Violations of Dulong-Petit rule as an upper limit to heat capacity

Does any known substance have a heat capacity at constant volume ($C_V$) **per mole of atoms** greater than $3k_B$ ~ 24.94 J/(mol K)? In order to count, the substance must actually be made of atoms, that is, ordinary nuclei and electrons. If so, what are the extra degrees of freedom responsible for this unusually large heat capacity?


Why doesn't string theory have a mass gap?

Gravitons are string modes which can have arbitrarily small energies. Just make the wavelength arbitrarily large. String theory has no mass gap, but why? A classical string can only have positive energies. The string tension makes a positive contribution. The string kinetic energy also makes a positive contribution. A classical string can have an arbitrarily small mass because it can be arbitrarily small in length. But quantum strings can't shrink beyond the string scale. The lack of a mass gap can be traced to a shift factor to the string energy in the Virasoro constraint. For a single string, this shift is unconstrained and arbitrary. Without interactions, this is none other than the "string chemical potential". But we can create/annihilate strings after second quantizing them. This shift becomes physical. Modular invariance of worldsheets of genus one fixes the shift, and it turns out to be negative. String theorists explain this as the Casimir effect on the worldsheet, but this is just a name, and not an explanation. Why is this Casimir shift negative? Only a negative value can cancel the mass gap. Don't say $ζ$-function regularization, because it's purely formal and doesn't explain a thing.


Quantum $n$-body problem

Is the quantum $n$-body problem as difficult as the classical $n$-body problem? Or quantum mechanics allows to get a simpler exact solution? Suppose there are 3 particles with uniform potential field. Can their wave functions be found exactly?


Dark Matter and Modified Gravity

Could someone please explain briefly or refer me to an article or manuscript that shows how f(R) modified gravity theories can be used to explain the problem of Dark Matter, particularly Galaxy rotation problem?


Transform hamiltonian

I have got the following Quantum Hamiltonian: $$H=\frac{p^{2}}{2m}+k_{1}x^{2}+k_{2}x+k_{3}$$ Which transformation can I use to change this Hamiltonian into an harmonic oscillator hamiltonian? Note: $k_{1}, k_{2}$ and $k_{3}$ are constants.


Matlab package: graphical calculus for quantum operations including beam splitters and polarizing beam splitters

I need a matlab package that will make my life easier. I have quantum circuits with optical beam splitters, polarizing beam splitters and photodetectors. These circuits are getting very complicated and I am wondering if anyone knows of a matlab package that would make my life easier. A graphical package would be excellent and I suppose any software will do.


Origin of Laue equations?

The Bragg condition (by Bragg in 1913) can be derived by the Laue equations that is making use of the Miller indices and all the latice/crystal stuff (so basically it's bringing Bragg's law to more fundamental ground). But what is interesting to me: When did the "Laue way" came up? Was it only named after Max von Laue or did he himself publish a paper introducing this formalism? Neither Wikipedia nor the books I have are citing it. The Laue diffraction/condition/law is always just presented without origin. But I'd like to find it's real origin.


$\pm$ (light-cone?) notation in supersymmetry

* I would like to know what is exactly meant when one writes $\theta^{\pm}, \bar{\theta}^\pm, Q_{\pm},\bar{Q}_{\pm},D_{\pm},\bar{D}_{\pm}$.  {..I typically encounter this notation in literature on $2+1$ dimensional SUSY like super-Chern-Simon's theory..}   * I guess that when one has only half the super-space (i.e only the $+$ of the above or the $-$) it is called the $(0,2)$ superspace compared to the usual $(2,2)$ superspace. In this case of $(0,2)$ SUSY I have seen the following definitions, $Q_+ = \frac{\partial}{\partial \theta^+} + i \bar{\theta}^+\left( \frac{\partial}{\partial y^0} + \frac{\partial}{\partial y^1} \right)$ $\bar{Q}_+ = -\frac{\partial}{\partial \bar{\theta}^+} - i \theta^+\left( \frac{\partial}{\partial y^0} + \frac{\partial}{\partial y^1} \right)$ which commute with, $D_+ = \frac{\partial}{\partial \theta^+} - i \bar{\theta}^+\left( \frac{\partial}{\partial y^0} + \frac{\partial}{\partial y^1} \right)$ $\bar{D}_+ = -\frac{\partial}{\partial \bar{\theta}^+} + i \theta^+\left( \frac{\partial}{\partial y^0} + \frac{\partial}{\partial y^1} \right)$   * I am guessing that there is an exactly corresponding partner to the above equations with $+$ replaced by $-$. Right?  How does the above formalism compare to the more familiar version as, $Q_\alpha = \frac{\partial}{\partial \theta^\alpha} - i\sigma^\mu_{\alpha \dot{\alpha}}\bar{\theta}^{\dot{\alpha}}\frac{\partial}{\partial x^\mu}$ $\bar{Q}_\dot{\alpha} = -\frac{\partial}{\partial \bar{\theta}^\dot{\alpha}} + i\sigma^\mu_{\alpha \dot{\alpha}}\theta^{\alpha}\frac{\partial}{\partial x^\mu}$ which commute with, $D_\alpha = \frac{\partial}{\partial \theta^\alpha} + i\sigma^\mu_{\alpha \dot{\alpha}}\bar{\theta}^{\dot{\alpha}}\frac{\partial}{\partial x^\mu}$ $\bar{D}_\dot{\alpha} = -\frac{\partial}{\partial \bar{\theta}^\dot{\alpha}} - i\sigma^\mu_{\alpha \dot{\alpha}}\theta^{\alpha}\frac{\partial}{\partial x^\mu}$ {..compared to the above conventional setting, in the $\pm$ notation among many things the most perplexing is the absence of the Pauli matrices!..why?..} I would be very grateful if someone can explain this notation. {..often it turns out that not just the Qs and the Ds but also various superfields also acquaire a $\pm$ subscript and various usual factors of Pauli matrices look missing..it would be great if someone can help clarify this..}


SUSY (Supersymmetric) Quantum Mechanics

I have seen some books, e.g. by Fred Cooper (Supersymmetry in Quantum Mechanics), define: $A = \frac{\hbar}{\sqrt{2m}} \frac{d}{dx} + W(x)$, $A^\dagger = \frac{-\hbar}{\sqrt{2m}} \frac{d}{dx} + W(x)$, $V_1(x) = W^2(x) - \frac{\hbar}{\sqrt{2m}} W'(x)$, $V_2(x) = W^2(x) + \frac{\hbar}{\sqrt{2m}} W'(x)$. In other places, I have seen it defined: $A = \frac{\hbar}{\sqrt{2m}} \frac{d}{dx} + W'(x)$, $A^\dagger = \frac{-\hbar}{\sqrt{2m}} \frac{d}{dx} + W'(x)$ $V_1(x) = W'(x)^2 - \frac{\hbar}{\sqrt{2m}} W''(x)$, $V_2(x) = W'(x)^2 + \frac{\hbar}{\sqrt{2m}} W''(x)$. It seems like these two definitions will give you completely different Hamiltonians and partner potentials. Could someone comment on if these two conventions are equivalent, and if so, how?


Nanotech - zero dimensional fullerene

Not really a physics junkie, and I think this is a chemistry question, but since there's no chemistry.stackexchange.com (yet) I think here is the best place. I was just reading up on nanotech and came across the phrase "can construct 0D fullerenes, 1D graphenes...". Flags went up at "0D fullerenes" and I can't find further information about it in many places. A fullerene is apparently any pure-carbon atomic structure, in various forms, but the zero-dimensional part is throwing me. What is a 0D fullerene?


Why does the coefficient of performance need to be calculated with $W + Q$?

> How much work must a heat pump with a COP of 2.50 do in order to extract > 1.00 MJ of thermal energy from the outdoors (the cold reservoir)? The first formula that came into my mind after reading this question is the $$COP=\frac{\lvert Q(\text{cool})\rvert}{\lvert W\rvert}$$ However, when I used the equation above, the answer that I got was 0.4 MJ instead of the answer given which is 0.67 MJ. And then I tried various other ways to try and get the 0.67 MJ and only when I used this equation that I finally get the answer 0.67 MJ. $$COP=\frac{\lvert W+Q(\text{cool})\rvert}{\lvert W\rvert}$$ What I don't understand is that why do we need to add $W$ with $Q$? Or is it not the correct way after all? Would somebody explain this to me, please?


What sort of thermodynamic process is this?

I have to calculate the work done in this thermodynamic cycle: LINK. ![enter image description here](http://i.stack.imgur.com/9l79q.jpg) But the bottom bit of the cycle is something I've never seen before. I guess the curve in $pV$ space is given by $$\left( V- \frac{V_2 -V_1}{2} \right) ^2 + \left( p-p_1 \right) ^2 = \left( \frac{V_2 -V_1}{2} \right) ^2$$ Which is physically meaningless because of units...


Is there any difference between massless Dirac fermions and Weyl fermions?

In graphene we call the low energy excitations around the Dirac point Dirac fermions, which are massless. Is this just by convention or is there any further differences between massless Dirac fermions and Weyl fermions


W + jets at NLO

I would like to calculate $pp \rightarrow W (\rightarrow \ell \nu)$ + n-jets (n=2, maybe also 3) at NLO, with some cuts and plot some distributions. I used MadGraph extensively for LO processes and I was wondering how reliable it is at NLO? Does anyone know any reference comparing multiple NLO codes? I have tried MCFM and I am not very experienced with it but the calculation seems to take a lot of time (obviously since it's NLO). Is there a way to make it run in parallel? I know that you can separate the real emission and virtual parts and then combine them later, but that doesn't seem to give a huge speed increase. Any help is appreciated. Thanks.


Does the induced charge on a conductor stay at the surface?

My textbook says that when a conductor is placed in an electric field, the electrons in it realign so that the net electric field inside the conductor is zero. There isn't a proof for this. It merely states that the electrons drift opposite to the applied electric field and create an opposite electric field inside the conductor canceling the external one. I imagine Gauss's Law would provide a simple proof. If we try enclose a point inside the conductor in a gaussian surface, and note that the enclosed charge is zero, so the flux through the surface is zero and so is the electric field at the surface. When I looked back at my reasoning, I realized that I had assumed that the charges induced on the conductor always stay at the surface, which I had no way of proving. How do I prove this formally?


How can a Photon have a "frequency"?

I picture light ray as a composition of photons with an energy equal to the frequency of the light ray according to $E=hf$. Is this the good way to picture this? Although I can solve elementary problems with the formulas, I've never really been comfortable with the idea of an object having or being related to a "frequency". Do I need to learn quantum field theory to _really_ understand this?


When a high speed neutrino just misses an old neutron star, why isn't it trapped?

