title: Is it physically meaningful to talk about the arrow of time in other universes?

text: The arrow of time is a concept that has been around for a while. In modern cosmology, it has been argued that there may be different arrows of time in other universes. Since these are also considered to be inaccessible, is it physically meaningful to talk of the arrow of time in other universes?


title: Physical consequences of non-abelian non-trivial holonomy

text: The Aharonov-Bohm effect (http://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect#Significance) can be well described and explained in terms of holonomy of the $U(1)$ connection of the electromagnetic field. What happens physically is that after parallel transport along a loop, the wave function acquires a phase difference, physically invisible, but that can lead to observable interference. Now, a "phase" in a $U(1)$ gauge theory can generalize in two different ways if we go to non-abelian gauge theories: 1) It could remain a phase, leading to no physical difference (except interference); 2) It could become a more general $G$ transformation, where $G$ is the gauge group (like $SU(2)$ or $SU(3)$), and for example change the color of a quark after a loop. What does quantum field theory predict? What would happen if we set a strong (or weak) equivalent of the Aharonov-Bohm effect (despite the obvious experimental difficulties)? If 1) is true: wouldn't it mean that the meaningful (curved) part of the holonomy is in fact Abelian? If 2) is true: wouldn't it violate (for quarks) conservation of color charge? Any reference would be good, as long as it gets to the point.


title: How is $\frac{dQ}{T}$ measure of randomness of system?

text: I am studying entropy and its hard for me to catch up what exactly is entropy. Many articles and books write that entropy is the measure of randomness or disorder of the system. They say when a gas system is let expand the randomness increases etc. But they end up saying $\frac{\mathrm dQ}{T}$ is the measure of increase in randomness and is called the entropy. Even if I believe that entropy is the measure of randomness of the system I don't understand:   1. How does $\frac{\mathrm dQ}{T}$ hold the information about the randomness of the system?   2. How is entropy independent property of any system. I suppose that any two parameter in the equation $PV=nRT$ should completely describe the system. Why would we need entropy? Thank you.


title: What do I need to do to find the stopping time of a decelerating car?

text: The question is: > A car can be stopped from initial velocity 84 km/h to rest in 55 meters. > Assuming constant acceleration, find the stopping time. Sorry for my ignorance, but I need to review physics knowledge. Can anyone explain me for this question?


title: Equation of progressive wave

text: A progressive wave traveling in positive x-direction given by $y=a\cos(kx-\omega t)$ meets a denser surface at $x=0,t=0$. The reflected wave is then given by: I know that since the wave changes its direction of motion the sign of the equation would change and as it gets reflected from a rigid surface there will be phase change of 180 degrees. So, is the answer $y=a\cos(kx-\omega t)$?


Why uncharged particles do not feel the Lorentz force? Please do not answer with the formula $ \vec F = q\left( \vec E + \vec v \times \vec B \right) $. Edit after an answer which is an circular reasoning. Let me explain this question with an example. Imagine, you nothing know about car traffic and you are standing at a traffic light junction. What is the law you could formulate? The green light moves the cars. In my question the moving charged particles are the cars, the light junction is the magnetic field. A running horse does not stop on red and the deeper answer is that the driver accelerate the car when he see green. So what "saw" charged particles what don't "saw" uncharged particles? Why an charged and not moving relatively to a magnetic field particle does not feel this force?


title: kinematics question about runner

text: Here's the question: Your goal is to complete your 10,000-m run in less than 30.0 min. Apparently you've started too slowly because at 27.0 min, you still have 1100 m to go! You can accelerate at 0.20 m/s2 . For how many seconds should you accelerate (and after this have constant velocity) in order to complete the run on time? So i can only think of using the equation x=xo+vot+0.5at^2 to solve for vinitial, but after that idk what to do, not sure if my first step is even correct


title: Free energy of the critical U(N) model

text: * Can someone help explain how the equations 30, 31 and 34 were obtained in this paper.  At a conceptual level I am wondering looking at equation 34 as to if they mean that $\lambda$ is somehow the anomalous dimension of the single-trace operators in this critical U(N) model - although looking at equation 30 it seemed that $\lambda$ is just the Hubbard-Stratonivich auxiliary field (though its not clear to me as to how is equation 30 derived). That factor of $(l + \frac{1}{2})$ is confusing me everywhere. Also what they call as $1/g$ in equation 30 is what I would think of as the threshold value of the translationaly invariant condensate at which the symmetry breaks. So it is like an order parameter and I can't see that as something that was there in the UV lagrangian as a coupling constant as they seem to imagine it here.   * Although I understand how equation 1 in this paper was derived but I still don;t see how equation 35 somehow follows immediately from it.


title: What are the fastest electron orbitals

text: I read that the mercury has a low melting point because its outer shell electrons are pulled in close by its nucleus (large nucleus, sparse outer shell) and because its outer shell electrons have significant relativistic mass. I think I remember them being roughly 1.28x the rest mass of an electron. I calculated that as meaning they must travel at roughly 60% of the speed of light. But are there faster electrons? (in orbitals, not just free electrons)


title: Does an electron move from one excitation state to another, or jump?

text: I'm wondering, when an electron changes state, does it move from one state to another over some (very small) time period? Or does it change from one state to another in no time? If the former, what does it mean for it to be in- between states (for however short a period of time)? If the latter, how does it _teleport_? (Does this question make sense?)


