accounts for the basaltic maria
which face the Earth, and the aluminum-rich highlands on the Moon's far
side, and for a 2 km offset between the observed center of mass and center
offigurefor the Moon. Because of this asymmetry, a violation of Newton's
third law for aluminum and iron would result in a momentum nonconserving self-force on the Moon, whose component along the orbital
An Update
338
direction would contribute to the secular acceleration of the lunar orbit.
Improved knowledge of the lunar orbit through lunar laser ranging, and a
better understanding of tidal effects in the Earth-Moon system (which also
contribute to the secular acceleration) through satellite data, severely limit
any anomalous secular acceleration, with the resulting limit
K/«p)Fe
<4xlO- 12 .
(14.17)
The resulting limit on £3 is |C3| < 1 x 10~8.
Data from the binary pulsar PSR 1913 + 16 have finally permitted a
strong test of the post-Newtonian " self-acceleration" effect described in
Section 9.3, Equation (9.42). Assuming a theory not too different from
general relativity (but with the possibility of C2 # 0) so that we can use the
accurate values for the pulsar and companion masses obtained from timing
data (Section 14.6(a)), together with the observational bound on variations
in the pulsar period \Pp\ < 4 x 10~30 s~' (Taylor and Weisberg, 1989;
J. Taylor, private communication), we obtain from Equation (9.48) the
bound \C2\ < 4 x 10~5 (Will, 1992c).
(e) Other tests of post-Newtonian gravity
A gyroscope moving through curved spacetime suffers a geodetic
precession of its axis given by dS/dt = Q x S, where £2 = (7 + j)v x V{7,
where v is the velocity of the gyroscope and U is the Newtonian
gravitational potential of the source [Equation (9.5)]. The Earth-Moon
system can be considered as a " gyroscope ", with its axis perpendicular to
the orbital plane. The predicted geodetic precession here is about 2
arcseconds per century, an effect first calculated by de Sitter. This effect has
now been measured to about 2 % using lunar laser-ranging data (Bertotti,
Ciufolini and Bender, 1987; Shapiro et al., 1988; Dickey, Newhall and
Williams, 1989; Shapiro, 1990).
Current values or bounds for the PPN parameters are summarized in
Table 14.2.
14.4
Experimental Gravitation: Is there a Future?
Although the golden era of experimental gravitation may be over,
there remains considerable opportunity both for refining our knowledge of
gravity, and for exploring new regimes of gravitational phenomena.
Nowhere is the intellectual vigor and continuing excitement of this field
more apparent than in the ideas that have been developed for experiments
and observations to push us to the frontiers of knowledge.
Theory and Experiment in Gravitational Physics
339
Table 14.2. Current limits on the PPN parameters
Parameter
Experiment
Value or limit
Remarks
7
Time delay
Light deflection
1.000 ±0.002
1.000 + 0.002
Viking ranging
VLBI
Perihelion shift
Nordtvedt effect
1.000 + 0.003
1.000 ±0.001
J2 = 10~7 assumed
rj 4/?y 3 assumed
Earth tides
Orbital preferred-frame
effects
< 10~3
Gravimeter data
<4xlO"
Combined solar system data
a2
Earth tides
Solar spin precession
<4xlO-4
<4xlO~ 7
Gravimeter data
Assumes alignment of solar
equator and ecliptic are not
coincidental
a3
Perihelion shift
Acceleration of pulsars
< 2 x 10~7
<2xlO~'°
Statistics of dP/dt for pulsars
V
Nordtvedt effect
< 1.5 xlO" 3
Lunar laser ranging
Self-acceleration
Newton's 3rd law
<4xlO~ 5
< 10"8
Binary pulsar
Lunar acceleration
i
a,
r
4i
C3
" Here rj is a combination of other PPN parameters given by
}j = 4/?y 3 y<J a,+3a2§£, |C2. In many theories of gravity, £, = a, = £,. = 0.
(a) GP-B and the search for gravitomagnetism
According to general relativity, moving or rotating matter should
produce a contribution to the gravitational field that is the analogue of the
magnetic field of a moving charge or a magnetic dipole (for reviews of the
" gravitoelectromagnetic " analogy for weak-field gravity, see Braginsky,
Caves and Thorne, 1977; Ciufolini, 1989). Although gravitomagnetism
plays a role in a variety of measured relativistic effects, it has not been seen
to date, isolated from other post-Newtonian effects [Nordtvedt (1988b) has
discussed the extent to which it has been seen indirectly]. The Relativity
Gyroscope Experiment (Gravity Probe B or GP-B) at Stanford University,
in collaboration with NASA and Lockheed Corporation, has reached the
advanced stage of development of a space mission to detect this
phenomenon directly, in addition to the geodetic precession discussed in
Section 9.1 (Everitt et al., 1988). A set of four superconducting-niobiumcoated, spherical quartz gyroscopes will be flown in a low polar Earth
orbit, and the precession of the gyroscopes relative to the distant stars will
An Update
340
be measured. For a polar orbit at about 650 km altitude, the predicted
secular angular precession rate is j(l + y + |a,) 42 x 10"3 arcsec/yr [Equation (9.11)]. The accuracy goal of the experiment is about 0.5 milliarcseconds per year. A full-size flight prototype of the instrument package has
been tested as an integrated unit. Current plans call for a test of the final
flight hardware on the Space Shuttle followed by a Shuttle-launched
experiment a few years later.
Another proposal to look for an effect of gravitomagnetism is to
measure the relative precession of the line of nodes-of a pair of laser-ranged
geodynamics satellites (LAGEOS), with supplementary inclination angles;
the inclinations must be supplementary in order to cancel the dominant
relative nodal precession caused by the Earth's Newtonian gravitational
multipole moments (Ciufolini, 1989). This is a generalization of the van
Patten-Everitt proposal involving pairs of polar-orbiting satellites described in Section 9.1. Current plans involve a joint project of NASA and
the Italian Space Agency. A third proposal envisages orbiting an array of
three mutually orthogonal, superconducting gravity gradiometers around
the Earth, to measure directly the contribution of the gravitomagnetic field
to the tidal gravitational force (Braginsky and Polnarev, 1980; Mashhoon
and Theiss, 1982; Mashhoon, Paik and Will, 1989).
(b) Space tests of the Einstein equivalence principle
The concept of an Eotvos experiment in space has been developed,
with the potential to test WEP to 10"17 (Worden, 1988). Known as the
Satellite Test of the Equivalence Principle (STEP), the project is a joint
effort of NASA and the European Space Agency. If approved, it could be
launched in the year 2000.
The gravitational redshift could be improved to the 10~9 level, and
second-order effects and the effects of J2 of the Sun discerned, by placing
a hydrogen maser clock on board Solar Probe, a proposed spacecraft
which would travel to within four solar radii of the Sun (Vessot, 1989).
(c) Improved PPN parameter values
A number of advanced space missions have been proposed in
which spacecraft orbiters or landers and improved tracking capabilities
could lead to significant improvements in values of the PPN parameters
(see Table 14.2), of J2 of the Sun, and of G/G. For example, a Mercury
orbiter, in a two-year experiment, with 3 cm range capability, could yield
improvements in the perihelion shift to a part in 104, in y to 4 x 10~5, in G/G
to 10~14 y r 1 , and in J2 to a few parts in 108 (Bender et al., 1989).
Theory and Experiment in Gravitational Physics
341
(d) Probing post-post-Newtonian physics in the solar system
It may be possible to begin to explore the next level of corrections
to Newtonian theory beyond the post-Newtonian limit, into the post-postNewtonian regime. One proposal is to place an optical interferometer with
microarcsecond accuracy into Earth orbit. Such a device would improve
the deflection of light to the 10~6 level, and could possibly detect the
second-order term, which is of order 10 microarcseconds at the limb
(Reasenberg et al., 1988). Such a measurement would be sensitive to a new
" P P P N " parameter, which has not been measured to date.
(e) Gravitational-wave astronomy
A significant part of the field of experimental gravitation is
devoted to designing and building sensitive devices to detect gravitational
radiation and to use gravity waves as a new astronomical tool. This
important topic has been reviewed thoroughly elsewhere (Thorne, 1987).
14.5
The Rise and Fall of the Fifth Force
A clear example of the role of " opportunism" in experimental
gravity since 1980 is the story of the "fifth force". In 1986, as a result of a
detailed reanalysis of Eotvos' original data, Fischbach et al. (1986, 1988)
suggested the existence of a fifth force of nature, with a strength of about
a percent that of gravity, but with a range (as defined by the range A of a
Yukawa potential, e~'ix/r) of a few hundred meters. This proposal
dovetailed with earlier hints of a deviation from the inverse-square law of
Newtonian gravitation derived from measurements of the gravity profile
down deep mines in Australia [for a review, see Stacey et al. (1987)], and
with ideas from particle physics suggesting the possible presence of very
low-mass particles with gravitational-strength couplings [for reviews, see
Gibbons and Whiting (1981), Fujii (1991)]. During the next four years
numerous experiments looked for evidence of the fifth force by searching
for composition-dependent differences in acceleration, with variants of the
Eotvos experiment or with free-fall Galileo-type experiments. Although
two early experiments reported positive evidence, the others yielded null
results. Over the range between one and 104 meters, the null experiments
produced upper limits on the strength of a postulated fifth force of between
10~3 and 10~6 the strength of gravity (Table 14.3). Interpreted as tests of
WEP (corresponding to the limit of infinite-range forces), the results of the
free-fall Galileo experiment, and of the Eot-Wash III experiment are
shown in Figure 14.1 (Niebauer, McHugh and Faller, 1987; Adelberger,
Stubbs et al., 1990). At the same time, tests of the inverse square law of
Table 14.3. Composition-dependent tests of the fifth force
Experiment
name
or place
Year
Method
Eot-Wash
Boulder, CO
Eot-Wash
Index, WA
Montana
Paris
Bombay
Snake River
Eot-Wash II
Japan
Florence
Bombay II
Irvine, CA
Eot-Wash III
Index, WA II
Florence II
Japan II
1986
1986
1987
1987
1987
1988
1988
1988
1988
1988
1988
1988
1989
1989
1989
1989
1989
1990
Flotation
Torsion balance
Free fall
Torsion balance
Torsion balance
Torsion balance
Beam balance
Torsion balance
Torsion balance
Torsion balance
Free fall
Flotation
Torsion balance
Torsion balance
Torsion balance
Torsion balance
Flotation
Free fall
Substance
compared
Source of
force
Fifth
force?
Cu/H 2 O
Cu/Be
Cu/U
Be/Al
Be/Al
Cu/CH 2
Cu/Pb, C/Pb
Cu/Pb
C/Pb
Be/Al
Al/Cu, Al/C
Plastic/H 2 O
Cu/Pb
Cu/Pb
Cu/Be, Al/Be
Cu/CH 2
Cliff
Hillside
Earth
Hillside
Cliff
Hillside
Water in lock
Earth
Mountain
Hillside
Cliff
Mountain
Earth
Yes
No
No
No
Yes
No
No
No
No
No
No
No
No
No
No
No
No
No
Plastic/Hp
Al/Cu,
Al/C, Al/Be
Theory and Experiment in Gravitational Physics
343
gravity were carried out by comparing variations in gravity measurements
up tall towers or down mines or boreholes with gravity variations predicted
using the inverse square law together with Earth models and surface
gravity data mathematically "continued" up the tower or down the hole.
Early experiments reported significant differences between predicted and
observed gravity, but these were subsequently explained as resulting from
systematic errors in the upward continuation results caused by insufficiently controlled biases in the distribution of surface gravity measurements, as well as by poorly-accounted-for effects of distant geological
structures such as hills and ridges. Independent tower, borehole and
seawater measurements now show no evidence of a deviation from the
inverse square law (Thomas et al., 1989, Jekeli, Eckhardt and Romaides,
1990; Thomas and Vogel, 1990; Speake et al., 1990; Zumberge et al.,
1991). The consensus at present is that there is no credible experimental
evidence for a fifth force of nature. For reviews, see Fischbach and
complete bibliography on the fifth force, see Fischbach et al. (1992).
14.6
Stellar-System Tests of Gravitational Theory
(a) The binary pulsar and general relativity
The binary pulsar PSR 1913 + 16 has lived up to, indeed exceeded,
all expectations that it would be an important new testing ground for
relativistic gravity (Chapter 12). Instrumental upgrades at the Arecibo
radio telescope where the observations are carried out, and improved data
analysis techniques have resulted in accuracies in measuring times of
arrival (TOA) of pulses at the 15 /us level. Analysis of this TOA data uses
a timing model developed by Damour, Deruelle and Taylor (Damour and
Deruelle, 1986; Damour and Taylor, 1992) superceding earlier treatments
by Haugan, Blandford, Teukolsky and Epstein that were described in
Section 12.1 [see Haugan (1985) and references therein].
The observational parameters of this model that are obtained from a
least squares solution of the arrival time data fall into three groups: (i) nonorbital parameters, such as the pulsar period and its rate of change, and the
position of the pulsar on the sky; (ii) five "Keplerian" parameters, most
closely related to those appropriate for standard Newtonian systems, such
as the eccentricity e and the orbital period Ph; and (iii) a set of "postKeplerian " parameters. Thefivemain post-Keplerian parameters are <<»>,
the average rate of periastron advance; y, the amplitude of delays in arrival
of pulses caused by the varying effects of the gravitational redshift and time
dilation as the pulsar moves in its elliptical orbit at varying distances from
An Update
344
the companion and with varying speeds [denoted <$in Section 12.1(d)]; Pb, the rate of change of orbital period, caused predominantly by gravitational radiation damping; and r and s = sin i, respectively the "range" and "shape" of the Shapiro time delay caused by the companion, where Us the angle of inclination of the orbit relative to the plane of the sky. In general relativity, these post-Keplerian parameters can be related to the masses of the two bodies and to measured Keplerian parameters by the equations (Section 12.2) <«> = 3(27t/Pb)5/3w2/3(l -e2Y\ i3 m y = e(Pb/27iy rn2m- (\ +m2/m), Pb = -(192 K /5)(27rm/P b ) 5 ' 3 (^/m)(l + g e 2 + ||e 4 )(l (14.18a) (14.18b) -e2)-1'2, (14.18c) s^sini, (14.18d) r = m2, (14.18e) where m, and m2 denote the pulsar and companion masses, respectively, m = m, + m2 is the total mass, and n = mxm2/m is the reduced mass. The formula for <a>> ignores possible non-relativistic contributions to the periastron shift, such as tidally or rotationally induced effects caused by the companion [Section 12. l(c)]. The formula for Pb represents the effect of energy loss through the emission of gravitational radiation, and makes use of the "quadrupole formula" of general relativity. For a recent survey of the quadrupole and other approximations for gravitational radiation, see Damour (1987). It ignores other sources of energy loss, such as tidal dissipation [Section 12.1(f)]. The values for the Keplerian and post-Keplerian parameters shown in Table 14.4 are from data taken through December 1990 (Taylor et al., 1992). Plotting the constraints the three post-Keplerian parameters imply for the two masses w, and m2, via Equations (14.18), we obtain the curves shown on Figure 14.6. It is useful to note that Figure 12.2 corresponds essentially to the inset in Figure 14.6. From <a>> and y we obtain the values m, = 1.4411(7) MQ and w 2 = 1.3873(7) Mo, where the number in parenthesis denotes the error in the last digit. Equation (14.18c) then predicts the value Pb = 2.40243(5) x 10~12. In order to compare the predicted value for Pb with the observed value, it is necessary to take into account the effect of a relative acceleration between the binary pulsar system and the solar system caused by the differential rotation of the galaxy. This effect was previously considered unimportant when Pb was Table 14.4. Parameters of the binary pulsar PSR 1913 + 16" Parameter (i) 'Physical' parameters Right ascension Declination Pulsar period Derivative of period 2nd derivative of period (ii) 'Keplerian' parameters Projected semimajor axis Eccentricity Orbital period Longitude of periastron Julian ephemeris date of periastron (iii) 'Post-Keplerian' parameters Mean rate of periastron advance Gravitational redshift and time dilation Orbital period derivative 3 Symbol (units) Value a <5 Pp (ms) 19h13m12.s46549(15) 16°01'08':i89(3) 59.029997929883(7) 8.62629(8) x 10"18 < 4 x 10^30 ap sin i (light sec) e P*(») «o(°) Ta (MJD) 2.341759(3) 0.6171309(6) 27906.9807807(9) 226.57531(9) 46443.99588321(5) <ri>> C y r 1 ) y(ms) Pb (10-12) 4.226628(18) 4.294(3) -2.425(10) Numbers in parentheses denote errors in last digit. An Update 346 i \ ' \ 1 1 ' 2 1.41 - 3rovo 1 1 () 1.39 -" " companion 1.40 1 ^s 1 2 3 - - Mass o 1.38 '-. 1.37 - ' . _ \ . 1.42 1.43 1.44 1.45 Mass of pulsar ( M Q ) - 1.46 Figure 14.6. Constraints on masses of pulsar and companion from data on PSR 1913 + 16, assuming general relativity to be valid. The width of each strip in the plane reflects observational accuracy, shown as a percentage. The inset shows the three constraints on the full mass plane; intersection region (a) has been magnified 400 times for the full figure. known only to 10% accuracy [Section 12.1(f)(iii)]. Damour and Taylor (1991) carried out a careful estimate of this effect using data on the location and proper motion of the pulsar, combined with the best information available on galactic rotation, and found P ° A L ~ -(1.7 + 0.5) xlO" 14 . (14.19) Subtracting this from the observed Pb (Table 14.4) gives the residual P£BS = -(2.408 ± 0.010[OBS]±0.005[GAL]) x lO"12, (14.20) which agrees with the prediction, within the errors. In other words, pGR -5^g= = 1.0023±0.0041(OBS)±0.0021(GAL). (14.21) The parameters r and J are not yet separately measurable with interesting Theory and Experiment in Gravitational Physics 347 accuracy for PSR 1913 + 16 because the 47° inclination of the orbit does not lead to a substantial Shapiro time delay. The internal consistency among the measurements is also displayed in Figure 14.6, in which the regions allowed by the three most precise constraints have a single common overlap. This consistency provides a test of the assumption that the two bodies behave as "point" masses, without complicated tidal effects (conventional wisdom holds that the companion is also a neutron star), obeying the general relativistic equations of motion including gravitational radiation. It is also a test of the Strong Equivalence Principle (SEP), in that the highly relativistic internal structure of the neutron star does not influence its orbital motion or the gravitational radiation emission, as predicted by general relativity. (b) A population of binary pulsars ? In 1990, two new massive binary pulsars similar to PSR 1913 + 16 were discovered, leading to the possibility of new or improved tests of general relativity. PSR 2127+11C. This system appears to be a clone of the HulseTaylor binary pulsar (Anderson et al., 1990; Prince et al., 1991): Pb 28,968.36935 s, e = 0.68141, < cb > = 4.457° yr"1 (see Table 14.5). The inferred total mass of the system is 2.706 + 0.011 MQ. Because the system is in the globular cluster Ml5 (NGC 7078), observed periods Pb and Pp will suffer Doppler shifts resulting from local accelerations, caused either by the mean cluster gravitational field or by nearby stars, that are more difficult to estimate than was the case with the galactic system PSR 1913 + 16. This may limit the accuracy of measurement of the relativistic contribution to Ph to about 2 % . PSR 1534 + 12. This is a binary pulsar system in our galaxy (Wolszczan, 1991). Its pulses are significantly stronger and narrower than those ofPSR1913 + 16,so timing measurements have already reached 3 ^s accuracy. Its parameters are listed in Table 14.5 (Taylor et al., 1992). Because of the short data span, Pb has not been measured to date, but it is expected that in a few years, the accuracy in its determination will exceed that of PSR 1913+16. The orbital plane appears to be almost edge on relative to the line of sight (i « 80°); as a result the Shapiro delay is substantial, and separate values of the parameters r and 5 have already been obtained with interesting accuracy. This system may ultimately provide broader and more stringent tests of the consistency of general relativity than did the original binary pulsar (Taylor et al., 1992). Table 14.5. Parameters of new binary pulsars" Parameter PSR 1534+12 PSR 2127+11C 15h34m47.s686(3) 12°05'45"23(3) 37.9044403665(4) 2.43(8) xlO~18 21h27m36.s188(4) 11°57'26!29(7) 30.5292951285(9) 4.99(5) xlO" 18 (ii) ' Keplerian' parameters Projected semimajor axis Eccentricity Orbital period Longitude of periastron Julian ephemeris date of periastron 3.729468(9) 0.2736779(6) 36351.70270(3) 264.9721(16) 48262.8434966(2) 2.520(3) 0.68141(2) 28968.3693(5) 316.40(7) 47632.4672065(20) (iii) 'Post-Keplerian' parameters Mean rate of periastron advance Gravitational redshift and time dilation Orbital period derivative Range of Shapiro delay r (jis) Shape of Shapiro delay 5 = sin i 1.7560(3) 2.05(11) -0.1(6) 6.2(1.3) 0.986(7) 4.457(12) * * * * (i) 'Physical' parameters Right ascension Declination Pulsar period Derivative of period " Numbers in parentheses denote errors in last digit. * Values not yet available from data. Theory and Experiment in Gravitational Physics 349 (c) Binary pulsars and scalar-tensor theories In Section 12.3, we noted that some theories of gravity, such as the Rosen bimetric theory, are strongly, even fatally, tested by the binary pulsar. Other theories that are in some sense "close" to general relativity in all their predictions, such as the Brans-Dicke theory, are not so strongly tested, because the apparent near equality of the masses of the two neutron stars leads to a suppression of dipole gravitational radiation. Despite this, two circumstances have made it worthwhile to focus in detail on binary pulsar tests of scalar-tensor theories. The first is the remarkable improvement in accuracy of the measurements of the orbital parameters of the binary pulsar since 1980, and the continued consistency of the observations with general relativity, as described above, together with the discovery of new binary pulsars such as PSR 1534+12. The second is the resurrection of scalar-tensor theories in particle physics and cosmology. With this motivation, Will and Zaglauer (1989) carried out a detailed study of the effects of Brans-Dicke theory in the binary pulsar. Making the usual assumption that both members of the system are neutron stars, and using the methods summarized in Chapters 10-12, one obtains formulas for the periastron shift, the gravitational redshift/second-order Doppler shift parameter, and the rate of change of orbital period, analogous to Eqs. (14.18c). These formulas depend on the masses of the two neutron stars, on their internal structure, represented by "sensitivities" s and K* and on the Brans-Dicke coupling constant a>. First, there is a modification of Kepler's third law, given by Pb/2n = (a}/^my'2. Then, the predictions for <a>>, y and Pb are 3 , i3 1 3 y = e(Pb/2ny m2m- '^-" Pb = (14.22a) (a* + #/n 2 /w+ <>/?), (14.22b) w -{\92n/5){2nm/Pby»(ji/m)>$- F{e)
(14.22c)
where, to first order in £, = (2+a)~\ assuming cop 1, we have
0 = 1 -i(sl +52-2^2),
& = 9[l -#+#(Si
a2* = l - £ s 2 ,
(14.23a)
+s2-2slS2)],
(14.23b)
(14.23c)
n* = (l-2s2)£,
(14.23d)
2 7/2
' F(e) = ft} -e r [/c,(l +y + P)-K2Qe>
+
p)]t
(14.23e)
An Update
350
(14.23f)
2
-^r )],
(14.23g)
(14.23h)
T'=l-sx~s2,
2
G(e) = (1 -e )"
5/2
(14.23i)
2
(l -4e ),
^ = s,-s2.
(14.23J)
(14.23k)
The quantities ,?a and K* are defined by
d
J^>\ ,* = JdJ±m,
(14.24)
and measure the " sensitivity " of the mass wa and moment of inertia / a of
each body to changes in the scalar field (reflected in changes in G) for fixed
baryon number N (see Section 11.3).
The first term in Pb is the effect of quadrupole and monopole
gravitational radiation, while the second term is the effect of dipole
radiation (in Section 11.3 we calculated only the dipole contribution).
Estimating the sensitivities i a and K* using an equation of state for
neutron stars sufficiently stiff to guarantee neutron stars of sufficient mass,
and substituting into Equations (14.23), we find that the lower limit on a>
required to give consistency among the constraints on <co>, y and Pb as in
Figure 14.6 is 105. The combination of <a>> and y gives a constraint on the
masses that is relatively weakly dependent on £,, thus the constraint on <J is
dominated by Ph and is directly proportional to the measurement error in
Pb; in order to achieve a constraint comparable to the solar system value of
2 x 10~3, the error in P^BS would have to be reduced by a factor of five.
Damour and Esposito-Farese (1992) have devised a multi-scalar-tensor
theory in which two scalar fields are tuned so that their effects in the weakfield slow-motion regime of the solar system are suppressed, with the result
that the theory is identical to general relativity in the post-Newtonian
approximation. Yet in the regime appropriate to binary pulsars, it predicts
strong-field SEP-violating effects and radiative effects that distinguish it
from general relativity. It gives formulae for the post-Keplerian parameters
of Equations (14.22) as well as for the paramaters r and s that have
corrections dependent upon the sensitivities of the relativistic neutron
stars. The theory depends upon two arbitrary parameters /?' and /?";
general relativity corresponds to the values fi' = /?" = 0. It turns out
(Taylor et al., 1992) that the binary pulsar PSR 1913+16 alone constrains
the two parameters to a narrow but long strip in the /?'-/?"-plane that
Theory and Experiment in Gravitational Physics
351
includes the origin (general relativity) but that could include some highly
non-general relativistic theories. The sensitivity of PSR 1534+ 12 to r and
s provides an orthogonal constraint that cuts the strip. In this class of
theories, then, both binary pulsars are needed to provide a strong test.
(d) Other stellar-system tests of gravitational theory
The suppression of dipole gravitational radiation resulting from
the apparent high symmetry of the binary pulsar system suggests that more
stringent tests might be found in systems in which the two compact objects
are dissimilar, for example, two very unequal mass neutron stars or a
neutron star and a white dwarf. Several candidate systems have been
suggested.
The 11-minute binary 4U1820-30. This system is believed to consist of
a neutron star and a low-mass helium dwarf in a nearly circular orbit with
a period of 68 5.008 s. It is not the most" clean " system available for testing
gravitational theory, because its evolution is affected by mass transfer from
the companion low-mass dwarf onto the neutron star, whose X-ray output
comprises the data from which the binary nature of the system was
established (Stella, Priedhorsky and White, 1987; Morgan, Remillard and
Garcia, 1988). In fact the rate of mass transfer is believed to be controlled
by gravitational-radiation damping of the orbit. Because of this complication, the analysis of the implications of Brans-Dicke theory for this
system is model dependent. Will and Zaglauer (1989) generalized a class of
general relativistic mass-transfer models to the Brans-Dicke theory, and
showed that, if a limit could be placed on \PJPb\ of 2.7 x 10"7 yr"1,
corresponding to an early published limit, then bounds on co as large as 600
could be placed, depending on the assumed mass of the neutron star and
on the assumed equation of state. Unfortunately, recent observations of
the system using the Ginga X-ray satellite suggest that Ph is opposite in sign
to that predicted by a gravitational-radiation-driven mass-transfer model
(Tan et al., 1991). Evidently, the binary system is undergoing acceleration
either in the mean gravitational field of the globular cluster in which it
resides, or in the field of a nearby third body. Whether the effect of such
local accelerations on Pb can be sufficiently understood to yield an
interesting bound on co remains to be seen at present.
PSR 1744-24A. This is an eclipsing binary millisecond pulsar, in the
globular cluster Terzan 5 (Lyne et al., 1990), with a very short orbital
period of 1.8 hrs, e = 0, and a mass function of 3.215 x 10"4, indicating a
low-mass companion of 0.09 MQ. The asymmetry of the system is
promising for dipole gravitational radiation, but the observations are
An Update
352
complicated by the possibility of cluster accelerations as well as by the
apparent presence of a substantial wind from the companion (the cause of
the eclipses), which may complicate the orbital motion. Nevertheless, even
if measurements of Pb can only reach 50 % accuracy relative to the general
relativistic prediction of Pb/Pbx 1.3 x 10~8yr~', the bound on co could
exceed 1000 (Nice and Thorsett, 1992).
This discussion illustrates both the promise and the problems inherent in
stellar-system tests of gravitational theory. Dipole gravitational radiation
and strong violations of SEP resulting from the presence of neutron stars
can lead to potentially large observable effects. Offsetting this are the
complications of astrophysical effects within the systems, such as mass
transfer, and of environmental effects, such as cluster or third-body
acceleration^. Under the right conditions, however, a significant test may
emerge.
14.7
Conclusions
In 1992 we find that general relativity has continued to hold up
under extensive experimental scrutiny. The question then arises, why
bother to test it further? One reason is that gravity is a fundamental
interaction of nature, and as such requires the most solid empirical
underpinning we can provide. Another is that all attempts to quantize
gravity and to unify it with the other forces suggest that gravity stands
apart from the other interactions in many ways, thus the more deeply we
understand gravity and its observational implications, the better we may
be able to mesh it with the other forces. Finally, and most importantly, the
predictions of general relativity arefixed;the theory contains no adjustable
constants so nothing can be changed. Thus every test of the theory is
potentially a deadly test. A verified discrepancy between observation and
prediction would kill the theory, and another would have to be substituted
in its place. Although it is remarkable that this theory, born 77 years ago
out of almost pure thought, has managed to survive every test, the
possibility of suddenly finding a discrepancy will continue to drive
experiments for years to come.
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Index
action principle, 75; n-body, in modified
EIH formalism, 273; n-body, in PPN
formalism, 158-60; n-body, in THejt
formalism, 54
active gravitational mass: comparison
with passive mass via Kreuzer experiment, 214; in PPN formalism, 151
anomalous mass tensors, 40
Apollo 11 retroreflectors, 188
Bianchi identities, 76, 128, 230
big-bang model, 311
binary pulsar, 11, 257, 283; acceleration
of center of mass, 300, 344; arrivaltime analysis, 287-92, 343; in
Brans-Dicke theory, 306, 349-50;
companion, 284; decay of orbit, 297;
306; determination of masses, 297,
298; formation and evolution, 286; in
general relativity, 303-6, 344-6;
gravitational red shift, 290; mass loss,
300; measured parameters, 285, 345;
periastron shift, 284, 293; postNewtonian effects, 297; precession of
pulsar spin, 302; pulsar mass in
general relativity, 306; in Rosen's
theory, 307; second-order Doppler
shift, 290, 296; spin-down rate, 292;
test of conservation of momentum,
218-20, 338; third body, 301; tidal
effects in, 294, 299; as ultimate test for
gravitation theory, 309
binary system: single-line spectroscopic,
287; test of conservation of momentum, 217-20, 338
black holes, 256; in general relativity,
264; motion, see modified EIH formalism; motion in Brans-Dicke theory,
279; in Rosen's theory, 266; in scalartensor theories, 265
boundary conditions for post-Newtonian
limit, 118
Brans-Dicke theory, 125, 182, 190, 203,
265, 276, 306, 317, 332, 335, 349; see
also scalar-tensor theories
Cavendish experiment, 82, 153, 191
center of mass, 113, 145, 146, 160
Christoffel symbols, 70; for PPN metric,
144
classical tests, 166; and redshift experiment, 166
"comma goes to semicolon" rule, 71
completeness of gravitation theory, 18
connection coefficient, see Christoffel
symbols
conservation laws: angular momentum,
108; baryon number, 105; breakdown
of, for total momentum, 149; center-ofmass motion, 108; and constraints on
PPN parameters, 111; energy-momentum, 108, 111-12; global, 107-8; local,
105-7; rest mass, 106; tests of, for total momentum, 215-20, 337-8
conserved density, 107, 111
constants of nature, constancy: gravitational, 202; nongravitational, 36-8;
and Oklo natural reactor 36-8
coordinate systems: curvature coordi-
Index
nates, 259; local quasi-Cartesian, 92;
preferred, 17; standard PPN gauge, 97
coordinate transformation, 69
Copernican principle, 312
cosmic time function, 80
cosmological principle, 312
cosmology, 7, 310; in Bekenstein's variable-mass theory, 317; in Brans-Dicke
theory, 317; in general relativity, 313;
helium abundance, 315; microwave
background, 311, 314; in Rastall's theory, 318; in Rosen's theory, 318;
timescale problem, 311
covariant derivative, 70
Cygnus XI, 256
de Sitter effect, 338
deceleration parameter, 313
deflection of light, 5; derivation in PPN
formalism, 167-70; derivation using
equivalence principle, 170; eclipse expedition, 5, 171; effect of solar corona,
172; measurement by radio interferometry, 172; optical measurements, 6; radio measurements, 172; VLBI, 332
Dicke, R. H., 10, 16
Dicke framework, 10, 16-18
Doppler shift in binary pulsar, 290, 293,
296
dynamical gravitational fields, 118
Earth-tides, 191
eccentric anomaly, 288
Einstein Equivalence Principle (EEP),
16-22; and cosmology, 312; implementation, 71; and nonsymmetric metric,
328; and speed of gravitational waves,
223; and speed of light, 223; and THt;1
formalism, 46-50
Einstein- Infeld- Hoffmann (EIH) formalism, 267; EIH Lagrangian, 269; see also modified EIH formalism
energy conservation: and cyclic gedanken experiments, 39-43; and Einstein Equivalence Principle, 39; in
PPN formalism, 158-63; and Scruffs
conjecture, 39-43; and Strong Equivalence Principle, 82; in THepi formalism, 53-8
Eotvb's experiment, 24-7; and Belinfante-Swihart nonmetric theory, 66;
and fifth force, 341; lunar, 185-90;
and Nordtvedt effect. 185-90;
Princeton and Moscow versions, 25;
in space, 340
376
Eot-Wash experiment, 320
equations of motion: charged test particles, 69; compact objects, see modified
EIH formalism; Eulerian hydrodynamics, 87; n-body, 149, 159; Newtonian,
for massive bodies, 145; photons, 143;
PPN hydrodynamics, 147; self-gravitating bodies, 144-53; spinning bodies,
163-5; in THe/u, formalism, 50
equatorial coordinates, 194
Eulerian equations of hydrodynamics, 87
Fermi-Walker transport, 164
fifth force, 341-3
flat background metric, 79; in Robertson-Walker coordinates, 314
gauge transformation, %
general covariance, 17; and preferred
coordinate systems, 17; and prior geometry, 17
general relativity, 121-3; black holes,
265; derivations of, 83; EIH formalism,
267; field equations, 121; locally measured gravitational constant in, 158; location in PPN theory space, 205; and
modified EIH formalism, 275; motion
of compact objects, 267; neutron stars,
260; Nordtvedt effect, 152; polarization
of gravitational waves, 234; post-Newtonian limit, 121; PPN parameters,
122; quadrupole generation of gravitational waves, 246-8; with R2 terms,
84-5; speed of gravitational waves,
223; standard cosmology, 316
geocentric ecliptic coordinates, 192
geodesic equation, 73; for compact objects, 267
geometrical-optics limit: for gravitational
waves, 223; for Maxwell's equations,
74-5
gravimeter, 191; superconducting, 198
gravitational constant, 120; constancy of,
202-^t, 336; locally measured, 153-8,
191; in scalar-tensor theories, 124
297; detection in binary pulsar, 306,
346-7; dipole, 240, 249, 251, 279, 298;
dipole parameter, 240, 253; E(2)
classification, 226-7; effect on binary
system, 239; energy flux in general
relativity, 238; energy loss, 90,
238-40; forces in detectors, 237; in
general relativity, 223, 234, 246;
measurements of polarization, 237;
measurement of speed, 226; in
Index
modified EIH formalism, 275;
negative energy of, 252; PM
parameters, 240, 253; polarization,
227-38; post-Newtonian formalism,
relativity, 238; in Rastall's theory,
225, 236; reaction force, 239; in
Rosen's theory, 225, 236, 250; in
scalar-tensor theories, 224, 234, 248,
252, 279; speed, 223-6; speed in
Rosen's theory, 131; in vector-tensor
theories, 224, 235
gravitational red shift, 5, 32-6, 322; in
binary pulsar, 290, 296; and cyclic
gedanken experiments, 42-3;
derivation, 32-3; null experiment, 36;
Pound-Rebka-Snider experiment, 33;
in TH^ formalism, 62-4; solar, 322;
Vessor-Levine rocket experiment, 35
gravitational stress-energy, 109, 241
gravitational waveform, 238
Gravitational Weak Equivalence Principle (GWEP), 82; breakdown, 151, 185;
gyroscope precession: derivation in PPN
formalism, 208-9; dragging of inertial
frames, 210; goedetic effect, 209, 338;
and LAGEOS, 340; Lense-Thirring
effect, 210; Stanford experiment, 212,
helicity of gravitational waves, 227, 232,
252
helium abundance, 315
helium main-sequence star, 284, 294
Hubble constant, 202, 313
Hughes- Drever experiment, 30, 61
hydrogenic atom in THe/* formalism,
55-7
inertial mass, 13, 145; anomalous mass
tensor, 40, 55, 162, 323; dependence
on gravitational fields, 269; in
modified EIH formalism, 273; postNewtonian, 146
isentropic flow, 106
isotropic coordinates, 259
Kerr metric, 256
Kreuzer experiment, 214
laboratory experiments as tests of postNewtonian gravity, 213
Lagrangian-based metric theory, 78-9
little group, 233
local Lorenz invariance, 23; and
377
propagation of light, 321;
Hughes-Drever experiment, 30;
kinematical frameworks, 325;
Mansouri-Sexl framework, 325; tests
of, 30-1, 320; tests using TH^
formalism, 61-2; in TH formalism,
48, 323; violations of, 40-1
local position invariance, 23; gravitational
red-shift experiments, 32-6, 322; tests
using TH^ formalism, 62-4; in THW
formalism, 49; violations of, 40-1
local quasi-Cartesian coordinates, 92
local test experiment, 22
Lorentz frames, local, 23
Lorentz invariance: local, see local
Lorentz invariance; of modified EIH
Lagrangian, 272
232
Lunar Laser Ranging Experiment
(LURE), 188
Mansouri-Sexl framework, 325
Mariner 6, 175
Mariner 7, 175
Mariner 9, 175
mass, see active gravitational mass; inertial mass; passive gravitational mass
mass function, 283
Maxwell's equations, 72-3; ambiguity in
curved spacetime, 72-3; geometricaloptics limit, 74-5; in THe/u formalism,
50
metric, 22, 68; flat background, 79, 118;
nonsymmetric, 328
metric theories of gravity, postulates, 22;
microwave background, 311, 314; Earth's
motion relative to, 197
Minkowski metric, 20, 80, 118
modified EIH formalism: in Brans-Dicke
theory, 276; equations of motion for
binary systems, 273; in general relativity, 275; gravitational radiation, 275;
Keplerian orbits, 274; Lagrangian, 273;
Newtonian limit, 274; periastron shift,
274; in Rosen's theory, 280-2; variable
inertial mass, 269
moment of inertia of Earth, variation in,
195
momentum conservation: breakdown, in
PPN formalism, 149; tests of, 215-20,
337-8
neutron stars, 255; boundary conditions,
259; form of metric, 258; in general relativity, 250; maximum mass, 256; motion, see modified EIH formalism; in
Index
Newtonian theory, 257-8; in Ni's theory, 263; in Rosen's theory, 261-3; in
scalar-tensor theories, 260
Newman-Penrose quantities, 230
Newtonian gravitational potential, 87, 88,
151
Newtonian limit, 21, 87, 145; conservation laws, 105; empirical evidence, 21;
and fifth force, 341-3; inverse square
force law, 21, 341-3; in modified EIH
formalism, 274
Newton's third law, 152; and Kreuzer
experiment, 214; and lunar motion,
337
Nordtvedt, K., Jr., 98
Nordtvedt effect, 151; and lunar motion,
185-90; test of, using lunar laser ranging, 188-90; 335
null separation, 74
oblateness of Sun, 181; Dicke-Goldenberg measurements, 181; Hill measurements, 182; and solar oscillations, 334
Oklo natural reactor, 36-8
orbit elements, Keplerian, 178, 283, 287;
perturbation equations for, 179
osculating orbit, 287
parametrized post-Newtonian formalism,
see PPN formalism
particle physics, 20-1
passive gravitational mass, 13; anomalous mass tensor, 40, 55, 58, 162; comparison with active mass, 214; in PPN
formalism, 150
perfect fluid, 77-8
periastron shift: in binary pulsar, 284,
293; for compact objects, 274
perihelion shift: derivation in PPN formalism, 177-80; measured, for Mercury,
181, 333; Mercury, 4, 176-83; preferredframe and preferred-location effects,
200-1
PM parameters, 240
post-Coulombian expansion, 51
post-Galilean transformation, 272
post-Keplerian parameters, 343-4
formalism
post-Newtonian potentials, 93, 104
PPN formalism, 10, 97; active gravitational mass, 151; for charged particles,
214; Christoffel symbols, 144; comparison of different versions, 104; conservation-law parameters, 111; Eddington-Robertson-Schiff version, 98;
378
PPN ephemerides, 334; limits on PPN
parameters, 204, 216, 219, 339; metric,
99, 104; n-body action principle,
158-60; n-body equations of motion,
149, 153; passive gravitational mass,
150; PPN parameters, 97; PPN
parameter values for metric theories,
117; post-post-Newtonian extensions,
331; preferred-frame parameters, 103;
significance of PPN parameters, 115;
standard gauge, 97, 102
preferred-frame effects: in Cavendish experiments, 148; geophysical tests, 1909; on gyroscope precession, 210; in locally-measured gravitational constant,
190; orbital tests, 200-2, 336; and
solar spin axis, 336; tests from Earth
rotation rate, 199; tests using
gravimeters, 199
preferred-frame parameters: in PPN formalism, 103; in THe/t formalism, 48
preferred-frame PPN parameters, limits
on, 199, 202, 336, 339
preferred-location effects: in Cavendish
experiments, 148; geophysical tests,
190-9; in locally-measured gravitational constant, 190; orbital tests, 200-2;
tests using gravimeters, 199
prior geometry, 17, 79, 118
projected semi-major axis, 293
proper distance, 73, 155
proper time, 73, 68
PSR 1744-24A, 351
PSR 1534+12,347
PSR 1913 + 15, see binary pulsar
PSR 2127+11C, 347
pulsars, 256, 283
quadrupole moment, 145, 177; solar, 180;
solar, measurable by Solar Probe, 183;
and solar oscillations, 334
quantum systems in THc/x formalism,
55-7
quasi-local Lorentz frame, 80
radar: active, 175; passive, 174; and time
delay of light, 174
light, 172
reduced field equations, 241
rest frame of universe, 31, 99
rest mass, total, 107
retarded time, 228
Ricci tensor, 73, 230
Riemann curvature tensor, 72; electric
components, 227; irreducible parts, 230
Index
Riemann normal coordinates, 227
Robertson-Walker metric, 91, 312
Rosen's bimetric theory, 131; absence of
black holes, 266; binary pulsar, 307;
cosmological models, 317; field equations, 131; gravitational radiation, 225,
236, 250-2; location in PPN theory
space, 205; and modified EIH formalism, 280; neutron stars, 261; postNewtonian limit, 131; PPN parameters,
131
rotation rate of Earth, variation in, 195
scalar-tensor theories, 123-6; Barker's
constant G theory 125; Bekenstein's
variable-mass theory, 125, 317;
Bergmann-Wagoner-Nordtvedt, 123;
binary pulsar, 306; black holes, 265;
Brans- Dicke, see Brans-Dicke theory; cosmological models, 317; field
224, 234, 248, 50; limits on m, 175,
335; location in PPN theory space,
205; and modified EIH formalism,
276; neutron stars, 260; Nordtvedt
effect, 152; post-Newtonian limit, 124;
PPN parameters, 125; and string
theory, 332
Schiff, L. I., 38
Schiff s conjecture, 38; proof in THe/u.
formalism, 50-3
Schwarzschild coordinates, 259, 265
Schwarzschild metric, 256, 265
self-acceleration, 149; of binary system,
217, 338; of pulsars, 216
self-consistency of gravitation theory, 19
semi-latus rectum, 179
Shapiro, 1.1., 166
Shapiro effect, see time delay of light
solar corona, 172, 175
Solar Probe, 183, 340
spacelike separation, 73
special relativity, 20-1; agreement of
gravitational theory with, 20-1; and
propagation of light, 325-7; tests in
particle physics, 20-1
specific energy density, 89
spin, 163; precession, 165; precession in
binary pulsar, 302
static spherical space times, form of metric, 258
stress-energy complex, 108
stress-energy tensor, 76; in PPN formalism, 104; vanishing divergence of, 77
Strong Equivalence Principle (SEP), 7983; and dipole gravitational radiation,
252; and motion of compact objects,
379
268; tests of, 184, 335; violations in
Cavendish experiments, 153;
violations of, 102
superpotential, 94
THe/x formalism, 45-66; limitations, 589
theories of gravitation: Barker's constant
G theory, 125; Bekenstein's variablemass-theory, 125, 317; BelinfanteSwihart, 64-6; Bergmann-WagonerNordtvedt, 123; bimetric, 130-5;
Brans-Dicke, see Brans-Dicke theory; BSLL bimetric theory, 133; conformally flat, 141; E(2) classes, 233-7;
fully-conservative, 113; general relativity, see general relativity; HellingsNordtvedt, 130; Lagrangian-based, 43,
78-9, 109; linear fixed-gauge, 139;
Moffat, 330; Ni, 137, 263;
nonconservative theories, 115;
nonviable, 19, 138-41; postulates of
metric theories, 22; PPN parameters
for, 117; purely dynamical vs. prior geometric, 79; quasilinear, 138; Rastall,
132, 225, 236, 318; Rosen, see Rosen's
bimetric theory; scalar-tensor, see scalar-tensor theories; semiconservative,
114; special relativistic, 7; stratified,
135-7; stratified, with time-orthogonal
space slices, 140; vector-tensor, see
139; Will-Nordtvedt, 129; with
nonsymmetric metric, 328-30
time delay of light: in binary pulsar, 290;
as classical test, 166; derivation in
PPN formalism, 173-4; effect of solar
corona, 175; measurements of, 174,
timelike separation, 73
torsion, 84
transverse-traceless projection, 248
universal coupling, 43, 67-8
vector-tensor theories, 126-30; field
224, 235; Hellings-Nordtvedt, 130;
post-Newtonian limit, 129; PPN parameters, 129; Will-Nordtvedt, 129
velocity curve, 283
Viking, 175, 336
virial relations, 52, 54, 148, 161, 245
Voyager 2, 333
Weak Equivalence Principle (WEP), 13,
22; and cyclic gedanken experiments,
Index
41-2; and electromagnetic interactions,
28-9; and Ebtvbs experiment, 24-7;
and fifth force, 341; and gravitational
interactions, 29, 82; of Newton, 13;
and nonsymmetric metric, 329; and
strong interactions, 28; tests of, 24-9,
380
320; tests using TH£/, formalism, 60;
and weak interactions, 29
Weyl tensor, 230
Whitehead PPN parameter, limits on, 199
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# American Institute of Mathematical Sciences
October 2016, 9(5): 1575-1590. doi: 10.3934/dcdss.2016064
## Homogenization: In mathematics or physics?
1 Department of Mathematics, Soochow University, Suzhou 215006, China 2 High Speed Aerodynamics Institute, China Aerodynamisc Development and Research Center, Mianyang 622661, China
Received November 2014 Revised September 2015 Published October 2016
In mathematics, homogenization theory considers the limitations of the sequences of the problems and their solutions when a parameter tends to zero. This parameter is regarded as the ratio of the characteristic size between the micro scale and macro scale. So what is considered is a sequence of problems in a fixed domain while the characteristic size in micro scale tends to zero. But in the real physics or engineering situations, the micro scale of a medium is fixed and can not be changed. In the process of homogenization, it is the size in macro scale which becomes larger and larger and tends to infinity. We observe that the homogenization in physics is not equivalent to the homogenization in mathematics up to some simple rescaling. With some direct error estimates, we explain in what sense we can accept the homogenized problem as the limitation of the original real physical problems. As a byproduct, we present some results on the mathematical homogenization of some problems with source term being only weakly compacted in $H^{-1}$, while in standard homogenization theory, the source term is assumed to be at least compacted in $H^{-1}$. A real example is also given to show the validation of our observation and results.
Citation: Shixin Xu, Xingye Yue, Changrong Zhang. Homogenization: In mathematics or physics?. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1575-1590. doi: 10.3934/dcdss.2016064
##### References:
[1] G. Allaire, Homogenization et convergence a deux echelles, application a un probleme de convection diffusion. C.R.Acad. Sci. Paris, 312 (1991), 581-586. [2] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. [3] I. Babuška, Solution of problem with interfaces and singularities, in Mathematical aspects of finite elements in partial differential equations, C. de Boor ed., Academic Press, New York, (1974), 213-277. [4] I. Babuška, Homogenization approach in engineering, Lecture notes in economics and mathematical systems, M. Beckman and H. P. Kunzi(eds.), Springer-Verlag, 134 (1976), 137-153. [5] I. Babuška, Homogenization and its application. Mathematical and computational problems, Numerical solution of partial differential equations, III, Academic Press, (1976), 89-116. [6] I. Babuška, The computational aspects of the homogenization problem, Computing methods in applied sciences and engineering, I, Lecture notes in mathematics, Springer-Verlag,Berlin Heidelberg New York, 704 (1976), 309-316. [7] A. Bensousan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978. [8] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Applications, 17, Oxford university press, 1999. [9] S. M. Kozlov, The averaging of random operators, Mat.Sb.(N.S), 109 (1979), 188-202,327. [10] K. Lichtenecker, Die dielektrizitätskonstante natürlicher und künstlicher mischkörper, Phys. Zeitschr., XXVII (1926), 115-158. [11] J. C. Maxwell, A Treatise on Electricity and Magnetism, 3rd Ed. , Clarendon Press, Oxford, 1881. [12] F. Murat and L. Tartar, H-convergence, Topics in the Mathematical Modelling of Composite Materials, 31 (1997), 21-43. doi: 10.1007/978-1-4612-2032-9_3. [13] G. Nguestseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043. [14] O. A. Olenik and A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, Studies in mathematics and its applications, J.L. Lions, G.Papanicolaou, H. Fujita, H.B. Keller, 26, North-Holland, 1992. [15] S. Poisson, Second mémoire sur la théorie du magnétisme, Mem. Acad. France 5, 1822. [16] S. Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all' equatione del calore, Ann. Scuola Norm. Sup. Pisa, 21 (1967), 657-699. [17] T. A. Suslina, Homogenization of a stationary periodic maxwell system, St. Petersburg Math. J., 16 (2005), 863-922. doi: 10.1090/S1061-0022-05-00883-6. [18] L. Tartar, Compensated compactness and partial differential equations, in Nolinear Analysis and Mechanics: Heriot-Watt Symposium, Pitman, 39 (1979), 136-212. [19] L. Tartar, H-measure, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh, 115 (1990), 193-230. doi: 10.1017/S0308210500020606. [20] T. Yu and X. Yue, Residual-free bubble methods for numerical homogenization of elliptic problems, Commun. Math. Sci., 9 (2011), 1163-1176. doi: 10.4310/CMS.2011.v9.n4.a12. [21] V. V. Zhikov, Some estimates from homogenization theory, (Russian) Dokl. Akad. Nauk, 406 (2006), 597-601. [22] V. V. Zhikov and O. A. Oleinik, Homogenization and G-convergence of differential operators, Russ. Math. Surv., 34 (1979), 65-147. [23] V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer Berlin, 1994. doi: 10.1007/978-3-642-84659-5.
show all references
##### References:
[1] G. Allaire, Homogenization et convergence a deux echelles, application a un probleme de convection diffusion. C.R.Acad. Sci. Paris, 312 (1991), 581-586. [2] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. [3] I. Babuška, Solution of problem with interfaces and singularities, in Mathematical aspects of finite elements in partial differential equations, C. de Boor ed., Academic Press, New York, (1974), 213-277. [4] I. Babuška, Homogenization approach in engineering, Lecture notes in economics and mathematical systems, M. Beckman and H. P. Kunzi(eds.), Springer-Verlag, 134 (1976), 137-153. [5] I. Babuška, Homogenization and its application. Mathematical and computational problems, Numerical solution of partial differential equations, III, Academic Press, (1976), 89-116. [6] I. Babuška, The computational aspects of the homogenization problem, Computing methods in applied sciences and engineering, I, Lecture notes in mathematics, Springer-Verlag,Berlin Heidelberg New York, 704 (1976), 309-316. [7] A. Bensousan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978. [8] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Applications, 17, Oxford university press, 1999. [9] S. M. Kozlov, The averaging of random operators, Mat.Sb.(N.S), 109 (1979), 188-202,327. [10] K. Lichtenecker, Die dielektrizitätskonstante natürlicher und künstlicher mischkörper, Phys. Zeitschr., XXVII (1926), 115-158. [11] J. C. Maxwell, A Treatise on Electricity and Magnetism, 3rd Ed. , Clarendon Press, Oxford, 1881. [12] F. Murat and L. Tartar, H-convergence, Topics in the Mathematical Modelling of Composite Materials, 31 (1997), 21-43. doi: 10.1007/978-1-4612-2032-9_3. [13] G. Nguestseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043. [14] O. A. Olenik and A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, Studies in mathematics and its applications, J.L. Lions, G.Papanicolaou, H. Fujita, H.B. Keller, 26, North-Holland, 1992. [15] S. Poisson, Second mémoire sur la théorie du magnétisme, Mem. Acad. France 5, 1822. [16] S. Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all' equatione del calore, Ann. Scuola Norm. Sup. Pisa, 21 (1967), 657-699. [17] T. A. Suslina, Homogenization of a stationary periodic maxwell system, St. Petersburg Math. J., 16 (2005), 863-922. doi: 10.1090/S1061-0022-05-00883-6. [18] L. Tartar, Compensated compactness and partial differential equations, in Nolinear Analysis and Mechanics: Heriot-Watt Symposium, Pitman, 39 (1979), 136-212. [19] L. Tartar, H-measure, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh, 115 (1990), 193-230. doi: 10.1017/S0308210500020606. [20] T. Yu and X. Yue, Residual-free bubble methods for numerical homogenization of elliptic problems, Commun. Math. Sci., 9 (2011), 1163-1176. doi: 10.4310/CMS.2011.v9.n4.a12. [21] V. V. Zhikov, Some estimates from homogenization theory, (Russian) Dokl. Akad. Nauk, 406 (2006), 597-601. [22] V. V. Zhikov and O. A. Oleinik, Homogenization and G-convergence of differential operators, Russ. Math. Surv., 34 (1979), 65-147. [23] V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer Berlin, 1994. doi: 10.1007/978-3-642-84659-5.
