title: If we have a current $I$ flowing down a wire, why must the net bound current be zero?

text: Say we are dealing with a wire that has a current $I$ flowing through it, i.e. $I$ is the free current. Why must it then hold that the net bound current, that is, the bound volume current, $J_b$, and the bound surface current, $K_b$, adds up to zero? I cannot get an intuitive grasp on why this is true, especially considering the fact that if you, on the other hand, have a uniformly magnetized cylinder, the net bound currents do not add up to zero, since there's no bound volume current, but there is a bound surface current. Similarly, in a solenoid you also get a bound surface current, but there's no bound volume current, so the two again do not add up to zero. Any help with this would be greatly appreciated, since I'm just stuck in trying to wrap my head around this. I can calculate the damn things, but I have no idea why this would be the case on a qualitative level.


title: Playground of Forces

text: Why is it that the gravitational force acts on large sized objects while the strong and weak nuclear forces act at subatomic levels only? What is that stops each other to enter each others domain?


title: Jerk Magnitudes and Earthquakes

text: Destruction from earthquakes depends on many factors, including magnitude, occurrence depth and closeness to epicenter. One measure that may relate destructive force for buildings and infrastructure would be jerk magnitudes and durations. Recall that jerk is the time derivative of acceleration, so destructive forces would likely be correlated to the product of building's jerk and mass. In addition, jerk from earthquakes would have horizontal as well as vertical components What are measured jerk magnitudes (via jerk-meters) for various recent earthquake magnitudes?


title: Introducing dimensionality of the $+$ and $-$ signs

text: Is it possible to introduce some dimensionalty as $\text{kg}$ for mass or $\text{m}/\text{s}^2$ for mathematical signs: plus - $+$ and minus - $-$. The main reason of this is to avoid some common mistake related to $\pm$ signs.


title: BPS states : Mathematical definition

text: First of all, let me congratulate the theoretical physics community for this site. I am a mathematics student with very little background in phyiscs. The question I want to ask is: What is the proper mathematical definition of BPS states? In string theory the BPS states correspond either to coherent sheaves or special Lagrangians of Calabi- Yau manifold depending upon the type of string theory considered. but in SUSY quantum field theories in 4d, there are no CYs as far I know (which is very little) and in gravity theories, these corresponds to some Black Holes. So what is the general mathematical definition of BPS states which is independent of the theory in consideration, say a general SUSY Quantum field theories, be it QFT, string theory, Gravity and in any dimension. I am very sorry, if the question is not framed properly due to my ignorance of the physics involved. Thanks


title: What is the meaning of $\%E$ on a graph?

text: I'm reading the book _The New Science of Strong Materials_ by J.E. Gordon. He writes > _when we plotted (...) breaking strain, against thickness, we found it did > not matter what the whiskers were made of, for they all plotted on the same > curve._ (p87, Penguin) He proves this point with a graph where the y-axis is "Fracture stress $\%E$". However, he previously stated that stress is very different to strain—stress is the load per area, while strain is the extension of the material under said load. Thus, the graph doesn't seem to correlate with the argument. I'm thinking that the $\%E$ is therefore relevant. Can someone clarify this for me? Here is the graph: ![enter image description here](http://i.stack.imgur.com/eArXf.jpg)


title: Why doesn't orbital electron fall into the nucleus of Rb85, but falls into the nucleus of Rb83?

text: Rb83 is unstable and decays to Kr-83. Mode of the decay is electron capture. Rb85 is stable. The nuclei Rb83 and Rb85 have the same charge. Rb85 is heavier than Rb85, but gravitation is too weak to take into account. So why orbital electron falls into the nucleus of Rb83? What is the reason?


title: Can you suggest me some good resource for learning Mathematica for physics?

text: I am into HEP. I know nothing in Mathematica and have to start from basics. HEP requires a lot of computation for Feynman diagrams etc could you please suggest some good resource to start learning Mathematica and some libraries that I would require?


title: How can I split a resultant force into its $x$ and $y$ components?

text: > Point charge 3.5μC is located at x = 0, y = 0.30 m, point charge -3.5μC is > located at x = 0 y = -0.30 m. What are (a)the magnitude and (b)direction of > the total electric force that these charges exert on a third point charge Q > = 5.0μC at x = 0.40 m, y = 0? ![Charges](http://i.stack.imgur.com/4sdvH.gif) I first drew a diagram, showing that it can be broken down into two .3, .4, .5 triangles, leaving me with a theta of 36.87°. I then calculated that because the charges are equal, each individual force enacted on charge Q is .63N. I know I need to split the force into it's x and y components to find the resultant vector using R=√Rx^2 + Ry^2, but I do not know how to split these forces up.


title: Quantum Yang-Mills Theory and AdS/CFT

text: I just read the first chapter of Becker-Becker-Schwarz. To quote: > A remarkable discovery made in the late 1990s is the exact equivalence (or > duality) of conformally invariant quantum field theories and superstring > theory or M-theory in special space-time geometries. Can this AdS/CFT duality be used to work in string theory or M-theory instead of quantum Yang-Mills theory, make the corresponding theory rigorous, and then go back to quantum Yang-Mills theory, thereby solving the open Millenium Prize question?


title: Does the pressure in a pressure tank depend on the amount of air?

text: I have a pressure tank which is filled with water and air. What want to know is: if I change the ammount of air in the tank would the ammount of water that leaves the tank, when pressurized, change? If the tank was initially pressurized and the water keep refilling as to keep the volume of water constant would the ammount of water expelled from ythe tank depend on the initial volume of pressurized air


title: Energy/Momentum required to deflect earth from its orbit

text: This question occurred to me when thinking about the firepower people have on earth. How hard is it to change the characteristics of the orbit of the earth when an explosion happens on its surface, using conservation of momentum? I know already that the orbit of the earth is not so accurate already and fluctuates a lot, causing the so-called "leap second". So my question is: How could such a thing be measured/quantified? Can we say something like: The explosion energy required to deflect the eccentricity of the earth's orbit by 1%. And how much would that be? Any ideas?


title: Solving the Schrödinger equation where the initial wave function is an energy eigenfunction

text: I was watching Allan Adams' lecture on energy eigenfunctions, and there's one part (around 43 minutes into the lecture) that confuses me. Suppose we have the initial wave function $\Psi (x,0)$ such that $\hat{E}\,\Psi (x,0)=E \,\Psi (x,0)$ for some constant $E$. Then, plugging this into the Schrödinger equation, we'd get: \begin{align} i \hbar \frac{\partial}{\partial t} \Psi (x, 0) &= E \, \Psi (x,0) \\\ \frac{\partial}{\partial t} \Psi (x, 0) &= \frac{E}{i \hbar} \, \Psi (x,0) \tag{1}\\\ \therefore \Psi (x, t) &= \exp\left({-i \frac{E\,t}{\hbar}}\right) \Psi(x,0) \tag{2} \end{align} I'm a bit confused about how to go from $(1)$ to $(2)$. Now if we make the additional assumption that $\hat{E}\,\Psi (x,t)=E \,\Psi (x,t)$ for all $t$, then the Schrödinger equation becomes: \begin{align} \frac{\partial}{\partial t} \Psi (x, t) &= \frac{E}{i \hbar} \, \Psi (x,t) \end{align} and I can solve this differential equation easily and get $(2)$. But from watching that part of the lecture, it seems we only need to assume a weaker statement - that the initial wave function is an energy eigenfunction. But then, it's not clear to me how I can get the solution $(2)$ from $(1)$. Am I missing something? Update: Thanks for all the answers. After reading through the accompanying lecture note, we indeed need to assume that the energy operator is a constant over time.


title: How to find a particular representation for the gamma matrices?

text: I asked this question as a subquestion in another thread, but got the answer below and thought it deserved a thread of its own. Two well-known representation of the gamma matrices are the Weyl and Dirac- Pauli reps. The Weyl rep is often used when dealing with ultra-relativistic systems (or massless ones) and the Dirac for the other case. But I've never come across in any QFT book how to derive these reps, they just give it to you. So **what is the procedure to derive a rep for the gamma matrices?**


title: Dirac equation on general geometries?

text: I have a numerical method for computing solutions to the Dirac equation for a spin 1/2 particle constrained to an arbitrary surface and am interested in finding applications where the configuration space has a complicated geometry, i.e., not just $R^2$ or the sphere, but a more general curved surface possibly with special boundary conditions. On the sphere, for instance, one can take advantage of symmetry and simply use the spherical spinors, but for a more general configuration space (or one described only by measurements, for instance) it may not be possible to come up with a nice, closed-form solution. However, I am not a physicist and would like to better understand where (or indeed, _if_ ) such problems arise. Any pointers are greatly appreciated. Thanks! Edit: Note that I _already have_ a numerical method for solving the Dirac equation -- I am not looking for information on how to solve it. (And, interestingly enough, Clifford algebra / geometric algebra is already the starting point for what we do.) Also, I am specifically interested in the case of _surfaces_ , i.e., 2-manifolds, so (many) applications in GR probably don't apply. Thanks!


title: If Betelgeuse were to go supernova

text: If Betelgeuse were to go supernova would the sky be twice as bright, or day time extended, depending on what time of year it happened in. Basically when it does supernova how bright and large will it appear in the sky, and how do you work out the apparent brightness and size for something that hasn't happened yet?


title: Types of Solitons

text: In the condensed matter literature, I have seen broadly two types of solitons which are dark and bright corresponding to fall and rise in density. (I know only the number density case ). But among the dark ones, there are black and grey. And there are probably a few types of bright ones. What are the black and grey solitons and what are the types of bright solitons ? Please explain.


title: home made atom destruction unit

text: Today we learnt at school that atoms can be destructed. I believe Physics is a great science to do experiment and I would like to try it at home. Could you tell me what I need to do it? and is it dangerous?


title: How does one define "Nuclear Harmonicity"?

text: How does one define _Nuclear Harmonicity_? Although the title is pretty much the entire case, the main question is, what exactly is "harmonicity" in terms of nuclear physics? What does it mean for a nucleus to be quasi-harmonic or nearly-harmonic? Is it perhaps in the sense that it approaches a harmonic oscillator, and if so, what's the intuition behind this?


title: Is sound relative?

text: Is sound relative? For example, if I and my friend are having a ride at 1000 mph and I shout towards him (speed of sound 700 mph). What would happen? Will the speed of the sound relative to the ground be 1700 mph? Or will it be something else? Why?


title: Absorption & emission spectra

text: The process of obtaining an absorption spectra involves passing a complete spectrum of light from the material under consideration. The material absorbs the specific wavelength and allows the rest to pass through. But the reason of this absorption is given as the specific quantum gaps between the various electronic energy levels in the material and when an electron jumps to a higher level only a specific wavelength photon is absorbed. But the higher energy state is considered to be unstable and thus the electron will fall back immediately and would again give the wavelength preventing any spectra to be formed. The same doubt is in emission spectra when used to describe various flame colours but the same doubts apply there too.