title: Do vortex tubes work with a reversed end plug?

text: Would a vortex tube still work if instead of a cone plugged into the 'hot' end you had a smaller hole on the 'cold' end? As I understand it, the point of the cone on the hot end is to only allow the fluid to escape from the outer-most part of the tube where it is hottest, so if you were to instead make the hole on the cold end smaller, as to only allow fluid to escape from the inner-most part of the tube where it is coldest, would it still work? \---Edit--- I've been thinking of building one of these for fun, but I was really hoping someone might be able to answer this question beforehand so I could avoid unnecessary work. As I understand it, these things have to be made very, very well to get a measurable effect. Oh well, I guess I'm just charging ahead on my own. Wish me luck. I'll post my results back here when I'm done.


title: overall energy transformation as a diver moves downwards through water

text: What is the answer to the following MCQ? > A swimmer dives into a very deep pool at high speed. He slows down as he > moves towards the bottom of the pool. What is the overall energy > transformation as the diver moves downwards through the water? > > **A)** gravitational potential energy → kinetic energy + thermal energy > (heat) > > **B)** gravitational potential energy → kinetic energy → thermal energy > (heat) > > **C)** kinetic energy + gravitational potential energy → thermal energy > (heat) > > **D)** kinetic energy → gravitational potential energy + thermal energy > (heat)


title: Linear decrease in temperature during a redox reaction?

text: This is physics-related, don't worry. To calculate the enthalpy change of a solution during a redox reaction, what we did in class was measure the temperature of the solution every 30 seconds (before, during, and after the reaction) and waited until it started decreasing. Once the temperature was decreasing, we made sure to get at least around 10 measurements (so 5 minutes of measuring) and then we plotted these. We made a linear fit for the temperature as a function of time but using $\textit{only the data from when the temperature decreased}$. Into this function we plugged in for time the initial time when the reaction began (we call it extrapolating, not sure if that's the correct term) to get the temperature that the solution _should have_ reached right after the reaction. Now, what I'm not sure about is this: we're using a linear function to model the decrease in temperature of the solution, but doesn't temperature decrease exponentially? (At least according to Newton's law of cooling). My only guess for why this could be correct is that maybe at short enough time scales, this exponential decrease can be approximated as linear, so it's justified that we're doing this. Is this the reason? Or is there some other justification for doing it this way?


title: What is special about a 126 GeV Higgs mass?

text: The 126 GeV Higgs mass seems to be a rather special value, from all the a priori possible values. One thing I've heard is that it is just at the edge of the mass range implying a stable vacuum, and that it allows vacuum stability all the way up to the Planck scale. This fact seems to be related to asymptotic safe gravity and the prediction of _m H_ = 126 GeV by Shaposhnikov and Wetterich. The other thing that makes me think 126 GeV is special is Supersymmetry. I was playing around with spectrum generators and exploring "simple" models like mSUGRA, and models with a bit of non-universality, and found that it is tricky to get a _h 0_ mass as high as 126 GeV. There seems to be a glass wall or asymptote there, in that you can get very close (e.g. by choosing extreme values of _A 0_), but often you can't arrange _m h0_ = 126 GeV without going to more complex models. I found a post by Lubos Motl from 2011 where he mentioned that and hinted to some interesting "marginal" properties. So what are all the things going on at 126 GeV? What coincidences, boundaries, magical effects are there? And what makes it so hard to reach with "simple" SUSY?


title: How can metal objects become electrically charged, if current flow means that an equal number of electrons enter/leave the object?

text: I am trying to answer to the question in the title. I am restricting my question to metal objects only. Here is my logical reasoning:   1. Current is the flow of charge over time.   2. In a circuit (simple series circuit, let's say), the flow of current is the same at every point in the circuit.   3. Therefore, the same # of coulombs of charge is flowing at every point in the circuit.   4. Electrons are the "material" of the charge that is flowing.   5. Therefore, equal flow of charge at every point in the circuit must mean equal flow of electrons at every point in the circuit.   6. Therefore, current can never cause a metal object to become positively or negatively charged, because the net number of electrons in the metal object will never change due to current flow. (!) Of course, objects CAN become electrically charged, gaining or losing electrons. So something is wrong with my reasoning or my premises. I just don't know what it is. Where am I going wrong?


title: How does Ptolemy's table, of angles in air and angles in water, fit perfectly on a parabola?

text: In "The Feynman Lectures on Physics" Vol.1 Optics Ch.26, Table 26-1 Ptolemy's list of angle in air and angle in water are given, in connection with Snell's law of refraction. I quote below values from that table: The angles are in degrees. Angle in air, Angle in water: (10, 8); (20, 15.5); (30, 22.5); (40, 29); (50, 35); (60, 40.5); (70, 45.5); (80, 50). In table 26-2 the corresponding angles according to Snell's law are given, to show the remarkable agreement with Ptolemy's list. In the text above the tables, a statement says all these values 'fit pefectly on a parabola'. I am unable to see the how these values fit on a parabola. How do you get the parabola from the listed values in the table?


title: Adjoint representations and bosons

text: Is there a deep mathematical reason why bosons should be in the adjoint representation of the gauge group rather than any other representation?