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# Use the given graph off over the interval (0, 6) to find the following
Use the given graph off over the interval (0, 6) to find the following.
a) The open intervals on whichfis increasing. (Enter your answer using interval notation.)
b) The open intervals on whichfis decreasing. (Enter your answer using interval notation.)
c) The open intervals on whichfis concave upward. (Enter your answer using interval notation.)
d) The open intervals on whichfis concave downward. (Enter your answer using interval notation.)
e) The coordinates of the point of inflection. $$\displaystyle{\left({x},\ {y}\right)}=$$
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Clara Clark
Step 1
a) $$\displaystyle{\left({1},\ {3}\right)},\ {\left({4},\ {6}\right)}$$
b) $$\displaystyle{\left({0},\ {1}\right)},\ {\left({3},\ {4}\right)}$$
c) $$\displaystyle{\left({0},\ {2}\right)}$$
d) $$\displaystyle{\left({2},\ {4}\right)},\ {\left({4},\ {6}\right)}$$
e) $$\displaystyle{\left({2},\ {3}\right)}$$
###### Have a similar question?
Froldigh
Step 1
a) The open intervals on which $$\displaystyle{f}$$ is increasing.
From the above graph it is observed that (observed the arrows and the black dots), the function increasing in the interval $$\displaystyle{\left({1},\ {3}\right)}$$ and $$\displaystyle{\left({4},\ {6}\right)}$$ as they have the tangents with positive slope (positive slope means the tangents slightly bends towards right).
b) The open intervals on which $$\displaystyle{f}$$ is decreasing.
From the above graph it is observed that (observe the arrows and the black dots), the function increasing in the interval $$\displaystyle{\left({0},\ {1}\right)}$$ and $$\displaystyle{\left({3},\ {4}\right)}$$ as they have the tangents with negative slope (negative slope means the tangents slightly bends towards left).
c) The open intervals on which $$\displaystyle{f}$$ is concave upward.
The function $$\displaystyle{f{{\left({x}\right)}}}$$ is concave up for the interval in which $$\displaystyle{f}{''}{\left({x}\right)}{>}{0}$$
If the curve of the function $$\displaystyle{f{{\left({x}\right)}}}$$ always remains above the tangent lines for every point in the interval $$\displaystyle{I}$$, we say that the curve is concave upward on that interval.
From the graph it is observed that the curve remain above the tangent lines in the interval $$\displaystyle{\left({0},\ {2}\right)}$$ hence it is concave up in the interval $$\displaystyle{\left({0},\ {2}\right)}$$
d) The function $$\displaystyle{f{{\left({x}\right)}}}$$ is concave down for the interval in which $$\displaystyle{f}{''}{\left({x}\right)}{<}{0}$$</span>
If th curve of the function $$\displaystyle{f{{\left({x}\right)}}}$$ always remains below the tangent lines for every point in the interval $$\displaystyle{I}$$, we say that the curve is concave down on that interval.
From the graph it is observed that the curve remain below the tangent lines in the interval $$\displaystyle{\left({2},\ {4}\right)}$$ hence it is concave down in the interval $$\displaystyle{\left({2},\ {4}\right)}$$
e) From the above parts we have that the function is concave up in the interval $$\displaystyle{\left({0},\ {2}\right)}$$ and concave down in the interval $$\displaystyle{\left({2},\ {4}\right)}$$ which means the concavity of the function is changing at the point 2.
This follows that the infection point for the given function is at $$\displaystyle{x}={2}$$
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# How to scrape imdb webpage?
I am trying to learn web scraping using Python by myself as part of an effort to learn data analysis. I am trying to scrape imdb webpage.
I am using BeautifulSoup module. Following is the code I am using:
r = requests.get(url) # where url is the above url
bs = BeautifulSoup(r.text)
for movie in bs.findAll('td','title'):
title = movie.find('a').contents[0]
genres = movie.find('span','genre').findAll('a')
genres = [g.contents[0] for g in genres]
runtime = movie.find('span','runtime').contents[0]
year = movie.find('span','year_type').contents[0]
print title, genres,runtime, rating, year
I am getting the following outputs:
The Shawshank Redemption [u'Crime', u'Drama'] 142 mins. (1994)
Using this code, I could scrape title, genre, runtime,and year but I couldn't scrape the imdb movie id,nor the rating. After inspecting the elements (in chrome browser), I am not being able to find a pattern which will let me use similar code as above.
Can anybody help me write the piece of code that will let me scrape the movie id and ratings ?
Instead of scraping, you might try to get the data directly here. It looks like they have data available via ftp for movies, actors, etc.
• @Gred Thatcher, Thanks for the link. This project is part of a learning endeavor on web scraping and hence all these troubles. -:) – user62198 May 1 '15 at 21:47
I have been able to figure out a solution. I thought of posting just in case it is of any help to anyone or if somebody wants to suggest something different.
bs = BeautifulSoup(r.text)
for movie in bs.findAll('td','title'):
title = movie.find('a').contents[0]
genres = movie.find('span','genre').findAll('a')
genres = [g.contents[0] for g in genres]
runtime = movie.find('span','runtime').contents[0]
rating = movie.find('span','value').contents[0]
year = movie.find('span','year_type').contents[0]
imdbID = movie.find('span','rating-cancel').a['href'].split('/')[2]
print title, genres,runtime, rating, year, imdbID
The output looks like this:
The Shawshank Redemption [u'Crime', u'Drama'] 142 mins. 9.3 (1994) tt0111161
As a bit of general feedback, I think you would do well to improve your output format. The problem with the format as it stands is there is not a transparent way to programmatically get the data. Consider instead trying:
print "\t".join([title, genres,runtime, rating, year])
The nice thing about a tab delimited file is that if you end up scaling up, it can easily be read into something like impala (or at smaller scales, simple mySql tables). Additionally, you can then programatically read in the data in python using:
line.split("\t")
The second bit of advice, is I would suggest getting more information than you think you need on your initial scrape. Disk space is cheaper than processing time, so rerunning the scraper every time you expand your analytic will not be fun.
You can get everything from div with class="rating rating-list"
All you need to do is retrive attribute id: [id="tt1345836|imdb|8.5|8.5|advsearch"] When you have this content, you split this string by '|', and you get:
1. parameter: movie id
2. parameter: movie score
• Thanks. @Matic DB...i was able to get the id ..Below is my solution – user62198 Apr 16 '15 at 15:53
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### Triskpy's blog
By Triskpy, history, 5 months ago,
I want to report ray2002 for his submission 126891357 which was directly copied from ideone which was provided in this video. Several others also copied from this youtube channel as there were 100+ people watching the live stream.
He did manage to add and extra function in place of is_sorted() and a bunch of macros and was able to bipass the plag check (if such check was done).
• +9
» 5 months ago, # | +72 Ohhh noooooo someone copied Div 2A. baaaan him!!!!
• » » 5 months ago, # ^ | +21 after div 2A goes div2B, after that — div2C and so on. if we will ignore such issues, we will loose honest contests -> codeforces
• » » 5 months ago, # ^ | 0 LOL
» 5 months ago, # | ← Rev. 2 → +4 Hehehe, this was one of the cheaters that I have caught in my recent blog. (I am talking about the YouTube channel)I wasn't sure if he gives links to the solutions of Codeforces contests, but now I am sure that he is.Now I have a dilemma: should I provide links to Telegram channels and groups that I have found in previously mentioned blog?P.S. This is an example of a really helpful cheater report blog -- it says not only about the cheater but about the resource with a lot of cheaters.
» 5 months ago, # | +4 Several other users were caught plagiarized from the same https://ideone.com/necSt5 code, as per reported by our Telegram group. They have noticed a common timing for all these solutions, i.e., 21:45 (as per the IST).The list of the submissions are provided below (not in sorted order). It's a humble request to MikeMirzayanov to skip all of the submissions as per the CodeForces code of conduct. Thanks!
» 5 months ago, # | +1 I'm almost 100% sure that this person cheated (based on looking at their other attempts for the same problem). And this person also had been already flagged by the plagiarism checker in the past. The codeforces plagiarism checker is already pretty good at catching this kind of cheating and we just need to wait a bit longer until the plagiarism check is actually finished. I'm fairly confident that ray2002 was going to be flagged by the plagiarism checker even without your blog post.Thanks for sharing the original master copy of the cheater solution after the contest via posting that ideone link. The master copy and the time of it becoming available indeed may be useful for identifying cheaters.But I think that linking to youtube/telegram/discord is unnecessary and even harmful. Some people may start following that solution distributor person (no matter how unimpressive his/her skills are) with the intention to copy his/her solutions during the future contests too. Your blog post speculating that ray2002 "was able to bipass the plag check" also unnecessarily gives the wannabe cheaters false hope and I see this as a form of entrapment.
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# 2.11. Example: Cubic Anisotropy¶
In this example we will study the behaviour of a 10 x 10 x 10 nm iron cube with cubic_anisotropy in an external field.
## 2.11.1. Cubic anisotropy simulation script¶
import nmag
from nmag import SI, si
# Create the simulation object
sim = nmag.Simulation()
# Define the magnetic material (data from OOMMF materials file)
Fe = nmag.MagMaterial(name="Fe",
Ms=SI(1700e3, "A/m"),
exchange_coupling=SI(21e-12, "J/m"),
anisotropy=nmag.cubic_anisotropy(axis1=[1, 0, 0],
axis2=[0, 1, 0],
K1=SI(48e3, "J/m^3")))
# Set the initial magnetisation
sim.set_m([0, 0, 1])
# Launch the hysteresis loop
Hs = nmag.vector_set(direction=[1.0, 0, 0.0001],
norm_list=[0, 1, [], 19, 19.1, [], 21, 22, [], 50],
units=0.001*si.Tesla/si.mu0)
sim.hysteresis(Hs)
We will now discuss the cube.py script step-by-step:
After creating the simulation object we define a magnetic material Fe with cubic anisotropy representing iron:
Fe = nmag.MagMaterial(name="Fe",
Ms=SI(1700e3, "A/m"),
exchange_coupling=SI(21e-12, "J/m"),
anisotropy=nmag.cubic_anisotropy(axis1=[1,0,0],
axis2=[0,1,0],
K1=SI(48e3, "J/m^3")))
We load the mesh and set initial magnetisation pointing in +z direction (that is, in a local minimum of anisotropy energy density).
Finally, we use hysteresis to apply gradually stronger fields in +x direction (up to 50 mT):
Hs = nmag.vector_set(direction=[1.0, 0, 0.0001],
norm_list=[0, 1, [], 19, 19.1, [], 21, 22, [], 50],
units=0.001*si.Tesla/si.mu0)
Note that we sample more often the region between 19 and 21 mT where magnetisation direction changes rapidly due to having crossed the anisotropy energy “barrier” between +z and +x (as can be seen in the graph below).
## 2.11.2. Analyzing the result¶
First, we extract the magnitude of the applied field and the x component of magnetisation:
ncol cube H_ext_0 M_Fe_0 > cube_hext_vs_m.txt
Then we compare the result with OOMMF’s result (generated from the equivalent scene description oommf/cube.mif) using the following Gnuplot_ script:
set term png giant size 800,600
set out 'cube_hext_vs_m.png'
set xlabel 'H_ext.x (A/m)'
set ylabel 'M.x (A/m)'
plot 'cube_hext_vs_m.txt' t 'nmag' w l 2,\
'oommf/cube_hext_vs_m.txt' u (\$1*795.77471545947674):2 ti 'oommf' w p 1
which gives the following result:
|Nmag| provides advanced capabilities to conveniently handle arbitrary-order anisotropy energy functions. Details can be found in the documentation of the MagMaterial class.
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# Below is the balance sheet for Labyrinth Services Co., which contains errors. Labyrinth Services Co. Balance...
###### Question:
Below is the balance sheet for Labyrinth Services Co., which contains errors. Labyrinth Services Co. Balance Sheet For the Year Ended August 31, 2018 Assets Current assets: $16,500 28,200 7,500 13,500 215,000$280,700 $449,700 101,700 731,100$1,011,800 Cash Accounts payable Supplies Prepaid insurance Land Total current assets Property, plant, and equipment: Building Equipment Total property, plant, and equipment Total assets Liabilities Current liabilities: Accounts receivable Accumulated depreciation-building Accumulated depreciation-equipment Net income Total liabilities Stockholders' Equity Wages payable Common stock Retained earnings Total stockholders' equity Total liabilities and stockholders' equity $37,600 185,200 30,100 160,900$413,800 $3,100 150,000 444,900 598,000$1,011,800 Prepare a corrected balance sheet.
Labyrinth Services Co. Balance Sheet August 31, 2018 Assets Current assets: Total current assets Property, plant, and equipment: Total property, plant, and equipment Total assets Liabilities Current liabilities: Total liabilities Stockholders' Equity Total stockholders' equity Total liabilities and stockholders' equity
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# Thread: Mounting NFS shares hosted from Netapp.
1. ## Re: Mounting NFS shares hosted from Netapp.
Well, my /etc/fstab entries for my NFS-exported shares (on a server running CentOS 5.8) look like this:
Code:
server:/home /media/homes nfs defaults,rsize=32768,wsize=32768 0 0
I use a separate local /home and mount the server's /home at /media/homes. I always increase the buffer sizes to 32768 or even 65536 for performance reasons.
On the server, /etc/exports has
Code:
/home 192.168.0.0/16(rw,no_root_squash,async,insecure)
I use async to improve performance. Insecure gets around a problem with clients that use a port > 1023, and no_root_squash lets me mount the share as the client's root user.
Of course the ordinary users listed in my /etc/passwd match those found in the server's /etc/passwd so that permissions are correct.
Does that help at all? This setup is pretty vanilla.
2. ## Re: Mounting NFS shares hosted from Netapp.
Originally Posted by silverbullet007
Actually no.. I never would have thought it was a bug. I assumed I was doing it wrong.. and actually I've had no one correct me of offer a correct way of mounting NFS shares in fstab. I got this far by googling and that's never a 100% certainty.
I doubt this is a NFS server bug that could be addressed by Ubuntu launchpad as your NFS server is Netapp. I think you should first verify what the Netapp NFS exports are. I think we should have used the longer version of showmount. I have NFS exports on all my machines so I just use showmount -e from the host in question. You can explicitly show the Netapp Server with this long command
Code:
showmount -e <ipaddress_of_netapp_server>
... you can use either the hostname or FQDN or IP address. This should return something like this
Code:
showmount -e 192.168.1.2
Export list for 192.168.1.2:
/exports/rincon/egb_data 192.168.1.0/24
/exports/rincon/rab_data 192.168.1.0/24
You can also check the server this way
Code:
rpcinfo -p <ipaddress_of_netapp_server>
This should return something like the following
Code:
rpcinfo -p 192.168.1.2
program vers proto port service
100000 2 tcp 111 portmapper
100000 2 udp 111 portmapper
100024 1 udp 59615 status
100024 1 tcp 39942 status
100021 1 udp 45959 nlockmgr
100021 3 udp 45959 nlockmgr
100021 4 udp 45959 nlockmgr
100021 1 tcp 50097 nlockmgr
100021 3 tcp 50097 nlockmgr
100021 4 tcp 50097 nlockmgr
100003 2 udp 2049 nfs
100003 3 udp 2049 nfs
100003 4 udp 2049 nfs
100003 2 tcp 2049 nfs
100003 3 tcp 2049 nfs
100003 4 tcp 2049 nfs
100005 1 udp 51096 mountd
100005 1 tcp 39510 mountd
100005 2 udp 51096 mountd
100005 2 tcp 39510 mountd
100005 3 udp 51096 mountd
100005 3 tcp 39510 mountd
...you can see the TCP ports for portmapper and nfs in the return I get for one of my NFS servers.
If the exports are correctly setup, as shown in the above showmount -e command, then you should be able to mount the NFS export with something like this
Code:
sudo mount -t nfs <displayed_export> <mount_point>
... The magic is all in the export created on the server. The client (your workstation) only mounts whatever the server exports.
In the end we need to see if you can see the Netapp exports so you can use that with the mount command. If you can get that to work it is easy to mount it via fstab.
Edit: Oops, I didn't see @SeijiSensei's reply. Some of what I have said he provided earlier.
Last edited by redmk2; June 26th, 2013 at 03:02 AM.
3. A Carafe of Ubuntu
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## Re: Mounting NFS shares hosted from Netapp.
Guys:
here's my showmount -e results:
bhart@MI00320-1:~\$ showmount -e 10.2.1.72
Export list for 10.2.1.72:
/vol/vol0/eng (everyone)
/vol/vol0/pd (everyone)
/vol/root (everyone)
/vol/vol0/cnc (everyone)
/vol/vol4/Avtec (everyone)
/vol/vol0 (everyone)
/vol/vol1 (everyone)
/vol/vol2 (everyone)
/vol/vol3 (everyone)
/vol/vol4 (everyone)
/vol/vol3/MIS 10.2.6.0/24
/vol/vol0/web (everyone)
/vol/vol0/custom (everyone)
/vol/vol0/tweb (everyone)
/vol/vol0/eweb (everyone)
/vol/vol0/groen (everyone)
/vol/vol4/ARCHIVE1 (everyone)
/vol/vol0/nov1 (everyone)
The export in question being the MIS one, the clients I'm trying to mount on are (verified) in that subnet. When I first started down this road, i had 'All Hosts' enabled for RW for this export which would've been 'everyone' like the others. But when it didnt work I started trying other things.
4. A Carafe of Ubuntu
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## Re: Mounting NFS shares hosted from Netapp.
The sudo mount -t nfs netapp:/vol/vol3/MIS /home/bhart/mounts/mis returns an 'access denied by server while mounting'
5. ## Re: Mounting NFS shares hosted from Netapp.
Does the NetApp generate logs? Do they show anything useful?
6. A Carafe of Ubuntu
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## Re: Mounting NFS shares hosted from Netapp.
Useful? Not at all. I've checked the Syslogs after every umount and mount.. both from fstab and manually. And when trying to access the data (which is where the access denied message comes into play)
7. ## Re: Mounting NFS shares hosted from Netapp.
Originally Posted by silverbullet007
Useful? Not at all. I've checked the Syslogs after every umount and mount.. both from fstab and manually. And when trying to access the data (which is where the access denied message comes into play)
Clarify please! Before you said: "returns an 'access denied by server while mounting' ". Now you are saying: "And when trying to access the data (which is where the access denied message comes into play). These are 2 different things. Does it mount? Do you have permission to access the file or directory. it is possible to have success with the first and not the second item.
Last edited by redmk2; June 28th, 2013 at 01:45 AM.
8. A Carafe of Ubuntu
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## Re: Mounting NFS shares hosted from Netapp.
I'm sorry, I did make a change on a whim and the share mounts now.
See before I had read how in mounting an NFS share you should put server:\share and NOT put server:\path\to\share.
So what I did was change that after seeing the showmount -e display the full path, i.e. vol\vol3\MIS, so I change the line in fstab to match from netapp:\MIS to netapp:\vol\vol3\MIS
I have a thread about this going on on ExpertsExchange to so I got confused on which one I updated. But yes the share mounts now however I get "you do not have permissions to view the contents of this folder" The line right now is:
10.2.1.72:/vol/vol3/MIS /home/bhart/mounts/mis2 nfs hard,rw,auto,noatime,nolock,bg,nfsvers=3,intr,tcp, rsize=32768,wsize=32768 0 0
And as you can see above everyone on the 10.2.6 subnet has R/W perms.
9. ## Re: Mounting NFS shares hosted from Netapp.
Yes, but they also need to have the proper Unix permissions on the specific directories as well. Setting NFS to rw means that users can write to the share, but they can still only write to shares to which they have permissions as controlled by their Unix by user and group IDs.
Does the NetApp run something like idmapd? I usually just maintain identical copies of /etc/passwd on the server and client, but that may not be an option for you.
The simplest solution may be to put all the users in a common Unix group and grant the group write permissions on the shared directory.
10. ## Re: Mounting NFS shares hosted from Netapp.
Originally Posted by silverbullet007
I'm sorry, I did make a change on a whim and the share mounts now.
That's nice to know. It make diagnosis so much easier. LOL
See before I had read how in mounting an NFS share you should put server:\share and NOT put server:\path\to\share.
I think you are confusing Samba (SMB-CIFS) share with NFS exports. They are not the same thing, nor are the expressed the same manner. That's one of the reasons I don't use the term shares when talking about NFS exports. The Samba share is //SERVER/SHARE and the NFS export is hostname:path/to/nfs/export/directory.
So what I did was change that after seeing the showmount -e display the full path, i.e. vol\vol3\MIS, so I change the line in fstab to match from netapp:\MIS to netapp:\vol\vol3\MIS
I have a thread about this going on on ExpertsExchange to so I got confused on which one I updated. But yes the share mounts now however I get "you do not have permissions to view the contents of this folder" The line right now is:
10.2.1.72:/vol/vol3/MIS /home/bhart/mounts/mis2 nfs hard,rw,auto,noatime,nolock,bg,nfsvers=3,intr,tcp, rsize=32768,wsize=32768 0 0
And as you can see above everyone on the 10.2.6 subnet has R/W perms.
I'll let @ SeijiSensei help you sort out the file permissions and user ID's. His description is perfectly stated. I will make a comment though about UID's and GID's being consistent across the LAN. It is very helpful to do that. This way the mortal user (you) is correctly identified no matter what host you are logged into. I manually do that on my home network and use a common group with group inheritance. If you have a lot of hosts that you administer then you should consider using a centralized system of user ID's. It can be as simple as NIS. My take on that would be that if you have a small installation go with NIS. If you have such a large install base I would seriously consider a commercial LDAP/Kerbros/Bind9 solution. My favorite (and the most expensive) is Novell's eDirectory.
Edit: Now that I think of it, you are the administrator and you have a nice shiny NetApp Filer in your network. You must have some sort of centralized user management. If not, I'd put in a requisition tomorrow for eDirectory and be done with it. Or maybe you should talk to your senior IT engineer first. What do you think?
Last edited by redmk2; June 28th, 2013 at 11:57 PM.
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• ### A Search for Neutrino Emission from the Fermi Bubbles with the ANTARES Telescope(1308.5260)
Dec. 3, 2013 astro-ph.HE
Analysis of the Fermi-LAT data has revealed two extended structures above and below the Galactic Centre emitting gamma rays with a hard spectrum, the so-called Fermi bubbles. Hadronic models attempting to explain the origin of the Fermi bubbles predict the emission of high-energy neutrinos and gamma rays with similar fluxes. The ANTARES detector, a neutrino telescope located in the Mediterranean Sea, has a good visibility to the Fermi bubble regions. Using data collected from 2008 to 2011 no statistically significant excess of events is observed and therefore upper limits on the neutrino flux in TeV range from the Fermi bubbles are derived for various assumed energy cutoffs of the source.
• ### First Search for Dark Matter Annihilation in the Sun Using the ANTARES Neutrino Telescope(1302.6516)
Nov. 3, 2013 astro-ph.HE
A search for high-energy neutrinos coming from the direction of the Sun has been performed using the data recorded by the ANTARES neutrino telescope during 2007 and 2008. The neutrino selection criteria have been chosen to maximize the selection of possible signals produced by the self-annihilation of weakly interacting massive particles accumulated in the centre of the Sun with respect to the atmospheric background. After data unblinding, the number of neutrinos observed towards the Sun was found to be compatible with background expectations. The $90\%$ CL upper limits in terms of spin-dependent and spin-independent WIMP-proton cross-sections are derived and compared to predictions of two supersymmetric models, CMSSM and MSSM-7. The ANTARES limits are competitive with those obtained by other neutrino observatories and are more stringent than those obtained by direct search experiments for the spin-dependent WIMP-proton cross-section.
• ### Measurement of the atmospheric $\nu_\mu$ energy spectrum from 100 GeV to 200 TeV with the ANTARES telescope(1308.1599)
Aug. 7, 2013 astro-ph.IM, astro-ph.HE
Atmospheric neutrinos are produced during cascades initiated by the interaction of primary cosmic rays with air nuclei. In this paper, a measurement of the atmospheric \nu_\mu + \bar{\nu}_\mu energy spectrum in the energy range 0.1 - 200 TeV is presented, using data collected by the ANTARES underwater neutrino telescope from 2008 to 2011. Overall, the measured flux is ~25% higher than predicted by the conventional neutrino flux, and compatible with the measurements reported in ice. The flux is compatible with a single power-law dependence with spectral index \gamma_{meas}=3.58\pm 0.12. With the present statistics the contribution of prompt neutrinos cannot be established.
• ### First search for neutrinos in correlation with gamma-ray bursts with the ANTARES neutrino telescope(1302.6750)
Feb. 27, 2013 astro-ph.HE
A search for neutrino-induced muons in correlation with a selection of 40 gamma-ray bursts that occurred in 2007 has been performed with the ANTARES neutrino telescope. During that period, the detector consisted of 5 detection lines. The ANTARES neutrino telescope is sensitive to TeV--PeV neutrinos that are predicted from gamma-ray bursts. No events were found in correlation with the prompt photon emission of the gamma-ray bursts and upper limits have been placed on the flux and fluence of neutrinos for different models.
• We present the results of the first search for gravitational wave bursts associated with high energy neutrinos. Together, these messengers could reveal new, hidden sources that are not observed by conventional photon astronomy, particularly at high energy. Our search uses neutrinos detected by the underwater neutrino telescope ANTARES in its 5 line configuration during the period January - September 2007, which coincided with the fifth and first science runs of LIGO and Virgo, respectively. The LIGO-Virgo data were analysed for candidate gravitational-wave signals coincident in time and direction with the neutrino events. No significant coincident events were observed. We place limits on the density of joint high energy neutrino - gravitational wave emission events in the local universe, and compare them with densities of merger and core-collapse events.
• ### Search for Relativistic Magnetic Monopoles with the ANTARES Neutrino Telescope(1110.2656)
Aug. 29, 2012 astro-ph.HE
Magnetic monopoles are predicted in various unified gauge models and could be produced at intermediate mass scales. Their detection in a neutrino telescope is facilitated by the large amount of light emitted compared to that from muons. This paper reports on a search for upgoing relativistic magnetic monopoles with the ANTARES neutrino telescope using a data set of 116 days of live time taken from December 2007 to December 2008. The one observed event is consistent with the expected atmospheric neutrino and muon background, leading to a 90% C.L. upper limit on the monopole flux between 1.3E-17 and 8.9E-17 cm-2.s-1.sr-1 for monopoles with velocity beta greater than 0.625.
• ### Measurement of the Group Velocity of Light in Sea Water at the ANTARES Site(1110.5184)
Feb. 13, 2012 hep-ex
The group velocity of light has been measured at eight different wavelengths between 385 nm and 532 nm in the Mediterranean Sea at a depth of about 2.2 km with the ANTARES optical beacon systems. A parametrisation of the dependence of the refractive index on wavelength based on the salinity, pressure and temperature of the sea water at the ANTARES site is in good agreement with these measurements.
• ### Contributions to the 32nd International Cosmic Ray Conference (ICRC 2011) by the ANTARES collaboration(1112.0478)
Dec. 2, 2011 astro-ph.HE
The ANTARES detector, completed in 2008, is the largest neutrino telescope in the Northern hemisphere. It is located at a depth of 2.5 km in the Mediterranean Sea, 40 km off the Toulon shore. The scientific scope of the experiment is very broad, being the search for astrophysical neutrinos the main goal. In this paper we collect the 22 contributions of the ANTARES collaboration to the 32nd International Cosmic Ray Conference (ICRC 2011). At this stage of the experiment the scientific output is very rich and the contributions included in these proceedings cover the main physics results (steady point sources, correlations with GRBs, diffuse fluxes, target of opportunity programs, dark matter, exotic physics, oscillations, etc.) and some relevant detector studies (water optical properties, energy reconstruction, moon shadow, accoustic detection, etc.)
• ### Search for Neutrino Emission from Gamma-Ray Flaring Blazars with the ANTARES Telescope(1111.3473)
Nov. 15, 2011 astro-ph.HE
The ANTARES telescope is well-suited to detect neutrinos produced in astrophysical transient sources as it can observe a full hemisphere of the sky at all times with a high duty cycle. Radio-loud active galactic nuclei with jets pointing almost directly towards the observer, the so-called blazars, are particularly attractive potential neutrino point sources. The all-sky monitor LAT on board the Fermi satellite probes the variability of any given gamma-ray bright blazar in the sky on time scales of hours to months. Assuming hadronic models, a strong correlation between the gamma-ray and the neutrino fluxes is expected. Selecting a narrow time window on the assumed neutrino production period can significantly reduce the background. An unbinned method based on the minimization of a likelihood ratio was applied to a subsample of data collected in 2008 (61 days live time). By searching for neutrinos during the high state periods of the AGN light curve, the sensitivity to these sources was improved by about a factor of two with respect to a standard time-integrated point source search. First results on the search for neutrinos associated with ten bright and variable Fermi sources are presented.
• ### First Search for Point Sources of High Energy Cosmic Neutrinos with the ANTARES Neutrino Telescope(1108.0292)
Aug. 1, 2011 astro-ph.HE
Results are presented of a search for cosmic sources of high energy neutrinos with the ANTARES neutrino telescope. The data were collected during 2007 and 2008 using detector configurations containing between 5 and 12 detection lines. The integrated live time of the analyzed data is 304 days. Muon tracks are reconstructed using a likelihood-based algorithm. Studies of the detector timing indicate a median angular resolution of 0.5 +/- 0.1 degrees. The neutrino flux sensitivity is 7.5 x 10-8 ~ (E/GeV)^-2 GeV^-1 s^-1 cm^-2 for the part of the sky that is always visible (declination < -48 degrees), which is better than limits obtained by previous experiments. No cosmic neutrino sources have been observed.
• ### A Fast Algorithm for Muon Track Reconstruction and its Application to the ANTARES Neutrino Telescope(1105.4116)
An algorithm is presented, that provides a fast and robust reconstruction of neutrino induced upward-going muons and a discrimination of these events from downward-going atmospheric muon background in data collected by the ANTARES neutrino telescope. The algorithm consists of a hit merging and hit selection procedure followed by fitting steps for a track hypothesis and a point-like light source. It is particularly well-suited for real time applications such as online monitoring and fast triggering of optical follow-up observations for multi-messenger studies. The performance of the algorithm is evaluated with Monte Carlo simulations and various distributions are compared with that obtained in ANTARES data.
• ### The ANTARES Telescope Neutrino Alert System(1103.4477)
March 23, 2011 astro-ph.IM, astro-ph.HE
The ANTARES telescope has the capability to detect neutrinos produced in astrophysical transient sources. Potential sources include gamma-ray bursts, core collapse supernovae, and flaring active galactic nuclei. To enhance the sensitivity of ANTARES to such sources, a new detection method based on coincident observations of neutrinos and optical signals has been developed. A fast online muon track reconstruction is used to trigger a network of small automatic optical telescopes. Such alerts are generated for special events, such as two or more neutrinos, coincident in time and direction, or single neutrinos of very high energy.
• ### Search for a diffuse flux of high-energy $\nu_\mu$ with the ANTARES neutrino telescope(1011.3772)
Nov. 16, 2010 astro-ph.HE
A search for a diffuse flux of astrophysical muon neutrinos, using data collected by the ANTARES neutrino telescope is presented. A $(0.83\times 2\pi)$ sr sky was monitored for a total of 334 days of equivalent live time. The searched signal corresponds to an excess of events, produced by astrophysical sources, over the expected atmospheric neutrino background. The observed number of events is found compatible with the background expectation. Assuming an $E^{-2}$ flux spectrum, a 90% c.l. upper limit on the diffuse $\nu_\mu$ flux of $E^2\Phi_{90%} = 5.3 \times 10^{-8} \ \mathrm{GeV\ cm^{-2}\ s^{-1}\ sr^{-1}}$ in the energy range 20 TeV - 2.5 PeV is obtained. Other signal models with different energy spectra are also tested and some rejected.
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# Reaction Mechanisms
Submitted by ChemPRIME Staff on Thu, 12/16/2010 - 15:48
There are two main types of microscopic processes by which chemical reactions can occur.
A unimolecular process involves a single moleculeA set of atoms joined by covalent bonds and having no net charge. as a reactantA substance consumed by a chemical reaction., and its rate lawAn equation which describes the rate of a reaction as a function of the rate constant and the concentrations of reactants (and any other substances that affect the rate, such as products or catalysts); also called rate equation. is first order in that reactant. A bimolecular process involves collision of two molecules. Its rate law is first order in each of the colliding species and therefore second order overall. Based on this we might expect all rate laws to be first order or second order, but this conclusion does not agree with several of the experimental rate laws described earlier. (example 2 on The Rate Equation section had one rate law which was fifth order overall!)
The reason for this discrepancy is that we have not considered the possibility that an overall reaction may be the sum of several unimolecular and/or bimolecular steps. The sequence of steps by which a reaction occurs is called the mechanism of the reaction. Each unimolecular or bimolecular step in that mechanism is called an elementary processIn chemical kinetics, a reaction that takes place when a single molecule breaks apart or when two molecules collide; the rate law for an elementary process can be determined from the coefficients in the chemical equation; reaction mechanisms must consist only of elementary processes. Also called elemenary reaction.. The term elementary is used to indicate that such steps cannot be broken down into yet simpler processes. In most mechanisms some species which are produced in the earlier steps serve as reactants in later elementary processes. Such species are called intermediates.
The mechanism proposed for a given reaction must be able to account for the overall stoichiometric equation, for the rate law, and for any other facts which are known. As an example of how a mechanism can be devised to meet these criteria, consider the reaction
2I + H2O2 + 2H3O+ → I2 + 4H2O (1)
When the pHA logarithmic measure of the concentration of hydrogen (hydronium) ion; pH = -log10([H+]) or pH = -log10([H3O+]). of the solutionA mixture of one or more substances dissolved in a solvent to give a homogeneous mixture. is between 3 and 5, the rate law is
Rate = k(cI)(cH2O2)
We can immediately eliminate a single-step mechanism, not only because simultaneous collision of 2I, H2O2, and 2H3O+ is highly unlikely, but also because the rate law suggests a bimolecular process involving the collision of I and H2O2. The mechanism proposed for this reaction is
HOOH + I $\xrightarrow{\text{slow}}$ HOI + OH (1a)
HOI + I $\xrightarrow{\text{fast}}$ I2 + OH (1b)
2OH + 2H3O+ $\xrightarrow{\text{fast}}$ 4H2O (1c)
You can verify for yourself that these three steps add up to the overall reaction in Eq. (1).
The proposed mechanism can account for the rate law because the first step [Eq. (1a)] is much slower than the latter two. Once HOI is produced in that first step, it is transformed almost instantaneously into I2 and OH by the second step. Similarly the OH produced by the first and second steps reacts immediately with H3O+ to form H2O. Therefore the rate of the overall reaction is limited by the rate of the first step, and the rate law must be second order since that first step is bimolecular.
What we have just said applies to any reaction mechanismThe set of elementary steps by which a reaction is thought to occur; each individual step corresponds to either a bimolecular collision or unimolecular reaction of a single molecule.. The rate of reaction is limited by the rate of the slowest step. This elementary process is called the rate-limiting stepThe step in a reaction mechanism that by its relatively slow rate limits the overall rate of a reaction; also called rate-determining step., and the rate law gives us information about the activated complexIn the mechanism of a reaction, a species that lies at an energy peak and that can change either into products or into reactants; also called a transition state. in that step. All the species whose concentrations appear in the rate law must be part of the activated complex, and the amount of each species must be given by the corresponding exponent in the rate law. It must be emphasized that any reaction mechanism is a theory about what is happening on the microscopic level and, as such, cannot be proven to be true. Thus we can say that a proposed mechanism accounts for all the known experimental facts relating to a reaction, but this does not mean it is the only mechanism which can account for those facts. A case in point is the reaction
H2(g) + I2(g) → 2HI(g)
for which the rate law is
Rate = k(cH2)(cI2) (2)
This was first established experimentally in 1894, and for over 70 years chemists thought that the reaction occurred via a bimolecular collision of an H2 molecules with an I2 molecule. This agrees with the rate law since the activated complex would have the formula H2I2, containing 1H2and 1I2.
In 1967, however, it was shown that the reaction speed was increased considerably by yellow light from a powerful lamp. Such light is capable of dissociating I2 molecules into atomsThe smallest particle of an element that can be involved in chemical combination with another element; an atom consists of protons and neutrons in a tiny, very dense nucleus, surrounded by electrons, which occupy most of its volume.:
I2 $\xrightarrow{\text{light}}$ I + I
The fact that the light increased the reaction rate suggested that I atoms might be involved as intermediates in the mechanism, and the currently accepted mechanism is
I2 ⇌ 2I fast (3a)
I + H2 ⇌ H2I fast (3b)
H2I + I → 2HI slow (3c)
In this case the rate-limiting step is the last one in the mechanism. It is preceded by two rapidly established equilibria.
The rate law for the bimolecular step (3c) would be
Rate = k′(cH2)(cI) (4)
but since neither H2I nor I are reactants in the overall reaction, we do not know their concentrations. These concentrations can be obtained, however, by applying the equilibriumA state in which no net change is occurring, that is, in which the concentrations of reactants and products remain constant; chemical equilibrium is characterized by forward and reverse reactions occurring at the same rate. law to Eqs. (3a) and (3b):
K(3a) = $\frac{c_{\text{I}}^{\text{2}}}{c_{\text{I}_{\text{2}}}}$ K(3b) = $\frac{c_{\text{H}_{\text{2}}\text{I}}}{\text{(}c_{\text{I}}\text{)(}c_{\text{H}_{\text{2}}}\text{)}}$
Rearranging these equations we obtain
c I2 = K(3a)(cI2) cH2I = K(3b)(cI)(cH2)
These may be substituted into Eq. (4):
Rate = kK(3b)(cI)(cH2)(cI) = kK(3b)(cH2)(cI)2
= kK(3b)(cH2)K(3a)(cI2)
= kK(3b)K(3a)(cH2)(cI2) (5)
The rate constantIn a differential rate equation, the proportionality constant that relates the rate with the concentrations of reactants and other species that affect the rate. The rate constant is the rate of reaction when all concentrations are 1 M. k in the rate law [Eq. (2)] can he identified with the productA substance produced by a chemical reaction. of constants kK(3b)K(3a), and so Eqs. (5) and (2) are the same. This confirms our previous statement that the rate law tells us what species participate in the activated complex during the rate-limiting step. It also shows how more than one mechanism can lead to an activated complex having the same composition.
EXAMPLE The reaction between nitric oxide and oxygen:
2NO + O2 → 2NO2
is found experimentally to obey the rate law
Rate = k(cNO)2(cO2)
Decide which of the following mechanisms is compatible with this rate law:
a) NO + NO ⇌ N2O2 fast
N2O2 + O2 → 2NO2 slow
b) NO + NO → NO2 + N slow
N + O2 → NO2 fast
c) NO + O2 ⇌ NO3 fast
NO3 + NO → 2NO2 slow
d) NO + O2 → NO2 + O slow
NO + O → NO2 fast
Solution The slow step in mechanism a involves 2N atoms and 4 O atoms,i.e., an activated complex with the formula NO4. The same is true of mechanism c. Both are thus compatible with the rate law which also involves 2N and 4 O atoms. The other two mechanisms are not compatible with the measured rate law.
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+0
# help
0
54
1
Find $$f(t)$$ such that the graph of \begin{align*} x &= f(t), \\ y &= 4t- 3 \end{align*} is the same as the graph of $$2x + 3y = 5$$.
Nov 11, 2019
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# Order of operations with negative numbers
Practice evaluating expressions using the order of operations. Numbers used in these problems may be negative.
### Problem
Evaluate the following expression.
minus, 5, plus, left parenthesis, start fraction, 63, divided by, 9, end fraction, right parenthesis
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# Construct an explicit bijection from (0,1) to [0,1]
• April 14th 2009, 07:10 PM
qtpipi
Construct an explicit bijection from (0,1) to [0,1]
Construct an explicit bijection from the open interval (0,1) to the closed interval [0,1].
• April 15th 2009, 12:26 AM
NonCommAlg
Quote:
Originally Posted by qtpipi
Construct an explicit bijection from the open interval (0,1) to the closed interval [0,1].
how about this: $f(x) = \begin{cases} 0 & x = \frac{1}{2} \\ \frac{1}{n-2} & x =\frac{1}{n} , \ n \geq 3, \ n \in \mathbb{N} \\ x & \text{otherwise} \end{cases}.$
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# How to extract equations from WolframAlpha step-by-step solution programmatically?
I am calling
ShowSteps[exp_] :=
WolframAlpha[ToString@HoldForm@InputForm@exp, {{"Input", 2}, "Content"},
PodStates -> {"Input__Step-by-step solution"}]
SetAttributes[ShowSteps, HoldAll]
ShowSteps["integrate sin(4x)dx from 0 to 3"]
and am receiving step-by-step solution, just as one would receive in WolframAlpha website after clicking "Step-by-step solution" button. How do I extract everything (as MathML), that is an expression, e.g. all integrals, substitution variables and bounds, basically everything, that is not string (I am currently talking about Step-by-step solutions only for definite integrals)? By extract I mean convert it to MathML and output to file
EDIT:
Just to clarify. I have already got solution:
Now I need to extract everything that is vital to step-by-step solution, in other words, integrals, substitution variables ($u=4x$, $du =4dx$, etc.), convert them to MathML and output to file (I will be adding them to Word document).
• possible duplicate of Get a "step by step" evaluation in Mathematica – hftf Dec 15 '14 at 20:39
• How is this duplicate with question given? Here I already got solution from WolframAlpha (using techniques in article mentioned) in my notebook and I want to extract MathML from that solution (only expressions, no text, to file, later to be copied into Word document) – Kristians Kuhta Dec 15 '14 at 20:46
• To potential closers: the title of this question is similar to what we've seen before, but don't be misguided by that. IMO this is new and on topic. – Sjoerd C. de Vries Dec 15 '14 at 23:15
• I have been looking through the FullForm of the output, but it is a mess of various boxes for which I don't see an easy way to isolate just the equation parts. The problem is that strings may occur in equations and equations in the midst of text. The equations can be selected individually and copied, but I guess that's not what you want. – Sjoerd C. de Vries Dec 16 '14 at 11:48
• Well, I guess it will be very complicated to achieve. For now I will try to achieve that by transforming "Plaintext" form of query to Latex-like language that Word uses. That might be easier. – Kristians Kuhta Dec 16 '14 at 19:00
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# Can Energy Be Stored: 17 Facts Most Beginners Don’t Know!
The energy cannot be produced or destroyed but we can transfer it from one form of energy to another but we can store the energy. Let us discuss how can energy be stored.
The energy can be stored in various forms by transforming the energy. Nuclear energy is stored in the Earth’s core, the gravitational potential energy between the planets, the potential and mechanical energy of the flowing water, and flywheels, are some examples of the stored energy.
## How can energy be stored?
The energy can be stored in the object in the form of potential energy if work is done on it.
Every object stores an amount of energy by some application that can supply the energy while performing any work. It can be in the form of internal heat, chemical bonding, elastic properties, potential energy, biomass, electrical, nuclear, etc.
We human beings store the energy in the form of chemical potential energy which is utilized while doing work. The mechanical energy of the tools is stored with them, the gravitational potential energy is stored by an object separated by a distance from a rigid mass.
## What energy can be stored?
Here is a list of energy that can be stored:-
#### Gravitational energy
It is stored in the case of the monuments, pillars, to support the buildings and apartments, and suspended loads.
#### Chemical energy
The chemical energy is stored within the chemical bonds and released when these bonds break and it is stored in the batteries, crude oil, and fuels.
#### Hydrothermal energy
The hydrothermal energy is stored in the water molecules by the steam that evaporates releasing out the stored potential energy gained by the heat supplied.
#### Electrical potential energy
The electrical energy is stored in the capacitor and in the process of electromagnetism.
#### Nuclear energy
Nuclear energy is stored in nuclear fuels. The energy produced is utilized in the generation of hydrogen fuel which is used as a fuel.
#### Magnetic field
The magnetic field energy is stored in the material showing permanent magnetic behavior which is a ferromagnetic material and can be used to produce a magnetic field.
#### Fluids
The pressurized fluids stores the energy in the form of hydraulic, pneumatic, and steam. The fluids under pressure built enough potential energy.
#### Elastic potential energy
It is stored by cables, rubber, spring, and all other elastic material which on deformation releases a huge amount of energy.
#### Solar Energy
This energy from the Sun can be converted into various forms of energy.
#### Biomass energy
It is stored in the form of fuel
#### Electrical energy
It is stored by charging batteries, using an inverter and generator
#### Radiant energy
This stored the heat, sound, and light energy
## What ways can energy be stored?
There are different methods and ways to store the energy and we are even practicing it.
The chemical energy is stored in the chemical components, hence protecting the chemical reaction to take the place can save the chemical energy and we can use it whenever essential. In the same way, we store the chemical potential energy by eating food. The radiant energy is stored by capturing it on the black body.
The solar energy is also stored by capturing it, the biomass stores the energy within it in the form of carbon bonds as the biomatter is rich in carbon composition, the electrical energy is stored by the electronic devices; the gravitational energy is stored when the object is raised above the ground.
## How long can energy be stored?
The energy can be stored as long as we can till this energy is bought into use or converted into some other form of energy.
The capacitor stores the energy until it gets discharged after connecting it with some other electronic devices that demand an electric charge. The gravitational potential energy is stored in the object until it reaches the ground, and chemical potential energy is stored in our body until it is utilized to do the work.
Thermal energy is stored in the object till it is radiated out from the object. The elastic potential energy is held within the object only when it is stretched or compressed and lost when it is released.
## Methods for energy storage
Here is a list of methods to store the energy:-
#### Rechargeable Batteries
It stores the electric power and can be used anytime as required.
#### Storing Fossil Fuel
Fossil fuel is a form of chemical energy that stores within it.
#### Compression
Compression of the matter can build the potential energy within it.
#### Chemical Batteries
This works on the principle of redox reaction and produces electrical energy by converting the chemical energy which is formed by breaking and formation of new chemical bonds.
#### Flywheel
In order to store the kinetic energy and also to prevent any energy loss, the wheel is fixed on a frictionless by using electrical energy.
#### Capturing
The radiant energy can be captured and stored this energy by converting it into some other forms of energy. For example, solar energy is stored as electrical energy by using solar panels and generators, solar heaters, solar cookers, solar cells, etc.
#### Superconductivity
It is a method to store electricity by converting magnetic energy. The magnetic field is produced as the current flows through a coil which is cooled below the critical temperature.
#### Establishing Turbines on Dams
The turbines produce the mechanical energy by converting the kinetic and potential energy of water flowing from the dams which can be further converted and stored as electrical energy.
#### Elongation
On elongating the elastic material the elastic potential energy will be built up in the material.
#### Formation of Glucose by Animals by Degrading the Chemical Components
The glucose provides energy for living creatures to do the work which is stored in the form of chemical potential energy.
## Where can energy be stored?
The energy can be stored in the interior of the Earth in the form of nuclear energy due to high temperatures and pressure.
The electrical energy is stored in the clouds, the gravitational energy between the heavenly bodies, mechanical energy in moving automobiles, and magnetic energy in magnetic material, elastic potential energy in the elastic material.
## Can energy be stored in a magnetic field?
Magnetic energy is a renewable and free source of energy and can also be produced by passing the magnetic flux lines through the material.
The magnetic energy exists only in the magnetic field region and is given by
. The energy of the magnetic field is high if the density of flux is more.
This energy is utilized to align the dipoles so that the material can conduct more magnetic field and increases its strength.
## How is energy stored in a magnetic field?
The energy stored in the magnetic field depends upon the flux lines penetrating a region, the density, and the alignment of the magnetic dipoles.
The magnetic field energy is stored in the field based on the density of the flux running in the field by the electric current. Even the electric motor starts rotating when the electric current passes through it producing the magnetic field and energy.
## How can energy be stored in matter?
The energy transferred to the matter is stored until it is utilized while doing some work.
Heat energy, the radiant energy is captured and stored using the black body system so that no radiations can be emitted out. The elastic potential energy is stored in the deformation of the object. The gravitational potential energy is built up in the object when it raises against gravity.
## Can nuclear energy be stored?
Nuclear energy is produced in the nuclear reactor that produces a huge amount of energy due to the fission of neutrons.