Suppose a neutrino is seen travelling so fast that its Lorentz gamma factor is 100,000. It races past an old, no longer active neutron star, narrowly missing it. As far as the neutrino is concerned, it is the NEUTRON STAR that is moving at extreme speed, & its mass is 100,000 times larger than 2 solar masses. Therefore, from the speeding neutrino's perspective, the neutron star should appear to be a black hole definitely large enough to trap the neutrino. So how come the speeding neutrino continues its travel right past the old stellar remnant? Is there an agreed name for this question or paradox?


Why is it thought that normal physics doesn't exist inside the event horizon of a black hole?

A black hole is so dense that a sphere around it called the event horizon has a greater escape velocity than the speed of light, making it black. So why do astronomers think that there is anything weird (or lack of anything Inc space) inside the event horizon. Why isn't simple the limit to where light can escape and in the middle of event horizon (which physically isnt a surface) is just a hyper dense ball of the matter that's been sucked in and can't escape just like light. Why is it thought that the laws of physics don't exist in the event horizon?


Moving Between Degenerate Vacua?

In spontaneous symmetry breaking, moving round the circular valley of Mexican hat potential doesn’t cost energy. These angular excitations are called Goldstone bosons. But doesn't the angular excitation imply that the system moves from one vacuum to the other because all points on the circular valley are degenerate vacua? If Goldstone excitations are like this then how can a symmetry remain broken?


Are combined masses in space, such as galaxies, considered to be uniform bodies? *In addition, a related question about force

If we were to calculate the force that one galaxy exerted onto another, would we consider the individual masses within the galaxies, or the masses of the galaxies as a whole? Do the individual stars within one galaxy have control over the individual stars within another? Or, do we consider the galaxies to be single bodies, each with it's own mass, and the distance between them as two uniform bodies? If so, could we call galaxy clusters uniform bodies as well? And if a galaxy cluster were to all of a sudden lose a star (hypothetically), would the net force exerted onto the other galaxy cluster be the same if you counted each star individually? As another curiosity, I would like to pose this question: if a galaxy cluster were to lose one fourth of its mass, $2.0\times10^3$ light years away from the other galaxy cluster's centre of mass, how would the effect be different than if the mass was lost only $0.50\times10^3$ light years away? What I mean is, the distance would be different, so the force would be different (**(if you can address this as well please)and information would have to travel faster than light for this to occur, yet with a mass loss this great, the other cluster would be sure to notice it almost immediately).


Does causality alone resolve the mathematical ambiguity of expressing physical systems?

Newton's 2nd law of motion is most often written in the differential form $\sum F = {dp \over dt} $ but can also be expressed in an integral form $ p = \int\sum F dt $ Each form of expressing the connection between force and momentum is mathematically correct, but if we adhere to causality we are limited to the second expression (Forces lead to momentum (movement) - not vice versa). This limitation is further exemplified when one attempts to simulate the equations in a digital computer. If feedback is involved the first expression can lead to algebraic loops, while the second gracefully evolves simulated motion. Algebraic loops do not represent physical reality (even the fastest of all physical systems are limited by the speed of light!) So other than what is defined to be the independent/dependent variables, and/or the operation of either differentiation vs integration, there is nothing in the mathematics that tells us which is the 'correct' formulation of the laws of motion. The presence of differentiation in the mathematics implies prediction - knowing the future and therefore expresses the physical system in a non-causal manner. Rewriting the system in an integral form expresses the system in a causal way. The integral form in a sense reveals the roles of each factor in cause and effect. Are there perhaps other ways (missing equation(s), other types of mathematics) that can help to resolve the ambiguity that calculus can cause? Is $F = ma$ incomplete? Perhaps some expression involving entropy is required for completeness. Physical reality favors memory and shuns prediction. BTW Newton's law just served as a simple example towards my question. The ambiguity can be realized in any other mathematics involving calculus that model physical systems (including Maxwell's equations).


Distribution of Kinetic energy after an Impact

How does the huge amount of kinetic energy of some fast object like a meteor or rocket etc. gets distributed into various kinds of energy like kinetic, sound, light etc after making an impact with another body. Is there a rule under which this happens or is it TOTALLY random, and if it is random then what kind of random process does it form?


Can a person at a photon sphere of a black hole decide where the black hole is?

A person at the photon sphere of a black hole will observe the following:   * The black hole surface will cover exactly half of the visible sky (say, the left half), and cosmic horizon will cover the other half.   * Looking in any direction between these hemispheres the person will see oneself.   * The person will observe no centrifugal force however fast he moves. Any object the person thowing to the left will experience the centrifugal force pulling it further left, while any object thrown to the right also will experience the centrifugal force pulling it to the right. Approaching the black hole in such circumstances may look quite like actually escaping it to the crew.   * The space to the right as well the space to the left seemingly includes objects of any length and dimensions, the further objects are, the more they are time-dilated and length-contracted. Is there any way for a spacecraft commander to decide (by an experiment conducted in a small$^1$ amount of time) which hemisphere is the sky and which hemisphere is black hole in such circumstances? \-- $^1$ Here _small_ means small compared to characteristic time scales of the problem.


On Aharonov–Bohm effect

Aharonov–Bohm effect in brief is due to some singularities in space. In books it's infinite solenoid most of the time, which makes some regions of space not simply connected. What intrigues me is the fact that in real experiment we can't use infinite solenoid. So even if we use one and say that locally it's good approximation it doesn't change the fact that whole space is still simply connected. But the fact is that the effect was experimentally observed. So the question arises - how one should describe this effect in more rigorous manner (or maybe not rigorous but possible in real world)?


Do Quantum Energy Generators really produce more energy than they consume?

I'm not a science expert, just someone whose interested. I was taught that you cannot produce more energy than the amount put in. I received a thing on facebook asserting that a Quantum Engery Generator can do this, so I'm naturally very cynical about the claims being made on www.collective- evolution.com. However, as someone without technical knowledge, I am unable to critically examine the claims being made. I'd be grateful if someone with more knowledge than me could take a look and let me know. Thank you.


How will a particle with energy less than $V_{\rm min}$ behave?

Consider e.g. the finite square well: $V = -V_o$ between $x=-a$ and $x=a$, $V=0$ elsewhere Now for scattering states, $E$ must be $> 0$. For normalizable bound states, $E$ must be $< 0$ and $> V_{\rm min}$ (=$-V_o$ in example). But if a particle in a lab has Energy which $< 0$ and $<V_{\rm min}$, is it bound or will it scatter? And I don't know why I haven't thought about this before but what does it mean for a photon to have negative energy?


What precisely, is the string theory landscape in 10 dimensions?

I was reading this Physics.SE thread. The OP said, (changed the last word to type HO) > > > For D = 10, there are 5 vacua. Or maybe it's more correct to say 4, > since type I is S-dual to type HO. This lead me to the 2 closely related questions:   1. When considering the landscape of string theories in some $D$ dimensions, are the S-dual theories considered as the same theory, as suggested by the quote? What about the T-dual theories? I don't think the T-dual theories should, because in that dimension itself, they are different until and unless one dimension is compactified, when they become approximately equal.   2. When it is said that the string theory/M-theory landscape in $4$ dimensions in $10^{520}$, are the S-dual theories counted as the same? I don't think they are, because there is no classification of this landscape yet.


2d or 1d conduction in this scenario?

There is a rectangular fin attached to a heat exchanger with a base temperature of 350K. The fin has uniform properties and experiencesa uniform heat generation. It also experiences heat transfer with the surroundings through convection and radiation. Dies the fin experience 1D or 2D conduction?


Gravity in stone

I want to ask is it possible for a stone to have some magnetic power different from power that we see in ordinary metal magnets (something like earth gravity that attract everything). Someone told me that he saw a stone that attract other smaller stone but did not attract metals. Is it practically possible in science?


Bubbles in zero gravity

What happens if you blow bubbles into a glass of lemonade on the international space station? Since you are weightless in orbit, there's no up, down, left nor right. We define down on the Earth surface as the direction things tend to fall, down. On Earth bubbles rise if blown into a glass of lemonade (with a straw and a cover).


Newton's Bucket

Newton's Bucket This thought experiment is originally due to Sir Isaac Newton. We have a sphere of water floating freely in an opaque box in intergalactic space, held together by surface tension and not rotating with respect to the distant galaxies. Now we set the box and water to rotate about some axis and we notice that the sphere flattens into an oblate spheroid. How does the water know it’s spinning? NOTE: Newton thought this proved the concept of absolute rotation with respect to a preferred spatial frame of reference. Perhaps these days we can do better, or different?


What's the meaning of dimensionality?

I got amazed with this answer and as you can see, I had a little talk with joriki. He suggested me some examples such as the 2DEG. I'm stuck with some aspects on dimensionality, see this line: > A two-dimensional electron gas (2DEG) is a gas of electrons free to move in > two dimensions, but tightly confined in the third. In mathematics, it seems that there could be $n$ dimensions, but how's it on the "real world"? I thought that such $n$-dimensional objects on the real world could exist on the absence or presence of a specific dimension, but at least in this wikipedia article, it seems that this object _have_ the third dimension - at least I conceive "confined" as something different of "abscence". Can you clarify this to me? I am open to both answers and references on this topic, I just hope that asking both is not being too broad.


Why is there a sharp cracking sound when we break a plastic ruler? The question appears to be simple and common but I need a apropriate answer.


Wave Particle duality because of discrete time?

If time is discrete, such as the Planck's length, would the transition from one frame of time to the next explain why it appears matter changes from a particle to a wave? During that infinitely small space between each frame we can not measure the particle and it appears as a wave?


Can matter leave the cosmic horizon?

Cosmic horizon in the de Sitter space is a sphere, centered at the observer with finite radius where the red shift due to cosmic expansion becomes infinite. Given that no information can be transmitted from behind the cosmic horizon, I wonder whether any matter can ever pass behind the event horizon, because this will give rise to an information paradox, similar to the black hole information paradox. The matter approaching the cosmic horizon seems to experience time dilation, so will that time dilation prevent it from crossing the horizon? To work around the paradox, one have to postulate that the information about outgoing matter should be encoded in the de Sitter radiation.


Chern-Simons theory

In Witten's paper on _QFT and the Jones polynomial,_ he quantizes the Chern- Simons Lagrangian on $\Sigma\times \mathbb{R}^1$ for two case: (1) $\Sigma$ has no marked points (i.e., no Wilson loops) and (2) $\Sigma$ has marked points and each point has attached a representation of the gauge group. In case (1), Witten shows that the vector space should be the space of holomorphic sections of a determinant line bundle over the moduli space of flat connections. For the second case he states that the vector space should be the $G$-invariant subspace of the tensor product of all the representations associated to the marked points; by this I mean, if $\Sigma$ has $r$ marked points and each point has a rep $R_i$ then the quantum Hilbert space is $(\bigotimes_{i=1}^r R_i)^G$. Does anyone know how to interpret this second case in terms of sections of some bundle? I mean, shouldn't the second case reduce to the first when you remove the marked points? Also, Witten states, immediately following case (2), that in the presence of no marked points the quantum Hilbert space is 1-dimensional. How can one see that from the formula for the quantum Hilbert space, $(\bigotimes_{i=1}^r R_i)^G$?