title: Does the speed of sand flow in a hourglass depend on a height of a sand column above the hourglass neck?

text: In a hourglass, does the sand flow through the neck depend on the amount of sand in the upper glass? If we consider a sand flow analogous to fluid flow, then it should depend linearly, but in that case amount of sand to represent the given time would rise squared depending on time?


title: Is the number of wavelengths of light spanning a distance invariant with respect to spacetime distortion?

text: I was recently asked by a friend how the expansion of spacetime affects photons. I gave him what I feel is a satisfactory general response, but it got me wondering how, exactly to calculate this effect. It occurred to me that a simple way to conceptualize the change in energy was considering comparing the number of wavelengths that would have spanned the distance originally to the number it would take to span the expanded distance. Given the time contraction of a photon, it seemed reasonable to me to assume that the number of wavelengths spanning the distance traveled would be a constant. Unfortunately, I never had the opportunity to take a GR class and I don't know if this simple concept is in any way valid. My question is, is this simple naive approach valid (even if it needs modification)? That is, given the relation between wavelength and energy $$E = \frac{hc}{\lambda}$$ where $E$ is the energy, $h$ is Plank's constant, $c$ is the speed of light, and $\lambda$ is the original wavelength of the light, can I make this $$E = \frac{nhc}{d}$$ where $d$ is the unexpanded distance of a path traveled by light, and $n$ is the number of wavelengths of the initial energy that would span the distance along the path traveled by light. From this point, is it proper to calculate the fractional change in energy due to a fractional expansion of spacetime by $$\frac{dE}{dd} = -\frac{nhc}{d^2} $$ where $dd$ is the change in distance due to expansion. Or, alternatively, $$E = E_0 + nhc \left(\frac{1}{d_f} - \frac{1}{d_i}\right).$$ There have been several questions about the temperature of the CMB and how the expansion of spacetime reduces it's temperature. In particular the following closely address the topic at hand:   * Have CMB photons "cooled" or been "stretched"?   * Effect of expansion of space on CMB though none of them addressed the issue in the way I have described. It is worth noting that in the article linked to in the first of these questions, the authors state "...'all wavelengths of the light ray are doubled' if the scale factor doubles." which seems to give the same proportionality described by my formulation. My intuition tells me that this is an inappropriate method at least in part because there is no observational frame stationary with reference to the path traveled, but I sense that there are other problems as well. * * * I just realized that I could equally well refrain from using $n$ and instead just use $d_i/\lambda_0$ or $d_i E_0 /hc$ which gives $$E = E_0 \left( \frac{d_i}{d_f} \right)$$


title: Why do some anomalies (only) lead to inconsistent quantum field theories

text: In connection with Classical and quantum anomalies, I'd like to ask for a simple explanation why some anomalies lead to valid quantum field theories while some others (happily absent in the standard model) seem to make the corresponding quantum field theory inconsistent. Edit: More specifically, in case of an anomalous gauge symmetry: Why can't one get a valid theory by using a central extension of the gauge group as the quantum version of the gauge group? Has this been tried and not found working, giving a no-go theorem? Or would that lead to a different classical theory in the limit $\hbar\to 0$?


title: Does a car use friction to move?

text: When a car's engine injects fuel into the cylinder chambers, the reaction creates a force that generates rotational momentum to the shaft and over the transmission, it translates that power to the wheels, right? But, something's bothering me, how does the car actually move, is it dependent on the force of friction? Wheels have a contact with the road in one exact point and when the drive gets to rotate them, a force is exerted on the road in the direction of movement and there is friction that is counteracting it. Is the key to get a car properly moving to have the drive force equal or less than the force of friction to keep that point locked to the ground and use the rolling of the wheel for translational displacement ie. moving forward? If it's bigger than the force of friction, that would cause the wheels to spin in place? Am I completely off or am I getting something right? If there's a knowledgeable person on this topic, I'd greatly appreciate some insight, perhaps even a bigger response with some basic vehicle physics.


title: Metric for infinite straight cosmic string

text: A string theory question on my general relativity problem set: Metric is given as $$\mathrm{d}s^2 = -A(r)\mathrm{d}t^2 + B(r)\mathrm{d}r^2 + r^2 \mathrm{d}\theta^2.$$ a) Solve the vacuum equations and find the metric around the massive source assuming that the Ricci curvature vanishes. Use a change of coordinates to put the metric in the form, $$ \mathrm{d}s^2 = -\mathrm{d}t^2 + \mathrm{d}r^2 + C(r)r^2 \mathrm{d} \theta^2$$ b) Consider the metric on a spatial slice ($t$ = constant). Use the relationship between curvature and parallel transport to determine the total curvature that must be contained in the region containing the source if parallel transport around a single cosmic string rotates a vector by $2\pi/N$. c) Write the metric at a large distance from a configuration of $k$ sources, assuming each separately gives rise to the metric described in the previous part. What is the maximum number of parallel such cosmic strings that can inhabit space-time? With this maximum number of cosmic strings, what is the total curvature over a 2D surface perpendicular to the strings? \--I attempted the first part, found all connections and Ricci tensor components. Requiring that the Ricci tensor be zero. I attempted to get what seemed to be a mess of algebra to solve for $B(r)$ and $A(r)$. But I didn't really get it to work. Regardless, I know that in the form C(r) is given by a few books to be something of the form $1-4\mu$, whose interpretation as canonical spacetime makes enough sense to me. The latter two parts are where I'm really at a loss.