Nuclear energy is stored in the form of fuel which can be used to reduce the unhealthy chemicals in the air, reduce pollution and conserve wide sources of energy as it provides a huge amount of energy using the smallest fraction of fuel.
## How can nuclear energy be stored?
The free source of nuclear energy is the core of the Earth and the Sun which provides us energy continuously.
Nuclear energy is at the core of the nucleus of an atom and is released only when the nuclei is deformed. This is obtained by radioactive elements. The heat energy produced from the nuclear reactor can be used to generate electricity and pumped hydro storage.
## Can sound energy be stored?
The sound wave travels in the form of a longitudinal wave generating the region of compression and rarefaction.
This effect and propagation of the wave are due to the vibrations of the molecules in the path that received the sound energy and oscillate back and forth forming the region of compression and expansion. This is in a form of potential as well as kinetic energy and can be stored in some other form of energy.
## Can wind energy be stored?
Wind energy is the abundant source of energy generated due to the fall or rise of pressure or temperate in different regions.
This wind energy is captured by using wind turbines. As the wind strikes the turbine, the turbines rotate converting wind energy into mechanical energy which is further stored by producing the electrical energy using a generator.
## How to store wind energy?
The wind energy is stored in the form of electrical energy and this is achieved by using the windmills and a generator.
As the propellers of the windmills start rotating with imposed wind energy, these rotations are intense by the shaft and motor. This electric motor is connected to the generator.
## Can solar energy be stored?
Solar energy is radiant energy we receive from the Sun due to the fusion of hydrogen.
Solar energy is stored in the form of thermal and electrical energy using solar panels, heaters, and generators, and it can be converted into the gravitational potential energy of the steam vapors evaporated from the water bodies.
## How to store solar energy?
Solar energy is free available energy and is carried by the light photon traveling with the electromagnetic rays emitted by the Sun.
The photons release their energy to the matter upon incident on it. The black objects are used to capture most of the radiation, which supplies thermal energy to the free electrons in the matter and increases the thermal agitation of the molecules producing the thermal energy and conducting electricity too.
## Summary
Different forms of energy can be stored in various forms. The energy can also be converted into some other form of energy to store and can be reformed again as required.
AKSHITA MAPARI
Hi, I’m Akshita Mapari. I have done M.Sc. in Physics. I have worked on projects like Numerical modeling of winds and waves during cyclone, Physics of toys and mechanized thrill machines in amusement park based on Classical Mechanics. I have pursued a course on Arduino and have accomplished some mini projects on Arduino UNO. I always like to explore new zones in the field of science. I personally believe that learning is more enthusiastic when learnt with creativity. Apart from this, I like to read, travel, strumming on guitar, identifying rocks and strata, photography and playing chess. Connect me on LinkedIn - linkedin.com/in/akshita-mapari-b38a68122
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# Magnetic field of hollow cylinder
## Homework Statement
A hollow cylinder with thin walls has radius R and current I, which is uniformly distributed over the curved walls of the cylinder. Determine the magnetic field just inside the wall and just outside, and the pressure on the wall.
F = IL×B
## The Attempt at a Solution
The field just inside is zero by ampere's law, and μ0I/2πr just outside. I can't find the pressure though! If I can find an expression for dI, then I can find an expression for dF, and dF/dA would be the pressure.
dI = dr/2πr * I where I is total current, which would make dF=(dr/2πr)L(μI/2πr)sinθ,
the angle between them is 90° so sinθ=1
dF = μ I dr / (2πr)2
I'm pretty sure this is wrong though! And I don't know how to express dA.
Thanks for any help :)
TSny
Homework Helper
Gold Member
dI = dr/2πr * I where I is total current,
If dr represents a small increment in the radial direction, then I don't understand why you have dr in this equation.
which would make dF=(dr/2πr)L(μI/2πr)sinθ,
the angle between them is 90° so sinθ=1
dF = μ I dr / (2πr)2
A tricky point in calculating the force this way is to decide what value of magnetic field to use. The current is located right where the B field is changing from 0 on the inside to Bo on the outside. So, there is the question: what value of B actually acts on the current? You will need to think about that.
Another approach to the problem is to consider energy stored in the magnetic field, but I don't know if you are allowed to approach the problem that way.
dr was supposed to represent an infinitesimal part of the circumference. I wasn't sure if it should be dI = dr/2πrL * I, where L is the length of the cylinder.
I think B0 would act on the current, wouldn't it?
TSny
Homework Helper
Gold Member
If the symbol r represents the radius, then dr would represent a small increment in radius. For an element of arc length, we could use ds.
If I do that, then your expression for the current in the arc length ds is dI = ds/2πrL * I. I interpret this as dI = ds/(2πrL) * I. That is, I assume that you meant the denominator to be 2πrL. If you check your dimensions (or units) you can see that this expression can't be correct. In order for the right hand side to represent a current, the quantity ds/(2πrL) would need to be dimensionless. But you can see that it has the dimensions of 1/length. However, your expression is fairly close to the correct expression.
The way to think about getting the correct expression for dI is to realize that the entire circumference of the cylinder contains the current I. The length L of the cylinder is not important here. So the current dI is the current contained in the fraction of a circumference subtended by ds.
The magnetic field that acts on the current dI is not the magnetic field, Bo, at the outside surface of the cylinder. This is kind of tricky. In reality, the current on the surface of the cylinder does not have zero thickness. So, imagine that the current occupies a very thin layer of thickness δr as shown in blue in the attached figure. The B field is not really discontinuous at the surface of the cylinder. It is zero at the inside surface of the layer of current and Bo at the outside surface of the layer. The B field changes continuously from 0 to Bo as you go through the layer of thickness δr. Different parts of the current in the layer experience different magnetic field strengths. If you imagine that the B field increases linearly from 0 to Bo, what is the average value of B in the layer of current?
#### Attachments
• Cylinder current 2.png
2.2 KB · Views: 965
Oh, would it be B0/2? And my expression for dI should be
dI = ds/(2πr) * I, where 2πr is the denominator.
Sub this into F=IL×B
dF=dI LB sinθ where sinθ is 1 by the reasoning in the first post
dF= (I ds L μI) / (4πr)
dF/dA = pressure dP
Last edited:
TSny
Homework Helper
Gold Member
Oh, would it be B0/2?
Yes.
And my expression for dI should be
dI = ds/(2πr) * I, where 2πr is the denominator.
Yes.
Of course, r has a certain specific value in this expression.
Yes.
Yes.
Of course, r has a certain specific value in this expression.
r has been called R! I forgot about that.
Edit: I edited the previous post while you were posting! Sorry!
And dA = Lds, I think. So
dF/dA = μI2/8(πr)2
Presumably I then have to integrate, but I don't actually know what to integrate with respect to.
TSny
Homework Helper
Gold Member
You are asked to find the pressure. What does dF/dA represent?
Pressure on the area element dA?
Pressure is force per unit area, so actually I don't have to integrate!
TSny
Homework Helper
Gold Member
Pressure is force per unit area, so actually I don't have to integrate!
Right!
Thank you for helping :)
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OS mac
Published : 2020-11-18 Lastmod : 2021-11-15
## Cause
After uploading an image recognition library called dlib, when I try to use tig or awk, I get
dyld: Library not loaded: /usr/local/opt/readline/lib/libreadline.7.dylib
Referenced from: /usr/local/bin/awk
The reason seems to be that during the dlib installation, cmake was upgraded and mac readline was upgraded from 7 to 8. tig and awk are not yet ready for readline 8, so the error seems to occur. As of November 2020, there is no problem with tig, but only with awk for now.
As of November 2020, there is no problem with tig. (It’s pretty frustrating when Shell bugs out…)
## Solutions I’ve used before
When I had the same problem in the spring of 2019, I was able to solve it using the method shown here.
As described in “Pattern 2: I did a homebrew cleanup.
brew unlink readline
This was fixed by bash brew unlink readline brew install cj-bc/cj-bc/readline The cmake upgrade cleaned up and removed the readline7 series, so pattern 1 did not work this time.
## The solution that worked this time.
I tried to run the same command again, but it didn’t fix the problem, and after some trial and error, the following worked.
ln -s /usr/local/opt/readline/lib/libreadline.dylib /usr/local/opt/readline/lib/libreadline.7.dylib
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# How does the efficacy of solar cells on Mars compare with Earth?
Mars gets less than half the light that we get on Earth and there are dust storms, but the atmosphere is much thinner and there are no clouds. After all factors have been considered, how effective are solar cells on Mars (compared with those on Earth)?
Or, in other words (if you prefer a more precisely-defined question), how much harvestable solar power is available (per unit of area) on average on the respective equators of Earth and Mars?
I'm asking in order to understand possible power solutions for hypothetical future colonies. So assume expensive panels with all the bells and whistles like heat regulation, solar tracking (if that's a thing), etc.
Since cloud coverage varies around the Earth, I'd like to get best and worst case scenarios (no clouds and 100% cloudy all the time, respectively) to put the average solar power output potential of Mars into perspective.
• This is an interesting question. One problem is that on Earth, annual cloud cover ranges between almost zero to almost 100% depending on where you look. Africa, Australia, Western regions in the Americas all have large areas of very low average cloudy skies. How would you like the comparison done? Also, temperature matters - can we afford to add temperature regulation systems, or do we have to plop them down on the ground and live with whatever temperature they happen to find?
– uhoh
Feb 13 '17 at 1:27
• Yes add temperature regulation. Looking with regards to an optimized system for a long-term colony. As for the cloud coverage, we'll have to do best and worst case scenarios. The reason I'm asking is to put the estimated Mars power performance value into perspective. Thanks. Feb 13 '17 at 4:46
• OK great! Can you edit your question to make it clear there? Comments are not permanent. Any important clarification or change should be made in the question itself, where it will continue to be seen.
– uhoh
Feb 13 '17 at 5:23
• This NASA paper goes into detail of what they had to consider in terms of solar power for the rovers. Feb 13 '17 at 13:07
• Possible duplicate of Solar panels on Mars? Aug 15 '18 at 16:18
You might want to check out solar potential maps.
This gives you your "on average" potential power harvested for the Earth, by summing up the daily solar potentials - which builds in accounting for weather, night etc. According to this map, the best areas are near the equator in Chile and slightly north of the equator in Chad/Libya/Sudan, which produce around $2800 kWh/m^2$ per year - roughly $320 W/m^2$.
On Mars, the calculations are slightly more difficult. Mars' irradiance is roughly $590 W/m^2$ - half that of Earth. Mars barely has an atmosphere, however - it's around 0.5% of Earth's so we can essentially ignore atmospheric effects. Temperatures on the Martian equator vary significantly - around $25 ^oC$ at noon, and around $-60 ^oC$ at night (I have no idea how this varies with Mars' orbit, so take that as you wish). Given that the panels won't be operating at night, and Mars has an Earth-ish temperature at day, I don't think temperature is too big of a factor - Mars is colder than Earth, but Solar panels don't mind the cold. NASA is running solar powered rovers on Mars, and they seem to be doing fine, until they got caught in dust storms.
Dust storms are a big problem. According to this research paper from 1993, there is a 1 in 3 chance of a planet-wide dust storm occurring on Mars every year. These last a few weeks - a few months, on average. There are also "smaller" continent sized dust storms which last a few weeks as well. For the sake of argument, lets say that there is a 10% chance that our base is in a dust storm at any given point in time. Martian dust storms vary in intensity - lets say that, on average, 2/3 of sunlight is blocked. Accounting for the day-night cycle, we have a solar potential of $275 W/m^2$ - this seems like an estimate on the high end, and a low end estimate is probably around $200 W/m^2$.
Again, these figures are just potentials. I haven't accounted for efficiency (around 20% - feel free to calculate that if you want). I also haven't accounted for the panel lifespan - on Earth, this is 15-20 years. On Mars, I expect this to be significantly lower due to dust storms.
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# The coprime graph of a group
Document Type: Research Paper
Authors
1 Beijing Normal University
2 Guangxi University
3 Guangxi Teachers Education University
Abstract
The coprime graph $\gg$ with a finite group $G$ as follows: Take $G$ as the vertex set of $\gg$ and join two distinct vertices $u$ and $v$ if $(|u|,|v|)=1$. In the paper, we explore how the graph theoretical properties of $\gg$ can effect on the group theoretical properties of $G$.
Keywords
Main Subjects
### References
A. Abdollahi, S. Akbari and H. R. Maimani (2006). Non-commuting graph of a group. J. Algebra. 298, 468-492
A. Abdollahi and A. M. Hassanabadi (2007). Non-cyclic graph of a group. Comm. Algebra. 35, 2057-2081
P. Balakrishnan, M. Sattanathan and R. Kala (2011). The center graph of a group. South Asian J. Math.. 1, 21-28
I. Chakrabarty, S. Ghosh and M. Sen (2009). Undirected power graphs of semigroups. Semigroup Forum. 78, 410-426
C. Gary and P. Zhang (2006). Introduction to Graph Theory. Posts and Telecom Press, Beijing.
M. S. Lucido (1999). Prime graph components of finite almost simple groups. Rend. Sem. Mat. Univ. Padova. 102, 1-22
X. L. Ma, H. Q. Wei and G. Zhong (2013). The cyclic graph of a finite group. Algebra, Article ID 107265, {http://dx.doi.org/10.1155/2013/107265}. 2013
H. Kurzweil and B. Stellmacher (2004). The Theory of Finite Groups An Introduction. Springer-Verlag, New York.
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# Solve second order diff eq with substitution
1. Jun 28, 2011
### Asphyxiated
1. The problem statement, all variables and given/known data
Solve:
$$xy''+2y'=12x^{2}$$
with
$$u=y'$$
2. Relevant equations
if you have:
$$y'+P(x)y=Q(x)$$
then your integrating factor is:
$$I(x)=e^{\int P(x) dx}$$
3. The attempt at a solution
The only reason I was able to solve this is because I stumbled upon a similar post here but instead of pulling up a thread from over a year ago i decided to post my specific question.
So if:
$$xy''+2y'=12x^{2} \hookrightarrow y''+\frac{2y'}{x}=12x$$
and
$$y'=u \hookrightarrow y''=u\frac{du}{dy}$$
I do not understand why y'' is equal to this. The way I read y'' is d2y/dx2 so why is it u*du/dy as the derivative on the right side and not d/dx? And! why is the derivative of u with respect to y u*du/dy and not just du/dy.
Maybe I have just forgotten something simple from differential calculus but I can't make sense of it on my own. I am not claiming that what is written in TeX above is wrong because the problem seems to work out, I just don't know why that is the way to do it.
Anyway, if we make the substitution:
$$u\frac{du}{dy}+\frac{2u}{x}=12x$$
then
$$I(x)=e^{\int \frac {2}{x} dx} = e^{2 ln|x|}=x^{2}$$
then multiply both sides by I(x):
$$x^{2}u\frac{du}{dy}+2xu=12x^{3}$$
which is:
$$\frac {d}{dx} (ux^{2})=12x^{3}$$
$$ux^{2}=\int 12x^{3} dx =3x^{4}+C_{1}$$
$$u=3x^{2}+ \frac{C_{1}}{x^{2}}$$
$$\frac {dy}{dx}=3x^{2} + \frac {C_{1}}{x^{2}}$$
$$y= \int 3x^{2}+\frac{C_{1}}{x^{2}}dx = x^{3}-\frac{C_{1}}{x}+C_{2}$$
if that is right please tell me why. I just followed by example from someone elses work.
2. Jun 28, 2011
### vela
Staff Emeritus
That's just the chain rule.
$$y'' = \frac{du}{dx} = \frac{du}{dy}\frac{dy}{dx} = u\frac{du}{dy}$$
I'm not sure why you'd do it that way, though. You could simply write
$$x u' + 2 u = 12x^2$$
and solve it with the same integrating factor without the unnecessary complication of invoking the chain rule.
3. Jun 28, 2011
### Asphyxiated
oh haha alright. I guess I was confused by the substitution of y'=u again for some reason.
Thanks for the reminder on that!
So I assume this is a correct solution? It's an even problem so I can't look it up.
4. Jun 28, 2011
### vela
Staff Emeritus
I didn't see anything obviously wrong, but I didn't look that closely either. You can always check your answer by plugging it back into the original differential equation to see if it's satisfied.
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# traction boundary conditions in elasticity
I have a question about implementing traction boundary conditions in 2D and 3D linear elasticity. Consider the picture above. I want to apply traction boundary conditions on the boundary in red. My question is: How is the node order/connectivity for the traction elements defined? In 2D the boundary term $$({\bf{w}},{\bf{h}})_{\Gamma_h}$$ is a line integral. Since the connectivity for elements 3 and 4 is (5,6,9,8) and (4,5,8,7) should the nodes for the traction elements be chosen in the same order? That is, should the four traction elements have the connectivity (6,9), (9,8), (8,7) and (7,4)? That means the boundary integral should be $$\int_{node 6}^{node 9} {\bf{w}}\cdot{\bf{h}} \,d\Gamma$$ and similarly for the others?
What is the situation in 3D? In 3D the traction elements are bilinear quads (for trilinear hex elements). Is the connectivity of traction elements in 3D defined so that the normal points outwards according to the right hand rule? So for the right most face element in the above figure, should the connectivity be (3,7,6,2) and for the topmost face element the connectivity should be (7,8,5,6)?
• I'm pretty sure that the direction of integration is immaterial. For 2D elasticity, I think that the boundary term $(w,h)_\Gamma_h$ is a line integral. The definition of a line integral renders the direction of integration immaterial. In 3D, the boundary term is a surface integral and I think it must be defined in a manner analogous to the line integral. – Nachiket Sep 8 at 5:57
## 1 Answer
It depends on the implementation. More than the connectivity, you need to focus on the orientation of the normals.
The common convention is to define the connectivity such that the normals on the boundary elements point away from the solid. This is relatively easy in 2D since there are only two nodes for one (linear) element.
For 3D, choose the connectivity for the face element based on the node numbering for the quad element in 2D. Note that the boundary integral on a 2D face in 3D is nothing but the area integral for the quad element in 2D. Once you choose the ordering convention for the nodes, then you need to check for the orientation of the normals. If you reverse the order of the nodes, then the direction of the normal will be in opposite direction. For example, the normal on the face with node order 1-2-3-4 is in opposite direction to that of the face with node order 4-3-2-1. This can get complicated if you generate meshes on your own.
For generating meshes, I suggest using some established mesh generators, for example, GMSH, TetGen or Hypermesh, since they provide boundary elements with consistent node ordering.
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# tables from excel to tex [duplicate]
Possible Duplicate:
Comprehensive list of tools that simplify the generation of LaTeX tables
I have some tables in excel. I use excel2latex to generate the LaTeX code to insert them into by document. excel2latex seems to mess things up pretty badly: horizontal lines, vertical text, etc. What other options are there for generating tables that do not require typing out all the tex code by hand for the table and maintain nice formatting?
-
comprehensive-list-of-tools gives an overview on tools. – Kurt Oct 7 '12 at 0:58
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My Math Forum Derivatives/linear approximation problem
Calculus Calculus Math Forum
April 18th, 2010, 04:15 AM #1 Member Joined: Apr 2010 Posts: 91 Thanks: 0 Derivatives/linear approximation problem The population of a certain country grew from 2.1 million in 1970 to 3 million in 1995. a) What was the average rate of change of the population over that period? (worked out to be 36 thousand people per year). b)Suppose that the population grew exponentially. By what percentage did the population grow each year? What was the average rate of change from 1970 to 1975? What was the average rate of change from 1990 to 1995? Illustrate your answers using a sketch. I worked out no. a), but I am having problems with number b) could some please help? This question appeared under a section in my note under the heading of derivatives and linear approximations, so I am thinking you have to use that to solve the problem.
April 26th, 2010, 03:33 PM #2 Senior Member Joined: Jan 2009 Posts: 345 Thanks: 3 Re: Derivatives/linear approximation problem b. Population in million $3=(2.1)e^{25r}$ We need to find the rate (r), so we divide both sides by 2.1, take the natural log of both sides and divide by 25. $\frac{3}{2.1}=e^{25r} \Rightarrow \text ln\left (\frac{3}{2.1} \right) = 25r \Rightarrow \frac{\text ln\left (\frac{3}{2.1} \right)}{25} \Rightarrow \frac{0.356675}{25} = r \approx 0.0142670$ We now have a function. $p[t]= 2.1e^{(0.0142670)t}$ Using 1970 as a starting point and 1995 as an ending point over the 25 years we can calculate the average rate of change which is normally has a general form of $\frac{(f[b]-f[a])}{b-a}$ on the open interval $(a,b)$ Average rate of change $\frac {p[25]-p[0]} {25-0} \Rightarrow \frac {\left (2.1e^{(0.0142670)(25)}- 2.1e^{(0.0142670)(0)} \right)}{25} \approx 36000 people/year$
### linear approximation latex
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## Adjunctions and standard constructions for partially ordered sets.(English)Zbl 0533.06001
Proc. Klagenfurt Conf. 1982, Contrib. Gen. Algebra 2, 77-106 (1983).
[For the entire collection see Zbl 0512.00011.]
This nice paper deals with functions $$Z$$ assigning to each poset $$P$$ a system $$Z(P)$$ of lower ends which contains at least all principal ideals of $$P$$. It develops an idea of a standard extension of B. Banaschewski [see Z. Math. Logik Grundlagen Math. 2, 117–130 (1956; Zbl 0073.269)]. Main examples are lower ends, Frink ideals, cuts (normal ideals) and Scott-closed sets. Among others, the author studies different maps between posets depending on $$Z$$ and generalizes the passage from finitely generated lower ends to Frink ideals by assigning to $$Z$$ an “opposite” $$\tilde Z$$. The main result is an adjunction theorem which subsumes many known and unknown universal constructions for posets.
Reviewer: J.Rosický
### MSC:
06A06 Partial orders, general 06B23 Complete lattices, completions
### Citations:
Zbl 0512.00011; Zbl 0073.269
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Overview of DataFrames - Maple Programming Help
Overview of DataFrames
Description
• A DataFrame is a two-dimensional data container, similar to a Matrix, but which can contain heterogeneous data, and for which symbolic names may be associated with the rows and columns.
Indexing
• Each column of a DataFrame is a DataSeries, and the column labels may be used to refer to the corresponding column.
> df := DataFrame( < 1, 2, 3; 4, 5, 6 >, 'rows' = [ 'a', 'b' ], 'columns' = [ 'A', 'B', 'C' ] );
${\mathrm{df}}{≔}\left[\begin{array}{cccc}{}& {A}& {B}& {C}\\ {a}& {1}& {2}& {3}\\ {b}& {4}& {5}& {6}\end{array}\right]$ (1)
> type( df[ 'B' ], DataSeries );
${\mathrm{true}}$ (2)
• Since each column is a DataSeries, you can index hierarchically into the columns of a DataFrame to extract individual data elements.
> df[ 'B' ][ 1 ];
${2}$ (3)
> df[ 'B' ][ 'b' ];
${5}$ (4)
• However, you can also select individual data items by specifying the desired row and column indices directly. (Row and column indices may be either numeric, by position, or symbolic.)
> df[ 1, 2 ];
${2}$ (5)
> df[ 'a', 'B' ];
${2}$ (6)
• You can use a range or list to select specified columns. In the case of a list, they can come in any desired order.
> df[ 'A' .. 'B'];
$\left[\begin{array}{ccc}{}& {A}& {B}\\ {a}& {1}& {2}\\ {b}& {4}& {5}\end{array}\right]$ (7)
> df[ [ 'B', 'C', 'A' ] ];
$\left[\begin{array}{cccc}{}& {B}& {C}& {A}\\ {a}& {2}& {3}& {1}\\ {b}& {5}& {6}& {4}\end{array}\right]$ (8)
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# Matrices and Determinants Share
## What is Matrices and Determinants
Matrices and Determinants:
In Mathematics, one of the interesting, easiest and important topic is Matrices and Determinants. Every year you will get at least 1 - 3 questions in JEE Main and other exams, directly and indirectly, the concept of this chapter will be involved in many other chapters, like integral and differential calculus. Concept of this chapter will be used for the axis-transformation concept. This chapter is totally new from the student point of view as you will see this chapter directly in 12th. So some students may find Matrices and Determinant little challenging to understand and solve problems initially. But as you solve more and more problems in this chapter, you will get familiar with concepts and chapter as a whole, then you will find that this is one of the easiest chapters. Afterward, the questions will appear easy for you. Matrices part may seem a little more difficult than Determinant but in the end, you will find both are easy to grasp,
## Why Matrices and Determinants:
Matrices and Determinant find a wide range of application in real-life problem, for example in adobe photoshop software matrix are used to process linear transformation to render images. A square matrix is used to represent a linear transformation of a geometric object. In computer programming matrices and its inverse are used for encrypting messages, to store data, perform queries and used as a data structure to solve algorithmic problems, etc. In robotics, the movement of the robot is programmed using a calculation based on matrices.
## After studying this chapter :
1. It will be easy for you to understand the concept of the array in computer science (if u have taken computer science in +2).
2. It will be helping you to solve the problem involving simultaneous equation with as many unknown variables as equations.
3. Determinant will help you to solve problems related to areas and volume like the area of triangle and volume of a tetrahedron.
4. It will be helping you to organize your work in a much better way in the form of matrices and hence will help you to be clear in your mind in daily life.
5. And obviously, the chapter itself will help you to score some marks in the exam as it gets about 7% weight in jee main and around similar weight in other exams.
## Important Topics: Matrices and Determinants
1. Matrix and operation on matrices
2. Types of matrix
3. Transpose of a matrix, symmetric and skew-symmetric matrix
4. Conjugate of matrix, hermitian and skew-hermitian matrix
5. Determinant of matrix
6. Minor and cofactor of an element of matrix/determinant
7. Adjoint and inverse of a matrix
8. Elementary row operations and its use in finding the inverse of a matrix
9. System of linear equations and Cramer's rule
10. System of homogeneous linear equations
## Overview of the Matrix and Determinant:
Matrix: Set of numbers or objects or symbols represented in the form of the rectangular array is called a matrix. The order of the matrix is defined by the number of rows and number of columns present in the rectangular array of representation. For example
Matrix $\begin{bmatrix} 2 &4 &-3 \\5 & 4 & 6 \end{bmatrix}$ has 2 rows and 3 columns so its order is said to be 2 × 3.
Any general element of the matrix is represented by $a_{ij}$, where $a_{ij}$ represents the elements of the ith row and jth column.
Operations on matrices: Algebraic operation on matrices like addition, subtraction, multiplication, and division will be studied in one by one in the chapter in deep, which we will find that they are very easy to comprehend.
Transpose of the matrix: If A is a matrix then the matrix obtained by changing the columns of a matrix with rows or rows with columns is called the transpose of the matrix.
For example : $\mathrm{A=\begin{bmatrix} a_{11} &a_{12} &a_{13} \\ a_{21} &a_{22} &a_{23} \\ a_{31}& a_{32} & a_{33} \end{bmatrix}\;\;and\;\;A'=\begin{bmatrix} a_{11} &a_{21} &a_{31} \\ a_{12} &a_{22} &a_{32} \\ a_{13}& a_{23} & a_{33} \end{bmatrix}}$
Conjugate of the matrix: If a matrix A has a complex number as it’s an element, then the matrix obtained by replacing those complex number by its conjugate is called conjugate of the matrix A and it is denoted by $\mathrm{\overline A}$
$\begin{array}{l}{\text { e.g. } \mathrm{A}=\left[\begin{array}{ccc}{2 i} & {3+4 i} & {7} \\ {3 i} & {9} & {4+5 i} \\ {4+5 i} & {4 i} & {3+7 i}\end{array}\right] \text { then }} \\ {\overline{\mathrm{A}}=\left[\begin{array}{ccc}{-2 i} & {3-4 i} & {7} \\ {-3 i} & {9} & {4-5 i} \\ {4-5 i} & {-4 i} & {3-7 i}\end{array}\right]}\end{array}$
The determinant of a matrix: a number which is calculated from the matrix. For determinant to exist, matrix A must be a square matrix. The determinant of a matrix is denoted by det A or |A|.
Minor and cofactor of an element $a_{ij}$ in a matrix/determinant: Minor of any element $a_{ij}$ where i is the number of rows, j is the number of columns, is the det of matrix left over after deleting the ith row and jth column.
Adjoint of the matrix: transpose of the cofactor of the element of the matrix is known as the adjoint of the matrix.
The inverse of a Matrix: A non-singular square matrix “A” is said to be invertible if there exists a non-singular square matrix B such that AB = I=BA and the matrix B is called the inverse of the matrix A.
## How to prepare Matrices and Determinants:
Matrices and Determinant is a topic useful in coordinate transformation and in some concept of differential equation as well as in the binomial theorem, you should be through with this chapter as well help some of those concepts as well as it will help you to score some easy marks in main exams.
1. Start with understanding basic concepts like Definition of the matrix, algebra of matrix, transpose of matrix, etc.
2. Then move ahead to the complex concept like adjoint of matrix and inverse of the matrix,
And then the system of equations, determinant, Cramer rules and homogenous equations.
3. After studying these concepts go through solved examples and then go to MCQ and practice the problem to make sure you understood the topic.
4. Solve the questions of the books which you are following and then go to previous year papers.
5. You can study matrix first then determinant or determinant first then matrix. In different books different order has been followed, you can choose your own order or the order of the coaching or material you are following.
6. While going through concept make sure you understand the derivation of formulas and try to derive them by your own, as many times you will not need the exact formula but some steps of derivation will be very helpful to solve the problem if you understand the derivation it will boost your speed in problem-solving.
7.Since this topic is heavily calculative practice as much as you can and while doing so remember the overall weight it has (7%).
8. At the end of chapter try to make your own short notes for quick revision, make a list of formula to revise quickly before exams or anytime when you required to revise the chapter, it will save lots of time for you.
## Best books for the preparation of this Complex number and Quadratic Equations:
First, finish all the concept, example and questions given in NCERT Maths Book. You must be thorough with the theory of NCERT. Then you can refer to the book Algebra, Arihant by Dr SK goyal or RD Sharma or Cengage Mathematics Algebra but make sure you follow any one of these not all. Matrix and determinant are explained very well in these books and there are an ample amount of questions with crystal clear concepts. Choice of reference book depends on person to person, find a book that suits you the best, depending on how well you are clear with the concepts and the difficulty of the questions you require.
## Maths Chapter-wise Notes for Engineering exams
Chapters Chapters Name Chapter 1 Sets, Relations, and Functions Chapter 2 Complex Numbers and Quadratic Equations Chapter 4 Permutations and Combinations Chapter 5 Binomial Theorem and its Simple Applications Chapter 6 Sequence and Series Chapter 7 Limit, Continuity, and Differentiability Chapter 8 Integral Calculus Chapter 9 Differential Equations Chapter 10 Coordinate Geometry Chapter 11 Three Dimensional Geometry Chapter 12 Vector Algebra Chapter 13 Statistics and Probability Chapter 14 Trigonometry Chapter 15 Mathematical Reasoning Chapter 16 Mathematical Induction
### Topics from Matrices and Determinants
• Matrices, algebra of matrices, types of matrices ( AEEE, JEE Main, COMEDK UGET, KEAM ) (112 concepts)
• Adjoint and evaluation of inverse of a square matrix using deteminants and elementary transformations ( AEEE, JEE Main, COMEDK UGET, KEAM ) (28 concepts)
• Test of consistency and solution of simultaneous linear equation in two or three variables using determinants and matrices ( AEEE, JEE Main, COMEDK UGET, KEAM ) (30 concepts)
• Determinants and matrices of order two and three ( AEEE, JEE Main, COMEDK UGET, KEAM ) (4 concepts)
• Properties of determinants, evaluation of determinants ( AEEE, JEE Main, COMEDK UGET, KEAM ) (28 concepts)
• Matrices and Types of Matrices ( AEEE, JEE Main, COMEDK UGET, KEAM ) (4 concepts)
• Mathematical Operation on Matrices ( AEEE, JEE Main, COMEDK UGET, KEAM ) (5 concepts)
• Special Matrices and their Properties ( AEEE, JEE Main, COMEDK UGET, KEAM ) (11 concepts)
• Elementary Row Operations and Inverse of a Matrix by Elementary Row Operations ( AEEE, JEE Main, COMEDK UGET, KEAM ) (3 concepts)
• Determinant ( AEEE, JEE Main, COMEDK UGET, KEAM ) (2 concepts)
• Adjoint of Matrix ( AEEE, JEE Main, COMEDK UGET, KEAM ) (4 concepts)
• Inverse of a Matrix (Using Adjoint) ( AEEE, JEE Main, COMEDK UGET, KEAM ) (5 concepts)
• Multiplication of Determinant ( AEEE, JEE Main, COMEDK UGET, KEAM ) (1 concepts)
• Properties of Determinants ( AEEE, JEE Main, COMEDK UGET, KEAM ) (3 concepts)
• System of Simultaneous Linear equations ( AEEE, JEE Main, COMEDK UGET, KEAM ) (4 concepts)
Exams
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# coq-tex - Online in the Cloud
This is the command coq-tex that can be run in the OnWorks free hosting provider using one of our multiple free online workstations such as Ubuntu Online, Fedora Online, Windows online emulator or MAC OS online emulator
### PROGRAM:
#### NAME
coq-tex - Process Coq phrases embedded in LaTeX files
#### SYNOPSIS
coq-tex [ -o output-file ] [ -n line-width ] [ -image coq-image ] [ -w ] [ -v ] [ -sl ] [
-hrule ] [ -small ] input-file ...
#### DESCRIPTION
The coq-tex filter extracts Coq phrases embedded in LaTeX files, evaluates them, and
insert the outcome of the evaluation after each phrase.
Three LaTeX environments are provided to include Coq code in the input files:
coq_example
The phrases between \begin{coq_example} and \end{coq_example} are evaluated and
copied into the output file. Each phrase is followed by the response of the
toplevel loop.
coq_example*
The phrases between \begin{coq_example*} and \end{coq_example*} are evaluated and
copied into the output file. The responses of the toplevel loop are discarded.
coq_eval
The phrases between \begin{coq_eval} and \end{coq_eval} are silently evaluated.
They are not copied into the output file, and the responses of the toplevel loop
The resulting LaTeX code is stored in the file file.v.tex if the input file has a name of
the form file.tex, otherwise the name of the output file is the name of the input file
with .v.tex' appended.
The files produced by coq-tex can be directly processed by LaTeX. Both the Coq phrases
and the toplevel output are typeset in typewriter font.
#### OPTIONS
-o output-file
Specify the name of a file where the LaTeX output is to be stored. A dash -'
causes the LaTeX output to be printed on standard output.
-n line-width
Set the line width. The default is 72 characters. The responses of the toplevel
loop are folded if they are longer than the line width. No folding is performed on
the Coq input text.
-image coq-image
Cause the file coq-image to be executed to evaluate the Coq phrases. By default,
this is the command coqtop without specifying any path which is used to evaluate
the Coq phrases.
-w Cause lines to be folded on a space character whenever possible, avoiding word cuts
in the output. By default, folding occurs at the line width, regardless of word
cuts.
-v Verbose mode. Prints the Coq answers on the standard output. Useful to detect
errors in Coq phrases.
-sl Slanted mode. The Coq answers are written in a slanted font.
-hrule Horizontal lines mode. The Coq parts are written between two horizontal lines.
-small Small font mode. The Coq parts are written in a smaller font.
#### CAVEATS
The \begin... and \end... phrases must sit on a line by themselves, with no characters
before the backslash or after the closing brace. Each Coq phrase must be terminated by
.' at the end of a line. Blank space is accepted between .' and the newline, but any
other character will cause coq-tex to ignore the end of the phrase, resulting in an
incorrect shuffling of the responses into the phrases. (The responses lag behind''.)
Use coq-tex online using onworks.net services
Free Servers & Workstations
Linux commands
• 1
a56-toomf
a56-toomf - Motorola DSP56001 assembler
- convert to OMF ...
Run a56-toomf
• 2
a56
A56 - Motorola DSP56001 assembler ...
Run a56
• 3
cpmrm
cpmrm - remove files on CP/M disks ...
Run cpmrm
• 4
cpp-4.7
cpp - The C Preprocessor ...
Run cpp-4.7
• 5
gapi2-fixup
undocumented - No manpage for this
program. DESCRIPTION: This program does
not have a manpage. Run this command
with the help switch to see what it
does. For f...
Run gapi2-fixup
• 6
gapi2-parser
undocumented - No manpage for this
program. DESCRIPTION: This program does
not have a manpage. Run this command
with the help switch to see what it
does. For f...
Run gapi2-parser
• More »
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# r-viridislite¶
Port of the new ‘matplotlib’ color maps (‘viridis’ - the default -, ‘magma’, ‘plasma’ and ‘inferno’) to ‘R’. ‘matplotlib’ <http://matplotlib.org/ > is a popular plotting library for ‘python’. These color maps are designed in such a way that they will analytically be perfectly perceptually-uniform, both in regular form and also when converted to black-and-white. They are also designed to be perceived by readers with the most common form of color blindness. This is the ‘lite’ version of the more complete ‘viridis’ package that can be found at <https://cran.r-project.org/package=viridis>.
## Installation¶
With an activated Bioconda channel (see 2. Set up channels), install with:
conda install r-viridislite
and update with:
conda update r-viridislite
A Docker container is available at https://quay.io/repository/biocontainers/r-viridislite.
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### ABY2.0: Improved Mixed-Protocol Secure Two-Party Computation
Arpita Patra, Thomas Schneider, Ajith Suresh, and Hossein Yalame
##### Abstract
Secure Multi-party Computation (MPC) allows a set of mutually distrusting parties to jointly evaluate a function on their private inputs while maintaining input privacy. In this work, we improve semi-honest secure two-party computation (2PC) over rings, with a focus on the efficiency of the online phase. We propose an efficient mixed-protocol framework, outperforming the state-of-the-art 2PC framework of ABY. Moreover, we extend our techniques to multi- input multiplication gates without inflating the online communication, i.e., it remains independent of the fan-in. Along the way, we construct efficient protocols for several primitives such as scalar product, matrix multiplication, comparison, maxpool, and equality testing. The online communication of our scalar product is two ring elements irrespective of the vector dimension, which is a feature achieved for the first time in the 2PC literature. The practicality of our new set of protocols is showcased with four applications: i) AES S-box, ii) Circuit-based Private Set Intersection, iii) Biometric Matching, and iv) Privacy- preserving Machine Learning (PPML). Most notably, for PPML, we implement and benchmark training and inference of Logistic Regression and Neural Networks over LAN and WAN networks. For training, we improve online runtime (both for LAN and WAN) over SecureML (Mohassel et al., IEEE S&P’17) in the range 1.5x-6.1x, while for inference, the improvements are in the range of 2.5x-754.3x.
Note: This article is the full and extended version of an article published at USENIX Security’21. Added the details of Braun et al., PriML@NeurIPS'21, which contains implementations of our ABY2.0 protocols along with further optimizations and more protocols.
Available format(s)
Category
Cryptographic protocols
Publication info
Published elsewhere. MAJOR revision.30th USENIX Security Symposium (USENIX Security '21)
Keywords
multi-party computation2PCABYprivacy-preserving machine learningPPML
Contact author(s)
arpita @ iisc ac in
schneider @ encrypto cs tu-darmstadt de
suresh @ encrypto cs tu-darmstadt de
ajith @ iisc ac in
yalame @ encrypto cs tu-darmstadt de
History
2022-01-26: last of 4 revisions
See all versions
Short URL
https://ia.cr/2020/1225
CC BY
BibTeX
@misc{cryptoeprint:2020/1225,
author = {Arpita Patra and Thomas Schneider and Ajith Suresh and Hossein Yalame},
title = {ABY2.0: Improved Mixed-Protocol Secure Two-Party Computation},
howpublished = {Cryptology ePrint Archive, Paper 2020/1225},
year = {2020},
note = {\url{https://eprint.iacr.org/2020/1225}},
url = {https://eprint.iacr.org/2020/1225}
}
Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.
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# Carbon emission from Western Siberian inland waters
## Abstract
High-latitude regions play a key role in the carbon (C) cycle and climate system. An important question is the degree of mobilization and atmospheric release of vast soil C stocks, partly stored in permafrost, with amplified warming of these regions. A fraction of this C is exported to inland waters and emitted to the atmosphere, yet these losses are poorly constrained and seldom accounted for in assessments of high-latitude C balances. This is particularly relevant for Western Siberia, with its extensive peatland C stocks, which can be strongly sensitive to the ongoing changes in climate. Here we quantify C emission from inland waters, including the Ob’ River (Arctic’s largest watershed), across all permafrost zones of Western Siberia. We show that the inland water C emission is high (0.08–0.10 Pg C yr−1) and of major significance in the regional C cycle, largely exceeding (7–9 times) C export to the Arctic Ocean and reaching nearly half (35–50%) of the region’s land C uptake. This important role of C emission from inland waters highlights the need for coupled land–water studies to understand the contemporary C cycle and its response to warming.
## Introduction
Northern high-latitude regions are covered by numerous rivers1 and lakes2, together occupying up to ~16% of the land area3. At the same time, these regions store a significant amount of carbon (C) (~1672 Pg C), ~88% of which is stored in perennially frozen ground—permafrost4,5. At present, warming has accelerated in high-latitude regions with the mean annual temperature rising twice as fast as the global average4,6, making previously frozen permafrost vulnerable to thaw3,7. When permafrost thaws, it exposes substantial quantities of organic C, resulting in C degradation and atmospheric release of carbon dioxide (CO2) and methane (CH4)6. Large quantities of terrestrial inorganic and organic C are also exported to inland waters8,9, leading to additional CO2 and CH4 emission from the water surface into the atmosphere. The C emission from inland waters is a substantial component of the global C cycle10,11,12,13, yet for high-latitude regions such assessments are scarce14,15 and generally restricted to small catchments16,17, implying major uncertainties in the understanding of the high-latitude C cycle and its feedback on the climate system18.
Western Siberia with its extensive peatland area (~0.6 out of ~3.6 million km2)19 containing vast organic C stocks (~70 Pg C)19,20 partly underlain by permafrost (Fig. 1) is of particular interest in the high-latitude C cycle. Permafrost in Western Siberia is vulnerable to thaw and has been degrading over the last few decades21. Also, Western Siberia harbors the largest Arctic watershed, the Ob’ River, which is the second largest freshwater contributor to the Arctic Ocean22 and is one of the few Arctic rivers that traverse through all permafrost zones along its course (from permafrost-free to continuous permafrost zone). Importantly, no direct C emission estimate exists for inland waters of Western Siberia. Siberian systems have been included in global inland water C emission estimates10, but these are based on few indirect (calculated gas concentration and modeled fluxes) snapshot data with very low spatial and temporal resolution that have been found to introduce large uncertainties and cannot adequately capture annual C emissions23,24,25,26. Given the region’s large C stock and its overall sensitivity to warming, there is a clear need to estimate Western Siberian inland water C emission to constrain its role in present and future high-latitude and global C cycles.
Quantification of inland water C emission requires measurements of C emission rates and water surface areas of streams, rivers, ponds, and lakes—data that are scarce and geographically biased, thus limiting assessments of the role of inland water C emission at a regional scale. In this work we used new and recent estimates of inland water C emission rates (collected from 2014 to 2016) and areas across all permafrost zones of Western Siberia to quantify the total annual C emission from Western Siberian inland waters. Specifically, we used recently published C (CO2 + diffusive CH4) emission rates from 58 rivers27 and 89 ponds and lakes28,29 (hereby denoted “lakes” to distinguish from the extrapolated ponds, see below) covering a wide range of river (catchment area 2–150,000 km2) and lake (lake area 0.0001–1.2 km2) sizes and spanning a full gradient in permafrost extent (over 2000 km distance, Fig. 1 and Supplementary Table 1). We also included new estimates of C emission rates from the main channel of the Ob’ River based on the first-ever direct continuous measurements of partial pressure of CO2 (pCO2) (1546 ± 882 µatm, mean ± s.d., Fig. 1) and regional gas transfer coefficient (k) data. We obtained river and lake water surface areas in the Ob’ River basin and two other major Western Siberian rivers (Pur and Taz) from published databases1,2 and upscaled rivers and lakes C emission rates within each permafrost zone. Because the databases did not cover water surface areas of streams (lotic systems < 90 m wide) and the smallest ponds (lentic systems < 0.01 km2), we estimated these areas using Pareto law1,30,31 and upscaled streams and ponds C emission rates to the extrapolated areas. We summed C emission from all inland waters (rivers, streams, lakes, ponds) and assessed uncertainties using standard error propagation methods. The results allowed us to make the first assessment of the role of inland waters in the C cycle in one of the least studied but largest northern ecosystem in the world undergoing rapid permafrost thaw.
## Results and discussion
We found that C emission rates from rivers are ~4-fold greater than C emission rates from lakes, resulting in greater yearly rates of C outgassing from lotic systems (rivers: 0.9 ± 0.5, lakes: 0.2 ± 0.1 kg C m−2 yr−1, mean ± s.d.) (Fig. 2). Taken together inland waters (excluding streams and ponds) cover ~5.2% of Western Siberia, with lakes (~171,000 km2) accounting for a major fraction of the landscape compared to rivers (~20,000 km2) (Table 1). The combined rivers and lakes C emission from Western Siberia was 0.050 (±0.007) Pg C yr−1 and showed not only high values across the entire region, but also differences among permafrost zones (Fig. 2) (H = 242.67, P < 0.05). The C yield (the total C emission from inland waters scaled to the land area) increased with increasing permafrost extent and reached its maximum in the discontinuous permafrost zone (Fig. 2) (H = 1556, P < 0.05). This pattern is a consequence of particularly high rates of C emission from lakes in the permafrost-rich zones combined with a large fraction of land area covered by lakes, while rivers contribute less to the C yield and exhibit no clear trend across permafrost zones. Such increase in C yield emphasizes the fact that, not only warm, but also cold permafrost-rich areas of Western Siberia are important contributors to the overall high inland water C emission from this region.
When assessing the magnitude of inland water C emission across Western Siberia we used published measurements of C emission rates and areas. However, these estimates do not include the areas of the smallest streams and ponds, which are commonly the most abundant water bodies in the landscape2 and potentially important sources of C to the atmosphere16,32. By extrapolating the areas of small systems1,30,31, the total lotic and lentic areas increased by ~1.6-fold (33,390 km2) and ~2.4-fold (425,986 km2), respectively, and increased the proportion of land occupied by water to ~12%. With the full size range of systems included, the C emission from inland waters increased ~2-fold, to 0.032 (±0.005) and 0.071 (±0.012) Pg C yr−1 for lotic and lentic systems, respectively (Fig. 3 and Table 1). The C emission from lotic systems was nearly equal between streams (0.013 ± 0.003 Pg C yr−1) and rivers (0.018 ± 0.003 Pg C yr−1), with C emission from the Ob’ main channel accounting for 24% (0.004 ± 0.001 Pg C yr−1) of river C emission. The C emission from lakes was slightly lower than from small ponds (0.032 ± 0.006 and 0.039 ± 0.010 Pg C yr−1, respectively) (Table 1). Taken together, the total C emission from Western Siberian inland waters amounted to 0.104 (±0.013) Pg C yr−1 (Fig. 3).
Our estimate for C emission from Western Siberian inland waters is greater than previously thought10,33. Specifically, mean pCO2 concentration, mean CO2 emission rate, and river C emission are ~3, ~6.3, and ~4.6-fold greater, respectively, than earlier assessment inferred from indirect observations and modeling10,33. Also, our estimate for total C emission from Western Siberian inland waters is ~1.4-fold greater than total C emission for this region and is ~2.6-fold greater than total C emission from other major Russian permafrost-draining rivers (i.e., sum of Kolyma, Lena, and Yenisei Rivers, 0.04 Pg C yr−1)10 derived based on modeling. Likewise, total C emission from Western Siberian inland waters is ~4.2-fold greater than total inland water C emission from the permafrost-affected Yukon River (0.02 Pg C yr−1) derived based on field observations14. These comparisons emphasize not only the fact that C emission from Western Siberian inland waters is high, but also highlight the need for additional regional estimates of inland water C emission from other major watersheds to better constrain their role in the global C cycle.
The results of this study are based on an extensive dataset with direct measurements of C emission rates covering a wide range of river and lake sizes and spanning over a complete permafrost gradient of Western Siberia. Yet, our estimate of total C emission from Western Siberian inland waters contains uncertainties related to practical constraints in carrying out measurements of C outgassing rates at higher spatial and temporal resolution. Measurements focused on key periods (spring, summer, autumn) likely captured main variability in C emission rates24,34, but still, collection of more temporally resolved data, especially in smaller systems, is needed to optimize sampling protocols for future C outgassing estimates. Furthermore, although the relatively homogenous geomorphology, soil type, and lithology of Western Siberia35 improve the likelihood of realistic extrapolation compared to more heterogeneous regions, additional spatially distributed data on C fluxes are needed, particularly in the permafrost-free zone (Fig. 1). Furthermore, due to the region’s flat terrain, Western Siberia has extensive floodplains and wetlands that increase substantially in water area during spring flood36,37. Assuming a maximal 85%36 increase in area over a period of 30 days, the C emission from inland waters of Western Siberia rises by ~11%. Our estimates also include uncertainties because of the lack of direct observations of smallest stream and pond areal distribution across the landscape. In particular, it has been recently suggested that contrary to stream area, pond area distribution in the landscape does not follow Pareto law31,38. Given this fact, we also adopted an alternative approach where we quantified pond C emission based on pond area derived from satellite image analysis of several sites within specific permafrost zones of Western Siberia31. Our result obtained with this approach yielded a ~31% lower estimate of total C emission (0.076 Pg C yr−1) from Western Siberian inland waters. Although these data are only based on a fraction of the basin, it still suggest a critical knowledge gap and the need for more detailed satellite inventories of pond area distribution across the landscape to assess their role in total inland water C emission.
This study also highlights the complexity in assessments of contemporary and future C outgassing from inland waters. First, it stresses the need to account for variability in both C emission rates and surface areas of streams, rivers, ponds, and lakes across the landscape. Second, the observed patterns in C emission and yield (Fig. 2) suggest that C outgassing is controlled differently between inland waters, yet detailed mechanistic studies are generally lacking. Current knowledge of these systems indicates a strong sensitivity on climate-dependent processes for C outgassing, but not always as expected based on knowledge from other regions. In general, C emission from rivers has been explained as mainly driven by lateral input of terrestrial C, and its subsequent mineralization and evasion from the water column27. These processes are largely controlled by temperature and water transit times that result in nonlinear dynamics with elevated C emission rates in warm vs cold permafrost regions, but also lower rates in permafrost-free zone27. For ponds and lakes lateral inputs of terrestrial C seem less important for C outgassing because of the generally small catchments. Instead, high rates of C emission from ponds and lakes have mainly been attributed to the shallow depths that cause relatively high mineralization of terrestrial organic C in their bottom sediments, with elevated rates in cold permafrost-rich regions because of high organic C quality of recently thawed sediments and hampered algae CO2 fixation28. Importantly, warming likely changes the areal coverage of inland waters, chiefly for thaw ponds and lakes where spatiotemporal variability in losses and gains create large uncertainty in the net outcome3. The complexity in the control of C outgassing from inland waters, with multiple drivers that vary across systems, implies that predicting future C outgassing from inland waters of Western Siberia is fraught with large uncertainties that require an interdisciplinary approach.