Estimate number of hairs on human head

A technique of vital importance at all levels in physics is estimation. This is obvious from the first chapter in any introductory physics textbook, but is also related to the working physicist. Checking orders of magnitudes during research presentations is common practice - I've seen many good questions with good followup answers that started with "If I estimated that value I would get something much different". In general, the actually result is not the interesting thing - it's what individual things will affect the result. There are even famous examples of this: Fermi's piano tuner problem, and the Drake equation. Apparently, Fermi was so good at this that he estimated the size of the Trinity nuclear bomb test to within a factor of 2 (see the wikipedia article for a discussion of that). In this spirit, I would like to see someone try and estimate the number of hairs on the human head. The answer must include the basic assumptions so we can see where the major unknowns lie, and the best answer is one which requires no specific knowledge


Why isn't this capacitor charging?

Let's say you have a parallel plate capacitor and you connect one plate to the positive terminal of a battery and the other plate to the negative end. So this is like a static situation, you have a large positive charge (the cathode in the battery, which keeps recharging)) connected to a metal plate, and a big negative charge (the anode) connected to another metal plate. From what I've learned about static situations, the charges should equalize so that the plate connected to the positive charge should have half the charge of the cathode, and the same for the negative side. However, my physics teacher said this would not happen and he even did an experiment to show me. However, I don't understand why my reasoning doesn't work. Could someone explain to me why I am wrong? Thanks!


Galilean invariance of the Schrodinger equation

I am only asking this question so that I can write an answer myself with the content found here: http://en.wikipedia.org/wiki/User:Likebox/Schrodinger#Galilean_invariance and here: http://en.wikipedia.org/wiki/User:Likebox/Schrodinger#Galilean_invariance_2 I learned about these pages in a comment to this answer by Ron Maimon. I think Ron Maimon is the original writer of this content. This is creative commons, so it's ok to copy it here. It is not on any textbook on non-relativistic quantum mechanics that I know of, and I thought it would be more accessible (if to no one else, at least for myself) and safe here. I hope this type of question is not in disagreement with site policy.


Is there any phenomenon in physics which is sensitive to irrational numbers?

We can measure only rational numbers by our scale. Here is an example where irrational numbers does makes sense. If so then this question may have some theoretical importance. Is irrational numbers important for physics?


Boundary conditions for crystals

As students on solid state physics, we are all taught to use the periodic boundary condition, taking 1D as an example: $\psi(x)=\psi(x+L)$ where $L$ is the length of the 1D crystal. My question is:   1. Why this boundary condition is acceptable? Only because we think we are dealing with the bulk, and the surface are not relevant too much?   2. How about other boundary condations, such as zero boundary condtion, grain boundary condition etc? If ignoring their complexity, can they get the same answer as the periodic boundary condition?   3. How about other artificial boundary conditions, for example, if I let $\psi(x)=2 \psi(x+L)$, could I get the same energy band?


How was Newton able to guess that gravitational force is inversely proportional to distance squared?

This question is puzzling me since I learnt about the gravitation law in school. Why did Newton guess/assume that gravitational force is inversely proportional to the square of distance? Did he verify that experimentally? (I remember hearing that the first experimental verification of the law of gravitation was after Newton's death.) If the answer to the above question is no, is it for example more plausible to suppose that $F\propto1/r^2$ than to suppose that $F\propto1/r^4$? Did Newton carry out a thought experiment that makes $F\propto1/r^2$ a plausible guess? So in summary: Why did Newton choose exponent of $-2$ instead of any other exponent? Was it a guess that depended on pure luck or an educated guess?


How does one get these definitions of the energy momentum tensor?

I was just reading a book - Mirror Symmetry by Clay Mathematics Institute, and on Page 402 of the book, the writer says that energy momentum tensor is defined classically by $$\delta S = -\frac{1}{4 \pi} \int \sqrt{h} d^2 x \delta h^{\mu\nu}T_{\mu\nu}$$ and quantum mechanically by $$\delta_h \langle O\rangle = \left\langle O \frac{1}{4\pi} \int \sqrt{h} d^2 x \delta h^{\mu\nu}T_{\mu\nu}\right\rangle$$ **Here is my question:** I know that by Einstein-Hilbert action one can get $T_{\mu\nu} = 2 \frac{\delta \mathcal{L}}{\delta h^{\mu\nu}} $. But how can I get the exact form of $\delta S$ as above? Or $\delta_h\langle O\rangle$.


Are $G$, $F$ and $H$ (thermodynamics potentials) extensive quantities?

Internal energy $E$ is an extensive quantity for most systems. But energy extensivity is not valid in systems with long-range interactions, like gravity (e.g. in astrophysical systems). For extensive systems: $$E=TS+\bf J. \bf x +\bf{\mu}.\bf N\tag{1}$$ where $\bf J$ and $\bf x$ are intensive and extensive thermodynamic variables. (like pressure and volume, respectively) Some other thermodynamic potentials like $G$, $F$, and $H$ are defined as : $$F=E-TS$$ $$G=E-TS-\bf J.\bf x$$ $$H=E-\bf J.\bf x$$ For extensive systems, physical meanings are obvious, as is written in Wikipedia: > we can say that ΔU is the energy added to the system, ΔF is the total work > done on it, ΔG is the non-mechanical work done on it, and ΔH is the sum of > non-mechanical work done on the system and the heat given to it. Apparently these interpretations are deduced from equation $(1)$, which doesn't hold in non-extensive systems. So it seems that these potentials are ,inherently, extensive quantities, and have no meaning in non-extensive systems. Do these potentials have a physical meaning (if applicable at all) in non- extensive systems like galaxies?


Definition and Measurment of Energy and Heat in Thermodynamics

The concept of work is clear from mechanics, where $dW=F\cdot dx$, leading to $dW=PdV$. Also different forms of energy are defined in mechanics, e.g., $E=\frac{1}{2}m v^2$, etc. How are the energy and heat defined and measured in thermodynamics? Because once we define them independently, we can state the first law of thermodynamics, $d E=\delta Q+\delta W$.


Why does your reflection suddenly turn upside down while slowly walking away from a curved mirror?

Many people should have realised, when looking into a concave curved mirror (or even a rather reflective spoon in that fact) at a close distance, you will see a slightly distorted reflection. But as you move further away, the image will suddenly become upside-down. why is this?


energy production in the nuclear fission

![enter image description here](http://i.stack.imgur.com/FEj0S.jpg) according to this image of nuclear fission, before fisson reaction, the total mass of uranium and the slow neutron is 236 amu. and after adding the masses of Barium, krypton and the 3 neutrons after the fission reaction, you will again get 236 amu. Question: if we get back all the mass, where did energy come from? is it the binding energy that is released?


Is Mach's Principle Wrong?

This question was prompted by another question about a paper by Woodward (not mine). IMO Mach's principle is very problematic (?wrong) thinking. Mach was obviously influenced by Leibniz. Empty space solutions in GR would result in a Minkowski metric and would suggest no inertia. Mach's principle seems incompatible with GR. Gravitational waves could also be a problem. I had thought that papers like one by Wolfgang Rindler had more or less marginalised the Mach Principle, but I see lots of Internet discussion of it. Is it correct? Wrong? Is there evidence? (frame dragging experiments)? Let's use this definition from ScienceWorld.Wolfram.com: > _In his book The Science of Mechanics (1893), Ernst Mach put forth the idea > that it did not make sense to speak of the acceleration of a mass relative > to absolute space. Rather, one would do better to speak of acceleration > relative to the distant stars. What this implies is that the inertia of a > body here is influenced by matter far distant._


What came first, Rice Crispy or "Snap," "Crackle," and "Pop"?

The fourth, fifth, and sixth derivatives of position are called "Snap" "Crackle" and "Pop". What came first, the rice crispy characters, or the physics units?


Alpha Centauri Bb: Comparing astrometric precision vs doppler precision

How do you compare astrometric precision vs doppler spectroscopic precision? In particular will the Gaia spacecraft to be launched next year be able to confirm the existence of Alpha Centauri Bb? I.e. is Gaia's astrometric precision at least as great as the precision of the doppler spectroscopy that was used to discover the planet?


How can the electromagnetic stress energy tensor be restricted to flat space-time

The Wikipedia article describing the electromagnetic stress energy tensor seems to suggest that this tensor can only be defined in flat space-time. How is it possible to define an electromagnetic stress energy tensor this way since any available electromagnetic energy/momentum must render the space-time curvature nonzero? How in practice would someone extract useful information with this stress energy tensor?


Energy density in Magnetic fields?

I don't understand what does magnetic field energy density stand for? The possible work that the magnetic field can do? So we can calculate the amount of "work" it can do just from the energy density? Example: A magnetic field is created by a solenoid, there is a energy density of 10J/m3. From that can I derive the magnitude of magnetic force using work = F*d Thus, it would be F = W/d?


Synchronization phenomenon: A simple explanation?

Being from a mathematical background, physicists' intuitive arguments always seemed challenging for me to follow. I am currently reading a book called "Synchronization: A Universal Concept in Nonlinear Sciences". The book is well-written but from time to time, I have hard time following the intuitive arguments. In its most simple case the authors discuss the effect of a periodic external force on a self-sustained quasilinear oscillator. I have to emphasize that these are arguments in chapter 3 of the book. Things get more formal in chapter 7. The only problem is that in chapter 7 there is no argument about potentials anymore. In chapter 3 the argument mainly relies on a few parameters, $\overline\phi_e$ being the initial phase of the forcing oscillator which has the following phase relation $\phi_e=\omega t+\overline\phi_e$ and the driven oscillator is based on $x(t)=A\sin(\omega_0 t+\phi_0)$ and the external force varies based on $\epsilon \cos(\omega t+\overline\phi_e)$. The following points are the main things that I don't understand:   1. What does potential mean?(I know that potential function defines an invariance most of the times, but how it is defnied for oscillators case?    2. What are big parallel vectors and the small ones in figure 3.3(a) representing?   3. Why they phase locking take place in $\phi_0$   4. Why point 1 is stable and point 2 is not? I imagine stability means the point with minimum level of potential. I hope the answer to this question becomes clear if I get an answer for the first question. The following figures might be helpful for this clarification: ![enter image description here](http://i.stack.imgur.com/pHyPj.png) ![enter image description here](http://i.stack.imgur.com/AH8ju.png)


Readings of the detectors at Japan and Izu-Bonin-Mariana Trenches

Given the relevance of this subduction system, I would expect that a wide range of detectors (temperature, vibration, seismometers, whatever) are deployed in the depth of these trenches. What would be the canonical source at which one could access the readings of these detectors?


Properties of material affecting magnetic refrigeration

I am reading about magnetic refrigeration at the moment - I see that two factors affecting the applicability of the material for use in magnetic refrigeration is that they must react strongly (i.e the magnetic moments) with temperature at the operational temperature of the refrigerator and that to maximize cooling, necessarily a small specific heat. My book says there are other properties that could be advantageous for this application, but I am not sure what they could be. Any ideas? Many thanks.


