Is the Higgs mechanism a fundamental interaction?

Is the Higgs mechanism a fundamental interaction of the same standing as the strong, weak and electromagnetic interactions? If not, is it mediated by the weak interaction? It seems that all the massive particles participate in the weak interaction, and the massless ones do not, but this might just be a coincidence.


Do any entanglement measures for mixed states exist that use only single site correlation functions?

For a pure state $\rho_{AB}$, the entropy of entanglement of subsystem $A$ is \begin{equation} S( \rho_A) = -tr (\rho_A \log \rho_A) \end{equation} where $\rho_A$ is the reduced density matrix of A. For a single site of a spin chain, $\rho_A$ can be written in terms of single site correlation functions $\langle \sigma_l^\alpha \rangle$ where $\alpha = x,y,z$. Are there any entanglement measures for mixed states that use only the same correlation functions, $\langle \sigma_l^\alpha \rangle$ where $\alpha = x,y,z$?


What do the BICEP2 results mean for string gas cosmology and the ekpyrotic universe?

The imprint of gravitational waves created shortly after the big bang may offer direct evidence for inflation theory, according to a discovery by the BICEP2 experiment at the South Pole and released today. What this means for those alternatives theories to inflation such as string gas cosmology and the ekpyrotic universe? What would happen with these two cases particularly?


Circuit breaker Trips during thunderstorm

The circuit breaker at the electrical mains trips at home when there is a thunderstorm outside. Why does this occur?


Why do these papers show the wrong concavity to the conductivity near the percolation threshold?

I'm looking at two papers in particular: A. L. Efros and B. I. Shklovskii, Critical Behaviour of Conductivity and Dielectric Constant near the Metal-Non- Metal Transition Threshold, Phys. Status Solidi B 76, 475 (1976) and Measurement of the conductivity exponent in random percolating networks of nanoscale bismuth clusters . In percolation theory, for concentrations $p$ higher than the critical concentration $p_c$, the conductivity $\sigma \propto (p-p_c)^t$ for $p>p_c$, where $t$ is 1.3 for 2 dimensions. It's often simplified so that the substrate is a perfect insulator, so you have $\sigma=0$ for $p<p_c$. So it should look pretty simple, something like: ![enter image description here](http://i.stack.imgur.com/VeCeo.png) My point is, concave up. However, looking at the two papers I mentioned above, they have these graphs: ![enter image description here](http://i.stack.imgur.com/FdxOx.png) with:![enter image description here](http://i.stack.imgur.com/azYOy.png) and: ![enter image description here](http://i.stack.imgur.com/FiDoY.png) Both of which are concave _down_ in the region $p>p_c$. Is there something going on, or is this a series of errors? I suspect the Efros paper may have just had an error, as the caption corresponding to the dotted line (1) points to the region of $x<x_c$, where they explicitly said that equation doesn't apply. It does have a very strange scale though. I have no idea what's going on in the graph of the second paper. Is there any explanation besides an error? Thank you!


Definition of causality relation

It seems to me that special relativity has a weird definition of the causality relation. In that theory, the only thing that matters is the space-time distance between events. But I don't think this is the case. It seems to me that if I am pressing a few keys on my keyboard and then causing this question to appear on your screen, my action has a greater locus of influence than just me sleeping in bed. So, what exactly is the definition of the causality relation?


Temperature loss of a moving object

I would like to know how to calculate the temperature change of a moving object. For example if I throw a ball that is warmer than the surrounding air, how much would it lose form it's temperature in every second? I searched for two days now, but I have only partial solutions :(. I would need a formula that takes in account the followings:   * The material of the ball (Iron, heat conductivity?)   * Area of the ball   * Speed of the ball (true airspeed)   * Material of the surrounding material (air, heat conductivity?)   * And ofc temperature of the ball and the surrounding material.   * anything else I forgot and neccessary :):) It dosen't need to take in account friction generated heat and other things, I only need the temperature loss :) BIG TANKS!


What exactly constitutes a measurement?

As far as I understand it, quantum mechanics only predicts the probabilities of finding a system in some state of some observable when we make a measurement. So, what exactly constitutes a measurement that will make the system to collapse?


Proper name for a thermodynamic process with constant internal energy $U$

Back in the day I learned that a few special thermodynamical processes have special names. For example, if one keeps $P$ constant, the process is called isobaric, if one keeps $T, V$ or $S$ constant, one gets, correspondingly, isothermic, isochoric or isentropic processes. Similarly, if one keeps $\dfrac{\mathrm{d} \ln P}{\mathrm{d} \ln \rho}$ constant during the process, it is called polytropic, and if $\delta Q = 0$ at any time, the process is called adiabatic. Now, the question: what is the process called, if one keeps internal energy $U$ constant?


Why aren't atoms affected by gravity, but molecules are?

I read this book here: http://tiny.cc/Gravity-Atom-Myth It claims that while gravity does affect individual atoms, it's not quantified like molecular mass due to lack of binding proteins which render air-oxygen trajectories improbable.


Will watering tea down make it clearer?

If I poured water into my tea, would I see more or less of the bottom of the tea-cup? Intuitively, there would be as many particles blocking as many photons, and so I'd see the bottom just as clearly as before.


What are common methods for calculating the time dependency of elliptical orbit?

After playing a game called "Kerbal Space Program" I got interested in orbital mechanics and started messing with simplified calculations to determine $\Delta v$ requirements. In which I compared two trajectories to get into orbit/land from orbit to see how much it would matter if you would burn vertical or horizontal (Hohmann) at the surface. Graphs of some results can be seen here. However after this I also wanted to know how much time it would take to perform these maneuvers. I assumed that changes in velocity are instantaneous, so this would be equal to the time between the two changes in velocity. For the Hohmann-like-transfer it is simple, since you travel from periapsis tot apoapsis, which takes half of the orbital period (due to symmetry). But for the other transfer, which applies a vertical change in velocity at the surface, it is very difficult (if there is a sidereal rotational velocity). I tried solving it using Kepler's second law, given this formula: $$ r=\frac{a(1-e^2)}{1+e\cos{\theta}} $$ Since the area underneath the path from $\theta_0$ to $\theta_1$ can be calculated like this: $$ A_{\theta_0,\theta_1}=\int_{\theta_0}^{\theta_1}{\frac{1}{2}r^2d\theta} $$ Which can be used to determine the amount of time it would take, since the total area of the ellipse would take one orbital period $T=2\pi\sqrt{\frac{a^3}{\mu}}$: $$ t_{\theta_0,\theta_1}=T\frac{\int_{\theta_0}^{\theta_1}{\frac{1}{2}r^2d\theta}}{\int_{0}^{2\pi}{\frac{1}{2}r^2d\theta}}=T\frac{\int_{\theta_0}^{\theta_1}{\frac{1}{2}r^2d\theta}}{\pi a^2\sqrt{1-e^2}} $$ I am not that good at solving integrals, so I both tried to solve it with MATLAB and WolframAlpha, which gave me this as the result: $$ \int{\frac{1}{2}r^2d\theta}=a^2\sqrt{1-e^2}\left(\tan^{-1}\left(\frac{\sqrt{1-e^2}\tan{\frac{\theta}{2}}}{1+e}\right)-\frac{e\sqrt{1-e^2}\sin{\theta}}{2(1+e\cos{\theta})}\right) $$ So: $$ t_{\theta_0,\theta_1}=\sqrt{\frac{a^3}{\mu}}\left[2\tan^{-1}\left(\frac{\sqrt{1-e^2}\tan{\frac{\theta}{2}}}{1+e}\right)-\frac{e\sqrt{1-e^2}\sin{\theta}}{1+e\cos{\theta}}\right]^{\theta_1}_{\theta_0} $$ However when I tried to determine the integral with symbolic parameters and bounderies, MATLAB returned "Warning: Explicit integral could not be found." probably because there is not a general solution for both closed and open orbits. But if I use this formula and filled in the boundaries myself, it seemed that I got wrong results. However when I let MATLAB calculate the integral for given values of $a$, $e$, $\theta_0$ and $\theta_1$ I do get results which seem correct. So is this a good approach to find a time dependency of an (elliptical) orbit. If so is there an (continuous) equation derivable from this. And what might be a better approach? **EDIT:** The unexpected results form this formula might be caused by the fact that $\tan^{-1}$ returns results from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$. Since when I calculated the time from $\theta=0$ to $\theta=\pi$ I got results which agree with the conservation of energy and angular momentum. For this is used the following equations to determine the time derivatives: $$ \omega=\frac{\delta\theta}{\delta t}=\frac{\delta \theta / \delta \theta}{\delta t / \delta \theta}=\sqrt{\frac{\mu}{a^3\left(1-e^2\right)^3}}\left(1+e\cos{\theta}\right)^2 $$ $$ \dot{r}=\frac{\delta r}{\delta t}=\frac{\delta r / \delta\theta}{\delta t / \delta\theta}=\omega\frac{\delta r}{\delta\theta}=\omega\frac{ae(1-e^2)\sin{\theta}}{\left(1+e\cos{\theta}\right)^2} $$ Which yields that the specific orbital energy $\epsilon$ and specific angular momentum $h$ are constant when substituting theses therms into there expressions: $$ \epsilon=\frac{v^2}{2}-\frac{\mu}{r}=\frac{\dot{r}^2+\omega^2r^2}{2}-\frac{\mu}{r}=-\frac{\mu}{2a} $$ $$ h=\omega r^2=\sqrt{a\mu\left(1-e^2\right)} $$


Question about derivation of tensor in Di Francesco's CFT

This is a question for anyone who is familiar with Di Francesco's book on Conformal Field theory. In particular, on P.108 when he is deriving the general form of the 2-point Schwinger function in two dimensions. He writes that the most general form of the tensor is $$S_{\mu \nu \rho \sigma} = (x^2)^{-4} \left\\{ A_1 g_{\mu \nu} g_{\rho \sigma} (x^2)^2 + A_2 (g_{\mu \rho}g_{\nu \sigma} + g_{\mu \sigma}g_{\nu \rho})(x^2)^2 + A_3(g_{\mu \nu}x_{\rho}x_{\sigma} + g_{\rho \sigma}x_{\mu}x_{\nu})x^2 + A_4 x_{\mu}x_{\nu}x_{\rho}x_{\sigma}\right\\}$$ This I understand and have obtained this result myself. What I don't understand however, is why he has neglected the following term since it seems to satisfy all the constraints presented on P.108: $$S_{\mu \nu \rho\sigma} = A_5 (x^2)^{-3} (g_{\mu \sigma} x_{\rho}x_{\nu} + g_{\mu \rho}x_{\sigma}x_{\nu} + g_{\nu \sigma}x_{\rho}x_{\mu} + g_{\nu \rho}x_{\sigma}x_{\mu})$$ In another thread I posted here, I wondered whether this could be reduced to terms already present in the form Di Francesco gave, but I was quickly reassured this to not be the case. So, if anyone is familiar with his book and would be willing to clarify this it would be great. I asked a professor at my university and he was not sure either why it has been neglected, so I thought I would pose the question here.


Why can't electrons fall into the nucleus?

