Would two astronauts one in a satellite, one on top of a tower have the same experience?

Imagine two astronauts. The first is in a satellite in geostationary orbit around the earth. The second is on top of a 35,786 km tall tower that has been built from the equator to space. This means that that the top of the tower is at the same height as the geostationary satellite. Naive logic would seem to indicate that the astronaut in the satellite would experience free-fall (as do all satellites in orbit), whereas the astronaut on top of the tower would feel the effects of gravity (albeit at a reduced level due to being 35,786 km above the surface of the earth. However, I can't believe that this would be the case. What's the flaw in my logic? Is the acceleration due to the earth's gravity zero at 35,786 km above the earth?


Calories burned doing 1 squat

To calculate the calories burned doing 1 squat, wouldn't it simply involve the distance that your center of mass is moving and your weight? I read some other article that said your legs act as levers, so the calculation has to involve that. Then of course you have to come up with the efficiency of your body.


Vorticity and circulation in fluid mechanics

Given a fluid element whose circulation was determined counter-clockwise, how can I calculate the Vorticity of the element? * * * The circulation is $$\Gamma = (\frac{\partial u_2}{\partial x_1}-\frac{\partial u_1}{\partial x_2})\delta x_1 \delta x_2$$ I thought this might give a better idea about the question. Here $u_1$ is the velocity along $x_1$ and $u_2$ is the velocity along $x_2$.


What is supersymmetry?

1. What is supersymmetry?    2. How is it useful in unifying physics? I have heard that supersymmetry states that every fermion is associated with a boson and vice versa. But I don't get hope this can help resolving problems in unifying physics.


Question on spin-orbit interaction

When you study the spin-orbit interaction in quantum mechanics, even for a simple hydrogen atom, you find only the electric field in the nucleus reference system, while in the electron reference system you find the magnetic field due to the relative orbit of the nucleus around it (through Lorentz Transformations). Why is that? Shouldn't there be a magnetic field in the nucleus reference system due to the orbit of the electron as well? Is it just a convention, since electric and magnetic field are actually just one field?


Why is the diffusion capacitance greater than the depletion capacitance in a p-n junction?

In a p-n junction the total capacitance is the sum of depletion capacitance and diffusion capacitance and it is mentioned that the diffusion capacitance is almost a million times greater than the depletion capacitance. Why is it so? Is this true that in a p-n junction diffusion capacitance dominates at all situations?


Does unitarity imply conservation of energy?

Not too long ago, someone began to discuss the thinking and motivation behind the Lagrangian and its formalism for the Newtonian framework and an intuitive understanding of such formalism. Somehow, it ended in the case that the Lagrangian can be understood in terms of conservation of information (largely) without having to rely on other laws. Is this line of argument correct? And, more importantly, would this mean that unitarity is a stronger statement than conservation of energy?


Determination of mass density distribution of an object

This is a follow-up to a previous question How can you weigh your own head in an accurate way?. My purpose is not to restart the flurry of more or less humorous jokes (which are not such a bad thing when infrequent) but to try to draw some conclusions, as I found the event instructive, and hopefully to get a real answer. Beyond the jokes, I think the question was also interesting from a more social point of view: how fallacies operate, how people vote, what is the effect how the fast answer inducement by site rules. It is clear now, at least to me, that the real physical problem was to determine the mass density distribution of a (partly ?) solid object without using destructive means. Of course the concept of destructiveness may depend on the concerned body, and that was one source of jokes. Still, it is a relevant topic to review the means, if any, for achieving such a purpose, as it seems that there are not that many. This is actually the question I am asking. But I would like also, at the same time, to review some of the answers given to the previous qiestion. One that attracted my attention was the micro-satellite solution. Does it really work ? So we could precisely ask: given a precise knowledge of the gravity field created by a solid object, can we deduce from it the mass density distribution in the solid. The answer seems to be no. Take the trivial case of a sphere with a uniform radial density distribution. The gravity outside the sphere depends only on the total mass, so that it tells us nothing about the internal density distribution. Is that an exceptional degenerate case, or is it a strong hint that the analysis of the gravity field is not enough ? The slicing solution I proposed was an interesting mistake, certainly a silly one. But it is ever so tempting to believe the solution is near when you have a set of equations, and easy ones to boot. A cute trap. It becomes more obvious when you try to do it in continuous rather than discrete form. It should have been obvious, as we know that the torque produced by weighing a massive object depend only on total mass and distance of the center of mass to the axis. What was less obvious, at least to me (but I have not done much physics since college), was the use of the moment of inertia. One could also think of measuring moments of inertia after virtual slicing of the object as previously with torque. Unfortunately the moment of inertia depends only on 3 values, the previous two and a reference moment of inertia (for a given direction of the rotation axis). It is a useful remark for computing moments of inertia, but shows that there is no hope there for solving our problem.   This was remarked by @Ben Crowell in another question (see below). More generally, it seems there is no hope from any discretized measurement of a quantity that depends polynomially on the distance. One cannot get more unknowns than the degree of the polynomial. I will not comment on Compton scattering, and other uses of indirect physical phenomena, if only for lack of competence. There is also the fact that I would like to know of a solution involving only mechanics. One technique I heard of is measuring wave propagation across the object. I hinted to that in a comment on the Compton scattering solution. I have no expertise on that, but I think that is how the structure of the planet is analyzed by geophysicists (earthquake waves). I also heard of underground analyses by similar means, using explosives. But I am not sure about the type of information that can be obtained in this way. Does one get density distribution. Does it always work ? The other technique I thought of is ultrasound based medical imagery. Could it measure densities. Now since we are considering waves, could we get something out of the mesurement of gravity wave propagation, assuming it is possible to do the necessary measurements. But I have no idea what this actually entails or means. Interestingly, a simpler version of the problem has already been discussed to some extent two months ago in How can I determine whether the mass of an object is evenly distributed? Some ideas were repeated for the recent question, but new ideas emerged too (gravity field analysis) even if they have weaknesses, and are somewhat hard to use in most situations. Then, what techniques have been used to know the density distribution of Earth, the Sun or possibly other bodies. Is there anything systematic that we overlooked. So, are there means of solving the problem purely with mechanics and gravity? A related question is whether it is easier, or as hard, to determine whether the mass density distribution is uniform. More generally, I am wondering whether there is a way of characterizing the properties of phenomena that can help determine the density distribution inside a solid object. Alternatively, could one prove it is not achievable by purely mechanical means.


Alpha Beta Gamma Biological Impact

Assume that alpha & beta particles and gamma photons each reach skin at the same energy. It's known that they penetrate most deeply in order by gamma, beta and alpha. How would they compare in terms of total biological impact? In other words, which would be most harmful?


Principle of superposition for capacitor

In a capacitor the parallel plates of opposite charge create equal electric fields in opposite directions. We know field outside the capacitor is zero but inside the capacitor it is non zero. My question is if we assume zero field because of superposition of the two electric fields, why can't we do the same in between the two plates, because here also the fields are opposite to each other and hence will cancel? Basically, I want to know when we can use superposition principle and when we can't.


Proof of diffraction equations

If light incident on a grating makes an angle φ with respect to the normal to the grating, show that dsin(θ)=mλ becomes d[sin(θ-φ)+sin(φ)]=mλ. I began by showing:               d[sin(θ-φ)+sin(φ)]=mλ, θi=φ=θ at m=0      (see my note after about why I think this line is wrong) then:               d(sin(θ-θ) + sin(θ))=mλ     d(sin(0)+sin(θ))=mλ     d(0+sin(θ))=mλ     dsin(θ)=mλ      It decomposes nicely to what I'm asked for but now that I look at it again I see that I've also declared m=0 which means that:               dsin(θ)=0λ     dsin(θ)=0      which implies that either d=0 or sin=0, both of which I don't find probable. I suppose I'm stuck on where or how to introduce φ to the equation. Any suggestions welcome.


Vector Maths to Simulate Car Physics

I'm working on a 3D program that involves controlling a Cube on-screen (which represents a car). I use C++, OpenGL and GLM (for vector and matrices etc, if you're familiar with programming; Once the car rotates (or steers), I wish to make its direction change and follow the new path... ). I would like to know the equations to simulate steering (one force at a time, starting with the basic)... I've read lots about velocity vectors and multiplying by the sin/cos of the angle to find it's new path... My question is, how can this be represented using vectors? And what are the equations/vectors need to simulate basic car movement?


Does an airplane's speed include the speed of the Earth?

After take off, does airplane's speed include Earth's movement/speed? Do airplanes turn with Earth movement/rotation?


Why isn't V. A. Fabricant given the credit for the discovery of the laser?

In another question here I asked for help in finding old Russian papers in physics and as an example I gave the founding papers on the laser by Basov and Prokhorov (from 1945 and 1954). In an answer @voix was so kind as to post this link http://www.lebedev.ru/data/50laser/Beginning_of_the_Laser_Era_in_the_USSR.pdf with a collection of some of the founding papers that have shaped the area of quantum electronics and, in particular, the idea of the laser. In this collection I found with astonishment that it was not Basov and Prokhorov, neither was it Townes who discovered the laser but the discovery of reverse absorption which is at the basis of the laser action is in fact due to V. A. Fabrikant in 1939, years before the ones I have always known to be the discoverers and who got the Nobel prize for it. Fabrikant even has a 1951 patent and a diploma No. 12 for a discovery in USSR regarding the phenomenon of reverse absorption.The astonishing fact is that Basov and Prokhorov do not even cite Fabrikant in their 1954 paper. Neither do Townes et al in their 1954 paper. What phenomenon, different from what Fabrikant has discovered did Basov, Prokhorov and Townes find to deserve all the credit for the discovery of the laser and have Fabrikant sent to obscurity?


What is a proton-rich atom?

http://wiki.chemprime.chemeddl.org/images/e/e4/Plot_of_Neutron_Number_vs._Proton_Number_.jpg The above graph shows that all elements have more neutrons than protons in this nucleus. So how is there such a thing as a proton rich atom? If all the heavy unstable atoms are attempting to balance the number of protons and neutrons (this is what I was taught) why do we get protons decaying to (eventually/after intermeditary particles) neutrons (and emitting a positron?), of which there are excess?


Could the Bernoulli effect be causing my ceiling to come down?

Very 'applied' question, but I have nowhere else to turn, so I'm asking the physics experts here: I have a carport whose ceiling is made of very lightweight paneling. I've had several times now that those panels have fallen down, without an evident reason. So I'm wondering: the wind (because of the placement of other buildings in the area) blows very strong underneath the carport. Could it be that strong wind gusts, through the Bernoulli principle, are causing 'pull' on the panels and tearing them from the staples they are attached with? Stated more generally, does a fluid flowing fast beneath a surface cause a force on that surface? Thanks.