title: Who formulated the idea of mass?

text: This may sound like a dumb question, but googling didn't seem to help. Mostly because I'm having a hard time formulating the question, and anything with "mass" seems to direct me to "Mass Effect". I am trying to find out if we know who (philosopher, scientists, or just civilization) formulated the idea that objects which are larger or heavier than others have more "stuff" in them which makes them heavier. I realize that cavemen would have known this since rock A is harder to move then rock B. But I want to know who is credited with actually formulating a theory of mass.


title: What is the difference between mechanical and thermodynamic pressure?

text: To start with I know thermodynamics deals with processes at equilibrium. Hence the thermodynamic pressure should most likely be the pressure of a fluid at equilibrium. I'm not sure if a fluid flow (in general unsteady) is in thermodynamic equilibrium (say flow in a channel which has a pressure gradient) and so would the static pressure at a point in the channel be different from the thermodynamic pressure? What does this entail about the ideal gas law $p = \rho RT$? can it be used for moving flow? What is the pressure in the equation referring to; mechanical or thermodynamic? EDIT: To clear up any confusion- In a given flow we can measure the pressure at any point, say using a pitot tube to get the stagnation and static pressure. My question is then, is the static pressure we measure (which is by definition an "$F/A$" (force / area) quantity any different from the thermodynamic pressure? The pressure in $P = \rho RT$ must be referring to the thermodynamic pressure, since the equation is derived purely from the laws of thermodynamics. However, in all literature I have encountered, compressible flows use the ideal gas equation to as a link between the incompressible variables ($p, \mathbf{V}$) and the full set of compressible variables ($p, \mathbf{V}, \rho, T $). So it seems the two pressures are equivalent?


title: Why uncertainty principle is not like this?

text: In Griffiths' QM, he uses two inequalities (here numbered as $(1)$ and $(2)$) to prove the following general uncertainty principle: $$\sigma_A^2 \sigma_B^2\geq\left(\frac{1}{2i}\langle [\hat A ,\hat B]\rangle \right)^2$$ defining $\lvert f\rangle=(\hat A-\langle A\rangle)\lvert \Psi\rangle$ and $\lvert g\rangle=(\hat B-\langle B\rangle)\lvert \Psi\rangle$, he uses   * Schwarz inequality: $$\langle f\lvert f\rangle\langle g\lvert g\rangle\geq|\langle f\lvert g\rangle|^2\tag{1}$$ and with $\sigma_B^2=\langle g\lvert g\rangle$ and $\sigma_A^2=\langle f\lvert f\rangle$, he arrives at $\sigma_A^2 \sigma_B^2\geq|\langle f\lvert g\rangle|^2$.   * the fact that for any complex number $z$ we have $$|z|^2\geq(Im(z))^2=[\frac{1}{2i}(z-z^*)]^2\tag{2}$$ here $z=\langle f\lvert g\rangle$ and so $z^*=\langle g\lvert f\rangle$ , and we find that $\langle f\lvert g\rangle=\langle\hat A \hat B\rangle-\langle A \rangle\langle B \rangle$ and $\langle g\lvert f\rangle=\langle\hat B \hat A\rangle-\langle A \rangle\langle B \rangle$, so $\langle f\lvert g\rangle-\langle g\lvert f\rangle=\langle [\hat A,\hat B]\rangle$. Replacing this into $(1)$ gives the uncertainty principle. Why he doesn't use $$|z|^2\geq(Re(z))^2=[\frac{1}{2}(z+z^*)]^2$$ instead of $(2)$? This could give us a (correct though different) relation between the $\sigma_A$ and $\sigma_B$ too: $$\boxed{\sigma_A^2 \sigma_B^2\geq\frac{1}{4}\left(\langle \hat A \hat B\rangle +\langle \hat B \hat A\rangle -2 \langle \hat A\rangle\langle \hat B\rangle\right)^2}$$


title: Were there time just after the Big Bang?

text: I have understood that just after the Big Bang there were no cesium atoms and the definition of time requires cesium atoms. So were there any time before cesium atoms appeared in the universe?


title: Conservation of momentum

text: I've got following problem. There are $N$ particles in an isolated system. The equation of motion for a particle $i$ is $$m_i\ddot{\vec{x}_i}(t)=\vec{F}_i(\vec{x}_1(t),...,\vec{x}_n(t),\dot{\vec{x}_1}(t),...,\dot{\vec{x}_n}(t),t),$$ and the force has a potential $$\vec F_i=-\nabla_{\vec{x}_{i}}V(\vec{x}_1(t),...,\vec{x}_n(t)).$$ The potential is translation-invariant. Now, the total force is $\vec F=\sum_i \vec F_i=0$. As a proof it is stated that for an arbitrary unit vector $\vec e$ holds: $$\vec e\cdot\vec F=-\frac{d}{d\lambda}V(\vec{x}_1(t)+\lambda\vec e,...,\vec{x}_n(t)+\lambda\vec e)|_{\lambda=0}=0.$$ This last equation I do not understand. Could anyone possibly give me a clue?


title: Cubic Gaussian Surface For Evaluating Electric Flux

text: i'm working through some problem sets about Gauss' law and all the examples I have come across so far require the use of a spherical gaussian surface for a point charge, so that it is possible to say the electric field lines are perpendicular to the surface. I feel confident in answering these. If my gaussian surface is a cube, I don't understand how to find $\cos \phi$, since the field lines radiating out of my point charge pass through the surface at different angles to the normal. Is there some assumption that can be made, or am I missing something obvious that allows me to evaluate this? One idea i've had (but can't find any justification for in my text book) is that if you consider angles on one side of the normal positive, and negative on the other, they cancel out and you can consider the field lines to be going perpendicular through all the surfaces. I hope my question is clear, thanks!


title: Turning off or tuning the Higgs field

text: What would happen to the Universe if we turned off the Higgs field? (Remark: "turning off the Higgs field" means to set its v.e.v. to zero-today its value is about 246 GeV.)   * Would the Z,W bosons be massive?   * Would atoms form?   * Would vacuum energy be zero? _Bonus_ : What advantages could we obtain if we could manipulate the Higgs field as we manipulate the electromagnetic field by tuning its v.e.v. as we wished? That is, imagine we were clever enough to change the Higgs v.e.v. to any finite value we wished. What could it be useful for?


title: What colour is the Earth?

text: More specifically, what shade of blue is the earth on average? I want to create an accurate earth-coloured blob in a simulation / presentation. A quick search didn't turn up anything. What I found was that it is blue and why it is blue. Many of the images available online appear filtered and colours vary wildly.


title: Physical Chemistry: What's the relationship between orbital overlap and barrier shape?

text: This is a question for the physical chemists out there. For a given chemical reaction there is a barrier to traverse in order to proceed from reactants to products. Reactants e.g. an organic base abstracting a proton can approach each other at different orientations - attack angles. At times, this orientation will lead to better orbital overlap between the lone pair on the base and the s orbital of the hydrogen. I have heard people talk about better orbital overlap, lowering/narrowing the barrier to reaction. Is this a valid way of conceptualizing this? Does better orbital overlap change the barrier? I personally have always associated a better overlap as starting further along the reaction coordinate/barrier, not the actual barrier itself being altered? Any thoughts from the physchems?


title: Are dangerous rays emitted during Solar Eclipse?

text: It is said one should avoid staring at Sun as it can damage the eyes, but it is also said that one should not come out in sun during eclipse as it emits dangerous rays. Is that true? If yes, why?


title: How/Why does voltage reflect?

text: I'm trying to understand antenna theory again, and I'm again stumped by the concept of VSW (voltage standing waves). I understand standing waves, I remember these from a physics clases, however I do not really understand how voltage can be a wave, and how it reflects. From my understanding of voltage it is really hard to understand how it can reflect. A voltage is the difference in electrical potential energy (per unit charge) between two places. This electrical potential energy (per unit charge) difference describes the force on that unit charge (due to the electric fields), and the distance this charge will travel. For example, with an electrical potential energy (per unit charge) difference of $10\, \mathrm{V}$ or $10\, \mathrm{J}/\mathrm{Q}$, an electron at the higher potential will travel to the lower potential. If the distance between these two potentials is $2\, \mathrm{m}$, the force on the electron will be $5\, \mathrm{N}$, as it travels from one potential to the other. _(Only if the electric field is constant between the two potentials. If it is not, the integral of the force across the $2\, \mathrm{m}$ will still equal $10$, however the force will vary at different distances)_ Voltage is just a made up concept to help explain the forces that charged particles experience. It isn't a real thing that can be reflected. So what **is** being reflected? I'm guessing it is the electric field, as this has an actual propagation speed, and even though the electric field is also a made up concept to describe the force a charged particle will experience at that point in the field, it is easier to imagine this being reflected. So can someone please explain to me how this all ties together? How does an electric field propagate through a conductor? Do the vectors in this vector field change direction when there is a bend in the wire, and why do they do this? Why wouldn't the electric field escape outside of the conductor? And how does an electric field (or voltage) reflect at a short?


title: Finite Square Potential Well Solving

text: I've got a question about solving the finite square well. Consider a 1D finite, square, potential well where: $$ V(x) = \begin{cases} 0&\,{\rm if} x<0 \\\ -V_0&\,{\rm if}\,x\in\left(0,L\right) \\\ 0&\,{\rm if} x>L \end{cases} $$ When applying the Schrodinger Equation to Zone II ($V(x)=-V_0$), it will be necessary to write a constant $k$, which my book has defined this way: $$k^2= - 2m\frac{E+V}{\hbar^2}$$ My Question is: Can we write this as $$k^2=2m\frac{E+V}{\hbar^2} $$ Shouldn't we get the same solution? If not, why not? The final solution should not depend on which way we write $k$, correct?


title: Eigenfrequencies of Normal Modes

text: I understand the whole deal with coupled oscillators and how to solve for normal modes and eigenfrequencies and such. But what is tripping me up is what these eigenfrequencies correspond to. If I have 4 masses along a ring connected to springs, I find that the frequencies are $$\omega^2=\frac km,\frac {2k}{m},0$$ But I do not know how to explain this. I assume 0 means they aren't moving. Does $k/m$ imply the system acts as a simple pendulum? What does this imply about $2k/m$? Thanks!


title: What is the tension in the string of a spherical pendulum?

text: Can some one solve it by using Lagrange's undetermined multiplier method or any other method that explains the physics in spherical pendulum system? book references: 1) Classical mechanics by Goldstein, poole & Safko 2) Classical dynamics by J.B Marrian


title: Problem with plane mirror focal length thoery: Is the focal length actually infinite?