I read a book on pop sci book on quantum mechanics and the author said that electrons do not fall into the nucleus due to quantum mechanics- which principles suggest this (I think it was Heisenberg's Uncertainty and Pauli's Exclusion Principle) and why? Also, I've heard that if Bohr's planetary model were correct, then electrons would lose energy/momentum and fall in- is this true and again, which physics principles say this?


Is there a classical analog to quantum mechanical tunneling?

In comments to a Phys.SE question, it has been written: > _'Tunneling' is perfectly real, even in classical physics. [...] For > sufficiently large temperatures this can put the system above a hump in its > potential energy._ and > _the only difference between the classical case and the quantum mechanical > one is that classical physics is a random walk in real time, while QM is a > random walk in imaginary time._ I understand that in a system of particles with finite temperature some particles can overcome a potential barrier. That's how I interpret the first statement. I don't understand the business of "random walk in imaginary time". Can someone explain? **Update** What I was originally looking for was 1.) classical system that can transport mass through a forbidden region and 2.) explanation of "random walk in imaginary time". So far, I don't see anything for question 1.), but I think I'll grok 2.) if I invest some time and energy.


How does Newton's first law asserts the existence of inertial frames?

Recently I've seem here one answer telling that Newton's first law really assures the existence of inertial reference frames. But how is that? I really can't see it. As I know, Newton's first law says: > _Every body continues in its state of rest, or of uniform motion in a right > line, unless it is compelled to change that state by forces impressed upon > it._ Now, what I've seem done is to use this to define that whenever this law holds in some reference frame we call it inertial. But how this law states that such a frame does exist?


Is Newton's first law something real or a mathematical formalism?

Why do objects always 'tend' to move in straight lines? How come, everytime I see a curved path that an object takes, I can always say that the object tends to move in a straight line over 'small' distances, but as you take into account the curvature of the path, a force acting on the particle appears. I mean, I can always take a small enough portion of the curve, zoom in enough, and conclude that the object is moving in a straight line, but then as I zoom out I find out that a force is acting on the particle. The force of gravity is everywhere and, no matter how weak it is, it will make the particle take a path which is different from a straight line. This is my question: since particles are, in reality, never moving in straight lines, is Newton's first law a mathematical formalism or some true property of material objects?


Why does only one side of a neon lamp glow?

When applying DC to a neon lamp, only one electrode glows: ![http://commons.wikimedia.org/wiki/File:Neonlamp3.JPG](http://i.stack.imgur.com/3poGs.jpg) The voltages across the lamps are left: DC (left lead positive), middle: DC (right lead positive), and right: AC. But... why? The electrodes are the same shape, so the electric field around them should be same shape, and the gas should break down at the same electric field strength. It seems like the fields would be symmetrical. Is there a difference in threshold between positive and negative coronas? If so, do both sides light up at high enough voltage? Or maybe only one type of corona is possible in neon since it's a noble gas? If it contained air would it glow at both electrodes?


Can a solar thermal collector designed for use in homes ever power a stirling engine designed for use in homes?

Can a solar thermal collector common on private roofs today designed for collecting heat for the household ever collect enough heat to power a stirling engine of say this model: http://www.whispergen.com/main/achomesspecs_info/ Normally a stirling engine of this model is powered by combustion of natural gas.


Temperature change effected by electric heater

A 40-gallon electric water heater has a 10kW heating element. What will the water temperature be after 15 min of heating if the start temp is 50F degrees. There must be an equation. I can't find it in notes or text.


How do we know the size of the universe?

Ok, from astronomical observations we can tell that the observable matter is separating - so rewind the clock about 13.7 billion years and it was all at a single point. However, how do we distinguish between the following two options:   1. Universe is expanding    2. Matter distribution is increasing into infinite void **Clarification 1:**   (My notion of) The traditional notion is that all time/space/matter was created at the instant of the big bang. I.e. BB was inital conditions of: $t=0$, $V=0$, $E=very big$ People say "the universe is expanding", rather than "observable matter is separating". Why is this? How do we know that the big bang event wasn't started by all matter condensed at a single point within a larger (otherwise empty) universe? How do we know that BB wasn't: $t=0$, $V(universe)>0$ but $V(matter)=0$, $E=very big$?


Plasma Treatment of LaAlO3: Surface Roughening

I'm trying to understand something I've observed in exposing LaAlO3 substrates to an oxygen plasma (yielding atomic oxygen). In literature, these substrates are frequently "cleaned" in a oxygen rich environment by exposing the surface to atomic oxygen (via plasma source) at substrate temperatures greater than 700 C. This typically results in a clear diffraction pattern indicative of a flat surface. In my experiments, I'm noticing that initially, the surface improves upon plasma exposure (enhanced diffraction intensity), but after a certain point, the intensity of the diffraction pattern begins to fade, and (presumably) the surface becomes disordered. I'm looking for insight into why this might be happening, because typically the surface will be exposed to the atomic oxygen plasma during growth, but this seems to be damaging my film. My first guess would be that initially, atomic oxygen reacts with surface species to produce a clean surface, and, after a point, begins to somehow erode the surface, reducing crystallinity.


How does the energy in a standing wave travel beyond a node?

In a standing wave, how does energy travel past a node? It should just get reflected. Assume the case of first overtone and you strike the string at a place. How will energy distribute itself? If it distributes over the whole string, then how is it the wave traveled past a node? Basically the problem is a standing wave has been set up in the string . Now you strike the string at a place, and this creates a new pulse. Now this pulse will travel past the next node in the string (not the boundary) or will it reflect off even that node and remain confined to a particular region of the string?


When will the Moon reach escape velocity?

From what I know, the Moon is accelerating away from the Earth. Do we know when it will reach escape velocity? How do we calculate this?


Rigid body problem

I have some doubts about the next excercise: > _A bar of length $2a$ and mass $m$ moves freely with both of its extremes on > a ring of radius $\sqrt2a$. The ring can rotate freely in a certain > diameter, remaining the center fix. Find the equations of motion._ I did it using the balance of angular momentum, but I want to try to do it with the Lagrangian. But I have a problem. The energy of the ring is easy to calculate (just a rotation), but the energy of the bar I am not sure how to do it. ¿Is a translation of the center of mass plus a rotation? ¿Or just translation? ¿Or just a complicated rotation? I did this picture to illustrate: ![enter image description here](http://i.stack.imgur.com/DhYke.png) Thanks!


Tensor Introduction

I have recently started learning about tensors during my course on Special Relativity. I am struggling to gain an intuitive idea for invariant, contravariant and covariant quantities. In my book, invariant quantities are described as being those physical quantities associated with a point in space e.g. temperature at point P. Thus, when changing coordinate system, the value associated with the point is unchanged: you may represent the point by a different set of coordinates, but on locating the point in the new coordinates, the temperature value will still be the same. Here comes my struggle: the archetypical example of a covariant quantity is the gradient vector. Is it not the case that we require it to transform between coordinate systems in a way that preserves the vector associated with each point? Thus, in a way, the gradient vector is an invariant quantity? Why then are many derivations in my course driven by the fact that physical quantities like Action, Lagrangians etc should be "Lorentz invariant"?


Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?

The standard treatment of the one-dimensional quantum simple harmonic oscillator (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\\{\vert n \rangle\\}_{n = 0}^{\infty}$ each with corresponding eigenvalue $E_n = \omega \left(n + \frac{1}{2}\right)$. Refer to this construction as the **abstract solution**. How does the abstract solution also prove uniqueness? Why is there only one unique sequence of countable eigenstates? In particular, **can one prove the state $\vert 0\rangle$ is the unique ground state without resorting to coordinate representation?** (It would then follow that the set $\\{\vert n \rangle\\}_{n = 0}^{\infty}$ is also unique.) The uniqueness condition is obvious if one solves the problem in coordinate representation since then one works in the realm of differential equations where uniqueness theorems abound. Most textbooks ignore this detail (especially since they often solve the problem both in coordinate representation and abstractly), however I have found two exceptions:   * Shankar appeals to a theorem which proves one-dimensional systems are non-degenerate, however this is unsatisfactory for two reasons:     1. Not every one-dimensional system is non-degenerate, however a general result can be proven for a large class of potentials (the SHO potential is in such a class).     2. The proof requires a departure from the abstract solution since it classifies the potentials according to their functional properties.   * Griffiths addresses this concern in a footnote stating that the equation $a \vert 0\rangle = 0$ uniquely determines the state $\vert 0\rangle$. Perhaps this follows from the abstract solution, however I do not see how.


Construction of Bound states in Lattice QCD

How the proton state is found and renormalized in lattice QCD? Is there any literature shows the method step by step? What about other bound states in QCD if we somehow know their quantum numbers?


Sign issue on electrostatic potential energy

I have been working on a problem about finding the electrostatic potential energy stored on a capacitor of concentric spheres with inner radius $a$ and outer radius $b$ and with charge $Q$. I've got to first calculate the energy using the capacitance and then integrating the energy density. I did the following: first of all I've used Gauss' Law to find the electric field a distance $r$ from the center of both spheres with $a < r < b$. As expected I've got the field of a point charge $Q$ at the center. Then I've integrated the field along a segment joining the two spheres and I've got the following difference of potential $$V=\frac{1}{4\pi\epsilon_0}\frac{Q(a-b)}{ab}$$ Then I've found the capacitcante using $Q=CV$ and finally I've found the energy using $U=Q^2/2C$. That's fine, I've got the value: $$U_1=\frac{1}{8\pi\epsilon_0}\frac{Q^2(a-b)}{ab}$$ Now the second part, I should find the same value integrating the energy density. So, the energy density is $\mathcal{u}=\epsilon_0E^2/2$ and hence it is: $$\mathcal{u}=\frac{1}{2}\epsilon_0 \frac{1}{16\pi^2\epsilon_0^2}\frac{Q^2}{r^4}$$ Since I must integrate on the region between the two spheres I've used spherical coordinates and computed the following integral getting the energy: $$U_2=\int_0^\pi\int_0^{2\pi}\int_a^b \frac{1}{2}\epsilon_0 \frac{1}{16\pi^2\epsilon_0^2}\frac{Q^2}{r^4}r^2\sin\phi dr d\theta d\phi$$ And this integral gave simply: $$U_2 =\frac{1}{8\pi\epsilon_0}\frac{Q^2(b-a)}{ab}$$ But wait a moment, I've got $U_2 = -U_1$ instead of $U_2 = U_1$ as expected. I've calculated it once again and once again and I've got the same problem. Can someone point out what's happening? Where's my mistake? Thanks in advance for your help!


Given a potential energy function, find expression of the force of a particle?

This comes from an AP review packet. I'm given a potential energy functon, $$U(r)=br^{-3/2} + c,$$ where $b$ and $c$ are constants, and need to find the expression for the force on the particle. There's a graph of $U(r)$ given with the problem, but I'm not sure if it's needed or not. I'm just looking for how to go about solving this problem.


Intro to Solid State Physics

I didn't see this listed on the books page so here it is. I'm currently in an introductory Solid State course, and we are using Kittel's book. I have been having a rough time with this book although I am starting to get used to it as we get farther in. What are good introductory solid state books?


What does Bell's Theorem really violate?

Well this is a fairly straightforward question. I know it states that either hidden variables is wrong or Quantum Mechanics. But indirectly hidden variables is a part of QM due to uncertainty. So what exactly does it violate. I am just a bit confused.