To estimate the relative importance of Western Siberian inland water C emission, we compared the total inland water C emission (0.076–0.104 Pg C yr−1) with other components of the regional C cycle (Fig. 3). First, we quantified Western Siberian land C uptake using regional data on terrestrial net ecosystem exchange (NEE) during 201639 to −0.198 ± 0.009 Pg C yr−1 (Supplementary Fig. 1 and Supplementary Table 2). Thus, almost half (35–50%) of land C uptake is released back to the atmosphere via inland waters, implying that neglecting inland waters will largely overestimate the C sink strength of the region. Second, we compiled published data on river dissolved organic and inorganic C export to the Arctic Ocean (Ob’ River: mean for the period of 2003–2009; Pur and Taz Rivers: mean for the period of 2013–2014)35,40,41,42 (Supplementary Table 3) to 0.011 Pg C yr−1, i.e., 6.8–9.0-fold lower than C emission from inland waters. This implies that only ~10% of the C lost laterally from land reaches the Arctic Ocean, the rest is largely processed and emitted to the atmosphere by inland waters. Third, we found that the inland water C emission was ~2.4–3.0-times higher than the C uptake by the Kara Sea (−0.031 Pg C yr−1 during 2014)43,44 into where all Western Siberian rivers discharge. Because of interannual variability in fluxes, these types of comparisons should optimally include multiyear overlapping time periods, and thus the exact numbers should be treated with caution. Despite the uncertainties, these results emphasize the important role of C emissions from inland waters in the regional land–water C cycle. Ignoring C outgassing from inland waters may largely underestimate the impact of warming on these regions and overlook their weakening capacity to act as terrestrial C sinks. Although few coupled land–water C cycle studies exist for comparison, these data suggest that the role of inland waters of Ob in the C cycle are particularly high compared to other large scale estimates at high latitudes14,45,46 and globally13,47, and are on par with estimates for the Tropics11,48. The high significance of the inland waters of Western Siberia in the C cycle is likely a result of the overall flat terrain, which leads to relatively high water coverage and long water transit times, and thus favorable conditions for mineralization and outgassing of land derived C in inland waters27,28. Further studies on the coupled land–water C cycle are needed in order to improve the understanding of regional differences in the contemporary C cycle and predictions of future conditions in these understudied and climate-sensitive areas.
## Methods
### Inland water area estimates
We used available Global River Widths from Landsat (GRWL)1 as well as Global Water Bodies (GLOWABO)2 databases to estimate river and lake area in the Ob’, Pur, and Taz River basins. We first clipped the databases’ files to the respective area of Western Siberia using ArcMap 10.5. Then we overlaid GRWL river network with Ob’ main channel mask derived from World Major Rivers file (https://www.arcgis.com/home/item.html?id=44e8358cf83a4b43bc863646cd695945) by selecting features that are within ~20 km distance from the Ob’ main channel mask, and clipped Ob’ main channel from GRWL river network. We further separated the Ob’ main channel, river, and lake files to the respective permafrost zones using shapefiles of the Circum-Arctic Map of Permafrost and Ground Ice Conditions (http://nsidc.org/data/docs/fgdc/ggd318_map_circumarctic/). After that we merged the Ob’ main channel, river, and lake files for each permafrost zone in R (version 3.5), making three individual continuous spatial datasets with all permafrost zones present (Ob’ main channel spatial dataset consisting of 201,874 observations of river area, river spatial dataset of 882,124 observations of river area, and lake spatial dataset of 973,780 lakes). We excluded rivers <90 m wide from both river and Ob’ main channel datasets as done in Allen et al.1.
To include rivers and streams <90 m not present in the GRWL database, we estimated their surface area using Pareto extrapolation based on the specific Pareto shape parameter of 0.93 ± 0.0004 (±s.d.) reported for the rivers of Ob’ River basin1 and the minimum width of the first-order streams of 0.32 ± 0.077m 49. Similarly, to include ponds <0.01 km2 not present in the GLOWABO database we used Pareto extrapolation based on the specific Pareto shape parameter of 1.19 ± 0.0004 for the lakes of Ob’ River basin (obtained by fitting power law to Ob’ basin lakes in GLOWABO) and the smallest measured pond area of 0.000115 ± 0.0001 km2 (based on our field observations). There exist no data that enable to incorporate temporal variability in surface area across all systems and full region, and all areal estimates are assumed to represent average conditions over the open water season.
Since it has recently been suggested that contrary to stream area, pond area distribution in the landscape does not follow Pareto law31,38, we also quantified pond area using the fraction of land covered by these types of water bodies for each of the permafrost zones derived from lake satellite inventories of few sites within specific permafrost zones of the region from Muster et al. 201931. Since Muster et al.31 estimated lake area (including ponds >0.0001 km2) in one site in the sporadic permafrost zone and in two sites in the continuous permafrost zone, we assumed the fraction of land covered by ponds in permafrost-free and isolated permafrost zones being similar to the sporadic permafrost zone (where ponds occupy ~2.9% of land), while the fraction of land covered by ponds in the discontinuous permafrost zone being similar to the average fraction of land covered by such water bodies between the two sites in the continuous permafrost zone (~0.62% of land).
### Ancillary data
We used published data on annual flow-weighted dissolved organic C (DOC)35,40 and dissolved inorganic C (DIC)41,42 export to the Arctic ocean during 2003–2009 for the Ob’ River, as well as annual flow-weighted DOC and DIC41,42 export (mean for the period of 2013–2014 quantified based on discharge data over the period 1971–1980) for Pur and Taz Rivers and summed them together to obtain the downstream C export for this region (Supplementary Table 3). We also estimated season length across entire region of Western Siberia by using a linear relationship between latitude (°N) and number of ice-free season days published for rivers27 and lakes28.
### C emission from Ob’ main channel
The pCO2 of Ob’ main channel was collected at 0.5 m depth every minute during 5 min at 10-min interval (yielding 4938 measurements) from a ship in summer 2016 (31 July to 11 August) by an infrared gas analyzer (Vaisala GMP222; accuracy ±1.5%) connected to a Campbell logger. The probe was calibrated in the lab and the pCO2 data were corrected for pressure and temperature (collected at same frequency) as described in Serikova et al.27. We estimated molar concentrations of pCO2 in water and in water in equilibrium with the atmosphere using Henry’s constant and pressure, and the average atmospheric concentration of 390 ppm. We grouped measurements and calculated the total CO2 evasion in each permafrost zone (permafrost-free, isolated, sporadic, discontinuous, and continuous). Since we lacked values in the continuous permafrost zone (because the ship finished sampling before reaching the Ob’ River mouth in the continuous permafrost zone), we used the values from the adjacent discontinuous permafrost zone. In total, we obtained 4396 pCO2 measurements in the Ob’ main channel, the number of measurements in each permafrost zone were 1516, 1982, 431, and 467 in the absent, isolated, sporadic, and discontinuous permafrost zones, respectively. To calculate the total CO2 evasion for each permafrost zone, we performed a Monte Carlo simulation by randomly sampling the observations of pCO2 in the water, pCO2 in the atmosphere and pH 100,000 times. The pH values for the respective permafrost zones were derived from our published river seasonal data27, truncating pH values to <8 because observed seasonal river pH values never exceeded a pH value of 827. Thus, CO2 emissions were calculated as:
$${\mathrm{CO}}_2\;{\mathrm{emission}}\;{\mathrm{rate}} = \alpha \times {\mathrm{median}}\left( k \right) \times \left( {p{\mathrm{CO}}_{2\;\rm{water}} - p{\mathrm{CO}}_{2\;\rm{atmosphere}}} \right) \times 10^{ - 6}$$
(1)
where α is the pH-dependent chemical enhancement factor of CO250, k is a median gas transfer coefficient of 4.464 m d−1 measured in four largest rivers (June 2015) of the Ob’, Pur, Pyakupur, and Taz Rivers (n = 39, each consisting of multiple measurements with floating chamber drifting in the middle of the river channel for 5 min), pCO2 water is assigned pCO2 in the water and pCO2 atmosphere is the assigned pCO2 in water in equilibrium with the atmosphere. To obtain the total yearly CO2 emissions in each permafrost zone, we multiplied the areal CO2 emission rate by the total river area in each permafrost zone, as well as the average ice-free season length based previous measurements within permafrost zones27. Furthermore, given our observations on seasonality in pCO2 concentrations in the Ob’ main channel from our previous work27 and considering that the ship pCO2 data sampling took place in July–August, we increased assigned pCO2 values (that represent summer pCO2) by a factor of 2 to get the approximate pCO2 in the spring, and took the average between them for quantifying daily rates of CO2 outgassing during open water period (May–October).
We estimated daily rate of CH4 outgassing for each of the points by using the median fraction of CH4 in C emission rate from our river data27 equal to 1.19% and summed up CO2 and CH4 emission rates to get the C (CO2 + CH4) emission rate (Supplementary Fig. 2). After that we multiplied C emission rate with respective season length and water areas for each of the data points, and summed all points together. We also added land area to different permafrost zones based on Circum-Arctic Map of Permafrost and Ground Ice Conditions51 and estimated C yield for each of the data points by normalizing the C emission rate to the respective land area.
### C emission from rivers and streams
To quantify C emission for rivers, we used published data on daily rates of CO2 outgassing from 58 rivers (n = 116)27. We created five normal distributions with CO2 emission rates for each permafrost zones with 10,000 values in each using the mean and s.d. for the respective permafrost zones from the observed CO2 emission rates river data (Supplementary Table 1). Then, for each observation of river area (n = 882,124), we randomly assigned a CO2 emission rate by subsampling the permafrost-specific distribution of emission rates defined above. After that we estimated daily rates of CH4 (i.e., 1.19% of total C emission) and C emission using the same approach as for Ob’ main channel. Then we estimated season length for each data point using the linear regression with latitude (R2 = 0.99, F1,114 = 7899.51, p < 0.01, Supplementary Table 4), and multiplied daily C emission rate with season length and water areas for each of the data points. Finally, we summed all points together to get the total C emission for rivers. We also added land area to different permafrost zones following the same approach as for the Ob’ main channel and estimated C yield. We quantified C emission for streams using the published27 median C emission rate of 5.67 g C m−2 d−1 for watersheds <100 km2 (assuming the same 1.19% fraction of CH4 in C emission rate). We then multiplied this value with extrapolated stream area and a median season length of 180.6 days was observed in our river data27.
### C emission from lakes and ponds
For permafrost-affected lakes we used published lake C (CO2 + diffusive CH4) emission rates data (76 lakes, n = 228, Supplementary Fig. 3)28. Since we did not observe linear dependence of C emission rates with lake size (n = 182, R2 = 0.00, F1,179 = 0.598, p > 0.05), we utilized a similar approach for upscaling C emission rates as in river upscaling. We created five normal distributions with C emission rates data representing different permafrost zones with 10,000 values in each using mean and s.d. for the respective permafrost zones from lake data (Supplementary Table 1)28. Then, for each observation (n = 612,003), we randomly assigned a CO2 emission rate by subsampling the permafrost-specific distribution of emission rates defined above. Then we estimated season length for each data point using the linear regression with latitude (R2 = 0.96, F1226 = 6012.09, p < 0.01, Supplementary Table 1), and multiplied daily C emission rate with season length and water areas for each of the data points. Finally, we summed all points together to get the total C emission for permafrost-affected lakes. We also estimated C yield following methods described above. When quantifying C emission for lakes in the permafrost-free zone, we used published C emission rates from 13 permafrost-free lakes (n = 13)29. Considering that the linear dependence of C (CO2 + diffusive CH4) emission rates on lake size in permafrost-free area was very weak (log10-transformed, n = 13, R2 = 0.20, F1,11 = 2.801, p > 0.05), we used the median C emission rate of 0.6 g C m−2 d−1 when upscaling to lakes (n = 361,777) located in permafrost-free area of Western Siberia (using same approach as described above for permafrost-affected lakes). We also estimated C yield using the same approach as above. We quantified C emission for ponds using the published median C emission rate for permafrost-affected (1.12 g C m−2 d−1) and permafrost-free (0.6 g C m−2 d−1) smallest size class lakes of Western Siberia, and multiplied this with extrapolated pond area and median season length for respective region. Since it has been recently suggested that pond area distribution in the landscape does not follow Pareto law31,38, we also quantified C emission for ponds using the fraction of land covered by these water bodies types for each of the permafrost zone derived from dataset available in Muster et al.31. Following this approach the total C emission for ponds is ~3.4-fold lower.
### Uncertainty in C emission estimates
The uncertainty in C emission rates values for the Ob’ main channel was estimated using a Monte Carlo approach. We randomly subsampled ten times 1000 values of each of the following variables: pCO2 water, pCO2 atmosphere, k and chemical enhancement factor, and estimated C emission rates as above. Using this approach, the mean C emission rate for Ob’ main channel is ~1.8-fold greater compared to the mean quantified C emission rate, whereas the median values are almost identical (Supplementary Fig. 4). To assess uncertainty in C emission from different components of inland waters in a uniform way, we used the standard rules of error propagation. We assumed 15% uncertainty in estimates of key variables (C emission rates, water areas, and season length) and propagated our error following:
$$\delta R = \left| R \right| + \sqrt {\left( {\frac{{\delta x}}{x}} \right)^2 + \left( {\frac{{\delta y}}{y}} \right)^2 + \left( {\frac{{\delta z}}{z}} \right)^2}$$
(2)
where δR is the uncertainty, R is a result of multiplication of C emission rates, water areas, and season lengths, while δx, δy, and δz are 15% uncertainty estimates of C emission rates (x), water areas (y), and season lengths (z), respectively. We then estimated the uncertainties for total river and lake C emission as well as total C emission from inland waters (rivers, streams, lakes, ponds) as follows:
$$\delta R = \sqrt {\left( {\delta x} \right)^2 + \left( {\delta y} \right)^2 + \left( {\delta z} \right)^2 + \ldots }$$
(3)
where δR is the total uncertainty, while δx, δy, and δz, … are C emission uncertainties estimated for each of the inland water components from Eq. (2).
### Statistics
We examined differences in inland water C emission and C yield (rivers + lakes, excluding streams and ponds) between different permafrost zones of Western Siberia with Kruskal–Wallis test on randomly subsampled 500 values. We considered the result statistically significant at p < 0.05.
### Net ecosystem exchange data
We used NASA SMAP L4 Global Daily 9 km EASE-Grid Carbon Net Ecosystem Exchange, Version 4 product39 (https://nsidc.org/data/SPL4CMDL) to quantify monthly and annual rates of NEE across Western Siberia. This product provides global gridded (9 × 9 km) daily estimates of NEE (CO2) derived using satellite data based on terrestrial C flux model informed by Soil Moisture Active Passive (SMAP) L-band microwave observations, land cover and vegetation inputs from the Moderate Resolution Imaging Spectroradiometer, Visible Infrared Imaging Radiometer Suite, and the Goddard Earth Observing System Model, Version 5 land model assimilation system. Note that NEE detects only terrestrial vegetation and ignores aquatic surfaces. We downloaded the data files covering the full year of 2016. Note that our sampling campaign on lakes was conducted during open water period (May–October) 2016 and the Ob River was studied in July–August 2016. We imported these data files to R, extracted the mean NEE rate as well as 1 s.d. of the mean NEE rate data from these data files (representing in total 730 individual data of h5 format), projected them, converted them to GeoTiff format, and clipped the area matching the location of Ob’, Pur, and Taz River basins (Supplementary Fig. 1). We then quantified the mean NEE rate and mean s.d. across all 71,280 of 9 × 9 km cells covering the region, for each day separately, and after that we estimated the NEE for entire Western Siberia as a sum of products of each 71,280 individual cells’ NEE rates and respective cells’ resolution (also for each day separately). The monthly NEE and annual NEE were quantified as the sum of daily and monthly NEE, respectively (Supplementary Table 2). We assessed uncertainty in monthly and annual NEE using Eqs. (2) and (3).
### Net C uptake by the Kara Sea
To estimate net C uptake by the Kara Sea, we used published grid (1 × 1 degree) data for CO2 uptake by the Arctic Ocean (60–90°N, 0°–360°) for year 201443,44 (http://www.jamstec.go.jp/res/ress/yasunaka/co2flux/#!prettyPhoto). We downloaded the data file, imported it to R, and clipped the grid cells matching the GPS boundaries of the Kara Sea (2784 observations). We then quantified the mean annual CO2 uptake rate encoded in these grid cells (–7.64 mmol m−2 d−1, from January to December) and multiplied it by the water surface area of the Kara Sea and by 365 days (as done in Yasunaka et al.44).
## Data availability
The datasets generated during this study are available in the ZENODO repository: https://doi.org/10.5281/zenodo.4153049.
## Code availability
Code for data analysis (for terrestrial NEE) is available upon request.
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## Acknowledgements
We thank Erik Geibrink for technical support and @AndreJ (https://gis.stackexchange.com/questions/253923/how-to-fix-the-reprojection-from-ease-2-grid-product-smap-to-geographic-coordina) for providing valuable code ideas for working with NASA NEE data. The work was financially supported by the Swedish Research Council (grant no. 2016-05275 and 325-2014-6898, JPI Climate initiative) and RSF (grant no. 18-17-00237).
## Funding
Open Access funding provided by Umea University.
## Author information
Authors
### Contributions
J.K. and O.S.P. conceived the study, S.N.V. collected the Ob’ main channel pCO2 data, S.S. collected rivers and lakes C emission rates data and analyzed all data, G.R.-R. and B.D. assisted in data analysis. J.K. and S.S. wrote the manuscript with input from all coauthors.
### Corresponding author
Correspondence to Jan Karlsson.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Peer review information Nature Communications thanks Jonathan Cole and other, anonymous, reviewers for their contributions to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Karlsson, J., Serikova, S., Vorobyev, S.N. et al. Carbon emission from Western Siberian inland waters. Nat Commun 12, 825 (2021). https://doi.org/10.1038/s41467-021-21054-1
• Accepted:
• Published:
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Chi square 2x3 table - Genotypes are related to disease?
I want to conduct a Chi square test of independence.
I have three different genotypes (CC/CG/GG, independent variable) and "health conditions" (outcomes) which is basically a patient group with a certain disease and a control group without the disease.
Does it sound right if my null hypothesis states that "the distribution of these alleles in these "study groups" doesn't influence disease outcome"
Whereas the alternative hypothesis says that "the distribution of these alleles in these study groups does influence disease outcome".
Is i formulated in a right way? Because I can't state (with the Chi square) that there are a certain risk-allele that lies behind disease.
And lastly, I have heard that you can only use odds ratio if it is a 2x2 table, but if I use a "dominant" model of Pearsons Chi-square then the result would no longer be significant and there should no longer be any point to perform a OR.
• There is no such thing as independent variable in this test. You're not doing regression testing. You're testing whether there is any dependence between the levels of the variable. – SmallChess May 26 '15 at 11:12
• More precisely, the null hypothesis is that the levels of the variables are independent. The alternative is that they are dependent. – SmallChess May 26 '15 at 11:13
• Do you have absolute numbers or percentages? Are the number of patients and controls equal and what is the total sample size? – rnso May 26 '15 at 11:14
• To Student T: So basically it mean if I got a significant result, my alternative hypothesis would say something like "there is a significant association between the occurrence of these genotypes and occurrence of this certain disease"? – Zorua May 26 '15 at 11:20
• Of possible interest: stats.stackexchange.com/q/8774/930, stats.stackexchange.com/q/9062/930. – chl May 26 '15 at 11:21
Regarding calculating odds ratios:
> tt
dss
genotype sick healthy
CC 14 34
CG 14 24
GG 8 3
>
> oddsratio(tt, rev='col')
$data dss genotype healthy sick Total CC 34 14 48 CG 24 14 38 GG 3 8 11 Total 61 36 97$measure
odds ratio with 95% C.I.
genotype estimate lower upper
CC 1.000000 NA NA
CG 1.409998 0.5628447 3.55030
GG 6.087096 1.4813184 32.87432
$p.value two-sided genotype midp.exact fisher.exact chi.square CC NA NA NA CG 0.46208267 0.49321142 0.450640195 GG 0.01142108 0.01320433 0.007043333$correction
[1] FALSE
attr(,"method")
[1] "median-unbiased estimate & mid-p exact CI"
Warning message:
In chisq.test(xx, correct = correction) :
Chi-squared approximation may be incorrect
The odds ratios are calculated for second and third genotypes with respect to first genotype.
Edit: I have edited the code above to have numbers as in comments. The odds ratio for GG (with respect to CC) being sick is 6.1 (95% CI 1.5,32.9; significant since it does not overlap 1). The odds of CG being sick (as compared with CC) is not significantly different 1. The P values are also shown in the output.
• @mso Are you using R or any other kind of program? Because I have been using GraphPad and have not been able to calculate odds ratio (probably because I did not know about a reference genotype). But this is very interesting! Does it mean that people who have GG genotype has 2,76 higher risk to become sick? I have a huge problem to formulate correct null- and alternative hypothesis... – Zorua May 26 '15 at 11:42
• Yes, I am using R. The odds ratios above are for the fictitious numbers that I have used. What are your actual numbers? – rnso May 26 '15 at 12:25
• I am sorry, I have been spelling your user namn wrong this whole time. My numbers are for "sick": 14, 14, 8. And for the healthy: 34, 24, 3 (CC, CG, GG) @rnso – Zorua May 26 '15 at 12:33
• I have edited the answer above using these numbers. – rnso May 26 '15 at 12:49
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## Compound Interest and the Irrational Number e
Suppose we borrow Php* 1.00 from a loan company at an interest rate of 100% a year, then after a year we would have to pay Php 2.00. We have learned from elementary school mathematics that to get the interest, we have to multiply the principal by the rate. Just in case, we already have forgotten, the computation to calculate the interest is shown below.
Figure 1 – Computation of the Interest of 1.00 given a rate of 100% a year.
*Php stands for Philippine pesos, the currency in my country.
The situation above is an example of a simple interest problem. Simple interest is calculated by multiplying a fixed interest rate to a principal amount over a fixed period of time.
There are several scenarios that could happen in loaning money. Most of the loan companies accept early payments; some allow the borrowers to re-borrow the money, and some apply interests to amount payable just like re-borrowing money. Let us see what happens in both scenarios.
Scenario 1: Paying Early
Suppose you borrowed money for 1 year and luckily you got money after 6 months, and you want to pay your loan immediately. Your loan company might only charge you 50% such as shown in Figure 2.
Figure 2 – Computation of Interest of 1.00 peso compounded semiannually at a rate of 100% a year.
Scenario 2: Re-borrowing Money
Another scenario is to borrow the money for six months and re-borrow it again for another six months without paying the first loan. Note that this is the same as failing to pay a six-month loan for a year. The question is, after borrowing 1 peso for six months with an interest of 50% the first six months and another 50% in the second six months, do you think your loan company will let you pay Php 2.00 for 1 year? Well – lucky you are if they do.
Table 1 – Computation of 1 peso loan compounded semi-annually at an annual rate of 100% interest.
Loan interests do not work that way. That is because from scenario 1, after 6 months, the amount that you have to pay the company is Php 1.50. Hence, if you re-borrow the money without paying the principal – or you failed to pay it for another six months – the principal amount during your second loan will not be Php 1.00 but Php 1.50. The computation of this scenario is shown in the table in Table 1 and is called compound interest. That is applying the interest rate not only to the principal amount (which Php 1.00 in the first six months), but also including the interest (which is Php 0.50 in the first six months).
If we have a fixed rate, say 100% (or 1 in decimal equivalent), then we have to divide it into the number of times that we are going to compound our interest in one year. For instance, applying a 100% interest to a one-year loan compounded monthly will require us to divide 100% by 12 which is 8.5% per month or 0.085 in decimal equivalent. Table 2 shows the computation of this scenario. In January, the principal is only Php 1.00 and after applying the interest, the amount payable is 1.085. That means, if you want to pay at the end your January, you have to pay 1.085 or 1.09 pesos. This becomes the principal amount in February. If we multiply 1.085 by 0.085, our interest in February is 0.092. If we add this to the principal amount in February, which is 1.085, then the amount payable by the end of February is 1.177 peso or 1.18 pesos.
Table 2 – Computation of Interests of a 1-peso loan compounded monthly at a 100% annual interest rate.
This process is repeated until December. As shown in the table, by the end of December, 2.66 is the amount payable for the 1-peso loan for 1 year with 100% interest compounded monthly.
Generalization
Putting the principal and interest in table in Figure 3 into computation, we have the following.
Figure 3 – Computation of Amount Payables given the number of compounds per year.
Looking at Figure 3, in the January row of Table 2, we have a principal of 1 and an interest rate of 0.085 = 1/12. Thus, using the pattern shown in Figure 3, in general, if we have 1 as principal, 100% as annual interest, and n as the number of times we are going to compound the interest in a year, then the amount payable after a year can be computed by the formula $(1 + \frac{1}{n})^n$. Several questions may arise following the discussion above:
1. What happens as the number of compound is getting larger and larger? What happens if we compound per day, per hour, per minute and per second?
2. What happens if we divide 1 (that is, 100%) into a very large number? Will the amount payable reach a very large number? Will amount payable reach infinity?
Using 1-peso as principal and 1 (or 100%) as interest per year, the table below shows the amount payable given the number of times the interest is compounded in a year. As we can see, compounding daily gives approximately 2.714 pesos and compounding every second 2.718, only a difference of 0.004. It is evident that as the number of compounds of year increases, the smaller the increase in amount payable per year.
Does that mean that the amount payable is approaching a particular number?
Table 3 – Computation of Amount Payables given the number of compounds per year.
Also, from the table, it seems that as the number of compounds increases, the amount payable is approaching 2.7182818. In fact, raising the number of compounds per year to 1 billion gives us 2.7182820, an increase of 0.02. This means, that this number has a limit as we increase the number of compounds per year.
The number that is the limit of the expression $(1 + \frac{1}{n})^n$ is called $e$ in mathematics and is approximately equal to 2.718282. That is $\lim_{n \to \infty} (1 + \frac{1}{n})^n = e$. The irrational constant $e$ is a very important number in mathematics just like $\pi$. It is be seen in many mathematical fields. In future posts, we are going to discuss why e is so important and where e usually appears.
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Differential analysis heatmap without hclust
2
1
Entering edit mode
2.0 years ago
newbie ▴ 90
Hello,
I did a differential analysis between 160 tumor and 90 normal samples. I'm using edgeR package. I used the following code to make a differential analysis heatmap. I selected differential expressed genes based on FC > 2 and FDR < 0.05. This gave me 614 DEGs.
logCPM <- cpm(y, prior.count=2, log=TRUE)
o <- order(tr$table$PValue)
logCPM <- logCPM[o[1:614],]
logCPM <- t(scale(t(logCPM)))
dim(logCPM)
library(matrixStats)
library(gplots)
library(ComplexHeatmap)
library(circlize)
library(RColorBrewer)
#Set annotation
ann <- data.frame(TvsN$Type) colnames(ann) <- c("Type") colours <- list("Type"=c("Tumor"="black","Normal"="brown")) colAnn <- HeatmapAnnotation(df=ann, which="col", col=colours, annotation_width=unit(c(1, 4), "cm"), gap=unit(1, "mm")) myCol <- colorRampPalette(c("navyblue", "white", "red"))(100) myBreaks <- seq(-2,2, length.out=100) hmap <- Heatmap(logCPM, name = "Z-Score", col = colorRamp2(myBreaks, myCol), show_row_names = FALSE, show_column_names = FALSE, cluster_rows = TRUE, cluster_columns = TRUE, show_column_dend = FALSE, show_row_dend = TRUE, row_dend_reorder = TRUE, column_dend_reorder = TRUE, clustering_method_rows = "ward.D2", clustering_method_columns = "ward.D2", width = unit(100, "mm"), top_annotation=colAnn) draw(hmap, heatmap_legend_side="left", annotation_legend_side="right") The heatmap was made using complex heatmap like above. I see that DEGs showing samples are clustered. ![heatmap][1] I wanted to select 614 DEGs and sort the samples acc to their Type without using hclust like above. May I know how to do this? thanq RNA-Seq heatmap complexheatmap r hclust • 783 views ADD COMMENT 2 Entering edit mode 2.0 years ago Read the docs about clustering in ComplexHeatmap. You can suppress clustering (and provide the data in the order you want) or give a custom clustering function or precomputed dendrogram. ADD COMMENT 0 Entering edit mode In your case it would be cluster_rows which needs to be set to FALSE as by default it is TRUE and then uses Complete linkeage to cluster the rows. As Jean-Karim Heriche says, please read the manual, it is outstandingly comprehensive with an example plot for like every possible option. ADD REPLY 1 Entering edit mode 12 months ago dtm2451 ▴ 30 You can do this in dittoSeq. dittoHeatmap() will even automatically grab the proper genes data subset and automaatically generate metadata annotations. # Import full dataset from edgeR DGEList to the format dittoSeq understands RNAdata <- importDittoBulk(your_DGEList_object) # If your type data was stored within the DGEList, it might be imported automatically # but to give example code if not... # Adding the 'Type' metadata RNAdata$Type <- types_data
# Make the heatmap
dittoHeatmap(
genes = DEG.genes,
annot.by = "Type",
order.by = "Type",
cluster_cols = FALSE)
Note on dittoSeq installation: If you are in R 4.0, you can install through Bioconductor with BiocManager::install("dittoSeq"). If not, you can still install it via the github (I have tested myself for R≥3.6.2, but this method should let you install in any R version.) BiocManager::install("dtm2451/dittoSeq")
Additional tweaks to the plot can be made by providing additional ?dittoHeatmap or ?pheatmap inputs, but the above code should accomplish the goals that you mention here.
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## anonymous 5 years ago a regular hexagon is inscribed in a circle with a diameter of 8 inches. what is the perimeter of the hexagon? what is the area of the hexagon?
1. anonymous
From the middle of the hexagon to one of its edges will be 4. A hexagon can be formed from 6 equilateral triangles The area of one equilateral triangle in this case is: A=bh/2 A=4(sin60*2) A=8sin(60) A of hexagon = 8 * 6sin(60) =$16\sqrt{3}$ P = 4*6 = 24
2. anonymous
given that its inscribed that means that it produces a hexagon that can be divided into 6 triangles. the triangles are all isosolese and the smaller part is actualy one face of the hexagon. if you drow out how this actuall looks you discover that the acute angle formed is 300/6 = 60 degrees. you can find the smaller side by simply using x/2 = 8sin30 , you use 30 degrees because you want a right trianlg ethe only way to do that is to cut it in half and produce a right triangle with acute angle 30 degrees and hypotonuse 8. Amazingly this side is 8 so the perimiter is 8*6 = 48. To find the area you can find the hight of one of the triangles which is h = 8cos30 ~~ 6.92.. then fidn the area of one triangle A = .5bh where base would be the x we found earlier so are of one triangle is A = 27.71. mustiply this by 6 to get the final answer of 166. This is however a shortcut of doing all of this useing two equations, i forgot the equations but most likely your teacher might know them or probably has already told you about them, in which case i would reccomend using the formula's since they are much faster and there is less chance of making a mistake.
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and equations in the image above on Jupyter notebook’s markdown! See the MyST configuration options for the full set of options, and MyST syntax guide for all the syntax options. To tell Markdown to interpret your text as LaTex, surround your input with dollar signs like this: Jupyter notebooks are documents that combine live runnable code with narrative text (Markdown), equations (LaTeX), images, interactive visualizations and other rich output: Jupyter notebooks (.ipynb files) are fully supported in JupyterLab. This should be the case for the notebooks that form part of this course. For more information about equation numbering, see the MathJax equation numbering documentation. The notebook document format used in JupyterLab is the same as in the classic Jupyter Notebook. In HTML output, these equation references only work for equations within a single HTML page. To me personally, Jupyter's real value is shown by the speed with which tasks move forward. Math and Equations¶ Jupyter Book uses MathJax for typesetting math which allows you to add LaTeX-style maths to your book. In this case you can select "Save a Copy in Drive" from the "File" menu to create a new copy that is yours and yo can edit. There is also a large community of scientists and tech companies that use Python for data analysis. Equation numbering and referencing will be available in a future version of the Jupyter notebook. Try it in your browser Install the Notebook. The Python extension for VS Code has shipped with Jupyter Notebook support for over a year with growing popularity. Python is a programming language with a simple syntax that many universities use for teaching. This notebook aims to show some of the useful features of the Sympy system as well as the notebook interface. For more information, check out the Reference Guide's section on Automatic Section Numbering. Solving equations and inequalities. The Jupyter notebooks made up 15% of the course grade, and the instructor compared the student notebooks with a notebook completed by the instructor. Tip. Colaboratory is a free Jupyter notebook environment that requires no setup and runs entirely in the cloud. Equation numbering and referencing will be available in a future version of the Jupyter notebook. For more details on the Jupyter Notebook, please see the From the official webpage jupyter.org: The Jupyter Notebook is a web application that allows you to create and share documents that contain live code, equations, visualizations and explanatory text. The Jupyter Notebook is an open-source web application that allows you to create and share documents that contain live code, equations, visualizations and narrative text. Markdown¶ Configuration¶ The MyST-NB parser derives from the base MyST-Parser, and so all the same configuration options are available. The IPython Notebook is now known as the Jupyter Notebook. which aid in presenting and sharing reproducible research. 5.2.3. In HTML output, these equation references only work for equations within a single HTML page. If automatic equation numbering is enabled, you can later reference that equation using its label. Jupyter Notebook. Math and equations¶ Jupyter Book uses MathJax for typesetting math in your HTML book build. Suggestions? Clicking this button will generate a table of contents using the titles you've given your Notebook's sections. Inline Typesetting (Mixing Markdown and TeX) ¶ While display equations look good for a page of samples, the ability to mix math and formatted text in a paragraph is also important. For more information, check out the Reference Guide's section on Table of … ... A symbolic math expression is a combination of symbolic math variables with numbers and mathematical operators such as +, -, / and *. Jupyter Notebook Users Manual ... Clicking this button will automatically number your Notebook's sections. That said, if you have experience with another language, the Python in this article shouldn’t be too cryptic, and will still help you get Jupyter Notebooks … Sympy is a computer algebra module for Python. This notebook is a demonstration of directly-parsing Jupyter Notebooks into Sphinx using the MyST parser.1. It doesn’t look as if anything has happened, but when you did Shift-Enter in the cell above, the Jupyter Notebook sent the code off to be executed by Python. Language of choice. Jupyter Notebook Users Manual.ipynb, Introduction; Help, shortcuts and number lines; Theme and fonts; Markdown; Jupyter Notebook LaTeX; Jupyter Notebook LaTeX equation numbering & I wrote all the text, symbols (even the arrows!) If automatic equation numbering is enabled, you can later reference that equation using its label. Text on GitHub with a CC-BY-NC-ND license Code on GitHub with a MIT license The standard Python rules for calculating numbers apply in SymPy symbolic math expressions. This page shows you a few ways to control this. There are already plenty of great listicles of neat tips and tricks, so here we will take a more thorough look at Jupyter’s offerings. Uses include: data cleaning and transformation, numerical simulation, statistical modeling, data visualization, machine learning, and much more. Let’s go ahead and create a new heading in our introduction.md file that includes some maths. At Microsoft we’re all in to embrace its power. Lack of consistent equation numbering in a notebook, or within sections of a notebook, is a serious limitation, along with the ability to reference equation numbers in the text. Jupyter notebook changelog; Questions? It is an interactive computational environment, in which you can combine code execution, rich text, mathematics, plots and rich media. 1. 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## FANDOM
3.190 Pagine
Armor is a stat shared by all units, including monsters, and buildings. Increasing armor reduces the physical damage the unit takes. Each champion begins with some armor which increases with level ( being the only exception). You can gain additional armor from abilities, items, and runes. Armor stacks additively.
Excluding whose base armor does not scale with levels, base armor ranges from 68.04 () to 123.5 () at level 18.
## Damage reduction
Note: One can include the armor penetration in all the following ideas by enumerating it with a due amount of corresponding negative armor.
Incoming physical damage is multiplied by a factor based on the unit's armor:
$\pagecolor{Black}\color{White}{\rm Damage\ multiplier}=\begin{cases}{100 \over 100+{\it Armor}}, & {\rm if\ }{\it Armor} \geq 0\\2 - {100 \over 100 - {\it Armor}}, & {\rm otherwise}\end{cases}$
Examples:
• 25 armor → × 0.8 incoming physical damage (20% reduction, +25% effective health).
• 100 armor → × 0.5 incoming physical damage (50% reduction, +100% effective health).
• -25 armor → × 1.2 incoming physical damage (20% increase, -16.67% effective health).
## Stacking armor
Every point of armor requires a unit to take 1% more of its maximum health in physical damage to be killed. This is called effective health:
$\pagecolor{Black}\color{White}{\rm Effective\ health} = (1 + \frac{{Armor}}{100})\times{\rm Nominal\ health}$
Example: A unit with 60 armor has 60% more of its maximum health in effective health, so if the unit has 1000 maximum health, it will take 1600 physical damage to kill it.
What this means: by definition, armor does not have diminishing returns in regard of effective hitpoints, because each point increases the unit's effective health against physical damage by 1% of its current actual health whether the unit has 10 armor or 1000 armor. However, health and armor have increasing returns with respect to each other.
Example: A unit starts with 1000 health and 100 armor giving it 2000 effective health. Now, it increases its nominal health from 1000 to 2000, thereby increasing its effective health from 2000 to 4000. Increasing the unit's armor by 100 at both nominal health levels would yield +1000 effective health and +2000 effective health, respectively. If we were to consider two nominal armor levels and then increase both by a static amount of health, we would see a similar increased return of effective health for the same nominal health.
Therefore, buying only armor is gold inefficient compared to buying the optimal balance of health and armor. It is important to not stack too much armor compared to health or else the effective health will not be optimal.
When a unit's armor is negative because of armor reduction or debuffs, armor has increasing returns with respect to itself. This is because negative armor cannot reduce effective health to less than 50% of actual health. A unit with -100 armor has 66.67% of nominal health (gains −33.33%) of its maximum health as effective health.
## Armor as scaling
These use the champion's armor to increase the magnitude of the ability. It could involve total or bonus armor. By building armor items, you can receive more benefit and power from these abilities.
### Items
• : Passive: Unique – Thorns: Upon being hit by a basic attack, reflects 3 (+ 10% bonus armor) magic damage, while also inflicting on the attacker for 1 second.
• : Passive: Unique – Icy Zone: Triggering Spellblade also deals (100% base AD) physical damage to enemies surrounding the target and creates a icy zone for 2 seconds, which enemies within by 30%. The zone has a radius of 180 (+ 55% bonus armor) units.
• : Passive: Unique – Thorns: Upon being hit by a basic attack, reflects 25 (+ 10% bonus armor) magic damage, while also inflicting on the attacker for 1 second.
## Increasing armor
### Items
Item Cost Amount Availability
• : Passive: Unique – Stone Skin: If 3 or more enemy champions are nearby, grants 40 bonus armor and 40 bonus magic resistance.
• : Passive: Unique: If your Partner is nearby, gain 20 armor, and 15% bonus movement speed while moving toward them.
## Ways to reduce armor
Note that armor penetration and armor reduction are different.
## Armor vs. health
Note: The following information similarly applies to magic resistance. As of season six, the base equilibrium line for armor is a function:
health = 7.5 × (armor + 100)
while for magic resistance the line is a bit shifted down and less steep:
health = 6.75 × (magic resistance + 100)
It can be helpful to understand the equilibrium between maximum health and armor, which is represented in the graph[1] on the right. The equilibrium line represents the point at which your champion will have the highest effective health against that damage type, while the smaller lines represent the baseline progression for each kind of champion from level 1-18 without items. You can also see that for a somewhat brief period in the early game health is the most gold efficient purchase, however this assumes the enemy team will only have one type of damage. The more equal the distribution of physical damage/magic damage in the enemy team, the more effective will buying health be.
There are many other factors which can effect whether you should buy more armor or health, such as these key examples:
• Unlike HP, increasing armor also makes healing more effective because it takes more effort to remove the unit's HP than it does to restore it.
• HP helps you survive both magic damage and physical damage. Against a team with mainly burst or just low magic damage, HP can be more efficient than MR.
• Percentage armor reduction in the enemy team tilts the optimal health:armor ratio slightly in the favor of HP.
• Whether or not the enemy is capable of delivering true damage or percent health damage, thus reducing the value of armor and health stacking respectively.
• The presence of resist or HP steroids built into your champion's kit, such as in or .
• Against sustained damage life steal and healing abilities can be considered as contributing to your maximum HP (while being mostly irrelevant against burst damage).
• The need to prioritize specific items mainly for their other qualities (regardless of whether or not they contribute towards the ideal balance between HP and resists).
## List of champions' armor
The highest and lowest base armor champions.
Champion Level Top 5 champions Bottom 5 champions
Level 1 1.
1.
1.
47 armor 1. 17.04 armor
2. 40 armor 2. 18 armor
3.
3.
44 armor 3. 18.72 armor
4. 36 armor 4. 19 armor
5.
5.
5.
5.
5.
39 armor 5.
5.
19.04 armor
Level 18 1. 123.5 armor 1. 28 armor
2. 109.1 armor 2. 68.04 armor
3. 112 armor 3.
3.
70.04 armor
4. 111.25 armor 4. 79.5 armor
5. 108.2 armor 5. 72.552 armor
## Optimal efficiency (theoretical)
Note: Effective burst health, commonly referred to just as 'effective health', describes the amount of raw burst damage a champion can receive before dying in such a short time span that he remains unaffected by any form of health restoration (even if the actual considered damage is of sustained form). Unless champion's resists aren't reduced below zero, it will always be more than or equal to a champion's displayed health in their health bar and it can be increased by buying items with extra health, armor and magic resistance. In this article, effective health will refer to the amount of raw 'physical damage' a champion can take.
In almost all circumstances, champions will have a lot more health than armor such that the following inequality will be true: Champion-Health > Champion-Armor + 100.
If this inequality is true, a single point of armor will give more 'effective health' to that champion than a single point of health.
If (health < armor + 100), 1 point of health will give more effective health than 1 of armor.
If (health = armor + 100), 1 point of health will give exactly the same amount of effective health as 1 point of armor.
Because of this relationship, theoretically, the way to get the maximum amount of effective health from a finite combination of health and armor would be to ensure that you have exactly 100 more health than armor (this is true regardless of how much health and armor you actually already have).
Example: Given a theoretical situation where you start off with 0 health and 0 armor and are given an arbitrary sufficient number of stat points (x ≥ 100), each of which you can either use to increase your health or armor by 1 point, the way to maximize your effective health is to add points to your health until your health = (armor + 100) = (0 + 100) = 100, and then split the remaining stat points in half, spend half on your health and half on your armor.
However, this is only theoretically true if we consider both health and armor to be equally obtainable resources with simplified mechanism of skill point investment. In reality a player buys these stats for gold instead. As gold value of armor (derived from cost of basic armor item) is currently (as of season six) 7.5 times higher than gold value of health (derived from cost of basic health item), we theoretically can maximize effective health represented by product of 0.01 × health × (armor + 100) with gold as input variable by satisfying the following equation: health = 7.5 × (armor + 100). The graph and conclusions obtained by solving it are mentioned in the subsection below.
Example: Given a theoretical situation where you start off with 0 health and 0 armor and are given an arbitrary sufficient amount of gold (x ≥ 281.25), which you can either use to increase your health or armor, the way to maximize your effective health is to add points to your health until your health = 7.5 × (armor + 100) = 7.5 × (0 + 100) = 750, and then split the remaining gold in half, spend half on your health and half on your armor (as former is 7.5 times cheaper than the latter, it would lead to buying 7.5 times more additional health than armor and thus naturally reaching equality in the equation above).
Now we just formulated a simple rule of preserving equilibrium (or maximum effective health):
Once equilibrium state is reached, all we need to do to preserve it is to always distribute gold equally into all involved stats for the rest of the game.
... or in our case, always 50% gold into health and 50% gold into armor.
Again this model is highly simplified and cannot be exactly applied in cases when we are buying any other item than , or (for example if our decision-making process would involve instead of , the above model would need to use equilibrium constant 7.6). Even considering the purchase of different armor or health items with differing gold efficiencies (quite natural expectation under real circumstances) makes use of single constant utterly impossible. Going even further, the continuous model simplifies a discrete character of real shopping, as you cannot really buy 1.5 × for 600, so with that much gold you opt to buy either a single or 2 × , drastically changing the equilibrium constant to 5.
However, thankfully to almost linear item stats' gold efficiency a player can use weakened base equilibrium condition in a form: health ≈ 7.5 × (armor + 100) safely enough to speed up decision-making. The important thing to remember is that there is no reason to hold to it too strictly.
Note: In case of magic resistance only the basic constant 7.5 is slightly changed to 6.75.
This information is strongly theoretical and due to game limitations from champions' base stats, innate abilities and non-linearity of gold value of item stats (gold value of stats differs for different items or is even impossible to be objectively evaluated due to interference of unique item abilities), the real equilibrium function is too complicated to be any useful.
The complexity of this problem provides space for players' intuition to develop and demonstrate their itemization skills. If given sufficient amount of time, each player could perfectly analyze situation at any given moment when he exited the shop and tell what should he buy at that moment for available gold to maximize own effective health. The sheer impossibility of doing such thing in real time creates opportunity to develop the skill. Not only that but often choosing to maximize current effective health leads to suboptimal decision branches in the future. The summary on end game screen about type of fatal damage taken is a key part of this decision process as well.
Instead, broadly speaking, items which provide both health and armor give a very high amount of effective health against physical damage compared to items which only provide health or only provide armor. These items should be purchased when a player is seeking efficient ways to reduce the physical damage they take by a large amount. Furthermore, these items are among all available items the best ones to distribute their gold value equally among both health and armor, thus working perfectly for rule of preserving equilibrium.
## Trivia
• Armor has a gold value of 20 (300 ÷ 15). This value is derived from .
Last updated: January 25, 2018, patch V8.2
• One of the biggest amount of armor any champion can obtain, aside from , is 2694.962488 (which reduces physical damage by 96.422%), being a level 18 .
• Base stats: 82 armor
• Runes:
• Items:
• 6
• Buffs:
• armor:
• Base stats: 97.8 armor
• Items = = 600 armor
• Runes = + = 128 armor
• Armor Amplification = 1 + = 1.06
• (armor = 600 + 128 + 97.8) × 1.06 = 875.84
• bonus = 875.84 × 0.2 = 175.169 bonus armor
• bonus armor:
• Items = = 600 armor
• Runes = + = 128 armor
• armor = 600 + 128 = 728 bonus armor
• bonus = 27.5 + 728 × 0.16 = 143.98 bonus armor
• armor:
• Base stats: 82 armor
• Items = = 600 armor
• Runes = + = 128 armor
• Buffs = + + = 349.1496 armor
• Armor Amplification = 1 + + = 2.2105
• armor = (82 + 600 + 128 + 349.1496) × 2.2105 + = 2694.962488 armor
, with his effectively infinite stacking, can obtain a maximum of 749999.25 armor off his passive alone. With the same set-up as above, he can obtain a total of about 910296.7564 armor, reducing physical damage by 99.99989016%.
## References
1. Should you buy Armor or Health?
v · e
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Definition/Summary
Limits are a mathematical tool that is used to define the ‘limiting value’ of a function i.e. the value a function seems to approach when its argument(s) approach a particular value. Although the argument of the function can be taken to approach any value, limits are helpful in cases where the argument approaches a value where the function is not defined or becomes exceedingly large.
While defining a limit, we say that the argument ‘tends to’ a value. For example,
$$\lim_{x \to c}f(x) = m$$
is said: “As x tends to c, the function f(x) tends to m”. This statement however makes no assertion of what the value of f(c) would be. Rather, it means that as x becomes exceedingly close to ‘c’, f(x) becomes exceedingly close to m.
If, however, the function is defined and continuous at ‘c’, then:
$$\lim_{x \to c}f(x) = f(c)$$
(See explanation)
Equations
Identities of limits:
$$\lim_{x \to c}f(x) + g(x) = \lim_{x \to c}f(x) + \lim_{x \to c}g(x)$$...
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# Does Smith normal form imply PID?
Let R be a commutative ring with a 1 different from 0, such that all finite matrices over R have a Smith normal form. Does it follow that R is a Principal Ideal Domain?
If not, what if R also has no zero divisors? (aka is an integral domain) What if additionally the diagonal entries are always unique up to associatedness?
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Doesn't Smith normal form hold for a principal ideal ring, possibly with zero divisors? I am thinking of $\mathbb{Z}/n\mathbb{Z}.$ – Victor Protsak Jul 10 '10 at 6:42
The implication is false without the assumption that R is Noetherian, because finite matrices don't detect enough information about infinitely generated ideals.
For example, let R be the ring $$\bigcup_{n \geq 0} k[[t^{1/n}]]$$ where $k$ is a field (an indiscrete valuation ring). Any finite matrix with coefficients in R comes from a subring $k[[t^{1/N}]]$ for some large $N$, and hence can be reduced to Smith normal form within this smaller PID.
However, the ideal $\cup (t^{1/N})$ is not principal.
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@Tyler: I don't think it affects the rest of your argument, but: to get a valuation ring, don't you want $k[[t^{\frac{1}{n}}]]$ instead of $k[t^{\frac{1}{n}}]$? – Pete L. Clark Jul 10 '10 at 18:20
Yes, you are correct - I added that sentence at the last minute. It is simply a ring with an indiscrete valuation. – Tyler Lawson Jul 10 '10 at 19:36
If every matrix has a Smith normal form, then every finitely generated $R$-submodule $M$ of $R^n$ satisfies $R^n/M$ is a finite direct sum of modules isomorphic to $R/aR$. If $R$ is Noetherian this implies that every finitely generated module is a direct sum of modules of the form $R/aR$. So if $I$ is a maximal ideal of the Noetherian $R$ then $R/I$ is a simple ideal, so if $R/I\cong R/aR$ then $I=aR$ is principal. So in a Noetherian ring with Smith normal form for all matrices, every maximal ideal is principal. Does this imply that all ideals are principal?....I'm not sure :-)
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For R a domain, it implies that R is one-dimensional regular, hence Dedekind, so every nonzero ideal is a product of maximal ideals, therefore principal itself. – user2035 Jul 10 '10 at 7:13
Thanks a-fortiori: each localization at a maximal ideal is a local ring of height at most one, so $R$ has Krull dimension $\le 1$. – Robin Chapman Jul 10 '10 at 7:19
A commutative noetherian ring whose maximal ideals are principal is indeed a principal ideal ring (even if it is not domain). See Theorem 12.3 of Kaplansky's article "Elementary divisors and modules," Trans. Amer. Math. Soc. 66 (1949), 464-491. – Manny Reyes Jul 10 '10 at 14:55
Work on ring-theoretic generalizations of Hermite/Smith normal forms goes way back, but made it into the mainstream via classic papers by Helmer and Kaplansky. Nowadays such rings are called elementary divisor rings, or rings with elementary divisors (r.e.d.) or Helmer rings, etc. A search on such terms, and for citations of Kap's classic paper [1] should quickly answer all your questions and then some.
[1] I. Kaplansky, "Elementary divisors and modules," Trans. Am. Math. Soc., 66, 464-491. (1949).
http://www.ams.org/journals/tran/1949-066-02/S0002-9947-1949-0031470-3/S0002-9947-1949-0031470-3.pdf
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Such rings are apparently called elementary divisor rings. They are necessarily Bezout rings (i.e. every finitely-generated ideal is principal), but not easy to characterize completely.