Spinning Tachyons

In all examples that I know, tachyons are described by scalar fields. I was wondering why you can't have a tachyon with spin 1. If this spinning tachyon were to condense to a vacuum, the vacuum wouldn't be Lorentz invariant---seems exotic but not a-priori inconsistent. Is there some stronger consistency requirement which rules out spinning tachyons? If someone could provide a reference that would be helpful too! Here's another confusion: I was reading Wikipedia, which claims that tachyons should be spinless and obey Fermi-Dirac statistics(?). (They reference an original paper by G. Feinberg which unfortunately I am not wealthy enough to download). The claim about Fermi-Dirac statistics is baffling---isn't the Higgs field a boson? Does anyone understand what they're talking about?


The Goos Hanchen shift mechanism

When the light is totally reflected in the interface between dense and less- dense medium, we know that the reflected beam will shift a little. Currently I have known the reflection coef r, will be a complex number and its phase angle will vary with the incident angle theta. Some paper explained this phenomenon as the light penetrates the less-dense medium a little, and re-emerge again, just like it is reflected by some virtual plane in the less-dense, but how can this be explained? Or by what mechanism?


Centrifugal force in inertial reference frame

Take for example the case of a rod rotating about an axis passing through its centre of mass and perpendicular to it. It has a ring hung from one of its sides. The rotation of the rod causes the ring to move out wards and ultimately fall off the rod. In the ground frame there is no real force acting on the body, but the ring still moves. If I am right, centrifugal force causes the motion of the ring. So does that mean that centrifugal force acts even in inertial reference frames? Also, I am finding it difficult to imagine the ring's reference frame? Is it a rotating reference frame? How exactly will u visualize motion in this frame of reference?


Constrained motion in a parabolic tube

> _A smooth parabolic tube is placed with vertex downwards in a vertical > plane. A particle slides down the tube from rest under the influence of > gravity. Prove that in any position, the reaction of the tube is equal to > $$2w({h+a})/\rho$$ where $w$ is the weight of the particle, $\rho$ the > radius of curvature of the tube, $4a$ the latus rectum and $h$ the initial > vertical height of the particle above the vertex of the tube._ I am not able to figure out how to write the equations of motion for motion in a parabolic tube. Have read constrained motion for a vertical cycloid. Is a tubular cycloid and parabolic tube the same? Please help!


Importance of phase in probability amplitude in QFT

I have started teaching myself QFT from the textbook by A. Zee. From reading that book, my idea of a path integral in field theory is the probability amplitude to go from a given field configuration to a different field configuration. At a point, he adds a source function $J(x)$ in the free field theory Lagrangian and evaluates the path integral. It turns out that $Z(J)=Ce^{iW(J)}$ where $W(J)$ is a term containing the integral of source function and the propagator. Then, he says > The overall factor C does not depend on $J$ and, as will become clear in the > discussion to follow, is often of no interest to us. Then, he goes on to consider $W(J)$ for some special localized functions $J$ and that leads to the interpretation of particles in field theory and a derivation of inverse square law. But, I am quite disturbed by this. QM teaches us that only probabilities are something that are physical and the absolute phase factors in the probability amplitude are of no interest to us; that we should just concentrate on $C$ above. Instead, Zee says that it is of no importance to us and such spectacular stuff comes out just by examining the phases. So, why is it that phase is so important in QFT? Why did people even try to analyze the phase and interpret the results in terms of particles and forces that can physically be observed?


Second Law of Black Hole Thermodynamics

I've been looking for a satisfying proof of this, and can't quite find it. I read the brief proof of the black hole area theorem in Wald, which is similar, but doesn't quite come down to the actual statement $\frac{dA}{dt} \geq 0$. I had pictured something fairly succinct relating $T_{ab}k^ak^b \geq 0$ (weak energy condition) directly to $\frac{dA}{dt}$, and I tried but I couldn't get anything to quite work out. Anyway- can anyone point me to a good source, or even just give the proof?


a priori validity of $W=\int Fdx$ in relativity?

There are lots of different ways of arriving at the relativistic relations involving mass, energy, and momentum such as $E=mc^2$ and $m^2=E^2-p^2$ (the latter with $c=1$). One that I've seen in some textbooks is to start with the equation $W=\int F dx$ for mechanical work. Here is a particularly careful and rigorous version, which makes it explicit that this is a nontrivial assumption, along with $F=dp/dt$: http://www.physicsforums.com/showthread.php?p=2416765 Historically, Einstein made use of $W=\int F dx$ in section 10 of his 1905 paper on SR http://www.fourmilab.ch/etexts/einstein/specrel/www/ . This was before the "Does the inertia of a body..." paper, in which he derived $E=mc^2$ and gave it its full relativistic interpretation. Einstein later decided that force wasn't a very useful concept in relativity and stopped appealing to it. What I have never seen in any of these treatments (neither the careful ones above nor the sloppy ones in some freshman physics texts) is any argument as to why the nonrelativistic relation $W=\int F dx$ should be expected _a priori_ to hold without modification in SR. If one has _already_ established the form and conservation of the energy-momentum four-vector, then I don't think it's particularly difficult to show that $W=\int F dx$. But what justification is there for assuming $W=\int F dx$ _before_ any of that has been established? Is there any coherent justification for it? (A secondary problem with some of these treatments based on $W=\int F dx$ is that they need to establish the constant of integration.)


Does wavefunction reach its largest peak near(not in) the classical forbidden region?

As we can see in the picture in this website: http://ctz116.ust.hk/xyli2/images/animation/quchem73.html It's strange that the bound state wavefunction always reach its largest peak near the boundary of its classically forbidden region(not in the region). Is it true that this phenomenom holds for all bound state wavefunction? I think that the reflected wave may interfere with the original one, thus creating the peak near the forbidden region.But I can't explain why it is the largest peak or there is no peak inside the classically forbidden region,Thanks for your attention.


Problem with a rotating frame of reference on the South pole

Consider this problem: > _A high-speed train is traveling at a constant 150 m/s (about 300 mph) on a > straight horizontal track across the south pole. Find the angle between a > plumb line suspended from the ceiling inside the train and another inside a > hut on the ground. In what direction is the plumb line on the train > deflected?_ I'm positively perplexed by it. I think that what we're looking for is a relationship $\tan(\theta)=\frac{a_n}{a_t}$, because for the plumb line in the hut $a_t=1$ but not for the plumb line in the train due to the Coriolis force. However,   1. What is the frame of reference here? Is it rotating, fixed so that the train is standing still? Is it rotating with the earth?   2. Since the train is on the south pole, isn't that the same to say that the earth's rotation doesn't affect it?   3. Can you show me how to get the right answer? The right answer is supposed to be 0.13 degrees. In which direction?


Lower limit of the size of the Universe? (WMAP)

The measurement of the WMAP satellite resulted a planar geometry of the universe with a 0.4% uncertainity (http://en.wikipedia.org/wiki/Shape_of_the_universe). If there is a little deviation from the measured zero curvature, I think it could give a lower limit to the size of the universe (in case of positive curveture and spherical geometry). How big is it? How could it be calculated?


How can a car's engine move the car?

Newton's First Law of Motion states that an object at rest or uniform motion tends to stay in that state of motion unless an unbalanced, external force acts on it. Say if I were in a car and I push it from the inside. It won't move. So how is the engine of a car capable of moving the car?


Human and Ultraviolet light

As I know, ultraviolet light can be created from burning something at high temperature. So I have a question: Can a human body become a source of ultraviolet light at high temperature? If the answer is yes, at what temperature does this happen in a human?


Finding Hamilton's equations given a Hamiltonian

I am trying to find Hamilton's equations for a general Hamiltonian given by $$H[u]=\int_\mathbf{R} \phi(u,u_x)dx$$ Suppose $$\frac{\delta f[u]}{\delta u(x)}\equiv \frac{\partial f}{\partial u}-\frac{\partial }{\partial x}\frac{\partial f}{\partial u_x}+\left(\frac{\partial }{\partial x}\right)^2\frac{\partial f}{\partial u_{xx}}+...$$ then Hamilton's equations are given by $$\frac{\partial u}{\partial t}=\\{u,H[u]\\}=\frac{\partial }{\partial x}\frac{\delta H[u]}{\delta u(x)}$$ But I don't know how to evaluate the last quantity. Would someone mind teaching me, please? * * * My guess would be $$\frac{\partial }{\partial x}\int_\mathbf{R} \frac{\partial \phi}{\partial u}-\frac{\partial }{\partial x}\frac{\partial \phi}{\partial u_x}dx$$ Then $$\left[\frac{\partial \phi}{\partial u}-\frac{\partial }{\partial x}\frac{\partial \phi}{\partial u_x}\right]_{-\infty}^{\infty}$$ Would that be right?


Estimating the Kolmogorov Complexity of the Standard Model

The Kolmogorov complexity of a hypothesis / theory / model is the shortest computer program that simulates it, regardless of how inefficient executing that program may be in terms of memory and time. I'm interested in how complex the standard model is, by this measure. For example, this MinutePhysics video notes that the standard model is (almost) one equation. That's pretty short (less than 50 characters), but of course in order to turn it into a computer program you also need to encode how to perform the underlying math. On the other end of the spectrum: teaching a human physics via text books can be done with millions of characters, but the majority of that "millions" is presumably due to the constraints of communicating to a human. I guess I expect the answer to be less than a million bytes, and _maaaaaaybe_ less than a kilobyte, but that's not really based on much except intuition. I searched google, and google scholar, and was surprised to fail to quickly find even a loose upper bound on the complexity of the known laws of physics. Has such an estimate been done? How difficult is it to do one? _How complicated is the standard model?_


Challenge: Answer this gedanken (PFP - Perpendicularly Fired Photon)

I'm challenging anyone who can answer the following question objectively: As usual, imagine a railway station and trains which are equipped with single photon sources (one each in the platform and inside the trains) and a narrow strip of detector which is positioned right above the photon sources. The photons are fired vertically upwards, perpendicular to the train’s direction of travel. In addition, let us assume that apart from the photo detectors on the train and the platform, the roof of the train is open, and that the ceiling of the station above the train also has photo detectors, corresponding to the position of the photo detectors on the platform. Now, the photon sources in the station and the trains are set to trigger at the same moment – i.e. when the train’s photon source passes in front of **Alice**. Now, **Bob** is sitting in front of the photon source in the train which is travelling at approximately half the speed of light, towards the station. When the train in which **Bob** travels crosses **Alice** , both photon sources emit a photon (on platform and on the train). Hence, one photon is emitted towards the roof of the railway station by the photon source stationed in the platform; another is emitted inside the train above which is the photo detector on the train (moving w.r.t. Alice) and the photo detector on the station’s roof (stationary w.r.t. Alice) above the train. Which means, **Alice** will (? might) record two photons hitting the photo detectors on the ceiling of the station (the photon from the platform and the photon from the train). The photon emanating from inside the train should hit the station’s ceiling since the train’s detector would have moved away by the time the photon travels upwards. Now from **Bob’s** view point, the photon emitted inside the train will be detected by the sensor IN the train; the photon emitted by the emitter on the platform will miss the detector on top of it (since the station is moving relative to Bob’s train). Either way, SR would not work for both observers at the same time. In case you are thinking that there is some trickery with which this question can be answered in a manner similar to the barn door paradox, let Charlie travel in the track parallel to Bob, in the opposite direction (both Bob and Charlie are equidistant from Alice’s view point and are approaching with the same speed). Now we will have three different observers with different predictions; but due to law of conservation of energy, the photons can be detected by only one of the sensors, which should settle the confusion (or if they do separate, we would be able to tell because of the lesser energy liberated by the photon). In general, the experiment suggests that depending on the relative speed of the train to the station, there are an infinite number of positions in which the photon can be found, which could either support the many worlds theory, or is at least at odds with the Special relativity. I have been digging around this for almost 13 months; still no solution. Any ideas? Would theory of General relativity help? Should I rephrase my question -or- add details?