title: How are electric flux calculations not double integrals?

text: > A disk of radius 0.10 m is oriented with its normal unit vector $\hat{n}$ at > 30$^{\circ}$ to a uniform electric field $\vec{E}$ of magnitude 2000 N/C. > What is the electric flux through the disk? I know that electric flux is given by the following equation: $$ \Phi_E = \oint_{S} \vec{E}\cdot \hat{n} \: dA $$ My question is that if you're integrating over a region $S$ and it's with respect to $dA$, how is this not a double integral? Why is $$ \Phi_E = \int_{0}^{0.10}\int_{0}^{2\pi}r \: \vec{E}\cdot \hat{n} \: d\theta \: dr $$ not right?


title: Why is Polarization of a wave important, and what happens when a wave is polarized in all three dimensions?

text: I don't quite understand this, I understand that given a wave traveling in a certain direction, it will be polarized in the direction its not travelling in, so if its travelling in the Z direction, k will be in x,y direction (polarization) Now what i don't understand is that, if the wave is polarized in all 3 directions, x,y,z. which way will it travel? also, for Tranverse magnetic, why is there no electric field normal to the plane of incidence? i know this might be stupid, but what is the purpouse or use of polarization of a wave, in class we were just taught that a wave can be polarized, they never indicated why is the use of it. Please help, thank you in advance


title: Scattering theory of Dirac equation in curved space-time in presence of a strong magnetic field

text: What is the exact solution of the Dirac equation in curved space-time in the presence of a strong magnetic field? The solution should be in momentum space for simplicity to calculate scattering cross sections.


title: Conservation Laws and Symmetries

text: Usually, in Quantum Mechanics, an observable is an operator on the space of the possible quantum states (labelled as $|\psi\rangle$). If this quantity is conserved, in the meaning that the associated operator $\hat D$ is constant: $$\frac{d\hat D}{dt}=0$$ it has to commute with the Hamiltonian operator $\hat H$. To prove this, in the Heisenberg picture of Classical QM, we use the Schroedinger Equation $$i\hslash\frac{\partial}{\partial t}|\psi\rangle =\hat H|\psi\rangle $$ On the other hand, the Noether theorem states that if a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time. And this is true either for classical, relativistic and quantum systems. Therefore, first question:   * Is it true, even in the relativistic case, when the Schroedinger eq. doesn't apply, that a conserved quantity has to commute with the hamiltonian to be conserved during the motion?  Then, one usually studies the transformations generated by $\hat D$, for instance if $\hat D$ is the angular momentum, the transformation $U(\theta)=e^{-i\hat D \theta}$ (a rotation of $|\psi\rangle$ with angle $\theta$). And assuming the angular momentum conserved, or verifying this by means of the commutation properties, we can construct the space of the eigenstates of $\hat H$ knowing that they are also eigenstates of $\hat D$. For instance we construct the eigenstates of a spin-1/2 particle using $\hat D = \hat S$, and knowing that $[S^2, S_z]=0$. Since it exists only two states: spin up and spin down, $|\psi\rangle$ will be a two-components spinor, and $\hat S$ will be a 2x2 matrix. We see that the Lie Group of the $\hat U$ transformation is generated by the Pauli's matrices throught the expressions $$\hat S_i=\hslash\sigma_i/s$$. So, I understand that, wanting to impose the conservation of $S$, we get into the $SU(2)$ Lie-group, because of the commutation relations satisfied by the $S_i$, and the infinitesimal decomposition of each $\hat U$. Final questions:   * If we want to conserve something else than the spin, as it happens in the strong interaction study, when we have the Isospin and the Strangeness to be conserved, we use $SU(3)$. Why?    * Who is the corresponding Casimir operator? (the analogous of $S^2$)   * Why in this case the generators should be 8, and not 3?   * What is the relationship between a normal rotation in the physical space, for which the generator is $\hat L$, and the rotation in the total space $\hat J$?    * Why in the case of the strong interaction we don't consider any spin or orbital angular momentum in the labeling and classification of the eigenstates?    * At last, is there any good textbook you would suggest to have a global view of this subject? Because I've been studying this problems in several courses during my undergrad courses, in QM classes, RQM classes, Nuclear Physics, and so on, but every time they've just mentioned some concepts without getting deeper, and this causes me a lot of confusion...


title: Matrix mechanics for those with wave mechanics background

text: Just curious: Is there any book or resource that teaches matrix mechanics (quantum mechanics) only without wave mechanics stuff - meaning that the book assumes wave mechanics background.


title: Cramer's rule, Origin of Quarks Fractional electric charge?

text: In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns. 2u+1d=1 1u+2d=0 $$a_1d+b_1u=c_1$$ $$a_2d+b_2u=c_2$$ $$u=\frac {c_1b_2-c_2b_1}{a_1b_2-a_2b_1}$$ $$d=\frac {a_1c_2-a_2c_1}{a_1b_2-a_2b_1}$$ u=+2/3 d=-1/3 there ain't no experiment that could be done, nor is there any observation that could be made, that would say, "You guys are wrong." The theory is safe, permanently safe. Is that a theory of physics or a philosophy? I ask you.