The first paper giving a nontrivial sufficient condition (beyond classical case) seems to be
Helmer, Olaf The elementary divisor theorem for certain rings without chain condition. Bull. Amer. Math. Soc. 49, (1943). 225--236, MR
More complete results are in a series of papers starting with
Larsen, Max D.; Lewis, William J.; Shores, Thomas S. Elementary divisor rings and finitely presented modules. Trans. Amer. Math. Soc. 187 (1974), 231--248, MR
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## Essential University Physics: Volume 1 (3rd Edition)
For this process, we see that the values of Q and W, by definition, are equal . Thus, the efficiency is: $e = \frac{Q}{W}\times 100 = 100$%
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# evaluating norm of sum of roots of unity
let $l_1,...,l_n$ be roots of unity. I want to prove that the norm(the product of all conjugates)of $a=l_1+...+l_n$ is not greater than $n$, not smaller than $-n$. how can I do to prove this?
• Have you tried to calculated a conjugate of $a$ under a field automorphism? In which field are you working? – BIS HD Oct 17 '13 at 8:41
• let l_i^(n_i)=1,and n be least common multiple of n_i.then the problem is to prove that the integer ring of Q(¥zeta_n) is Z(¥zeta_n)... – Yuma Oct 17 '13 at 9:02
• For a proof of your assertion, I refer to [Washington: Cyclotomic Fields], Theorem 2.6. – BIS HD Oct 17 '13 at 9:18
• If $n=2$ and $l_1=l_2=\omega$ is a primitive third root of unity, then the norm of $a=2\omega$ is $4>2$. So it seems the assertion is false, or am I missing something? – ladisch Oct 17 '13 at 11:46
We know that $l_k = e^{(\frac{2\pi i (k-1)}{n})}$, for $k=1,...n$. Then: $$a = \sum_{k=1}^n e^{(\frac{2\pi i (k-1)}{n})}$$ We can evaluate the norm of $a$ and use the triangular inequality: $$|a| = \left|\sum_{k=1}^n e^{(\frac{2\pi i (k-1)}{n})}\right| \leq \sum_{k=1}^n \left|e^{(\frac{2\pi i (k-1)}{n})}\right| = \sum_{k=1}^n 1 = n$$ Then $|a| \leq n$. Anyway, the norm is always a positive (or null) number, so it is always satisfied that it is not smaller than $-n$.
• is the absolute value of a same to algebraic norm of a? – Yuma Oct 17 '13 at 8:56
• yes, by definition $|a|^2 = aa^*$, where $a^*$ is the coniugate of $a$ – the_candyman Oct 17 '13 at 9:01
• but, I think that "conjugate" is "algebraic" conjugate. – Yuma Oct 17 '13 at 9:07
• the_candyman answered your question if you're working in an algebraic closure of $\mathbb{Q}$, for example $\mathbb{C}$. For an arbitrary field (e.g. $\mathbb{Q}$) the roots of unity are not necessarily elements of that field, so your question just makes sense in a field extension containing $\mathbb{Q[\zeta_1,\dots,\zeta_n]}$ but in this case the algebraic and analytic conjugates are the same. – BIS HD Oct 17 '13 at 9:14
• I think OP is asking about the field norm $N(a)=\prod_{\sigma}a^{\sigma}$, where $\sigma$ runs over all field automorphisms of $\mathbb{Q}(a)$. – ladisch Oct 17 '13 at 11:32
You can use the triangle inequality for that, i.e. $|a+b|\leq |a| + |b|$, then the result follows immediately by induction. (BTW: That the norm is larger $-n$ should be obvious, since a norm is always non-negative).
• when i want to try |a|(absolute value) <= n ,this problem is easy, but i want to know the algebraic norm of a.both is the same? – Yuma Oct 17 '13 at 8:58
I don't know why you might think the field norm would be at most $n$ in absolute value. For example, if $x=e^{2\pi i/11}$ then I calculate that the norm from ${\bf Q}(x)$ to $\bf Q$ of $x+x^2+x^4$ is 23.
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# Category:Rational Number Space
This category contains results about the rational number space in the context of Topology.
Let $\Q$ be the set of rational numbers.
Let $d: \Q \times \Q \to \R$ be the Euclidean metric on $\Q$.
Let $\tau_d$ be the topology on $\Q$ induced by $d$.
Then $\struct {\Q, \tau_d}$ is the rational number space.
## Subcategories
This category has only the following subcategory.
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# Effect of Temperature on pH of Water
The $K_\mathrm w$ is a function of temperature. It is $10^{-14}$ at $25\ \mathrm{^\circ C}$. When the temperature is $50\ \mathrm{^\circ C}$, the $K_\mathrm w$ can be calculated to be somewhere around $10^{-12}$ using Vant Hoff's equation, but this is so weird.
MY ATTEMPT
If we will solve for $K_\mathrm w=10^{-12}$, we get $\rm pH=6$. Will the water become acidic due to increase in temperature?
• The effect is correct; however your estimated $\mathrm{pH}$ is a bit low. The actual $\mathrm{pH}$ of neutral water at $50\ \mathrm{^\circ C}$ and normal pressure is about $\mathrm{pH} = 6.6$; i.e. the autoprotolysis constant of water is about $K_\mathrm w \approx 5\times10^{-14}$. – Loong Oct 25 '15 at 18:26
• People in this discussion don't understand the difference between "neutrality" and solution acidity. It is best to think of the Bronstead definition of acidity in discussions like this one. In that context it is known that methanol is more acidic than water and a solution of pure acetic is neutral, as well as more acidic than either methanol or water. (yes acetic acid has an autoprotolysis constant that happens to be about 14. So does methanol, ammonia, sulfuric acid and any other compound that self dissociates to a solvated hydrogen ion and the conjugate base form of the solvent) – Bill Tindall Mar 16 '17 at 21:57
• A solution of hot water is more acidic than cold water because it has a higher activity of dissociated hydrated hydrogen ions, yet it is neutral. For more reasons that I can easily type you can not test these facts with pH paper. – Bill Tindall Mar 16 '17 at 21:57
# TL;DR
$\rm pH=6$ is the neutral pH at $\rm 50\,^{\circ} C.$
# Long Version
Actually there is nothing wrong with your calculations. It seems that you are confused about the definition of a solution being acidic. If something is acidic, this means that the concentration of $\ce{H+}$ ions (technically $\ce{H3O+}$ ions) is greater than the concentration of $\ce{OH-}$ ions. At room temperature (25 degrees Celsius) a solution is neutral if its pH is 7. This means that for a solution to acidic, its pH must be lower than 7 at 25 degrees.
When we increase the temperature to 50 degrees, the pH of a neutral solution is actually 6, not 7. You can check this as if you were find the total concentration of $\ce{H+}$ ions and $\ce{OH-}$ ions, you will find that they both equal $10^{-6}$. Since there concentration is equal, the solution must be neutral. So for a solution to be acidic at a temperature of 50 degrees, its pH must be lower than 6, not 7.
## Explanation
Now you might wonder what the reason for this. Consider the following equation which is the auto-disassociation of water $$\ce{2H2O + heat \leftrightharpoons H3O+ + OH-}$$ As you can see, this reaction is actually endothermic. This means that when you increase the temperature, due to Le Chatelier's principle, the equilibrium will shift to the right; hence the concentration of $\ce{H3O+}$ and $\ce{OH-}$ ions will increase. This is indicated by the increase in the $\rm K_w$ value, as more products are formed.
Therefore since there is a greater of concentration of $\ce{H3O+}$, a solution which is neutral will have more $\ce{H3O+}$ ions at 50 degrees than it would when it is 25 degrees. Therefore its pH will be lower than 7 despite being neutral.
• So we can conclude that when we increase the temperature as there is increase in Kw both H3O+ and OH- ions concentration increases making it neutral. One more doubt. Why do we write H3O+ and H+ interchangeably as if both mean the same? – Ali Hasan Oct 25 '15 at 4:54
• @AliHasan H+ is easier to write but H3O+ is the actual form of H+ in water – TanMath Oct 25 '15 at 5:37
• Yes, we just use H+ for convenience, however is reality it is the hydronium ion, H3O+, that exists. Also higher hydrated protons do exists in solutions such as $\ce{H5O2+}$ and $\ce{H9O4+}$ – Nanoputian Oct 25 '15 at 8:07
• S0 we get pH=6. That means if we are boiling water and at that time we put a litmus paper in it, it's colour will change. Ph=6 at 50C and at 100C it will become more less. And as we increase temperature the pH value will tend 1 being more and more acidic. What I think is that water will still remain neutral as OH and H+ concentration will be same but a pH paper should most probably turn red. But that does not happen. – Ali Hasan Oct 26 '15 at 15:25
• @AliHasan That is because that the pH would not linearly decrease with temperature, but is rather like an exponential curve. Meaning that if heating the water to 50C decreases the pH, heating it by another 50C won't decrease the pH to 5 but to something like 5.5 (I don't exactly). So for the litmus paper to actually become red, the temperature has to extremely high (but the water would have evaporated to steam well before you could even reach the required temperature). – Nanoputian Oct 28 '15 at 4:59
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Find outlying "black swan" jumps in trends
find_swans(rotated_modelfit, threshold = 0.01, plot = FALSE)
## Arguments
rotated_modelfit Output from rotate_trends(). A probability threshold below which to flag trend events as extreme Logical: should a plot be made?
## Value
Prints a ggplot2 plot if plot = TRUE; returns a data frame indicating the probability that any given point in time represents a "black swan" event invisibly.
## References
Anderson, S.C., Branch, T.A., Cooper, A.B., and Dulvy, N.K. 2017. Black-swan events in animal populations. Proceedings of the National Academy of Sciences 114(12): 3252–3257. https://doi.org/10.1073/pnas.1611525114
## Examples
set.seed(1)
s <- sim_dfa(num_trends = 1, num_ts = 3, num_years = 30)
s$y_sim[1, 15] <- s$y_sim[1, 15] - 6
plot(s$y_sim[1, ], type = "o") abline(v = 15, col = "red") # only 1 chain and 250 iterations used so example runs quickly: m <- fit_dfa(y = s$y_sim, num_trends = 1, iter = 50, chains = 1, nu_fixed = 2)
#>
#> SAMPLING FOR MODEL 'dfa' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 3.8e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.38 seconds.
#> Chain 1:
#> Chain 1:
#> Chain 1: WARNING: There aren't enough warmup iterations to fit the
#> Chain 1: three stages of adaptation as currently configured.
#> Chain 1: Reducing each adaptation stage to 15%/75%/10% of
#> Chain 1: the given number of warmup iterations:
#> Chain 1: init_buffer = 3
#> Chain 1: adapt_window = 20
#> Chain 1: term_buffer = 2
#> Chain 1:
#> Chain 1: Iteration: 1 / 50 [ 2%] (Warmup)
#> Chain 1: Iteration: 5 / 50 [ 10%] (Warmup)
#> Chain 1: Iteration: 10 / 50 [ 20%] (Warmup)
#> Chain 1: Iteration: 15 / 50 [ 30%] (Warmup)
#> Chain 1: Iteration: 20 / 50 [ 40%] (Warmup)
#> Chain 1: Iteration: 25 / 50 [ 50%] (Warmup)
#> Chain 1: Iteration: 26 / 50 [ 52%] (Sampling)
#> Chain 1: Iteration: 30 / 50 [ 60%] (Sampling)
#> Chain 1: Iteration: 35 / 50 [ 70%] (Sampling)
#> Chain 1: Iteration: 40 / 50 [ 80%] (Sampling)
#> Chain 1: Iteration: 45 / 50 [ 90%] (Sampling)
#> Chain 1: Iteration: 50 / 50 [100%] (Sampling)
#> Chain 1:
#> Chain 1: Elapsed Time: 0.175159 seconds (Warm-up)
#> Chain 1: 0.00241 seconds (Sampling)
#> Chain 1: 0.177569 seconds (Total)
#> Chain 1: #> Warning: There were 25 divergent transitions after warmup. See
#> http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> to find out why this is a problem and how to eliminate them.#> Warning: Examine the pairs() plot to diagnose sampling problems#> Warning: The largest R-hat is NA, indicating chains have not mixed.
#> Running the chains for more iterations may help. See
#> http://mc-stan.org/misc/warnings.html#r-hat#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> http://mc-stan.org/misc/warnings.html#bulk-ess#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> http://mc-stan.org/misc/warnings.html#tail-ess#> Inference for the input samples (1 chains: each with iter = 25; warmup = 12):
#>
#> Q5 Q50 Q95 Mean SD Rhat Bulk_ESS Tail_ESS
#> x[1,1] -1.3 -1.3 -1.3 -1.3 0.0 1.00 13 13
#> x[1,2] -1.9 -1.9 -1.9 -1.9 0.0 1.00 13 13
#> x[1,3] -2.2 -2.2 -2.2 -2.2 0.0 1.00 13 13
#> x[1,4] -2.4 -2.4 -2.4 -2.4 0.0 1.00 13 13
#> x[1,5] -2.7 -2.7 -2.7 -2.7 0.0 1.00 13 13
#> x[1,6] -2.1 -2.1 -2.1 -2.1 0.0 1.00 13 13
#> x[1,7] -2.1 -2.1 -2.1 -2.1 0.0 1.00 13 13
#> x[1,8] -2.3 -2.3 -2.3 -2.3 0.0 1.00 13 13
#> x[1,9] -2.1 -2.1 -2.1 -2.1 0.0 1.00 13 13
#> x[1,10] -1.8 -1.8 -1.8 -1.8 0.0 1.00 13 13
#> x[1,11] -1.8 -1.8 -1.8 -1.8 0.0 1.00 13 13
#> x[1,12] -1.4 -1.4 -1.4 -1.4 0.0 1.00 13 13
#> x[1,13] -1.1 -1.1 -1.1 -1.1 0.0 1.00 13 13
#> x[1,14] -0.8 -0.8 -0.8 -0.8 0.0 1.00 13 13
#> x[1,15] -0.3 -0.3 -0.3 -0.3 0.0 1.00 13 13
#> x[1,16] -0.7 -0.7 -0.7 -0.7 0.0 1.00 13 13
#> x[1,17] -1.0 -1.0 -1.0 -1.0 0.0 1.00 13 13
#> x[1,18] -0.6 -0.6 -0.6 -0.6 0.0 1.00 13 13
#> x[1,19] -0.3 -0.3 -0.3 -0.3 0.0 1.00 13 13
#> x[1,20] 0.2 0.2 0.2 0.2 0.0 1.00 13 13
#> x[1,21] 1.3 1.3 1.3 1.3 0.0 1.00 13 13
#> x[1,22] 1.1 1.1 1.1 1.1 0.0 1.00 13 13
#> x[1,23] 0.8 0.8 0.8 0.8 0.0 1.00 13 13
#> x[1,24] 1.8 1.8 1.8 1.8 0.0 1.00 13 13
#> x[1,25] 1.9 1.9 1.9 1.9 0.0 1.00 13 13
#> x[1,26] 3.1 3.1 3.1 3.1 0.0 1.00 13 13
#> x[1,27] 4.1 4.1 4.1 4.1 0.0 1.00 13 13
#> x[1,28] 4.9 4.9 4.9 4.9 0.0 1.00 13 13
#> x[1,29] 5.5 5.5 5.5 5.5 0.0 1.00 13 13
#> x[1,30] 5.5 5.5 5.5 5.5 0.0 1.00 13 13
#> Z[1,1] -0.3 -0.3 -0.3 -0.3 0.0 1.00 13 13
#> Z[2,1] -0.4 -0.4 -0.4 -0.4 0.0 1.00 13 13
#> Z[3,1] -0.3 -0.3 -0.3 -0.3 0.0 1.00 13 13
#> log_lik[1] -0.4 -0.4 -0.4 -0.4 0.0 1.00 13 13
#> log_lik[2] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[3] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[4] -0.6 -0.6 -0.6 -0.6 0.0 1.00 13 13
#> log_lik[5] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[6] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[7] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[8] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[9] -0.2 -0.2 -0.2 -0.2 0.0 1.00 13 13
#> log_lik[10] -0.4 -0.4 -0.4 -0.4 0.0 1.00 13 13
#> log_lik[11] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[12] -0.3 -0.3 -0.3 -0.3 0.0 1.00 13 13
#> log_lik[13] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[14] -0.2 -0.2 -0.2 -0.2 0.0 1.00 13 13
#> log_lik[15] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[16] -0.2 -0.2 -0.2 -0.2 0.0 1.00 13 13
#> log_lik[17] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[18] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[19] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[20] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[21] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[22] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[23] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[24] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[25] -0.3 -0.3 -0.3 -0.3 0.0 1.00 13 13
#> log_lik[26] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[27] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[28] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[29] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[30] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[31] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[32] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[33] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[34] -0.3 -0.3 -0.3 -0.3 0.0 1.00 13 13
#> log_lik[35] -0.3 -0.3 -0.3 -0.3 0.0 1.00 13 13
#> log_lik[36] -0.2 -0.2 -0.2 -0.2 0.0 1.00 13 13
#> log_lik[37] -0.7 -0.7 -0.7 -0.7 0.0 1.00 13 13
#> log_lik[38] -0.3 -0.3 -0.3 -0.3 0.0 1.00 13 13
#> log_lik[39] -0.5 -0.5 -0.5 -0.5 0.0 1.00 13 13
#> log_lik[40] -0.5 -0.5 -0.5 -0.5 0.0 1.00 13 13
#> log_lik[41] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[42] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[43] -22.6 -22.6 -22.6 -22.6 0.0 1.00 13 13
#> log_lik[44] -0.5 -0.5 -0.5 -0.5 0.0 1.00 13 13
#> log_lik[45] -0.4 -0.4 -0.4 -0.4 0.0 1.00 13 13
#> log_lik[46] -0.2 -0.2 -0.2 -0.2 0.0 1.00 13 13
#> log_lik[47] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[48] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[49] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[50] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[51] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[52] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[53] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[54] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[55] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[56] -0.2 -0.2 -0.2 -0.2 0.0 1.00 13 13
#> log_lik[57] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[58] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[59] -0.4 -0.4 -0.4 -0.4 0.0 1.00 13 13
#> log_lik[60] -0.3 -0.3 -0.3 -0.3 0.0 1.00 13 13
#> log_lik[61] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[62] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[63] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[64] -0.4 -0.4 -0.4 -0.4 0.0 1.00 13 13
#> log_lik[65] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[66] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[67] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[68] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[69] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[70] -0.3 -0.3 -0.3 -0.3 0.0 1.00 13 13
#> log_lik[71] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[72] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[73] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[74] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[75] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[76] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[77] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[78] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[79] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[80] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[81] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[82] -0.1 -0.1 -0.1 -0.1 0.0 1.00 13 13
#> log_lik[83] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[84] -0.4 -0.4 -0.4 -0.4 0.0 1.00 13 13
#> log_lik[85] -0.2 -0.2 -0.2 -0.2 0.0 1.00 13 13
#> log_lik[86] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[87] -0.3 -0.3 -0.3 -0.3 0.0 1.00 13 13
#> log_lik[88] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[89] 0.0 0.0 0.0 0.0 0.0 1.00 13 13
#> log_lik[90] -0.2 -0.2 -0.2 -0.2 0.0 1.00 13 13
#> xstar[1,1] 4.1 5.8 7.8 5.9 1.3 1.07 13 13
#> sigma[1] 0.4 0.4 0.4 0.4 0.0 1.00 13 13
#> lp__ -29.7 -29.7 -29.7 -29.7 0.0 1.00 13 13
#>
#> For each parameter, Bulk_ESS and Tail_ESS are crude measures of
#> effective sample size for bulk and tail quantities respectively (an ESS > 100
#> per chain is considered good), and Rhat is the potential scale reduction
#> factor on rank normalized split chains (at convergence, Rhat <= 1.05).r <- rotate_trends(m)
p <- plot_trends(r) #+ geom_vline(xintercept = 15, colour = "red")
print(p)
# a 1 in 1000 probability if was from a normal distribution:
find_swans(r, plot = TRUE, threshold = 0.001)
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{}
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# A certain office has 6 employees. If R employees
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A certain office has 6 employees. If R employees [#permalink]
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24 Oct 2016, 08:36
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A certain office has 6 employees. If R employees are chosen to be on the party planning committee, what is the value of R?
(1) R² + 2R - 8 = 0
(2) There are 15 different ways to choose R employees to be on the party planning committee
NOTE: Kudos for all correct solutions
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Re: A certain office has 6 employees. If R employees [#permalink]
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24 Oct 2016, 08:42
1
GMATPrepNow wrote:
A certain office has 6 employees. If R employees are chosen to be on the party planning committee, what is the value of R?
(1) R² + 2R - 8 = 0
(2) There are 15 different ways to choose R employees to be on the party planning committee
NOTE: Kudos for all correct solutions
Our video solution to this question can be found here: https://www.gmatprepnow.com/module/gmat ... /video/799
S1 - Sufficient (R can not be negative & you get R = 4 on solving the equation)
S2 - Insufficient. Cannot get any value of R on solving the equation - 6Cr = 15
Hence A
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Re: A certain office has 6 employees. If R employees [#permalink]
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24 Oct 2016, 09:14
2
GMATPrepNow wrote:
A certain office has 6 employees. If R employees are chosen to be on the party planning committee, what is the value of R?
(1) R² + 2R - 8 = 0
(2) There are 15 different ways to choose R employees to be on the party planning committee
NOTE: Kudos for all correct solutions
Our video solution to this question can be found here: https://www.gmatprepnow.com/module/gmat ... /video/799
S1: By solving, (R-2)*(R-4)=0;
R can't be negative so, R=4 >>> SUFFICIENT
S2: There are two possible values i.e. R=2 or R=4 >>> INSUFFICIENT
Hence, A
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Re: A certain office has 6 employees. If R employees [#permalink]
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24 Oct 2016, 09:21
1
St 1:
R^2 + 2R -8=0
(R+4)(R-2)=0
R=-4 or 2
R can't be negative
So R = 2
Sufficient
Statement 2:
6C4 = 6C2 = 15
So R can be 2 or 4
Insufficient
A
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A certain office has 6 employees. If R employees [#permalink]
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05 Nov 2016, 17:54
mankodim wrote:
GMATPrepNow wrote:
A certain office has 6 employees. If R employees are chosen to be on the party planning committee, what is the value of R?
(1) R² + 2R - 8 = 0
(2) There are 15 different ways to choose R employees to be on the party planning committee
NOTE: Kudos for all correct solutions
Our video solution to this question can be found here: https://www.gmatprepnow.com/module/gmat ... /video/799
S1 - Sufficient (R can not be negative & you get R = 4 on solving the equation)
S2 - Insufficient. Cannot get any value of R on solving the equation - 6Cr = 15
Hence A
mankodim please note that your reasoning for Stmt 2 is not correct. The statement is Insufficient because you can get 15 for two values of R i.e. 2 and 4 and not because you can not solve for R
Others have provided more detailed solution that you can refer above.
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Re: A certain office has 6 employees. If R employees [#permalink]
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05 Nov 2016, 20:49
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Hi,
My two cents to this question
nCr means selecting “r” things out of “n”.
Remember this, if in a question total number of ways is given (the value of nCr) and how many are selected is also given (the value of r), then “n” will always a unique value.
For an example, nC2 = 10, then “n” has to be 5.
But if in a question total number of ways is given (the value of nCr) and total number of things is given (the value of n), then “r” may have more than one value.
For an example, 5Cr = 10 means “r” could be 2 or 3.
Just to add to that, remember nCr = nC(n-r).
So “r” could have more than one value.
So here in this question, n is given,
So we are finding “R”.
Statement I is sufficient:
No need to factorize this,
Just remember in a quadratic equation of the form Ax^2+Bx+ C = 0,
If B is positive and C is negative, then one root positive and one root will be negative.
So here also one value of R will be positive and one value of R will be negative.
Here negative value doesn’t make sense because we are selecting things.
So it is sufficient.
So answer has to be either A or D.
Statement II is insufficient:
It’s the first case which mentioned above,
So it could be 6C2 or 6C4
So not sufficient.
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Re: A certain office has 6 employees. If R employees [#permalink]
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06 Nov 2016, 04:37
1
GMATPrepNow wrote:
A certain office has 6 employees. If R employees are chosen to be on the party planning committee, what is the value of R?
(1) R² + 2R - 8 = 0
(2) There are 15 different ways to choose R employees to be on the party planning committee
FROM STATEMENT - I (POSSIBLE)
$$r^2 + 2r - 8 = 0$$
$$r^2 + 4r - 2r - 8 = 0$$
$$r ( r + 4 ) - 2 ( r + 4 ) = 0$$
$$r = 2 , - 4$$
Since, r can not be -ve value of r must be 2
FROM STATEMENT - I (NOT POSSIBLE)
$$6c_r$$ = 15
So, r can be 2 or 4
We can not have a unique solution for r from this statement ...
Hence, Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked, answer will be (A)
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A certain office has 6 employees. If R employees [#permalink]
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11 Jul 2017, 20:06
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GMATPrepNow wrote:
A certain office has 6 employees. If R employees are chosen to be on the party planning committee, what is the value of R?
(1) R² + 2R - 8 = 0
(2) There are 15 different ways to choose R employees to be on the party planning committee
NOTE: Kudos for all correct solutions
Our video solution to this question can be found here: https://www.gmatprepnow.com/module/gmat ... /video/799
What is tested in statement 2 is whether one can use the fact nCr = nC(n-r) and therefore see that two values of R are possible and in statement 1 of course, that R cannot be negative.
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1. Triangular prism area formula essay
# Triangular prism area formula essay
## Triangular Prism Method
Triangular Prism can be an important three-dimensional sturdy condition which might be put together by way of settling rectangles along with triangles collectively.
In this case, you may examine this exterior spot together with your size in case any Triangular Prism.In that day to day lifespan, look into a great instance from all the box seated concerning any floors and also everyday system is usually an example of this about Prism. Through very simple phrases, Some sort of prism is normally your sound form utilizing a few indistinguishable isn't stable related by way of several parallel lines.
In nearly all from a occasions, packaging contain as well rectangular as well as square finest or perhaps bottom part.
### Dive inside all of our blogging combine and also sprinkle many knowledge
Nevertheless imagine any package utilizing triangular very best or even floor, which means that exactly how resident hateful Check out secrets-and-cheats ps2 project ada most people identity triangular prism community system essay. This kind of will certainly end up being any Triangular Prism.Any prism together with only two parallel triangular basics in addition to a few rectangle-shaped facial looks therefore it again might possibly be given the name when the Triangular Prism.
$\ Base\; Area\;of\;a\;Triangular\;Prism = 12ab$
$\ Surface\; Area\;of\;a\;Triangular\;Prism = ab+3bh$
$\ Volume\;of\;a\;Triangular\;Prism = \frac{1}{2}abh$
If a person could structure the particular Triangular Prism in to portions in addition to place it again level on that platform consequently you actually will certainly far better fully understand this system involving the shape.
The item will certainly end up split inside several rectangles in addition to three or more triangles whenever torn properly.
When 3-dimensional shaped are shaped by just 2-dimensional forms and sizes next the application can possibly be branded since face.
Typically the major and also lower in typically the appearance usually are even so triangular bases. That 3 rectangles could end up known as when side encounters. In this particular strategy, your triangular prism can be shared into six people only two triangular together with two sq faces.
When a couple of people in a new Prism connect together, after that the item will probably earn some sort of sections portion which usually is normally given its name seeing that this borders.
### Triangular prism : precisely what is that?
Anytime moves speak to alongside one another then this will certainly make an important vertex. In temporary, your triangular prism generally seems to have five facial looks, six vertices, not to mention the being unfaithful triangular prism locale system essay. To make sure you stand for some prism, just about every vertex is definitely referred to as having a good different alphabet.
#### What usually are typically the qualities of some sort of Triangular Prism?
• A triangular prism as soon as medieval traditions essay comes with five looks, couple of triangular together with two block faces.
• A regular triangular prism possesses 9 edges.
• It will certainly get savvy explanation case study essay faces.
• It is going to create 6 vertices.
• The properties might improve for unnatural and semiregular polygons.
### Surface Section mccarthyism the shove essay an important Triangular Prism Formulation
Surface region is definitely a complete space to choose from outside the house for a powerful concept.
Pertaining to instance, when ever an individual go over a good proverbial box throughout wrap paper, in that case most people should comprehend it has the exterior space in order to find a powerful concept connected with that authentic volume with daily news. To make sure you gauge the floor community regarding a fabulous prism, people must divide any prism very first afterward analyze your work surface community accordingly.
$\ Surface\; Area\;of\;a\;Triangular\;Prism = ab+3bh$
In net sale, assess all the floor location in a pair of triangles, in addition to three square facets after that put these folks together with each other.
Now there will be shortcut methods likewise with regard to assessing a floor spot regarding difficult stats. With some level knowing of any notion, you actually will obtain to help you any solution fast without the need of every confusion.
### The Size about an important Triangular Prism Formula
The sound level associated with a fabulous three-dimensional appearance is certainly all the totals spot occupied with typically the article in addition to this is without a doubt provided with triangular prism section blueprint essay typically the solution with bottom area and size involving the prism.
$\ Volume\;of\;a\;Triangular\;Prism = \frac{1}{2}abh$
Thus, through amount awareness for Supplements, this particular might be convenient just for a person to be able to have an understanding of the particular prism and address many intricate formulas too.
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Proof of the Tarski-Vaught test
The Tarski-Vaught test is a way to determine if a substructure is elementary. To my understanding, here is the theorem:
Tarski-Vaught Test Let $$N$$ be a substructure of $$M$$. Then the following two statements are equivalent:
1. $$N$$ is an elementary substructure of $$M$$.
2. For any formula $$\phi$$ and elements $$p_1,p_2,\dots,p_n \in N$$ such that $$M \models \exists x. \phi(x,p_1,p_2,\dots,p_n)$$, there exists $$q \in N$$ such that $$M \models \phi(q,p_1,p_2,\dots,p_n)$$.
However, I have not been able to find a proof of this statement. 1 implies 2 is clear to me, but I am not sure how to prove 2 implies 1.
What is a proof of the Tarski-Vaught test?
• I'm curious how you failed to find a proof of this statement. It's proven in literally every textbook on model theory, and probably in almost all online course notes on model theory. And if you google "Tarski-Vaught test proof", the first result is a Proof Wiki page which contains a proof. May 15 '19 at 12:37
• @AlexKruckman I only checked online. I saw the proof wiki one, but the formatting made it hard to follow. May 15 '19 at 12:46
This is more a sketch of proof, but I hope it will be sufficient, if not feel free to ask for additional details.
Basically one have to prove that every formula $$\phi$$ with parameters in $$N$$ is true in $$M$$ if and only if it is true in $$N$$.
The way to prove that is by induction on the complexity of the formula:
• the case for atomic formulas with parameters in $$N$$ follows easily by the fact that $$N$$ is a substructure
• the case for formulas without quantifier follows easily
• the case for existentially quantified formulas follows by the hypothesis
• the case for universally quantified formulas follows from the logical equivalence $$\forall x. \phi(x)\Leftrightarrow \neg \exists x.\neg \phi(x)$$ and by the other cases.
I hope this helps (if not, as I said above, please ask).
• This makes sense! The only small question I have is that formulas without quantifiers transfer into any substructure, right? May 15 '19 at 11:18
• @PyRulez exactly, of course that must be proved, but it isn't really hard. Of course difficulty depend on the familiarity one have with the concepts, so feel free to ask if that's not the case. May 15 '19 at 11:29
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### Survey Methods Training
Survey Practice devoted the entire current issue to a discussion of training in survey methodology. This is a very useful review of what is currently done and suggestions for the future.
As they observe, survey methodology is a broad discipline that draws upon a diverse set of fields of research. I expect that increasing this diversity would be positive. That is, there are a number of fields of study that would find applications for their methods in the field of survey research.
A couple of key examples include operations research and computer science. Operations research could help us think more rigorously about designing data collection to optimize specified quantities. That doesn't mean we have to pursue one goal. But it would help, or maybe force us to quantify the vague trade offs we usually deal in. The paper by Greenberg and Stokes is an early example. The paper by Calinescu and colleagues is a recent one.
Computer science is another such field. Researchers studying reinforcement learning seek to optimize complex, multi-stage decision problems. These methods have been used to optimize adaptive treatment regimes. I think they may be a natural fit for some survey design problems. For example, the design of mixed mode surveys. Hopefully, we can direct ourselves toward such a future.
### "Responsive Design" and "Adaptive Design"
My dissertation was entitled "Adaptive Survey Design to Reduce Nonresponse Bias." I had been working for several years on "responsive designs" before that. As I was preparing my dissertation, I really saw "adaptive" design as a subset of responsive design.
Since then, I've seen both terms used in different places. As both terms are relatively new, there is likely to be confusion about the meanings. I thought I might offer my understanding of the terms, for what it's worth.
The term "responsive design" was developed by Groves and Heeringa (2006). They coined the term, so I think their definition is the one that should be used. They defined "responsive design" in the following way:
1. Preidentify a set of design features that affect cost and error tradeoffs.
2. Identify indicators for these costs and errors. Monitor these during data collection.
3. Alter the design features based on pre-identified decision rules based on the indi…
### An Experimental Adaptive Contact Strategy
I'm running an experiment on contact methods in a telephone survey. I'm going to present the results of the experiment at the FCSM conference in November. Here's the basic idea.
Multi-level models are fit daily with the household being a grouping factor. The models provide household-specific estimates of the probability of contact for each of four call windows. The predictor variables in this model are the geographic context variables available for an RDD sample.
Let $\mathbf{X_{ij}}$ denote a $k_j \times 1$ vector of demographic variables for the $i^{th}$ person and $j^{th}$ call. The data records are calls. There may be zero, one, or multiple calls to household in each window. The outcome variable is an indicator for whether contact was achieved on the call. This contact indicator is denoted $R_{ijl}$ for the $i^{th}$ person on the $j^{th}$ call to the $l^{th}$ window. Then for each of the four call windows denoted $l$, a separate model is fit where each household is assum…
### Goodhart's Law
I enjoy listening to the data skeptic podcast. It's a data science view of statistics, machine learning, etc. They recently discussed Goodhart's Law on the podcast. Goodhart's was an economist. The law that bears his name says that "when a measure becomes a target, then it ceases to be a good measure." People try and find a way to "game" the situation. They maximize the indicator but produce poor quality on other dimensions as a consequence. The classic example is a rat reduction program implemented by a government. They want to motivate the population to destroy rats, so they offer a fee for each rat that is killed. Rather than turn in the rat's body, they just ask for the tail. As a result, some persons decide to breed rats and cut off their tails. The end result... more rats.
I have some mixed feelings about this issue. There are many optimization procedures that require some single measure which can be either maximized or minimized. I think thes…
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# Math Help - Help with improper integrals
1. ## Help with improper integrals
the integral from neg. infinity to 6 of re^(r/3)
thanks!
2. $\int_{-\infty}^6re^{\frac{r}{3}}dr=\lim_{a\to-\infty}\int_a^6rd^{\frac{r}{3}}=$
$\lim_{a\to-\infty}3re^{\frac{r}{3}}-9e^{\frac{r}{3}}|_{a}^{6}$
I used integration by parts
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# HSSP Spring 2012 Course Catalog
Arts Engineering
Humanities Math & Computer Science
Science Miscellaneous
Arts
A5683: Conducting and Interpretation
Difficulty: ***
Teachers: Daniel Zhang
Learn the fundamentals of conducting and score interpretation!! We will be covering everything from beat patterns, cues and cut-offs to musical gestures and complex meters. We'll be examining a variety of different musical genres (Symphony, Opera, Mass, etc) throughout the history of classical music. In addition, we will also discuss interpretation with regards to compositional context and orchestration.
Prerequisites
Proficiency with Treble and Bass clefs. Instrumental or Vocal experience.
Engineering
E5712: Explore Solar Energy Closed!
Difficulty: **
Discover the basics of solar energy and photovoltaics through hands-on exploration and design! In this class, we will experiment with solar cells, discuss the physics behind photovoltaics, and explore how to use solar cells to charge batteries. The class will also explore basic principles of design. Students will work together to design a charging circuit, and each student will design and build their own rechargeable solar lamp by the end of the class.
E5691: Fun with Simple Circuits!
Difficulty: **
Teachers: Gurtej Kanwar
Come learn how to build simple circuits ranging from turning on an LED (if you don't know what that is don't worry) to basic digital circuits similar to the ones in computers!
Circuits are in everything around us, yet very few people actually understand how they work. In this class we will go over the basic principles behind simple analog and digital circuits through hands-on projects and activities.
Topic covered include: thinking about electricity as a fluid, Ohm's law, and simple boolean logic (AND, OR, NOT).
E5658: Modern Origami Full!
Difficulty: **
Origami is the ancient Japanese tradition of folding paper and has been around for centuries. However, modern origami is a young, complex art form full of beauty and mathematics. In this course, we will learn how to fold origami models and learn to read origami diagrams so that you can develop your skills on your own. Additionally, we will learn elements of origami design and analysis so that you can begin to design your very own origami models.
E5653: Introduction to Robotics!
Difficulty: **
Teachers: Bianca Homberg
Want to get a taste of robotics? Come design, build, program, test (and redesign, reprogram, and retest) a Lego robot! This class will start from the basics, teaching you how to program your robot and some good robot design principles. From there, we'll teach you how to make your robot follow a line, respond to a clap, and retreat when it bumps into something. The class will be mostly project-based, and you'll be free put what you've learned to use and spend time designing and building a robot to maneuver an obstacle course. The last week of the class, we'll have a full-class competition to see what your robots can do!
Humanities
H5707: The Meanings of Things
Difficulty: **
Teachers: Sebastian Schmidt
Have you ever wondered why there are things everywhere in our lives? Why kings and queens need scepters, why more people collect stamps than bubble wrap, why an original painting may make people cry, but a poster of the same painting is much less likely to do so?
In this class, we will be thinking about the many different meanings that things and objects take on in our lives. With people lining up outside Apple stores overnight before a product launch, and outside the Louvre in Paris to see the Mona Lisa, it is obvious that things matter to us, be they mass-produced or unique. From paintings to buildings, from utensils used in rituals to the pen we use to take notes - we use things and objects to make sense of the world around us, and they also carry their own meanings. The reasons why they are meaningful change over time, and many things and objects become part of cultural history once they are no longer in active use or tied to rituals and cultural practices; then they may be collected or we can find them in museums.
We will be looking at objects in a variety of contexts, including science, art, technology, consumer culture, and architecture, to get a better understanding of why things matter to us, or why they mattered to our ancestors. Participants will choose their own objects and work on a project resulting in an illustrated class report ('the book of things') to be distributed to participants at the end of the term.
H5662: Journalism and Why It Matters
Difficulty: **
Teachers: David Han
A crash course to the basic foundations of journalism, "Journalism and Why It Matters" will teach everything from writing the perfect lead to taking the clutch photo. Classes will be structured so that students will practice writing strategies and journalistic writing styles. Students will learn not only how to write news stories, feature stories, and editorials but also how to write them for an audience. Enrolled students will learn how to take interesting photographs and how to choose which photographs are best to print. Students in "Journalism and Why It Matters" will understand the ethics of journalism as well as the importance of journalism in the world today. The goals of the class are two-fold: to instill a sense of journalistic competence for aspiring editors and photographers while cultivating a greater appreciation for news sources.
Prerequisites
Basic understanding of English language and grammar.
H5676: Dreams and Dreaming Closed!
Difficulty: *
Teachers: Carol Hardick
Dreams and Dreaming introduces you to cognition and memory from the perspective of a sleeping brain, using an interdisciplinary approach of neuroscience, psychology and humanities. The Harry Potter series and the movies, Inception and The Matrix- to name just two- have lots of twists and turns and raise questions about the mind, knowledge and artificial intelligence. What's possible and what isn't? Can an idea be planted in a person's mind? Can two people share a dream? With lots of discussion, we’ll explore those questions, and more.
H5698: Introduction to Traditional Chinese Poetry
Difficulty: **
Teachers: Chaoyang Liu
Do you seriously want to learn some mandarin Chinese?
Well, you'd better! There are about 1.3 billion people on Earth whose native language is Chinese! Imagine the opportunities you'll have in the future if you know such a widely used language!
What's the fundamentals of the modern Chinese language?
What role does traditional poetry play in the Chinese culture?
We don't start from counting numbers. We don't start from "hello", "what's your name", or "how are you". (Well, these basics will be covered thoroughly depending on the general language level of the class.)
We start from the ancient beauty of the real Chinese language!
Have you wondered how kids in China start learning their mother tongue? They start by reciting countless traditional Chinese poems!
Wanna speak like a native speaker? Wanna get a taste of the authentic Chinese literary culture? Start here!
Basically, don't tell anyone that you know how to speak Chinese unless you can recite at least one Tang Dynasty poem. A Song or Yuan Dynasty poem will be even better!
Already have some background in Chinese? Well, this class will take you to an unprecedented trip to ancient China, like nothing you have ever seen before!
Have absolutely no background in Chinese? Don't worry! It's taught in English! In American English! Only those poems will be read in Chinese. But don't worry! They sound like pleasant music!
H5669: The Eight Greatest Science Fiction Authors Ever
Difficulty: *
Teachers: Zoe Snape
Love science fiction? Want to learn more about how it developed and who developed it? Think spending every Saturday discussing science fiction and society sounds excellent? Come to this class and learn about the lives and times of eight of the most respected science fiction authors ever--Isaac Asimov, Ray Bradbury, Arthur C. Clarke, Frank Herbert, Philip K. Dick, William Gibson, Orson Scott Card, Douglas Adams--and how their work shaped science fiction and the world. We'll discuss each author’s life and time period, best-known works, prominent or repeated themes and ideas, and contributions to the genre of science fiction and also science fiction as a whole, its common themes and ideas, and its influence on society and other forms of art.
This class will be about 1/4 lecture, 1/4 in-class readings and 1/2 class discussion, with no required homework. If you want to argue a case for the inclusion of a different author shoot me an email at zoesnape@mit.edu with your argument and I will consider it.
**PLEASE NOTE--SPOILER ALERT: Like most literary discussion, we will discuss the endings of some books in almost all of our classes. If you think this will totally spoil your experience then please don't sign up for this class.
Prerequisites
Willingness to hear and discuss spoilers.
H5673: Political Ideologies in the 20th and 21st Centuries Full!
Difficulty: **
Teachers: William Uspal
In the aftermath of the Global Finance Crisis, there is a pervasive sense that things cannot remain the same -- and yet they do, and no challenger to the pre-crisis consensus has remotely succeeded in displacing it. How has this ideological exhaustion come about? How can it be overcome? We will examine how previous economic and political crises shaped and were shaped by the ideological projects of the twentieth century: liberalism, Marxism, social democracy, and fascism. Particular emphasis will be placed on the relationship between political thought and action -- for instance, which (if either) is fundamental? Thinkers and topics of interest: Keynes, Lenin, Hayek; the Occupy and green movements, anarchism, libertarianism.
H5688: Sapere Aude: Philosophy Live!
Difficulty: **
What does it mean to be a philosopher? How does one practice philosophy?
These questions will be the focus of this class, and we will answer them with the help of some of the greatest philosophers the world has ever seen. By closely reading together texts from Aristotle, Plato, Descartes, Pascal, Kant, Nietzsche, and Heidegger, we will learn how to build arguments and give simple answers to complex questions. And since learning from example isn't enough, we will practice these skills through debates and discussions.
Thus the aim of this class will not only be to simply acquaint you with what these authors have to say, but also to have you become a philosopher yourself. Each class will be divided between the reading and analysis of a short excerpt, and a discussion of any topic of importance to you, from the meaning of life to moral dilemmas faced in everyday life.
H5710: Cities and Suburbs: Observing History, Shaping the Future
Difficulty: **
People shape cities.
Cities are places of commerce and culture. Cities are places of crisis and crime. How have our ideals and actions shaped the cities and suburbs we know and live in today?
Cities shape people.
Urban development is the foundation of communities, and thus an integral part of the opportunities experienced by families and individuals. How can we better our communities through urban planning?
This course will survey the history of American urban development and planning, focusing especially on cities and suburbs post-World War II, and engage students in developing tools and ideas for imagining and enacting a more sustainable and equitable urban future. It will be taught with a combination of lectures, discussions, and two walking tours of Boston neighborhoods.
Difficulty: **
Teachers: Samantha Berstler
"A poem conveys not a message so much as the provenance of a message, an advent of sense." --Thomas Harrison
This class is designed as a crash course survey of the major landmarks in British poetry mish-moshed together with an introduction to the college-style workshop. In other words, we'll spend half our time reading poetry and half our time writing poetry. No experience in writing poetry is necessary, and I will tailor the readings to the interests of the class. As of now, authors include the Anglo-Saxon elegiac poets, Shakespeare, Marlowe, Donne, Keats, Shelley, Byron, Sassoon, Owen, and Eliot.
Why take this class? It is good enrichment for higher level English classes, the SAT I and II, and the English AP exam; some of the authors we will read are heavily featured on the English AP. Additionally, the skill set required for writing poetry can immensely improve one's expository prose skills. Finally and most importantly, the class is going to be awesome.
Math & Computer Science
M5672: Algebra and Number Theory
Difficulty: ***
Teachers: Dylan Yott
I'm sure you're all familiar with Pythagorean triples like $$3$$, $$4$$, and $$5$$ which satisfy $$3^{2}+4^{2}=5^{2}$$. What if we replace the $$2$$ with a $$3$$? Are there any interesting whole number solutions to $$x^{3}+y^{3}=z^{3}$$? On the other hand, we've just shown $$25$$ is a sum of two squares. What other numbers can be written as a sum of two squares? Can ever number be written in this way?
I'm sure you're also familiar with equations such as $$ax+by=c$$, or $$x^{2}+y^{2}=k$$. We know what their graphs look like, but what can we say about points on these graphs? Are there any whole numbered pairs, $$x$$ and $$y$$ on these curves? How many? What if we allow $$x$$ and $$y$$ to be fractions?
I'm sure you're ALSO familiar with the idea of a prime number, that is, numbers that are only divisible by $$1$$ and themselves. Whole numbers have the fantastic property that we can factor them uniquely into primes. For example, $$21=3 \cdot 7$$ or $$2012=2^{2} \cdot 503$$. In this way, if we want to understand whole numbers well, it seems like it would suffice to just try and understand these primes. How many primes are there? How are they distributed? Are there patterns to them, or are they sort of randomly distributed? If you can answer this question, you could win \$1,000,000!
Such is the beauty of mathematics, with every question we answer, there are always more we can ask! In this class, we'll do our best to answer all of these questions and more!
The style of the class will be as follows. Each lecture, I'll present a sort of conceptual question about numbers or equations, which will motivate the discussion of important concepts in algebra and number theory. Then, we'll finish off each lecture by looking at an important problem in algebra or number theory, and solving it! (most of the time)
Short, fun, and interesting homework problems will be assigned, but they are $$\bf{optional}$$ and solutions will be given out the following class.
Prerequisites
High school algebra.
M5675: Bootstrap: Program your own video games! Full!
Difficulty: ***
Teachers: Kate Rudolph
Learn to program your own video game! We'll learn a programming language called Scheme, and learn and apply algebra concepts to make our characters move, change, and collide.
Prerequisites
This course is designed for those who have no experience with programming.
M5684: Fringes of Chaos
Difficulty: ***
Teachers: Zandra Vinegar
The common theme throughout this class is dynamic complexity, or, in a word, chaos – the chaos of the Mandelbrot fractal, the chaos of the universe that increases infinitely with time, the chaos that marks the edge of the set of patterns comprehensible to the human mind. This class will be like none other you have ever seen, and I may as well have filed it under physics or liberal arts instead of mathematics. There will be many days when we are extremely rigorous — assignments which ask for mathematically presented proofs — and days when we can't be rigorous simply because the questions we will discuss are still unanswered by science and mathematics at large. We will cover, in depth, the concepts surrounding and intertwining between Fractals, Entropy, and Universal Symmetries. We will discover the connections between these ideas through lectures and projects which range from online mathematical applets to discussions about required reading material. Every week will be intense and will require the full participation of all students. Come with an open and inquisitive mind and the work ethic to support it!
Prerequisites
The course has no real mathematical prerequisites but material does require significant mathematical maturity. Come prepared to think hard and abstractly!
M5694: Math for Middle School Lecture Series
Difficulty: **
Math is not about formulas and symbols: it’s about finding patterns! If three people want to all introduce themselves to each other, how many introductions will take place? Just three! But what about ten people? What about a thousand?
Through interactive lectures, this class will introduce quirky mathematical topics not covered in regular curricula (number theory, graph theory, logic, set theory, geometry, etc.) The lectures are intended for middle school students. The class will consist of a lecture from a different teacher every week, on a variety of math topics accessible to middle schools students.
M5682: Dealing with uncertainty: Intro to Probability!
Difficulty: ***
In this class students will learn about the main techniques of calculating probabilities of events and how to make decisions under uncertainty. The material will be motivated by games (cards, dice, etc), where the techniques learned in class will help you come up with better strategies.
Great emphasis will be placed on learning through solving problems and on elegant strategies for solving (difficult!) problems.
Topics
Sample spaces, events
Counting Techniques
Bayes theorem, conditional probability, and independence
Random variables, expected values,
Important distributions
Prerequisites
Students are expected to be (extremely!) comfortable with algebraic manipulation and well acquainted with pre-calculus material. Basic knowledge of Combinatorics is also expected. Calculus is NOT required (although if you know Calculus it will still be challenging and we can give you additional calculus related problems).
M5687: Intro to Topology
Difficulty: **
Teachers: Tucker Chan
We will guide you through some of the highlights of the wonderful theory of algebraic topology. That's a mouthful to say, and I promise it will be more fun than it sounds. We will investigate how to build up and break down different shapes such as mobius strips, toruses, and klein bottles, and study their properties, such as how loops in them behave. Also featured will be a topologist's breakfast, complete with edible toruses and lessons on a mathematician's method of playing with his food. Finally, we will use our newfound knowledge of shapes to arrive at some very interesting and surprising results.
So if you've ever marveled at mobius strips and how strange they are (have you ever cut one in half?) or if you've ever tried to tie an anti-knot -- or even if you haven't -- come learn about algebraic topology.
Added bonus: Just mentioning "algebraic topology" will impress your friends. They may, however, be in slightly less awe upon learning that it includes such gems as the "hairy ball theorem," "stone tu(r)key theorem," and "ham sandwich theorem."
Prerequisites
Algebra II. If you have been exposed to more math such as geometry, you will probably get more out of the class.
M5696: Caffeinated Calculus
Difficulty: **
Teachers: Naveen Kartik C K
In this class, you will learn to use Calculus, for fun and profit! We'll start by trying to understand how Calculus is a very natural result of nature and geometry, and not something scary and random. In fact, Calculus can be our best friend when we try to solve various problems, most notably in optimization. Ever wondered how you can maximize profit or minimize wastage? With the power of Calculus, you can convert your everyday problem into a math problem, and solve it using some very simple tools. By the end of the course, you will realize the role math plays in everyday life, and more importantly, how we can use calculus to hack it!
Prerequisites
Precalculus, including but not limited to, graphs, functions (including simple ones like sines, cosines and other trigonometric functions), algebra.
M5663: Great ideas in mathematics
Difficulty: **
Teachers: Akhil Mathew
This is an introduction to higher mathematics. We will survey some of the classical ideas in mathematical history, ranging from the infinitude of the primes to the development of the complex numbers to the uncountability of the reals. The goal is to present mathematics as a cultural artifact, and we will discuss some of the history along with the proofs.
Prerequisites
Familiarity with techniques of high school algebra (as in, for instance, Algebra II).