Direction of current in concentric cylinders

Example 7.2 in David Griffiths E & M book (3rd edition) has a side view of 2 concentric cylinders, with smaller radius $a$ and larger radius $b$. The region in between $a$ and $b$ has conductivity $\sigma$. "If they are maintained at a potential difference $\textit{V}$, what current flows from one cylinder to the other for a given length $L$?" The E field is pointing radially outward along $\textit{s}$. My question is: what direction is the current? Do electrons flow in the opposite direction of an E field? If so, does that mean the current is flowing radially inward, along $\textit{-s}$, from $b$ to $a$?


The implication of anti-commutation relations in quantum mechanics

All the textbooks I saw are very clear about the implications of commutating operators in quantum mechanics. However, much less is said about anti- commutation relations. Does it have a general implication if two operators anti-commutate?


Why does physics not attempt to explain observations in addition to describing them?

Physics are there to describe observations of things' behaviours to recreate them or basically use nature for your goals. Which is not enough. Why do physicists not try to explain, why things are the way they are? In other words: > Why does it work? instead of > How does it work?


Can we describe the classical laws of physics in a frame-of-reference-independent way?

First of all, I am not a physicist, so I cannot guarantee things I say will make sense. I will try my best, though. In classical mechanics we have the notion of inertial frame of reference. If my understanding is correct, such frames are exactly those for which Newton's laws of motion have the usual form. I believe I have also heard, that they are the frames that are "moving at a constant speed" with respect to some distinguished point in space. So we have to be careful whether our frame is accelerated or not. Which makes me wonder: > Does there exist a set of physical laws (of same descriptive power Newton's > laws have, but perhaps expressing relations in some other set of physical > quantities) such that the laws from this set look the same, no matter how > the frame of reference moves? > > Does such a set of physical laws exist if we restrict to polynomially moving > frames of reference? (By this I mean that the motion of the frame is > described by a polynomial in $t$, where $t$ is time.) I imagine this could simplify some calculations.


Empirical meaning of relativity in the context of QM

In special relativity an event E is mapped to coordinates (x,t) in one inertial frame, and to coordinates (x',t') in another, and SR provides the relation between (x,t) and (x',t'). What is the empirical content of this theory (e.g. how would you operationally test it)? Since there are two frames, two observations are required - one that measures the (x,t) of E in one frame, and another that measures (x',t') of E in the other frame. Classically this presents no problem, since we can assume an ideal measurement exists that would not disturb whatever is being measured (the event E), and so two of these measurements can be made in rapid succession, one in each frame. But in QM, the result of measurement is intrinsically tied to the measurement apparatus and also the very act of measurement affects what is being measured. So when you observe an event E (say, particle position x at time t) in the first frame, using the apparatus in that frame, you can no longer observe that same event E in the second frame. Your second observation will at most be measuring the position of the particle after it has been observed in the first frame. So, in effect, a single event E does not have well defined (empirically/operationally) coordinates in all frames of reference at once. How then can we even talk about a relation between (x,t) and (x',t'), when these are not well defined?


Can colliders detect B violation?

I think there is some theoretical uncertainty whether high-energy collisions can violate B. It is known that at high temperature (higher than the Higgs scale) you violate B by SU(2) instantons. But in a situation where you have a very energetic 2-particle collision at arbitrarily high energy, I am not sure if there is a non-negligible probability of producing a Baryon violating configuration. I don't know of any calculation of B violation expected in accelerators, although there might be an argument that it should be very close to zero, because of the non-thermalizing nature of 2-particle collisions. Can you detect standard model B violation in colliders? Does LHC look for rare B violating events, or would such rare events be indistinguishable from a proton or neutron escaping undetected?


Solid objects inside the event horizon - can they remain "solid"?

So, once something is inside a black hole's event horizon, it can only move towards the center. This is fine for a point-object. But 3D solid objects rely on molecular forces to stay in one piece. These forces act in all directions inside the solid. But there could not be any interaction between atoms at different distances from the singularity, right? So, what happens to a solid (let's assume a crystalline lattice, for simplicity) hypothetically placed inside the event horizon? Does it get cleaved in ultra-thin concentric layers, centered on the singularity?


Functional Derivative of Convolution

How to carry out the following functional derivative? $$\frac{\delta F}{\delta n(r)}$$ where $$F=\int dr n(r) \int C(|r-r'|) n(r') dr'$$ is it simply: $$2 \int dr' C(|r-r'|) n(r')$$?


Why do car keys have longer range when held next to your head?

I first saw this on Top Gear, and assumed it was a load of rubbish, but today I tried it out and it actually works, if you walk away from the car continuously unlocking/locking it until it stops, and go a bit further away so that the car will not un/lock. Then out the car keys next to your head and point them directly at the car it will un/lock. How does your head/brain/etc increase the range of the keys? And why does it only work next to your head if you point the keys directly at the car, when you do not have to point towards the car when in normal range?


Does coffee with cream cool slower than black coffee?

At Modernist Cuisine, Nathan Myhrvold states that "Coffee with cream cools about 20% slower than black coffee". He then gives three reasons, two of them related to radiation and the third one to the influence of viscosity on the evaporation rate. My question is twofold:   1. **Is his conclusion correct?**   2. **Are those reasons relevant?** The question is somewhat open-ended, because "20% slower" could refer to the instantaneous cooling rate at $t=0$ or to the time needed to reach some lower temperature. It would be great to consider both parameters in the answer (will they behave the same?)


Commutator of parity and Hamiltonian operators under even potential function

I need to show what is $[H,P]$ where $H$ is the Hamiltonian and $P$ the parity operator. $V(\underset{\sim}x) = V(-\underset{\sim}x)$ in this case. I start off with $$ \langle \underset{\sim}x|HP|\psi\rangle = \langle \underset{\sim}x|(\frac{p^2}{2m}+V)P|\psi\rangle = \langle \underset{\sim}x|(\frac{p^2}{2m}+V)P|\psi\rangle = \langle \underset{\sim}x|\frac{p^2}{2m}P|\psi\rangle+\langle \underset{\sim}x|VP|\psi\rangle $$ and since $\langle \underset{\sim}x|V = V(\underset{\sim}x)\langle \underset{\sim}x|$ (is this step valid?) and $ \langle \underset{\sim}x|P = \langle -\underset{\sim}x|$, the above equation becomes $$ \langle \underset{\sim}x|HP|\psi\rangle = \langle \underset{\sim}x|\frac{p^2}{2m}P|\psi\rangle + V(\underset{\sim}x)\psi(-\underset{\sim}x) $$ Similarly I have $$ \langle \underset{\sim}x|PH|\psi\rangle = \langle \underset{\sim}x|P\frac{p^2}{2m}|\psi\rangle + V(-\underset{\sim}x)\psi(-\underset{\sim}x) = \langle \underset{\sim}x|P\frac{p^2}{2m}|\psi\rangle + V(\underset{\sim}x)\psi(-\underset{\sim}x) $$ Taking the difference of the two, I find that $$ \langle \underset{\sim}x|HP-PH|\psi\rangle = \langle \underset{\sim}x|\frac{p^2}{2m}P-P\frac{p^2}{2m}|\psi\rangle = -\langle \underset{\sim} x|\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x_i^2}P|\psi \rangle +\langle -\underset{\sim} x|\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x_i^2}|\psi\rangle $$ which I had trouble evaluating. Any hints?


Was Sun a part of a binary system?

I've read that many stars and other celestial bodies are found to constitute binary systems where the two bodies spin around each other. But our Sun is one of the exceptions. Could it be possible that the Sun was also a part of a binary system long ago ?


Can exergy and exergy destruction be understood through thermodynamical and/or statistical-mechanical principles?

My textbook _Fundamentals of engineering thermodynamics, Moran and Shapiro_ states: > The exergy is the maximum theoretical work obtainable for an _overall_ > system consisting of a system and the environment as the system comes into > equilibrium with the environment (passes to the dead state). The exergy of a system (which I'll call $\chi$), at a specified state is given by the expression: $$ \chi = (U-U_o)+p_o(V-V_o)-T_o(S-S_o) +KE +PE$$ Where $S_o, U_o$ etc. denote the dead states of a system. An Exergy change is : $$\chi_2 - \chi_1 = (U_2-U_1)+p_o(V_2-V_1)-T_o(S_2-S_1) +(K.E._2-K.E._1) +(P.E._2-P.E._1)-^?T_o\sigma_o$$ Where $\sigma_o$ is the "entropy production" and I'm not sure if it belongs there or not. The equation is clearly related to the Gibbs free energy, but the gibbs free energy is for an isothermal and isobaric process _(natural variables are T and P)_. It seems obvious to me that if a system is passing to the dead state it's temperature and pressure could change, for example: as a cup of coffee comes into equilibrium with the environment it's temperature will eventually be the same as the temperature of the "reservoir".(Note that these equations can also be expressed as rates) I haven't really been satisfied with the depth of the physics in my textbook and have had trouble finding good conceptual and mathematical explanations of exergy and exergy destruction. Can someone present a thermodynamic or statistical-mechanical description of exergy and exergy destruction? The intuitive mechanical engineer explanation I've come across is that it is about the economics of energy use ( _for example: heating your house with a radiator would be a poor use of exergy_ ), but I'd like something more mathematically rigorous if possible. Online references or a generalized example would be great too.


What is the area in Faraday's law if we have only a piece of metal moving in a magnetic field?

If a piece of metal of length $l$ is moving with a speed $v$ in a region where there is a uniform magnetic field $B$ perpendicular to it, there will be a potential difference across its terminals equal to $lvB$ which is known as motional EMF. This can be shown and understood in terms of magnetic and electric forces on the free charges in the metal. How can one calculate such EMF from Faraday's law, $\displaystyle\mathcal{E} = \left|\frac{d\Phi_B}{dt}\right|$? (where $\Phi_B$ is the magnetic flux $\int \bf{B}\cdot d\bf{a}$) (If $B$ is not changing, then the change in the magnetic flux must be due to change in an area, but the area of what? What are the boundaries of this area?)


Tight binding operators in 2D lattice system

I have a very naive problem about lattice system, how to translate common operators defined in the bulk ($\hat{x}$, $\hat{p}$...) into their lattice analogues. In a single- band tight-binding description, the hopping Hamiltonian is taken in the form $$\hat{H}_0=-J(\sum_{m,n} a^{\dagger}_{m+1,n}a_{m,n}+a^{\dagger}_{m,n+1}a_{m,n}+H.c)$$ Then in the limit of lattice constant $a\rightarrow0$, how can we recover the usual kinetic energy term? $$\hat{H}_0=\frac{1}{2m^*}(\hat{p}_x^2+\hat{p_y}^2),$$ with a effective mass $m^*=1/(2Ja^2)$.


Quantum Entanglement - an illusion based on a wrong assumption?