text: We all learned that the focal length of plane mirrors is considered to be infinite, as the radius of curvature is infinite. However, imagine this scenario: You have a perfectly flat (no one get mad, this is theoretical)plane mirror, and the earth is perfectly flat as well(theoretical). If you looked into this mirror with a telescope (imagine it is powerful enough), would you be able to see as far away as you wanted to, just as if you were looking into reality? I know that spherical apertures are limited by Lord Rayleigh's criterion (after all, only so many photons can be recorded onto a surface), so are plane mirrors limited by this as well? If so, then we should call the focal length of a plane mirror something besides infinite, right?


title: Why does our voice sound different on inhaling helium?

text: This question (and answer) is an attempt to clear the air on what appears to be a very simple issue, with conflicting or unclear explanations on the internet. Arguments, negations, etc are invited. I'm classifying this as a physics question, since it has to do with resonances, attenuation, etc.         Question:   I recently Googled this, and found a ton of articles with confusing explanations:     1. One of the top Google results from a site called Live"Science", says - "....their wavelengths stay the same regardless of whether the tract is filled with helium gas or air....That means the frequencies of the resonant harmonics must increase in a helium-filled cavity instead."   2. The above appears to completely contradict the source university article as well as this other article in Scientific American that say the pitch of the sound (and the actual frequency of oscillations) doesn't change, only the timbre (and the distribution of power between low and high frequencies) changes.   3. Additionally, neither the university article quoted nor Scientific American explain HOW the presence of helium leads to the presence of higher frequencies. For example, the university article simply says - "Inhaling helium changes the frequencies of the resonances, and therefore of the formants they produce" --- Okay, how?   4. There also appears to no consensus on whether a LISTENER in a helium atmosphere would hear the same frequencies as normal (it's helium all the way so it shouldn't make any difference right?) OR hear the squeaky voices associated with helium. This should be explainable by physics, what is the answer?


title: Electromagnetic radiation and health issue

text: A lots of article even wiki tells that Cell phone radiation is very dangerous for health. Some people tell that it can be reason for brain cancer. A article Mobile phone subsides man’s sperms! astonished me that, if I use Cellphone in my pocket it might kill my sperms. Can I trust this article? I am saying this because, after reading this article I am very much worried. Another thing is, is there any way evaluate the amount of heat release the mobile phone which could be harmful for our health?


title: How to account for the huge difference in susceptibility of liquid and gaseous oxygen?

text: I noticed this while studying magnetic fields in matter from Griffiths' Electrodynamics book. A table is provided in the chapter which shows the materials with their respective susceptibilities. Under paramagnetic materials, there is oxygen and liquid oxygen(-200 degree celsius) with their susceptibilities 1.9* 10^(-6) and 3.9*10^(-3). My question is that why is there such a huge difference between their susceptibilities when they are the same element? To be precise, is there any theory to account for this property?


title: Computing 2d Orbital Parameters

text: When using formulas that describe a 2d orbit around a planetary body:   * is it actually OK for the semi-major axis $a$ to be negative?   * when $a$ is negative, it is impossible to compute mean anomaly from time, so what is the equivalent of mean anomaly for parabolic or hyperbolic orbits, assuming that $a$ can be negative?


title: B3LYP vs PBE functionals for conjugated organic systems

text: Two of the most popular (exchange and correlation) functionals for density functional theory are B3LYP and PBE. Out of the people I've worked with / learned from, mostly the computational chemists were using B3LYP, and the physicists were using PBE. Now, different functionals have different strengths and weaknesses, but these two functionals seem to have a lot of overlap in their applicability. Right now I'm interested in ground-state properties of molecules with extended pi- frameworks in the gas-phase. For example, I'd like to compute the geometries, ground-state electron densities, and molecular polarizabilities of the acenes. Does anyone have concrete information or references on the performance of B3LYP or PBE in this situation, or in general?


title: Infinite Mirror Face to Face With Infinite Two Way Mirror

text: If you are in deep space, and there is an infinite plane mirror, and in front of it there is another infinite mirror that is two way, with the see through side towards you, what do you see? Is it the image the universe behind you?


title: Why doesn't relativistic momentum appear conserved in this frame?

text: Suppose I have an inelastic head on collision between two idential particles of mass $m$ that come to rest in the centre of momentum frame where relativistic momentum is obviously conserved. If I now switch to the proper frame of one of the particles, the velocity of the other before the collision is $2v\gamma^2$, and after they both travel with velocity $v$. ![collision](http://i.stack.imgur.com/92BsJ.gif) This would appear to suggest that relativistic momentum isn't conserved in the proper frame of either particle if I write the initial relativistic momentum as $(\gamma_{2v\gamma^2}) (m)(2v\gamma^2)$ and the final as $(\gamma_v) (2m)(v)$. So where is the error in this reasoning?


title: How fast do you need to travel to go 35 light-years in 2 (apparent) years?

text: Time slows the faster you go, so perhaps going 35 light-years in "2 years" is actually do-able, but how fast would you actually be going? If $$t' = \frac{t}{\sqrt{1-\frac{v^2}{c^2}}}\\\ $$ If we say $t' = 2 \text{ years}$, and $x = 35\text{ light-years}$ (and $t = x/v$), it's fairly basic to get to $$\left(\frac{x}{t'}\right)^2 = {v^2\left(1-\frac{v^2}{c^2}\right)} = v^2-\frac{v^4}{c^2}$$ However, time has worn on my math to such a degree that I'm now miffed. I can't figure out how to get an equation in terms of $v$.


title: How to find the Stress-Energy tensor?

text: I am a bit at loss about how to proceed to find the stress-energy tensor given some distribution of matter. The Wikipedia page gives some examples, and some (inequivalent) definitions for it:   * Using the Einstein-Hilbert action we get the Hilbert stress-energy tensor $$T^{\mu\nu} = \frac{2}{\sqrt{-g}}\frac{\delta (\mathcal{L}_{\mathrm{matter}} \sqrt{-g}) }{\delta g_{\mu\nu}} = 2 \frac{\delta \mathcal{L}_\mathrm{matter}}{\delta g_{\mu\nu}} + g^{\mu\nu} \mathcal{L}_\mathrm{matter}$$ With this formula, my problem reduces to: what is the matter Lagrangian $\mathcal{L}_\mathrm{matter}$ given a configuration of matter?   * The canonical stress-energy tensor is the Noether current associated to translations in spacetime. This works very well in Minkowski spacetime, but our manifold could be something different. In that case, what would a "translation in spacetime" be?


title: When an electron absorbs a photon doesn't that change it's mass?

text: When an electron absorbs a photon it leaps to a higher energy level, what exactly happens when an electron absorbs a photon? By the mass-energy equivalence doesn't that changes the electron's mass and thus alters it?


title: How much does increased world population contribute to global warming?

text: In 1974 there where 4 billion people on earth. Now in 2013 we passed 7 billion people. So the world population is nearly doubled in 40 years. Every living human being also haves a body temperature of $37.5^\circ \: \mathrm{C}$ not to mention all the animals it need to feed on. Not taking into consideration the heat generated by machines and animals required to keep the population alive; by how much will 3 billion people at $37.5^\circ \: \mathrm{C}$ potentially affect the world temperature?


title: Simple pendulum and planet mass

text: A simple pendulum, $20cm$ in length, has the period of $2.7s$ on a certain planet. Find the mass of the planet if its diameter is $18000km$. $G=6.67\times10^{-11}Nm^2/kg^2 $ I have no idea how to get to the solution except for the formula $$g=(GM)/(R^2) $$


title: How does a minority carrier diffuse?

text: I have gone through a lot of questions but none of them ask how do the minority carriers approach the depletion layer in the first place. When a p-n junction is formed, negative space charge accumulates at p side and positive space charge at n side. A minority carrier (say electron) from the periphery of the p side, when approaching the depletion region, should experience repulsion from the accumulation of negative space charge or ions in the depletion region, before it 'slides' down the potential barrier. The question is, minority carriers being so little in number, the probability of having thermal energy sufficient to overcome that repulsion is extremely low. So how is diffusion possible? Is it because the space charge has no component of electric field away from the depletion region? Plus, why doesn't applied bias have any effect on the minority carrier current (according to the text books, this current is only a function of temperature). Shouldn't the reverse bias diffusion current be more than that during forward bias?


title: How to find $\tau$ in this circuit?

text: 1)![circuit 1](http://i.stack.imgur.com/VNFAx.png) 2)![circuit 2](http://i.stack.imgur.com/vzCwB.png) I know, that $$ τ = \frac{L}{R} $$ but what is $R$ in this formula? It seems to be the total resistance, but how to find it in 1) and 2)?


title: Integrating equations with units

text: I was looking through an old copy of Barron's AP Physics and found this problem relating to impulse which I was initially confused about how to integrate. > **Example 6.1** During a collision with a wall lasting from $t=0$ to > $t=2\text{ s}$, the force acting on a $2\text{-kg}$ object is given by the > equation $\mathbf{F} = (4\mathrm{\ kg\ m/s^4})t(2s-t)\hat{i}$ They work out that the integral is equal to: $$\frac{16}{3}\hat{i}\frac{\text{kg m}}{\text{s}}$$ I am confused about the role of the units in the problem. Looking at the answer, it seems that if I were to just ignore all the units and simply integrate $4t\cdot(2-t)$ that would give me $16/3$, and because that is a force I know it to be $\mathrm{kg\,m/s}$ or $\mathrm{N}$. Why it would be OK to ignore the units in the integral though is somewhat unintuitive to me (other than that I know the end result must be a force) and I feel like it may get me into trouble with other problems. Can someone explain how it is that the scalar values of the original unit quantities still give you the correct answer?


title: Astronomical-wavelength radio (AWR) transmissions between cosmic plasmas?