Introduction to string theory

I am in the last year of MSc. and would like to read string theory. I have the Zwiebach Book, but along with it what other advanced book can be followed, which can be a complimentary to Zwiebach. I would like a more mathematically rigorous book or lecture notes along with Zwiebach. Specifically, mention whether the book discusses string theory   * Rigorously?   * Intuitively? What's the scope of the book? Does it cover the advanced materials, e.g. Matrix string theory, F-theory, string field theory, etc. Maybe even String Phenomenology?


Prerequisites and introduction to string theory

Can someone please give me the prerequisites and mathematics required for string theory? Are there some good references to study it, both online and in a book? Please consider I am a newbie in string theory, and only about up to the level explained in Hawking's Brief History of Time and Penrose's Road to Reality. I would prefer something which does not omit rigor, but builds it up.Also, are there good online lectures on this? (I have covered most of GR, but I am not really comfortable with advanced Quantum mechanics. I have studied QM only from Griffith's book, and a few McGraw hill lecture notes.)


Gravitational potential energy

Consider two places next to each other: Place 1, where there is a gravitational field whereas Place 2 - there's no field. Now if we lifted a box in place 1, it gains potential energy. Then, we move this box horizontally to place 2. What happens to this energy?


Does quantum reversibility require many worlds?

The source S sends a photon into the beam splitter below. There is a 50% chance that it will be detected at A and a 50% chance it will be detected at B.               S -----------\---------> A                  |                  |                  |                  v                  B      Now if we assume that physics is (generally) reversible we should be able to time-reverse this process. This would imply that if we send in a photon at A or B it should _always_ appear back at S. How can this happen? We know that if we send in a photon at A (or B) we would expect it to appear 50% of the time at S and 50% at C as below.                            C                                         C                  ^                                         ^                  |                                         |                  |                                         |                  |                                         |     S <----------\--------- A               S <------------\                                                            |                                                            |                                                            |                                                            B      In order that the photon always appears back at S and never at C we need a superposition of the two situations above such that the paths to S constructively interfere and the paths to C destructively interfere. Thus it seems that in order to retain time-reversibility we need to assume a many-worlds view. As a photon is sent in at A (or B) it must be assumed that another photon is simultaneously sent in at B (or A) and both states weighted with the correct amplitude such that a photon appears with certainty back at S. What do people think?


How light know which path is smallest?

We know from fermat's principle that light follows the smallest path. But how light know that which path is smallest?


Pressure in waves on a string

We know that when we speak sound waves are created. The air particles compress and rarefy and pressure is more at the nodes and less at anti-nodes. But can we say the same thing about waves on a string,- that pressure is more at the nodes than the anti-nodes?


Energon: is it possible?

I'm always quite curious about the "Energy cube" in Transformers, or namely Energon. Is it really possible to store energy, such as electricity, into such a compact form? safe to distribution, and seems nothing left after being consumed? ps. Wikipedia has a page for Spark, which is more for transformer's soul. I'm not asking for that yet.


Does humidity make winter air feel colder?

I sometimes hear stories where people compare their feelings in winter in different places in the world. It goes like > in city X the temperature was the same as in city Y, but the humidity made > me feel much colder... or > oh well, -20°C would be cold, but the humidity was low, so it felt OK so it implies that humidity somehow makes it feel colder. I am talking about temperatures below freezing (-30...0°C). Does this have any physical explanation, or is it some sort of psychological phenomenon?


Looking for the actual reason of refraction explained precisely without analogies

I'm a high school teacher trying to teach my students (15year olds) about refraction. I've seen a lot of good analogies to explain why the light changes direction, like the marching band analogy, that the light "choose" the fastest way etcetc, and for most of my students these are satisfying ways to explain the phenomenon. Some students, however, are able to understand a more precise and physically correct answer, but I can't seem to find a good explanation of why the lightwaves actually changes direction. So what I'm looking for is an actual explanation, without analogies, of how an increase/decrease in the speed of a lightwave cause it to change direction. Thanks a lot


If you put a latex balloon in a vacuum, how much would it expand? And would it pop? Assume it doesn't leak. EDIT: Some numbers: Ambient pressure is 100 KPa, balloon is perfectly spherical with a diameter of 300 mm, deflated it has a diameter of 25mm, temperature is always at equilibrium. Question components stated more formally:   * What pressure does the balloon need to have been inflated to to reach the 300 mm radius?   * What is the relationship between the (gauge) pressure and volume? _I think this simply comes down to PV=nRT_   * If there is some sort of spring constant involved, what is this value for a typical latex balloon?   * What is the relationship between the pressure (or volume) and the tension on the material the balloon is made from?      * What is the limit the tension can reach before popping? Semi-related: how much would a balloon expand if you sealed it deflated and put it in a vacuum?


Are there entire anti-matter galaxies?

Would we be able to tell if an entire neighboring galaxy was made up of anti- matter? What would we see in its signals? It seems like the anti-matter could be stable if it was completely surrounded by other anti-matter.


Definition of scattered particle?

Compare the number of scattered particles:   $N_s=Fa\int\sigma(\theta)d\Omega$   With the total number of incident particles:   $N_{in}=Fa$ Here, F is the flux of incoming beam, a the area. sigma the crossection and omega the solid angle. Why isnt $N_s=N_{in}$? How does one define which particles are scattered and which are not, arent they all interacting with the target to some degree? Isnt particles conserved normally? Do almost all the particles either pass right through almost undetected or are scattered signficiantly, so what we are really integrating over is a sphere surrounding the target except a spot of area $a$ where the beam exits?


A theory about a travelling wave due to a vibrating rope seems to be contradictory, please spot what is wrong?

Consider a travelling wave produced by vibrating one end of a rope while the other end is made to freely move along a vertical line. Mathematically, the equation of the traveling wave that also represents the equation of motion of each point on the rope is given as follows. $$y(x,t)=A\sin(kx-\omega t)$$ In the book I read, a certain particle on the rope moves up and down perpendicular to the propagation vector as wave does not transfer material but energy. I don't agree with the statement that "a certain particle on the rope moves up and down perpendicular to the propagation vector". It is because I think to maintain the same point on the rope to move up and down, the arc length of $y(x,t)$ must be equal to that of $y(x,t+\Delta t)$ for $0\leq x \leq a$ and any $\Delta t$. However, according to the Notes below, their arc length are not generally equal. ![enter image description here](http://i.stack.imgur.com/8B30K.png) From the figure above, how can $P$ and $Q$ be the same point of the rope while the red and blue curves have different length? So my question are: **Does each point of the rope move up and down?** ## Notes From $x=0$ to $x=a$, generally the arc length of $y=\sin x$ is not equal to that of $y=\sin(x-x_0)$. Let $S(x_0)$ be the arc length of $y=\sin(x-x_0)$ for the given interval. \begin{align} S(x_0) &= \int_0^a\sqrt{1+\left(\frac{dy}{dx}\right)^2}\, \textrm{d}x\\\ &= \int_0^a\sqrt{1+\cos^2(x-x_0)}\, \textrm{d}x\\\ \end{align} If $S(0)=S(x_0)=\text{const}$ for any $x_0$ then $\frac{\textrm{d}S(x_0)}{\textrm{d}x_0}=0$. Thus I have to check whether or not $\frac{\textrm{d}S(x_0)}{\textrm{d}x_0}=0$. \begin{align} \frac{\textrm{d}S(x_0)}{\textrm{d}x_0} &= \frac{\textrm{d}}{\textrm{d}x_0}\int_0^a\sqrt{1+\cos^2(x-x_0)}\, \textrm{d}x\\\ &= \int_0^a\frac{\textrm{d}\left(\sqrt{1+\cos^2(x-x_0)}\right)}{\textrm{d}x_0}\, \textrm{d}x\\\ &= \int_0^a\frac{\cos(x-x_0)\sin(x-x_0)}{\sqrt{1+\cos^2(x-x_0)}}\, \textrm{d}x\\\ &= \int_{-x_0}^{a-x_0}\frac{\cos y\sin y}{\sqrt{1+\cos^2y}}\, \textrm{d}y\\\ &= \sqrt{1+\cos^2 x_0}-\sqrt{1+\cos^2(a-x_0)}\\\ &\not = 0 \end{align} It implies that generally they are not equal.


Which symmetry for which distance function

For evaluating the electric field of some charge distribution one can use $$\phi(r):= \frac{1}{4 \pi \varepsilon_0}\int_{\mathbb{R}^3} \frac{\rho(r')}{||r-r'||_2} dr'.$$ My question is: What symmetry do we need to have that we can write in spherical coordinates $$||r-r'||_2 = \sqrt{||r||_2^2+||r'||_2^2-2||r||_2||r'||_2\cos(\theta')}~?$$ This is of course not the most general way to express this distance, as the $\phi$ dependence is missing. So, under what conditions can the distance be expressed like this? Notice that $\theta'$ is the respective angle in spherical coordinates, so it's NOT the angle between $r$ and $r'$. So in particular, your answer should clarify, why we can evaluate for example the electric potential of a sphere by integrating: $$\frac{1}{4 \pi \varepsilon_0} \int_0^{2\pi}\int_0^\pi \int_0^{\infty} \frac{\rho(r')||r'^2|| sin(\theta')}{\sqrt{||r||_2^2+||r'||_2^2-2||r||_2||r'||_2\cos(\theta')}}d||r'||_2 d\theta'd\phi',$$ but need to refer to a more general equation in this example, where the $\phi$ angle is used too: excercise 14b)


Turbulence parameterization from gravity - fluid dynamics correspondence

I`m looking for a nice introductary reference that explains how the turbulence coefficient or any kind of turbulence parameterization (in view of applications to atmospheric turbulence for example) can be derived from the gravity - fluid dynamics correspondance, such that even I can get it. I mean, if something like this exists ... I`m basically quite familiar with the hydrodynamic part (NS equation, etc) of this correspondance whereas about the other side I feel a bit more shaky ... I`m finally looking for a citable reference, but any "reasonable" source (slides of a talk, video, ect) that explains how a turbulenc coefficient / parameterization can be obtained would be welcome and appreciated. Edit To clarify what I mean, relevant papers for the topic are for example here, here, and jep this one linked to by Mitchell.


A tunnel inside the Earth (but not an ordinary tunnel)

I decided to dig a tunnel inside the Earth. In equatorial plane. It should be designed in such a way that it follows the Coriolis effect. That means if, say a stone is dropped from rest into the tunnel, then it travels in the tunnel without touching the walls of the tunnel until it reaches the tunnel's lip on the other side somewhere near the surface of the Earth. So the question itself is an equation of the curve of the tunnel. I suspect that there is no nice closed form for the curve. But who knows? Nevertheless, it would be good to know how deep the tunnel goes and where it rises again to the surface and the travel time. You could say that it would be impossible to do this on Earth, but you actually could do this on the Moon.


Can space-time be defined by the requirement that the physical laws are simple?