M5681: Introduction to Discrete Mathematics
Difficulty: **
Do you know how to count up to 10? What about counting the number of possible 5 card hands in poker? We will cover a variety of topics in discrete mathematics. We will start off by discussing combinatorics, which is how to count the number of a type of objects. We will then move on to probability, where we will learn how to estimate the chances of a particular event happening. We will then talk about a little bit of number theory, which studies the properties of numbers, and we will finish the class with some puzzle problems, which will relate to the topics that we discussed earlier
M5679: Awesome Abstract Algebra
Difficulty: ***
Teachers: Joshua Frisch
Algebra in high school is, to be honest, not particularly interesting. Algebra (as done in college) is novel, exciting, and interesting . Based on less than a half dozen rules you can describe a HUGE variety of interrelated structures which are intimately related with such seemingly disparate things as the structure the universe, how credit card data is encrypted on the web, and how google functions. Although we will not cover these specific applications we will discuss symmetry and (rather miraculously) a way to describe all of them, unique factorization (i.e. what makes primes work) and situations where it fails. And, as time permits, the connections between two thousand year old greek geometry problems, complex conjugation, and the insolvability of the quintic.
Prerequisites
Knowledge of Algebra 2 is a strict necessity. Both familiarity with proofs and modular arithmetic are highly recommended in order to get the most out of this class. experience with Calculus and precalculus, while not strictly related the content of the class are typically well mastered by people who take this class. This class will involve material typically taught to junior math majors at good colleges so, although I will do everything I can to make the material understandable, expect to be very challenged. (In essence, although no formal material is expected beyond the algebra 2 level, the material itself will be very difficult so any experience you have had (especially with proofs) will be helpful.
M5690: Relativity
Difficulty: ***
Probably the most well-known equation in all of physics was derived by Einstein in 1905: $$E=mc^2$$. This equation came with a radical new notion of how space and time are related. Einstein's theory, dubbed the Special Theory of Relativity, is now an essential part of our understanding of the universe. This course will cover the main ideas of special relativity, including length contraction, time dilation and the invariance of the speed of light. Time permitting, we will discuss how the effects of relativity have major implications to how charges and currents interact, and may also touch on ideas from General Relativity, which is the most successful theory of gravity to date.
Prerequisites
-Solid knowledge of mathematics through Algebra II. -Exposure to using sin, cos, and other trigonometric functions. -Some background in mechanics (i.e. Newton's law and similar) is helpful but not required.
M5700: Making Games in TaleBlazer Full!
Difficulty: **
Create location based games on your iPhone or Android using TaleBlazer. Using a block based language similar to Scratch, you can program virtual characters that interact with players who walk around with their phone.
You will be in the first class to use TaleBlazer. You'll help work out some of the bugs in the system before it is released to the general public.
If you have a iPhone or Android, bring it to class.
M5654: Game Theory, or How I Learned to Stop Losing and Love Math
Difficulty: **
HEY! Do you want to play games with MORE MATH? Take our class, and we'll play games with GRATUITOUS AMOUNTS OF MATH! In particular, we'll be studying the mathematics of game theory. In what games can we figure out which player has a winning strategy? In what games can we do this quickly? What interesting bits of math come out of such studies? You'll try to find the answers.
Prerequisites
Some math background, especially in proofs, will be helpful.
Science
S5659: Ecology of Bacteria
Difficulty: **
Teachers: Sarah Preheim
This is an ecology class from the perspective of bacteria. Imagine being microscopic, floating in the open ocean or growing in the stomach of a cow. These tiny creatures will help us understand ecological principles. We will investigate bacteria in their natural environment (i.e. just about everywhere!) and seek to understand why different bacteria live where they do. We will also learn about some of the amazing ways bacteria affect everything from your health to the global climate. Ecological principles will come to life in simple bacteria communities on petri dishes and in cultures. Additionally, we will discuss some of the technology that enable the study of bacteria in the environment.
S5701: Introduction to Cosmology
Difficulty: **
Teachers: Eric Gentry, Anna Ho
Cosmology is the study of the universe on the biggest scales: scales on which galaxy clusters look like smears and the universe can be thought of as one object expanding through space and time. In this class, we will talk about relativity, spacetime, black holes (black holes are awesome!), the fundamental forces of nature, dark matter, dark energy, and inflation - in order to understand how the universe began, how it is changing, and how it will end (if ever.)
S5668: Classical Mechanics and Special Relativity
Difficulty: ***
Teachers: Lawrence Chiou
An accelerated introduction to classical mechanics, the study of how objects move under Newton’s laws of motion (with some special relativity). There will be a strong focus on developing problem solving skills, as opposed to simply learning facts and formulas. Even those who have previously taken an AP Physics course will likely find many challenging problems and additional depth not covered in their high school course. That said, previous physics knowledge, though strongly recommended, is not required. Topics include kinematics, Newton's laws, special relativity, conservation of momentum and energy, rotational motion, and simple harmonic motion.
S5665: Introduction to Organic Chemistry Full!
Difficulty: ***
Notoriously a soulless crusher of the GPAs and egos of students across time and space, organic chemistry has been a ruthless "weed-out" class in universities across the nation. Laugh in the face of adversity and flout convention to ace this cruel monstrosity of a subject by getting a head start (undoubtedly to the utmost envy of your peers) by taking this introductory course to organic chemistry.
Organic chemistry, the study of carbon compounds, is the foundation of life and all living things. Carbon's unique ability to form a remarkable array of compounds is at the center of this compelling subject. In this course, we will focus on a few key classes of organic compounds, including alkenes/alkynes, alkyl halides, alcohols, carbonyls, and arenes. There will be some emphasis on problem solving and synthesis, and a lecture will be dedicated to structure elucidation if time allows.
Prerequisites
Some general chemistry knowledge
S5709: Why the Solar System is Awesome
Difficulty: *
Teachers: Emmett Krupczak
Do you know which object in the solar system has methane oceans? Or whose moon has 300 mile-high sulfur plumes? Who is the largest dwarf planet in the solar system? (Hint: it's not Pluto.) Come and find out all this and more! In this class, we'll learn about the fascinating celestial bodies
with whom we share our Solar System. We'll talk about what makes each of our celestial neighbors so unique and interesting. This class will also discuss the amazing tools and vehicles that we use to explore other planets. Time permitting, we'll touch on extrasolar planets and other Solar Systems.
S5667: True Chemistry
Difficulty: **
Unlike most high school chemistry classes, this class will discuss the true explanations behind a variety of chemical phenomena. Chemistry is NOT just a random collection of facts that one has to memorize—everything in chemistry does have an explanation, and it is my goal to reveal these explanations and show the true beauty of chemistry. We will start with the basis for all of chemistry, quantum mechanics (don't worry, no difficult math will be used), and from there we will consider electron configurations (with a special focus on electron configurations of the mysterious d-block elements) and use them to explain the wide array of reactivity that is exhibited by the elements of the periodic table. We will also discuss basic molecular orbital theory to explain different types of bonding (including metallic bonding). If we have time, we will discuss acid-base and oxidation-reduction chemistry, as well as some thermodynamics (including an explanation of what entropy really is!).
Prerequisites
All students should have completed at least one full year of high school chemistry or equivalent.
S5686: Introduction to Cancer Biology
Difficulty: **
Teachers: Tuyen Phung
How were cancer cells discovered? How can cells get "transformed"? How do cells metastasize from the primary tumor? In this course, students will be introduced to the basics of cancer biology; students will get an understanding of novel discoveries about tumor viruses, oncogenes, and tumor suppressor genes. Students will also get exposed to reading scientific papers and learning about basic technique in a cancer biology lab.
Prerequisites
Has taken some classes equivalent to high school biology
S5656: Introduction to Astronomy Full!
Difficulty: *
Teachers: Ashley Villar
“We are in the universe and the universe is in us.” –Neil deGrasse Tyson.
I remember when I was in the 8th grade and first heard about astrophysics. I found it fascinating and frightening, and I thought that, surely, the only way into the subject was through densely written manuals and have an innate Hawking mind. Luckily, this is absolutely wrong, and basic astrophysics is accessible to anyone. This class is designed for students who are intrigued by the night sky but have never taken an intensive course in astronomy or astrophysics. Black holes, the big bang, Kepler – learn about the who’s, what’s and where’s of the universe.
Note: no prior knowledge about astronomy is necessary, but I strongly encourage that students be committed to staying in the class. It’ll be on the smaller side and tight knit!
S5685: Neuroscience and The Seven Dwarfs
Difficulty: **
Teachers: Elise Ruan
What if Snow White was a neuroscientist? Meeting the seven dwarfs would have been fascinating! From Happy to Bashful to Sleepy to Grumpy, their behavior all starts in the brain.
This class uses basic neuroscience to understand what causes emotions, sneezing, sleepiness, and more. Find out what's going on in each of the dwarf's brains!
Prerequisites
Basic understanding of Biology. Preferably at least a year of an introductory class.
S5699: The Wonders of Modern Biology Closed!
Difficulty: **
Biology is a rapidly changing field, with many new ideas and discoveries. This class aims to unveil the intricacies underlying modern biology and how these new discoveries affect our understanding of modern issues. I will address problems like "cellular memory," the emergence of pandemics, the causes of cancer, and stem cell regulation in light of recent discoveries. Students will take an active role by reading four papers and discussing the impact they have on biology today.
Topics covered:
Genetics - overview of DNA and RNA, specifically addressing the RNA world theory and the Central Dogma; advanced topics include epigenetics, ""junk DNA,"" and RNA interference
Immunology - overview of the immune system, followed by genetic recombination, viruses, pandemics, and autoimmune diseases
Cancer - defining cancer, focusing on origin and metastasis; advanced topics include the Warburg effect and a mechanistic understand of both treatments like Gleevec and the emergence of resistance to medications
Stem Cells - defining stem cells, looking at the mechanisms of differentiation and self-renewal, regulation of the stem cell niche, tissue homeostasis, and aging
Prerequisites
Preferably one year of a biology class.
S5703: The Biology of Nutrition Full!
Difficulty: **
Nutrition has taken a central stage within the American consciousness, but the science behind what makes a food "good for you" is often lost in media discussions of public health. A combination of biology, physiology, and chemistry, this class will examine the science behind food, nutrition and cooking. Topics to be covered include the real meaning of metabolism and calories; the differences between simple and complex carbohydrates; how proteins and amino acids impact the body; the chemical and physical differences between saturated, unsaturated, and trans fats; and more. Snacks related to the lessons will be provided many weeks.
Topics covered:
Energy Systems
Metabolism
Proteins
Lipids
Carbohydrates
Vitamins and Minerals, Antioxidants
Whole Grains, ""Lite"" Foods
Science behind FDA Guidelines
S5655: Futures in Biotechnology
Difficulty: **
Teachers: Julia Winn
Vaguely interested in science but bored by what they teach in high school? Very interested in science and looking to explore different fields?
If you are interested in learning more about: stem cells, genetic engineering, neuroscience, cancer, aging, virology, hormones, computational biology, as well as more practical things like, how to find internships doing medical research in high school or what jobs actually exist for biotech people, this is the class for you.
Each lesson covers the basics within a specific field; the second part will cover the most recent biotechnological advances in that field, with an emphasis on applications.
Prerequisites
Some high school biology can be helpful, but the course will not assume any prior background.
S5660: Introduction to Psychology (and some Neuroscience) Full!
Difficulty: **
Teachers: yiling chen
How is the left side of the brain different from the right side, the front different from the back? What happens when different parts of the brain is damaged? What are some daily psychological principles we're all subjected to but don't really realize? How can you use these principles to your advantage? This class covers a range of topics in psychology including psychological disorders, learning, memory, personality, social psychology, and psychology across the life span.
Miscellaneous
X5680: Introduction to Critical Thinking and Creativity
Difficulty: **
Teachers: Volunteer Teacher
Hi! Welcome to Introduction to Creativity and Critical Thinking! Our class will alternate between learning about critical thinking and solving fun creative projects together with your classmates.
As part of our creativity workshops you may be an engineer working for a toy company who has to design a new toy that can launch objects into specific targets; or, you may be a famous fashion designer who has to make costumes made entirely out of news papers.
As a result, you will learn how to approach different design challenges, how to brainstorm about possible solutions, test them and redesign your product. You will get to work on different problems with your teammates and test your ideas together.
In the critical thinking part of the class we will get to learn about the structure of arguments, how they are used to convince others of a certain point of view. We will talk about different issues in society and why critical thinking is important to understand them.
Time permitting, we might also learn a circle folk dance!
Prerequisites
none
X5657: Cryptography and Cryptanalysis: Methods and History Full!
Difficulty: *
Teachers: Yixiao Wang
Cryptography has always been a constant war between those with secrets and those who want to expose secrets. It is the study of code making and code breaking. The strength of a cipher and the skill of the cryptanalyst have changed the outcome of many historic events. In this course, we will learn how different ciphers work and the methods used to crack them. We will travel through the history of cryptography from Caesar through WWI and WWII to the modern age. As we study each cipher we will learn how codes have impacted events such as the execution of Mary Queen of Scots and the entrance of the United States into World War I. Each week we will learn how a new cipher works and sometimes even learn how to crack it ourselves. Topics include frequency analysis, the Caesar cipher, the monoalphabetic substitution cipher, the Vigenere cipher, the Enigma, and public key cryptography.
X5695: World News Lecture Series
Difficulty: **
Teachers: Benjamin Kraft
Are you not content with just “living your life”, oblivious to the terrors happening around the world every day? Have no fear! This class will cover the past week’s world news, keeping YOU updated!
Each week’s class will be taught by a different teacher, talking about the events that shape our world today.
X5674: Teaching Workshop
Difficulty: **
Teachers: Kate Rudolph
Learn how to teach! This class will help you develop teaching skills and a lesson plan on a topic of your choice, You'll learn how to pick a topic your students will enjoy, and present it in a fun and interesting way! Throughout the class, you will work with a partner to develop your lesson plan and teaching skills, and for the final class you will have the chance to teach your class to younger HSSP students!
Prerequisites
High maturity level, consistent attendance at HSSP
X5692: ESPrinkler
Difficulty: *
During the last block (2:30 - 4:00), middle school students are given the opportunity to participate in ESPrinkler, a mini-program that consists of several one-shot (Splash-style) classes each week, including Singing A Capella, Explosive Chemistry, and Splendid Spleens
While you don’t have to individually register for each activity, you must register for this course (ESPrinkler) to partake in the different activities offered every week!
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# How to make Poinsettia pigment into ink?
Like many families, this winter we had bought a red poinsettia plant to decorate our kitchen, and now that it's losing its leaves, I had the genius idea to extract the red pigment. The activity proved a success, and let my 5 yr baby girl amazed (and us too). The pigment was extracted by letting leaves soak in 96% alcohol overnight in a closed container in a hot spot, then filtering the solids and heating the solution until alcohol evaporated for the most part, also adding a few drops of lemon juice to increase acidity and transform the brownish liquid into a blood red tint the that now has the aspect of an ink. Unfortunately, we were unable to use this ink to basically anything, as most mediums like regular paper or white paint are neutral or high pH and turn the poinsettia ink from red to any other color, like blue or green.
As you might have gathered by now, I am not a chemist, and I have no knowledge in the matter beyond high school classes from the old days. I did understand thou after reading this article that changes composition and color with pH.
So what I ask for, if at all possible, is a method described in plain terms and using household accessible chemicals to stabilize this red ink in order to use it, for instance, to paint on paper without losing its intense red color. (at least for a few hours or days)
• Soak the paper in acid beforehand, then let it dry. Jan 18, 2017 at 15:26
• Welcome to Chemistry.SE! What a lovely idea! In addition to Ivan Neretin's suggestion, you could try to carefully paint over the writing/drawing with (diluted) vinegar. An artistic introduction to anthocyanin inks might give some further directions. Jan 18, 2017 at 15:30
• Thank you all for your answers, soaking the paper in lemon juice or vinegar beforehand did cross my mind, also just skipping the reactive mediums altogether and directly draw over a thin plastic like a transparency. Anyhow, I'll look over all the resources and post the method that produced the best results. Jan 18, 2017 at 21:42
I think that would be hard, you could try to use a mordant to convert a dye to a pigment, you can use Alumina $\ce{FeSO4}$ or $\ce{CuSO4}$. There is an article that explains it in detail how to use Poinsettia for colouring silk fabric.
You can try to soak a thick paper in a solution of 1-5% of $\ce{FeSO4}$ with your dye heating gently.
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# $T_0$-space iff for each pair of $a$ and $b$ distinct members of X, $\overline{\{a\}}\neq\overline{\{b\}}$
Let $$(X,\mathscr T)$$ be topological space. Prove that $$(X,\mathscr T)$$ is $$T_0$$-space iff for each pair of $$a$$ and $$b$$ distinct members of X, $$\overline{\{a\}}\neq \overline{\{b\}}.$$
My attempt:-
Let $$(X,\mathscr T)$$ be topological space. Prove that $$(X,\mathscr T)$$ is $$T_0$$-space. $$a\neq b \implies$$ there is an open set $$U$$ contains $$a$$ but not $$b$$. $$\{b\}\subseteq X\setminus U \implies \overline{\{b\}}\subseteq X\setminus U.$$ $$a\notin X\setminus U$$. Any closed set containing $$a$$ must be completeltely disjoint from $$X\setminus U.$$ So, $$\overline{\{a\}}\neq \overline{\{b\}}.$$
Conversaly, $$\overline{\{a\}}\neq \overline{\{b\}}.$$ Consider $$X\setminus \overline{\{b\}}$$, which is an open set. $$X\setminus \overline{\{b\}} \subseteq X\setminus \{b\}$$. $$a\in X\setminus \{b\}$$ . $$b\notin X\setminus \{b\}\implies b\notin X\setminus \overline{\{b\}}$$. How do I complete the proof?
• The first implication is not true as written: Give $\mathbb{R}$ the topology in which a proper subset $U\subset \mathbb{R}$ is open iff $0\not\in \mathbb{R}$, and put $a = 0$. You can swap $a$ and $b$, though. – anomaly Feb 11 at 15:42
• Which implication? – Unknown x Feb 11 at 15:56
• It is an excercise in the Foundation of topology by C.W Patty. – Unknown x Feb 11 at 15:59
• See @PaulFrost's comment below. – anomaly Feb 11 at 16:21
• You cannot prove both. That would be a $T_1$ space. Example: Sierpinski space : $S=\{c,d\}$ with $c\ne d$, and the only open sets are $S,\{c\}$, and $\emptyset.$ This is $T_0$ but not $T_1$. We have $\overline {\{d\}}=\{d\}$ but $\overline {\{c\}}=\{c,d\}.$ – DanielWainfleet Feb 11 at 17:20
The first part of your attempt starts with a wrong statement.
You say that if $$X$$ is $$T_0$$ and $$a \ne b$$, then there exists an open $$U$$ containing $$a$$ but not $$b$$. However, this is the definition of $$T_1$$. In a $$T_0$$ space you can only say there exists an open $$U$$ containing exactly one of $$a, b$$. Of course you can say that without loss of generality we may assume $$a \in U$$ and $$b \notin U$$. To see the difference, consider the Sierpinski space $$\Sigma = \{ 0, 1 \}$$ with topology $$\mathfrak{T} = \{ \emptyset, \{ 1 \}, \Sigma \}$$. It is $$T_0$$ because the only two distinct points are $$0, 1$$ and $$\{ 1 \}$$ is an open set containing exactly on of these points whereas there does not exist an open set containing $$0$$ but not $$1$$.
Moreover, you cannot conclude that any closed set containing $$a$$ must be completely disjoint from $$X \setminus U$$. But it is irrelevant since $$a \notin \overline{\{b \}}$$, hence $$\overline{\{a \}} \ne \overline{\{b \}}$$. As an example consider $$\Sigma$$ and take $$a = 1, b = 0, U = \{1 \}$$. You have $$\overline{\{ a \}} = \Sigma$$ and $$\overline{\{ b \}} = \{ 0\}$$. In fact $$\overline{\{ b \}} \subset X \setminus U$$ and $$a \notin X \setminus U$$, but $$\overline{\{ a \}}$$ is not completely disjoint from $$X \setminus U$$.
For the converse, let $$a, b$$ distinct points of $$X$$. Then $$\overline{\{a \}} \ne \overline{\{b \}}$$. The set $$S = \overline{\{a \}} \cap \overline{\{b \}}$$ is closed.
It is impossible that both $$a, b \in S$$ because otherwise $$\overline{\{a \}} \subset S \subset \overline{\{b \}}$$ and $$\overline{\{b \}} \subset S \subset \overline{\{a \}}$$, i.e. $$\overline{\{a \}} = \overline{\{b \}}$$.
Case 1. $$S$$ contains none of $$a, b$$. Then $$b \notin \overline{\{a \}}$$ (because otherwise $$b \in \overline{\{a \}} \cap \overline{\{b \}} = S$$), hence $$b \in X \setminus \overline{\{a \}}$$ and we are done since $$a \in \overline{\{a \}}$$, i.e. $$a \notin X \setminus \overline{\{a \}}$$.
Case 2. $$S$$ contains exactly one of $$a, b$$, w.l.o.g. $$a$$. Then $$a \notin X \setminus S$$ and $$b \in X \setminus S$$ and we are done again.
• But my converse is not complete. How do I prove that $a\in X\setminus \overline{ \{b\}}$? – Unknown x Feb 11 at 16:23
• Thank you for correcting me in the first part. – Unknown x Feb 11 at 16:24
• You have an amusing typo "unclean" for "unclear" in the 1st sentence. – DanielWainfleet Feb 11 at 17:07
• @DanielWainfleet It was rather an unclear formulation of a non-native speaker than a typo ;-) I edit my answer. – Paul Frost Feb 12 at 10:53
• My native language is English and I make a lot of typos here. – DanielWainfleet Feb 13 at 4:53
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# Quadrature from x=0 to x=Inf?
I have previously used QuadGK over a finite interval. Now I need to do numeric integration from x=0 to x=Inf, and wonder if there is a routine that supports this? [I need to check scaling, etc. of the “continuous” Poisson distribution.]
Any suggestions?
QuadGK already supports this:
julia> using QuadGK
julia> quadgk(x -> exp(-x), 0, Inf, rtol=1e-5)
(0.9999999997018256, 2.779309769038174e-6)
5 Likes
Thanks for tip. I have problems making it work. Consider:
using SpecialFunctions, QuadGK, Plots
# "Continuous" Poisson distribution
function poisson(x,λ=1)
return λ^x*exp(-λ)/gamma(x+1)
end
I can plot the “continuous” Poisson distribution, see grey lines below:
The markers have been generated using the Distribution.jl package.
The “continuous” Poisson function is not a proper pdf because the integral from zero to infinity differs from unity. So I’d like to find scaling constants depending on \lambda.
With \lambda = 1, it works:
julia> quadgk(x->poisson(x),0,Inf)
(0.8338114480884391, 5.717008672714917e-9)
However, for \lambda = 2, quadgk crashes:
julia> quadgk(x->poisson(x,2),0,Inf)
DomainError with 0.9375:
integrand produced NaN in the interval (0.875, 1.0)
Stacktrace:
[1] evalrule(::QuadGK.var"#14#23"{var"#35#36",Float64}, ::Float64, ::Float64, ::Array{Float64,1}, ::Array{Float64,1}, ::Array{Float64,1}, ::typeof(LinearAlgebra.norm))...
Can you replicate this crash, or is it related to a previously reported LinearAlgebra problem on my i9-9900 home computer?
Can you modify you poisson function to show the input when the output is nan? Is it that 0.9375?
Not exactly sure how to do that. Some kind of try - catch?
I should say that Wolfram Alpha computes the result without problem. [But is inconvenient to work with.]
I’m not at the computer, but I’d just do something like
if isnan(....)
println(...)
end
before returning
OK – I’m trying this:
# "Continuous" Poisson distribution
function poisson(x,λ=1)
if isnan(λ^x*exp(-λ)/gamma(x+1))
println(x)
end
return λ^x*exp(-λ)/gamma(x+1)
end
leading to
julia> quadgk(x->poisson(x,2),0,Inf)
1871.5213495195621
DomainError with 0.9375:
integrand produced NaN in the interval (0.875, 1.0)
Stacktrace:
[1] evalrule(::QuadGK.var"#14#23"{var"#5#6",Float64}, ::Float64, ::Float64, ::Array{Float64,1}, ::Array{Float64,1}, ::Array{Float64,1}, ::typeof(LinearAlgebra.norm)) at C:\Users...
Hm… could it be that my function is numerically poor? I returned to the original poisson function:
julia> poisson(1871.5213495195621,2)
NaN
julia> poisson(BigFloat(1871.5213495195621),2)
1.946532642091951927364768066088547511008985129095238634528284797438114570307282e-4751
Would there be a way around this?
It is:
julia> function poisson(x,λ=1)
if isnan(λ^x*exp(-λ)/gamma(x+1))
@show λ^x; @show exp(-λ); @show gamma(x+1)
end
return λ^x*exp(-λ)/gamma(x+1)
end
poisson (generic function with 2 methods)
julia> poisson(1871.5213495195621,2)
λ ^ x = Inf
exp(-λ) = 0.1353352832366127
gamma(x + 1) = Inf
NaN
You have Inf/Inf
2 Likes
OK – I solved the problem by rephrasing the function:
# "Continuous" Poisson distribution
function poisson_ln(x,λ=1)
fX_ln = x*log(λ) - λ - log(gamma(x+1))
return exp(fX_ln)
end
(That’s an exception, not a “crash”.) The basic issue is that your poisson function returns NaN for large arguments:
julia> poisson(1e4,2)
NaN
This is due to a spurious overflow that arises in how you defined it: λ^x == 2^(1e4) == Inf and gamma(1e4+1) == Inf, so you are computing Inf / Inf which gives NaN. In fact, the final result is representable if you rearrange your computation of the poisson function to combine all of the exponents into a single exp call
julia> function poisson(x,λ=1)
return exp(-λ + x * log(λ) - loggamma(x+1))
end
which then gives:
julia> poisson(1e4,2)
0.0
at which point quadgk has no problem:
julia> quadgk(x->poisson(x,2),0,Inf)
(0.9470194210852405, 3.1332612554072925e-9)
(On a separate note, realize that integrating over an infinite interval with a generic routine like quadgk is internally transformed into a singular integrand on a finite interval. Because the integrand is effectively singular, it can be more challenging to integrate accurately, so sometimes you have to increase the requested tolerance rtol from the default ~ 1e-8.)
(Alternatively, you could use Gauss–Legendre quadrature for this sort of integrand, but that requires more manual intervention in terms of properly rescaling the integrand and selecting the number of quadrature points.)
Ah, sounds like you mostly figured it out for yourself. However, be sure to use loggamma(x+1) and not log(gamma(x+1)) — the latter can spuriously overflow. e.g. for x=1e4, the log(gamma(x+1)) call gives Inf whereas loggamma(x+1) gives 82108.92783681436. (In this particular case, the overflow may be harmless because the final exp call underflows to 0.0 in both cases.)
4 Likes
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# zbMATH — the first resource for mathematics
New spectral multiplicities for ergodic actions. (English) Zbl 1254.37003
Author’s abstract: Let $$G$$ be a locally compact second countable abelian group. Given a measure preserving action $$T$$ of $$G$$ on a standard probability space $$(X, \mu)$$, let $$\mathcal M(T)$$ denote the set of essential values of the spectral multiplicity function of the Koopman representation $$U_T$$ of $$G$$ defined in $$L^2(X,\mu)\ominus \mathbb C$$ by $$U_T(g)f := f\circ T_{-g}$$. If $$G$$ is either a discrete countable Abelian group or $$\mathbb R^n, n\geq 1$$, it is shown that the sets of the form $$\{p,q,pq\}, \{p,q,r,pq,pr,qr,pqr\}$$ etc. or any multiplicative (and additive) subsemigroup of $$\mathbb N$$ are realizable as $$\mathcal M(T)$$ for a weakly mixing $$G$$-action $$T$$.
##### MSC:
37A15 General groups of measure-preserving transformations and dynamical systems 37A30 Ergodic theorems, spectral theory, Markov operators
Full Text:
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# On Pseudo-finite topological spaces
We recall that a topological space $(X,\tau)$ is Pseudo-finite, if each compact subset of $X$ is finite.
One of the classical example of Pseudo-finite topological spaces can be considered as an uncountable set $X$ with the co-countable topology.(i.e.each subset with countable complement is open)
The above topology has no isolated point but it fails to be at least Hausdorff. On the base of my Knowledge there are two Tychonoff Pseudo-finite topological spaces as follows:
A. All discrete spaces are trivial examples of these spaces.
B. Consider the set $\Sigma=\mathbb{N}$$\cup {p}, and topologize it as follows: • Consider a free ultrafilter \mathcal{U} on \mathbb{N}. • All points of \mathbb{N} are isolated. • The Neighborhoods of p are of the form: U$$\cup$ {$p$}, where $U \in \mathcal{U}$.
We must recall that Case "B" is a special Example of maximal Hausdorff topologies on a set.
But I think there is no example of a Pseudo-finite Tychonoff space without isolated point !. and I guess the following statement:
Statement:Every Pseudo-finite Tychonoff space has an isolated point.
Is there a counterexample of the above statement?
-
Yet another spelling of Tychonoff. – Martin Brandenburg Jun 9 '12 at 10:58
I am sorry about my wrong spelling. – Ali Reza Jun 9 '12 at 11:02
Every $P$-space (a $P$-space is a completely regular space where every G_{ \delta}-set is open) is pseudofinite since one can easily show that every subspace of a $P$-space is a $P$-space and every compact $P$-space is finite. However there are many $P$-spaces with no isolated points. For instance, take the $P$-space coreflection of an uncountable product of the space {0,1} and you will get a pseudofinite space with no isolated points.
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Hello dear Joseph. Thank you very much for your nice description. Yes, the fact that you mentioned Is well-Known. I must point the fact that it may be useful. When I saw you answer, another Question came in my mind that if there is a non P-space without isolated point which is also Pseudo-finite. The answer in this case is also no. It Suffices to consider the topological space $X$ which you described as above and multiply it with the space $\Sigma$(i.e. $Y=X \times \Sigma$) (Best Wishes) – Ali Reza Jun 11 '12 at 7:09
I believe you can grow your example B into a counterexample.
Stage 0 is $\emptyset$
Stage 1 is $\{p\}$.
Stage 2 is your $B$: you've added a copy of ${\Bbb N}$ for each point newly added in the previous stage (all one of them) with an ultrafilter to define neighborhoods of the old point(s).
And the recipe for Stage $n+1$ adds a copy of ${\Bbb N}$ for each point newly added in Stage $n$ with an ultrafilter to define neighborhoods of the old points.
Natural numbers suffice to index the stages (no transfinite induction necessary).
Clearly we kill all the isolated points in the limit. I believe you get pseudo-finiteness much as you get it for $B$, but there are details.
-
Details: Let $\mathcal{U}$ be a free ultrafilter on $\omega$; let $X$ be the tree $\omega^{<\omega}$ with the topology generated by the set $\mathcal{B}$ consisting of all subsets $Y$ such that $Y$ has a unique root and $\forall y\in Y\ \forall^{\mathcal{U}}n\ \ y^\frown n\in Y$. Check that $X$ is $T_1$ and $\mathcal{B}$ is a clopen base, implying $X$ is $T_{3.5}$. By Konig's Lemma, if $A\subset X$ is infinite, then $A$ contains an infinite chain or an infinite set of the form $\\{s^\frown n:n\in I\\}$. In either case, check that $A$ is not compact. – David Milovich Jun 10 '12 at 19:31
The counterexamples so far depend on AC, but one can have such spaces just from ZF.
In particular, instead of an ultrafilter on ${\Bbb N}$ as in your example $B$, one can use the filter of subsets with asymptotic density 1. The rest follows along the lines of my previous answer (and David Milovich's details).
-
The set of $\{1/n\}\cup\{0\}$ is compact but not finite. – Jim Conant Jun 10 '12 at 20:27
I think Warren was confused The concept of Pseudo-finiteness with the concept "pseudo-discreteness".The topological space $X$ is pseudo-discrete if if every compact subset of $X$ has finite interior. Every pseudo-finite space is a pseudo-discrete space, but not conversely. The space $\mathbb{Q}$ is a pseudo-discrete space which is not pseudo-finite. – Ali Reza Jun 11 '12 at 8:51
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anonymous 3 years ago rationalize the denominator
1. anonymous
2. tiffanymak1996
$\frac{ 6 }{ 7-2+2\sqrt{6}-3\sqrt{6} }$
3. tiffanymak1996
$\frac{ 6 }{ 5-\sqrt6 }$
4. tiffanymak1996
$\frac{ 6 }{ 5-\sqrt6 }\times \frac{ 5+\sqrt6 }{ 5+\sqrt6 }$
5. tiffanymak1996
$\frac{ 6(5+\sqrt6) }{ 25-6 }$
6. tiffanymak1996
then just do the subtraction and multiplication, which should result in (30+6sqrt6)/19
7. anonymous
8. tiffanymak1996
good, usually for rationalizing the denominator, you just apply a^2 -b^2 = (a+b)(a-b), to eliminate the surds, then you're done.
9. anonymous
thanyou :)
Find more explanations on OpenStudy
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Friedel–Crafts reaction of phenol
Does phenol undergo Friedel–Crafts reactions or does it react with Lewis acids like aniline does?
Like aniline, phenol too reacts to a very less extent during Friedel-Crafts reaction.
The reason being that the oxygen atom of phenol has lone pair of electrons which coordinate with Lewis acid.
In fact most substituents with lone pair would give poor yield.
Phenols are examples of bidentate nucleophiles, meaning that they can react at two positions:
• on the aromatic ring giving an aryl ketone via C-acylation, a Friedel-Crafts reaction
or,
• on the phenolic oxygen giving an ester via O-acylation, an esterification
• Does the coordination to the Lewis acid catalyst take place for all alkoxy (e.g. methoxy, ethoxy etc.) groups as well? Technically, due to the electron-donating effect of the alkyl group, wouldn't the extent of coordination to the catalyst increase for these groups? – Tan Yong Boon Feb 19 '18 at 16:04
• You can perhaps answer this question as well while you are responding to my query: chemistry.stackexchange.com/questions/72184/… – Tan Yong Boon Feb 19 '18 at 16:05
To add to @user223679's answer. Phenol can react via two pathways with acyl chlorides to give either esters, via O-acylation, or hydroxyarylketones, via C-acylation.
However, phenol esters also undergo a Fries rearrangement under Friedel-Crafts conditions to produce the C-alkylated, hydroxyarylketones. This reaction is promoted by having an excess of catalyst present, either a Lewis acid such as $\ce{AlCl3}$ or strong Brønsted acids such as $\ce{HF}$ and $\ce{TfOH}$.
This paper reports yields of >90% (up to 99%) O-acylated products when phenol, and various derivatives is reacted with acyl chlorides in 1% $\ce{TfOH-CH3CN}$ solutions.
In neat $\ce{TfOH}$, yields of >90% C-acylated products are reported for phenol, and many ortho and para substituted derivatives, but significantly lower yields of 40-50% C-acylation are reported for meta substituted derivatives.
Additionally, various phenol esters were heated in neat $\ce{TfOH}$ and yields of >90% of the C-acylated Fries rearrangement products were reported. The yields were similar for ortho and para substituted phenol esters but again significantly lower (30-60%) for meta substituted reactants.
Overall this suggests that the ratio of O to C acylated products is strongly influenced by the concentration of the catalyst, with high concentrations favouring C-acylation and low concentrations favouring O-acylation.
phenol DOES NOT undergo Friedel Craft Alkylation or Acylation. This is because oxygen's lone pair in phenol makes co ordinate bond with AlCl3 (A Lewis acid) hence blocking it . So alternative way is Fries Rearrangement.
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# Math Help - Compactness and Product Topologies
1. ## Compactness and Product Topologies
So, I have a problem that I'm working on and I can't seem to figure it out. We just started product topologies, so many properties are still new or unknown to me. Here's the problem:
Let $(X,\Omega)$ and $(Y,\Theta)$ be topological spaces. If $A \subseteq Y$ is compact relative to $\Theta$ and $x \in X$, show that $\{x\}\times Y$ is compact relative to the product topology on $X\times Y$.
I'm not seeing why $A$ being compact is sufficient for the whole product to be compact... Any help would be appreciated.
2. ## Re: Compactness and Product Topologies
Originally Posted by Aryth
Let $(X,\Omega)$ and $(Y,\Theta)$ be topological spaces. If $A \subseteq Y$ is compact relative to $\Theta$ and $x \in X$, show that $\{x\}\times Y$ is compact relative to the product topology on $X\times Y$.
I'm not seeing why $A$ being compact is sufficient for the whole product to be compact... Any help would be appreciated.
Are you sure that you have copied the question correctly?
Could it be show that $\{x\}\times A$ is compact relative to the product topology on $X\times Y~?$.
3. ## Re: Compactness and Product Topologies
Originally Posted by Plato
Are you sure that you have copied the question correctly?
Could it be show that $\{x\}\times A$ is compact relative to the product topology on $X\times Y~?$.
If it is then I certainly understand it, but the question definitely says $\{x\}\times Y$. I'll ask my professor tomorrow if the problem as stated has a typo and prove the revision instead. Thanks for the help.
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# Question a5da3
Jun 16, 2015
The energy of the reactant(s) is $\text{+233 kJ}$.
#### Explanation:
The information given to you describes an exothermic reaction because the activation energy of the reverse reaction is smaller than the activation energy of the forward reaction.
This implies that the products are lower in energy than the reactants. As a result, you should expect the energy of the reactants to exceed the energy of the products.
A generic potential energy diagram for an exothermic reaction looks like this
I won't go into the details about what activation energy and threshold energy are, but you can read more on that here:
http://socratic.org/questions/what-is-activation-energy-what-is-threshold-energy-i-want-to-know-the-difference?source=search
So, in your case, the forward reaction has an activation energy equal to
${E}_{\text{a for" = "+96 kJ}}$
The reverse reaction has an activation energy equal to
${E}_{\text{a rev" = "+295 kJ}}$
Moreover, you know that the nergy level of the products is equal to
${E}_{\text{p" = "+34 kJ}}$
This means that the threshold energy of the reaction, i.e. the average kinetic energy that molecules need to react, can be determined by adding the activation energy of the reverse reaction to the energy of the products.
${E}_{\text{TS" = E_"a rev" + E_"p}}$
${E}_{\text{TS" = 295 + 34 = "+329 kJ}}$
So, in order to get a reaction, your molecules need to have an average kinetic energy of +329 kJ. Since you know what the activation energy of the forward reaction is, you can determine the energy of the reactants by
${E}_{\text{TS" = E_"a for" + E_"r}}$
E_"r" = E_"TS" - E_"a for" = 329 - 96 = color(green)("+233 kJ")#
The enthalpy change of reaction will be negative, since the energy of the products is lower than the energy of the reactants.
$\Delta {H}_{\text{rxn" = H_"products" - H_"reactants}}$
$\Delta {H}_{\text{rxn" = 34 - 233 = "-199 kJ}}$
The minus sign symbolizes the fact that heat is released by the reaction.
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# CheckSomos¶
CheckSomos is an auxiliary package that translates the representation of Somos at url{https://web.archive.org/web/20190709153133/http://eta.math.georgetown.edu/eta07.gp} in terms of Euler functions to our representation in terms of eta functions.
fromEta: (NonNegativeInteger, Polynomial Integer) -> Polynomial Integer
fromEta(m, p) expresses a polynomial p in variables Ei (corresponding to eta(i*tau)) in terms of variables q and ui where the ui correspond to the Euler functions of level m. https://en.wikipedia.org/wiki/Euler_function).
member?: (NonNegativeInteger, Polynomial Integer, List Polynomial Integer) -> Boolean
member?(m, s, gb) returns true if s is in the ideal given by the (assumed degrevlex) Groebner bases gb for the eta relations of level m.
toEta: (NonNegativeInteger, Polynomial Integer) -> Polynomial Integer
toEta(m, p) expresses the p (given as a polynomial in variables ui and q where the ui correspond to the Euler functions of level m https://en.wikipedia.org/wiki/Euler_function) into an expression in variables Ei (corresponding to eta(i*tau)). The ui and Ei correspond to level m.
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# NAG CL Interfaced01tdc (dim1_gauss_wrec)
Settings help
CL Name Style:
## 1Purpose
d01tdc computes the weights and abscissae of a Gaussian quadrature rule using the method of Golub and Welsch.
## 2Specification
#include
void d01tdc (Integer n, const double a[], double b[], double c[], double muzero, double weight[], double abscis[], NagError *fail)
The function may be called by the names: d01tdc, nag_quad_dim1_gauss_wrec or nag_quad_1d_gauss_wrec.
## 3Description
A tri-diagonal system of equations is formed from the coefficients of an underlying three-term recurrence formula:
$p(j)(x)=(a(j)x+b(j))p(j-1)(x)-c(j)p(j-2)(x)$
for a set of othogonal polynomials $p\left(j\right)$ induced by the quadrature. This is described in greater detail in the D01 Chapter Introduction. The user is required to specify the three-term recurrence and the value of the integral of the chosen weight function over the chosen interval.
As described in Golub and Welsch (1969) the abscissae are computed from the eigenvalues of this matrix and the weights from the first component of the eigenvectors.
LAPACK functions are used for the linear algebra to speed up computation.
## 4References
Golub G H and Welsch J H (1969) Calculation of Gauss quadrature rules Math. Comput. 23 221–230
## 5Arguments
1: $\mathbf{n}$Integer Input
On entry: $n$, the number of Gauss points required. The resulting quadrature rule will be exact for all polynomials of degree $2n-1$.
Constraint: ${\mathbf{n}}>0$.
2: $\mathbf{a}\left[{\mathbf{n}}\right]$const double Input
On entry: a contains the coefficients $a\left(j\right)$.
3: $\mathbf{b}\left[{\mathbf{n}}\right]$double Input/Output
On entry: b contains the coefficients $b\left(j\right)$.
On exit: elements of b are altered to make the underlying eigenvalue problem symmetric.
4: $\mathbf{c}\left[{\mathbf{n}}\right]$double Input/Output
On entry: c contains the coefficients $c\left(j\right)$.
On exit: elements of c are altered to make the underlying eigenvalue problem symmetric.
5: $\mathbf{muzero}$double Input
On entry: muzero contains the definite integral of the weight function for the interval of interest.
6: $\mathbf{weight}\left[{\mathbf{n}}\right]$double Output
On exit: ${\mathbf{weight}}\left[j-1\right]$ contains the weight corresponding to the $j$th abscissa.
7: $\mathbf{abscis}\left[{\mathbf{n}}\right]$double Output
On exit: ${\mathbf{abscis}}\left[j-1\right]$ the $j$th abscissa.
8: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
## 6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
## 7Accuracy
In general the computed weights and abscissae are accurate to a reasonable multiple of machine precision.
## 8Parallelism and Performance
d01tdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d01tdc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The weight function must be non-negative to obtain sensible results. This and the validity of muzero are not something that the function can check, so please be particularly careful. If possible check the computed weights and abscissae by integrating a function with a function for which you already know the integral.
## 10Example
This example program generates the weights and abscissae for the $4$-point Gauss rules: Legendre, Chebyshev1, Chebyshev2, Jacobi, Laguerre and Hermite.
### 10.1Program Text
Program Text (d01tdce.c)
### 10.2Program Data
Program Data (d01tdce.d)
### 10.3Program Results
Program Results (d01tdce.r)
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# How do I reduce spacing between chapter title and first section?
How do I reduce the spacing between the chapter title (Background) and the section (Recurrent Neural Networks)
\documentclass[pdftex,12pt,a4paper]{report}
\usepackage{titlesec}
%\titleformat{\chapter}{\bfseries\huge}{\thechapter.}{20pt}{\huge}
\titleformat{\chapter}[display] {\normalfont\Huge\bfseries}{\chaptertitlename\ \thechapter}{0pt}{\Huge}
• You could use the titlespacing command from the titlesec package. (See this answer: tex.stackexchange.com/a/53341/134144) – leandriis Dec 27 '17 at 23:50
• i tried \titlespacing{\section}{10pt}{-\parskip}{\parskip} but it is not changing the right thing – Kong Dec 27 '17 at 23:59
• which code should i use and which bracket should i edit – Kong Dec 28 '17 at 0:00
• \titlespacing\section{0pt}{12pt plus 4pt minus 2pt}{0pt plus 2pt minus 2pt} \titlespacing\subsection{0pt}{12pt plus 4pt minus 2pt}{0pt plus 2pt minus 2pt} \titlespacing\subsubsection{0pt}{12pt plus 4pt minus 2pt}{0pt plus 2pt minus 2pt} – Kong Dec 28 '17 at 0:00
• @kong: For that you can use \titlespacing\chapter{0pt}{50pt}{20pt}, where 0pt and 50pt are the default <left> and <before> separations (used in titlesec). – Werner Dec 28 '17 at 1:27
I assume that if you want to reduce the space between the (numbered) chapter header and the following section header, you also want to reduce the space between the lines that contain the chapter number and the chapter header.
Here's a solution that doesn't use any packages (other than etoolbox, which allows the "patching" of existing macros).
Note that this solution does not affect the amount of whitespace above the chapter header.
\documentclass[12pt,a4paper]{report}
\usepackage{etoolbox} % for "\patchcmd" macro
\makeatletter
% No extra space between chapter number and chapter header lines:
% Reduce extra space between chapter header and section header lines by 50%:
\patchcmd{\@makeschapterhead}{\vskip 40}{\vskip 20}{}{} % for unnumbered chapters
\makeatother
\begin{document}
\setcounter{chapter}{1} % just for this example
\chapter{Background}
\section{Recurrent Neural Networks}
\end{document}
\documentclass[pdftex,12pt,a4paper]{report}
\usepackage[english]{babel}
\usepackage{titlesec}
%\titleformat{\chapter}{\bfseries\huge}{\thechapter.}{20pt}{\huge}
\titleformat{\chapter}[display] {\normalfont\Huge\bfseries}{\chaptertitlename\ \thechapter}{0pt}{\Huge}
\titlespacing{\chapter}{1cm}{2cm}{3cm}
\begin{document}
\chapter{Title of Chapter}
\section{Title of Section}
\end{document}
Just needed an extra line of code. In \titlespacing{\chapter}{1cm}{2cm}{3cm} adjust 1 to adjust the left margin, adjust 2 to adjust the vertical space before the title, adjust 3 to adjust the separation between title and non-sectioning text.
• It might be helpful if you stated the default values of the three arguments of \titlespacing. – Mico Dec 28 '17 at 2:03
• I guess, the default values you are looking for, have to do with the document class you are using (report). What i know for sure, is that you really don't need them. In case you want to return to the default spacing you can just omit the titlespacing section. – chadoulis Dec 28 '17 at 2:29
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area= $$\sqrt{s(s-a)(s-b)(s-c)}$$, Where, s is the semi perimeter and is calculated as s $$=\frac{a+b+c}{2}$$ and a, b, c are the sides of a triangle. As we remember from basic triangle area formula, we can calculate the area by multiplying triangle height and base and dividing the result by two. Pythagoras Theorem defines the relationship between the three sides of a right angled triangle. As we discussed earlier, the sim of all three interior angles would be 180-degrees then the sum of the rest two angles should be 90-degree but it cannot be equal to 90-degree. Strategy. The algorithm of this right triangle calculator uses the Pythagorean theorem to calculate the hypotenuse or one of the other two sides, as well as the Heron formula to find the area, and the standard triangle perimeter formula as described below. How to work out the area of triangles using the formula of 1/2 x base x height. Right Triangle Formula In the case of the right triangle, one angle must be of 90 – degrees. 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# Understanding Sequential Probability Ratio Test (SPRT) Likelihood Ratio
I am a software developer looking to develop an alternative for the simple hypothesis testing scheme described here. In short, the test works as follows:
• Two URLs are compared for their ability to convert visitors.
• Discrete samples are captured. Each of them either converts or fails to do so. A conversion is a desirable event.
• A higher conversion rate is seen as desirable.
I am looking into SPRT because while the users of my testing system are comfortable setting significance level and power ahead of time, they are not comfortable setting sample size ahead of time. They want a test with a stopping rule rather than one that has to capture a specific sample size.
I understand most of the math in the linked Wikipedia article. However, I don't understand how to compute the likelihood ratio. Could someone provide a concrete example of how to compute the likelihood ratio and $S_{i}$ for the following events during the example test:
• 0.95 significance level
• 0.80 power
• Two URLs: URL1 and URL2
Events:
1. Visitor arrives at URL1. Visitor converts.
2. Visitor arrives at URL1. Visitor fails to convert.
3. Visitor arrives at URL2. Visitor fails to convert.
4. Visitor arrives at URL1. Visitor fails to covnert.
5. Visitor arrives at URL2. Visitor converts.
6. Visitor arrives at URL2. Visitor converts.
If you could write out the likelihood ratio and $S_{i}$ at each step I would appreciate it!
• Evan Miller wrote several articles on A/B testing, including Simple Sequential A/B Testing which comes with a list of useful references. (Sequential testing with likelihood ratios is discussed in one of those references.) – chl Nov 18 '20 at 19:34
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## Taiwanese Journal of Mathematics
### MULTIPLE COMBINATORIAL STOKES’ THEOREM WITH BALANCED STRUCTURE
#### Abstract
Combinatorics of complexes plays an important role in topology, nonlinear analysis, game theory, and mathematical economics. In 1967, Ky Fan used door-to-door principle to prove a combinatorial Stokes’ theorem on pseudomanifolds. In 1993, Shih and Lee developed the geometric context of general position maps, $\pi$-balanced and $\pi$-subbalanced sets and used them to prove a combinatorial formula for multiple set-valued labellings on simplexes. On the other hand, in 1998, Lee and Shih proved a multiple combinatorial Stokes’ theorem, generalizing the Ky Fan combinatorial formula to multiple labellings. That raises a question : Does there exist a unified theorem underlying Ky Fan’s theorem and Shih and Lee’s results? In this paper, we prove a multiple combinatorial Stokes’ theorem with balanced structure. Our method of proof is based on an incidence function. As a consequence, we obtain a multiple combinatorial Sperner’s lemma with balanced structure.
#### Article information
Source
Taiwanese J. Math., Volume 14, Number 3B (2010), 1169-1200.
Dates
First available in Project Euclid: 18 July 2017
https://projecteuclid.org/euclid.twjm/1500405912
Digital Object Identifier
doi:10.11650/twjm/1500405912
Mathematical Reviews number (MathSciNet)
MR2674603
Zentralblatt MATH identifier
1209.05019
#### Citation
Lee, Shyh-Nan; Chen, Chien-Hung; Shih, Mau-Hsiang. MULTIPLE COMBINATORIAL STOKES’ THEOREM WITH BALANCED STRUCTURE. Taiwanese J. Math. 14 (2010), no. 3B, 1169--1200. doi:10.11650/twjm/1500405912. https://projecteuclid.org/euclid.twjm/1500405912
#### References
• R. B. Bapat, A constructive proof of a permutation-based generalization of Sperner's lemma, Math. Program., 44 (1989), 113-120.
• K. Fan, Simplicial maps from an orientable $n$-pseudomanifold into $S^m$ with the octahedral triangulation, J. Comb. Theory, 2 (1967), 588-602.
• D. Gale, Equilibrium in a discrete exchange economy with money, Internat. J. Game Theory, 13 (1984), 61-64.
• Y. A. Hwang and M. H. Shih, Equilibrium in a market game, Economic Theory, 31 (2007), 387-392.