text: My son asked me if electromagnetic waves longer than radio exist. I told him that even though physics permits such waves, there are no antennas long enough to radiate or detect them. However, on further thought I realized that is not quite correct. Supernovas like the Crab Nebula, as well as other astronomical phenomena such as cosmic jets, routinely create continuous plasmas with conductive lengths ranging from planetary distances to light years across. Such plasmas are also often in violent motion. Without (yet) attempting any detailed calculations, I'm pretty confident that violent motion in such giant conductive constructs should cause them to radiate in astronautical wavelength radio bands (AWR, and yes, I just now made that up), possibly powerfully. But how would one detect such emissions, even if they do exist? That's where the hair-of-the-dog-that-bit-you principle comes in: A sufficiently powerful AWR broadcast should be received and converted into large-scale plasma currents by any nearby cosmic-scale plasmas of the same general type. So, my question if anyone who is willing to take a whack at something this speculative: If AWR emissions exist and are powerful enough to create slowly alternating, long-duration current flows within cosmic-scale plasmas, are there experimentally detectable optical or short-radio phenomena -- e.g. polarization effects, or faster cooling of transmitting plasmas due to unexpected energy losses, or unexpected energy gains in receiving plasmas -- by which the emission or receipt of AWR could be detected? I realize this is as much an astronomy question as a physics question, but if my physics argument is drastically wrong somewhere, or is correct but does not produce experimentally detectable phenomena, the astronomy part doesn't matter much. * * * 2013-06-08 -- The discussion is interesting, but perhaps drifted a bit towards the question of much shorter (and directly detectable) VLF waves. Eduardo make the excellent point that the interstellar medium is opaque to VLF. By simple generalization of VLF behavior to far lower frequencies, that may also mean that astronomical-scale EM waves are also not possible. It's worth emphasizing that even if something within the astronomy zoo is able to generate powerful AWR band emissions, and even if such transmissions could make it to earth, I'm pretty sure no antenna on earth would be able to pick up those transmissions. That in turn emphasizes that the only readily apparent way to detect AWR band energy transfers, if they exist at all, would be by them enabling unexpected modes of transfer of energy between visible astronomical plasmas that would act as both AWR generators and receivers. Eduardo's excellent point about the opacity of the interstellar plasma may provide the simplest answer: AWR cannot exist because interstellar plasmas will never allow it to. However, that same VLF opacity issue brings up at least two more questions:   1. Opacity at VLF may not _necessarily_ guarantee interstellar opacity in the AWR band, which after all would be orders of magnitude lower in frequency. Speculating wildly, AWR might for example be so gentle in its ion-level impact that the interstellar plasma ends up being largely transparent to it. And yes, that really is just a wild speculation, nothing more.   2. Even if the interstellar plasma is highly absorbing of AWR, any major generator of AWR band energy would still likely result in some sort of unexpectedly higher or faster outward transfer of energy surrounding the AWR generator. More specifically: if (big if_ AWR exists and can be generated at high power levels by self-excitation of currents in violently moving astronomically sized plasmas (think Crab Nebula), but is also absorbed very quickly by the interstellar medium, it should still be indirection detectable by its enabling of otherwise inexplicable increases in the rate or speed at which energy dissipates outward from the event into the surrounding interstellar plasma. Such an effect would be indirectly detectable by a lack of explanation for it from other known and well-modeled energy transfer modes. I realize this has likely turned into an unfair question at this point, since the range of issues that would need to be analyzed is pretty high. It seems likely the idea of a AWR band that _could_ (maybe!) exist, and even have indirectly observable effects on visible phenomena such as supernova cloud expansions, has not been explored much, if at all. That's interesting all by itself. I'll probably either award Eduardo the answer soon for best coverage of the issue so far, or perhaps I'll even try a bonus. I am a bit intrigued by this one.


title: heat spread from hot coil: diffusion, radiation?

text: When modeling the heat spread from a hot object through a vacuum (or a gas), what is the dominant physical process for heat energy transportation? Thermal diffusion, thermal radiation, a mix of the two, or something else? What properties of the object or the gas may influence this?


title: Measurement in reciprocal metres

text: I'm trying to name a measurement that is measured in reciprocal length, which is in a draft document for vehicle risk management. It currently says:               Crash rate = number of crashes / million kilometres driven      where the measurements are taken annually. So the units are in reciprocal length. I'm not happy with the word "rate" because that seems to imply the number of events over time, whereas the measurement is the number of events over distance. Existing names of units of reciprocal length in various disciplines don't seem to work - it's not a measurement of any cyclic phenomenon. Is there an better word for this measurement?


title: Computing average force exerted by ground when jumping

text: Problem: A man of mass $m$ jumps vertically into the air from a sitting position in which his center of mass (CoM) is at a height $h_1$. When his feet are just about to leave the ground his CoM is at height $h_2$. When he is at the highest position in the jump, his CoM. is at height $h_3$. What is the average force exerted by the ground on him? My solution: The average force must be equal to the amount of work done divided by distance i.e. $m \cdot g \cdot (h_3 - h_1) / (h_3 - h_1) = m \cdot g$ But the actual solution involves all three heights, which I don't understand; if he is sitting on a chair, the force exerted by the chair must also be included in any `force exerted by the ground` shouldn't it? Actual answer: > $m \cdot g \cdot (h_3 - h_1) / (h_2 - h_1)$


title: First-order and second-order wave equations, versus the uncertainty principle

text: In classical physics, we have second-order equations like Newton's laws, so we need to specify both position (zeroth order) and velocity (first order) of a particle as initial conditions, in order to pick out a unique solution. In non-relativistic quantum mechanics, we have Schrödinger's equation, which is first-order. As initial data we can therefore choose only the wavefunction's value at each point in space, but not its time derivative. This seems to dovetail with the uncertainty principle, which says a quantum particle does not have independent position and momentum degrees of freedom. We can choose the wavefunction to specify _either_ a position _or_ a momentum, but not both. (Or we can also choose a wavefunction that has uncertain position and momentum, within the bounds of the uncertainty principle.) In quantum field theory of fermions, we have the Dirac equation which is again first-order like Schrödinger's. But not every quantum particle is a fermion, and, AFAIK relativistic non-fermion particles obey the Klein-Gordon equation, which is second-order! So with the Klein-Gordon equation we can apparently choose both the wavefunction and its time derivative at each point in space, giving more freedom than the Schrödinger equation. Why do we have this extra freedom, and how can it be reconciled with the uncertainty principle as applied to relativistic bosons?


title: Heat engine efficiency, compression ratio vs temperatures

text: I want to understand the role or even existence of a cold reservoir in the Otto cycle gasoline engine. The efficiency of the Otto engine is given as a function of compression ratio $$ \eta_\text{Otto} = 1 - \frac{T_1}{T_2} = 1 - r^{1 - \gamma} $$ where $\gamma = c_p/c_v$, $r$ is the compression ratio $r = \frac{V_1}{V_2}$ and $T_1,T_2$ are the initial and final temperatures in the adiabatic compression stroke. (all is taken from the Wikipedia article on the Otto cycle). The temperature changes during the cycle due to the compression, which apparently is isoentropic, so $$ \frac{V_1}{V_2} = \bigg(\frac{T_2}{T_1}\bigg)^{\frac{1}{\gamma - 1} } $$ But the efficiency is still a function of the compression ratio which is just a geometric aspect of the engine design. So, is it wrong to say that the radiator is helping the thermal efficiency by dropping the temperature of the cold reservoir? I want to hold up the radiator as a real world example of Carnot's theorem, but I'm not sure it really applies. ## Edit Just found this document that says that an IC engine is not actually bound by Carnot efficiency because it is an "open cycle" rather than a "closed cycle". That would agree with the efficiency formula because I suppose if you made the engine compress all the way to 0, then the engine would operate at 100% efficiency. But wait, it says the Otto cycle is just an idealization because it is actually a closed system. I'm getting really confused here...


title: Why does a minor vertical vibration cause my coffee to spill?

text: When I walk my kid sometimes I put a cut of coffee in a cupholder attached to the stroller. When I push the stroller over a brick pavement it seems to vibrate vertically. That is enough for the coffee to spill, even if the cup is only 3/4 full. It looked like waves were forming in the cup, beating its forward and backward sides, and they quickly grew large enough to spill. Thus the question: is it possible to quantitively estimate the size of the waves caused by the vibration, so that one could figure out the level one can safely fill the cup to?


title: Action and Action integral: Different kinds of variational principles

text: What are the difference between:   1. the action $\int_{t_{1}}^{t_{2}}(L+H) dt$ that we use in the _principle of least action,_ and    2. the action integral $\int_{t_{1}}^{t_{2}}L dt$ that we use in _Hamilton's variational principle?_ References:   1. H. Goldstein, _Classical Mechanics,_ Section 8.6.   2. H. Goldstein, _Classical Mechanics,_ Chapter 2.


title: The physical definition of work seems paradoxical

text: So this is possibly a misunderstanding of the meaning of work, but all the Physics texts, sites, and wiki that I've read don't clear this up for me: In the simplest case with the simplest statement, work is force times distance. If you push with a force $F_{1}$ on an object that doesn't budge because of friction, you do no work. If your friend helps push and you still apply the same force $F_{1}$ and the thing moves, all the sudden you're doing work and it's not really because of what you're doing. Moreover, if you continue applying the same force, and your friend increases her force so that the thing moves faster, and covers a greater distance, again you're doing more work and by no fault of your own. This just seems paradoxical, and maybe the only sensible answer to this paradox is "Well, the physical notion of work is not the same as the everyday notion of work," but I'm wondering if anyone can say anything about this to make it feel more sensible than just accepting a technical definition for a word that doesn't seem like the right word to use.


title: Adding versus multiplying identical photons' wavefunctions?

text: I am currently confused with understanding many identical photons' wavefunctions. I think that photon wavefunctions are supposed to be multiplied together to describe the total state of all bosons. However, through some magic I am confused with, the total boson wavefunction must yield a field strength proportional to the sum of all of the boson wavefunctions?


title: Mapping a continuum XY model to a discrete one

text: A low energy expansion of a system I am currently investigating is described by this XY-like model: $$ H_1 = \int_0^\beta \mathrm{d} \tau \int_{\left[ 0, L \right]^2} \mathrm{d}^2 x \left( J \left( \nabla \theta \right)^2 + K \left( \partial_\tau \theta \right)^2 \right) $$ It is essentially an anisotropic XY model. It is better analyzed, as far as my question is concerned, by considering $\tau$ a space dimension, so the model is effectively 3D. If we take $\beta=L$ and $J=K$ to make it isotropic, then it undergoes a phase transition for $T=2.20$; a mean field model is presented by Kleinert ("Gauge fields in Condensed Matter", World Scientific, 1989), there the critical temperature is estimated to be $3$. I have some Montecarlo results for a similar lattice model: $$ H_2 = - \sum_{\langle ij \rangle} J_{ij} \cos \left( \theta_i - \theta_j \right) $$ in which $J_{ij}$ can take different values if the link is on the $x-y$ plane or along the $z$ direction, I'll call them $J_{xy}$ and $J_z$; these features should account for the anisotropy of the first system, because I'd like to map it $H_2$ to $H_1$. The system described by $H_2$ lives on a cube lattice. How do I map the first Hamiltonian into the second one? By knowing for which values of $J_{xy}$ and $J_z$ the system $H_2$ is at the critical point, I would like to know the critical values of $J$ and $K$. I observed that a naive mapping $J_{xy} \to J$, $J_z \to K$ doesn't work, because I can always rescale $\tau$ and $(x,y)$ in the first Hamiltonian. This is mathematically equivalent, but leads to a different mapping, so it shows that this naive mapping doesn't work. I think that the reason is that by rescaling the length I am also modifying the limits of integrations, and this cannot be ignored by just "naively" imposing periodic boundary conditions: the system is anisotropic as far as the coupling is concerned, but also as far as the box it's contained is concerned. My guess: maybe I should rescale the lengths in the first Hamiltonian so that the system lives on a cube, i.e. $\beta \to \beta' = L$ and _then_ maybe I am allowed to map the first Hamiltonian to the first one. Also in this case the $L \to \infty$ limit seems particularly badly behaved, I can provide more details if needed. However I am not sure the this is the way to go, and I would like some advice about this issue. Thank you in advance!