How do particles become entangled?

A person asked me this and I'm just a lowly physical chemist. I used a classical analogy (how good or bad is this and how to fix?) Basically, light has a net angular momentum of zero, insofar as it is not polarized into its left and right plane polarized forms until it hits a crystalline structure. However, once it does hit such a structure, we have left and right plane polarized light--that is left and right photon beams. Since the original light was not polarized, this polarization (left right) be inherent in the light. The original light is a superposition of left and right polarized light, each with a total angular momentum of -1 and 1, so that they result in the total zero polarized incident light. Thuse they are entangled to lower their spin (quantum angular momentum). Once we measure one of the particles in the superposition, we know the other by conservation of angular momentum. Is this close?


Minimum atomic clearance permitting motion

Suppose you were to build the piston and cylinder in a car engine atom-by- atom. Let's just say carbon, since you can make a lot of different shapes due to it's high valence. So assuming you make the most perfect piston possible, a perfect cylinder accurate to within an atom; and you make a perfect cylinder wall to go around the piston, again accurate to within an atom; and the diameter of piston has a single atom clearance from the cylinder wall. What would happen if you tried to move the piston up and down inside the cylinder? Would there be significant force due to the Casimir effect? Would there be friction? The two objects aren't strictly touching, but then again the atoms in my fingers aren't strictly touching the atoms in the keys on my keyboard. It's all just the electro-magnetic force. What is the minimum distance required between the piston and cylinder to allow smooth motion with no friction? (Assume we have a way of moving the piston up and down perfectly along the common axis, and it's not "wobbling" about)


Grover algorithm $R_D$ Circuit

I need sketch two circuits to understand Grover algorithm. The first is the operator $R_f$ and another is the operator $R_D = H^{\otimes n}(2|0\rangle\langle0|-I)H^{\otimes n}$. I get the first operator. How I will be able to sketch the operator $R_D$ using Toffoli and Hadamard gates? I need a hint to solve this exercise please


What is the current rate of change of the volume of the Universe?

Is it possible to know the current rate of change of the volume of the Universe? If we can't see past the cosmic horizon, can we still get a value for this rate of change, and if so, how would it be calculated?


covarient derivative of electromagnetic field tensor

I'm trying to prove the energy momentum tensor in curved spacetime for Electromagnetic field is **Divergence-less** directly(Without using general lie derivative method which can prove **any** energy momentum tensor obtained from varying $g^{ab}$ . From free lagrangian $$ L = -1/4 F_{ab}F_{cd}g^{ac}g^{bd} $$ We get $$ T_{ab} = 1/4(F_{ac}F_{bd}g^{cd} -1/4g_{ab}F_{mk}F_{nl}g^{mn}g^{kl})$$ Now what we have in hand is $$\nabla_aF^{ab} = 0$$ But To prove $\nabla_aT^{ab} = 0 $ from the above expression of $T_{ab}$ We have to compute stuff like $\nabla_aF_{bc} $ But I'm actually getting no where, i have started like following: $$ \nabla_a F_{bc}$$ $$ = g_{bn}g_{cn}\nabla_a F^{bc}$$ $$= g_{bn}g_{cn}( \partial_aF^{bc} + \Gamma^b_{af}F^{cf} + \Gamma^c_{af}F^{bf})$$ From the divergence relation I can get $$ \partial_aF^{ab} + \Gamma^a_{af}F^{bf} + \Gamma^b_{af}F^{af} = 0 $$ $$ \Rightarrow \partial_aF^{ab} + \Gamma^a_{af}F^{bf} = 0$$ What should i do from here?


Does an Ising lattice that returns to equilibrium create a current by induction?

Consider you have an Ising lattice with a dominant up component out of thermal equilibrium, that's your initial state. The down spins want to flip up and align with the ups, and they'll do so until a thermal distribution is reached. While they flip, the overall magnetization increases. Does that create a current by induction? I haven't been able to find a good reference.


Two dimensional elastic collisions with varying angle of incident

If in an elastic collision I know all initial values and that mass for each object remains constant throughout the collision (but different from one another) how can I determine their final velocity vectors when the angle of incident is also a variable? I've tried decomposing the vectors and have determined that for perpendicular collisions the following formula will work: $$v_1=\frac{u_1(m_1−m_2)+2m_2u_2}{m_1+m_2}.$$ Will this work for variable angles as well?


What are natural sources of longwave radiation

I am studying now the Earth radiation balance and I came across the question regarding the sources of the long-wave radiation on earth surface. The only source of theses wavelengths indicated in the book are clouds, which have high reflectivity and reflect a part of the long-wave radiation. I am looking for some other sources of it. I was thinking that CO2 can be another source because of this greenhouse effect. Are there any other major global sources which can bring long-wave radiation into earth system>


What exactly is $\hat{\psi}^\dagger(x)$? An operator or a function?

I've recently read Cohen-Tannoudji on quantum mechanics to try to better understand Dirac notation. A homework problem is giving me some trouble though. I'm unsure if I've learned enough yet to understand this. The creation operator is defined as: $$\hat{a}_k^\dagger = (2 \pi)^{-1/2}\int dx\,e^{ikx} \hat{\psi}^\dagger(x)$$ From my notes, I have $\hat{\psi}(x)$ defined (in one dimension) as: $$\hat{\psi}^\dagger(x) \lvert 0\rangle = \lvert x\rangle $$   * Cohen-Tannoudji says linear operators take a ket vector and associate them with another ket vector within $\mathscr{E}$ space. $\hat{\psi}^\dagger(x)$ does not look like a linear operator to me. It looks like an operator-function hybrid. What do you call this thing? How is it rigorously defined? Why is $x$ not a subscript? Assuming it's a linear operator and proceeding anyway, let's apply $\hat{a}_k^\dagger$ to some ket vector: $$\hat{a}_k^\dagger = \left( (2 \pi)^{-1/2}\int dx\,e^{ikx} \hat{\psi}^\dagger(x) \right)\lvert\phi\rangle$$ I assume the order of operation is to apply $\hat{\psi}^\dagger(x)$ first to $\lvert\phi\rangle$, then multiply by $e^{ikx}$, then integrate, then multiply by $(2\pi)^{-1/2}$   * How do I integrate over the ket vector $e^{ikx}\hat{\psi}^\dagger(x)\lvert\phi\rangle$? Such an operation does not seem to be defined. I can integrate over complex numbers, but not kets. I still don't understand why $k$ is a subscript but $x$ is an operator-function-hybrid variable. What am I missing here?


What happens if the energy has no lower bound?

In almost all theories that we investigate there is an lower bound on the energy. But is this necessary for a theory to be a good description of reality? I know from quantum field theory, that in its early days there was the idea of a Dirac sea of negative energy particles. This interpretation was replaced and is not used today, was this also due to the fact that there has to be a lower bound to the energy?


Why does gravity assist transfer twice the planet's velocity?

In orbital mechanics and aerospace engineering a gravitational slingshot (also known as gravity assist manoeuver or swing-by) is the use of the relative movement and gravity of a planet or other celestial body to alter the path and speed of a spacecraft, typically in order to save propellant, time, and expense. Gravity assistance can be used to accelerate (both positively and negatively) and/or re-direct the path of spacecraft. Over-simplified example of gravitational slingshot: the spacecraft's velocity changes by up to twice the planet's velocity. ![SHOOOT](http://upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Gravitational_slingshot.svg/200px- Gravitational_slingshot.svg.png) Where does the factor of $2$ come from in this relation?


How can I physically interpret voltage and current?

My understanding of an electrical current is that it is the flow of electrons through a material. The only magnitude I can fathom for this process is the number of electrons which are flowing. I know ohms law, and how some of the basic components of a circuit work (resistors and capacitors, specifically). If there is a circuit attached to a 120V power source, the voltage at the beginning of the circuit will be 120V, and at the end of it will be 0V. When the current experiences resistance, voltage is lost, so more current has to be supplied to the circuit so that exactly all of the voltage is lost by the end of the circuit. My question is: what is happening when voltage is lost? When current is increased? When learning about capacitors in university, I understood that voltage is built up on each plate by electrons flowing in and spreading out across the surface of the plate. But if voltage is the number of electrons in a circuit, what is current? And if all voltage is dropped at the end of a circuit, where do the electrons go? I figured that since electrons won't build up ad infinitum throughout the circuit (obviously since everything would build up an increasingly negative charge), the magnitude of the current must be the number of the electrons. But then what is voltage? What is lost? How should I interpret the stored charge on a capacitor?


Symmetry factor of tree diagram

In Mark Srednicki's Quantum field theory(page 89) it says > This is a general result for _tree diagrams_ (those with no closed loops): > once the sources have been stripped oﬀ and the endpoints labeled, each > diagram with a distinct endpoint labeling has an overall symmetry factor of > one. The tree diagrams for a given process represent the lowest-order (in g) > nonzero contribution to that process. So the question is why the symmetry factor of the tree diagram is one with each endpoint labeled? Could you kindly provided a proof or some links including the proof?


Laws of Motion: Acceleration to be applied on a free falling object to reduce velocity to 0

Assume that an object of mass 'm' is falling to the earth.      The force acting on the same would be F = m*g = 10m (assuming g = 10m/s^2).   In this case the velocity of the object at time 't' = 10 seconds will be 100 m/s (v = u + at).   At this point, if an opposing force of F' = -10m is applied on the falling object, how long would it take for the velocity of the falling object to reduce to 0?      Would it be valid to say that the opposing force of F' = -10m has to be applied for time 't' = 10 seconds to reduce the velocity of the falling object to 0?


Acceleration: Value Disparity?

If we consider a ball moving at an acceleration of $5ms^{-2}$, over a time of 4 seconds, the distance covered by the ball in the first second is $5m$. In the 2nd second will $5 + 5 = 10m$. In the third second it will cover a distance of $5 + 5 + 5 = 15m$ and so on and so forth. Now, when we substitute this answer in the equations of motion derived from the area under velocity- time and distance-time graphs, we see a variation: $$s = 1/2at^2$$ $$ = 1/2*5*4^2$$ $$ = 1/2*5*16$$ = 40 metres is the distance covered. Now if we go back to our initial description of acceleration we that in the 1st sec = 5m 2nd sec = 10 m 3rd sec = 15 sec 4 sec = 20 sec. Total distance covered in this case is 5 + 10 + 15 + 20 = 50 metres? 40 != 50? Why this disparity between the values? Can someone please explain?!