When I was student I was told that time is defined by the requirement that the physical laws are simple. For example, in classical mechanics time can be defined by the requirment that the velocity of an isolated body is constant. One could generalize this approach by assuming that _space-time_ is defined by the requirement that the physical laws are simple. This can be easily expressed in mathematical language as follows. In general relativity the universe is represented by the triple $(M, g, T)$ where $M$ is a four-dimensional differentiable manifold, and $g$ and $T$ are a Lorentzian metric and a tensor field on $M$ satisfying Einstein's equation. According to the above approach, we could say that the "true" physical reality of the universe is represented by the pair $(M, T)$, while the metric $g$ emerges as an appeareance from the requirement that evolution appears simple, i.e., that $g$ and $T$ satisfy Einstein's equation. From the mathematical point of view, this approach poses for example the following problems: does any tensor field $T$ admit a metric $g$ such that Einstein's equation is satisfied? To what extent a tensor field $T$ univocally determines the metric $g$? Does this approach make sense, at least from the mathematical point of view?


About the Hayden-Preskill circuit

Can someone summarize as to what are the problems and/or the open questions with the Hayden-Preskill circuit? (in the context of understanding black-holes or as a computer science question)It gives a framework to see how the black- hole can thermalize in a $log (entropy)$ time scale. What next? To start off, The sort of problem with the Hayden-Preskill circuit is that it works via random unitary transformations on disjoint pairs of qubits and hence doesn't have an Hamiltonian interpretation. I believe one has always believed that matrix models (of matrix size $\sqrt{entropy}$) can saturate this logarithmic thermalization bound via Hamiltonian evolution.   * Has this above belief about the matrix models been proven yet?    * Is there an intuitive explanation as to why a theory with finite dimensional matrices should behave like an infinite dimensional system?


Did cosmological natural selection get a reprieve?

The March 2014 issue of Physics Today has an article by Lee Smolin in which he argues that natural laws must change over time. As examples of such theories, he gives Penrose's CCC and his own cosmological natural selection (CNS): http://arxiv.org/abs/hep-th/0612185 My understanding was that CNS had already been falsified in 2010 by the discovery of a 2-solar-mass pulsar: http://arxiv.org/abs/1010.5788 But in the Physics Today article, Smolin says that CNS makes predictions and that, "The two main predictions,[4] first published in 1992, have survived despite several chances to falsify them since.[7] One of those is actually easy to state: The upper mass limit of neutron stars is at most two solar masses." The footnotes are: [4] L. Smolin, Class Quantum Grav 9, 173 (1992); The Life of the Cosmos, Oxford U Press, New York (1997) [7] L. Smolin, http://arxiv.org/abs/1201.2632 ;http://arxiv.org/abs/0803.2926 . The two arxiv papers are from 2012 and 2008, respectively. Their abstracts don't mention anything about empirical tests of the theory. I searched the text of the 2012 paper for the word "neutron," and found only one mention of neutron stars, which is in a reference to this 1997 paper:http://arxiv.org/abs/astro-ph/9712189 ; since it's from 1997, it's from before the discovery of the 2-solar-mass pulsar. What's up here? Did Smolin move the goalposts? Did he decide that the 2-solar- mass limit was fuzzy rather than sharp? Was the mass of this neutron star found to be a mistake? Are the error bars on its mass too big to allow it to falsify CNS? The article also describes Penrose's CCC as alive and kicking, whereas in fact I think it's clearly been dead for several years. Can anyone edit my tag


What is the meaning of pressure of a solid?

Currently I'm taking an introductory course on thermodynamics. I've got a problem with understanding what is the meaning of pressure of a solid body. The question arose when I looked at phase diagram in P, T coordinates. When we're talking about pressure of gas or liquid, it can be defined thus: it is the force per unit area with which the molecules of the substance hit the surface. This force is due to the molecular motion. But in the case of solids, molecules don't "move", they just oscillate about their fixed positions. Therefore, they don't hit (in usual meaning) a testing surface and we have problems with pressure's definition. How then the pressure can be defined?


Problem about thermodynamic relations

We know that for constant pressure thermodynamic processes, $dH=dQ_p$. My question is, does it implies that only reversible work is possible in this processes so that $dw=0$ because $dv$ is zero? In addition, does $Q_v$ necessarily be reversible heat transfer in this case? What if the heat transfer is irreversible? Similar question for $dU=dQ_v$, does the process need to be reversible? Another question is, do the above relations have anything to do with whether or not the system is an ideal gas? I have heard from a lecture that $du≠dQ_v$ in Joule's free expansion for non-ideal gas. Consider reversibility and whether it's ideal gas there are four combinations of situations (ideal gas reversible,ideal gas irreversible, non-ideal gas reversible, non-ideal gas irreversible). I get confused with how these factors affect the thermodynamic relations.


Showing that the Ricci scalar equals a product of commutators

I have to compute the square of the Dirac operator, $D=\gamma^a e^\mu_a D_\mu$ , in curved space time ($D_\mu\Psi=\partial_\mu \Psi + A_\mu ^{ab}\Sigma_{ab}$ is the covariant derivative of the spinor field and $\Sigma_{ab}$ the Lorentz generators involving gamma matrices). Dirac equation for the massless fermion is $\gamma^a e^\mu_a D_\mu \Psi=0$. In particular I have to show that Dirac spinors obey the following equation: $$(-D_\mu D^\mu + \frac{1}{4}R)\Psi=0 \qquad (1)$$ where R is (I guess) the Ricci scalar. Appling to the Dirac eq, the operator $\gamma^\nu D_\nu$ and decomposing the product $\gamma^\mu \gamma^\nu$ in symmetric and antisymmetric part I found: $$D_\mu D^\mu\Psi + \frac{1}{4}[\gamma^\mu,\gamma^\nu][D_\mu, D_\nu]\Psi=0$$Now I have troubles to show that this last object is related with the Ricci scalar. Can somebody help me or suggest me the right way to solve Eq. $(1)$?


Are the speeds of the different wavelengths of visible light different or varying in a medium such as air? If so, please inform by how much? Also, even if the wavelength speeds vary minimally, please inform.


In what ways can a lunar eclipse occur?

In what ways can a lunar eclipse occur? Also, on what percentage of the Earth are they usually viewable? I am aware that there are multiple configurations that constitute a lunar eclipse (umbral, penumbral, partial) and would like more information about each and how they occur.


Power dissipated through resistor in RC circuit with charged Capacitor

so here's the problem: A simple RC circuit where the capacitor has been charged, and two resistors are in parallel configuration. How does one find the power (as function of time) through any of the two resistors? ![enter image description here](http://i.stack.imgur.com/mOGjJ.jpg)


Ensuring globally hyperbolic geodesically-complete spacetimes

Let's say we have an incomplete spacetime A that is globally hyperbolic, does there necessary exist a globally hyperbolic completion? My guess is no, in which case what further restrictions can be placed on A to ensure that it can always be extended to geodesically-complete globally hyperbolic spacetime?


Frequency of rotating coil

Given a coil initially in the x-y plane, rotating at angular frequency $ \omega $ about the x-axis in a magnetic field in the z-direction. This uniform time varying magnetic field is given by $B_z (t)=B(0)cos(\omega t) $ I am required to show that there is a voltage of frequency $2\omega $ across the loop. Clearly when t=0 the flux is at a maximum, but I dont understand how to relate to the frequency? If the frequency is just the inverse of the period then $f=\omega / 2\pi $ ? Clearly I am not understanding something. How does the voltage affect the frequency?


Why don't tuning forks have three prongs?

I was reading Why tuning forks have two prongs?. The top answer said the reason was to reduce oscillation through the hand holding the other prong. So if having 2 prongs will reduce oscillation loss, surely a 3-pronged tuning fork would be even more efficient. Why don't you see more 3-pronged tuning forks?


Does thermodynamic processess affect phase transition in solid state metals

I read an article about phase transitions and I read about thermodynamic processes such as Adiabatic process, Isochoric process, Isobaric process, Isothermal process Do these processes affect phase transition in solid state metals? The reason I ask is because when these thermodynamic processes are being defined or explained, a fluid is used to explain the concept behind them. Take for example when a solid state metal is in use (say semiconductor chip) a lot of heat is evolved or given out, if the heat is too much, and the metal is considerable small, it might melt, thus going from solid of a liquid form. How are these thermodynamic processes modeled? Is there any reference that you can provide?


Deriving Feynman rules from Renormalized Lagrangian

In the context of Renormalized Pertubation Theory Peskin Schröder says:   The Lagrangian $$ \mathcal{L}=\frac{1}{2} (\partial_\mu\phi_r)^2-\frac{1}{2}m^2\phi_r^2-\frac{\lambda}{4!}\phi_r^4 + \frac{1}{2} \delta_Z(\partial_\mu\phi_r)^2 -\frac{1}{2}\delta_m^2\phi_r^2-\frac{\delta_\lambda}{4!}\phi_r^4 $$ gives the following set of Feynman rules:   \------------>------------ = $\frac{i}{p^2-m^2+i\epsilon}$   \------------X------------ = $i(p^2\delta_Z-\delta_m)$   and the two 4-vertices.      The question is: Why look the Feynman rules for the first and the fourth term of the Lagrangian look so different? I believe the answer is connected to the fact that one has to bring the kinetic term of the Lagrangian to its canonical form $\frac{1}{2} (\partial_\mu\phi_r)^2$ and has to interpret everything else as (possibly momentum dependent) vertices. How does this look in formulae?


Collision between a photon and an electron

Looking through this AP Physics question, I was struck by how the 'collision' between a photon and electron looks so much like a macroscopic collision. Is this even physically possible? Look at the last page of this pdf: http://lodischool.tripod.com/dovesol/DOVE02SOL.pdf EDIT: Some more questions: How could a photon _collide_ with an electron, when their positions cannot be determined exactly? Also considering how very small the electron is, I doubt that it is even possible to make the two collide; and if it is, how could you possible detect that? It also seems as though the photon and electron are acting as particles, which seems to me not to be the whole story. What if I put the electrons behind a double slit apparatus, and treat individual photons as particles? Based on this "compton scattering," it's possible for the photon to be deflected any which way. I could claim that the diffraction pattern observed in the double-slit experiment is due to compton scattering, among other factors. Prove me wrong!


Distribution of pressure inside a capsule

How would pressure of an ideal gas be distributed over the inside of a capsule (a cylinder with semi-spheres on the ends)? What about the strain on the material? Is there a general formula for how much?


![enter image description here](http://i.stack.imgur.com/qJfUU.png) "Fermi level" is the term used to describe the top of the collection of electron energy levels at absolute zero temperature. Why does the Fermi level for the bulk of topological insulator fall within the bulk band gap? There is no energy level in the band gap.


Confusion about the probability cloud

What is the meaning of the electron probability cloud? I understood it to mean that the electron has a probability to be found in a certain postion before measurement, but now after reading experiments involving Schrödinger's cat type states (with bonding and antibonding gaps in $\mathrm{H_2}$ and molecule and squid measurements where the wavefunction interferes and separates out the bonding band from the antibonding band) and that this physical effect of the supercurrent moving in both directions across the squid junction proves the electron is really everywhere in the cloud, smeared out and not just a probability cloud, which interpretation is right? Does the above experiement prove the electron is everywhere and smeared out in the wavefunction?