title: Where does the electron get its high magnetic moment from?

text: I have always found the concept of spin a little weird. I had read somewhere that for the charge or size of electrons, their magnetic field is very high. In order to produce such fields, they must be spinning faster than the speed of light which is not possible. So where do the electrons get their high magnetic field from? An answer without too much math would be preferred. So I quote this source http://scienceblogs.com/principles/2010/07/26/electron- spin-for-toddlers/ :- > If we want to say that the magnetic moment of the electron is due to the > motion of a spinning ball of charge, then we can easily calculate what the > spin rate should be, given what we know about the size of an electron. If > you use the maximum size you could possibly associate with the electron, the > “classical electron radius”, and calculate how fast a sphere of that size > would need to be spinning to produce the observed magnetic moment, you find > that a point on the surface would need to be moving at a speed several times > the speed of light in vacuum, which is impossible. That’s also a gross > overestimate of the size of an electron– as far as well can tell, the > electron has no physical size. It’s a point particle, and thus doesn’t have > a surface that can be physically rotating. > > OK, then, maybe the magnetic moment is just one of those things, you know? > Maybe the “spin” angular momentum isn't really angular momentum at all. This > is also false– spin angular momentum is real angular momentum. We know this > because you can drive transitions from one spin state to another using > polarized light, and we know from careful experiments done in 1936 that the > angular momentum carried by light is real angular momentum. The angular > momentum of a polarized photon can be used to make physical objects rotate, > and it can also be used to make electron spins change states; this at least > strongly suggests that spin angular momentum is real angular momentum > comparable to that of spinning basketballs and all the rest. So I do not understand what all this is suppose to mean. It first says maybe it is not spinning at all and then says that it is real angular momentum. Please clarify this for me. Or is this topic yet to be researched upon? **EDIT after answers** I think I have nearly got the answer but it was not what I was expecting (I am not talking about the quality of the answers). In the end I just want to ask as the electron actually does not spin but has angular momentum. I have pretty much understood that actual spinning is not the reason for its angular momentum. So is it safe to say that the actual reason behind the angular momentum is unknown?


title: What is the range of the validity of Coulomb's law?

text: What is the smallest and biggest distance in which Coulomb's law is valid? Please provide a reference to a scientific journal or book. Just saying that this law is valid from this range to that range is not enough. Which experiments did result to these ranges?


title: Why does a CD work as a diffraction grating even with light from a light bulb?

text: I have a really hard time solving the following problem I accidentally came across today. Looking at the back of a usual CD one sees coloured bands. This is explained by the fact that the surface (pits and bumps) is effectively a diffraction grating. But why is this effect already visible when one uses _incoherent_ light, e.g. from a light bulb, whereas the interference pattern of a single or double slit would be missing with such illumination?


title: Why can coercivity be used in place of magnetization when modelling permanent magnets?

text: I'm trying to understand how permanent magnets can be modeled in finite element methods. FEMM's manual, here, states that it treats a PM as a solenoid with the same surface current as the PM, and then solves the field from there, which I understand. However, they use the coercivity $H_c$ in place of the magnetization $M$ (I might be dropping factors of $\mu$ here.) to calculate this surface current, i.e., $H_c (\hat{m} \times \hat{n})$. The argument given is that this is the amount of current you would need to surround the magnet in (in opposite direction) to drive the field to zero (this is the definition of coercivity), which is the same field you get with no current at all. Therefore, the surface current of the original PM can be physically cancelled out with $-H_c (\hat{m} \times \hat{n})$, so this must be it's actual value. My problem is that when we model a permanent magnet in absence of any external currents, we are at the $H=0$ point of the $B-H$ diagram (i.e., $B = B_r$), and this is not the same physical point. Intuitively I would want to use $B_r$ in place of $M$, modulo some factors of $\mu$ again. I know that this makes assumptions about linearity of the magnet, but I'm not exactly sure where they creep in. Thanks in advance for any input.


title: Weird Behaviour of the act of measurement to a quantum system

text: I and my friend were disputing about some weird behaviour of the act of measuring some observables quantities e.g. Energy, position. But I still don't think what he said is strictly true. He said" each observable has its own Eigenstate, and when you measure it, the wave function will collapse to that eigenstate , giving its eigenvalue for that particle observable quantity. However ,I think some observables do share same eigenstate. i.e. a free particle with zero potential, you can prove this by Solving the TISE. When you now measure and obtained an exact momentum of a particle of a system, by uncertainty principle, the uncertainty of $x$ will be infinite, then the wave function will be spread everywhere,therefore it has a well define wavelength, thus you have well defined $k$ for wavenumber, hence a well defined energy by $p^2 / 2m$. To conclude, He said: is it true that every time you measure a quantity, it only produces a eigenstate for that particular quantity? I.e. measure position gives position eigenstate, momentum for momentum eigenstate. My argument shown above claimed the above is not strictly true: So my argument is energy and momentum do share same eigenfunction, when potential is zero. Who is correct? I also mentioned something called "Conjugate variables" e.g. position and momentum, you can only know one at a time. But I think energy and momentum are not, hence resulting in the reasoning i wrote above!


title: Writing a Wavefunction as a Linear Combination of Eigenstates

text: We have the following wavefunction for the hydrogen atom: $$\psi(r,\theta,\phi)=\frac{1}{\sqrt{4\pi}}\frac{1}{(2a)^{3/2}}\frac{r}{a}e^{-r/2a}\sin(\theta)\sin(\phi)$$ where $a$ is the Bohr radius. `Question`: How can I express the wavefunction above as a linear combination of the eigenstates of the hydrogen atom's Hamiltonian? In other words, we need to express the above in terms of a linear combination of $$\psi_{nlm}=\sqrt{\left(\frac{2}{na}\right)^3 \frac{(n-l-1)!}{2n[(n+l)!]^3}}e^{-r/na}\left(\frac{2r}{na}\right)[L_{n-l-1}^{2l+1}(2r/na)]Y_l^m(\theta,\phi)$$ For some $n$, $l$, and $m$'s. How can I find such a linear combination without just a simple guess and check; i.e., what is a systematic way to find the linear combination?


title: What laws (formulas) govern forces between atoms?

text: What laws (formulas) govern the fundamental forces of nature? For example, gravity is governed by the inverse-square law. I am thinking about how particles attract each other, but also repel. All matter attracts via gravity, but also attract and repel each other due to the electromagnetic force, also governed via the inverse-square law, and then there's the strong and weak force. I understand the idea that two hydrogen atoms repel, but when pushed together hard enough will attract and form helium and release energy. So this "upwards slope" of repulsion which is eventually "snaps" into attraction, like trying to roll a ball up a smooth hill with a hole in the center of it, is explained, likely, by varying forces governed by various laws. Can you help me understand it, **on the level of particles**? Where I am stuck is thinking about two particles, each being attracted and repelled by two different forces each governed by the inverse-square law. This doesn't explain the "snapping" together of atoms. Do I need different math in my forces, or is it ultimately explained by more than just two particles? Here is a diagram of what I mean, describing H + H -> H2: ![enter image description here](http://i.stack.imgur.com/3YesO.gif) _Image source and article:http://employees.csbsju.edu/hjakubowski/classes/ch111/olsg- ch111/equilibkinetics/equilbkin.htm_ Please feel free to reword my question if you understand what I am asking but am not making myself clear.


title: How does the research in theoretical physics differ from mathematics

text: I would like to know what is the difference between research in theoretical physics and pure mathematics. In particular, what does a theoretical physicist actually do all day long for his research? In other words, what does research in theoretical physics involve? Do theoretical physicists work only on big problems like string theory or any other major topics or there are other topics which are more fit for, say, PhD dissertations?


title: Irreducible tensor representations with "covariant" indices

text: As a follow-up of my question on the "most general" $\mathrm{SU}(2)$-symmetric interaction of two spin 1/2 particles, I ponder the following question: Consider an operator acting just on one particle of spin 1/2, and just on the spin part. In second quantization, it can be written as $$\sum_{\alpha\beta} \psi^\dagger_\alpha V_{\alpha \beta} \psi_\beta$$ Now, under $SU(2)$, the operators will transform as $$\psi^\dagger_\alpha = \sum_{\alpha'} \psi^\dagger_{\alpha'} D_{\alpha\alpha'}$$ with $D \in SU(2)$. I can move this transformation to the operator and conclude that it transforms like $$V_{\alpha\beta} \rightarrow \sum_{\alpha'\beta'} D_{\alpha\alpha'} V_{\alpha'\beta'} D_{\beta\beta'}^\dagger$$. (I might have gotten some of the daggers and indices backwards, but the important point is that one matrix is the adjunct of the other). Now if I understand correctly, this allows me to conclude that $V_{\alpha\beta}$ transforms "reducibly like a tensor product of two spin 1/2 representations", i.e., like $\frac{1}{2} \otimes \frac{1}{2}$. From that, I conclude that $V$ should decompose into one component that transforms like a spin $0$ particle and one component that transforms like a spin $1$ particle. Indeed, as $V$ is a $2\times 2$ matrix, I can write it as a linear combination of the unit matrix and the pauli matrices. The former transforms like a scalar, i.e., like spin $0$, whereas the latter transform like vectors. However, I have trouble relating this to what I know about combining spin $1/2$ particles using Clebsh-Gordan coefficients, where I have, e.g., for the singlet $$|0 0\rangle = \frac{1}{\sqrt{2}} \left( \uparrow \downarrow - \downarrow \uparrow\right)$$ Because of this conceptual problem, I also have trouble generalizing this to the case of two interacting spin 1/2 particles, which then should lead to an interaction that transforms reducibly as $1/2 \otimes 1/2 \otimes 1/2 \otimes$ and gives rise to two different singlets. I'd appreciate it if someone could disentangle my misconceptions... EDIT: I forgot to specify what I mean with "covariant" in the title: I think an important thing to notice is that the matrix elements $V_{\alpha\beta}$ are **not** the 4 elements of a "cartesian" tensor of rank 2. The entire operator $V$ might be an element of a tensor of higher rank, or even the sum of elements of tensors of different rank. A simple example for such a thing would be an operator "1 + x", which is the sum of a rank zero tensor (scalar) and the element of a cartesian tensor of rank 1. Now, what I mean with covariant is that one of the indices of $V_{\alpha\beta}$ transforms with matrix $D$ and the other with matrix $D^\dagger$. Also important is that the components of the operator transform with the actual $SU(2)$-matrices and not with some rotation matrix $R \in SO(3)$. I guess this is why I have trouble translating the standard literature on tensor operators to my situation...