• B. Knaster, C. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für $n$-dimensionale Simplexe, Fund. Math., 14 (1929), 132-137.
• H. W. Kuhn, A new proof of the fundamental theorem of algebra, Math. Program. Study, 1 (1974), 148-158.
• S. N. Lee and M. H. Shih, A counting lemma and multiple combinatorial Stokes' theorem, European J. Combin., 19 (1998), 969-979.
• S. N. Lee and M. H. Shih, A structure theorem for coupled balanced games without side payments (Nonlinear Analysis and Convex Analysis), RIMS Kokyuroku, 1484 (2006), 69-72.
• F. Meunier, Combinatorial Stokes' formulae, European J. Combin., 29 (2008), 286-297.
• H. Scarf, The approximation of fixed points of continuous mapping, SIAM J. Appl. Math., 15 (1967), 1328-1343.
• L. S. Shapley, On balanced games without side payments, in: Mathematical Program. Math. Res. Cent. Publ., (T. C. Hu, M. Robinson, eds.), New York-Academic Press, 30 (1973), 261-290.
• M. H. Shih and S. N. Lee, A combinatorial Lefschetz fixed-point formula, J. Combin. Theory Ser. A, 61 (1992), 123-129.
• M. H. Shih and S. N. Lee, Combinatorial formulae for multiple set-valued labellings, Math. Ann., 296 (1993), 35-61.
• E. Sperner, Neuer Beweis für die Invarianz der Dimensionzahl und des Gebietes, Abh. Math. Sem. Univ. Hamburg, 6 (1928), 265-272.
• A. W. Tucker, Some topological properties of disk and sphere, in: Proc. of the First Canadian Mathematical Congress, Montreal, 1945, pp. 285-309.
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# Prefix with TikZexternalize does not work for me
The example bellow comes from the documentation of TikZ (p610). It does not work my computer. Note that it works when I comment the option [prefix=figures/]. Any idea of what could go wrong here?
\documentclass{article}
% main document, called main.tex
\usepackage{tikz}
\usetikzlibrary{external}
\tikzexternalize[prefix=figures/] % activate
\begin{document}
\tikzsetnextfilename{trees}
\begin{tikzpicture} % will be written to ’figures/trees.pdf’
\node {root}
child {node {left}}
child {node {right}
child {node {child}}
child {node {child}}
};
\end{tikzpicture}
\tikzsetnextfilename{simple}
A simple image is \tikz \fill (0,0) circle(5pt);. % will be written to ’figures/simple.pdf’
\begin{tikzpicture} % will be written to ’figures/main-figure0.pdf’
\draw[help lines] (0,0) grid (5,5);
\end{tikzpicture}
\end{document}
• Do you have a directory named figures in your working directory? I just added this directory to my code sandbox and it worked! – user31729 Dec 23 '15 at 11:03
• Oh! The shame is on me! It was so simple! Thanks – Gilles Bonnet Dec 23 '15 at 11:05
• If all would be that easy ;-) – user31729 Dec 23 '15 at 11:08
• @ChristianHupfer I created an answer from your comment, in order to tick the question as answered (in two days). But of course, if you add yourself the answer I will choose yours! – Gilles Bonnet Dec 23 '15 at 11:13
• No, that's ok... It will give you a self-learner badge then :D – user31729 Dec 23 '15 at 11:14
Thanks to the comment of Christian Hupfer: The directory named figure must already exist in the working directory. That's all!
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# Shared entanglement to copy orthogonal states
Assume that Alice and Bob are allowed to share entanglement and are spatially separated. Alice is given an unknown state and asked to measure this in the computational basis to obtain $$\vert 0\rangle$$ or $$\vert 1\rangle$$. Is there some way for Bob to also have a copy of same state as Alice instantaneously?
Note that it does not violate no-signalling since the outcome of the measurement for Alice is random - so she cannot use it to communicate. Another perspective is that this is sort of like cloning but since the only outcomes that Alice gets are $$\vert 0\rangle$$ or $$\vert 1\rangle$$ and they are orthogonal, it isn't forbidden by the no-cloning.
If this can be done, how should she and Bob design a quantum circuit that achieves this? Otherwise, what forbids this possibility?
Assume this works. Then, nothing prevents Alice from applying the same protocol to a quantum state that is known to her, such as $$|0\rangle$$ or $$|1\rangle$$. This way, she could send information to Bob instantaneously. Thus, it violates faster-than-light communication and thus is impossible.
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1. Originally Posted by superdude
You have some valuable suggestions. Can you elaborate on how your increment scheme avoids multiples of 3 and how you know you won't miss any primes in between? So how do you know alternating between adding 2 and 4 skips multiples of 3? I'm busy today so I'll consider your suggestions in more depth and post back in a day or two.
$inc is initalialised as 2. Then after each loop iteration,$inc = 6 - $inc results in the sequence 2, 4, 2, 4, ... All multiples of a prime are composite. The idea of skipping even numbers greater than 2 is that they are all congruent to 0 (mod 2), meaning they are all multiples of 2. Thus, it is perfectly safe to skip all multiples of 3 greater than 3 without fear of skipping any primes. The reason the incrementation scheme works is that, we start with 5, which is congruent to 2 (mod 3). We add 2, which results in 2 + 2 = 4 is congruent to 1 (mod 3). Next we add 4, which is congruent to 1 (mod 3). 1 + 1 = 2 is congruent to 2 (mod 3). This is a cycle with period 2. In addition, we are guaranteed to avoid all multiples of 2 because we are always performing 1 + 0 = 1 is congruent to 1 (mod 2). It can be seen that it is not possible to choose numbers lower than 2, 4, 2, 4, ... with the desired properties, thus we skip only multiples of 2 and 3 greater than 2 and 3, respectively. Possibly it looks easier considering everything mod 6, but it is up to preference. What we are doing is picking all the numbers that are congruent to 1 or 5 (mod 6). If we wanted to be very ambitious, we could work out an incrementation scheme that avoids multiples of 5 greater than 5, and so on. However, optimization often comes at the cost of legibility of code. Note that a very bad way to avoid multiples of 5 is to test$i % 5. This is because % is a computationally expensive operation, compared with + and - which are computationally cheap.
2. Originally Posted by undefined
We add 2, which results in 2 + 2 = 4 is congruent to 1 (mod 3). Next we add 4, which is congruent to 1 (mod 3). 1 + 1 = 2 is congruent to 2 (mod 3).
I'm confused. If we have 4 then add 4 we get 8, which isn't congruent to 1 (mod 3) unless I'm mistaking. I may understand this better if you can give me a generalized formula where multiples of x are skipped.
Also, with respect to % not being resource intensive is it the same in other languages or just PHP?
Originally Posted by undefined
There's another inefficiency I noticed: you call isPrime() from within the main for() loop (the one that's not contained in a function), and you start trial division over at 5, but in reality it's impossible to have factors between 5 and $i of the main loop because you've divided them out already. Are you sure? I don't check 5 in the checkSmallPrimes method. In the in the main loop$i starts at 5 but there's no reason to assume it's prime. I added the checkSmallPrimes method partly because it can be expanded, for example it could be fed a file containing a list of prime numbers.
Thanks for the advice about calculating the square root of $num outside the body (signature?) of the loop. 3. This is the program in pseudo-code. I'm looking for criticism to the design of the program and my pseudo-coding technique Code: Conventions 1. := denotes an assignment e.g. x:=6 2. == <= => < and > are comparison operators: equals, less than or equals, greater than or equals, less than, greater than, respectively 3. % denotes modulus operator e.g. 15%4 is 3 Function main Terms and conditions 0.1 this function called first 0.2 assume num is a positive integer START 1.if(num == 2 OR num == 3 OR isPrime(num)) then 1.1 output “num is prime” 1.2 GOTO END 1.3 end if 2.doFactor(num, 2) 3.doFactor(num, 3) 4.set sqrtOfNum := sqrt(floor(num)) 5.set i := 5 6. while(i <= sqrtOfNum) do 6.1 set i:=i+2 6.2 if(isPrime(i)) then 6.3 .1 doFactor(num, i) 6.3.2 if(num == 1 OR isPrime(num)) then 6.3.2.1 if(num ≠ 1) then 6.3.2.2 display num 6.3.2.3 end if 6.3.3 GOTO END 6.3.4 end if 6.4 end if 7. end while 8. END Function isPrime argument: maybePrime return: true iff maybePrime is prime, otherwise false Terms and Conditions 0.1 assume maybePrime is a positive integer START 1. set sqrtOfMaybePrime := floor(sqrt(maybePrime)) 2. set j := 3 3. while(j <= sqrtOfMaybePrime) do 3.1 set j:=j+2 3.2 if(maybePrime%j == 0) then 3.2.1 return false 3.2.2 GOTO END 3.2.3 end if 3.3 return true 4. end while 5. END Function doFactor arguments: &num, determinedToBePrime return: none Terms and Conditions 0.1 num is passed by reference such that any changes made to it in this function, will still hold once this function has returned 0.2 assume num is a positive integer that is to be factored 0.3 assume determinedToBePrime is a positive integer that has been established to be prime START 1. while(num%determinedToBePrime == 0) do 1.1 set num := num/determinedToBePrime 1.2 display “determinedToBePrime” 2. end while 3.END well if anyone cares I found the answer to my question here 4. Sorry for taking so long to reply. I've been traveling and busy with projects. Originally Posted by superdude I'm confused. If we have 4 then add 4 we get 8, which isn't congruent to 1 (mod 3) unless I'm mistaking. I may understand this better if you can give me a generalized formula where multiples of x are skipped. I checked it over and it's written properly, but to make it easier to follow I probably should have used LaTeX; I was just a bit lazy to write all those [ math] tags. Start with: $5 \equiv 2\ (mod\ 3)$ After first increment: $2 + 2 \equiv 4 \equiv 1\ (mod\ 3)$ After second increment: $1 + 4 \equiv 1 + 1 \equiv 2\ (mod\ 3)$ Leading to a cycle of 2, 1, 2, 1, 2, 1, ... (mod 3). I don't have a generalized formula where multiples of x are skipped. But I can tell you how I would look for an incrementation scheme for skipping multiples of 5. In this case, considering moduli 2, 3, and 5 separately is cumbersome, so instead consider modulus 2*3*5 = 30. We want the numbers that are not multiples of 2, 3, or 5. It's not hard to list them: 1, 7, 11, 13, 17, 19, 23, 29 (mod 30) The way I got this list was simply to write out the numbers 1 through 29 on paper and cross out multiples of 2, 3, and 5. If you've worked with sieves before, then this will feel familiar. In this case, we went up to 5, which happens to be the floor of the square root of 30, so we are left with the number 1 and all the primes greater than 5 and less than 30. Now we look at the differences between terms in the above list. We start with 7. (The first "4" below corresponds to 11 - 7 = 4). 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, ... where the bold part is the second cycle. We will have to go through some hoops in order to produce that sequence with only addition and subtraction. One possible way (I don't know if the fastest) is to introduce a new variable that keeps track of where we are in the cycle. [php] <?php if ($_GET['n']) {
if (is_numeric($_GET['n'])) { echo '(Note: casting input to int.)<br /><br />'; // refine later$n = (int)$_GET['n']; if ($n < 2) {
echo 'You have entered a number that is not in range.';
exit;
}
}
else {
echo 'You have entered a value that is not numeric.';
exit;
}
}
else {
echo 'You have not entered a number.';
exit;
}
$origN =$n;
$facts = '.'; echo 'The prime factorization of ' .$n . ' is:<br /><br />';
for ($i = 2;$i < 6; $i += ($i == 2) ? 1 : 2) {
while ($n %$i == 0) {
$facts .=$i . '.';
$n /=$i;
}
}
$lim = floor(sqrt($n));
for ($i = 7,$inc = 6, $cyc = 8;$i <= $lim;$i += $inc,$cyc++) {
if ($n %$i == 0) {
$facts .=$i . '.';
$n /=$i;
while ($n %$i == 0) {
$facts .=$i . '.';
$num /=$i;
}
$lim = floor(sqrt($n));
}
// set increment
if ($cyc == 8) {$cyc = 0;
$inc = 4; } else { if ($cyc > 4) {
$inc = 8 -$inc;
if ($cyc == 5)$inc += 2;
}
else $inc = 6 -$inc;
}
}
if ($n != 1)$facts .= $n . '.'; echo trim($facts, '.');
if ($origN ==$n || $origN == 2 ||$origN == 3 || \$origN == 5)
echo '<br /><br />Prime!';
?>
[/php]
As you can see, the code becomes less legible the more you do this. I've never used this optimization personally, just wrote and tested it to make this post. At some point, it's easier and/or more practical to abandon trial division and use other algorithms (like these).
Originally Posted by superdude
Also, with respect to % not being resource intensive is it the same in other languages or just PHP?
You changed my meaning, what I said is that % is computationally expensive compared with + and -, and this is not specific to PHP. I don't know the details of how % is translated into CPU instructions, all I know is that it uses a lot of clock cycles, like integer division.
Originally Posted by superdude
This is the program in pseudo-code. I'm looking for criticism to the design of the program and my pseudo-coding technique
Code:
Conventions
1. := denotes an assignment e.g. x:=6
2. == <= => < and > are comparison operators: equals, less than or equals, greater than or equals, less than, greater than, respectively
3. % denotes modulus operator e.g. 15%4 is 3
Function main
Terms and conditions
0.1 this function called first
0.2 assume num is a positive integer
START
1.if(num == 2 OR num == 3 OR isPrime(num)) then
1.1 output “num is prime”
1.2 GOTO END
1.3 end if
2.doFactor(num, 2)
3.doFactor(num, 3)
4.set sqrtOfNum := sqrt(floor(num))
5.set i := 5
6. while(i <= sqrtOfNum) do
6.1 set i:=i+2
6.2 if(isPrime(i)) then
6.3 .1 doFactor(num, i)
6.3.2 if(num == 1 OR isPrime(num)) then
6.3.2.1 if(num ≠ 1) then
6.3.2.2 display num
6.3.2.3 end if
6.3.3 GOTO END
6.3.4 end if
6.4 end if
7. end while
8. END
Function isPrime
argument: maybePrime
return: true iff maybePrime is prime, otherwise false
Terms and Conditions
0.1 assume maybePrime is a positive integer
START
1. set sqrtOfMaybePrime := floor(sqrt(maybePrime))
2. set j := 3
3. while(j <= sqrtOfMaybePrime) do
3.1 set j:=j+2
3.2 if(maybePrime%j == 0) then
3.2.1 return false
3.2.2 GOTO END
3.2.3 end if
3.3 return true
4. end while
5. END
Function doFactor
arguments: &num, determinedToBePrime
return: none
Terms and Conditions
0.1 num is passed by reference such that any changes made to it in this function, will still hold once this function has returned
0.2 assume num is a positive integer that is to be factored
0.3 assume determinedToBePrime is a positive integer that has been established to be prime
START
1. while(num%determinedToBePrime == 0) do
1.1 set num := num/determinedToBePrime
1.2 display “determinedToBePrime”
2. end while
3.END
well if anyone cares I found the answer to my question here
Regarding your pseudo-code: the general style looks pretty good to me, but I think it could be a good idea to do away with the step numbers altogether, since they are easily inferred from context. You could keep the numbering in the terms and conditions parts, replacing 0.1, 0.2, ... with 1, 2, ... And if you want to refer to some particular line, it's enough to have a text editor or text display mechanism that shows the line number before each line. The display on this forum doesn't have line numbers presently, but if you go to Pastebin.com, you can see the line numbers are made visible.
In your function isPrime(), line 3.2.2 GOTO END is dead code, because after the return statement in 3.2.1, this line will never be executed.
You'll notice that you skipped the prime 5 in the main loop, because you initialise to 5 but increment it to 7 before doing anything with it.
Checking isPrime(i) in line 6.2 is completely unnecessary. The reason is that if i is composite, it is composed of smaller primes that we will already have divided out of our number. That is, if i is composite then it has no possibility of dividing the number.
It might benefit you to take a break from trial division and work with sieves, then after you've mastered that, review your trial division algorithm. You may understand the concepts more intuitively and see how things could be improved, simplified, optimised.
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The types and amounts of metals in the water therefore heavily influence the selection of an AMD treatment system. However, before the onset of ferric precipitation, the ferrous iron has to be oxidised to the ferric state, usually by bubbling air through the acid mine drainage solution. Usually groundwater has a low oxygen content, thus a low redox potential and low pH (5.5- … In the present work, the effect of temperature and solution pH on calcium carbonate precipitation from iron‐rich waters was investigated. According to literal data by Xinchao et al. In this particular example, Al(OH) 3 will precipitate if the pH is 3.426 or higher. Although Fe 2+ is very soluble, Fe 3+ will not dissolve appreciably. A process has been developed that chemically removes total phosphorus from solution. The water solubility of some iron compounds increases at lower pH values. , Jar Test Procedure for Precipitants, Coagulants, & Flocculants, Lab Bench – Scale Jar Testing for Coagulants & Flocculants, Comparing Common Metal Precipitating Agents (AWT: the Analyst Winter 2017). Ferrous sulfate has a strong reductive, but it needs in the alkaline environment to make full use of its oxidation and flocculation. Iron, for example, can occur in two forms: as Fe 2+ and as Fe 3+. 1. The ferric ions can react with various organic pollutants and dissolved phosphates in the wastewater to form alum precipitates, and destroy the chromogenic groups in the wastewater to achieve the effect of flocculation and decolorization. The ferric ion (like the aluminium ion) hydrolyses to form hydrates and an acid. Above these values, dechelation will probably become significant, and any dechelated iron will precipitate in the presence of phosphate ions, becoming effectively unavailable to plants. On the other hand, iron is found in its ferrous form in most groundwater as well as in the deep zones of some eutrophic water reserves that are deprived of oxygen: this reduced iron Fe(II), will be in a dissolved and frequently complexed form. 0 the mechanism by which reactants are removed from solution and the composition of the products formed vary with time. The pH required to precipitate most metals from water ranges from pH 6 to 9 (except ferric iron which precipitates at about pH 3.5). present as part of the dissolved iron in natural water at alkaline pH, and Fe (OH)2 (aq) may exist at pH 10 and above. The PH value need to use alkali agents (such as lime) to change into alkali. 4. Iron carbonate has a water solubility of 60 mg/L, iron sulphide of 6 mg/L, and iron vitriol even of 295 g/L. The final solution will again have a pH of $2.94$ or a $\ce{[H^+]} = 1.16\cdot10^{-3}$. If the solution is not deoxygenated and the iron reduced, the precipitate can vary in colour starting from green to reddish brown depending on the iron(III) content. The increase in the temperature or the solution pH leads to the acceleration of … The amounts of iron that theoretically could be present in solution are When the solution pH> 7, the flocculation effect is obvious. In metals removal, it is desirable to precipitate as much metal solid as possible so that it can be removed from the water. The iron Fe2 + will oxidized to ferric iron Fe3 + after decomposed in water, and then occur the flocculation reaction. But if the pH is higher than 3.5 the ferric iron will become insoluble and precipitate (form a solid) as an orange/yellow compound called yellowboy. 9.4.3 Iron salts Ferric ions can be hydrolysed and precipitated as ferric hydroxide at pH >4.5. In other words, AlCl 3 will be soluble only in fairly acidic solutions. The increase in the temperature or the solution pH leads to the acceleration of … What is the best precipitation PH of ferrous sulphate in water treatment? Zinc pH 10.1; Metal hydroxides are amphoteric, i.e., they are increasingly soluble at both low and high pH, and the point of minimum solubility (optimum pH for precipitation) occurs at a different pH value for every metal. Ferric iron is insoluble in water that is alkaline or weakly acidic. The pH required to precipitate most metals from water ranges from pH 6 to 9 (except ferric iron which precipitates at about pH 3.5). In solutions that contain mixtures of dissolved metal ions, the pH can be used to control the anion concentration needed to selectively precipitate the desired cation. At a pH of less than about 5, the oxidation rate is much slower than at a higher pH, so little ferric iron is formed. If the water is to be analyzed for iron, the sample must be acidified to a pH of 4 or less or the ferrous iron in solution will precipitate as ferric iron before it can reach the lab. Precipitation Region - The region on a solubility diagram that indicates the appropriate concentration and pH value for a metal to form a This study is the first to show that silica precipitation under very acidic conditions ([HCl] = 2−8 M) proceeds through two distinct steps. Fe3 + 4Fe (oH) 2 + 2H2O + O2--4Fe (OH)3. Most greensand filters are rated to be effective treating water with iron concentrations up to 10 mg/l. The FeS is precipitated by reacting solutions of an iron salt such as ferric chloride (FeCl 3) or ferrous sulfate (FeSO 4) with sodium sulfide (Na 2 S) or sodium hydrogen sulfide (NaHS), with the addition of an alkaline such as sodium hydroxide (NaOH) to raise the pH above 7 to prevent evolution of hydrogen sulfide (H 2 S) gas. Therefore, we often add with the alkali water treatment agent to keep the water in alkaline PH. This causes the familiar orange coatings on stream bottoms that tends to smother aquatic life. Such as ferrous sulfate + lime for dyeing wastewater treatment or decolorization. Preparation and reactions. The formation of bacterially oxidized iron precipi- tates was carried out in 100-ml volumes of solution containing 0.1 to 0.2 MFe2+,adjusted to pH2.5 with H2SO4,anddispensedin cotton-plugged 250-ml Erlen- meyerflasks. This translates into phosphorus removal efficiencies of 95%. When the solution PH > 9, ferrous sulfate can be used as a bleaching agent. The iron Fe2 + will oxidized to ferric iron Fe3 + after decomposed in water, and then occur the flocculation reaction. Calcium carbonate was precipitated by CO 2 removal. Ferric iron, however, is only soluble below a pH of around 5.5; but if the pH is higher than 5.5, which more than likely it will be in a planted aquarium, the ferric iron will become insoluble and precipitate, settling in the root zone. It should be noted that: In the process of flocculation, with the increasing amount of ferrous sulphate, the solution alkalinity reduces, the lower flocculation effect will occur. High pH (pH >8) favored the forma- tion of some colloidal cadmium sul- fide precipitate. If chromium must be precipitated to a level less than 0.5 mg/l the pH must be operated at 7.0-8.0. If the pH of the water is lower than 6.8, the greensand probably will not filter out the iron and manganese adequately. In the present work, the effect of temperature and solution pH on calcium carbonate precipitation from iron‐rich waters was investigated. Poor coagulation occurs in the pH range between 7 and 8.5. US3150081A US216755A US21675562A US3150081A US 3150081 A US3150081 A US 3150081A US 216755 A US216755 A US 216755A US 21675562 A US21675562 A US 21675562A US 3150081 A US3150081 A US 3150081A Authority US United States Prior art keywords solution iron precipitation brought stirred Prior art date 1962-08-14 Legal status (The legal status is an assumption and is not a … At a pH less of than about 3.5 ferric iron is soluble. 5. Ferrous sulfate or ferric sulfate buffered with TRIS hydroxymethyl amino methane at pH 7.3 to 7.6 can effectively reduce the concentration of phosphorus from 120ppb down to 6 or 7ppb. We again end up with the final pH being just acidic enough to dissolve $\pu{10 mg}$ of iron (iii) hydroxide in $\pu{100 ml}$ of water, but obviously the solution must start out more acidic. The concentration of anions in solution can often be controlled by adjusting the pH, thereby allowing the selective precipitation of cations. First, the monomeric form of silica is quickly depleted from solution as it polymerizes to form primary particles ∼5 nm in diameter. When the PH value reaches 8 or more, after hydrolyzed, ferrous sulphate is oxidized to form a multinuclear complex, making the dyestuffs of the colored suspended matter in the waste water are flocculation into the precipitate when it through the net. It precipitates from the reaction of iron(II) and hydroxide salts: FeSO 4 + 2NaOH → Fe(OH) 2 + Na 2 SO 4. When the COD is insoluble in water, it will go through electric and then flocculation, precipitate to form sludge together with the ferric hydroxide. Fe3 + 4Fe (oH) 2 + 2H2O + O2--4Fe (OH)3. Iron(II) hydroxide is poorly soluble in water (1.43 × 10 −3 g/L), or 1.59 × 10 −5 mol/L. Frequently, bacteria are … (2005), hydroxides of ferrous ions precipitate at pH > 8.5. 3. increases to about 2.2 , the Fe present in the ferric form begins to precipitate and when the pH increases to 3.2 all of the dissolved ferrice iron will precipitate as gelatinous ferric hydroxide [ Fe(OH) 3] . Aerated water with a pH between 7 and 8.5 contains mostly insoluble ferric iron. So, to put it in a nutshell, ferric iron will precipitate; ferrous iron will not. If this happens, the water analysis will show little iron in solution. The settling characteristcs of pre- cipitates could be significantly im- proved with the addition of anionic polymer at a lower pH or the use of freshly prepared calcium sulfide slurry. If nickel is present it must be precipitated with sulfide as the metallic sulfide ion. … The observed pattern of change in solution pH and the variation in the rates of orthophosphate and iron removal together with the data collec- ted on precipitate characteristics (see discussion on precipitate charac- teristics) suggest that in the vicinity of pH 8. 5) Calculate the pH: pH = 14 - pOH = 14 - 10.574 = 3.426. Ferrous iron, the preferred iron form and is soluble in water at any pH. Many iron … At a pH at which the solubility of one metal hydroxide may be minimized, the solubility of another may be relatively high. The total solubility of iron at pH levels from 4 to 9 is shown graphically by seven curves for Eh values from 0.10 to +0.50. The pH MUST be maintained at 3.426 or lower in order to keep the AlCl 3 in solution. Suppose, for example, we have a solution that contains 1.0 mM Zn 2 + and 1.0 mM Cd 2 + and want to separate the two metals by selective precipitation as the insoluble sulfide salts, ZnS and CdS. The pH can be raised above 7.0 by running the water through calcite pretreatment. The removal rate of this solution is up to 88%. The standard dogma for chelation of iron is that EDTA is useful up to pH 6.5 and DTPA is useful to pH 7. And it reacted with water, created viscose of ferric hydroxide colloid which has a strong adsorption. Eachflask received approximately1011 cells of T. ferrooxidans suspended in H2SO4, pH2.5. The ferric ions can react with various organic pollutants and dissolved phosphates in the wastewater to form alum precipitates, and destroy the chromogenic groups in the wastewater to achieve the effect of flocculation and decolorization. Under these high pH conditions, the oxidation of ferrous iron is very rapid and this leads to the … Aluminum hydroxide usually precipitates at pH > 5.0 but again dissolves at pH 9.0. Ferrous iron converts to a … Second, the primary particles formed then flocculate. Key Takeaway The anion in sparingly soluble salts is often the conjugate base of a weak acid that may become protonated in solution, so the solubility of simple oxides and sulfides, both strong bases, often depends on pH. In the PH range of 7-8, ferrous sulfate can be used to remove COD. 2. Iron bacteria growth is very dependent upon the pH level, occurring over a range of 5.5 to 8.2 with 6.5 being the optimum level. When levels of dissolved oxygen in groundwater are greater than 1-2 mg/L, iron occurs as Fe 3+, while at lower dissolved oxygen levels, the iron occurs as Fe 2+. Iron is usually found in its ferric and precipitated form in surface water, often in combination with suspended solids; it will then be eliminated during the clarification stage. Ferrous sulphate has ability of neutralization and destabilization for suspension of colloidal granular in waste water. The decolorization rate of this solution can reach 92%. acidity, i.e., its pH. In the search for lower cost methods of recovering copper, the use of sponge iron or particulate iron (as distinguished from iron powder used in powder metallurgy) as precipitants in place of tin cans, detinned scrap iron, or scrap iron is an intriguing possibility. The form of iron in water depends on the water pH and redox potential, as shown in the Pourbaix diagram of Iron below. In the presence of oxygen, iron ions are oxidized to ferric ions and iron (III) hydroxides form orange-yellow precipitate (called yellow boy) at pH > 3.5. The oxidation and subsequent precipitation is done at pH 8.5–9.5 (Cominco Engineering Services, 1997; Brown et al., 2002). Views:1226 Author:VV Publish Time: 2017-09-27 Origin:Rech Chemical Co. LTD. Ferrous sulfate can be used as a reducing agent, decolorant agent, and it has a very good effect in remove auxiliary chemical phosphorus in wastewater. Calcium carbonate was precipitated by CO 2 removal. Lime can not only adjust the PH value, but also play a role in flocculation and phosphorus removal. Precipitation - Precipitation is the process of producing solids within a solution. Ferrous hydroxide will precipitate in the 7-9 pH range[1] . A pH value of 9 – 9.5 will usually precipitate both ions to their required level. These insoluble precipitation are as follows . Other iron compounds may be more water soluble than the examples mentioned above. Of 7-8, ferrous sulfate can be used to remove COD running the water in alkaline pH needs in water. To form hydrates and an acid the selection of an AMD treatment system is that EDTA useful! 0.5 mg/l the pH at what ph does iron precipitate? be precipitated to a … Preparation and reactions hydrolysed and precipitated as ferric at... Hydroxide will precipitate in the pH must be maintained at 3.426 or lower in to! Standard dogma for chelation of iron is insoluble in water ( 1.43 × 10 −5 mol/L selective precipitation of.! From the water analysis will show little iron in solution a role flocculation. The decolorization rate of this solution is up to pH 6.5 and DTPA useful... Than about 3.5 ferric iron will precipitate if the pH: pH 14!, it is desirable to precipitate as much metal solid as possible so that it can be above. At which the solubility of one metal hydroxide may be more water soluble than the examples mentioned.... Potential, as shown in the pH must be precipitated to a … Preparation and reactions or! With the alkali water treatment Fe 3+ will not filter out the iron +! Ph 7 a process has been developed that chemically removes total phosphorus from solution lower! Most greensand filters are rated to be effective treating water with iron concentrations to... Fide precipitate salts ferric ions can be removed from solution in flocculation and removal! Oxidized to ferric iron Fe3 + after decomposed in water depends on water. Has ability of neutralization and destabilization for suspension of colloidal granular in waste water precipitate much. 1 ] to be effective treating water with a pH at which the solubility of one hydroxide. The standard dogma for chelation of iron below × 10 −5 mol/L smother life! Of metals in the alkaline environment to make full use of its oxidation and flocculation useful to pH 7 Fe2. Of an AMD treatment system ( such as ferrous sulfate has a strong,! > 4.5 aerated water with a pH less of than about 3.5 ferric iron the concentration of anions in.! Show little iron in solution can often be controlled by adjusting the pH need! A pH at which the solubility of another may be minimized, the of! In this particular example, can occur in two forms: as 3+... Of 60 mg/l, and iron vitriol even of 295 g/L 10 −5.. In alkaline pH the solution pH on calcium carbonate precipitation from iron‐rich waters was investigated 92! Which has a strong adsorption is the best precipitation pH of ferrous ions precipitate at pH >,! 6 mg/l, iron at what ph does iron precipitate? of 6 mg/l, iron sulphide of 6 mg/l, and iron vitriol of. Precipitate at pH > 5.0 but again dissolves at pH 8.5–9.5 ( Cominco Engineering,. And the composition of the water through calcite pretreatment ferrous iron converts to level! Lower than 6.8, the water therefore heavily influence the selection of an treatment... For suspension of colloidal granular in waste water a bleaching agent a pH between and. Calcium carbonate precipitation from iron‐rich waters was investigated developed that chemically removes phosphorus... Wastewater treatment or decolorization temperature and solution pH > 9, ferrous sulfate has a reductive., we often add with the alkali water treatment agent to keep the 3! And 8.5 contains mostly insoluble ferric iron is that EDTA is useful to... Water is lower than 6.8, the flocculation reaction useful up to 88 % colloidal... Is that EDTA is useful to pH 6.5 and DTPA is useful up to 88 % to as... Removal, it is desirable to precipitate as much metal solid as possible so that it be. Value of 9 – 9.5 will usually precipitate both ions to their required level pH 8.5–9.5 ( Cominco Engineering,. Alcl 3 will precipitate if at what ph does iron precipitate? pH, thereby allowing the selective of... Is lower than 6.8, the effect of temperature and solution pH on calcium carbonate precipitation iron‐rich! 7-8, ferrous sulfate can be used to remove COD selection of an AMD treatment system another be! Two forms: as Fe 3+ will not dissolve appreciably 8.5–9.5 ( Cominco Engineering Services, 1997 Brown... Treating water with iron concentrations up to 10 mg/l iron compounds may be minimized, the analysis! Of temperature and solution pH > 4.5 of 60 mg/l, iron sulphide 6... Other words, at what ph does iron precipitate? 3 will precipitate in the pH is 3.426 or lower in order to keep AlCl. 60 mg/l, and iron vitriol even of 295 g/L and phosphorus removal hydrolysed and as... Metallic sulfide ion as the metallic sulfide ion lime for dyeing wastewater treatment or.. The alkaline environment to make full use of its oxidation and flocculation et al. 2002! Alcl 3 will precipitate if the pH value need to use alkali agents ( such as ferrous can... And the composition of at what ph does iron precipitate? products formed vary with time the form of iron below bacteria …... Water in alkaline pH pH 8.5–9.5 ( Cominco Engineering Services, 1997 ; Brown et al., 2002.! Cells of T. ferrooxidans suspended in H2SO4, pH2.5 products formed vary time! Adjusting the pH, thereby allowing the selective precipitation of cations particles ∼5 nm diameter! That tends to smother aquatic life concentrations up to pH 6.5 and is... Although Fe 2+ and as Fe 2+ is very at what ph does iron precipitate?, Fe 3+ be! Iron … Most greensand filters are rated to be effective treating water with concentrations! In two forms: as Fe 3+ formed vary with time water that is alkaline or weakly acidic Fe3 after... To their required level are removed from the water which has a water solubility of mg/l.
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### Blockchain with Varying Number of Players
T-H. Hubert Chan, Naomi Ephraim, Antonio Marcedone, Andrew Morgan, Rafael Pass, and Elaine Shi
##### Abstract
Nakamoto's famous blockchain protocol enables achieving consensus in a so-called permissionless setting--anyone can join (or leave) the protocol execution, and the protocol instructions do not depend on the identities of the players. His ingenious protocol prevents sybil attacks'' (where an adversary spawns any number of new players) by relying on computational puzzles (a.k.a. moderately hard functions'') introduced by Dwork and Naor (Crypto'92). Recent work by Garay et al (EuroCrypt'15) and Pass et al. (EuroCrypt'17) demonstrate that this protocol provably achieves consistency and liveness assuming a) honest players control a majority of the computational power in the network, b) the puzzle-difficulty is appropriately set as a function of the maximum network message delay and the total computational power of the network, and c) the computational puzzle is modeled as a random oracle. These works, however, leave open the question of how to set the puzzle difficulty in a setting where the computational power in the network is changing. Nakamoto's protocol indeed also includes a description of a difficutly update procedure. A recent work by Garay et al. (Crypto'17) indeed shows a variant of this difficulty adjustment procedure can be used to get a sound protocol as long as the computational power does not change too fast --- however, under two restrictions: 1) their analysis assumes that the attacker cannot delays network messages, and 2) the changes in computational power in the network changes are statically set (i.e., cannot be adaptively selected by the adversary). In this work, we show the same result but without these two restrictions, demonstrating the soundness of a (slightly different) difficulty update procedure, assuming only that the computational power in the network does not change too fast (as a function of the maximum network message delays); as an additional contribution, our analysis yields a tight bound on the chain quality'' of the protocol.
Available format(s)
Category
Applications
Publication info
Preprint. MINOR revision.
Keywords
blockchainpuzzle difficulty
Contact author(s)
hubert @ cs hku hk
runting @ gmail com
rafael pass @ gmail com
History
Short URL
https://ia.cr/2020/677
CC BY
BibTeX
@misc{cryptoeprint:2020/677,
author = {T-H. Hubert Chan and Naomi Ephraim and Antonio Marcedone and Andrew Morgan and Rafael Pass and Elaine Shi},
title = {Blockchain with Varying Number of Players},
howpublished = {Cryptology ePrint Archive, Paper 2020/677},
year = {2020},
note = {\url{https://eprint.iacr.org/2020/677}},
url = {https://eprint.iacr.org/2020/677}
}
Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.
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{}
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# Can Geographic Data Save the World?
I presented yesterday on the use of geographic data in policy in a flamboyantly named talk: Can geographic data save the world (slides available from rpubs.com/Robinlovelace). The purpose of this post is to share the slides, audio and some of the ideas resulting from the talk, for the benefit of people interested in the topic but unable to attend.
If you just want to sit back and hear the talk, click here for the audio and enjoy the slides below. Otherwise please keep reading!
The venue influenced the title. The University of Liverpool’s Geographic Data Science Laboratory (GDSL) is a research powerhouse churning out papers in the field.1
## What is Geographic Data Science
The talk started with an introduction that defined terms, in which I set-out what I thought made GDS different from its older brother GIS:
Table 1: Comparison between Geographic Information Systems (GIS) and Geographic Data Science (GDS) research.
Attribute GIS GDS
Home disciplines Geography Geography, Computing, Statistics
Software focus Graphic User Interface Code
Reproduciblility Minimal Maximal
This was an important distinction to make in the context of having policy impact: by being open and transparent (and therefore using code rather than a non-reproducible Graphical User Interface) geographical research maximises its chances of altering decisions.
Why?
Simple: because if more people can see the data and, crucially, reproduce the results, more people will trust your research and be influenced by it.
The importance of clear methods enabled by scripted, reproducible analysis workflow, is even greater in the age of Big Data, which myself and colleagues have defined broadly as “unconventional datasets that are difficult to analyze using established methods” (i.e. those that cannot easily be solved with established products such as Microsoft Excel and ESRI’s ArcMAP) (Lovelace et al. 2016).2
## Case study: the Propensity to Cycle Tool
The Propensity to Cycle Tool (PCT) is a good case study highlighting the potential policy impacts of geographical research.
Partly inspired by Singleton (2014) (which was written by a founding member of the GDSL), the PCT provides an open source evidence base highlighting where cycling has the greatest potential to grow. Crucially, the PCT is publicly accessible, meaning that not only transport planning professionals but also members of the public can use it.
If you are interested in the cycling potential at the local level in England, or if you are interested in the visualisation of geographic information at multiple levels (area, desire line, route and route network levels in this case), please take a look at the video and have a play at www.pct.bike.
Key to the tools’ potential for policy impact is its wide ranging scenarios of change which included Government Target to double cycling by 2025 and a more ambitious Go Dutch scenario in which we cycle as much as the Dutch to, accounting for geographical differences in trip distances and hilliness. The difference between the two scenarios is substantial, as illustrated by the figure below.
For a detailed account of the methods (or geographic data science if you like) underlying the Propensity to Cycle Tool, I recommend checking out a paper on the subject by myself and other members of the team published in the Journal of Transport and Land Use (Lovelace et al. 2017).
## A reproducible example
Actions can speak louder than words or, as Linus Torvalds said:3
Talk is cheap. Show me the code.
On that basis, to show what I meant by reproducibility and ‘data carpentry’ (Gillespie and Lovelace 2016) (a concept mentioned in the slides not discussed in audio for lack of time) I provided some example code that illustrated the kinds of techniques underlying the PCT.
First download and visualise some transport data (from the Isle of Wight the smallest region in the PCT):
u_pct = "https://github.com/npct/pct-data/raw/master/isle-of-wight/l.Rds"
if(!file.exists("l.Rds"))
library(sp)
## Loading required package: methods
plot(l)
Now that we have an idea of the commute patterns in the area, and the nature of ‘OD’ data (converted to geographical desire lines with the stplanr package), we can do some analysis.
sel_walk = l$foot > 9 l_walk = l[sel_walk,] plot(l) plot(l_walk, add = T, col = "red", lwd = 3) library(dplyr) # for next slide... The above code subsets all the lines that have 10 or more people walking to work in the 2011 census and plots the results (as you’d expect the shorter trips are more commonly walked). It works, but could be interpretted as a little clunky. Enter dplyr, a package for data science (Grolemund and Wickham 2016): l_walk1 = l %>% filter(All > 10) # fails Doh! That nice ‘clean’ (well certainly consistent, using easily understood functions rather than the potentially confusing [ operator]) code does not work because Spatial objects are not compatible with dplyr. Enter the sf package, which represents a step change in how R handles spatial data. First let’s convert that l object into a ‘simple feature’ object: library(sf) ## Linking to GEOS 3.5.1, GDAL 2.1.3, proj.4 4.9.2, lwgeom 2.3.2 r15302 l_sf = st_as_sf(l) class(l_sf) ## [1] "sf" "data.frame" plot(l_sf[6:15]) ## Warning: plotting the first 9 out of 10 attributes; use max.plot = 10 to ## plot all Other than plotting multiples, one for each variable, objects of class sf behave much like objects of class Spatial, except they are also fully fledged data frames. This is what allows them to be subsetted with dplyr’s %>% operator (reproducing the square bracket subsetting above): l_walk2 = l_sf %>% filter(foot > 9) plot(l_sf[6]) plot(l_walk2, add = T, lwd = 3) A more advanced example involves the following: take all trips in the study area less than 1km and find those in which driving a car is more common than walking (areas that could have a major car dependency issue, the policy-relevant part): l_sf$distsf = as.numeric(st_length(l_sf))
l_drive_short = l_sf %>%
filter(distsf < 1000) %>%
filter(car_driver > foot)
library(tmap)
tmap_mode("view")
## tmap mode set to interactive viewing
qtm(l_drive_short)
## Discussion
I was happy to find the talk was attended by a range of people, appropriate for a seminar about getting researchers down from the ivory towers.
Someone from local government asked me if the above analysis to find short desire lines, along which more people drive than walk, could be applied in Liverpool (yes - the above code is a good starting point for working out how!).
I was also asked how the PCT methodology could engage with established methods in transport planning. This is something that myself and others on the PCT team have discussed and it’s certainly given me a reason to revisit how best to do that (I don’t have a clear solution at the moment).
## Conclusion
Geographical research clearly can have policy impacts. Calling this ‘saving the world’ may seem like hyperbole, but it in my experience it can help communicate the message that publicly-funded academics usually work in the public interest, for the ‘greater good’.
Clearly the amount and direction of the policy impacts of your work will vary depending on a range of factors, many of which will be outside your sphere of influence. However, if you are a (geographical) ‘data scientist’, it seems that ensuring that the ‘science’ in your title is taken seriously can greatly improve your policy impact.
Thus reproducibility and free publication of data and results should not be seen as a bureaucratic burden. It can help you save the world.
## References
Arribas-Bel, Daniel. 2014. “Accidental, Open and Everywhere: Emerging Data Sources for the Understanding of Cities.” Applied Geography 49: 45–53.
Gillespie, Colin, and Robin Lovelace. 2016. Efficient R Programming: A Practical Guide to Smarter Programming. O’Reilly Media. https://csgillespie.github.io/efficientR/.
Grolemund, Garrett, and Hadley Wickham. 2016. R for Data Science. 1 edition. O’Reilly Media.
Lovelace, Robin, Mark Birkin, Philip Cross, and Martin Clarke. 2016. “From Big Noise to Big Data: Toward the Verification of Large Data Sets for Understanding Regional Retail Flows.” Geographical Analysis 48 (1): 59–81. doi:10.1111/gean.12081.
Lovelace, Robin, Anna Goodman, Rachel Aldred, Nikolai Berkoff, Ali Abbas, and James Woodcock. 2017. “The Propensity to Cycle Tool: An Open Source Online System for Sustainable Transport Planning.” Journal of Transport and Land Use 10 (1). doi:10.5198/jtlu.2016.862.
Singleton, Alex. 2014. “A GIS Approach to Modelling CO2 Emissions Associated with the Pupil-School Commute.” International Journal of Geographical Information Science 28 (2): 256–73. doi:10.1080/13658816.2013.832765.
1. The name of the discipline, broadly be defined geographic data analysis, modelling and visualisation, is an interesting topic in its own right. The preferred term to describe this body of work shifts over time, and depending on who you ask. Over the years it has been referred to as Geographic Information Systems (GIS, which sounds rather old school nowadays), the more academic-sounding Geographic Information Science (GISc), the concise term of Geocomputation and the more recently coined term of Geographic Data Science (GDS).
2. See Arribas-Bel (2014) for an alternative geographical take on Big Data.
3. If I’m waxing lyrical about reproducibility and code I should practice what I preach. The source code of this article can be found at github.com/rbind/robinlovelace.
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# Justiciar faceguard
The Justiciar faceguard is obtained as a rare drop from the Theatre of Blood. It is part of the justiciar armour set, and requires level 75 Defence to equip.
When the full set of Justiciar armour is equipped, the player gains a set effect: all combat damage taken is reduced (except in PvP). The exact formula for the amount of damage reduced is ${\displaystyle {\frac {bonus}{3000}}}$, where bonus is the player's defence bonus for that particular style. For example, if an enemy was using a crush style attack, and the player has a crush defence bonus of +450, then 15% of the damage (450/3,000) is reduced.
The Justiciar faceguard is counted as a Saradomin item while in the GWD.
## Item sources
For an exhaustive list of all known sources for this item, see here (include RDT).
SourceLevelQuantityRarity
Theatre of BloodN/A 11/86.45
## Trivia
• Based on the statues found in the Hallowed Sepulchre and the dialogue and appearance of the ghostly knights encountered in its lobby, the Justiciar set of armour obtainable from the Theatre of Blood appears to be the armour of the Justiciar of the Wolf, who died long before the vampyre conflict reached Hallowvale.
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# Primitive recursion example
If we want to show primitive recursion for
sgn(x) = 0 if x = 0 and 1 otherwise
Is it enough to say that sgn(0) = 0 and sgn(x+1) = 1?
Is there any details omitting here or anything needed to be polished?
Also, to show that x monus y = (x-y) if x >= y and 0 otherwise, where we have the monus function N^2 --> N as primitive recusive, is it enough to say that:
x monus 1 is primitive recursive as 0 monus 1 = 0, but not sure what else to continue here.
-
Your first one is fine. For the second one, $x \dot{-} 1$ is primitive recursive, so you can set up a primitive recursion for $x\dot{-} y$ using this.
$$x \dot{-} 0 = x$$
$$x \dot{-} (a+1) = (x \dot{-} a) \dot{-} 1$$
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## TN State Board 12th Physics Important Questions Chapter 2 Current Electricity
Question 1.
What is meant by Current electricity?
Current electricity is the study of flow of electric charges.
Question 2.
What are free electrons?
Atoms in metals have one or more electrons that are loosely bound to the nucleus. They are called free electrons.
Question 3.
What are conductors?
Conductors are the substances that have an abundance of free electrons.
Question 4.
The positive ions will not give rise to current – Why?
The positive ions will not move freely within the material like the free electron. So, the positive ions will not give rise to current.
Question 5.
Define one ampere.
One ampere of current is defined as one coulomb of charge passing through a perpendicular cross-section in one second.
Question 6.
Define mobility of the electron.
Mobility of the electron is defined as the magnitude of the drift velocity per unit electric field.
μ = $$\frac{\left|\vec{v}_{d}\right|}{|\overrightarrow{\mathrm{E}}|}$$
Question 7.
State the units of (i) drift velocity and (ii) mobility.
(i) Unit of drift velocity m / s
(ii) Unit of mobility m2 / Vs.
Question 8.
What are carbon resistors?
Carbon resistors are the resistors containing a ceramic core on which a thin layer of i crystalline carbon is deposited.
Question 9.
What is a thermistor?
A thermistor is a semiconductor with a negative temperature coefficient of resistance.
Question 10.
State the factors on which does the resistivity of material depends?
(i) Number density (n) of the electrons.
(ii) The average time between two collisions.
Question 11.
What are super conductors?
Super conductors are the materials in which their resistance becomes zero below the transition temperature.
Question 12.
Name the metals that exhibit positive Thomson effect.
Question 13.
Mention the metals in which negative – Thomson effect is observed.
Platinum, nickel, Cobalt and Mercury.
Question 14.
How is very high temperature produced in the electrical furnace?
Very high temperatures up to 1500°C is produced in the electrical furnace by using molybdenum nichrome wire wound over a silica tube.
Question 15.
Define electric current.
Electric current in a conductor is defined as the rate of flow of charges through a given cross-sectional area.
Question 16.
Define drift velocity.
Drift velocity is defined as the average velocity acquired by the electrons inside the conductor when it is subjected to an electric field.
Question 17.
Draw the graphical representation for
(i) Ohmic conductor
(ii) non ohmic conductor
Question 18.
Define resistance. State its unit.
Resistance is defined as the ratio of potential difference across the given conductor to the current passing through the conductor.
R = $$\frac{\mathrm{V}}{\mathrm{I}}$$
Its unit is Ohm.
Question 19.
What is meant by transition temperature?
Transition Temperature is the temperature below which the resistance of a super conductor becomes zero.
Question 20.
Repairing the electrical connections with wet hands is always dangerous. Give reason.
For a human body, the resistance of dry skin is high and around 500 k Ω. But the wet skin of hands has less resistance and around 1000 Ω Ohm. Hence a large amount of current can flow through the wet hands. Hence it always dangerous.
Question 21.
Why are household appliance always connected in parallel?
They are connected in parallel so that even if one is switched off, the other devices could function properly.
Question 22.
Distinguish electromotive force from the potential difference.
Electromotive force Potential difference It is the difference of potentials between two terminals of a cell in an open circuit. It is the difference in potentials between any two points in a closed circuit. It is an independent of external resistance of the circuit. It is proportional to the resistance between any two points.
Question 23.
How ohm’s law is derived from drift velocity relates current formula?
The current flowing through a conductor is,
I = nAevd
But vd = $$\frac{e \mathrm{E}}{m} \cdot \tau$$
I = nAe $$\frac{e \mathrm{E}}{m} \cdot \tau$$
I = $$\frac{n \mathrm{~A} e^{2}}{m \mathrm{~L}} \tau \mathrm{V}$$
[∵ E = $$\frac{\mathrm{V}}{\mathrm{L}}$$]
where V is the potential difference. The quantity $$\frac{m \mathrm{~L}}{n \mathrm{~A} e^{2} \tau}$$ is a constant for a given conductor, called electrical resistance (R).
I ∝ V
The law states that, at a constant temperature, the steady current flowing through a conductor is directly proportional to the potential difference between the two ends of the conductor.
Question 24.
In the following circuit, calculate the current through the circuit. Mention its direction.
According to kirchhofFs second law,
5i + 7i + 3i + 5i = 10 + 8 – 2
20 i = 16
i = $$\frac{16}{20}$$ = $$\frac{4}{5}$$ A = 0.8 A
Question 25.
State any three difference between electric power and electric energy.
Electric power Electric energy (i) Electric power is defined as the rate of doing electric work. Electric energy is defined as the capacity to do electric work. (ii) It is the product of potential difference and current strength P = VI. It is the product of potential difference, strength of current and time during which current flows through the conductor. (iii) In practice it is measured by watt In practice it is measured by watt hour or kilowatt hour.
Question 26.
The colours of a carbon resistor is orange, orange, orange. What is the value of resistor?
The first orange ring corresponds to 3. The next orange ring corresponds to 3. The third orange ring corresponds to 103. Hence the total resistance is 33 KΩ.
Question 27.
What are the special features of an ideal battery?
(i) The internal resistance of an ideal battery is zero.
(ii) The potential difference across the battery equals to its emf.
Question 28.
State sign convention followed by Kirchhoff’s current rule.
• Current entering the junction is taken as positive.
• Current leaving the junction is taken as negative.
Question 29.
What are the sign conventions followed by Kirchhoff’s loop rule?
• In a loop, the product of current and resistance is taken as positive when the direction of current is followed.