Finding Time & Required Angle for Projectile Motion with Quadratic Air Resistance

This is a physics/calculus/computer science problem, but I think I'll get better results in the Physics SE. I have a fun little project I've been working on (hobby, not homework/production), that models the flight of a projectile in various environments (2 Dimensional). Everything is nice and shiny with gravity and no air resistance. But of course, I've run into the classical Quadratic Air Resistance problem. Now I have no problem with just accurately calculating various position: I have a 4th Order Runge-Kutta Method which works just fine for that. What I would like is, given an (x,y) coordinate pair, find the Required Angle needed to hit that coordinate, as well as the time needed to get there. Currently I'm working on an algorithm that starts by finding two angles, one which overshoots, and another that undershoots, then using a binary search to narrow the range between them by plotting using the Runge-Kutta method, and increasing the stepsize until I've reached an acceptable accuracy. Unfortunately I fear this will be much slower than I like, not to mention that one angle might overshoot one time, then when I increase the stepsize, could undershoot it, which would just make things crazier than I think they should be. I've googled this quite a bit, but sadly couldn't find anything that, you know, _helps_. If you guys know of any algorithms that might be helpful, I would greatly appreciate it. Even better would be a way to rework the Differential Equations to simplify the search, but sadly I am no where near good enough at Calculus for that to be a viable option. Thanks, Jacob


All matter has a mass but does all matter have a gravitational pull?

_I know that all planets and stars have a gravitational pull but does a simple much smaller object have a gravitational pull for example a pebble?_


Does the mass distribution matter in (Schwarzschild) black holes?

Is it possible that from the same initial mass different black hole radius will be created due to different mass distribution during black hole creation? If mass is concentrated more on the outside bigger event horizon will be created? If mass is concentrated more in the center smaller black hole will be created?


Rubbing a rod with silk?

> Rubbing a glass rod with silk causes charges to be exchanged and > consequently both objects get charged. Why do the objects have to be "rubbed"? I get that one has a stronger pull on the electrons than the other, but shouldn't just allowing the objects to make contact be enough.? I would appreciate a "visualization" of whats happening. Similar questions: Why two objects get charged by rubbing? How does rubbing cause the transfer of electrons from one object to the other? **Neither question addresses why the objects need to be rubbed instead of just making contact.**


What's the dimensionality of a solid angle?

I haven't seen this explained clearly anywhere. Solid angles are described usually as a fraction of the surface area of a unit sphere, similar to how angles are the fraction of the circumference of a unit circle. However, I don't know how solid angles are actually quantified. Are solid angles just a single number, the describes this fraction of the area? It's confusing to me since often times, I've seen integrals that integrate over a sphere using solid angles, which seems to imply that solid angles are multi-dimensional quantities (e.g. when integrating using spherical coordinates, the solid angle would have to consist of the azimuthal and polar angles covered by the differential solid angle). Following from this, how would you write down a solid angle that covers the entire surface of a unit sphere?


How do the single photon energy and em-signal energy correlate?

If the photon (as a quantum of the electromagnetic field) has no defined(?) amplitude, how does (or where from?) the electromagnetic wave's amplitude appear? The formulation of the question is not quite precise, but the main idea is if we cannot define the single photon's amplitude, since $E=hf$ only (are there any other equations?), how can we talk about an amplitude of an EM wave, which is carried via photons (EM quanta)?


Derivation of Noether's theorem - A problem with physical significance

My question is about the field theoretic version of Noether's theorem. I am deeply troubled by one of the hypotheses of the theorem. As it is the standard textbook for Lagrange mechanics, I'll follow Goldstein's account (starting p. 588 in the second edition of "Classical Mechanics"). I have no problem with condition 1 since I work in Minkowski space. I am completely okay with condition 2, which amounts to asking that the equations of motion be the same for two observers who use different systems of coordinates to describe the same spacetime and different functions to describe the same fields. However, I can't make any sense of condition 3. I don't see what its physical meaning can be. I haven't seen it explained convincingly anywhere, and can't seem to figure it out for myself. For those who don't have any access to Goldstein's book but feel they might be able to help, condition 3 is the requirement that the action integrals be equal for the two aforementioned observers. I hope someone has some fantastic insight on this! :-)


Motion of mercury

I studied that mercury motion around the sun slightly displace by a certain value in each year. But, this is not predicted by kepler until general theory of relativity. What does general theory does with mercury.


What did general relativity clarify about Mercury?

I frequently hear that Kepler, using his equations of orbital motion, could predict the orbits of all the planets to a high degree of accuracy -- _except Mercury_. I've heard that mercury's motion couldn't be properly predicted until general relativity came around. But what does general relativity have to do with Mercury's orbit?


General Relativity & Kepler's law

According to Kepler's law of planetary motion, the earth revolves around the sun in an elliptical path with sun at one of its focus. However, according to general theory of relativity, the earth revolves around the sun in the curved space and it revolves in a circular path. Which is correct? Does the planet moves in a circular path or an elliptical path?


What is information?

We're all familiar with basic tenets such as "information cannot be transmitted faster than light" and ideas such as information conservation in scenarios like Hawking radiation (and in general, obviously). The Holographic Principle says, loosely, that information about a volume of space is encoded on its two-dimensional surface in Planck-sized bits. In all these contexts, I can take "information" to mean predictive or postdictive capability, i.e. information is what enables us to state what the outcome of a measurement was or will be (locally). But what _is_ information, exactly? Do we have any kind of microscopic description of it? Is it just a concept and, if so, how can we talk about transmitting it? I suspect this is probably as unanswerable as what constitutes an observer/measurement for wave function collapse, but I'd love to know if we have any formulation of what information is made of, so to speak. If I'm talking nonsense, as I suspect I may be, feel free to point this out.


2d soft body physics mathematics

> **Possible Duplicates:**   >  Modern references for continuum mechanics   >  Good books on elasticity The definition of **rigid body** in Box2d is > A chunk of matter that is so strong that the distance between any two bits > of matter on the chunk is completely constant. And this is exactly what i don't want as i would like to make **_2D_** (maybe 3D eventually), elastic, deformable, breakable, and even sticky bodies. What I'm hoping to get out of this community are resources that teach me the math behind how objects bend, break and interact. I don't care about the molecular or chemical properties of these objects, and often this is all I find when I try to search for **how to calculate what a piece of wood, metal, rubber, goo, liquid, organic material, etc. might look like after a force is applied to it.** Also, I'm a very visual person, so diagrams and such are EXTREMELY HELPFUL for me.


Does time really exist?

Does time really exist? or is it a human invention and convention? What is the truth? are there time particles? please remember, I'm looking for constructive answers, not condescending and void ones.


What makes us move in time?

Time is considered to be a dimension, and we are moving at certain rate in one direction in time. **What force makes us move in time?** I mean it must be ether time moving or us moving in time so there has to be some force that 'pushes'/'pulls'? Was this 'time inertia' acquired during big bang, or nothing is moving and I am just being silly?


What are quarks made of?

So atoms are formed from protons and neutrons, which are formed from quarks. But where do these quarks come from? What makes them?


What is the smallest existing thing in theory and law?


Are quarks and leptons actually fundamental particles?

> **Possible Duplicate:**   >  What are quarks made of? Are quarks and leptons actually fundamental, or are they made up of even more fundamental particles? And is it true that many consider quarks and leptons are so small that they may be thought of as geometrical points in space with no spatial extension at all?


Non-dimensionalization for spatially varying material parameters

For a homogeneous material of length $L$, we can write the heat equation as $$\rho c\frac{dT}{dt}=k\frac{du^2}{dx^2}, \text{ } x\in (O,L)$$ where $T$ is the temperature, $\rho$ is the thermal conductivity, $c$ is specific heat, and $k$ is the thermal conductivity. To non-dimensionalize it, I know that I can define a new variable for space $\hat{x}=\frac{x}{L}$, time $\hat{t}=\frac{kt}{L^2\rho c}$ and temperature $\hat{T}=\frac{T}{T_0}$ where $T_0$ is the initial temperature, using the chain rule, obtain the simplified form $$\frac{d\hat{T}}{dt}=\frac{du^2}{dx^2}.$$ When the material is heterogeneous, we often assume that the thermal conductivity varies in space and is modeled by the function $k=k(x)$. When this is the case, I'm not quite sure what is the standard approach non- dimensionalization. I conjecture that the process should still be similar even though $k(x)$ is not a constant. Should I choose a reference thermal conductivity $k$ from the range of values $k(x), x\in (0,L)$ follow the same proceedure? Is there an alternative methodology to non-dimensionalize in the case of spatially varying material parameters?


Angular Momentum Addition Theorem - Sanity Check

Looking back at my quantum mechanics notes, the angular momentum addition theorem is listed as: $j=j_1+j_2,j_1+j_2-1, ..., |j_1-j_2| $ (Using conventional notation) , but I'm a little unsure how to interpret the introduction of the modulus operation ($|...|$) and couldn't easily find any examples. I'm assuming you apply the modulus to any expression which would otherwise yield a negative value for $j$? I'd appreciate a nod from someone in the know :-).


Prove that a derivative with respect to a covariant 4-vector is a contravariant vector operator

In special relativity, I know you can prove that the derivative with respect to a contravariant 4-vector component transforms like a covariant vector operator by using the chain rule, but I can't work out how to prove the inverse, that the derivative with respect to a covariant 4-vector component transforms like a contravariant vector operator.


Work Done On a Body When It is Not in Contact with the agent of the Force

_Is it necessary for work to be done on a body that the agent of the force remains in contact with the body?_ For example, if I hit a football with my foot with a small amount of force and it moves a certain distance on the ground, then what could we say about the 'work' in this case if there would be no friction and the air resistance to stop the ball?


Improvement of microscope resolution with Oil

Why and when do we need to place oil over the sample to achieve higher optical resolution ? Is this idea is valid for the enhancement of all optical microscopy techniques and magnification scales ?


Analytic continuation of imaginary time Greens function in the time domain

Consider the imaginary time Greens function of a fermion field $\Psi(x,τ)$ at zero temperature $$ G^τ = -\langle \theta(τ)\Psi(x,τ)\Psi^\dagger(0,0) - \theta(-τ)\Psi^\dagger(0,0)\Psi(x,τ) \rangle $$ It is well known that we can obtain the retarded Greens function by performing Fourier transformation into frequency space and performing the analytic continuation $iω \to ω + i\eta$. What I would like to do is to perform the analytic continuation directly in the form $iτ \to t$, but I don't know how to deal with the $\theta(τ)$ terms. > How to perform the analytic continuation $iτ \to t$ of the step function > $θ(τ)$? In my case, I am dealing with a chiral Luttinger liquid, giving something like $$ G^τ(x,τ) = -\left[\theta(τ)\frac i{iλ + ivτ - x} - \theta(-τ)\frac i{iλ - ivτ - x}\right] $$ where $λ \approx 0$ is an infinitesimal but important regularization. Of course, the analytic continuation into the time domain is going to look something like $$ \frac1{iλ + vt - x} $$ but I'm interested in the precise form. Also, I'm ultimately interested in the spectral function, so I don't mind if analytic continuation gives me yet another variant of a Greens function, but I would like to obtain it precisely from the imaginary time Greens function without going through a tedious Fourier transform. For instance, Giuliani and Vignale's book "Quantum Theory of the Electron Liquid" uses the Greens function $G_{>}(x,t)$ to great effect (equation (9.133)).