Almost all resources I've read about Quantum Entanglement speak about how 'amazing' it is that two entangled particles are bound over any distance, and that the state of one particle determines the state of the other. I believe that there is possibly a profoundly wrong assumption here that doesn't get addressed properly. The assumption is that when the state of one particle is observed, it is then, and only then, determined, exiting it's super position state (and thus the state of the other particle also being determined, over any distance, instantly - which is where most of the focus lies when talking about entanglement). But here is my problem - why is the assumption that the state of the first particle is being _determined_ on observation so easily accepted ? It seems to me much more logical and absolutely free of unexplained voodoo that:   1. the particles are entangled (have opposite symmetrical states).   2. The state of the first and second particles is _unknown_ and unknowable until observed, but is predetermined from the moment of the particle's inception.   3. Upon observation, nothing in the particle changes, except that our knowledge of the first particle's state leaves a "super-positioned" state into a specific one.   4. Basic logical consistency dictates that we "instantly" know the state of the second particle, without the need for "spooky action at a distance". So I guess that my basic premise is - it seems much more reasonable that our (my?) understanding of superposition is wrong, rather than that particles exchange state information instantly across any space. Please help me understand where I am wrong ?


Why do cars gain lift while going at a high speed?

I've been researching the physics involved with spoilers and I've just been turning up the same information about spoilers giving the back wheels of a car more traction by acting as an up side down airfoil. So my question is, what exactly causes the car to apparently gain lift while moving quickly, so as to merit the use of a spoiler, and is this entirely due to the aerodynamics of the vehicle in question? Edit: Clarity of question.


Why are Saturn's rings so thin?

Take a look at this picture (from APOD http://apod.nasa.gov/apod/ap110308.html): ![Saturn's rings at APOD](http://apod.nasa.gov/apod/image/1103/titansaturn_cassini_900.jpg) I presume that rocks within rings smash each other. Below the picture there is a note which says that Saturn's rings are about 1 km thick.   Is it an explained phenomenon?


Resampling spectral data?

I want to use the CIE CMFs (Color Matching Functions) ![CMFs](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8f/CIE_1931_XYZ_Color_Matching_Functions.svg/446px- CIE_1931_XYZ_Color_Matching_Functions.svg.png) provided here: http://www.cvrl.org/cmfs.htm With data from my spectrometer in 10 nm increments. However, the CMFs are only provided at 5 nm increments. What would be the best way to do this? Would it be okay to resample the spectral data with a linear interpolation? Any ideas?


What's the Standard Model width of a 125 GeV Higgs?

There's a fairly broad mass spread in the new results out of Atlas and CMS. I'm curious how this fits with the expected SM width.


Noether Charge For Scalar Fields Under Lorentz Transformations

The conserved charge associated with the Lorentz transfomation of a **scalar field** is given by $Q^{\alpha\beta}=\int d^3x\frac{1}{2}(x^\alpha T^{0\beta}-x^\beta T^{0\alpha})$. The quantities $Q^{ij}$ is recognized as the components of orbital angular momentum but what physical quantities does $Q^{0i}$ correspond to for $i=1,2,3$? This cannot be intrinsic angular momentum or spin because we are considering a scalar field. What is the significance of these quantities?


When is temperature not a measure of the average kinetic energy of the particles in a substance?

I had always thought that temperature of a substance was a measure of the average kinetic energy of the particles in that substance: $E_k = (3/2) k_bT $ where $E_k$ is the average kinetic energy of a molecule, $k_b$ Boltzmann's constant, and $T$ the temperature. (I'm not sure of the 3/2 coefficient.) Then I heard from several folks that this is a simplistic notion, not strictly true, but they didn't explained what they thought was flawed with this idea. I'd like to know what (if anything) is objectionable about this idea? Is it that the system must be macroscopically at rest? Is it that it ignores the quantum mechanically required motion of particles that persists at low temperatures? When is it not valid?


The Einstein-Klein-Gordon (EKG) equations

I am a little confused about a few papers I read on the Einstein-Klein-Gordon (EKG) equations. From what I understood one takes the energy-stress-tensor of the scalar field: $$T_{\mu\nu } = −\partial_\mu \varphi ∂_\nu \varphi − \frac12 g_{\mu\nu}∂_\alpha\varphi∂_\alpha\varphi − V(φ )$$ $$V(φ) = −\frac12 (mφ)^2 + \fracκ4φ^4$$ Where $κ$ is the usual self-interaction coupling constant . Then this stress-tensor is plugged into Einstein's equation and solved usually with the Schwarzschild or other convenient metrics. Now from what I know about QFT, isn't $φ$ an operator that maps the Hilbert space $H$ of particle states to $H$ itself? Doesn't that make the components of the energy-stress-tensor observables and thus operators as well? If so, then how can one equate the components of the Einstein tensor (which are purely geometric tensor fields) to operators? I once read that people (as of today) usually plug in < $T_{μν} $ > in Einstein's equation, but in the papers I read they directly used the operator itself and not it's expectation value. I also wondered how people plug the Maxwell stress tensor into Einstein's equations in a similar way? I get that the EM and scalar field are real valued fields, but shouldn't we use the expectation values in Einstein's equation? What am I missing? Here is a link to one such paper: http://arxiv.org/abs/0805.3211


Maximal Gravity

I found this interesting problem in Introduction to Classical Mechanics with Problems and Solutions by David Morin: > Given a point $P$ in space, and given a piece of malleable material of > constant density, how should you shape and place the material in order to > create the largest possible gravitational field at $P$? Any ideas?


What does it mean to be stationary?

I'm looking for a simple answer. What do we regard a stationary. Do we mean an object that is not moving noticeable from the viewers perspective because then a parked car would be considered stationary, but (assuming it's on earth) the earth is rotating and those at the circumference where the spin is greatest are traveling 1 038 mph and the earth itself is moving with the Milkyway at 600 000 mph. I have a bunch of other questions but I rather not put them all in one post, but do we really account for theses other speeds or do we ignore them.


What's the typical material used as a low-k ILD in semiconductors?

Semiconductor companies (namely IBM, Intel, TSMC and etc.) may use different low-k materials as inter-layer dielectric (ILD) in their fabrication process. I'm wondering what is the most typical one. Where can I find their thermo- electro-mechanical material properties? I need these information for FEM simulations.


If nothing weighs something then doesn't that mean there is mass associated with spacetime?

Lawrence M. Krauss says, "If you removed all of the particles, all of the radiation,absolutely everything from space and all that remained was nothing that nothing would weigh something." What weighs something has mass. If you remove everything else then all that is left is spacetime. Meaning, there is mass associated with spacetime. Mass physically occupies three dimensional space. A particle moving through the mass associated with spacetime would displace the mass analogous to the bow wave of a boat. If mass is associated with spacetime then wouldn't it be the mass associated with spacetime which waves in a double slit experiment?


Does Hamilton Mechanics give a general phase-space conserving flux?

Hamiltonian dynamics fulfil the Liouville's theorem, which means that one can imagine the flux of a phase space volume under a Hamiltionian theory like the flux of an ideal fluid, which doesn't change it's volume. But is it possible to reproduce every phase space conserving flux with an appropriate Hamiltonian? So can I simply imagine the entity of all possible Hamiltonian dynamics as all possible phase space conserving fluxes? Or are Hamiltonian dynamics a special case for phase space conserving fluxes? If they are a special case, what would be an example for a phase space conserving flux for which there is no Hamiltonian that can produce it?


Is the Ground State in QM Always Unique? Why?

I've seen a few references that say that in quantum mechanics of finite degrees of freedom, there is always a unique (i.e. nondegenerate) ground state, or in other words, that there is only one state (up to phase) of the Hamiltonian with the minimum eigenvalue. My questions: Q1) Is it true? Q2) Under what condition is it true? Q3) I can easily construct a Hermitian operator, in a finite dimensional space, which has two lowest eigenvectors. For example, if $ \left\\{ {\left| a \right\rangle ,\left| b \right\rangle ,\left| c \right\rangle } \right\\} $ is an orthonormal basis of a 3-dimensional Hilbert space, define a Hamiltonian $H = 1 \cdot \left| a \right\rangle \left\langle a \right| + 1 \cdot \left| b \right\rangle \left\langle b \right| + 2 \cdot \left| c \right\rangle \left\langle c \right|$ . Then $\left| a \right\rangle $ and $\left| b \right\rangle $ are two ground states. How is that consistent with the answer to Q1 (if that answer was positive...)?


Do laser beams interfere noticeably with each other?

I want to arrange some red lasers along a 90 degree arc of a circle, aimed towards the center of the circle. When their beams hit the opposite side of the circle, will I notice any interference effects? What if I use dozens of lasers? ![Laser arc](http://i.stack.imgur.com/Ok0m9.png) If they do interfere, how do I calculate the interference pattern?


Identification of extended quantum objects?

In some physics theories like string theory we have notion of spatially extended quantum objects - strings, membranes etc. Assuming that such objects exist, how would they appear in experiments ? More precisely are there any thought experiments for determining whether a quantum thing being observed has spatial extension or not, and if it has spatial extension then what is its dimension, topology etc.


Why can radio wave travel through walls while imfrared can't?

> **Possible Duplicate:**   >  If both radio waves and gamma rays can travel through walls Does it have anything to do with the materials used to build the wall? What determines if a material can absorb an (x) wavelength?


Mirror image at relativistic speeds

Imagine moving parallel to the surface of a very large flat mirror at relativistic speeds. What would be the effect of viewing yourself in the mirror? At non-relativistic speeds your image would be similar to what you would see when stationary. Assume you are moving to your right with respect to the mirror at high speeds. Without doing the math I would expect your image would start moving to your left. Also, your image would start to distort because the path lengths of the photons from various parts of your body would take different times to arrive back at your eye. So what arrives simultanously to your eye would be different regions of yourself at different times in the past. Also, assuming that the light source that is illuminating you is moving with you, the photons would transfer some of their momentum and energy to the mirror, and so the light from the image would be red-shifted. The last part I am not sure about, is reflection a process of absorption and re-emission from the surface atoms of a mirror?


Minimal Maxwell's Demon

I would like to understand where the waste heat is generated in the Maxwell's demon problem. To this end I've come up with the simplest scenario I can think of. If my scenario is workable I am hoping someone will be kind enough to apply the right math to it to show what is going on and why it does not work to provide a "free lunch." My scenario is as follows: Take a 1 dimensional container of length l. Start with an evenly distributed population of particles; slow red ones and fast blue ones, traveling at +v or -v, and +2v or -2v respectively. The demon sits at the boundary at l/2. He lets slow red particles at +v and fast blue particles at -2v through and keeps the opening between the two halves of the container closed otherwise. The idea of using a one dimensional container is to keep the math as simple as possible. In order for this to work the particles have to be able to go right through one another without interacting. I am imagining that if one can work out what happens in this simplest case then it should be possible to extend the scenario to a 2-dimensional container in which the line forming the barrier between the two halves of the container is made up of discrete copies of the Maxwell's Demon in my 1-dimensional scenario.