title: low frequency permittivity of metals from Drude's model

text: In reference to the Optical constants of noble metals: the Drude model for microwave modelling regarding Drude's model these parameters were listed $$ \omega_P=1.36\times 10^{16} \text{ rad/s} $$ $$ \gamma=1.05\times 10^{14} \text{ rad/s} $$ If I substitute these values into Drude's formula $$ ϵ(ω)=1−ω_p ^2 /(ω(ω+i/τ)) $$ then near dc $\omega$~0 the real part of $ \epsilon\ $ will be very negative $ Re(\epsilon(\omega ))$ ~ $-10^4$. What might this mean physically, how one would interpret the negative energy $W = (1/2) \epsilon\ E^2 $ of a piece of metal placed in electrostatic field? Or the answer is just that Drude's model breaks down at very low frequencies; if so, why, where and how?


title: Question about canonical transformation

text: I was going through my professor's notes about Canonical transformations. He states that a canonical transformation from $(q, p)$ to $(Q, P)$ is one that if which the original coordinates obey Hamilton's canonical equations then so do the transformed coordinates, albeit for a different Hamiltonian. He then considers, as an example the Hamiltonian $$H=\frac{1}{2}p^2,$$ with a transformation $$Q = q,$$ $$P = \sqrt{p} - \sqrt{q}.$$ The notes state that _this transformation is locally canonical with respect to $H$,_ and that in the transformed coordinates the new Hamiltonian is $$K = \frac{1}{3} \left( P + \sqrt{Q} \right)^3.$$ I don't understand how we know that this is locally canonical, or what it really even means to be _locally_ canonical. Also, where do we get K from. Since the inverse transformation would be $$q=Q,$$ $$p=\left( P + \sqrt{Q} \right)^2,$$ why isn't the new Hamiltonian $$K= \frac{1}{2} \left(P + \sqrt{Q} \right)^4,$$ where all I've done is plug the inverted transformation into the original Hamiltonian? I'm a bit confused by all this. Would appreciate any help.


title: Friction at zero temperature?

text: By the fluctuation-dissipation theorem (detailed-balance for Langevin equation), $$\sigma^2 = 2 \gamma k_B T$$ where $\sigma$ is the variance of noise, $\gamma$ is a friction coefficient, $k_B$ is Boltzmann's constant, and $T$ is temperature. So in principle, one can have $\gamma\neq 0$ while $T=0$ and $\sigma=0$. Is it indeed possible to experimentally achieve a system whose temperature and noise approach zero, but whose friction coefficient $\gamma$ does not approach zero?   * If yes, what would be an example of such a system? What is the physical source of friction for such a system?   * If not, why not? Is there some sort of "quantum" correction to the fluctuation-dissipation theorem that rules out such zero-noise, non-zero friction systems?


title: Phase space volume and relativity

text: Much of statistical mechanics is derived from Liouville's theorem, which can be stated as "the phase space volume occupied by an ensemble of isolated systems is conserved over time." (I'm mostly interested in classical systems for the moment.) It's clear that special relativity doesn't change this, since relativity just adds a different set of invariances to the Hamiltonian. So under special relativity, the phase space volume of an ensemble must remain constant over time, as long as we're consistent in using a given inertial reference frame. However, this doesn't tell us how the phase space volume would change if we boosted to a different reference frame. So I'd like to know the following:   1. Can anything useful be said about how the phase space volume occupied by an ensemble of systems changes under a Lorentz boost? I guess this would entail taking a different time-slice through the system, as explained below. I would suspect that phase space volume doesn't behave very nicely under Lorentz transforms, since phase space is defined in terms of 3-position and 3-momentum.   2. If phase space volume doesn't behave nicely under Lorentz boosts, is there a generalisation (e.g. in terms of 4-position and 4-momentum) that does? And if so, where can I go to read more about it?   3. Optional bonus questions: what effect do general relativity and quantum mechanics have on all of the above? **Additional clarification** Below are some diagrams which should make it clearer what I'm asking for. **(a)** shows a space-time diagram of a system consisting of several classical particles. We take a simultaneous time-slice though the system (red line) and note the 3-position and 3-momentum of each particle. This gives us a point in a phase space (diagram **(b)** ), which I've drawn as 2 dimensional for convenience. An ensemble of similar systems (meaning systems with the same dynamics but different initial conditions) can be thought of as occupying a region of this phase space. Liouville's theorem tells us that if we do the same thing at different values of $t$ (for the same ensemble of systems), the shape of this region may change but its volume will be the same. **(c)** shows a Lorentz-transformed version of the same system as (a). We can take a simultaneous time-slice through the system in the new $(x',t')$ reference frame, but it will not be parallel to the one in the $(x, t)$ frame. The particles' 3-positions and 3-momenta will also be different. We can plot the system's position in the new phase space generated by doing the same procedure in the new reference frame **(d)**. We can also plot the region of phase space occupied by the ensemble. Doing the same thing at different values of $t'$ will produce different regions with the same volume as each other. However, my question is about whether volume of the region plotted in (d) must equal the volume of the region in (b). ![enter image description here](http://i.stack.imgur.com/9wn0C.png) **Progress** In this document (J. Goodman, Topics in High-Energy Astrophysics, 2012, p.12-13; unfortunately Goodman gives no further reference) there is a proof that infinitesimal phase space volumes are Lorentz invariant. It looks legit, but the author assumes that every particle's $x$ position is within the same small interval $[x, x+dx]$, which means that he doesn't have to take account of the fact that you take a different time slice when you change the reference frame. Additionally, I'm using an integrated version of Liouville's theorem, in which the ensemble has a finite rather than infinitesimal phase space volume, and it isn't immediately clear to me whether this makes a difference. So this seems to suggest that the phase space volumes sketched in figures (b) and (d) above will be equal, but I'm still not sure and would like to know where I can find the full proof, if it exists. **Further Progress** I'm awarding a 100 point bounty to Qmechanic for his helpful answer, which expresses Goodman's argument (discussed above) in more formal language. However, I don't think the question has been answered yet. This is because the argument doesn't just assume that the system occupies an infinitesimal volume of phase space. On its own this would be fine, since you can just integrate over phase space to get the finite phase-space volume version. The problem is that this argument _also_ assumes the system occupies only an infinitesimal volume of real space. In general, a system that occupies an infinitesimal volume of phase space can occupy a finite area of actual space (think of a system composed of multiple, spatially separated, classical particles, as illustrated above) and so this argument seems to cover only a restricted subset of the cases I'm interested in. Goodman's argument is fine if you're considering non-interacting particles or fluids in equilibrium (which makes sense given its origins in astrophysics) but I'm also interested in multi- particle systems that may or may not be in thermal equilibrium. It is my strong intuition that the argument _can_ be extended to deal with all cases, and I'll happily award an additional 100 point bounty to anyone who can show how to do so. I think one route to an answer might be to note that, in special relativity, classical particles cannot interact unless they actually collide. This probably means that in the times in between collisions they can be thought of as occupying their own individual phase spaces after all, and perhaps with the aid of the proof that's been presented and some careful accounting about what happens during collisions, it will all work out OK. But of course then it has to be extended to deal with (classical) fields, which could be tricky. One gets the feeling there ought to be a nice simple way to derive it from Lorentz invariance of the Lagrangian, without worrying about what type of system we're dealing with. (I'll award a 200 point bounty to a satisfactory answer that takes this latter approach!)