Photon Angular Momentum

Essentially I am wanting to evaluate $$\langle j\, m \mid a^\dagger(\mathbf{k}, \lambda) \mid 0 \rangle \,,$$ where $\lambda$ indicates the circular polarization (about $\mathbf{k}$). We have that $\mathbf{J}= \mathbf{L} + \mathbf{S}$. It's straightforward to show that circular polarization corresponds to definite spin projections along the $\hat{\mathbf{k}}$ axis (the direction of propagation). However, I don't yet know how to find the $\mid \ell\, m_\ell \rangle$ projections. I don't know how how to express $\mid \ell\, m_\ell \rangle$ states in terms of Fock states, for example. With $\hbar = c = 1$, $$\mathbf{L} = \frac{1}{4\pi}\int \mathop{d^3r} \sum_i E_i\left(\mathbf{r} \times \boldsymbol{\nabla} \right)A_i $$ $$\mathbf{S} = \frac{1}{4\pi}\int \mathop{d^3r} \mathbf{E} \times \mathbf{A} = -i \int \mathop{d^3k} \mathbf{a}^\dagger\left(\mathbf{k}\right) \times \mathbf{a}\left(\mathbf{k}\right)\,. $$ The above uses the following definition $$\mathbf{a}\left( \mathbf{k}\right) = \sum_{\lambda=\pm1} \boldsymbol{\epsilon}_\lambda\left( \mathbf{k}\right) a_\lambda\left( \mathbf{k}\right)$$ So, any help determining the angular momentum of a photon would be appreciated (incuding mention of references dealing with this subject). I have been using the Coulomb gauge.


Thermodynamic entropy vs. quantum mechanical entropy

Is there a fundamental difference in the definition of entropy when considering the classical thermodynamic picture vs. the quantum mechanical picture, or are they both fundamentally equivalent?


Dipole matrix elements through parity argument

I am trying to find the following dipole moment matrix element $(|n,\ell,m\rangle)$. $$e\langle1,0,0|\vec r|2,0,0\rangle$$ I believe that I can say this matrix element is zero because of parity. The wavefunctions have parity $(-1)^\ell$ and seeing as each has $\ell =0$, they are even parity. Then r is odd, as it sends $\vec r \rightarrow-\vec r$. This means the entire expression is odd, therefore the matrix element is 0. Is my reasoning sound?


Constant Acceleration Question

A Cessna 150 aircraft has a lift-off speed of approximately 125 $kmh^{-1}$. What minimum constant acceleration does this require if the aircraft is to be airborne after a take-off run of 129 m? So I wanted to use the formula that $x=\cfrac{(v_{final}^2 - v_{initial}^2)}{2a}$ To solve for the constant acceleration but I don't have a final velocity from this problem? My initial velocity is 125 $kmh^{-1}$ and my $x$ (displacement) would be 129 meters but how do I find the constant acceleration?


Equivalent Rotation using Baker-Campbell-Hausdorff relation

Is there a way in which one can use the BCH relation to find the equivalent angle and the axis for two rotations? I am aware that one can do it in a precise way using Euler Angles but I was wondering whether we can use just the algebra of the rotation group to perform the same computation?


Why $\frac{d}{dt}r_{a}\nabla_{a}U_{ab}+\frac{d}{dt}r_{b}\nabla_{b}U_{ba}=\frac{d}{dt}U_{ab}?$

In classical mechanics for two mass particles $a$,$b$ we assume the symmetric potential arising from $F_{ab}$ and $F_{ab}$ given by $$U_{ab}(r)=-\int^{r}_{r_{0}}F_{ab}(r')dr'$$ and $$U_{ba}(r)=-\int^{r}_{r_{0}}F_{ba}(r')dr'$$ The book **mechanics** by _Florian Scheck_ gives $$\frac{d}{dt}r_{a}\nabla_{a}U_{ab}+\frac{d}{dt}r_{b}\nabla_{b}U_{ba}=\frac{d}{dt}U_{ab}$$ because $$[\frac{d}{dt}r_{a}\nabla_{a}+\frac{d}{dt}r_{b}\nabla_{b}]U_{ab}=\frac{d}{dt}U_{ab}$$ I am confused how we get the summation form. My derivation goes as follows: Notice $F_{ab}=-\nabla_{b} U_{ab}$. Thus we should have $$\frac{d}{dt}U_{ab}=\frac{d}{dr}*\frac{dr}{dt}U_{ab}=\frac{dr}{dt}\frac{d}{dr}U_{ab}=\frac{dr}{dt}[-F_{ab}]=\frac{d}{dt}[r_{a}-r_{b}][-F_{ab}]=[\frac{d}{dt}r_{a}\nabla_{a}+\frac{d}{dt}r_{b}\nabla_{b}]U_{ab}$$ My simple question is just whether my derviation is correct, for I assume $r=r_{a}-r_{b}$ at here.


Is imperative magnetic flux the capacity of individual atoms, or the constituent valency of chemically-bonded molecules within a vacuum?

If magnetic energy depends on the electron poles within two-fields within a permanent magnets void, how do invidiual atoms react within the attraction or repulsion of poles, and what incurrence does this live by in regards to molecular struction?


Why is $\vert I=1,I_3=1\rangle = -p\bar n$

My book doesn't explain well how to build a doublet of antiparticles that transforms the same way the particle doublet $(p,n)^T$ (proton neutron) does. They claim $$\tag 1 \vert I=1,I_3=1\rangle = -p\bar n$$ for a composite nucleon-antinucleon system. **Why is $(1)$ true?** Perhaps it's just bad notation in the book? I got confused because the Clebsch-Gordan for $(1)$ comes with a $+1$ and not a $-1$, but perhaps one should insclude this negative sign into the CG coeff? That is $(1)$ should be $$\tag 2 \vert I =1,I_3=1\rangle =\underbrace{\vert 1/2,1/2\rangle}_{p}~\Big(\underbrace{-\vert 1/2,1/2\rangle}_{-\bar n}\Big)?$$


Efficiently distinguishing mixed quantum states?

Assume we know two different mixed states, p and q, and an efficient (quantum) algorithm for creating such two. Does it follow that there exists a computationally efficient method/measurement for optimally(that is, according to the trace norm distance) distinguishing between the two?


How could there be a truly "pure" state?

If the Universe did start from a single point, then wouldn't all particles be fundamentally entangled? How then could there be a truly "pure" state?


Behaviour of liquid in vaccume

Is it possible for a liquid to exist in a high quality vacuume? For example, a few Torr. If so what are the methods for doing this?


What is "thermal undulation" in the context of lipid bilayers?

What is thermal undulation in the context of lipid bilayers? Is it another word for "thermal fluctuation"?


How does this formula for calculating the "mass sum" in a collision translate to 3D?

According this tutorial, formula number 5: $$j = \frac{-(1 + e)((V^{A} - V^{B}) * t)}{\frac{1}{mass^{A}} + \frac{1}{mass^{B}}}$$ translates into formula number 6: $$j = \frac{-(1 + e)((V^{A} - V^{B}) * t)}{\frac{1}{mass^{A}} + \frac{1}{mass^{B}} + \frac{(r^{A} \times t)^{2}}{I^{A}} + \frac{(r^{B} \times t)^{2}}{I^{B}}}$$ when dealing with oriented bodies. $j$ is a scalar by which you divide normal and friction impulses to simulate them. This is only valid for the 2D case, though, where the inertia is a scalar. What if I have the inertia as a tensor (more specifically a 3x3 matrix)?


Gauss's / Divergence theorem in Classical electrodynamics for the Electric field

Can somebody explain the proof of Gauss's theorem / divergence theorem taking the vector as electric field $$\iiint(\nabla\cdot\vec E)\mbox{ d} V=\iint \vec E \cdot\hat{n} \mbox{ d} A?$$


Bass and Treble-Car Steroes

In a car which phenomenon, diffraction or the resonant frequency of the car, lends itself more to the ability of bass to go farther? Related Answer: Why do bass tones travel through walls?


The discontinuity of Electric Field

''electric field always undergoes a discontinuity when you cross a surface charge $\sigma$'' GRIFFITHS In the derivation; Suppose we draw a wafer-thin Gaussian Pillbox, extendind just barely over the edge in each direction. Gauss law states that: $\int_{S} E \cdot A = Q_{enc}/ \epsilon $ and so $E_{above}^{perp} - E_{below}^{perp} = \sigma/ \epsilon $ My question is why not $2A$ ? $\int_{S} E \cdot A = 2EA$ because the top area of pillbox and the bottom area of pillbox, just as because the 2 parts of the flux... SO.. WHY NOT : $E_{above}^{perp} - E_{below}^{perp} = \sigma/ 2\epsilon $ And why there is tangencial component of electric field; not just perpendicular to the surface, which can be seen as flat just looking very close to the surface.


Quantum Theory as a framework for other theories of nature

We know that Quantum Theory should be considered as a framework in which all other theories/forces (Strong, Weak, EM and Gravity) exist. For example, we have the Quantum Chromodynamics, Quantum Flavordynamics (Electroweak), Quantum Electrodynamics (but still no Quantum GR). When I think about this, it then strikes me why gravity, and specifically special relativity, is part of the framework itself (because QFT is based on QM and SR). Why would a theory like SR (and maybe in future GR) be part of the framework? This looks like a circular logic. I would appreciate it if someone can explain? My understanding is that gravity is different because it exist everywhere, and are not a result of a charge (electric, color or flavor). That's why we cannot speak about a quantum theory of EM without considering gravity, but we can do it without considering strong or weak force. Is this correct? EDIT: it would make more sense to me to see Gravity/GR considered only as a framework (a geometric one) rather than a force, or it can be considered a fictitious force, as with the centrifugal force. see first paragraph of first answer here.


Upward bouyant forces?

When an object is submerged in a fluid (e.g., water), there is a pressure on the object due to weight of water by p=mgh, but why is there an upward buoyant force?


Does existence of magnetic monopole break covariant form of Maxwell’s equations for potentials?

Absence of magnetic charges is reflected in one of Maxwell's fundamental equations: $$\operatorname{div} \vec B = 0 \text{ (1).}$$ This equation allows us to introducte concept of vector potential: $$\vec B = \operatorname{rot} \vec A.$$ Using this concept, it is possible to express Maxwell's equations in a graceful symmetric form: $$\nabla^2 \vec A - \frac{1}{c^2}\frac{\partial^2 \vec A}{\partial t^2} = = - \frac{\vec j}{\epsilon_0 c^2} \text{ (2)}$$ $$ \nabla^2 \phi -\frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = - \frac{\rho}{\epsilon_0} \text{ (3)} $$ Noticing, that vector $\vec A$ and scalar $\phi$ potentials, as well as electric current density $\vec j$ and charge density $\rho$, form a 4-vector in Minkovsky space-time. Therefore, Maxwell's equations can be expressed in a covariant form, using dalambertian: $$\nabla_{\mu}\nabla^{\mu} A_{\nu} = \frac{j_{\nu}}{\epsilon_0} \text{ (4)}$$. If magnetic monopols exist, Maxwell's equation (1) will look as: $$\operatorname{div} \vec B = \mu_0 c \rho_{magnet}$$ As the divergence of $\vec{B}$ isn't equal to zero, it impossible to introduct concept of vector potential. Thus, the equation in the form of (4) will not be possible to express.