Simple circuit with resistances

I have a question about a circuit that I made in class! So, we had a circuit with 2 resistors, one of $R_1 = 330 \Omega$ and one of $R_2 = 470 \Omega$. The tension applied to the circuit should be $10V$ but we applied $9.99V(V_f)$. When measuring the resistances with a multimeter the value for the $R_1$ was $328 \Omega$ and for the $R_2$ was $470 \Omega$. The value for both $R_1$ and $R_2$ in series was $R_1 + R_2 = 799 \Omega$. The tension measured in $R_1$ was $4.02V(V_{r1})$ and in $R_2$ was $5.74V(V_{r2})$ The intensity of the current ($I$) was $12.34 mA$ With that said, we were asked to calculate $R_1$, $R_2$ and $R_1+R_2$ theoretically using the values of $V_{r1}$, $V_{r2}$ and $V_f$. I used Ohm Law: $R = \frac VI$ $R_1 = 326 \Omega$ and $R_2 = 465 \Omega$ **My question is: should be $R_1+R_2$ be calculate using $V_f (9.99/0.01234)$ as it was asked or calculating an equivalent resistance? By the first method the result is $810$ and be the second is $791$. The ammeter's resistance is $11 \Omega$ at the used scale $(20mA)$.** **I'm asking cause I found it weird that $810$ is bigger than $799$(experimental value) given that the experimental should be bigger because of ammeter's resistance.** Sorry if the question is dumb, but I'm not a Physics student though I attend Physics classes.


Action principle for the Electromagnetism and Gravity

Here is the formula for the stress energy tensor: $$ T_{\mu\nu} = - {2\over\sqrt{ |\det g| }}{\delta S_{EM}\over \delta g^{\mu\nu}} $$ (This follows from varying the total action $S = S_H + S_{EM}$, where $S_H={c^4\over 16\pi G} \int R \sqrt{ |\det g_{\mu\nu}| } d^4 x$ is the Hilbert action and it gives the Einstein's equations, and $S_{EM}$ are other terms in the Lagrangian, that contribute to the right hand side of the Einstein's equations in form of the $T_{\mu\nu}$ above.) The Lagrangian of the electromagnetic field is: $$ S_{EM1} = -\int {1\over 4\mu_0} F_{\alpha\beta} F^{\alpha\beta} \sqrt{ |\det g| } d^4 x $$ and using the formula above, we get for the stress energy tensor: $$ T_{\mu\nu} = {1\over \mu_0} \left( F_{\mu\beta} F_\nu{}^\beta -{1\over 4} F_{\alpha\beta} F^{\alpha\beta} g_{\mu\nu} \right) $$ which is the correct elmag. stress energy tensor. However, the interaction part of the elmag. Lagrangian is $$ S_{EM2} = -\int j_\mu A^\mu \sqrt{ |\det g| } d^4 x $$ and if we interpret this as a function of $g^{\mu\nu}$ it would also contribute to the stress energy tensor like this: $$ \delta S_{EM2} = -\delta\int j_\mu A^\mu \sqrt{ |\det g| } d^4 x =-\int \delta (g^{\mu\nu} j_\mu A_\nu \sqrt{ |\det g| }) d^4 x = $$ $$ =-\int (\delta g^{\mu\nu}) (j_\mu \sqrt{ |\det g| })) A_\nu d^4 x $$ where the $j_\mu \sqrt{ |\det g| }$ is treated as the current _density_ (and thus not depending on $g^{\mu\nu}$ when varying), however, clearly the stress energy tensor corresponding to this would be: $$ T_{\mu\nu} = 2 j_\mu A_\nu $$ (Possibly only the symmetric part contributes, because the antisymmetric part cancels with $g^{\mu\nu}$, so we would get $T_{\mu\nu} = j_\mu A_\nu + j_\nu A_\mu$.) In either case, such terms should then appear on the right hand side of the Einstein's equations. However, I don't think that this is correct. Does anybody know what is wrong here?


Guessing what a simple partial differential equation is describing physically

Is there an easy way to look at a partial different equation and get a sense of what kind of phenomena it is physically describing? I have an equation that looks like this: $\partial_tA=C_3\partial^2_{xx}A+C_2\partial_x\left(vA\right)+C_1A+C_0$ From inspecting it and doing some searching, it seemed like it might be trying to describe a phenomena that had advection and diffusion occurring at the same time. Is this a correct interpretation? Note that $C_3>0$, $C_2<0$, $C_1<0$, $C_0>0$ and $v(x,t)$ is velocity. I'm not sure, but it seems reasonable to assume velocity can be either negative or positive. Also, $A(x,t)$ should be positive. Note that I'm not looking for a solution to the equation, just the physical interpretation of what kind of physics would be involved.


Meaning of the word "Moment"?

This question is more of a question about the origin of a physical term moment used in many contexts. My question is about the linguistic or historical meaning of the word "moment". Please don't provide the mathematical equations for the moment of inertia or the electric of magnetic dipole moment. My question is about the essence of the word. I'm asking this question because as a non-native English speaker, I find it confusing as how did this word got to be used in the different contexts.


Is scattering cross section determined by mass?

This is just for visible light/photon scattering. Let's say I have small spherical particles in water, if I keep the weight concentration of glass the same but change the diameter of glass sphere from 0.5$\mu$m to 1$\mu$m, how will the scattering power, or the total scattering cross section change for visible light, e.g., at 632nm? I was told as long as the mass doesn't change, the scattering cross section won't change, the size of diameter only modifies the phase function of the scatter. is that true?


Thought Experiment: Force on magnets in a Stern Gerlach Experiment

Background: In the SG experiment, an inhomogenous magnetic field affects a force on particles passing between two magnets. "Measurement" takes place when a screen is placed on one end, blocking one of the states. The final measurement we perform (spin state of particles exiting) is heavily dependent on whether we perform a measurement or not, whether by blocking particles exiting the device or by blocking particles by setting up a screen within a modified SG device ![Modified SG device](http://i.stack.imgur.com/motIz.png) The interaction between the particles and magnets should also exert a force on the magnets, causing them to move slightly. The motion of these magnets seems to not affect the outcome of the experiment significantly, yet that would imply that the motion of the magnets could be measured without affecting the outcome of the experiment. This seems at odds with Heisenberg's uncertainty principle, that we are getting information for free. My Professor tells me that the force exerted on the magnets by the particles is so minuscule that it cannot be measured (I assume it is similar to how we can't measure particles' momentum and position simultaneously, because any photons we shoot at the particle will end up imparting change?) What's going on here? I wanted to double check here and see where I went wrong with my assumptions.


Do I need to take elastic potential energy into account?

> Let's say I have a vertical spring with end points $A$ and $B$ and length > $a$ and a mass attached to the endpoint $B$. The mass is dropped from the > point $A$ and I need to find the kinetic energy of the mass at a point $ak$ > where $k>1$. Do I need to include the Elastic Potential Energy to find the kinetic energy? If yes, why can't I just use the Kinetic Energy and Gravitational Potential Energy? I know $\text{Work Done}=mgka$ by the particle and the change in GPE is $mg(ka-a)$ but I don't get what else am I missing?


Why is an electric and magnetic field present in all universe?

the Earth has its fields due to the motion of molten metals in the core but how does this whole universe has its fields without any discontinuity? And we can say for sure that fields are present in whole universe as we can see light traveling thousands of light year and as light is an EM wave it can't travel without the presence of fields.


Observations used to derive orbital and timing information required for GPS satellites

Although I have studied GPS receiver position finding procedure in some detail, I am struggling to find enough information to allow me to deduce how the essential highly accurate orbital and time information required for each GPS satellite is actually obtained. I can only surmise that the information is somehow slowly and iteratively derived, for each satellite, from all the satellite signals received (probably receiver time-tagged) by the earthbound GPS monitoring stations, that the position information is resolved in an earth-centred inertial frame-of-reference (ECIF) and that the time information (in the ECIF) is synchronized to that from the US Navy master time clock value referred to the earth's geoid. I realise that relativistic corrections are also required at the receiver for orbit ellipticity and to refer the satellite positions to the earth-centred, earth-fixed (ECEF) WGS84 frame of reference. I am particularly wondering whether any other types of observations of the GPS satellites are also involved and I'm hoping someone could kindly give me a rigorous-descriptive rather than mathematical explanation.


Invariance of a tensor under coordinate transformation

I know, that a tensor is a mathematically entity that is represented using a basis and tensor products, in the form of a matrix, and changing a representation doesn't change a tensor, is kind of obvious. So does the invariance of a tensor under coordinate transformation mean what I stated above or does it mean that under a set of particular transformation the representation of a particular tensor also doesn't change. Quoted from Wikipedia: > _A vector is invariant under any change of basis, so if coordinates > transform according to a transformation matrix $L$, the bases transform > according to the matrix inverse $L^{−1}$, and conversely if the coordinates > transform according to inverse $L^{−1}$, the bases transform according to > the matrix $ L$._ Can someone please shed some light on this?


Is there any idea why the electric charges of electron and muon are equal?

Is there any idea explaining why the electric charges of electron and muon are equal? Edit: The total charge of a particle is proportional to the integral of its own electric field flow through the sphere of a big radius surrounding the particle at rest. The free Dirac equation describes charged fermion. It contains the mass term $m$. If $m$ tends to zero, Dirac equation tends to the pair of Weil equations that describe electrically neutral particles. Does it mean that charge somehow depends on mass? If yes, why do the electron and muon (both described by Dirac equation, but with different mass terms) have the same electric charges?


Statics software for structural engineering

I'm attempting to expand my knowledge of engineering software. I've found comsol and ansys for acoustics and thermodynamics/fluid dynamics (not necessarily in that order), now I'd like to see if I can find something for architectural statics/load calculations. Which packages are in common industry use?


Newton's third law of motion, and the collapse of objects under heavier weight

Suppose a table is placed on earth. The weight of the table exerts a force on the earth say $10N$, then the earth will also exert a force of $10N$ on the table. If we place a truck on the table weighing $4000N$, then the total force exerted by the table and the truck is $4010N$ on the earth. The earth would also exert a force of $4010N$ on the table. Therefore, the net force on the table is $10N$ and the table colapses. Is this what happens to a table scientifically when it gets broken by something having a relatively a much greater weight? Whereas, my textbook says: > Consider a book of weight $W$ which is placed on a horizontal table, it > pushes the table with a force in downward direction in response to this > action the table also pushes the book in the upward direction the resultant > force acting the book must be zero that is, > > $F = 0 => F = 0$ > > Two forces are acting on the book, > > (a) the force of gravity acting downward which we call the weight of book. > > (b) the other force is the push of table on the book in the upward > direction. > > These two forces are equal and opposite to keep the book in the stationary > position. If this is so, then the truck should also have remained in the stationary position. However, what we normally see is that the truck will crush the table.