title: Why only one prong in the plugs are not showing response to tester pen?

text: I have 2 pin outlet for A.C current . Now when I connect a TV plug in that and when I touch one of the prongs with my tester, the tester is showing response. But on the other prong the tester shows no response. I am using an A.C current(50 amps)(230 volts). As far as I know, Both the two prongs will act as a phase for every 1/50th second alternatively. i.e For the first 1/50 second one hole in the outlet must act as a phase and in the next 1/50 second other outlet hole must act as a phase. And this cycle must continue... So , If i touch any prongs my tester should show response. But only one is showing response. If my concept is wrong please ignore my mistake and explain where Ive gone wrong. Someone please explain. Thanks in advance.


title: How does the critical density decide the fate of the Universe?

text: Trying to teach myself cosmology. Given the Friedmann equation $$\left[\frac{1}{R}\frac{dR}{dt}\right]^{2}=\frac{8\pi G}{3}\rho-\frac{kc^{2}}{R^{2}}$$ and the critical density $$\rho_{c}\left(t\right)=\frac{3H^{2}\left(t\right)}{8\pi G}$$ I can see why, if $\rho>\rho_{c}$ then $k=+1.$ But I can't see why this leads to a collapsing universe? Conversely, if $\rho<\rho_{c}$ then $k=-1.$ But why does this lead to an expanding universe? I'm possibly confused because I've seen graphs of other FRW models with no correlation between the sign of $k$ and whether the universe is contracting or expanding. Do I need to learn about the deceleration parameter (I've heard of it and that's all) in order to understand this? Thank you


title: Why muonium is unstable?

text: This question is closely related to my previous question Bound states in QED. Muonium is a system of electron and anti-muon. This article in wikipedia claims that muonium is unstable. QUESTION: Why it is unstable? Is it due to presence of weak interactions, or it can be explained only in the framework of QED (involving both electrons and muons and their anti-particles). To compare with, the positronium, consisting from electron and positron, in an unstable particle (see e.g. this article in wikipedia). It can annihilate to two photons. This effect can be explained by the usual QED. This explanation does not seem to work for the muonium.


title: Does a photon travel through space exactly at the speed of light?

text: In practice, given that in the vacuum of the universe is not perfectly empty, does the photon actually reach exactly c? Is it even possible? If it travels at exactly c, then from the photon perspective, it arrives at the same time it leaves its point of origin, so for him the distance between the 2 points is zero? Many thanks.


title: Centrifugal force?

text: Everyone calls it a pseudo force and that it is not really present, even there are so many machines listed to advantage of centrifugal force ( centrifuge, washing washing drier),and also it has the same magnitude as centripetal force, so that they cancel cancel each other out but i can definitely feel it every time a sharp turn is taken in the car. So can you explain what centrifugal force is exactly, i don't understand it.


title: Additive constants of motion

text: I've read in a book, that in general case energy $E$, momentum $\textbf{p}$ and angular momentum $\textbf{M}$ of a closed system are _the only_ additive constants of motion, that is, if I have $N$ closed systems that are not interacting with each other then $$ E = \sum_{i=1}^N E_i, \;\;\;\;\; \textbf{p} = \sum_{i=1}^N \textbf{p}_i,\;\;\;\;\; \textbf{M} = \sum_{i=1}^N \textbf{M}_i $$ It is obvious that indeed they are additive constants of motion, but how can I prove that there are no other?


title: Fractal nature of turbulence

text: Someone described to me the difficulty of numerically simulating turbulence as that as you look at smaller length scales you see more structure like you do in a fractal. Searching on google for 'fractal turbulence' does seem to bring up quite a few hits. But since fractals are self repeating as you go to smaller and smaller scales, doesn't this mean that we would already know all about the turbulence if we calculate it at one scale, and then add in the all the other scales assuming they are self similar? I am not a physicist, so please answer with simple language and not overly mathematical.


title: Prefactor for phase space integration

text: When calculating the canonical partition sum, we had the following: $$ Z_\text C = \sum_{\vec p} \sum_{\vec x} \exp(-\beta H(\vec p, \vec x)) $$ Now, since $\vec p$ and $\vec x$ are pretty much continuous when taking the limit $V \to \infty$, we can write it as integrals. However, there are certain prefactors now. The following makes sense to me: Look at the wavenumber of the $x$-direction $k_x$. If we (without loss of generality) assume that the volume is a cube, the system has the volume $V = l^3$. The smallest wave number which is possible by quantum mechanics is $k_0 = 2\pi / l$. All other wavenumbers are $n k_0$ then. Then I can write: $$ \sum_{k_x} f(k_x) = \sum_{n = 1}^\infty f(n k_0) $$ Now I can write the sum over $n$ with an integral, like so: $$ \int \mathrm dn \, f(n k_0) $$ To get back an integration over $k_x$, I do a substitution, giving me a prefactor: $$ \frac{l}{2\pi} \int \mathrm dk_x \, f(k_x) $$ With momentum $p = \hbar k$, it will be $l/(2\pi\hbar)$. Doing this for all three dimensions, I get $V / (2\pi\hbar)^3$, which is the same in my textbooks. But what happens when I do another sum/integration in space? With the same quantisation, $n x_0$, with $x_0 = l / N$, where $N$ is just some large number would give me another prefactor $N / l$. For three dimensions, this is $N^3 / V$. Now I do have this huge number $N$ in the formula, which does not belong there. If I omit this, and just integrate over $x$, $Z_\text C$ will have the dimensions of a volume, but it should have no dimension. Where is the mistake?


title: Why does the water rise?

text: It's a very popular experiment (eg), from elementary school : put a burning candle on a dish filled with water, cover the candle with an inverted glass: after a little while, the candle flame goes out and the water level inside the glass rises. The standard explanation (as I recall it) was that combustion "burns" oxygen, and the consummed volume accounts for the extra water that goes inside the glass. Is this correct? I remember feeling (years later) uncomfortable with the explanation, because "to burn" is certainly not "to dissapear": I thought that oxygen combustion produces (mainly) $CO_2$ and hence one oxygen molecule would produce another $CO_2$ molecule, and the volume would remain basically the same. Perhaps $CO_2$ dissolves into the water? I would doubt that. To add to my confusion, others state that the main cause is not the oxygen combustion but the changes of air temperature, that decreases when the flame goes out and makes the air inside the glass contract... which would rather invalidate the experiment as it was (and is) traditionally taught to students. What is the right explanation? ![enter image description here](http://i.stack.imgur.com/Ed2fz.jpg) (image from here) _Update_ : As from webpage linked in accepted answer, there are several effects here, but I think it's safe to say that the "traditional" explanation (consumption of oxygen) is just wrong: oxygen turns into $CO_2$ and water and -even assuming that this is the complete chemical picture (it's not) and that all water condenses- that would produce a slight decrease of volume ($CO_2$ dissolves in water poorly and very slowly). The main effect here is thermal expansion-contraction of air.


title: How much energy is in a lightning strike?

text: According to Wikipedia an average lightning has 1TW, the whole world used 16TW in 2006. (I suppose this means the same as 16TWh in one year?) Sometimes the lightning reaches 100kA. This peak last for 30 microseconds. Does this mean that you get 100TW in 30 microseconds, and this is enough to power a 100 watts lightning bulb for two months, and that if the lightning lasted for about 1/6th of a year, it would have provided enough energy for the whole world?


title: How can Drift Velocity of an electron causes change?

text: Assume we have a conductor that has current flowing and its placed in a magnetic field so that it experiences the Lorentz force, and it gains Kinetic energy due to that force. Would drift velocity change the electron's direction thus the current's total direction changes, therefore, changing the Lorentz force direction applied on the conductor?


title: Ghost Number Conservation

text: I've been reading about gauge theory quantization, and understand it mostly. The only thing I don't get is why people talk about "ghost number conservation". As far as I can tell, the ghost number is a quantity defined to be $+1$ for ghost and $-1$ for antighost fields. It is then claimed that total ghost number is **conserved** for the action. I don't understand what is means for something to be "conserved for an action". I only know what it means for something to be conserved in classical or quantum evolution. Alternatively I know what it means for an action to be **invariant** under a transformation. Is this just bad nomenclature? I'm worried about it for the following reason. In the quantum theory there is a quantum "ghost number" operator which operates on the Fock space of the theory and tells you the cumulative ghosts $-$ antighosts in your state. I've seen it argued that this must commute with the Hamiltonian because "ghost number is conserved in the action". This seems pretty fundamental, but the logic seems flawed because I don't understand the terminology. Maybe everyone just means that the ghost number is $0$, which is obvious. But this seems like too simple an explanation (why would people call it conserved, when $0$ is significantly shorter?!)


title: Can a super-extremal charged black hole be made out of electrons only?

text: In a previous Question it was argued that it would be impossible to add enough charge to a black hole to make it pass the extremal black hole limit since adding charge would increase the mass of the black hole due to the electrostatic field energy (and thus mass) that would be added as the charge is added. Note that an electron **cannot** be a black hole but **if** an electron were a black hole it would be a super-extremal black hole per this wikipedia article: Basically the Schwarzchild radius for the electron's mass is $r_s = 1.35 \times 10^{-57} m$ whereas the charge radius of the electron is $r_q = 9.15 \times 10^{-37} m$. So since $$\frac{r_q}{r_s} \approx 10^{21} \gt 1$$ an electron, if it was a black hole, would be a super-extremal black hole by a large margin. In words, the electrons mass is completely negligible compared to its charge. An uncharged black hole can be constructed out of matter at any given mass density by simply constructing a big enough sphere of that matter. This is true because a sphere of radius $R$ with a constant (low?) density will have a mass $M$ that is $\propto R^3$ whereas the Schwartzchild radius is $\propto M \propto R^3$. So as $R$ increases the radius of the sphere will eventual be less than the Schwartzchild radius. So, can we make a super-extremal charged black hole by using a very large sphere of radius $R$ that is made out of electrons uniformly distributed with (low) charge density $\rho$?


title: Newton's original proof of gravitation for non-point-mass objects

text: Suppose we have two bodies, one very large (Earth), and one very small (a cannon ball). If the cannon ball is some distance away from the Earth, to find out the force produced on the cannot ball, we need to compute an integral in _three dimensions_ of how each infinitesimal piece of Earth pulls on the ball, i.e. $$\int_{\Omega}G \frac{m\delta(\mathbf{r})}{||\mathbf{r}||^2}dV,$$ where $\Omega$ is region of the large mass, $\delta$ is the density of the large mass, and $\mathbf{r}$ is the distance from the small mass (which we consider a point object) to each point of the large mass. Newton showed that, provided that the large mass is a perfect sphere with uniform density, we can treat it as a point mass as well. This is not hard to do with an iterated triple integral. Newton did not know about triple iterated integrals, and according to a physics book I have, devised a clever proof of this using a single integral only. Does someone know how Newton did this?