Atoms' excitation energies as derived from Frank-Hertz experiment data

I'm analyzing the experimental data obtained during Frank-Hertz experiment (conducted with Hg atoms): ![enter image description here](http://i.stack.imgur.com/D3K82.jpg) Accelerating voltage values were multiplied by 0.1 during measurement (i.e. the mean value of energies differences if not 0.508 eV, but 5.08 eV). The output voltage was measured on resistive load of the anode, therefore it is proportional to anode's current. The first five minimas in output voltage (top graph) have constant voltage difference - this value is used to calculate the first excitation energy of Hg atom (bottom graph). The sixth minima, however, can be seen to have larger increase in output voltage(top graph - the first five voltage peaks may be fitted by a linear graph, whereas the sixth is no longer fits to this graph) and a higher voltage offset from the previous minima (bottom graph). **Question 1:** Is the sixth minima corresponds to the higher order excitation of Hg, or ionization, or else? **Question 2:** The curve is clamped at $V_{out} = 5V$. Is there a physical effect which causes this, or it is just the limit of measurement equipment, or else?


How to work out momentum when there are velocity and mass changes

I have a pretty simple homework question, but I can't rap my head around it. In the question a swimmer of $55 \mbox{ } \mathrm{kg}$, jumps off a stationary raft of $210\mbox{ }\mathrm{kg} $. The swimmer jumps off the raft with a speed of $4.6 \mbox{ } \mathrm{ms}^{-1} $. I need to work out the recoil velocity of the raft. So because Momentum before = Momentum after, I went: $p_i = 0$ Therefore $0 = p_f$ and $p_f = mv$, $m = 210\mbox{ }\mathrm{ } $, so the $v$ would have to equal $0$. Making the recoil velocity equal $0$. However that doesn't seem right. Could use some clarification or help, thanks.


Why is Titan able to maintain an atmosphere, and not Luna?

I just read stability of hypothetical lunar atmosphere. From the correct answer, i understand, the low escape velocity from Luna is part of the reason it is unable to retain an atmosphere. Titan apparently has a comparable escape velocity   * Titan = 2.65 km/sec    * Luna = 2.4 km/sec ; yet Titan maintains an atmosphere. Why? What have I missed? Does Luna's relative proximity to Sol make the difference?


Is a photovoltaic cell on an artificial satellite the same construction as one used on Earth's surface?

Are photovoltaic cell arrays on a satellite the same that are used within Earth, or is there some difference in their construction given the differing environment in which they are to operate? Does the efficiency/life of a photovoltaic cell change depending upon whether it is within/without Earth's atmosphere?


Explain Turbulence

I'm a high school student. I still don't understand what turbulence is. Please can you explain what it really is? This is what I think it is: rotating motion of water when a particle travels at a velocity of $V$.


Consistency of equation with special relativity?

The following is the equation which, I want to know, if it is valid in relativistic domain. Consider two equal charges moving in same direction with velocity $v$ and charge $q$ at a separation of $d$. The magnetic force acting between them is $(1/4\pi\epsilon_0 c^2)v^2q^2/d^2$ and the electrostatic force is $(1/4\pi\epsilon_0 )q^2/d^2$.the ratio of the magnetic (attractive) force and the electrostatic (repelling) force is $v^2/c^2$ and therefore the conclusion provided was that two individual charges with same (or any) velocity can never attract. Is this equation valid relativistically. Moreover, since magnetic field is just the relativistic counterpart of electrostatic force, can we find a frame of reference wherein in the case stated, all the force is magnetic and must be attractive? I am comfortable only with basic special relativistic mechanics but an intuitive understanding would be better.


Do we have 2 minutes of extra morning?

my physics teacher told me about the refraction and its applications one of them was   **2 minutes of early sunrise** and after she explained this effect she concluded that days are   **2 minutes longer** than one would naively presume. However, I think that **her conclusion is wrong** because   **if sunrise is considered then sunset should also be considered** and according to me **sunsets should be 2 minutes late** therefore **the day time is increased by 4 minutes and not 2 minutes** over the naive calculation. According to me, the situation looks something like this:   ![enter image description here](http://i.stack.imgur.com/O1YUG.jpg)   Is this idea of mine correct or not? And if we both are wrong, then what should be the right conclusion and why ?


What was Feynman's "much better way of presenting the electrodynamics" -- which did **not** appear in the Feynman lectures?

Does anyone know what Feynman was referring to in this interview which appears at the beginning of The Feynman Tips on Physics? Note that he is referring to something that did not appear in the Feynman lectures. > I didn't like to do the second year, because I didn't think I had great > ideas about how to present the second year. I felt that I didn't have a good > idea on how to do lectures on electrodynamics. But, you see, in these > challenges that had existed before about lectures, they had challenged me to > explain relativity, challenged me to explain quantum mechanics, challenged > me to explain the relation of mathematics to physics, the conservation of > energy. I answered every challenge. But there was one challenge which nobody > asked, which I had set myself, because I didn't know how to do it. I've > never succeeded yet. Now I think I know how to do it. I haven't done it, but > I'll do it someday. And that is this: How would you explain Maxwell's > equations? How would you explain the laws of electricity and magnetism to a > layman, almost a layman, a very intelligent person, in an hour lecture? How > do you do it? I've never solved it. Okay, so give me two hours of lecture. > But it should be done in an hour of lecture, somehow -- or two hours. > > **Anyhow I've now cooked up a much better way of presenting the > electrodynamics, a much more original and much more powerful way than is in > the book.** But at that time I had no new way, and I complained that I had > nothing extra to contribute for myself. But they said, "Do it anyway," and > they talked me into it, so I did. Did this approach to teaching electrodynamics appear in any of his later writing?


Rotating a complex number

Let us begin in a two-dimensional Euclidean plane. The vector is e.g. $\vec{V}(x,y)$ It is often useful – but in this case, it's just a mathematical trick that doesn't make the complex numbers "fundamental" – to combine the components into a complex number, $z=x+iy$. The two-dimensional rotations in $SO(2)$ are fully determined by the angle $δ$. And the matrix acting on the vector $\vec{V}(x,y)$ $M= \begin{bmatrix}+\cosδ & −\sinδ\\\\+\sinδ & +\cosδ\end{bmatrix}$ may be fully replaced by the complex coefficient $e^{iδ}$ that multiplies our complex coordinate $z$. Instead of $z$, however, we could have dealt with its power $z^p$. The rotation could be described by the complex transformation $z^p→z^{′p},z^{′p}=e^{ipδ}z^p$. In this context, the complex number $e^{ipδ}$ plays the role of the "transformation matrix" that rescales the one and only component of our tensor-spinor-whatever, $z^p$. In our overly trivial two-dimensional context, the coefficient or exponent $p$ may be anything you want. But the value $p=1/2$ may be identified with the spinors in two dimensions. The spinor in two dimensions may be represented as $\sqrt{z}$ where $z=x+iy$ encodes a vector; so the spinor $z^{1/2}$ is literally the square root of a vector (translated to a complex number) in this case. So I got all this from a blog that tries to explain spinors which unfortunately I don't remember. I don't understand why the $p$ in $e^{ipδ}$ has to be there since any complex number can be rotated without the need for that coefficient. Is there some rule for rotating complex numbers that are raised to higher powers $p=2,...,n$ or am i missing something?


Textbook on the Geometry of Special Relativity

I am looking for a textbook that treats the subject of Special Relativity from a geometric point of view, i.e. a textbook that introduces the theory right from the start in terms of 4-vectors and Minkowski tensors, instead of the more traditional "beginners" approach. Would anyone have a recommendation for such a textbook ? I already have decent knowledge of the physics and maths of both SR and GR ( including vector and tensor calculus ), but would like to take a step back and expand and broaden my intuition of the geometry underlying SR, as described by 4-vectors and tensors. What I do **_not_** need is another "and here is the formula for time dilation..." type of text, of which there are thousands out there, but something much more geometric and in-depth. Thanks in advance.


Direction of expansion of the universe

From what I understand the expansion of the universe has no "center". If we're flying through space away from the "center of the big bang", there's basically no way to tell. Every two given points in space gets farther away from each other, and we can pick any point as center if we like. I also understand that the speed of light is not relative to the speed of the source emitting the light. If I go on a train in $c/2$, turn on a flash light pointing forward, the light emitted from the flashlight will still travel at the speed of $c$. Now here's my question: Why can't we set up a sphere with photodetectors with synchronized clocks on the inner walls, turn on a light in the center, record the exact time at which each photodetector detects the light, and compare the times to figure out if the sphere was traveling in some certain direction? I mean if we turn on a lightbulb and light travels with the speed of $c$ in all directions at the same time, my intuition tells me that we should be able to figure out some form of "reference stand still". (Tagging this with general relativity because I _suspect_ that it's impossible to set up the experiment the way I like due to relativity.)


Why is the inner product between divergence-free current$\vec{J}$, (satisfied $\nabla\cdot\vec{J}=0$) and a gradient field$\nabla \varphi$ zero?

I read a statement saying that the inner product between divergence-free current and a gradient field is zero. Divergence-free surface current is $\nabla\cdot\vec{J}=0$, and $\vec{J}$ could be represented as $\vec{J}=\nabla\times(\psi\hat{n})$, where $\hat{n}$ is the normal vector of the surface. So the statement becomes: $\nabla\times(\psi\hat{n})\cdot \nabla \varphi=0$. I think according to the identity: $$\nabla\cdot(\vec{A}\times\vec{B})=\vec{B}\cdot(\nabla\times\vec{A})-\vec{A}\cdot(\nabla\times\vec{B})$$ we have $$\nabla\times(\psi\hat{n})\cdot \nabla \varphi=\nabla\cdot(\psi\hat{n}\times\nabla\varphi)+\psi\hat{n}\cdot\nabla\times\nabla\varphi=\nabla\cdot(\psi\hat{n}\times\nabla\varphi),$$ but what next? **Update** Thank you Luboš Motl. I suppose I now understand why, but I have no enough points to reply below, so just update here my answer. Target is to prove $\int_s \vec{J}\cdot\nabla\varphi ds=0$ The whole process is as follows: First, $\vec{J}$cannot go across the surface edge, so $\vec{J}\cdot\hat{t}=0$, where $\hat{l}$ is the surface edge direction and $\hat{t}=\hat{l}\times\hat{n}$ is the edge out direction Second, according to the identity $\nabla\cdot(\vec{A}\times\vec{B})=\vec{B}\cdot(\nabla\times\vec{A})-\vec{A}\cdot(\nabla\times\vec{B})$, we have $\vec{J}\cdot\nabla\varphi=\nabla\times(\psi\hat{n})\cdot \nabla \varphi=\nabla\cdot(\psi\hat{n}\times\nabla\varphi)+\psi\hat{n}\cdot\nabla\times\nabla\varphi=\nabla\cdot(\psi\hat{n}\times\nabla\varphi)$ since $\nabla\times(f\vec{A})=\nabla{f}\times\vec{A}+f(\nabla\times A)$ $\psi\hat{n}\times\nabla\varphi=-\nabla\times(\varphi\psi\hat{n})+\varphi\nabla\times(\psi\hat{n})$ Then $\nabla\cdot(\psi\hat{n}\times\nabla\varphi)=\nabla\cdot(-\nabla\times(\varphi\psi\hat{n})+\varphi\nabla\times(\psi\hat{n}))=\nabla\cdot(\varphi\nabla\times(\psi\hat{n}))$ finally, $\int_s \vec{J}\cdot\nabla\varphi ds=\int_s\nabla\times(\psi\hat{n})\cdot \nabla \varphi ds =\int_s \nabla\cdot(\varphi\nabla\times(\psi\hat{n}))ds=\oint_l \varphi\nabla\times(\psi\hat{n})\cdot\hat{t}dl=\oint_l \varphi\vec{J}\cdot\hat{t}dl=0$ I think here the important things are:   1. Generally speaking, divergence-free current usually can be expressed as $\vec{J}=\nabla\times\vec{T}$, and $\vec{J}=\nabla\times(\psi\hat{n})$ is specially for surface current.   2. the $\hat{n}$ is only valid on the surface(there is no meaning of $\hat{n}$ for point in side of a body). the integral is on the surface rather than on the body. According to the original article, it is just talking about PEC and surface current.