How can I create hindrances to radio waves?


What is the equation for the scale factor of the universe, a(t), for the best fit of data to the $\Lambda CDM$ Model of Cosmology?

Ideally I like a single formula or multiple formulas for different time ranges that would cover the time from the end of inflation through 100+ billion years after the big bang using the $\Lambda CDM$ Model. I know that from the end of inflation back to the time of the big bang would be much more speculative, but some wild estimate would be appreciated for that time range also! The Friedmann–Lemaître–Robertson–Walker metric defines the scale factor, a(t), from the metric: $$-c^2d\tau^2 = -c^2dt^2+a(t)^2d\Sigma^2$$ where $d\Sigma^2$ ranges over the 3 dimensional space of the universe and does not depend on time. Usually the scale of the scale factor is set by defining $a(t_{now}) = a(13.78 B yr) \equiv 1$


Why does breaking one bulb on a string of Christmas Lights cause all the rest to break in some lights?

If I break one bulb on a string of Christmas lights, or the filament goes out, then all of the rest seem to stop working. Why is this? However, I've also seen the opposite happen in some newer lights -- when one breaks, the others stay on. What makes this one different than the others?


About the work done by gravity to pull the object down to the earth surface

The following question is from an physics exercise and I know the answer and the way to solve the problem but just curious why my own way doesn't work. The question is asking "A 2500kg space vehicle, initially at rest, falls vertically from a height of 3500km above the Earth's surface. Determine how much work is done by the force of gravity in bringing the vehicle to the Earth's surface." The way to solve that is to integrate the differential work from the force $F_g = GM_\text{earth}m/r^2$ such that $$W = \int_{6400 \times 10^3}^{(6400+3500)\times 10^3}F_g dr$$. It does solve the problem but my question is why can't I use the $mgh$ to estimate the work done by gravity?


Ask for help: the Reaction-Diffusion related problem

In container there are two kinds of molecules A and B which are distributed uniformly. Initially the quantities of A and B are $N_A$ and $N_B$ respectively A and B are distributed uniform, i.e., in any small space (of cause the space should be large enought that there are thousands of molecules in the space), the ratio of number A and B is equal to the ratio of total number of A and B. Assume the molecules are fixed all the time (i.e., we don't consider diffusion) and in every peroid, there is at most one molecule $B_j$ around molecule $A_i (i=1,2,...,N_A)$ be transformed to molecule A: if the molecule $B_j$ is on the boundary/neighbor of molecule $A_i$, the molecule $B_j$ would be transformed to A, otherwise (i.e., all molecules neighboring $A_i$ are molecule A) no molecule around $A_i$ be transformed. In addition, if there are several $A_k$ and $B_j$ on the neighbor of molecule $A_i$, the closest $B_j$ would be transformed to A. Now the problem is: Is there any analytical expression to show how many molecules (approximately) would be transformed to A in every peroid until all molecule B are transformed to molecule A? Thanks, Tang Laoya


Spin matrix for various spacetime fields

Let $V^{\mu}$ be a vector field defined in a Minkowski spacetime and suppose it transforms under a Lorentz transformation $V'^{\mu} = \Lambda^{\mu}_{\,\,\,\nu}V^{\nu}$. We can write this like $V'^{\mu} = (e^{i\omega})^{\mu}_{\,\,\,\nu}V^{\nu}$ I think where $\omega$ denotes a rotation in some plane spanned by indices $\left\\{\rho \sigma\right\\}$, say. In 2D Euclidean space time, we can write the matrix representation of $\Lambda$ as $$\begin{pmatrix} \cos \omega & \sin \omega\\\\-\sin \omega&\cos \omega\end{pmatrix}$$ and in Minkowski space this changes to the 'hyperbolic' rotation. Linearising the above yields $$\begin{pmatrix}1&\omega\\\\-\omega&1\end{pmatrix} = \text{Id} + \begin{pmatrix} 0&\omega\\\\-\omega&0\end{pmatrix} = \text{Id} + \omega \begin{pmatrix} 0&1\\\\-1&0\end{pmatrix}$$ Now compare with the more general treatment: $V'^{\mu} = \Lambda^{\mu}_{\,\,\,\nu}V^{\nu} \approx (\delta^{\mu}_{\nu} + \omega^{\mu}_{\,\,\,\nu})V^{\nu}$, where $\omega^{\mu}_{\,\,\,\nu} \equiv (\omega^{\rho \sigma} S_{\rho \sigma})^{\mu}_{\,\,\,\nu}$ In 2D, the spin matrix $S$ when acting on vectors in 2D Euclidean space time is therefore the matrix multiplying $\omega$ above, which agrees with the single generator of the SO(2) group. If we continue with the general analysis, we obtain $V'^{\mu} - V^{\mu} = \omega^{\mu}_{\,\,\,\nu}V^{\nu} = \eta^{\mu \rho}\omega_{\rho \nu} V^{\nu} = \omega^{\rho \sigma} \delta^{\mu}_{\rho} \eta_{\sigma \nu} V^{\nu}$ Now use the antisymmetry of $\omega_{\rho \sigma}$ gives $$2(V'^{\mu} - V^{\mu}) = \omega^{\rho \sigma}(\delta^{\mu}_{\rho} \eta_{\sigma \nu} - \delta^{\mu}_{\sigma} \eta_{\rho \nu})$$ from which we can identify $S$ to be $\delta^{\mu}_{\rho} \eta_{\sigma \nu} - \delta^{\mu}_{\sigma} \eta_{\rho \nu}$. I am wondering how this agrees with the matrix I obtained above. Many thanks.


Why must phonon-phonon interaction be anharmonic?

I'm learning about phonons and this is really interesting. Why can't I describe two phonons colliding as a harmonic oscillation? Why does it have to be anharmonic for the phonon model to work?


Edge states at high magnetic field (Quantum ballistic transport)

I am learning about edge states at high magnetic field (Quantum ballistic sample). What I understand so far from it is that at high magnetic field, Landau states arise. Then (I don't know exactly why?), edge states are formed. To enhance my understanding, can someone help me with these questions: *Why must the chemical potential cross a Landau level? *Why is the net current carried solely by these edge states? (I am only interested in intuition, no derivations)


Where are the time dilatational effects of orbital motion and gravitational acceleration equal?

Nearly four years ago, upon hearing of the observation of time dilation in two optical atomic clocks at an elevation one metre apart, due to acceleration towards earths centre of gravity by Chou, C. W.; Hume, D. B.; Rosenband, T.; Wineland, D. J. I wondered where the point would be, at which time dilation caused by acceleration toward the earth gives way to time dilation caused by acceleration around the earth, via orbital velocity. (I mean the point at which the two effects are at equilibrium with each other.) I asked a friend who is a physicist, he got back to me almost immediately, fascinated by his back of an envelope calculations that this point is the Clarke orbit, or the geosynchronous orbit. If for instance the rotation velocity of earth were a slightly different speed, the result would be different, and no relationship apparent. I think there may be a very interesting relationship waiting to be revealed. Is there a physical or mathematical relationship? And if so, what is it? Not being a mathematician (I'm an artist) I'm not capable of doing the math, but this has fascinated me ever since.


Curious about this perpetuum concept

I know it's not possible to build a perpetual motion machine, but I still got this concept running in my mind.   If you take a (longer)copper coil, and insert an axis through it(right-angled) with a magnet fixed on it,   it would generate electricity when you spin the axis. The axis and magnet rotate, thus alteration of the flux, thus electricity. Of course normally there is a soft-iron core inserted in the coil to achieve better efficiency,   but it also brakes the movement. If you don't use a core and insert the axis+magnet into the coil,   you'd still generate a certain amount of electricity. The magnet is **only** braked by the friction of the axis and the air, but this friction is almost insignificant. So if you use good bearings to reduce this friction to a minimum, and you take a coil with as much as possible windings, and the strongest magnet you can find, you could _in theory_ build a perpetuum mobile right? I'm almost sure I'm not right but I'm asking anyway. Just curious.   Any comments or responses are welcome.


What does the appearance of a classical particle fundamentally reduce to?

I've been reading an article that describes what seems to be a classical particle as a regularity in the global wavefunction over a quantum configuration space: > When you actually see an electron trapped in a little electron trap, what's > really going on, is that the cloud of amplitude distribution that includes > you and your observed universe, can at least roughly factorize into a > subspace that corresponds to that little electron, and a subspace that > corresponds to everything else in the universe. So that the physically real > amplitude distribution is roughly the product of a little blob of amplitude- > factor in the subspace for that electron, and the amplitude-factor for > everything else in the universe. ...which looks sensible enough, but I cannot figure out what is actually meant by this configuration space; specifically, what is the identity of an individual point in that space? A simplified picture - mentioned in the above article - tells us that an individual point corresponds to something like "a photon here, a photon there (etc., for every photon), an electron here, an electron there, etc. for every species of elementary particles". But this can't be quite right: if an electron corresponds to a blob of amplitude in the configuration space, a single point in that space can't correspond to a position of this blob... and besides, what would a dimension of that space even be, since the number of particles in the universe isn't fixed. An "explanation" offered in the article is as follows: > A single point in quantum configuration space, is the product of multiple > point positions per quantum field; multiple point positions in the electron > field, in the photon field, in the quark field, etc. But what does this actually _mean_? As I'm given to understand, in quantum field theory one has a wavefunction(al) defined over all configurations of a specific field, so if one jumps over to the real world where space-time is filled with many fields, is it possible to speak of the quantum configuration space over which the universal wavefunction is defined as a space where every point corresponds to some configuration of every quantum field, and if so, can this picture still be used to see an electron as a roughly independent part/blob of this giant amplitude distribution or not? I've been gnawing at this for two days now and I'd really appreciate some help here...


Decoherence without time?

Decoherence is a phenomenon that provides a part of the explanation of why quantum systems and classical systems behave differently. What I understood from decoherence so far is that it requires time because it simply takes a (somehow very short) while for a large quantum system to transit to classical behaviours through decoherence. Now, the status of time in quantum gravity is not clear, and it could very well occur that time is an emergent quantity. So I thought decoherence should occur at time scales where time has already emerged. Yet I found this assertion on www.decoherence.de > Decoherence explains also how the Schrödinger equation of general relativity > (the Wheeler-DeWitt equation) may describe the appearance of time in spite > of being time-less" I would like to understand how can decoherence be active in the timeless domain of the Wheeler-deWitt equation to allow the emergence of time, as decoherence seems to require time itself to happen. (Handwaving arguments are OK, but references are welcome!)


Why are binary objects so critical to astronomy?

There are a lot of astronomical objects out there to study but binary objects seem to get more attention. Why is this? What makes binary objects (stars, Kuiper Belt Objects, black holes, galaxies, etc.) more important or special than lone objects?


Would a solution to the Navier-Stokes Millennium Problem have any practical consequences?

I know the problem is especially of interest to mathematicians, but I was wondering if a solution to the problem would have any practical consequences. Upon request: this is the official problem description of the aforementioned problem.