Does photon possesses no time to cover any arbitrary distance?

Photon travel 8 minutes (with speed $c$) from the sun to reach the earth. Any particle (or space-ship) with velocity $0.99 c$ covers the same distance (93 millions km) within less than 2 minutes (according to SR). Does this make sense?


What happens when you heat vodka in a microwave?

Since ethanol has a lower dielectric constant than water would the water heat up and boil before the ethanol? Would the water transfer heat to the ethanol and, since ethanol has a lower boiling point, bring it to a boil before the water?


Statistical physics of molecular dissociation of a diatomic gas

Say there are $N$ atoms of type $A$ in a box of volume $V$ and they are undergoing a reversible association-dissociation reaction $A + A = A_2$. Let an $A$ atom have mass $m$, and hence the molecule $A_2$ has mass $2m$. Assume that the temperature $T$ is high-enough that Boltzmann statistics applies. Say $N_1$ be the number of $A$ atoms and say $N_2$ be the number of $A_2$ molecules at equilibrium. Then $N_1 + 2N_2 = N$. If there had to be $N$ $A$-type atoms in the the box then I guess the partition function $Z$ would be given by, $$Z = \frac{1}{N!} [ \int e^{-\frac{\beta p^2}{2m}} \frac{d^3p d^3q}{h^3} ] ^N = \frac{1}{N!} [ \frac{V}{h^3} (2\pi m kT)^{\frac{3}{2}} ] ^N .$$ Say that the dissociation energy of the reaction is $E_0$. Say $N_1$ be the number of $A$ atoms and say $N_2$ be the number of $A_2$ molecules at equilibrium. Then $N_1 + 2N_2 = N$. I want to write the partition function for the above mixed system in equilibrium. Using the expression for $Z$ as stated above, I am guessing that this partition function (say $Z_N$) would look like, $$Z_N = \sum _ {N_1 + 2N_2 = N} e^{-\beta N_2 E_0} \frac{1}{N_1 !} [ \frac{V}{h^3} (2\pi m kT)^{\frac{3}{2}} ] ^{N_1} \frac{1}{N_2 !} [ \frac{V}{h^3} (4\pi m kT)^{\frac{3}{2}} ] ^{N_2}. $$ (In the above I am using the idea that for $N_2$ molecules to have formed $N_2$ $E_0$ (binding) energy would have been released and hence that much of extra energy has to be accounted for which is not captured in the partition functions over the $N_1$ atoms and $N_2$ molecules...)   * Is the above expression for the partition function correct ?  One further introduces the grand canonical partition function (say $Z_{\mu}$ ) for the above system with a chemical potential $\mu$ to "take care of the constraint $N_1 + 2N_2 = N$", $$Z_{\mu} = \sum _ {N=0} ^\infty e^{\beta \mu N} Z_N.$$   * Firstly I don't understand the meaning of the $\mu$ in the above. (I can't relate it to what I would usually think as chemical potential) I mean..isn't the above expression for $Z_{\mu}$ independent of $N_1$ and $N_2$?    * How does the $Z_{\mu}$ help find the expectation value for $N_1$ and $N_2$?   * If my expressions for $Z_N$ is correct then does $Z_{\mu}$ have a closed form expression?  All this is to understand the claim that as expectation values the following relationship apparently holds, $$\frac{N_1 ^2} {N_2} \sim e ^{-\frac{|E_0|}{kT}} .$$ I would be glad to know of other insights along the above.


QM interpretation of Larmor precession

Typically when I think of a spin moment in an external field, I visualize it classically in terms of a magnetic moment precessing around the external field vector. Now how is this described quantum mechanically? My guess would be that the external field causes transitions between the up-state and down-state at the Larmor frequency. However if the Zeeman energy was much greater than the thermal energy, wouldn't the spin need to stay in the low energy state? Does that mean the spin ceases precession in the classical picture?


How to interpret the magnetic vector potential?

In electromagnetism, we can re-write the electric field in terms of the electric scalar potential, and the magnetic vector potential. That is: $E = -\nabla\phi - \frac{\partial A}{\partial t}$, where $A$ is such that $B = \nabla \times A$. I have an intuitive understanding of $\phi$ as the electric potential, as I am familiar with the formula $F = -\nabla V$, where $V$ is the potential energy. Therefore since $E = F/q$, it is easy to see how $\phi$ can be interpreted as the electric potential, in the electrostatic case. I also know that $F = \frac{dp}{dt}$, where $p$ is momentum, and thus this leads me to believe that $A$ should be somehow connected to momentum, maybe like a "potential momentum". Is there such an intuitive way to understand what $A$ is physically?


Is it theoretically possible for the orientation angle of a projectile to remain exactly equal to the orientation of velocity?

This question is sparked by my answer to this question: Is this simulation following real physics? After examining the math, I don't see how it is theoretically possible for the situation simulated to happen. The situation in question is whether or not a spear in projectile motion can stay oriented over the entire path such that the angle is equal to the angle of the velocity vector. I start with proposing that the rate of change of the angle of the velocity vector must be a constant - there can be no torque applied during the projectile motion, only the initial angular velocity imparted at the release. So taking a projectile, with say an initial velocity of <20, 20> - this would be at a 45 degree angle, the case where the condition intuitively seems most likely: $\frac{dy}{dt} = -9.8t + 20$, $\frac{dx}{dt} = 20$ We find $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{-9.8t + 20}{20}$ And an expression for the angle, then: $\theta = \arctan{\frac{dy}{dx}}$ Graphing the derivative of this function over the time span of the projectile shows that it is NOT a constant. This implies that there would have to be some sort of torque being applied mid-air if the angular displacement is directly proportional to $\theta$. However, it IS a very shallow bowl. First of all, is this analysis correct? Second, if this implies that the condition is impossible, what happens when one throws a javelin or spear perfectly? Why is it so natural to do so?


How to calculate the intensity of the interference of two waves in a given point?

There are two different point sources which produce spherical waves with the same power, amplitude, ω, wavenumber and phase. I can calculate the intensity of each wave in a point: $$ I_1 = P / (4 \pi r_1^2) $$ $$ I_2 = P / (4 \pi r_2^2) $$ But how can I calculate the intensity of the resulting wave in that point?


How would I explain a farad being a second per ohm?

The farad has many alternative representations in SI. I can comprehend some of them. For example, a capacitance of 1 farad means that it will take a single coulomb to create a single volt between the capacitor plates. It could also mean that it will take one second of continuous 1A current to create that volt between the plates. In this manner, how would I imagine the second/ohm representation?


Permanently charging a sphere by induction using a high voltage

I want to charge a sphere using a van de graff generator , it would sound easy , but using a 1 MV or a 900 KV is really hard, it is enough to break like 1 meter of air so whenever i try to ground the sphere there is spark . So i wanted to know if it will help if i covered the sphere with a layer of plastic or an insulator , or if there was a vacuum between the the generator and the sphere. And another thing, can a graphite rod be charged like a metal one ?


Light Ray Reflection from concave mirror

Suppose a ray of light hits a concave mirror and is parallel to principle axis but far away from it such that it doesn't follow _paraxial ray approximation_. Will it pass through focus or between focus and radius of curvature or between pole and focus? Here pole, focus and radius of curvature mean the same thing as in paraxial ray approximation .


Will a black hole eventually turn into a neutron star?

As far as i understand, black holes radiate away energy in form of Hawking Radiation. Thus, they lose mass, i suppose. Is there a point where the mass becomes too small for the object to still be a black hole? What happens then, will it turn into a neutron star supported by degeneracy pressure? Or is it all about density, and once a black hole is formed, it will stay a black hole (i guess until it completely radiates away)?


Double decay $\beta\beta$ observation and neutrino mass

If the double decay $\beta\beta$ will be detected this means the neutrino is a Majorana particle coincident with its antiparticle. At the moment the half life of this decay is put to $\tau\ge10^{25}$years. The limit of the Majorana neutrino mass is about 0.2–0.4 eV. What is the relation between the probability of $\beta\beta$ decay and the neutrino mass? Thanks.


Magnetic field and AC current

If I run an ac current under a plate of iron filings, what pattern should the filings form? If I use the plug for an appliance, will the magnetic fields in the two wires cancel each other out (more or less)?


Steel filings just like iron filings?

I am having problems with an experiment involving field lines. I filed a piece of steel to get "iron filings." Ought these steel filings be vastly different from the iron filings typically used in this sort of experiment?


Why are anti-de Sitter spaces so interesting when we believe the universe is expansionary?

Perhaps this is a naive question, but in my recent (admittedly limited) readings about AdS spaces, I keep wondering why they seem to be such a hotbed for theoretical research (AdS/CFT correspondence, etc.). To my understanding, an AdS space has constant negative curvature in a vacuum, which should yield an attractive universe, not one with accelerating expansion. An AdS space can be thought of as having a negative cosmological constant, while a universe with accelerating expansion would imply that such a constant be positive. Since we observe that our universe's expansion is accelerating, it seems that if anything, we should be seeking to model it as a de Sitter space. Am I mistaken? What aspects of our universe do AdS spaces attempt to model?


Does this equation mean anything?

I saw this equation on a coffee mug, and I am wondering what it means. I am not far enough along in physics to understand it. Does it actually mean anything or is it gibberish? ![enter image description here](http://i.stack.imgur.com/ifd4q.jpg) I can't embed the image because I am a new user. (But I can, L.M.)