title: Classical mechanics: Generating function of lagrangian submanifold

text: I have a short question regarding the geometrical interpretation of the Hamilton-Jacobi-equation. One has the geometric version of $H \circ dS = E$ as an lagrangian submanifold $L=im(dS)$, which is transverse to the fasers of $T^*Q$ and lies in the coisotropic submanifold $H^{-1}(E)$. Furthermore the pullback of the canonical one-form $\theta$ to L is exact. The question is: Is L an **embedded** submanifold? I think if I have a solution S of the Hamilton-Jacobi-equation, then it generates an embedded submanifold $dS(Q)$. But have I have to use an embedded langrangian submanifold to get an "analytical" solution or is an "normal" submanifold enough? Thx!


title: How is stable levitation possible?

text: This question is with reference to the video in this blog post: http://www.universetoday.com/90183/quantum-levitation-and-the-superconductor/ My question is the following: how is the disc stable in its levitated position? Specifically, 25 seconds into the video, the exhibitor turns the entire thing upside down, and the disc doesn't fall. This contradicts two intuitive ideas I have:   1. Right-side-up, gravity is counteracting the repulsive effect of the magnet. Upside down, gravity is working with it. Unless there's some adaptation going on somewhere else, shouldn't "gravity switching sides" be enough to tear the disc away from the magnet?   2. I remembered something about "inverse-square" forces not permitting stable equilibria from college physics - sure enough Wikipedia calls it Earnshaw's theorem. I don't see how this is exempt from that rule. I guess I don't understand diamagnetism.


title: What are the requirements on conditional unitaries for overcomplete bases?

text: On way to describe "pure" decoherence (that is, decoherence with respect to a basis that doesn't involve transitions between basis states) between a system $\mathcal{S}$ and an environment $\mathcal{E}$ is with a unitary operator that acts _conditional_ on the state of $\mathcal{S}$: $$ U=\sum_i \vert S_i \rangle \langle S_i \vert \otimes U^\mathcal{E}_i. $$ Here, $U^\mathcal{E}_i$ describes the evolution of $\mathcal{E}$ conditional on $\mathcal{S}$ being in state $\vert S_i \rangle$. A CP map on $\mathcal{S}$ constructed by applying $U$ and tracing out $\mathcal{E}$ will leave fixed the diagonal matrix elements of the initial state $\rho_0^\mathcal{S}$ in the basis $\\{\vert S_i \rangle\\}$, but will in general add phases and decoherence (i.e. suppression of the norm) to off-diagonal elements. Importantly, $U$ is a valid unitary for any choice of unitary $U^\mathcal{E}_i$. Now suppose I want to construct a unitary that acts like this on an overcomplete basis. I'll specialize to the coherent states $\vert \alpha \rangle$ where $\alpha = x + i p$ is a point in phase space, but the solution for a general basis would be interesting. If I write down $$ U= \int \mathrm{d}\alpha \vert \alpha \rangle \langle \alpha \vert \otimes U^\mathcal{E}_\alpha. $$ one can check that this is _not_ a valid unitary for arbitrary $U^\mathcal{E}_\alpha$. (This can be easily seen when $\mathcal{S}$ has only two dimensions using an overcomplete basis of 3 or 4 vectors, and choosing random conditional unitaries.) Is there a compact way to write down the requirements on the $U^\mathcal{E}_\alpha$ for $U$ to be unitary? Obviously one can expand $U^\dagger U = I$ using the above definition, but I'm unable to transform this to something with a clear interpretation. My intuition is that $U^\mathcal{E}_\alpha$ and $U^\mathcal{E}_\beta$ should be "close" for $\vert \alpha - \beta\vert^2 \ll 1$ since they ought to be unrestricted for $\vert \alpha - \beta\vert^2 \gg 1$, but I can't formalize this in a useful way. (Incidentally, this is closely related to my previous questions which did not generate much interest. I'm giving this one last shot.) **Edit** : In the case where the unitary is taking the environment from a certin initial state $\vert E_0 \rangle$ to a conditional state $\vert E_\alpha \rangle$, one can see that the associated CP map is well behaved (i.e. that $\sum_i K_i^\dagger K_i = I$, where the $K_i$ are the Krauss operators) iff the operator $$ \int \mathrm{d}\alpha\mathrm{d}\beta \vert \alpha \rangle \langle \alpha \vert \beta \rangle \langle \beta \vert \cdot f(\alpha,\beta) $$ equal the identity, where $f(\alpha,\beta) = \langle E_\alpha \vert E_\beta \rangle$ is the Gram matrix of inner products of the (normalized) conditional states. A _sufficient_ condition for this is that $f(\alpha,\beta)$ depends only on $\alpha - \beta$ but not $\alpha + \beta$. I believe this is a necessary condition, but I am unable to prove it (or find a counter-example). This condition, along with the fact that $f(\alpha,\beta)$ is a Gram matrix, makes $f(\alpha-\beta)$ a positive-definite function, which I presume is important but don't know how to exploit it.


title: Why is E6 favored over E8 for GUT building?

text: What is the simplest explanation on why is E6 gauge group more favored as a group for Grand unified theory builders, while E8 is not? What about other exceptional groups? Which of them originate from the string theory in the most natural way, and why?


title: How does the star that has collapsed to form a Schwarschild black hole appear to an observer falling into the black hole?

text: I understand that to an outside observer, the light from a star that is collapsing into a black hole will become more and more red-shifted as the surface of the star appears to approach the black hole event horizon. The outside observer will never actually see the surface of the star cross the black hole event horizon. This applies to all outside observers: at infinity, in orbit around the star/black hole or those using a rocket to hover above the black hole. Conversely, I know that for someone on the surface of the star that is collapsing to form a black hole it will appear quite different. The observer on the surface will not see anything unusual happen as they cross the event horizon and in a finite time they will reach the singularity at the center of the black hole where we do not know what will happen since general relativity breaks down in a singularity. So, now consider an observer that starts at a great distance from the star who is continually falling directly into the star that has formed a black hole. Assume that he is falling in an exactly radial direction with no angular momentum. While the observer is still very far from the black hole, he will see the (original) visible light of the star get red-shifted to infrared, to microwaves and then to longer and longer radio waves. But as he approaches the black hole, he starts to fall faster and faster, so I assume he starts to see that red-shifted photons from the star surface will begin to be blue-shifted by his increasing speed as he falls. So, I think that the red-shift photons will be blue-shifted such that when the observer crosses the event horizon he will see the "visible light" photons that were sitting there at the horizon waiting for him. Since he is falling at the speed of light (in some sense) when he crosses the horizon these photons will become "visible light" again. So the first question is this true? When he crosses the event horizon will the surface of the star be back to having no net red or blue shift? Will a visible light photon emitted by the star as it crosses the horizon be once again a visible light photon? The second part of my question is what about the number density of photons? Will it look as "bright" as it would have looked for an observer falling with the surface of the black hole or will it look "dimmer" as if the surface was further away? Finally, what happens as the falling observer continues his fall past the event horizon of the black hole. Will the photons form the original in-falling star be red or blue shifted. What will the observer see during the short time before he himself hits the singularity? This is a follow on to my previous question since this was not answered there.


title: Operator-state correspondence in QFT

text: The operator-state correspondence in CFT gives a 1-1 mapping between operators $\phi(z,\bar{z})$ and states $|\phi\rangle$, $$ |\phi\rangle=\lim_{z,\bar{z}\mapsto 0} \phi(z,\bar{z}) |0\rangle $$ where $|0\rangle$ is the $SL(2,\mathbb{Z})$ invariant vacuum. Why can't we have a similar operator-state correspondence in non-CFT QFTs? Can't we just map operators to states by acting with the operator on the vacuum state?


title: Boundary Element Method or Boundary Integral Method Computational Aspects

text: I have to solve a Helmholtz equation inside a simply connected domain. I know that in general the boundary integral can be written as, $$\phi(x)=\int_V G(x,x') \rho(x')\ d^3x'+\int_S \left[\phi(x')\nabla' G(x,x')-G(x,x')\nabla'\phi(x')\right] \cdot d\hat\sigma'$$ Since my problem is to solve the Helmholtz equation with the Dirichlet boundary condition the boundary integral can be written as ($\rho$ is zero inside the boundary and $\phi(x)=0$ on the boundary), $$\phi(x)=-\int_SG(x,x') \nabla'\phi(x') \cdot d\hat\sigma'$$ My problem is, how to implement this boundary integral numerically in order to get the eigenvalues and eigenfunctions inside the entire boundary. How can I transform this into a matrix form to obtain the eigenvalues and eigenfunctions? By the way, the domain is two-dimensional and I am solving this equation in connection with quantum chaos (Quantum chaotic billiard).


title: How does a vacancy become mobile during annealing?

text: The Nitrogen Vacancy (NV) centre is a defect in diamond consisting of a substitutional nitrogen atom accompanied by a vacant nearest-neighbour lattice site. Substitutional nitrogen impurities are common in diamond, and vacancies can be introduced with radiation damage (there are other methods). The NV centre is formed by annealing the diamond: during this process the vacancy becomes mobile and is eventually 'trapped' by the substitutional nitrogen. Why does the vacancy become mobile at high temperature, and why is it trapped by the nitrogen?


title: Hamilton-Jacobi formalism and on-shell actions

text: My question is essentially how to extract the canonical momentum out of an on- shell action. The Hamilton-Jacobi formalism tells us that Hamilton's principal function is the on-shell action, which depends only on the coordinates $q$. Therefore, if $S$ refers to the off-shell action with Lagrangian $L$, a result for the canonical momentum is $$ \frac{\partial L}{\partial \dot q} = p = \frac{\partial S^\text{on- shell}}{\partial q} $$ In theory, when evaluating the on-shell action, one plugs in the EOM into $S$ and all that's left are boundary terms. If we illustrate this with the harmonic oscillator in 1D, one gets \begin{align} S^\text{on-shell} &= \int_{t_i}^{t_f} L dt \\\ &= \frac{m}{2} \int_{t_i}^{t_f} \left( \dot q^2 - \omega^2 q^2 \right) dt\\\ &= \frac{m}{2} \int_{t_i}^{t_f} \left( \frac{d}{dt} (q \dot q) - q \ddot q - \omega^2 q^2 \right) \\\ &= \frac{m}{2} \left[ q(t_f) \dot q(t_f) - q(t_i) \dot q(t_i) \right] \end{align} Now, this depends on $\dot q$, whereas it should depend only on generalized coordinates. Also, the on-shell action depends on the boundary values of our coordinate $q$, so we can't differentiate w.r.t. $q(t)$ to get a conjugate momentum $p(t)$ for all times, like it is usually done in the Lagrangian formalism. What am I not seeing?


title: Radiation emission and absorption

text: Any object can emit and absorb radiation and the power of emission can be represented by the Stefan-Boltzmann law: $$P=A\epsilon\sigma T^4$$ In many texts the net power radiated is the difference between the power emitted and the power absorbed: $$P_{net}=A\epsilon\sigma (T^4-T_s^4)$$ where $$T_{s}$$ is the temperature of the surroundings. Why can the surrounding and the object share the same $\epsilon$ ? If we try to find out the radiation emitted from the surrounding it should be $P_s=A\epsilon_s\sigma T_s^4$, and if $\epsilon_s<\epsilon$, we will get a strange result that energy radiated from the surrounding is less than the radiation absorbed by the body from the surrounding. What am I missing?