Quick question on perturbation theory

Suppose we have a particle in an infinite potential well, with $V(x) = 0,\space 0< x < a $ and infinity everywhere else. Now suppose we have a perturbation on the LHS of the well: $V_1(x) = v, \space 0 < x < \frac{a}{2}$. By first order perturbation theory, all energy levels are shifted by the same amount: $$\Delta E = \langle E_n|v|E_n\rangle = \frac{v}{2} $$ Ok all is good. By considering the perturbed wavefunction: $$ \psi_n = \phi_n + \sum_{n \neq k} \frac{\langle \phi_k|V_1|\phi_ n\rangle}{E_n - E_k} $$ How do I make use of the condition that $v << E_2 - E_1 $ to show the results above is correct?


The Berry Connection is $$A_\mu(R)=-i \langle \Psi(R) |\partial_\mu \Psi(R) \rangle$$ which allows us to parallel transport a state indexed by $R$. We can integrate the Berry Connection to get the Berry Phase, and we can differentiate the Berry Connection to get the Berry Curvature. Can the Berry Connection be derived from a metric? As a prototypical example, I'm thinking about how the Christoffel Symbols in General Relativity (GR) can be derived from the metric tensor. Also, I think for every connection, there exists a metric for which the connection is a Levi-Civita connection. However, is there a natural and physical metric that induces the Berry Connection? In this presentation, the great and powerful Haldane relates "quantum distance" to the Berry Curvature, but it doesn't look like you can derive the Berry Curvature from the quantum distance in the same way curvature and the metric are related in GR.


Calculation of the differential of the entropy

In this review (for those who wants a precise reference see page 8 eq 21), the Author says that: \begin{equation*} S=-\sum_{i}P\left(i\right)\ln P\left(i\right) \end{equation*} and using the Maxwell-Boltzmann distribution \begin{equation*} P\left(i\right)=P_{MB}\left(i\right)+dP\left(i\right) \end{equation*} \begin{equation*} dS=-\sum_{i}dP\left(i\right)\ln P_{MB}\left(i\right)=-\sum_{i}dP\left(i\right)\ln e^{-\beta E\left(i\right)}=\beta\sum_{i}dP\left(i\right)E\left(i\right) \end{equation*} However I tried to do the detailed calculation but here is what I get: \begin{equation*} dS=-\sum_{i}dP\left(i\right)\ln\left[P\left(i\right)\right]-\sum_{i}P\left(i\right)\frac{1}{P\left(i\right)}dP\left(i\right)=-\sum_{i}dP\left(i\right)\ln P\left(i\right)-\sum_{i}dP\left(i\right)= \end{equation*} \begin{equation*} =-\sum_{i}dP\left(i\right)\ln\left[P_{MB}\left(i\right)+dP\left(i\right)\right]-\sum_{i}P\left(i\right)+\sum_{i}P_{MB}\left(i\right) \end{equation*} Can someone help me to get the right result? Thanks!


Mirror problem of radiation pressure

If two perfect mirrors are placed facing one another and they are in proximity, and photons (don't ask me how) are traveling between them and toward one of them, what is to keep the radiation pressure from reaching incredible amounts? I might be way off of base here because I am new to this field. I heard that the radiation pressure doubles on a mirror. If this is the case, a laser beam focused on a mirror at any angle will cause the mirror to accelerate one of two ways-- no more. This also violates energy conservation and thus isn't valid.


Infinite and Finite Square Wells

For the **infinite square well** in the first region, **outside the well** : $$\frac{-\hbar^2}{2m}\frac{d^2 \psi}{dx^2} + V(x) \psi (x) = E \psi (x),$$ where you set **$V = 0$**. Rearranging gives $$\frac{d^2 \psi}{dx^2} = -\frac{2mE}{\hbar^2} \psi.$$ We identify $$k = \sqrt{\frac{2mE}{\hbar^2}}.$$ For the **finite square well** we have the same situation (for bound solutions) but we set: $$\alpha = \sqrt{\frac{-2mE}{\hbar^2}}.$$ **Why do we absorb the minus sign into the square root now and not before?** How does this tie in with bound/unbound solutions and parity?


Why does a car bonnet (hood) rise when you connect the clutch with a brake on?

Is the rotational force to overcome the brakes moved to the opposite effect of moving the car chassis, until the brake is released?


Where / what is the source of the force that causes refraction?

So we know that as light moves from, say, a glass block to air, it speeds up and refracts, but if there is to be acceleration then surely there must be a resultant force to cause it? So where does that come from? Secondly as the light enters the glass block it slows down, but where does the excess energy go?


Band gaps: are they at the centre or at the edge of the Brillouin zone?

Reading about electronic band structures, I came across the following: > Band gaps open at the edges of the Brillouin zone (BZ), since that is where > the Bragg scattering occurs. I am slightly confused, as most band structures show a gap opening at $\Gamma$ which is at the centre of the BZ. What am I missing?


The viscous force between the layers of liquid is same, then why there is variation in the velocities of its layers?

I have learned in my textbook that when the liquid flows the bottom layer of the liquid never moves because of friction, but the upper layers move with increasing velocities how it is possible if the viscous force between all these layers is same


Why is second moment of area often referred as moment of inertia?

In many references, mostly civil engineering, the second moment of area is referred as Moment of Inertia. Is that "really" correct? From my understanding, moment of inertia is analogous to mass in rotational motion, not the second moment of area.


Electrostatics kinematics equations (Coulombs Law)

If there are two similarly charged particles $Q$ and $q$ of masses $M$ and $m$, a Coulomb force is applied on each one (action-reaction). Suppose that $Q$ is stationary and we are shooting $q$ with an initial velocity $\vec{v_0}$ from $\infty$ into the field. I have, as well, figured out that by rearranging Coulombs Law and the Fundamental Law of Mechanics we end up with the following differential equation: $$x''(t) [x(t)]^2 = \frac{kQq}{m}$$ because: coulombs law states that $$F = \frac{kQq}{[x(t)]^2}$$ and 2nd Newton's Law: $$F = ma = m x''(t)$$ By rearranging we get: $$m x''(t) = \frac{kQq}{[x(t)]^2} \Leftrightarrow x''(t) [x(t)]^2 = \frac{kQq}{m}$$ We also know that its velocity is zero when $x = \frac{2kQq}{m{u_0}^2}$ (from Work-Energy Thm) Is the general solution of this differential equation (2nd order non-linear ODE) the solution of the displacement over time function? If yes, what is it!?


What causes acceleration of particles in the expansion section of a De Laval nozzle?

A De Laval nozzle has a compression section, where the propellant is compressed (and thereby accelerated) as it moves towards a narrow section (the throat). After the throat, the nozzle widens out again. Here's a reference image. ![](http://i.stack.imgur.com/6Gv6t.png) According to what I've been reading, the propellant is moving at subsonic speeds before the throat, and is accelerated to supersonic speeds as it passes the throat. Furthermore, the propellant _continues to accelerate_ as the nozzle expands again. What I don't understand is: what force continues to accelerate the propellant after it passes through the throat? I've been trying to imagine a single molecule of propellant and thinking about the forces from the particles around it, but can't come up with a force "pushing" from upstream (because it's moving at supersonic speeds), or "pulling" from downstream (because at the minimum there would be 0 opposing force in a vacuum, but certainly not a negative attractive force from downstream). Any ideas?


Einstein gravity versus Newton's gravity

What's the basic difference between the gravity as seen by Einstein, and that by Newton?


Is there a good explanation for the observation of Martian canals?

Martian "canals" have been observed by independent observers after their first description. Now, they are attributed to "optical illusion", but I think that this is not a good choice of word, because an optical illusion should be visible today as well. It would have to be a psychological effect, but it is rather astonishing to have people draw whole maps of non-existent canal systems. Is there a good explanation (optical or psychological) for the observation of Martian canals?


"Dear radioactive ladies and gentlemen" - Letter by Wolfgang Pauli

In 1930, **Wolfgang Pauli** wrote a letter to Lise Meitner for a convention in Tübingen, considering the problem of beta decay. Does anybody know, where to find the original letter online ?


What is Dark Energy, and how was it discovered?

I've heard a lot about Dark Energy, and how it's supposed to be one of the most powerful forces in the universe. What is it, and how was it discovered?


What is a proportionality constant? (Planck's constant)

I understand that Planck's constant is essentially the ratio between the energy of a photon and its frequency. There are 2 things that im trying to verify:   1. isn't the number that Planck's constant represents just an arbitrary result of the measurement system we use? As in, Planck's constant represents the ratio between 1 Joule and 1 Hertz? So if i assume correctly, Planck's constant is the increase in energy (in joules) that you would get from an increase in frequency of 1hz? Thus if i wanted to I could make an alternative unit for energy which had a 1 to 1 correspondence with hz, and no proportionality constant would be required?   2. So regardless of units used to measure, a fixed percentage increase in one quantity results in an equal percentage increase of the other, correct?