1-form formulation of quantized electromagnetism

In a perpetual round of reformulations, I've put quantized electromagnetism into a 1-form notation. I'm looking for references that do anything similar, both to avoid reinventing the wheel and perhaps to cite. Taking the electromagnetic field $\hat A$ to be an (operator-valued distributional) 1-form, we can write a smeared operator $\hat A_H$ in terms of the inner product on the exterior algebra as $\bigl<\hat A,H\bigr>$ (taking the test function $H$ also to be a 1-form, but satisfying Schwartz space-like smoothness conditions in real space and in fourier space instead of being a distribution). $\hat A$ projects into annihilation and creation parts, $\hat A^+$ and $\hat A^-$ respectively, $\hat A=\hat A^+ +\hat A^-$, for which the action on the vacuum is defined by $\hat A^+\left|0\right>=0$, and we have the commutation relations $\Bigl[\bigl<\hat A^+,H\bigr>,\bigl<\hat A^-,J\bigr>\Bigr]=\left<H,E(J)\right>$, where $\widetilde{E(J)}(k)=2\pi\delta(k^2)\theta(k_0)\tilde J(k)$ projects to the positive frequency forward light-cone. The commutation relations are not positive semi-definite for arbitrary test functions $H$, $\Bigl[\bigl<\hat A^+,H^*\bigr>,\bigl<\hat A^-,H\bigr>\Bigr]\not\ge 0$, which is fixed by the Gupta-bleuler condition, which can be stated in this formalism as $\delta\hat A^+\left|\psi\right>=0$, for _all_ states $\left|\psi\right>$, not just for the vacuum state. By the Hodge decomposition theorem, we can write the test function $H$ uniquely as $H=d\phi+\delta F+\omega$, where $\phi$ is a 0-form, $F$ is a 2-form, and $\omega$ is a harmonic 1-form, so we can write $$\bigl<\hat A^+\\!,H\bigr>\left|\psi\right>\\! =\bigl<\hat A^+\\!,d\phi+\delta F+\omega\bigr>\left|\psi\right>\\! =\Bigl(\\!\bigl<\delta\hat A^+,\phi\bigr>+\bigl<\hat A^+\\!,\delta F+\omega\bigr>\\!\Bigr) \left|\psi\right>\\! =\bigl<\hat A^+\\!,\delta F+\omega\bigr>\left|\psi\right>.$$ The harmonic 1-form $\omega$ has to be zero to satisfy the Schwartz space condition, and by the left action of $\hat A^-$ on arbitrary states we have the same projection of the arbitrary test function $H$ to $\delta F$, so all we are left with is the electromagnetic field observable $\hat\Phi_F=\hat A_{\delta F}=\bigl<\hat A,\delta F\bigr>$. Unsurprisingly, in the free field case, in the absence of charges, the electromagnetic potential observable is exactly equivalent to just the electromagnetic field observable, for which we can verify that $\Bigl[\bigl<\hat A^+,\delta F^*\bigr>,\bigl<\hat A^-,\delta F\bigr>\Bigr]\ge 0$, using which we can use the GNS construction of a free field Hilbert space. So this is a reference request. Is there any literature that uses this kind of mathematical formalism for the quantized electromagnetic field? Even _vaguely_ the same! My sense is that AQFT has moved to more abstract methods, while more practical interacting QFT has become historically committed to index notations that are little changed from 50 years ago, even though the methods of interacting quantum fields have changed in many other ways, and that mathematicians who take on the structures of QFT, as Folland does in http://www.amazon.com/Quantum-Theory-Mathematical-Surveys- Monographs/dp/0821847058, make strenuous efforts not to let their notation and methods stray too far from the mainline.


Loop of wire in a varying magnetic field

Let us suppose we have a loop of wire with some definite resistance in a magnetic field. Let the magnetic field be varying. This varying magnetic field will st up an EMF in the wire. My questions are as follows: 1). Energy is dissipated across the wire resistance. Where does this energy come from? 2). The current generated in the loop will also generate a varying magnetic field. Will this magnetic field induce another EMF in the loop setting up another varying magnetic field? Will this go on indefinitely?


What happens to speed and frequency of a light beam moving in transparent medium when observed from different inertial frame of reference?

Suppose a transparent medium where speed of light is $c/n$, an inertial frame of reference $K$ which is stationary relatively to the medium and an inertial frame of reference $K'$ which is moving with speed $V$ according to $K$ along the $x$ axis. Also suppose that a light beam is travelling through the medium and its frequency measured from $K$ is $f$ and from $K'$ is $f'$ and the angle between the $x$ axis and the bean is $\theta$ and $\theta'$ respectively. If we know $n,f,\theta'$(or $\theta$ or both) and $V$ how can we calculate $f'$? * * * From a book I have read the transformation was the following: $$f'~=~\frac{f}{\gamma (1+\beta n\cos\theta')},$$ where $$\gamma~=~\frac{1}{\sqrt{1-\beta^2}},\qquad \text{and}\qquad \beta~=~v/c.$$ I can prove it with momentum-energy Lorentz transformations only if I suppose that light is moving with same speed ($c/n$) both in $K$ and $K'$ which (I think) is wrong according to speed transformation. Does the speed of light when moving through a transparent medium change when observed form a different inertial frame of reference or not?


Any difference between "Mueller matrix" and "Scattering matrix"?

I find in some references 4x4 Mueller matrix and in other references 4x4 Scattering matrix. Are they different or identical? If they are different, I would like to know the book or any research paper describing the relation between them.


Would a craft travelling increasingly close to the speed of light appear to be decelerating?

I've been pondering the implications of time dilation. Is it right to assume that if a craft was travelling at a speed very close to the speed of light (>= 0.9999c for example), that to an external observer it would appear to be moving very much slower? Example: at 0.9999c, time dilation is 70.71, so while the craft is travelling at 186,263.37 miles/s, doesn't that mean to a (stationery) observer it's only moving at 2,634.10 miles/s? And at 0.99999c, time dilation is 223.61, so while the craft is now travelling at 186,280.14 miles/s, doesn't that mean to the observer it's now only moving at 833.07 miles/s? The obvious implication is that as the craft approaches the asimptote of the speed of light, external observations (beyond around 0.9c) see it decrease in velocity towards the asimptote of being stationary. Is this reasoning correct?


How about retrodiction, if climate prediction is difficult?

What I mean by the question is if _retrodiction_ has been attempted back to, say, 50 or 100 years starting with the presently known state of climate as initial conditions using the same equations and methods that are popularly used to _predict_ climate 50 or 100 years ahead? If yes, how well retrodiction turned out? If not, why not?


What if all the electrons leave a metal?

I was studying photoelectric effect. Then I thought that what will happen if all the electrons from a metal piece come out as photoelectrons by using a light source of particular frequency? Will the metal still exist?


Cycling down a mountain

I am Dutch, so I cycle a lot, hence the following question. Say I am standing at the top of a mountain on my bicycle. I have an amount of energy $E$ available. Say that after the slope of the mountain ends, there is an infinite long, straight, smooth road ahead. How do I get the furthest on this flat road? Options: a) using all my energy $E$ to peddle while I am coming down the mountain b) roll gently down the mountain and start peddling precisely when I am at the end of the mountain c) roll gently down the mountain and down the straight road and only when I am starting to get to a hold, I start cycling very slowly, keeping my acceleration and speed to a minimum d) some kind of hybrid of the above. In this story I would like to keep it as realistic as possible: there's friction, both from my tires and from air resistance. I once heard that air resistance goes $\propto v^2$, such that I am tempted to opt for option c). I am interested in your opinions/thoughts. Help me spend as little energy as possible while cycling!


State-dependent diffusions: Fick's law vs. Fokker-Planck's, which and why?

Consider a "state-dependent diffusion": a diffusion process for which the diffusion coefficient $D(x)$ depends on the (stochastic) state $x$ of the system. (An example is provided by the diffusion of tracers in a spatially inhomogeneous bath, e.g. with varying viscosity or temperature.) What is the associated flux? If $c(x)$ is the probability to find the system in the state $x$ at a given time, should the flux read $j(x)=-D(x)\nabla c(x)$, as in Fick's law? Or rather $j(x)=-\nabla(Dc)(x)$, as in the Fokker-Planck equation? (I'm using the terms "Fick" and "Fokker-Planck" here only as tags for each alternative.) It is well known that this question does not have a definite answer, and should be answered case by case. My actual question is threefold:   1. Do you know examples of physically relevant state-dependent diffusions? What category do they fall in? Fick, or Fokker-Planck?   2. Do you have an intuition for the physics underlying this alternative?    3. The Fokker-Planck case has the peculiarity to lead to equilibrium distributions which are not of the Boltzmann-Gibbs form (the steady-state $c(x)$ depends on $D(x)$ and not just on the state's energy). What does this tell us about the foundations of statistical mechanics?


Dimensionial Regularization (integrating over extra dimensions)

I tried to replicate a result about dimensional regularizations in the review article of Howard Georgi, _Effective Field Theory_ at: people.fas.harvard.edu/~hgeorgi/review.pdf. On page 17, he writes about Integrals of the form: $$I = \int \frac{d^4l}{(4\pi)^4}\frac{1}{(l^2 + A^2)^\alpha}$$ He then regularizes the integral in the form $$I = \int \frac{d^{4 + \delta}l}{\mu^\delta(4\pi)^{4+\delta}}\frac{1}{(l_\delta^2 +l^2 + A^2)^\alpha}$$ and integrates seperately over the extra dimensions $\delta$ getting: $$I = \int \frac{d^4l}{(4\pi)^4} \left(\int \frac{d^{\delta}l_\delta}{(4\pi\mu)^{\delta}}\frac{1}{(l_\delta^2 +l^2 + A^2)^\alpha}\right) = \\\ \int\frac{d^4l}{(2\pi)^4}\left(\frac{1}{(l^2 + A^2)^\alpha}\frac{\Gamma(\alpha - {\delta\over 2})}{\Gamma(\alpha)}\left(\frac{l^2 + A^2}{4\pi\mu^2}\right)^{\delta\over 2}\right)$$ This seems to make perfect sense to me, but when I try to calculate the integral over the extra dimensions $\delta$, I run into problems. First of all, the angular part of the Integration over $\delta$ is $$\frac{\pi ^{\delta /2} \delta }{\Gamma \left(\frac{\delta }{2}\right)}$$ The radial part of the integral is: $$\int \frac{d^{\delta}l_\delta}{(4\pi\mu)^{\delta}}\frac{l_\delta^{\delta-1}}{(l_\delta^2 +l^2 + A^2)^\alpha} = \frac{\Gamma \left(\frac{\delta }{2}\right) \Gamma \left(\alpha -\frac{\delta }{2}\right) \left(A^2+k^2\right)^{\frac{1}{2} (\delta -2 \alpha )}}{2 \Gamma (\alpha )}$$ Together I get: $$\frac{\pi ^{\delta /2} \delta \Gamma \left(\alpha -\frac{\delta }{2}\right) \left(A^2+k^2\right)^{\frac{1}{2} (\delta -2 \alpha )}}{2 \Gamma (\alpha )}$$ All in all I get a extra factor of $\delta$ which makes the integral zero in the limit $\delta\rightarrow 0$. But it should not give zero? I can calculate dimensionally regulated integrals the usual way, integrating over all dimensions d at once, reproducing all the known results. But I fail at the Integration over the extra dimensions $\delta$. Where is my error? (How) Is it possible to integrate over extra dimension $\delta$ seperately?


Rotation in an 'empty' universe

> **Possible Duplicate:**   >  Is rotational motion relative to space? Assume a universe with the same physics as ours, but containing only one rotating (charge-free) body - let's say the size of the Earth. Would the rotation cause a bulge at the equator of the rotating body?


Apollo and orbital mechanics: orbital decay if the Trans Earth Injection (TEI) burn had failed

I'm reading Jim Lovell (Apollo 8 and 13) and Jeffrey Kluger's book _Apollo 13_ , which is a fantastic read about a long past era I only have kindergarten memories of. On page 54 there is a paragraph that reads (emphasis mine): > Also, unlike the LOI burn, during TEI there would be no free-return > slingshot to send the ship home in the event that the engine failed to > light. If the hydrazine, the dimethylhydrazine, and nitrogen tetroxide did > not mix and burn and discharge just so, Frank Borman, Jim Lovell, and Bill > Anders would become permanent satellites of Earth's lunar satellite, > expiring from suffocation in about a week and then **continue to circle the > moon, once every two hours, for hundreds -- no thousands; no, millions -- of > years.** Apollo 8 was in an elliptical orbit of 169.1 × 60.6 miles. I am skeptic about how long such an orbit would be stable. My skills in orbital mechanics are rudimentary at best and I know there is no atmosphere to cause orbital decay due to atmospheric drag. But I have heard about other effects causing orbits to decay over time, such as variations in the moon's gravitational field (density anomalies, mascons). This may cause lunar satellites to eventually crash into the moon. Can someone estimate how long this would be for the case of the Apollo 8 lunar orbit?