Could spontaneous symmetry breaking happen again in our universe?

It is generally believed that $10^{-35}$ seconds after the Big Bang, the symmetry of a GUT was broken and after $10^{-12}$ seconds the electroweak force was broken: \begin{equation} \mathrm{SU(2)} \times \mathrm{U(1)} \rightarrow \mathrm{U(1)} \end{equation} This symmetry breaking is a result of the universe cooling down and undergoing a phase transition. I'm aware that the temperature of the universe it about $2.7$ Kelvin, so the temperature of the universe cannot decrease much more, but I was wondering if there is a chance that another phase transition might happen again in the future?


Good introductory books on AdS/CFT correspondence

> **Possible Duplicate:**   >  Introduction to AdS/CFT Since my question in a similar topic was deleted, I'll ask away and hope ppl won't come here telling me: this was already asked! :\ I have a course in advanced gravitation, and I was thinking of doing this for a seminar, and maybe something about it for a masters thesis (but there is still time for that), so I'd need something that's readable and something that can introduce someone into this field of physics. I have some knowledge of particle physics (mostly what I've learned through Griffith's book), and I've started to read Introduction to AdS-CFT, lectures by Horatiu Nastase that I found on ArXiv, and so far I can follow what he's talking about (I am familiar with path integral notation, Feynman diagrams...). But I'd like some 'simple' books, so that when I'd be giving presentation, my colleagues won't be blindly looking at me and wonder what the hell I'm talking about :D Any suggestion is appreciated :)


electric potential energy symbol in Schrodinger equation

In my introductory physics class, $V$ is the symbol for electric potential (joules per coulomb) and $U$ is the symbol for electric potential energy (joules). Since the Schrodinger equation is the sum of Kinetic and Potential energies in the system, $V(r)$ must represent $U$... if so, is there any particular reason why $V(r)$ is used as opposed to $U$?


Why a system should be at its lowest energy state for its stability?

Every possible reaction in chemistry is to attain stability. In physics, the alignment of an electric dipole in an external electric field and in all other physical systems (at least those I study in high school) attains stability at the lowest energy state. But, why is it so?


References for ADM formalism and cosmological perturbation theory

What would you consider the best online resources for learning the 3+1 ADM formalism and gauge invariant perturbation theory in cosmology? (Assuming intermediate level GR and QFT familiarity)


What is the relationship between the verbal definition and the mathematical definition of some quantities?

I know this is probably an easy question, but it's been a while since I've studied physics and I've started reading some circuit analysis textbooks. I'm finding hard to understand the relationship between between the verbal definition of quantites and the mathematical definitions. For instance, in the Sadiku's book "Fundamentals of electic circuits", I've got the following verbal definiton for voltaje (literal) "Voltage (or potential difference) is the energy required to move a unit charge through an element, measured in volts (V)." And then, it says that this mathematically "means" $$ v_{ab} \triangleq \frac{dw}{dq} $$ I can't understand well the relationship between this two "definitions" could someone explain further the relationship?


Kepler Orbital Elements to Cartesian (xyz)

I'm not really sure if this is the place to ask this, but anyway here is my question: Let's say I have the Kepler orbital elements of the ISS, for example, (NASA stuff). Now I want to compute the coordinates relative to the earth at a specific time so that it can be displayed, like in a sky map (I am actually experimenting with Google Sky Map). I found many websites discussing Kepler orbital elements, but I have only found 2 pdf's that talk about this conversion: http://www.phas.ubc.ca/~newhouse/p210/orbits/cometreport.pdf and I can't post a 3rd link yet. I do not fully understand which values I need to calculate, and which are given by NASA. I am also assuming that the data given by NASA is valid for the vector time provided, meaning that time obviously has to come into the calculations somewhere. Thanks, I would appreciate some clarification.


Gauge Invariance of Yang Mills Lagrangian

I am trying to show the invariance of the following Yang Mills Lagrangian: $$L= -\frac{1}{4} F^a_{\mu \nu} F_a^{\mu\nu} + J_a^\mu A_\mu^a$$ under the following gauge transformation ($\theta$ being a rotation in color space and $g$ related to the structure constant): $$L \rightarrow -\frac{1}{4} \left( F^a_{\mu\nu} +\epsilon ^a_{jk} \theta^j F^k_{\mu \nu}\right)\left(F_a^{\mu\nu} + e_a^{jk}\theta_j F_k^{\mu\nu}\right) + \left( J_a^\mu + \epsilon_a^{jk} \theta_j J_k^\mu \right) \left( A_\mu^a + \epsilon^a_{jk}\theta^jA_\mu^k -\frac{1}{g} \partial^\mu \theta^a \right),$$ where each term is now transformed accordingly. I was able to simplify the above to and obtain: $$L \rightarrow -\frac{1}{4} \left( F^a_{\mu\nu}F_a^{\mu\nu} + \epsilon^a_{jk} \theta^j F^k_{\mu\nu} \epsilon_a^{j' k'} \theta_{j'} F_{k'}^{\mu\nu}\right) + J_a^\mu A_\mu^a - J_a^\mu \frac{1}{g} \partial^\mu \theta^a + \epsilon_a^{jk} \theta_j J_k^\mu \epsilon^a_{j'k'} \theta^{j'}A_\mu^{k'} - \epsilon_a^{jk} \theta_{j} J_k^{\mu}\frac{1}{g} \partial^\mu \theta^a.$$ How could I possibly reduce it to a form similar to the original, untransformed Lagrangian? There are about 4 terms I can't get rid of, though it has been suggested to me that I use the equation of motion of YM, which I have handy but can't seem to use them appropriately. Any help would be greatly appreciated. Also note that I may end up with a boundary term which would vanish when varying the action, thus possibly giving me say 3 terms instead of the original 2 (which is fine, though I can't identify them yet).


What is the length of 1 second in meters

If time is treated as a fourth dimension of spacetime, what is relation between length and time units? Or in other words, how can I convert time units to length units, for instance seconds to meters?


Why the energy of a marshmallow is so huge?

In this comment in a blog kudzu computes the energy of a marshmallow with mass $M=25 grams$ by using $E=mc^2$: $E=Mc^2 = 2.247\times 10^{+15} Joules$ I may be wrong but this seems like a huge energy for a marshmallow. Can anybody help me put this in context.


Do we have control on what an electron transition emits: light or heat?

I don't know quantum mechanical model. So, I'm referring to just bohr's model of atom. Any atom emits energy when it makes transition from higher excited state to lower excited state. Now, some times they say that this energy is light energy and some times heat energy. I'm confused. What decides the emitted energy will be light or heat? Do we have control over what kind of energy it emits? I mean can I make an atom emit light and not heat? or heat and not light? What are the factors influencing this? **EDIT:** To clarify the issue of what exactly do I mean by heat. Well, I myself am not quite sure. I just thought heat as just heat. This question araised from my previous question where I discovered that in a filament bulb 98% of energy is converted into heat energy and rest into light energy. But where as in CFL about 40% to light and rest into heat. I know these conversion is because of electron transistions. But whats controlling light & heat components?


Exercise about Lagrange-Euler equations

I'm solving an exercise about the Lagrange-Euler equations, that states the following: > _Let $\gamma (t) = \\{ (t,q) : q = q(t), t_0 \leq t \leq t_1\\}$ be a curve > in $\mathbb{R} \times \mathbb{R}^2$. Further let $F(q,\dot{q},t)$ be the > function from $\mathbb{R}^2 \times \mathbb{R}^2 \times \mathbb{R} > \rightarrow \mathbb{R}$ for which the functional $\Phi = \int_{t_0}^{t_1} > F(q,\dot{q},t) dt$ is the length of the curve._ > > _(a) Which is the form of $\Phi$ in cartesian coordinates? Which is its form > in polar coordinates?_ > > _(b) Give the Euler-Lagrange equations in both coordinate systems._ > > _(c) Solve the differential equations in both coordinate systems and show > that the solutions are the same._ Now, my problem begins with giving the form of $\Phi$. I found that the element of length in cartesian coordinates is $ds = \sqrt{dx^2 + dy^2}$, so with $$\int ds = \int \frac{ds}{dt} dt = \int \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dx}{dt}\right)^2} dt,$$ We find that $\Phi = \int_{t_0}^{t_1} ||\dot{\gamma}(t)|| dt$. Now, my plan is finding the element of length in polar coordinates, and plugging in the respective expressions in terms of $\gamma$. The problem is that I don't see how to find the element of length in polar coordinates. I looked it up on Wikipedia, and found $ds^2 = dr^2 + r^2 d\theta^2$. Now, for $dr^2$ I would plug in $||\dot{\gamma} (t)||^2$, for $r^2$ I'd set $||\gamma (t)||^2$, and for $d\theta$ I have no idea. Can you help me, especially with the derivation of the polar line element and the form of $\Phi$ in polar coordinates?


**Why don't most physics programs study the primary sources?** In other words: Why don't they include Newton's _Principia_ , Lagrange's _Analytical Mechanics_ , etc., in the curricula?


Classical Information carrying capacity of two states

What classical information is carried by $\alpha|0\rangle+\beta|1\rangle$ and $\alpha|00\rangle+\beta|11\rangle$? How to quantify it? To be specific, A GHZ state, $\frac{1}{\sqrt 2}[|000\rangle+|111\rangle]$ can deterministically teleport both the states. Both the states appears to be same from the teleportation point of view. Does they carry same information?


What does really mean by- power of a number or an exponential function is dimensionless?

Is power of only a number or an exponential function is dimensionless? If power of any other thing can also be dimensionless then please explain with examples.


Problem in Youngs double slit experiment

![double slit ray diagram](http://i.stack.imgur.com/Whguh.gif) This is from Young Double slit experiment. But How to prove the the two $\theta$ are equal, I meant, how $\angle EAD= \angle PEC$? I see from the both triangle have $90^0$ but what about others?


Why do wind power plants have just 3 blades? It seems that adding more blades would increase the area that interacts with the wind and gather more energy. ![enter image description here](http://i.stack.imgur.com/KOC1w.jpg) (Image from Wikipedia.)