The Dual Cloud Chamber Paradox

**2012-04-07 Addendum: The Dual Cloud Chamber Paradox** Two 10m diameter spheres $A$ and $B$ of very cold, thin gas have average atomic separations of 1nm. Their atoms are neutral, but ionize easily and re- radiate if a relativistic ion passes nearby. Both clouds are seeded with small numbers of positive and negative ions. The clouds collide at a relative speed of $0.994987437c$, or $\gamma=10$. Both are sparse enough to minimize collisions and deceleration. Both show passage of each other's ions. (a) What do distant observers moving parallel to $A$ and $B$ observe during the collision? (b) Are their recordings of this event causally consistent? The answer to (a) requires only straightforward special relativity, applied from two perspectives. For the traveler moving in parallel to cloud $A$, cloud $B$ should appear as a Lorentz contracted oblate spheroid with one tenth the thickness of cloud $A$, passing through cloud $A$ from right to left. If you sprinkled cloud $A$ with tiny, Einstein-synchronized broadcasting clocks, the $A$-parallel observer would observe a time-stamped and essentially tape- recording like passage of the $B$ spheroid through cloud $A$. The $B$-parallel observer sees the same sort of scenario, except with cloud $A$ compressed and passing left-to-right through cloud $B$. If $B$ is sprinkled with its own set of Einstein synchronized broadcasting clocks, the $B$p-parallel observer will record a time-stamped passage of the compressed $A$ through $B$. So, am I the only one finds an _a priori_ assumption causal self-consistency between these two views difficult to accept? That is, while it may very well be that a recording of a flattened $B$ passing through cloud $A$ in $A$-sequential time order can always be made causally self-consistent with a recording of a flattened $A$ passing passing through through cloud $B$ in $B$-sequential order, this to me is a case where a mathematically precise proof of the information relationships between the two views would be seem like a good idea, if only to verify how it works. Both views after all record the _same_ event, in the sense that every time a clock in $A$ or $B$ ticks and broadcasts its results, that result becomes part of history and can no longer be reversed or modified. It's tempting to wonder whether the Lampa-Terrell-Penrose Effect might be relevant. However, everything I've seen about L-T-P (e.g. see the video link I just gave) describes it as an optical effect in which the spheres are Lorentz contracted at the physical level. Since my question deals with fine-grained, contact-like interactions of two relativistic spheres, rather than optical effects at a distance, I can't easily see how L-T-P would apply. Even if it did, I don't see what it would mean. So, my real question is (b): Does an information-level proof exist (not just the Lorentz transforms; that part is straightforward) that the $A$-parallel and $B$-parallel recordings of dual cloud chamber collision events will always be causally consistent? * * * **2012-03-03: This was my original version of the question, using muonium clocks** My SR question is how to predict length contraction and time dilation outcomes for two interacting beams of neutral particles. The two beams are:   1. In different inertial (unaccelerated) reference frames for the duration of measurement part of the experiment.   2. Intermixed at a near-atomic level so that frame-to-frame photon exchange times are negligible. An example would be two intersecting beams of relatively dense, internally cold muonium atoms.   3. Clock-like even at the atomic level (e.g., decaying atoms of muonium).   4. Part of a single causal unit. By this I mean that there is a sufficient short-range exchange of photons between the two beams to ensure that state changes within each beam have an entropic effect, however small, on nearby components of the other beam. This makes their interactions irreversible in large-scale time. An example would be decay of an anti-muon in one frame transferring energy to nearby muonium atoms in the other frame. Simply illuminating the intersection region with light frequencies that would interact with muonium in both of the beams would be another option.   5. Observed by someone who is herself isolated from the rest of the external universe. Muons generated by cosmic rays and traveling through earth's atmosphere provide a helpful approximation the above experiment. The muons provide the first beam, and the atmosphere forms the second one, which in the case of muons is shared by the observer. Such muons have tremendously extended lifespans, which is explained by invoking time dilation (but not length contraction) for the muon frame as viewed from the atmosphere frame. Conversely, length contraction (but not time dilation) is invoked to describe the view of the atmosphere from the perspective of the muon frame. Since this results in the atmosphere appears greatly compressed in the direction of travel, the muons simply travel a shorter distance, thereby ensuring the same decay result (causal symmetry) for both views of the interacting frames. My question then is this: _For the thought experiment of intersecting, causally linked beams of muonium, what parameters must the observer take into account in order to predict accurately which of the two intersecting muonium beams will exhibit time dilation, and which one will exhibit length contraction?_ **2012-03-04 Addendum by Terry Bollinger (moved here from comment section):** Sometimes asking a question carefully is a good way to clarify one's thinking on it. So, I would now like to add a hypothesis provoked by own question. I'll call it the local observer hypothesis: _Both beams will be time dilated based only on their velocities relative to the observer Alice; the beam velocities relative to each other are irrelevant._ Only this seems consistent with known physics. However, it also implies one can create a controllable time dilation ratio between two beams. I was trying to avoid that. So my second question: Are physical time dilation ratios ever acceptable in SR? **2012-03-06 Addendum by Terry Bollinger:** Some further analysis of my own thought problem: A $\phi$ _set_ is a local collection of clock-like particles (e.g. muons or muonium) that share a closely similar average velocity, and which have the ability to intermix and exchange data with other $\phi$ sets at near-atomic levels, without undergoing significant acceleration. A _causal unit_ $\chi = \\{\phi_0 ... \phi_n\\}$ is a local collection of $(n+1)$ such $\phi$ sets. By definition $\phi_0$ contains a primary observer Alice, labeled $\aleph_0$, where $\aleph_0 \subset \phi_0$. Each $\phi_i$ has an associated XYZ velocity vector $\boldsymbol{v_i} = (\phi_0 \rightarrow \phi_i$) that is defined by the direction and rate of divergence of an initially (nominally) co-located pair of $\phi_0$ and $\phi_i$ particles, with the $\phi_0$ particle interpreted as the origin. The vector has an associated magnitude (scalar speed) of $s_i=|\boldsymbol{v_i}|$. Theorem: If $\phi_i$ includes a subset of particles $\aleph_i$ capable of observing Alice in $\phi_0$, then $\aleph_i$ will observe $\phi_0$ as length contracted along the axis defined by $\boldsymbol{v_i}$. Conversely, Alice ($\aleph_0$) will observe each $\phi_i$ as time dilated (slowed) based only on its scalar speed $s_i$, without regard to the vector direction. This dependence of time dilation on the scalar $s_i$ means that if $\phi_a$ has velocity $\boldsymbol{v_a}$ and $\phi_b$ has the opposite velocity $\boldsymbol{-v_a}$, both will have the _same_ absolute time dilation within the causal unit (think for example of particle accelerators). Analysis: This $\chi = \\{\phi_i\\}$ framework appears to be at least superficially consistent with data from sources such as particle accelerators, where time dilation is observable in the variable lifetimes of particles at different velocities, and where absolute symmetry in all XYZ directions is most certainly observed. Applying observer-relative time dilation to fast particles (e.g. in a simple cloud chamber) is in fact such a commonly accepted practice that I will assume for now that it must be valid. (It is worth noting that while particle length contraction is typically also assumed to apply to fast-moving particles, serious application of length contraction in a causal unit would cause the particles to see longer travel paths that would shorten their lifespans. This is the same reason why muons are assumed to see a shorter path through a length-contracted atmosphere in order to reach the ground.) My updated questions are now: (1) Is my $\chi = \\{\phi_i\\}$ framework for applying SR to experiments valid? If not, why not? (2) If $\chi = \\{\phi_i\\}$ is valid, _what property makes the observer $\aleph_0$ in $\phi_0$ unique from all other $\aleph_i$?_


Angular acceleration and linear acceleration

I have a small confusion. I learned very recently that all particles of a rotating body have the same angular acceleration but different linear acceleration (same for velocity as well). But how is this possible? I find difficulty understanding this though I know that it is almost simple. Can you please help me understanding this?


Why does the Hubble parameter keep getting smaller if the expansion of the universe accelerates?

Suppose a region of space at a distance D from Earth is escaping from us with the velocity v. Since it seems like the expansion of the universe is accelerating, things at D from Earth should be receding faster and faster. It would seem from the v = HD equation, that H is then getting bigger in time. However, in other sources we can read, that the Hubble "constant" it's getting smaller with time. How come?


Why is force described as rate of change of momentum?

> momentum = mass * velocity Differentiating both sides leads to > force = mass * acceleration since the mass doesn't participate in the differentiation as it is constant. * * * **Is this a sound definition** , or a pointless use of differentiation? Or are there systems (like rockets) where the mass is also a variable wrt time?


Super-String and Anti-Particle

I'm afraid this is quite basic question . What I learned from the String theory is that the theory unify the Path-integral, Superposition Principle, and commutation relation .   In that context , I wondered whether the Anti-Particle can be described by the similar way the Superposition Principle is expressed by the string theory . How can the anti-particle be described by the String Theory ?


Confusion about Lie derivative on metric

According to this site, the Lie derivative of a $(0,2)$-tensor is $$ \mathcal{L}_XT_{ab}=\partial_XT_{ab}+T_{cb}\partial_aX^c+T_{ac}\partial_bX^c $$ However, according the same website, the Lie derivative of the metric is $$ \mathcal{L}_Xg_{ab}=\nabla_Xg_{ab}+g_{cb}\nabla_aX^c+g_{ac}\nabla_bX^c $$ and if the connection is metric compatible, the first term vanish. My question is, why do we change $\partial$ into $\nabla$ here? In fact, in a note, I see both two expressions for the Lie derivative of the metric. Are they really equivalent? * * * Update: I tried to expand the second equation, and turns out it indeed is the first one. In fact, $$ \begin{align} &\nabla_Xg_{ab}+g_{cb}\nabla_aX^c+g_{ac}\nabla_bX^c\\\ =&X^c(\partial_cg_{ab}-\Gamma^{d}_{ac}g_{db}-\Gamma^d_{cb}g_{ad})+g_{cb}\partial_aX^c+g_{ac}\partial_bX^c+g_{cb}\Gamma^{c}_{ad}X^d+g_{ac}\Gamma^c_{ad}X^d\\\ =&\partial_XT_{ab}+T_{cb}\partial_aX^c+T_{ac}\partial_bX^c \end{align} $$ All the $\Gamma$'s cancel out. But now my question is, is it generally true that when expanding $\mathcal{L}_XT$ for some tensor $T$, we can use either partial derivative or covarient derivative?


String breaking in QCD

I have trouble understanding string breaking in QCD. I have read an article on arxiv (http://arxiv.org/abs/hep-lat/0505012), and I still don't understand what truly happens. My understanding of string breaking (of a meson - let's say bottomonium) is as follows: We have a quark $b$ and an anti-quark $\bar b$ on distance of $R$ at temperature $T=0$ (this is probably to limit the effects of quark-gluon plasma?). As we "pull" them apart the potential energy increases. At some crytical distance $R_c$ it is more favourable that a pair of quark-antiquark is created from vacuum and thus the final state are two mesons. For calculating the static potential between a quark and antiquark we use correlation function for creating a meson at $t=0$ and destroying it at $t=\tau$. We then use Wick rotation $t \rightarrow -it$ which then transforms oscilating functions into exponentially decaying ones: $$C(R,t)=a_0e^{-\frac{E_0}{\hbar}t}+a_1e^{-\frac{E_1}{\hbar}t}+\ldots \hspace{2 mm}.$$ We interpolate ground state with $e^{-\frac{E_0}{\hbar}t}$ which is just the static potential $V(R)$, since we neglect kinetic energy. My first question is: Does the potential static potential $V(R)$ at temperature $T=0$ saturate or not. Many articles say it follows a Cornell potential, which is $$V(r)=-\frac{e}{r}+\sigma r\hspace{1mm},$$ which grows linearly with distance (and doesn't saturate and thus doesn't show string breaking). My second question is about the first excited state, which should be a final state of two mesons (in this case probably $B$ and $\bar B$ meson). Can we get the energy of this excited state with simple correlation function which I mentioned (energy $E_1$), or must we use the correlation function (which is a matrix) in the article (defined on the first page). They also calculate mixing of states which I don't understand fully. Where do they get the energy of eigenstates? The eigenstates they calculate are:$$|1\rangle=\cos(\theta)|b\bar b\rangle + \sin(\theta)|B\bar B\rangle \\\ |2\rangle=-\sin(\theta)|b\bar b\rangle + \cos(\theta)|B\bar B\rangle\hspace{1 mm}.$$ PS The picture of string breaking is showed on page 19 in the article. I would be extremely thankfull if someone showed me an article which explains (qualitatively and quantitatively) string breaking in simpler language.


How can a body be transparent?

How can a body be transparent? I need the theoretical explanation for the same. Has anybody succeeded in doing that practically?


Can "state" be considered a 5th dimension?