Is there any way to anticipate exceptional Aurora Borealis viewing conditions?

I've always wanted to see the Aurora Borealis, but never had the opportunity, since it is so rarely visible in my area (Pennsylvania). However, last week was apparently a rare opportunity of exceptional visibility. Unfortunately, I missed my opportunity, as I did not find out about it until the next day. Are there any good resources or strategies for anticipating opportunities like this? Are coronal mass ejections like the one that precipitated this event predictable? If so, are such predictions available to the amateur public?


Show that the electric field, $\mathbf{\vec{E}}$ cannot be generated by any static distribution of charges

> Show that the electric field $$\mathbf{\vec{E}}=\begin{pmatrix}0 \\\ E_0x > \\\ 0\end{pmatrix}$$ where $E_0$ is a constant, cannot be generated by any > static distribution of charges. I understand that an electrostatic field is irrotational and the divergence of the electric field is the charge density over epsilon 0 but I have no idea why that makes the answer.


How 4-vector nature of the value is connected with it's conservation law?

In electrodynamics Poynting vector and energy flux of field don't create 4-vector. Also they aren't conserved independently from substance (conservation law includes summand connected with current density). In linearized gravity mass density and mass current density as components of stress-energy tensor also aren't conserved. And they also aren't components of 4-vector. Is facts of non-conservation of values above connected with the absense of 4-vector nature?


Can a super conducting magnet & gravity create a cheap source of energy?

I have a question about whether this would produce a cheap source of energy. If you created two superconductive magnets configured in a figure of eight layout, in a cold enough environment so you needed no extra power for cooling and the magnets would be in persistent state (so after initial charge would need no extra power). If you then made these magnets levitate a very heavy piece of iron could you then use the gravity of the earth (or moon if the experiment was set up on the cold moon) to generate power when then power was switched from one magnet to the other magnet. E.g. The iron block 1 would rise, power would move to magnet 2, iron block 1 would fall (maybe turning a flywheel on the way down), iron block 2 would rise etc. In my mind this configuration would not cost any additional energy ( as long as the environment was cold enough) and you could get virtually free energy by utilising the gravitational field of the earth or moon... Has this idea got legs?


Choosing the right point when calculating moments

A body is composed of two straight pins that are joined at a right angle. They have lengths $a$ and $b$ and the mass per unit length is $\rho$. When the body is balanced on a flat surface, as shown, how large is the normal force against the ground in the right point of contact? 4 options as can be seen in the picture. ![picture](http://i.imgur.com/Hr1RBRF.jpg) Let me point out that this is a conceptual question. When I first tried to solve this problem I decided to choose to calculate the moment around the left contact-point in order to reduce one term(left normal force). This seems like a natural way but gives a false answer($N_2= \rho g$. If I instead calculate about the vertex of the triangle and use Newton's second law I get the correct solution.(answer is D). So how should I choose the "correct" point? **Around left contact point:** By dropping an altitude **h** at the right angle we get that: Notice that: $\cos\alpha=\frac{a}{\sqrt{a^2+b^2}}$, $\cos\beta=\frac{b}{\sqrt{a^2+b^2}}$ and $m_ag=\rho ag$ and $m_b=pbg$ $-\frac{\rho a^2g}{\sqrt{a^2+b^2}}-\frac{\rho b^2g}{\sqrt{a^2+b^2}}+N_2\sqrt{a^2+b^2}=0$. Simplifying gives $N_2=\rho g$


Poseuille number for flat channel?

For a flat channel with fully developed flow, the Poseuille number when using the hydraulic diameter is $$Po=C_fRe_{D_{H}}=24$$ But when using 2b as the length scale, the Poseuille number is $$Po=C_fRe_{2b}=12$$ Why is this? I thought that the hydraulic diameter of a flat channel is 2b. Shouldn't they be the same? In the derivation used, b is half the channel width


How to measure vector potential such as that of magnetic field?

If we want to measure gravitational potential energy of a body-earth system, we can simply do it by measuring the body's height from ground. How could we can measure vector potential such as that of magnetic field?


What happens if you let a cable roll slip into a black hole?

1. Does the cable roll spin faster the more cable goes into the black hole in reference of a observer standing next to it?   2. Can gravity pull the cable that it exceed the speed of light inside a black hole in reference to the center of the black hole?


Translation invariance in crystals

**Context** : solid state physics - dynamics of atoms in crystals. Coupling constants are defined as: \begin{align*} \frac{\partial^2\Phi}{\partial r_{nai}\partial r_{mbj}} = \Phi_{nai}^{mbj} \end{align*} where $\Phi$ is the total energy of the crystal expressed as a function of nuclei coordinates. $r_{nai}$ is the position vector of the $n$-th cell, $a$-th atom at $i$-th spatial direction. My book says that coupling constants have some properties. One of them is the "translation invariance", which is stated as: \begin{align} \Phi_{nai}^{mbj} = \Phi_{0ai}^{(m-n)bj} \end{align} I don't understand the zero index in RHS. Since the $n$ or $m$ represent the cell, what does it mean to refer to zero-th cell ?


Where do our 4 macroscopic spacetime dimensions reside in multidimensional models of the universe?

In models such as M-theory with 7 'higher dimensions' plus the 4 macroscopic spacetime dimensions, where do our 4 macroscopic spacetime dimensions reside _ordinally_? My reason for asking is TV shows such as the 'Fabric of the Cosmos' that alludes to 'lower' dimensions in string theory. I can understand if our 4 are, say the 7th, 8th, 9th and 10th so there are dimensions lower than us but that we cannot readily detect.


Are we so sure about superposition?

Apparently particles can be anywhere when not observed. How strong is this theory really? Okay the wave-function can be collapsed through observation but how are we so sure that when an object is not being observed that it is in potential states? How are we so sure whether or not the wave function is collapsed by a conscious observing force other than us? And is superposition an assumption due to the fact that when we observe waves they act as particles, so we assume that everything has different potential until measurement; when really we don't know for sure whether it is that an object is in potential states or a conscious observance by a higher power already puts things in order already eliminating this potentiality?


matrix elements of the electronic molecular Hamiltonian between a hartree product and a Slater determinant

This may belong in Chemistry, but I thought I might try my luck here first. In Szabo's book, an exercise requires a proof that = (N!)^(1/2) * given that |K(HP)> is the Hartree product wave function corresponding to the N-electron Slater determinant |K>. I've been trying for a couple days now, but the only idea I have is to evaluate both integrals explicitly by throwing away all terms arising from the Slater determinant that would vanish by orthonormality (all but one for the one) then evaluating the integrals over the sums of one and two electron operators as a sum over allowed orbital permutations. This would defeat the point of having the theorem to simplify the derivation. Does anyone have a better more clever way or a hint to point me in the right direction? Thanks


How to measure rainfall? And how is it determined for a specific location?

How is rainfall measure in a pinpoint location? This is what I think:   There are automatic rainfall measuring stations at: A, B, C. But say I want to know point D, located between A and B, its rainfall level. How do I figure that out? Do I take the average? Do I go to the nearest neighbor on the map? Thanks in advance!


Time reversal symmetry in the presence of friction

I was reading a paper on time reversal symmetry, and came across an example of a pendulum swinging in the presence of friction: > When we consider the more realistic physical situation of a swinging > pendulum in the presence of friction, we can tell the difference between a > forward and a reverse film of this pendulum. Namely, the original (forward) > film will show the swinging pendulum losing amplitude until it comes to a > standstill. However, the film in reverse direction shows a swinging pendulum > whose amplitude is increasing in time. The latter film is clearly unphysical > as it does not satisfy the natural laws of motion anymore (assuming there is > no hidden source of energy feeding the pendulum). The presence of friction > breaks the time-reversal symmetry of the ideal pendulum. I don't understand this. If I am not wrong, we are only reversing the direction of time, not that of the frictional force (the nature and cause of this force is not relevant either). In the movie played backwards, the frictional force will be along the direction in which the bob swings. This force would be proportional to $v^2$, and would give the bob more momentum consistent with the observation. So what am I missing? How does the presence of friction break time reversal symmetry?


Energies and numbers of bound states in finite potential well

Hello I understand how to approach finite potential well (I learned a lot in my other topic here). However i am disturbed by equation which describes number of states $N$ for a finite potential well ( $d$ is a width of a well and $W_p$ is potential ): $$ N \approx \dfrac{\sqrt{2m W_p}d}{\hbar \pi} $$ I am sure it has something to do with one of the constants $\mathcal L$ or $\mathcal K$ defined this way: \begin{align} \mathcal L &\equiv \sqrt{\tfrac{2mW}{\hbar^2}} & \mathcal{K}&\equiv \sqrt{ \tfrac{ 2m(W_p-W) }{ \hbar^2 }} \end{align} and the transcendental equations for ODD and EVEN solutions: \begin{align} &\frac{\mathcal K}{\mathcal L} = \tan \left(\mathcal L \tfrac{d}{2}\right) &&-\frac{\mathcal L}{\mathcal K} = \tan \left(\mathcal L \tfrac{d}{2}\right)\\\ &\scriptsize{\text{transc. eq. - EVEN}} &&\scriptsize{\text{transc. eq. - ODD}} \end{align} **QUESTION:** Could anyoe tell me where does 1st equation come from?


Why does a car engine not do work if the wheels don't slip?

I saw this mind boggling result that if the tires don't slip then the work done by an engine to move a car is zero. Why is this true? Moreover, what does this truly mean? **Update:** Sorry about not being clear, but I was talking about an idealised case where air resistance is negligible and the tires are perfectly rigid and there is no internal friction at play.


Does the covariant derivative of a derivative of the metric vanish?

The title holds the main question. For a little more backgroud: In a calculation I keep coming across terms of the form $$ D_M \partial_N e_P^A \overset{?}{=} 0$$ where $e_P^A$ is the vielbein and $D_M$ is the covariant derivative _including_ both the spin _and_ the affine connection, i.e. the vielbein postulate reads $D_M e_P^A = 0$. Generally, I would not expect the expression to vanish, but the again I know little about the more involved machinery of differential geometry.


Phase Space dimension of Lorenz Strange Attractor

It is often discussed in 3 spatial dimensions and the need for third dimension to prevent self intersection is mentioned. But shouldn't the phase space of the Lorenz system be 6 dimensional, i.e., the 3 momenta also?