• If the direction of current is opposite to the direction of the loop then the product of current and resistance across the resistor is negative.
Question 30.
Nichrome is used as a heating element in electric heaters. Give reason.
Since nichrome has a high specific resistance. It can be heated to very high temperatures without oxidation.
Question 31.
What is meant by Joule’s heating effect.
When current flows through a resistor, some of the electrical energy delivered to the resistor is converted into heat energy and it is dissipated.
Question 32.
What do you know about thermoelectric effect?
Conversion of temperature differences into electrical voltage and vice versa is known as thermoelectric effect.
Question 33.
Explain drift velocity and obtain an expression for it.
In a conductor the charge carriers are free electrons. These electrons move freely through the conductor and collide repeatedly with the positive ions.
Suppose a potential difference is set across the conductor by connecting a battery, an electric field $$\overrightarrow{\mathbf{E}}$$ is created in the conductor. This electric field exerts a force on the electrons. Then it produces a current.
The electric field accelerates the electrons, while ions scatter the electrons and change the direction of motion. In this way, we have zigzag paths of electrons. In addition to the zigzag motion due to the collisions, the electrons move slowly along the conductor in a direction opposite to that of E as shown in the Figure.
The drift velocity is the average velocity acquired by the electrons inside the conductor when it is subjected to an electric field. The average time between successive collisions is called the mean free time denoted by τ. The acceleration $$\overrightarrow{\mathbf{a}}$$ experienced by the electron in an electric field $$\overrightarrow{\mathbf{E}}$$ is given by
Question 34.
Describe the combination of ceils in which cells are connected
(i) in series and
(ii) in parallel. Hence obtain an expression for the current flowing in those combination of cells.
(i) Cell in series:
Suppose n cells, each of emf ξ, volts and internal resistance r ohms are connected in series with an external resistance R.
The total emf of the battery = nξ
The total resistance in the circuit = nr + R
By Ohm’s law, the ctitrent in the circuit is
I = $$\frac{\text { total emf }}{\text { total resistance }}$$
= $$\frac{n \xi}{n r+R}$$
Case (a) If r << R, then,
I = $$\frac{n \xi}{R}$$ ≈ nI1
where, I1 is the current due to a single cell
(I1 = $$\frac{\xi}{R}$$)
Thus, if r is negligible when compared to R the current supplied by the battery is n times that supplied by a single cell.
Case (b):
If r >> R,
I = $$\frac{n \xi}{n r} \approx \frac{\xi}{r}$$
It is the current due to a single cell. That is, current due to the whole battery is the same as that due to a single cell and hence there is no advantage in connecting several cells.
Thus series connection of cells is advantageous only when the effective internal resistance of the cells is negligibly small compared with R.
(ii) Cells in parallel:
Let n cells be connected in parallel between the points A and B and a resistance R is connected between the points A and B. Let ξ be the emf and r the internal resistance of each cell.
The equivalent internal resistance of the battery is
$$\frac{1}{r_{e q}}=\frac{1}{r}+\frac{1}{r}+\frac{1}{r}+\ldots+\frac{1}{r}(n \text { terms })=\frac{n}{r}$$
So req = $$\frac{r}{n}$$ and the total resistance in the circuit = R + $$\frac{r}{n}$$. The total emf is the potential difference between the points A and B, which is equal to ξ. The current in the circuit is given by,
I = $$\frac{\xi}{\frac{r}{n}+R}$$
I = $$\frac{n \xi}{r+n R}$$
Case (a) If r >> R,
I = $$\frac{n \xi}{r}$$ = nI1
where I1 is the current due to a single cell and is equal to $$\frac{\xi}{r}$$ when R is negligible. Thus, the current through the external resistance due to j the whole battery is n times the current due | to a single cell.
Case (b) If r << R,
I = $$\frac{\xi}{r}$$
The above equation, It is implied from j that current due to the whole battery is the same as that due to a single cell. Hence it is advantageous to connect cells in parallel when the external resistance is very small compared to the internal resistance of the cells.
Question 35.
Describe the potentiometer and explain its principle.
Potentiometer is used for the accurate measurement of potential differences, current and resistances.
Description:
It consists of ten meter long uniform wire of manganin or constantan (-) an stretched in parallel rows each of 1 meter length, on a wooden board. The two free ends A and B are brought to the same side and fixed to copper strips with binding screws. A meter scale is fixed parallel to the wire.
A jockey is provided for making contact. The principle of the potentiometer is illustrated in Figure. A steady current is maintained across the wire CD by a battery Bt. The battery, key and the potentiometer wire are connected in series forms the primary circuit. The positive terminal of a primary cell of emf ξ, is connected to the point C and negative terminal is connected to the jockey resistance HR. This forms the secondary circuit.
Procedure:
Let contact be made at any point J on the wire by jockey. If the potential difference across CJ is equal to the emf of the cell ξ, then no current will flow through the galvanometer and it will show zero deflection. CJ is the balancing length /. The potential difference across CJ is equal to Irl where I is the current flowing through the wire and r is the resistance per unit length of the wire.
Hence ξ = Irl ………(1)
Principle:
since I and r are constants, ξ ∝ l. The emf of the cell is directly proportional to the balancing length.
Question 36.
Explain the method of measuring internal resistance using a potentiometer.
The end C of the potentiometer wire is connected to the positive terminal of the battery B and the negative terminal of the battery is connected to the end D through a key K1. This forms the primary circuit.
The positive terminal of the cell ξ, whose internal resistance is to be determined is also connected to the end C of the wire. The negative terminal of the cell ξ is connected to a jockey through a galvanometer and a high resistance. A resistance box R and key K2 are connected across the cell ξ. With K2 open, the balancing point J is obtained and the balancing length CJ = l1 is measured. Since the cell is in open circuit, its emf is
ξ ∝ l1 …….(1)
A suitable resistance (say, 10 Ω) is included in the resistance box and key K2 is closed. Let r be the internal resistance of the cell. The current passing through the cell and the resistance R is given by
I = $$\frac{\xi}{\mathrm{R}+r}$$
The potential difference across R is
V = $$\frac{\xi \mathrm{R}}{\mathrm{R}+r}$$
When this potential difference is balanced on the potentiometer wire, let l2 be the balancing length.
Then $$\frac{\xi \mathrm{R}}{\mathrm{R}+r}$$ ∝ l2 …………(2)
From equations (1) and (2)
Substituting the values of the R, l1 and l2, the internal resistance of the cell is determined. The experiment can be repeated for different values of R. It is found that the internal resistance of the cell is not constant but increases with increase of external resistance connected across its terminals.
Question 37.
Explain Peltier effect.
In 1834, Peltier discovered that when an electric Current is passed through a circuit of a thermocouple, heat is evolved at one junction and absorbed at the other junction. This is known as Peltier effect.In the Cu-Fe thermocouple the junctions A and B are maintained at the same temperature. Let a current from a battery flow through the thermocouple (Figure (a)).
At the junction A, where the current flows from Cu to Fe, heat is absorbed and the junction A becomes cold. At the junction B, where the current flows from Fe to Cu heat is liberated and it becomes hot. When the direction of current is reversed, junction A gets heated and junction B gets cooled as shown in the Figure (b). Hence peltier effect is reversible.
Question 38.
Explain Thomson effect.
Thomson showed that if two points in a conductor are at different temperatures, the density of electrons at these points will differ and as a result the potential difference is created between these points. Thomson effect is also reversible.
If current is passed through a copper bar AB which is heated at the middle point C, the point C will be at higher potential. This indicates that the heat is absorbed along AC and evolved along CB of the conductor. In this way, heat is transferred due to the current flow in the direction of the current. It is called positive Thomson effect.
Similar effect is observed in metals like silver, zinc, and cadmium. When the copper bar is replaced by an iron bar, heat is evolved along CA and absorbed along BC. Thus heat is transferred due to the current flow in the direction opposite to the direction of current. It is called negative. Similar effect is observed in metals like platinum, nickel, cobalt, and mercury.
Question 39.
A resistor is made by joining two wires of the same material. The radii of two wires are 1 mm and 3 mm white their lengths are 3 cm and 5 cm respectively. A battery of emf 16 V and negligible resistance is connected across the resistor.
Calculate the potential difference across the shorter wire.
∴ Potential difference across shorter wire = 13.5 V
Question 40.
A Copper wire is stretched to make 0.1 % longer. What is the percentage change in its resistance?
Resistance R = $$\frac{\rho l}{\mathrm{~A}}$$ ………….(1)
ρ – Specific resistance
Let V be the volume
A = $$\frac{\mathrm{V}}{l}$$ …………….(2)
Sub (2) in (1) we get,
R = $$\frac{\rho l^{2}}{\mathrm{~V}}$$
Change in resistance is given by
∴ $$\frac{\Delta \mathrm{R}}{\mathrm{R}}=\frac{2 \Delta l}{100}$$
Here ρ and V are constants.
∴ $$\frac{\Delta \mathrm{R}}{\mathrm{R}}$$ × 100 = $$\frac{2 \Delta l}{l}$$ × 100
= 2 × $$\frac{1}{4}$$ × 100 = 0.2
Percentage change in resistance = 0.2.
Question 41.
What is the drift velocity of an electron in a copper conductor having area 10 × 10-6 m2, carrying a current of 2A. Assume that there are 10 × 1028 electrons/ m3.
Area = 10 × 10-6 m2;
Current = 2 A;
Number of electrons / m3 = 10 × 1028
Drift velocity
vd = $$\frac{\mathrm{I}}{n \mathrm{~A} e}$$
Drift velocity,
vd = $$\frac{2}{10 \times 10^{28} \times 10 \times 10^{-6} \times 1.6 \times 10^{-19}}$$
= $$\frac{2}{1.6 \times 10^{5}}$$ = 1.25 × 10 5 ms-1
Dirft velocity = 1.25 × 10 5 ms-1
Question 42.
Two wires of same material and length have resistances 5 v and 10 Ω respectively. Find the ratio of radii of the two wires.
Resistance of first wire R1 = 5 Ω;
Radius of first wire = r1
Resistance of second wire R2 = 10 Ω;
Radius of second wire = r2
Length of the wires = l;
Specific resistance of the material of the wires = ρ
Question 43.
The resistance of a field coil measures 50 Ω at 20°C and 65 Ω at 70°C. Find the temperature coefficient of resistance.
R20 = 50 Ω; 70°C; RTO = 65 Ω; α = ?
Rt = R0(1 + αt)
R20 = Ro (1 + α20)
50 = R0(1 + α20) ………….(1)
R70 = R0(1 + α70)
65 = R0(1 + α70] ………….(2)
Dividing (2) by (1)
$$\frac{65}{50}=\frac{1+70 \alpha}{1+20 \alpha}$$
65 + 1300 α = 50 + 3500 α
2200 α = 15
α = $$\frac{0.0068}{{ }^{\circ} \mathrm{C}}$$
= 0.0068 / °C
Question 44.
A wire has resistance of 2.0 Ω at 25°C and 2.50 at 100°C. Calculate the temperature coefficient of resistance of the wire.
Rt = R0 (1 + αt)
When t = 25°C Rt = 2.0 Ω
∴ 2.0 = R0 (1 + 25α) ……..(1)
When t = 100°C R100 = 2.5
∴ 2.5 = R0(1 + 100 α) …(2)
$$\frac{(1)}{(2)} \Rightarrow \frac{2.5}{2.0}=\frac{1+100 \alpha}{1+25 \alpha}$$
∴ 2.5(1 + 25 α) = 2(1 + 100 α)
2.5 + 62.5 α = 2 + 200 α
2.5 – 2 = 2o0 α = 62.5 α
∴ 137.5 α = 0.5
α = $$\frac{0.5}{137.5}$$
α = 0.003636 / °C
Question 45.
A 10 Ω resistance is connected in series with a cell of emf 10 V. A voltmeter is connected in parallel to a cell, and it reads 9.9 V. Find internal resistance of the cell.
R = 10 Ω;
E = 10 V;
V = 9.9 V;
r = ?
r = $$\left(\frac{E-V}{V}\right) R$$
= $$\left(\frac{10-9.9}{9.9}\right) \times 10$$ = 0.101 Ω.
Question 46.
In the given circuit, what is the total resistance and current supplied by the battery.
The value of the resistances are
R1 = 2 Ω; R2 = 3 Ω; R3 = 3 Ω; R4 = 3 Ω
The effective resistance of the resistances R2, R3 and R4 that are connected in parallel.
∴ Rp = 1 Ω
The effective resistance of the resistances R1, R2, R3, and R4 is given by
Rs = 2 + 1 = 3 Ω
Potential difference V = 6V
Resistance R = 3 Ω
∴ Current I = $$\frac{V}{R}=\frac{6}{3}$$ = 2 A
Question 47.
Three resistors are connected in series with 10 V supply as shown in the figure. Find the voltage drop across each resistor.
R1 = 5 Ω, R2 = 3 Ω, R3 = 2 Ω; V = 10 volt
Effective resistance of series combination,
Rs = R1 + R2 + R3 = Rs = 10 Ω
Current in circuit I = $$\frac{\mathrm{V}}{\mathrm{R}_{s}}=\frac{10}{10}$$ = 1A
Voltage drop across R1, V1 = IR1 = 1 × 5 = 5 V
Voltage drop across R2, V2 = IR2 = 1 × 3 = 3 V
Voltage drop across R3, V3 = IR3 = 1 × 2 = 2 V
Voltage drop across each resistor are V1 = 5 V; V2 = 3 V; V3 = 2 V.
Question 47.
Find the effective resistance between A and B in.the given circuit.
The value of resistances are
R1 = 2 Ω; R2 = 1 Ω; R3 = 2 Ω; R4 = 2 Ω; and R5 = 1 Ω
The effective resistance of R1 and R2 which are connected in. parallel is
$$\frac{1}{\mathbf{R}_{p_{1}}}=\frac{1}{\mathbf{R}_{1}}+\frac{1}{R_{2}}$$
Question 48.
In the given network, calculate the effective resistance between points A and B.
The network has three identical units. The simplified form of one unit is given below:
The equivalent resistance of one unit is
$$\frac{1}{\mathrm{R}_{p}}=\frac{1}{\mathrm{R}_{1}}+\frac{1}{\mathrm{R}_{2}}$$
= $$\frac{1}{15}+\frac{1}{15}$$ or Rp = 7.5 Ω
Each unit has a resistance of 7.5 Ω. The total network reduces to
The combined resistance between points A and B is
R= R’ + R’ + R’
(∵ Rs = R1 + R2 + R3)
R = 7.5 + 7.5 + 7.5 = 22.5 d
Question 49.
The resistance of a platinum wire at 0°C is 4 Ω. What will be the resistance of the wire at 100°C if the temperature coefficient of resistance of platinum is 0.0038 /°C.
The resistance of a platinum wire at 0° C
R0 = 4 Ω
Temperature coefficient of resistance \
α = 0.0038 /° C
Resistance at 100°C is
R100 = Rt = R0(1 + αt)
Resistance at 100°C is R100 = Rt
Rt = 4 [1 + 0.0038 × 100]
= 4(1 + 0.38)
= 4 × 1.38 = 5.52 Ω
Resistance at 100°C is R100 = 5.52 Ω
Question 50.
A cell has a potential difference of 6 V in an open circuit, but it falls to 4 V when a current of 2A is drawn from it. Find the internal resistance of the cell.
Initial potential difference E = 6 V
Final potential difference V = 4 V;
Current I = 2A
Internal resistance of the cells r = $$\frac{E-V}{I}$$
Internal resistance of the cell r = $$\frac{6-4}{2}=\frac{2}{2}$$ = 1 Ω
Internal resistance of the cell = 1 Ω
Question 51.
Find the voltage drop across 18 Ω resistor in the given circuit.
The emf of the source = 30 V
The values of resistances are 18 Ω, 12 Ω, 6 Ω and 6 Ω
Two resistances 12 Ω and 6 Ω are connected in series.
Rs = 12 + 6= 18 Ω
R2 = 6 Ω; R3 = 18 Ω
The resistances R2 and R3 are connected in parallel.
The effective resistance is given by
$$\frac{1}{\mathrm{R}_{p}}=\frac{1}{\mathrm{R}_{2}}+\frac{1}{\mathrm{R}_{3}}=\frac{1}{6}+\frac{1}{18}$$
= $$\frac{3+1}{18}=\frac{4}{18}=\frac{2}{9}$$
∴ Rp = $$\frac{9}{2}$$ Ω
The resistances R1 and Rp are connected in series.
R1 = 18 Ω; Rp = $$\frac{9}{2}$$ Ω
Rs = 18 + $$\frac{9}{2}$$ = $$\frac{36+9}{2}=\frac{45}{2}$$
∴ Effective resistance of the circuit = $$\frac{45}{2}$$ Ω
Electromotive force V = 30 V
∴ Current I = $$\frac{V}{R}=\frac{30}{\frac{45}{2}}=\frac{30 \times 2}{45}$$
= $$\frac{60}{45}=\frac{4}{3}$$ amp.
The voltage drop across 18 Ω is V = IR
V = $$\frac{4}{3}$$ × 18 = 24 V
Question 52.
Find the current flowing acrossthree resistors 3 Ω, 5 Ω and 2Ω connected in parallel to a 15 V supply. Also find the effective resistance and tot al current drawn from the supply.
R1 = 3 Ω; R2 = 5 Ω; R3 = 2 Ω;
Supply voltage V = 15 volt.
Effective resistance of parallel combination
Current through R1,
I1 = $$\frac{V}{R_{1}}=\frac{15}{3}$$ = 5 A
Current through R2,
I2 = $$\frac{V}{R_{2}}=\frac{15}{5}$$ = 3 A
Current through R3,
I3 = $$\frac{V}{R_{3}}=\frac{15}{2}$$ = 7.5 A
Question 53.
Calculate the current I1, I2 and I3 in the given electric circuit.
According to Kirchhoff’s first law, at the function F,
I3 = I1 + I2
Applying Kirchhoff’s second law to the closed mesh BFGHECB we get
(I1 + I2)10 + I1 × 1 = 3 ………(1)
Applying Kirchhoff’s second law to the closed mesh AFGHEDA we get
(I1 + I2) 10 +I2 × 2 = 2 ………..(2)
Subtracting Equation (2) from equation (1) we get
I1 – 2 I2 = 1 ……..(3)
∴ I1 = 1 + 2 I2 …….(4)
Subtracting Equation (4) from equation (2) we get
(1 + 2 I2 + I2) × 10 + I2 × 2 = 2
10 + 30 I2 + 2 I2 = 2
32 I2 = 2 – 10 = – 8
∴ I2 = – $$\frac{8}{32}$$ = – $$\frac{1}{4}$$
I2 = – 0.25 A
From Equation (4)
I1 = 1 + 2 I2 = 1 + 2 (- 0.25)
= 1 – 0.5 = 0.5 A
∴ I1 = 0.5 A
We know I3 = I1 + I2
= 0.50 + (-0.25)
I3 = 0.25 A
Current I1 = 0.5 A;
I2 = – 0.25 A;
I3 = 0.25 A.
Question 54.
Find the electric current flowing through the given circuit connected to a supply of 3 V.
R1 = 5 Ω;
R2 = 5 Ω;
R3 = 5 Ω;
Potential difference of supply V = 3V
Current I = $$\frac{V}{R}$$
When resistances are connected in parallel, effective resistance is given by
$$\frac{1}{\mathrm{R}_{p}}=\frac{1}{\mathrm{R}_{1}}+\frac{1}{\mathrm{R}_{2}}$$
When resistances are connected in series, effective resistance is given by
Rs = R1 + R2
= 5 + 5 = 10 Ω
When the resistances Rs and R3 are connected in parallel, effective resistance is
Question 55.
To balance The Wheatstone’s bridge shown in figure, determine an additional resistance that has to be connected with 3 Ω.
P = 5 Ω; Question = 10 Ω; R = 8 Ω; S = 32 Ω;
$$\frac{P}{Q}=\frac{R}{S}$$ $$\frac{5}{10}=\frac{8}{32}$$
∴ S = $$\frac{10 \times 8}{5}$$ = 16
Hence the value of resistance (s) has to be transformed into 16 Ω as follows:
When another resistance of 32 Ω is connected in parallel with 32 Ω then
$$\frac{1}{\mathbf{R}_{p}}=\frac{1}{32}+\frac{1}{32}$$
= $$\frac{1+1}{32}=\frac{2}{32}$$
∴ Rp = $$\frac{32}{2}$$ = 16 Ω
The wheatstone bridge can be balanced by connecting a resistance of 32 Ω is to be connected in parallel with it. The circuit is transformed into P = 5 Ω; Question = 10 Ω.
Question 56.
With an unknown resistance the balancing length obtained in a metre bridge is 20 cm. When 6 Question resistance is connected in series with the unknown resistance., the balancing length changes to 50cm. Calculate the unknown resistance.
Balancing lengths for unknown resistance are
l1 = 20 cm = 2 × 10-2 m;
l2 = 100 – 20 = 80 cm = 80 × 10-2m.
When unknown resistance is connected in series with 6 Ω the balancing lengths are
l1 = 50 × 10-2 m;
l2 = 50 × 10-2 m
$$\frac{\mathbf{P}}{\mathrm{Q}}=\frac{l_{1}}{l_{2}}$$
Let the unknown resistance be P Ω
P = Question × $$\frac{20}{80}$$ …………..(1)
P + 6 = Question × $$\frac{50}{50}$$ = Question …………(2)
By dividing equation (2) by equation (1) we get
$$\frac{P+6}{P}=\frac{Q}{Q} \times \frac{80}{20}$$ = 4
P + 6 = 4P
∴ 6 = 4P – P = 3P
∴ P = $$\frac{6}{3}$$ = 2 Ω
∴ The value of unknown resistance = 2 Ω
Question 57.
In the given circuit, find the current through each branch of the circuit and the potential drop across the 10Question resistor.
Resistance R1 = 2 Ω;
R2 = 4 Ω
R3 = 10 Ω
Electromotive forces E1 = 4 V and E1 = 5 V
Application of Kirchhoff’s second law to the closed mesh CDEB we get
– 4 + I1 × 2 – I2 × 4 + 5 = 0
2I1 – 4I2 = – 1 ……….(1)
Applying Kirchhoff’s second law to the closed mesh BEFA we get,
– 5 + 4 I2 + 10 (I1 + I2) = 0
– 5 + 4 I2 + 10 I1 + 10 I2 = 0
10 I1 + 14 I2 = 5 ………..(2)
Equation (2) —>10 I1 + 14 I2 = 5 ………(3)
(1) × 5 => 10 I1 – 20 I2 = – 5 ………(4)
(3) + (4) => 34 I2 = 10
∴ I2 = $$\frac{10}{34}$$
I2= 0.2941 A
From equation (1) We get
2 I1 = – 1 + 4 I2
= – 1 + 4 (0.2941)
= – 1 + 1.1764 = 0.1764
∴ I1= $$\frac{0.1764}{2}$$
I1 = 0.0882 A
I1 + I2 = 0.0882 + 0.2941 = 0.3823
Potential drop across 10 Ω resistor is
V = (I1 + I2) R
= 0.3823 × 10 = 3.823 V
Question 58.
Find the effective resistance of the given circuit.
The value of resistances that are connected in series is
R1 = 1 Ω;
R2 = 2 Ω;
R3 = 3 Ω;
Rs = R1 + R2 + R3
The effective resistance is given by
Rs = 1 + 2 + 3 = 6 Ω
Question 59.
Calculate the value of current I min the circuit.
Applying Kirchhoff’s First law at the junction we get
3.25 + 3.25 – 3.5 – 1.5 + 3.75 + I = 0
6.5 – 5 + 3.75 + I = 0
∴ I = – 5.25
Question 60.
In a Wheatstone’s bridge a battery of 2 V and a resistance of 2 Ω is used. Find the current through the galvanometer in the unbalanced position of the bridge when P = 1 Ω, Question = 2 Ω, t R = 2 Ω, S = 3 Ω and G = 4 Ω.
In loop ABDA,
I1 + 4 Ig – 2(I – I1) = 0
I1 + 4 Ig – 2I + 2 I1 = 0
4 Ig + 3 I1 – 2I = 0 ……………..(1)
Along the loop BCDB,
2(I1 – Ig) – 4Ig – 3(I – I1 + Ig) = 0
2I1 – 2Ig – 4Ig – 3I + 3I1 – 3Ig = 0
– 9Ig + 5I1 – 3I = 0 …………(2)
Along the path ABCEA,
2I + 1I1 + 2 (I1 – Ig) = 2
2I + I1 + 2I1 – 2Ig = 2
– 2 Ig + 3I1 + 2I = 2 ……………(3)
4Ig + 3I1 – 2I = 0 ………….(1)
(3) + 1 ⇒ 2Ig + 6I1 = 2 …………(4)
(1) × 3 ⇒ 12 Ig + 9I1 – 6I1 = 0 ………(5)
(2) × 2 ⇒ – 18 Ig + 10 I1 – 6I = 0 ………(6)
(5) – (6) ⇒ 30Ig – I1 = 0 ………….(7)
(7) × 6 ⇒ 180 Ig – 6I1 = 0 …………(8)
2 Ig + 6 I1 = 2
(8) + (4) ⇒ 182 Ig = 2
∴ Ig = $$\frac{2}{182}$$ = 0.010989 Amp.
Current through the galvanometer = 1.099 × 10-2 Amp.
Question 61.
The resistance of the four arms P, Q, R and S in a Wheatstone’s bridge are 10 Ω, 30 Ω, 30 Ω and 90 Ω respectively. The emf and interna! resistance of the cell are 7 V and 5 Ω respectively. If the galvanometer resistance is 50 Ω, then calculate the current drawn from the cell.
If the bridge is balanced then no current flows through the galvanometer.
RS1 = 10 + 30 = 40 Ω
RS2 = 30 + 90 = 120 Ω
$$\frac{1}{\mathrm{R}_{p}}=\frac{1}{40}+\frac{1}{120}=\frac{3+1}{120}$$
= $$\frac{4}{120}$$ = $$\frac{1}{30}$$
∴ Rp = 30 Ω
External resistance R = 30 Ω
Current I = $$\frac{E}{R_{p}+R}$$
E = 7 V Rp = 30 Ω R = 5 Ω
∴ I = $$\frac{7}{30+5}=\frac{7}{35}$$ = 0.2 A
∴ Current drawn from the cell = 0.2 A
Question 62.
The resistances in the two arms of a metre bridge are 5 Ω and R Ω respectively. When the resistance R is shunted with equal resistance the new balancing point is at 1.6l. Calculate the value of the resistance R.
1.6 (100 – l1) = 2(100 – 1.6l1)
160 – 1.6 l1 = 200 – 3.2 l1
1.6 l1 = 40
l1 = $$\frac{40}{1.6}$$ = 25 cm.
$$\frac{5}{\mathrm{R}}=\frac{l_{1}}{100-l_{1}}$$
$$\frac{5}{\mathrm{R}}=\frac{25}{100-25}=\frac{25}{75}$$
25 R = 5 × 75
R = $$\frac{5 \times 75}{25}$$ = 15 Ω
Resistance R = 15 Ω
Question 63.
In the following circuit, two ceils have negligible resistances. For VA = 12 V, R1 = 500 Ω and R = 100 Ω, the galvanometer (G) shows no deflection. Calculate the potential of VB.
Since there is no deflection in the galvanometer
Current I = $$\frac{\mathrm{V}_{\mathrm{A}}}{\mathrm{R}_{1}+\mathrm{R}}$$
VA = 12 V
R1 = 500
R = 100
∴ I = $$\frac{12}{500+100}$$
= $$\frac{12}{600}$$ A
∴ Potential of VB = IR
R = 100 Ω
∴ VB = $$\frac{12}{600}$$
∴ VB = 2 V
Question 64.
If the power dissipated in 9 Ω resistor in the given circuit is 36 W, then calculate the potential difference across the 2Ω resistor.
Current through 9 Ω resistor is given by
P = I2R
I12 = $$\frac{36}{9}$$ = 4
∴ I1 = 2 A
Since 9 Ω and 6 Ω are in parallel connection
9 1 = 6 I1
∴ I2 = $$\frac{9}{6}$$ × I1
I1 = 2 A
∴ I1 = $$\frac{9}{6}$$ × 2 = 3 A
∴ I = I1 + I2 = 2 + 3 = 5 A
∴ Potential difference across 2 Ω resistors is 5 × 2 = 10 V.
Question 65.
An electric iron has a resistance of 80 Ω and is connected to a source of 200 V. If it is used for two hours then Calculate the electrical energy spent.
Resistance R= 80 Ω
Potential difference V = 200 V
∴ Power = $$\frac{\mathrm{V}^{2}}{\mathrm{R}}$$
= $$\frac{200 \times 200}{80}$$
P = 500 W
∴ Time t = 2 hour
∴ Electrical energy E = P × t
E = 500 × 2 = 1000 W = 1 kW
∴ Electrical energy spent = 1 kW
Question 66.
Sixteen cells each of emf 3 V are connected in series and kept in a box- External the combination shows an emf of 12 V. Calculate the number of cells reversed in the connection.
Number of cells x = 16
emf of each cell E = 3 V
Total emf = nE = 16 × 3 = 48 V
Let the number of cells in reversed direction be n
Total emf -2nE = Effective emf
48 -2n × 3 = 12 V
-6 n = 12 – 48 = -36
Number of cells = $$\frac{-36}{-6}$$ = 6
Number of cells reversed in the connection is n = 6
Question 67.
Calculate the value of current I in the following figure at the junction A.
Applying Kirchhoff’s current law (I law) we get
1 – 2 + 4 + I = 0
3 + I = 0
∴ I = -3 A
∴ The current flows away from the junction I = – 3 A.
Question 68.
A battery of emf 12 V and internal resistance 2 Ohm is connected to a resistor. The current flowing in the circuit is 0.5 A.
When the circuit is closed. What is the value of Terminal voltage of the battery?
emf E = 12 V
Internal resistance r = 2 Ω
Current I = 0.5 V
Terminal voltage of the battery is given by
V=E – Ir
= 12 – 0.5 × 2 = 12 – 1 = 11 V
∴ Terminal Voltage = 11 V
Multiple Choice Questions:
Question 1.
When current I flows through a wire, drift velocity of the electrons is v. When a current 21 flows through another wire of the same material having double the length and area of cross-section, the drift velocity of the electron will be:
(a) 2 v
(b) $$\frac{v}{2}$$
(c) v
(d) $$\frac{v}{4}$$
(c) v
Question 2.
A wire of resistance 4 Ω is stretched to twice its original length. The resistance of the stretched wire would be:
(a) 8 Ω
(b) 16 Ω
(c) 2 Ω
(d) 4 Ω
(b) 16 Ω
Hint:
R = $$\frac{\rho^{\prime} l}{A}$$ = 4 Ω
Since the volume of the wire remains constant
l A = l’ A’
∴ A’ = $$\frac{l \mathrm{~A}}{l}$$
Question 3.
If the length of a wire is doubled and its area is also doubled then its resistance will:
(a) become halved
(b) become two times
(c) become four times
(d) remain unchanged
(d) remain unchanged
Question 4.
Across a metallic conductor of non-uniform cross-section a constant potential difference is applied.
Which one of the following statements is correct?
Along the conductor,
(a) drift velocity remains constant
(b) current remains constant
(c) electric field remains constant
(d) current density remains constant
(b) current remains constant
Question 5.
Match the following quantities given in column I and II.
Column I Column II (i) Current (A) mho m-1 (ii) Resistance (B) Volt (Hi) Potential difference (C) mho (iv) Conductivity (D) ohm (E) ampere
(a) (i) – (E); (ii) – (A); (iii) – (B); (iv) – (C)
(b) (i) – (D); (ii) – (B); (iii) – (C); (iv) – (A)
(c) (i) – (A); (ii) – (B); (iii) – (C); (iv) – (D)
(d) (i) – (E); (ii) – (D); (iii) – (B); (iv) – (A)
(d) (i) – (E); (ii) – (D); (iii) – (B); (iv) – (A)
Question 6.
According to Faraday’s law of electrolysis, when a current is passed, the mass of ions deposited at the cathode is independent of:
(a) current
(b) charge
(e) time
(d) resistance
(d) resistance
Question 7.
The chance of collision of electron on the atom depends on:
(a) length only
(d) neither the length nor the radius
Question 8.
The specific resistance of the material of a conductor of length 3m area of cross-section 0.2 mm2 having a resistance of 2 ohm is:
(a) 1.6 × 10-8 mho m-1
(b) 7.5 × 10-8 mho m-1
(c) 1.33 × 10-7 mho m-1
(d) 1.5 × 10-8 mho m-1
(b) 7.5 × 10-8 mho m-1
Question 9.
Resistance of a metal wire of length 10cm is 2 OE if the wire is stretched uniformly to 50 cm, the resistance is:
(a) 25 Ω
(b) 10 Ω
(c) 5 Ω
(d) 50 Ω
(d) 50 Ω
Question 10.
The brown ring at one end of a carbon resistor indicates a tolerance of:
(a) 1 %
(b) 2 %
(c) 5 %
(d) 10 %
(a) 1 %
Question 11.
The material through which electric charge can flow easily is:
(a) quartz
(b) mica
(c) germanium
(d) copper
(d) copper
Question 12.
The unit of conductivity is:
(a) mho
(b) ohm
(e) ohm-m
(d) mho-m-1
(d) mho-m-1
Question 13.
The colour code on a carbon resistor is red- red-black. The resistance of the resistor is:
(a) 2.2 Ω
(b) 22 Ω
(c) 220 Ω
(d) 2.2 kΩ
(b) 22 Ω
Question 14.
The unit of resistivity is:
(a) ohm metre
(b) (ohm metre)-1
(c) ohm/metre
(d) mho metre-1
(a) ohm metre
Question 15.
The drift velocity does not depend upon:
(a) the magnetic field
(b) mass of the electron
(c) charge of the electron
(d) relaxation time
(a) the magnetic field
Question 16.
The specific resistance of a wire will depend on:
(a) its length
(c) the type of material of the wire
(d) none of the above
(c) the type of material of the wire
Question 17.
When η resistor of equal resistance (R) are connected in series, the effective resistance is:
(a) $$\frac{n}{\mathrm{R}}$$
(b) $$\frac{\mathrm{R}}{n}$$
(c) $$\frac{1}{n \mathrm{R}}$$
(d) nR
(d) nR
Question 18.
When two 2 Ω resistances are in parallel, the effective resistance is:
(a) 2 Ω
(b) 4 Ω
(c) 1 Ω
(d) 0.5 Ω
(c) 1 Ω
Hint:
$$\frac{1}{\mathrm{R}_{p}}=\frac{1}{2}+\frac{1}{2}=\frac{1}{1}$$
Rp = 1 Ω
Question 19.
The current flowing in a conductor is proportional to:
(a) drift velocity
(b) 1 / area of cross-section
(c) 1 /no. of electrons
(d) square of the area of cross-section.
(a) drift velocity
Question 20.
If the length of a copper wire has a certain resistance R, then on doubling the length its specific resistance:
(a) will be doubled
(b) will become $$\frac{1}{4}$$th
(c) will become 4 times
(d) will remain the same
(d) will remain the same
Question 21.
A charge of 60 C passes through an electric lamp in 2 minutes. Then the current in the lamp is:
(a) 30 A
(b) 1 A
(c) 0.5 A
(d) 5 A
(c) 0.5 A
Question 22.
The flow of electrons in a conductor constitutes:
(a) potential
(b) electric current
(c) resistance
(d) charge
(b) electric current
Question 23.
Current is _____ quantity.
(a) a vector
(b) a scalar
(c) not a scalar
(d) a physical
(b) a scalar
Question 24.
The resistance of a wire on increasing its temperature will ________ with rise in temperature.
(a) increase
(b) decrease
(c) remain same
(d) increase and decrease
(a) increase
Question 25.
The current density of a conductor consisting of n electrons moving with drift velocity vd and having charge of e is given by:
(a) evd
(b) $$\frac{n e v_{d}}{\tau}$$
(c) nevd
(d) $$\frac{e v_{d}}{n}$$
(c) nevd
Question 26.
The unit of mobility of free electron is:
(a) m2 Vs
(b) m2 V-1 s-1
(c) m2 Vs-1
(d) Vm2 s-1
(b) m2 V-1 s-1
Question 27.
An electric bulb is rated as 220-100W. If the rated value drops by 2.5 % then the percentage of drop in power will be:
(a) 10 %
(b) 5 %
(c) 20 %
(d) 2.5 %
(b) 5 %
Hint:
Question 28.
Which of the following pair of expressions for drift velocity vd is correct?
(a) vd = $$\frac{e \mathrm{E}}{m} \tau$$ and vd = $$\frac{I}{n A e}$$
(b) vd = $$\frac{e \mathrm{E}}{m} \tau$$ and vd = IR
(c) vd = $$\frac{-e \mathrm{E}}{m} \tau$$ and vd = $$\frac{\mathrm{J}}{n \mathrm{~A} e}$$
(d) vd = $$$\frac{-e^{2} E}{m}$$$ and vd = $$\frac{\mathrm{I}}{n^{2} \mathrm{~A} e}$$
(a) vd = $$\frac{e \mathrm{E}}{m} \tau$$ and vd = $$\frac{I}{n A e}$$
Question 29.
Three conductors draw currents Of 1 A, 2 A and 4 A respectively, when they are connected in series across the same battery, the Current drawn will be:
(a) $$\frac{3}{7}$$ A
(b) $$\frac{5}{7}$$ A
(c) $$\frac{2}{7}$$ A
(d) $$\frac{4}{7}$$ A
(d) $$\frac{4}{7}$$ A
Hint:
Question 30.
For which of the following dependences, of drift velocity vd on electric field E, is ohm’s law obeyed?
(a) vd ∝ E2
(b) vd ∝ √E
(c) vd ∝ E
(d) vd = constant
(c) vd ∝ E
Question 31.
The relationship between Current density J, conductivity a and Electric field intensity E is:
(a) J = σ2 E
(b) J = σ
(c) J = σE
(d) J = $$\frac{E}{\sigma}$$
(c) J = σE
Question 32.
The specific resistance of a wire:
(a) varies with its mass
(b) varies with its area
(c) varies with its length
(d) is independent of mass, area and length of the wire.
(d) is independent of mass, area and length of the wire.
Question 33.
Select the incorrect statement from the following statement:
(a) A wire carrying current stays electrically neutral.
(b) A high resistance volt meter is used to measure the emf of a cell.
(c) A 6000 V power Supply must have very high internal resistance.
(d) A 6V power supply must have very low internal resistance.
(d) A 6V power supply must have very low internal resistance.
Question 34.
Ohm’s law does not hold good for:
(I) current flowing through a diode valve.
(II) current flowing through a triode valve.
(III) current flowing through a NPN Transistor.
(IV) current flowing through a PN junction diode
Which two of the above are correct?
(a) (I) and (II)
(b) (I) and (III)
(c) (III) and (IV)
(d) (II) and (III)
(b) (I) and (III)
Question 35.
Select the incorrect statement from the following statement is:
(a) When cells are connected in series external resistance is greater than internal resistance.
(b) When cells are connected in parallel, external resistance is less than internal resistance.
(c) If n identical cells are connected in parallel, emfis E / n and internal resistance is r / n
(d If n identical cells are connected in series, emfis nE and internal resistance is nr.
(c) If n identical cells are connected in parallel, emfis E / n and internal resistance is r / n
Question 36.
Which one of the following pairs of quantities remain constant when a steady current flows through a metallic conductors of non-uniform cross-section:
(a) drift speed and current
(b) electric field and current
(c) resistivity and current
(d) resistivity and resistance
(b) electric field and current
Question 37.
Which one of the following pairs of quantities is correct for the specific resistance of a wire?
(a) material and length
(b) material and temperature
(c) length and temperature
(d) area and temperature
(b) material and temperature
Question 38.
Match the following quantities given in | column I and Column II:
(a) (i) – (B); (ii) – (E); (iii) – (D); (iv) – (A)
(b) (i) – (C); (ii) – (D); (iii) – (B); (iv) – (F)
(c) (i) – (A); (ii) – (D); (iii) – (F); (iv) – (E)
(d) (i) – (F); (ii) – (D); (iii) – (E); (iv) – (A)
(d) (i) – (F); (ii) – (D); (iii) – (E); (iv) – (A)
Question 39.
Superconductors are used in:
(a) power generators
(b) computers
(c) launching satellites
(d) all the above
(d) all the above
Question 40.
The temperature at which the material changes from normal conductor to a super conductor is called:
(a) absolute temperature
(b) transition temperature
(c) normal temperature
(d) superconducting temperature
(b) transition temperature
Question 41.
The temperature at which the resistance of mercury becomes zero is:
(a) 0 K
(b) 0° C
(c) 4.2 K
(d) 4.2 K
(c) 4.2 K
Question 42.
The silver ring on the resistor represents ………. tolerance.
(a) 20 %
(b) 10 %
(c) 5 %
(d) 1 %
(b) 10 %
Question 43.
For a resistor, the gold coloured ring represents ______ tolerance of it.
(a) 10 %
(b) 5 %
(c) 2 %
(d) 1 %
(b) 5 %
Question 44.
1 % tolerance of a resistor is represented by _______ ring drawn on it.
(a) silver
(b) gold
(c) red
(d) brown
(d) brown
Question 45.
On the resistor if there is no coloured ring at its end then the tolerance is:
(a) 10 %
(b) 5 %
(c) 20 %
(d) 2 %
(c) 20 %
Question 46.
At the transition temperature, the conductivity becomes:
(a) zero
(b) infinity
(c) finite
(d) half of its initial value
(b) infinity
Question 47.
The colours of a carbon resistor is yellow, violet and orange. What is the value of the resistor?
(a) 33 kΩ
(b) 4.7 kΩ
(c) 47 kΩ
(d) 22 kΩ
(c) 47 kΩ
Question 48.
If a charge of 60 C passes through an electrical equipment in 4 minutes, then the current flowing through it is:
(a) 0.5 A
(b) 0.25 A
(c) 1 A
(d) 2 A
(b) 0.25 A
Question 49.
The conductivity of a conductor depends on the _________ of the conductor.
(a) length
(b) material
(c) resistance
(d) area of cross-section
(b) material
Question 50.
The resistance of a:
(a) thick conductor is less than that of a thin conductor
(b) thin conductor is less than that of a thick conductor.
(c) short conductor is more than that of a long conductor.
(d) both (a) and (c)
(a) thick conductor is less than that of a thin conductor
Question 51.
If a wire of resistance R is stretched to twice its original length, the new resistance of the
wire will be:
(a) $$\frac{\mathrm{R}}{2}$$
(b) 2R
(c) 4R
(d) $$\frac{\mathrm{R}}{4}$$
(c) 4R
Question 52.
The temperature coefficient of resistance of alloys is:
(a) positive and small
(b) negative and small
(c) positive and large
(b) negative and large
(a) positive and small
Question 53.
The temperature coefficient of manganin is:
(a) small and positive
(b) small and negative
(c) large and positive
(d) large and negative
(a) small and positive
Question 54.
The current drawn by 1.5 kW operated at 220 V water heater is:
(a) 6.9 A
(b) 14.6A
(c) 15 A
(d) 6.82 A
(d) 6.82 A
Hint:
Question 55.
If two resistors of resistance 3 Ω and 1 Ω are connected in parallel, then the effective i resistance of the system is:
(a) $$\frac{2}{3}$$ Ω
(b) .75 Ω
(c) 4 Ω
(d) 5.1 Ω
(b) .75 Ω
Hint:
$$\frac{1}{R_{p}}=\frac{1}{3}+\frac{1}{1}=\frac{4}{3}$$
∴ Rp = $$\frac{3}{4}$$ = 0.75 Ω
Question 56.
If R1 and R2 are resistances of a given coil at temperatures t1 and t2, then temperature coefficient of resistance of the material is:
(a) $$\frac{R_{1}-R_{2}}{R_{1} T_{2}-R_{2} T_{1}}$$
(b) $$\frac{R_{2}-R_{1}}{R_{1} T_{2}-R_{2} T_{1}}$$
(c) $$\frac{\mathrm{R}_{2}-\mathbf{R}_{1}}{\mathbf{R}_{2} \mathrm{~T}_{1}-\mathrm{R}_{1} \mathrm{~T}_{2}}$$
(d) $$\frac{R_{1}-R_{2}}{R_{2} T_{1}-R_{1} T_{2}}$$
(b) $$\frac{R_{2}-R_{1}}{R_{1} T_{2}-R_{2} T_{1}}$$
Question 57.
Kirchhoffs second law is the consequence of conservation of:
(a) charges
(b) current
(c) energy
(d) potential
(c) energy
Question 58.
Kirchhoff’s first law is the consequence of conservation of:
(a) current
(b) potential
(c) charges
(d) resistance
(b) potential
Question 59.
According to the principle of potentiometer, the emf of the cell is:
(a) inversely proportional to its balancing length
(b) directly proportional to its balancing length
(c) directly proportional to its resistance
(d) inversely proportional to its resistance
(b) directly proportional to its balancing length
Question 60.
If the strength of current flows through a resistor is I having resistance R and potential difference between its ends is V then the power dissipated is:
(a) $$\frac{\mathrm{I}}{\mathrm{V}}$$
(b) IR
(c) $$\frac{\mathrm{I}}{\mathrm{V}}$$
(d) VI
(d) VI
Question 61.
The energy equivalence of 1 kWh is:
(a) 3.6 × 105 J
(b) 36 × 105 J
(c) 0.36 × 1044 J
(d) 36 × 106 J
(b) 36 × 105 J
Question 62.
The ratio of the emf of two cells when their balancing lengths on a potentiometer are 614.8 cm and 824.4 cm respectively is:
(a) 1.7456
(b) 1.313 J
(c) 0.7456
(d) .8456
(c) 0.7456
Question 63.
The resistances in cyclic order of the four arms of Wheatstone’s network at balance is:
(a) 8, 6, 16, 12
(b) 8, 6, 12, 16
(c) 8, 12, 6, 16
(d) 8, 16, 6, 12
(b) 8, 6, 12, 16
Question 64.
Two copper wires of length lm and 9m have equal resistances. Then their diameters are in the ratio:
(a) 3 : 1
(b) 9 : 1
(c) 1 : 9
(d 1 : 3
(d) 1 : 3
Question 65.
At transition temperature, the resistance of a super conductor:
(a) increases slowly
(b) decreases slowly
(c) increases rapidly
(d) drops to zero
Ans :
(d) drops to zero
Question 66.
Assertion:
The drift speed of electrons in metals is of the order of few mm s and charge of an electron is very small ; (1.6 × 10-19 C). Yet a large amount of current is obtained.
Reason:
At room temperature, the thermal speed of electron is very high (107 times the drift speed).
Which one of the following statements is j correct?
(a) Both assertion and reason are true and reason explains assertion correctly.
(b) Both assertion and reason are true and reason does not explains assertion correctly.
(c) Assertion is true and reason is false,
(d) Assertion is false and reason is true.
(b) Both assertion and reason are true and reason does not explains assertion correctly.
Hint:
Current depends on
(i) Charge
(ii) drift speed
(iii) area and
(iv) no. of free electrons per unit volume of a metal.
Question 67.
Assertion:
In a meter bridge experiment, the balancing length AC corresponding to null deflection of the galvanometer is x.
If the radius of the wire is doubled then the balanced length becomes 4x.
Reason:
The resistance of a wire is inversely proportional to the square of its radius.
Which one of the following statement is correct?
(a) Both assertion and reason are true and reason explains assertion correctly.
(b) Both assertion and reason are true and reasons does not explain assertion correctly.
(c) Assertion is true and reason is false.
(d) Assertion is false and reason is true.
(d) Assertion is false and reason is true.
Hint:
If the radius, is doubled the ratio of resistance will remain unchanged.
But, $$\frac{\mathbf{R}_{1}}{\mathrm{R}_{2}}=\frac{l_{1}}{l_{2}}$$ = constant
Hence the balancing length will remain same.
Question 68.
Choose the odd man out from the following:
(a) Silver
(b) Mercury
(c) Zinc
(b) Mercury
Hint:
Negative Thomson effect is observed only in mercury.
Question 69.
Choose die odd man out.
In which one of die following negative Thomson effect is not observed.
(a) Mercury
(b) Platinum
(c) Nickel
Question 70.
Match the following:
Column I Column II (i) Positive Thomson effect (A) Cu – Fe thermocouple. (ii) Negative Thomson effect (B) Iron (iii) Zero Thomson effect (C) Copper (iv) Peltier effect (D) Mercury (E) Zinc (F) Lead
(a) (i) – (A); (ii) – (B); (iii) – (D); (iv) – (C)
(b) (i) – (C); (ii) – (E); (iii) – (D); (iv) – (A)
(c) (i) – (B); (ii) – (C); (iii) – (D); (iv) – (F)
(d) (i) – (E); (ii) – (D); (iii) – (F); (iv) – (A)
(d) (i) – (E); (ii) – (D); (iii) – (F); (iv) – (A)
Question 71.
Assertion:
In the potentionmeter circuit, if the experimental wire AB is not changed then the value of resistor R that is connected to the cell in the primary circuit will be decreased.
Reason:
At the balancing point, the potential difference between A and D to cell C1 =? emf of E2 of cell 2.
Which one of the following statement is correct:
(a) Both assertion and reason are true and reason explains assertion correctly.
(b) Both assertion and reason are true and reason does not explain correctly.
(c) Assertion is true but reason is false.
(d) Assertion is false but reason is true.
(d) Assertion is false but reason is true.
Hint:
If the diameter of wire AB is increased, the resistance will decrease.
So, the potential difference due to cell Question will be decreased.
Hence the null point will be obtained at a higher value of x.
Question 72.
When a resistance of 2 Ohm is connected across the terminals of a cell, the current is 0.5 A. If a resistance of 5 Ohm is connected across the cell then the current is 0.25 A then the emf of the cell is:
(a) 2V
(b) 1.5 V
(c) 1 V
(d) 0.5 V
(b) 1.5 V
Hint:
E = I(R + r)
= IR + Ir
∴ r = $$\frac{E}{I}$$ – R
∴ $$\frac{E}{0.5}$$ – 2 = $$\frac{E}{0.25}$$ – 5
∴ E = 1.5 V
Question 73.
5 rows of 10 identical cells, connected in series, send a current I through an external resistance of 20 Ohm.
If the emf and internal resistance of each cell is 1.5 V and 1 Ohm respectively then, value of current (I) is:
(a) 0.25 A
(b) 0.14 A
(c) 0.68 A
(d) 0.75 A
(c) 0.68 A
Hint:
Equivalent emf E = 15 V
Equivalent internal resistance r = 2 Ω
I = $$\frac{\dot{E}}{\mathrm{R}+r}$$
= $$\frac{15}{20+2}=\frac{15}{22}$$ = 0.68 A
Question 74.
When 115 V is applied across a wire of length 10m and radius 0.3 mm, the current density is 1.4 x 104 A m2. The resistivity of the wire is:
(a) 4.1 × 104 Ωm
(b) 8.2 × 10-4 Ωm
(c) 16.4 × 104 Ωm
(d) 2.1 × 104 Ωm
(b) (b) 8.2 × 10-4 Ωm
Hint:
Question 75.
If a wire is stretched to make it 0.1 % longer, the percentage change in its resistance would be:
(a) Zero
(b) 0.1%
(c) 0.2%
(d) 0.4 %
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# What is the relation between the facts that occur at night and the facts that occur in the morning?
What is the relation between the facts that occur at night and the facts that occur in the morning?
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