I searched for an answer to this question on Google but just found articles that mention either string theory or a 5th dimension in passing (such as Maxwell equations as they relate to Riemann curvature tensor.) I stumped myself while driving to school today thinking about this... We can explain an objects position in the universe by describing its spacial and time locations, such as _at 2nd and 3rd street on the fifth floor at 10:00am_ But is that enough to fully describe an object when you take into account Schrödinger's cat? For example, what if, Schrödinger's cat is the object that is at that location and time. It seems like you would have to have a 5th _state_ dimension to fully explain its position, as in _at 2nd and 3rd street on the fifth floor at 10:00am with a probability of .5_ It seems like if we had _state_ as an additional dimension it would help explain things like quantum entanglement as the particles could be moving away from each other in space-time but standing still in state. Am I merely relating things that have no business being related, or is there a connection between state and dimension?


Accessible microstates of harmonic oscillator in microcanonical enemble

While reading up on statistical physics, I am going through the calculation of the partition function of the harmonic oscillator in the microcanonical ensemble. The result for the partition function is: $\Omega(E) = \frac{E_o}{\hbar \omega}$, i.e. all one-dimensional harmonic oscillators of the same frequency $\omega$ have the same number of accessible microstates. Consequently, as $\omega$ grows, $\Omega(E)$ goes to zero. This is somewhat counter-intuitive to me. I would have expected that an oscillator with a higher frequency $\omega$ would have more accessible microstates than one with a lower frequency. What is wrong with my intuition? Note that I am going through the classical calculation, the partition for the harmonic oscillator is then given as: $\Omega(E) = \frac{E_0}{h}\int dp \int dx \delta \left( \frac{p^2}{2m} + \frac{1}{2}k x^2 - E\right) $


Why do rocket engines have a throat?

Diagrams of rocket engines like this one, ![rocket engine](http://i.stack.imgur.com/EZWGCm.png) (source) always seem to show a combustion chamber with a throat, followed by a nozzle. Why is there a throat? Wouldn't the thrust be the same if the whole engine was a U-shaped combustion chamber with a nozzle?


Short term track of a hand held sensor (gyro, accelerometer) data with rough precision

I have acceleration and attitude data for the movement of a device in 3d space. what transformations, before numerical integration, are required, in order to produce a 3D track of the the movement in the earth's frame of reference. I know about the precsion and error issues. And I am not looking for commercial or mil spec accuracy. I want to try. I have seen the transformations in various papers. Can anyone suggest links or code that could accelerate my efforts?


Possible Error in deriving conformal generator

My professor gave me the following derivation for the full generator of the Lorentz transformations. The starting point is to consider a subgroup of the conformal group that leaves the origin fixed (as done so in Di Francesco, for example). Denote by $J_{\mu \nu}$ the full generator and so $J_{\mu \nu}\Phi(0) = S_{\mu \nu}\Phi(0)$, where $S_{\mu \nu}$ is the component of the full generator transforming only the field spin components. Infinitesimally, $\Phi'(0) = (1 + \epsilon S_{\mu \nu})\Phi(0)$ but also $\Phi'(0) = e^{-ix' \cdot P}\Phi'(x')$. This I understand. The next line is the first one I am not so sure of. He then writes $$\Phi'(0) = e^{-ix' \cdot P}(1+\epsilon J_{\mu \nu})\Phi(x)$$ How is this true? He seems to be using the transformation $\Phi'(x') = D \Phi(x)\,\,\,(1)$, but using $D$ as the **_full_** generator. If I had read that statement a few months back, I would have accepted it fine, but I posted a question here earlier that discussed the fact that in $(1)$, $D$ is the **_spin_** component of the generator because $x'$ and $x$ represent the same physical point in the space, they are just two different representations of the same point relative to two different coordinate systems. So the statement seems to be contradicting what I thought I understood. That was my first concern. My second concern is less fundamental (more of a computational problem) and I can edit it into my answer later. Many thanks! Edit: So assuming the last line to be correct, we proceed: $$\Phi'(0) = e^{-ix' \cdot P}(1 + \epsilon J_{\mu \nu})e^{i x \cdot P}\Phi(0) \approx (1 + \epsilon \underbrace{e^{-ix \cdot P}J_{\mu \nu}e^{i x \cdot P}})\Phi(0)$$ using the fact to zeroth order $x' \approx x$. The brace can be evaluated using the Hausdorff formula: \begin{align}e^{-ix \cdot P}J_{\mu \nu}e^{i x \cdot P} &\approx J_{\mu \nu} + [J_{\mu \nu}, ix^{\sigma}P_{\sigma}] \\\&= J_{\mu \nu} + i\left[x^{\sigma}[J_{\mu \nu},P_{\sigma}] + [J_{\mu \nu}, x^{\sigma}]P_{\sigma}\right] = J_{\mu \nu} + x_{\mu}P_{\nu} - x_{\nu}P_{\mu},\end{align} where the second commutator on the RHS there vanishes. Now compare with $\Phi'(0) = (1+\epsilon S_{\mu \nu})\Phi(0)$ as written at the top and we have the result $J = L + S$. My question is: why does this second commutation relation vanish? I should say I am going through 'Conformal field theory' Di Francesco section 4.2.1. The above argument was an alternate argument my professor posed when we sat down and agreed that equation (4.26) in Di Francesco is simply wrong.


Antipodes are mostly ocean - so what happens after digging that hole through earth?

Digging a hole through earth is a common **thought experiment** , often used to explain effects of gravity. But what would happen if someone finally **dug the hole**?   Sure, he took care to stabilize and isolate it inside with two and a half layer of unobtanium. But he forgot to check what exactly is on the other side of the earth, the antipode.   And, just like surprising 85% of earth [travel.SE: antipodes], it turns out to be ocean. The tunnel is under vacuum, and about the diameter of a two track railway tunnel. In the last step building it, obviously the ocean floor broke, and water is streaming in.   What happens then? I'm aware that a body would **oscillate** through the core - the water can not, if the ocean does not run dry assume - at least if the water coming in can mostly fill the diameter. First, I thought the water will just fill the hole, and directly end up in an **equilibrium regarding gravity and pressure**. But that assumes the water, filling the diameter at the ocean ground will do so in the whole tunnel; But doesn't the **stream** of water **get thinner** , as it accelerates up to the center of the earth, and thicker from there? So will it oscillate for a while?   Would it change if we assume the hole in the ocean floor is much smaller than the tunnel's diameter?


What is the difference between Quantum teleportation and Radio waves?

Quantum teleportation over 143 kilometres using active feed- forward What is the difference between Quantum teleportation and radio waves?


Must string models that describe 4d effective field theories always have D-branes that extend in the 4 non-compact spacetime dimensions?

In string theory the D-branes give those directions that the strings are allowed to move along. The string excitations give the fields that we detect. Is it correct to think of a particle propagating in 4d spacetime as a string propagating in the same 4d spacetime (in other words if we could zoom in sufficiently far a photon for example would reveal itself to be a string). If so, this implies that a realistic model should always have D-branes that extend in these 4 spacetime directions. Otherwise how would the strings, and their associated excitations, propagate in the 'normal' spacetime we see at low energies?


fusing two air containers with fixed pressures $p_1,p_2$ and temperatures $T_1, T_2$, what end-result $T$ and $p$ will be?

Let's say that I have two boxes with both volumes equal $V$, filled with an air. Air in first box have pressure $p_1$ and temperature $T_1$, likewise air in second box have pressure $p_2$ and temperature $T_2$. Suppose this is ideal gas, then from Clapeyron equation we have: $$ p_1 V = n_1 R T_1 $$ $$ p_2 V = n_2 R T_2 $$ Thus, after simple transformation: $$ \frac{p_1 V} {R T_1} = n_1 $$ $$ \frac{p_2 V} {R T_2} = n_2 $$ Now, let's say that we want to fuse both boxes and remove bulkhead, so we will have instead 1 box with volume equal $2V$. What will be temperature $T$ and pressure $p$ of air in the new box? We can use the same law as above to get: $$ 2 p V = ( n_1 + n_2 ) R T = \left( \frac{p_1 V} {R T_1} + \frac{p_2 V} {R T_2} \right) R T $$, thus: $$ p = \frac { T } { 2 } \left( \frac { p_1 } { T_1 } + \frac { p_2 } { T_2 } \right) $$ **And there's my question:** If I had heated air after fusion of boxes to temperature $T'$, in other words I had known temperature, I could say what pressure is. Likewise, knowing pressure $p'$ I could say the temperature. But if I just fuse boxes of air and "let it be" (wait very long for air to mix), what the end result should be? Shouldn't (statistically) the pressure and temperature always fix to some certain, unambiguous state? If yes, how I can find them? I mean that I have one equation $ \quad $ $ p = \frac { T } { 2 } \left( \frac { p_1 } { T_1 } + \frac { p_2 } { T_2 } \right) $, $ \quad $ and two variables: $T, p$, so how can I find both of them? Did I missed something? Or I must to make some more assumptions?


Wavefunctions of different observable values are orthogonal because for any state one of them equals zero. Right?

I'm trying to understand correspondence between observables and quantum operators. Let's assume that $\phi_1$ and $\phi_2$ are wavefunctions that can be obtained after the measurement of some observable (in other words - quantum states of the system immediately after measurement). $\phi_1$ is a quantum state if the measurement result equals $\lambda_1$, and $\phi_2$ is a quantum state if the measurement result equals $\lambda_2$ ($\lambda_1 \neq \lambda_2$). Let's assume that $x$ is a classical state of the system, i.e. point in position and momentum space of the system ($x = \\{p_i,q_i\\}$). And $f(x)$ is a value of the aforementioned observable in state $x$. If $\phi_1^*(x) \phi_1(x) \ne 0$ than $f(x) = \lambda_1$ (i.e. there is non- zero probability for the system in quantum state $\phi_1$ to be in point $x$). If $\phi_2^*(x) \phi_2(x) \ne 0$ than $f(x) = \lambda_2$ (i.e. there is non- zero probability for the system in quantum state $\phi_2$ to be in point $x$). It means that, for any $x$ either $\phi_1^*(x) \phi_1(x) = 0$ or $\phi_2^*(x) \phi_2(x) = 0$ (or both). So we can suppose (can we?) that for any $x$ either $\phi_1(x) = 0$ or $\phi_2(x) = 0$ (or both). Therefore for any $x$ $\phi_1^*(x) \phi_2(x) = 0$ , and therefore $\langle\phi_1 | \phi_2 \rangle = 0$. The last equality means orthogonality of $\phi_1$ and $\phi_2$. Having full set of possible observable values and corresponding orthogonal wavefunctions by spectral theorem we can build an operator $A$ for this observable as following: $$ A = U\Lambda U^*$$ where $\Lambda$ - diagonal matrix with $\Lambda_{ii} = \lambda_i$ and $U_{ij} = \phi_i(x_j)$. Is this derivation correct?


What are particle multiplets in the Standard Model?

The particles of the standard model are often displayed in groupings known as multiplets. I know that this somehow relates to the underlying symmetries of the standard model, which can be viewed as tranforming these particles into each other. I have still struggled to pin down exactly what is meant by these multiplet groupings however, and to get a good 'feel' of the concept. Could somebody elaborate please?