title: Will an entangled idler electron induce a current in a conductor if the signal elctron's spin is measured?

text: I'm assuming a hypothetical setup as follows: Two labs (Alice and Bob) exist. Each has one electron of an entangled pair. At Alice, the electron travels through free space towards a magnetic field of fixed orientation. When it is sufficiently close to the field, it's dipole moment will align itself to match the magnetic filed, thus being measured. This will cause the electron at Bob to anticorrelate and assume the orthogonal dipole moment. If the electron at Bob was traveling through space sandwiched between two conductors of no net charge, will the electron induce a current (even a very tiny one) in the conductor(s) as it realizes its new spin? Please correct me as to any false assumptions in the setup I have described. Thanks


title: Supersymmetry algebra

text: * Let the $i=x,y,z$ components of the angular momentum have the commutation relations with the supersymmetry generators (also called supercharges?) $Q_a$ ($a = \pm \frac{1}{2}$) as, $[J_i , Q_a] = -\frac{1}{2} (\sigma _i )_{ab}Q_b$. Now if I want to calculate the $J^2$ eigenvalue of $Q_a$ then I might have thought of doing the following calculation using properties of Lie bracket, (using the summation convention) $[J_i^2,Q_a] = J_i[J_i,Q_a] + [J_i,Q_a]J_i = -\frac{1}{2} (\sigma_i)_{ab}\\{ J_i, Q_b \\} $ But this is apparently the wrong answer! The right answer is gotten by doing the following calculation where by the action of $J^2$ on $Q_a$ is defined as (using the summation convention) $[J_i,[J_i,Q_a]] = [J_i,-\frac{1}{2} (\sigma _i )_{ab}Q_b] = -\frac{1}{2} (\sigma _i )_{ab} [J_i, Q_b] = \frac{1}{4} (\sigma _i)_{ab} (\sigma _i)_{bc} Q_c = \frac{3}{4}Q_a$ I would like to know why is the above the correct action and not the former.   * But when lets say one wants to evaluate the action of a supercharge on the square of say the scalar component of a chiral superfield then the right action is of the former kind i.e, $[Q,\phi ^2] = \phi [Q,\phi] + [Q,\phi]\phi$ I don't understand why here what was "wrong" earlier is now the right thing.   * When the supercharge acts on the fermion say $\psi$ then in terms of the auxiliary field in the same multiplet (say $F$) the action as derived by Weinberg in his book is, $\\{ Q_b, \psi_a^*\\} =2i \delta _{ab}F^*$ Some people tell me that the above can't be right because the RHS does not have the same spinorial symmetry as the LHS. Also they say that the above implies that irrespective of the nature of $F$ the RHS will go to $0$ if $a\neq b$. But apparently there are theories with superpotential known where even if the spinor indices don't match the supercharge has non-trivial anti- commutation relation with the fermion. I can't see anything wrong about Weinberg's argument and nor do I see Weinberg using any assumption about the superpotential in deriving the above. I would like to know what is the right way to think about the above.   * If $A$ is the gauge field component of a vector superfield and the chiral superfield to which it couples has $\phi$ and $\psi$ as scalar and spinor components then the following seems to be true, $[Q,A] = [\phi, \psi]$ I would like to know if there is a general argument for the above and if it is always true. And also what is the space in which the above commutators on the both sides are defined.