A problem in Carroll (a general relativity textbook) asks if a certain metric is regular. What does it mean for a metric to be regular?


title: Why do bubbles make a sound?

text: I have an understanding of how bubbles work. They encapsulate air (or other fluids) in a membrane caused by surface tension. When they pop, there is often a sound. Sound is a type of energy, kinetic to be precise, that usually occurs from collisions. When a bubble pops I would assume that a sound implies that air rushes out due to a pressure change. Why is there a pressure change? I wouldn't expect the bubble to exert enough pressure to compress air. If the sound is caused by the air now being able to move into the rest of the room due to Brownian motion, then why wouldn't I hear air moving in a still room? Perhaps I'm over-thinking this. Just curious. Thanks.


title: How do I know which force to use in moments calculation?

text: I'm working on some Statics problem, with the method of joints, etc and for all of them, we use sum of moments = 0. However, I'm very confused about which forces I should use to find the sum of moments and which forces shouldn't be used. Can someone give me some tips/tricks to this?


title: Fierz identity for Weyl spinors in tensor currents

text: Using Fierz identity I found that certain four-fermion operator with left $l_i$ and right-chiral $r_i$ Weyl spinors vanish $\bar{l}_1\sigma_{\mu\nu} r_2 \bar{r}_3 \sigma^{\mu\nu} l_4 =$ $ -\frac{3}{2} \bar{l}_1 l_4 \bar{r}_3 r_2 - \frac{1}{2}\bar{l}_1\sigma_{\mu\nu} l_4 \bar{r}_3 \sigma^{\mu\nu} r_2$ $ - \frac{3}{2} \bar{l}_1 (-1) l_4 \bar{r}_3 (+1) r_2 =$ $= -\frac{1}{2}\bar{l}_1\sigma_{\mu\nu} l_4 \bar{r}_3 \sigma^{\mu\nu} r_2 = 0$ Why does this operator vanish? Is that true? Why is the product of tensor currents expressed in tensor and scalar currents only, while for such combinations of Weyl spinors $l_1$, $r_2$, $r_3$ and $l_4$ in the Fierz identity all of them vanish?


title: Does a current carrying wire produce electric field outside?

text: In the modern texts of electromagnetism in the presence of stationary currents the electric field is assumed conservative $\nabla \times E =0 $. Using this we get $E_{||}^{out}=E_{||}^{in}$ which means we have the same amount of electric field just outside of the wire! Is this correct? Is there any experimental proof?


title: Does electrically heated water have an adverse effect on hair?

text: I know I should have asked this question on a different site. but this was the most suitable site available right now for my question. Perhaps after **this proposed site** goes on Beta, we can move it there. Coming to the question, Some of my well-wishers always say that electrically heated (specifically using a coil heater) when used to wash hair, affects the hair adversely. Though I always thought it was a myth, lately I remembered the concept of electrolysis that I had studied in school. According to what's running on my mind, in this setup, the electric heater may act as anode and the water container may act as cathode or vice-versa and the minerals in water may act as electrolyte. So, my fear is that there may be some chemical reactions going on that may result in different compounds which affect the hair. How far is my fear justified?


title: Production vs. Collection, and Contaminants vs. Depositions, what might be missing in cold fusion research

text: I though cold fusion and LENR were discredited, but just a few days ago I found out that NASA is claiming LENR is real. So I thought if they're detecting something, what could it be, and why wasn't it there before? Then it hit me -- what if this new phenomenon is not producing the rare isotopes, but attracting them out of the air? (Credit to reading Kirk Shanahan's essay for the inspiration.) Is there a good enough record of global wind patterns to determine where the fallout from the Fukushima disaster might have blown? Because perhaps NASA's new experiments have been unintentionally collecting it? This could be a breakthrough technology. A large version of the device could be placed at the site of a nuclear accident, bomb, etc. and collect the unstable atoms from the air into depositions. And then whatever part of the machine got coated with the depositions could just be tossed into a radioactive waste barrel.


title: Geometrical Representation Grover algorithm

text: I am studying the Grover algorithm and in my and others lectures, I've come across this picture. ![](http://upload.wikimedia.org/wikipedia/commons/1/16/Grovers_algorithm_geometry.png) If the dimension of the computational basis is greater than 2, why does the evolution algorithm have this geometrical representation in the plane?


title: Tensor equations in General Relativity

text: In the context of general relativity it is often stated that one of the main purposes of tensors is that of making equations frame-independent. **Question** : why is this true? I'm looking for a mathematical argument/proof about this fact.


title: Black hole thermodynamics in a time dependent metric

text: For a time dependent space time metric, to get the thermodynamics, does the standard procedure of Wick rotating the time, and then calculating the free energy, work ?


title: Flux Quanta in the Arahanov-Bohm effect

text: I have been reading about the quantum hall effect during which i had to read about the AB effect used in the Laughlin gauge argument. In many sources, it is directly assumed that the flux quantum in the AB effect is $\frac{\hbar c}{e}$. How do I prove this? How do I prove this is the smallest change in magnetic flux that can occur? Why is it called the flux quantum if $\Phi$ is not quantized? How can I prove that it is the smallest flux change that would leave the physics same?


title: Molecular dynamics: Thermostat via damping modulation

text: Molecular dynamics (MD) often use a thermostat so that they can control the temperature. Temperature tends to drift due to endo/exothermic reactions (that would normally absorb/emit heat to a large thermal bath) and numerical errors. Thermostats are a bit cumbersome, each has significant disadvantages (i.e. "flying ice cube" effect). How about a thermostat that simply sets a global damping coefficient on the short-range interactions? If we are too cold, it would set a negative damping to add energy and warm us up, and visa-versia if we are too hot. This would inject or remove energy mostly from the higher frequency vibrations, which is where numerical error tends to originate. This idea seems good at first glance, but it must have important issues of it's own. What are the problems with using this thermostat, and does this thermostat have a name in the literature?


title: How do I relate $ds$ to $dx$?

text: > A block of mass $5\text{ kg}$ is kept on an inclined plane with angle of > inclination $37°$, attached to a spring with spring constant $10\text{ N/m}$ > kept at the base of the incline. > > The coefficient of friction between the inclined plane and block is $0.2$. > > At $t = 0$, the block is at rest. > > Find the minimum work done by an agent to pull the block up the incline by > $1\text{ m}$. My approach to the question: As the block is at rest, we can find the compression in the spring. $$Mg\sin(37°) = 10k + N(0.2)$$ where $k$ is the compression in spring We can find $k = 2.2\text{ m}$. Then we can find the force required to pull the block up by $1\text{ m}$ $$F_a = 38 - 10x$$ Now, using $\mathrm{d}w = \vec{F}\cdot\mathrm{d}\vec{s}$, $$\begin{align} \mathrm{d}w &= \vec{F}_a\cdot\mathrm{d}\vec{s} \\\ \mathrm{d}w &= (38-10x)\mathrm{d}s \end{align}$$ Here is my problem. How do I integrate the function. That is how do I express $\mathrm{d}s$ in terms of $\mathrm{d}x$? $\mathrm{d}x$ is the compression in spring whereas $\mathrm{d}s$ is the displacement. How do I relate displacement with the compression in spring.


title: How to use an accelerometer to correctly quantify the magnitude of vibrations

text: Consider the following setup, an accelerometer is placed inside a helicopter, measuring "g" values along 3 axes (x,y,z) every 10 ms (100 times a second), during a helicopter flight. If one, wants to obtain some measure of the maximum g force exerted on an object inside the helicopter, what would be the best way to do this? Make the analysis for each axis independently? Or treat x,y,z as components of a vector ? Is it correct to consider the maximum difference between two consecutive measurements as an approximation of the maximum g force exerted on an object inside the helicopter


title: How does (or can) SR/GR extend to phase space or symplectic manifolds?

text: I'm asking this question because of an article in New Scientist about a recent preprint by a group including Lee Smolin. I haven't taken the time to comprehend the paper completely. My knowledge of differential and pseudo- Riemannian geometry is out of practice, so it'd take me a while to do so. Even so, the paper put some ideas in my head about phase space and symplectic manifolds that can be put as simple questions but I couldn't answer with some basic Googling. Also, this is resting on the (mis)understanding that phase space is a symplectic manifold. If this is wrong, these questions might be meaningless.   1. Is there a canonical symplectic manifold that is associated with a given pseudo-Riemannian manifold? I'm guessing yes, because a pseudo-Riemannian manifold is itself a differential manifold, and I think the tangent bundle is a symplectic manifold.   2. If the space we started with has a metric, does this imbue the associated symplectic manifold with any structure? e.g. is the symplectic form limited by or related to the metric or something similar?   3. Could any ideas from SR/GR be extended to this symplectic space such that we learn something new? e.g. could there be something like the Einstein Field Equations (EFEs), written for the 8-dimensional phase space, that have new solutions that are realistic but wouldn't be found using the EFEs? Or could there be non-canonical symplectic manifolds associated with a given pseudo-Riemannian space? (I suspect Darboux's theorem says "no".) This is mostly crazy speculation based on knowledge I haven't exercised in a few years now. For all I know, there's some underlying concept that makes this pointless or I've just totally misunderstood the paper in the first place. Also, I've posted here because of the theoretical physics nature of the problem, but I accept it may be better served on another SE.


title: Naïve relativistic schrodinger equation

text: > **Possible Duplicate:**   >  Why are higher order Lagrangians called 'non-local'? Bjorken and Drell presents the equation: $$i\hbar\frac{d\psi}{dt}=H\psi=\sqrt{p^2 c^2+m^2 c^4}\psi=\sqrt{-\hbar^2 c^2 \nabla^2+m^2 c^4}\psi$$ The squareroot can be expanded to obtain an equation with all powers of the derivative operator. What do they mean when they say this leads to a non-local theory? And is this equation incorrect or just impractical?


title: Is there an equation for convective heat transfer?

text: Is there an equation I can use to calculate the temperature (as a function of time) of an object which is gaining or losing heat by convection? Or equivalently, the rate of energy transfer from the object to the surrounding fluid (or vice-versa)? It can involve constants representing properties of the object and the surrounding fluid. I know about the heat equation for conduction and the Stefan-Boltzmann law for radiation, but I can't think of one for convection.