title: How can I tell which end of a wire will have a higher potential?

text: I have the following setup:               A <---- wire ----> B      $V_b - V_a = \Delta v = \text{a positive value}$ I have two questions:   1. Which end of the wire has a higher potential?   2. What is the direction of the electric field inside the wire?


title: What is a continuous superselection sector?

text: I'm studying the terrible subject of continuous superselection rules and I faced with the following problem. Usually (continuous or discrete) superselection rules are defined involving a direct integral (with respect to some measure) of Hilbert spaces, each one to be interpreted as a "sector": an Hilbert space corresponding to the system for which some physical property (the one generating the superselection rules, e.g. the charge) is fixed. This interpretation is traditionally valid in the discrete case (direct sum) and I agree with it. But if the cardinality of the decomposition index set is the continuum one, I find some troubles trying to extend it. In general, indeed, a point of the index set can have null measure, hence the corresponding Hilbert space cannot be embedded in the global one, because the direct integral is defined with an "almost everywhere" construction. So what is a "sector" in this case? It seems to me that it would be more correct to consider again a direct sum, even if with a non countable index set. Now the sectors can be interpreted in the usual way, but the global space ceases to be separable. So my last question: could the Hilbert space associated with a physical system be non separable? Is there some physical (non mathematical) reason for the separability assumption in the quantum mechanics axioms?


title: Graphene and Klein bottle?

text: I am trying to understand graphene as a topological insulator. The spin orbital interaction in graphene is very small (~10mK?). But if we consider that, then graphene should be a topological insulator. And at the edge electrons should move in opposite direction according to their spin as shown in the figure for graphene flake with zigzag edges! ![enter image description here](http://i.stack.imgur.com/UM7sq.png) Then the map from Brillioun zone to the energy space should be nontrival. The energy space should be topologically equal to a Klein bottle instead of sphere. Is it the right picture? update1: Thanks for the comments. Indeed the question is not well defined and I will try to refine it a bit as further as I could. If you glue the 2D Brillouin zone properly you will get a torus. And as I understood there are two cases shown in figure below that create a gap in the 'bulk' graphene. Case 1 show two valleys without gap. Case 2 is the normal insulator case. Case 3 is the TI case. ![enter image description here](http://i.stack.imgur.com/h044h.png) For the energy space I mean the band structure (I don't know its relation with Hilbert space.) Not sure about how to glue it and if a Klein bottle will be formed. update2: Just realized that last update without considering the spin. If we do that, then the momentum space should be 2 torus each for one spin instead of one torus. Then the problem is how to glue (topologically map) the band structure on that. update3: Just got some ways of mapping 2 tori to a Klein bottle. (shown below, sorry for the bad drawing) ![enter image description here](http://i.stack.imgur.com/pMdoZ.png) The cover of the book shows how to map a torus to a sphere. Is this the way to understand mapping from momentum space to band structure (or topology of the system)? update4: Well, the next question is how this relates with Berry flux and Chern number. The band structure I mentioned above should be $<\Psi(k)|H|\Psi(k)>$. And here $H$ is proportional to $\nabla_k$. Berry flux ~ $\nabla_k\times <\Psi(k)|\nabla_k|\Psi(k)>$. With it one can calculate the Chern number. Is the right way to link those two pictures? How do this relate with critical points in 2D vector field? **update after 4tnemele's nice answer** Thanks for the answer! I must say it is a very nice one. But may be it is still worth to clear why I asked this question. I am not in the TI field and have no strong background in condensed matter physics, algebra topology, differential geometry etc. Once I read that graphene is a TI. I just cannot **see** how to construct the calculations of those topological invariants from what I knew at that time. But I have to start from somewhere and do it in 'my own way'. I intuitively think that the 'energy space' could be a 2D non-orientational close surface. The only thing come into view is a Klein bottle. I thought it could related with the picture below. If you glue fiber on a torus, you can have no critical point. But it shall be different for the non- orientational surface case (here I call it a Klein bottle.) I thought those fiber could be related with Berry phase. Then I reached the 'energy space', I knew it might be somewhere in between the links. I don't know how to imagine the shape of related Hilbert space. ![enter image description here](http://i.stack.imgur.com/S2dbP.png) I hoped I can find a very intuitive way of 'seeing' how to construct the calculation. But now I think I need to read a bit more and it may be other easier ways (like begin from equations) to understand the problem. And I think that to clear the background of the question will help other people to answer it. It is a lesson to me. Thanks a lot.


title: Mathematical Formulation for Accelerating Charges Generating Light?

text: I haven't been formally taught the fact that accelerating charges generate light, and I have seen the picture of an accelerating charge sending ripples down its field lines, but is there a mathematical formulation someone can provide that show that an acceleration on a charge produces E&M waves?


title: Stokes-Einstein's formula results in incorrect units for rotational drag coefficient

text: The Stokes-Einstein-Sutherland relationship, $$D = \frac{kT}{ 6 \pi \eta a}$$ where $D$ is the translational diffusivity is well known. A similar relationship is used to calculate the rotational diffusivity, specifically, $$D_{rot} = \frac{kT}{8 \pi \eta a^3},$$ where $a$ is the radius of the object and $\eta$ is the viscosity of the medium. However, the drag coefficient in the denominator, $(8\pi\eta a^3)$ has units, $Js/rad$. But $D_{rot}$ has the units $rad^2/s$. Now $kT$ is in Joules, so does anyone know how this can be resolved?


title: Calculating the electric potential in cylindrical coordinates from constant E-field

text: I am having so much trouble with this problem. I feel like I _shouldn't_ be, but I am. > _A uniform electric field, $\vec{E} = E_0\hat{x}$. What is the potential, > expressed using cylindrical coordinates, $V(s,\phi,z)$?_ I asked a question related to this yesterday, and got a solid answer, but when I carry things out, it still doesn't make sense physically (though, after a bit of studying, I now understand the Gradient Theorem). We know that: $$\vec{E} = E_0\hat{x} = -\nabla V \implies -\int_{C[x_0,x]} E_0\hat{x}\cdot dl = V(x) - V(x_0)$$ Since the electric field only has quantity in the $\hat{x}$ direction, the dot product inside of the integral comes out to $E_0~dx$, giving us: $$E_0(x_0 - x) = V(x) - V(x_0)$$ Now, this makes sense. Since the electric field is a conservative one, the path we take shouldn't depend on any other variable but $x$. If we move in some $y$ direction and some $z$ direction, there will be zero change in potential. Let $V(x_0) = 0$ be our point of reference, so: $$V(x) = E_0(x_0 - x)$$ So far (I think) this all seems fine and dandy. The reason I don't set $E_0x_0$ to zero is because there is no $1/x$ term; otherwise I'd be able to make the reference point at $x=\infty$, and things would cancel out nicely. Now, I try to convert to cylindrical coordinates. We know: $$r = \sqrt{x^2 + y^2 + z^2} = \sqrt{x^2} = x \\\ \theta = \tan^{-1}{\frac{y}{x}} = 0 \\\ z = 0 $$ So $V(r) = E_0(r_0 - r)$ But... this doesn't make any sense. With this potential function, moving $r$-distance in the $-x$ direction will give you the same potential as moving $r$-distance in the positive $x$ direction (which will give you the same potential as moving $r$-distance in the $y$ direction, even, since $r$ is just a radial distance from, say, the $z$-axis. I am doing something terribly wrong and I have no idea what it is, haha.


title: Density as a function of Temperature?

text: I know that volume = mass/density....and density is the function of temperature. Is there any standard equation that describes density as a function of temperature? The system I am interested in is like a liquid tank or reservoir


title: Actions at a distance vs. contact interaction

text: The ancestors could not imagine an action at a distance (in German: "Fernwirkung"). Today physicists don't take serious its _opposite_ anymore (in German: "Kontaktwechselwirkung"). So my first question is: > What is/was the official _english_ name of the opposite of an "action at a > distance"?   >  Something like "contact interaction"? Next to terminology there is the matter itself: The prototypical _repelling_ "contact interaction" has always been collision. > But has there ever been a prototypical _attracting_ "contact interaction"?   > Something like agglutination? What have been prototypical examples, eventually?


title: Would dropping large blocks of ice into the ocean mitigate global warming?

text: This amusing Futurama sketch 'solves' global warming by dropping increasingly larger blocks of ice into the ocean. Presumably the blocks of ice are harvested from a different planet. Would this theoretically be effective? My understanding is that by global warming, we mean an increased amount of energy in Earth's atmosphere. Adding ice to the planet wouldn't remove energy from the planet, it would just add matter to the planet. However, possibly you could argue that energy is transferred from the atmosphere to the hydrosphere. However, even if that's the case, would that constitute a reduction in global warming?


title: What is the relationship between Maxwell–Boltzmann statistics and the grand canonical ensemble?

text: In the grand canonical ensemble one derives the expectation value $\langle \hat n_r\rangle^{\pm}$ for fermions and bosons of sort $r$: $$ \langle \hat n_r\rangle^{\pm} \ \propto \ \frac{1}{\mathrm{exp}[(\varepsilon_r-\mu)/k_B T] \mp 1} . $$ For $(\varepsilon_r-\mu) / k_B T\gg 0$, we find $$ \langle \hat n_r\rangle^{\pm} \ \approx \ \frac{1}{\mathrm{exp}[(\varepsilon_r-\mu)/k_B T]} \ \propto \mathrm{exp}[-(\varepsilon_r-\mu)/k_B T].$$ The same motivation seems to be found in this Wikipedia article. However, on the same page, right at the beginning, that intuitive statement is made: > In statistical mechanics, Maxwell–Boltzmann statistics describes the average > distribution of non-interacting material particles over various energy > states in thermal equilibrium, and is applicable when the temperature is > high enough or the particle density is low enough to render quantum effects > negligible. **Now from my derivation above, it seems that " _temperature is high enough_ " does the opposite of helping $(\varepsilon_r-\mu) / k_B T\gg 0$ to be fulfilled. What is going on?**


title: Does classical axial current conservation prevent the spontaneous emission of a photon by an electron?

text: I understand the axial current conservation (in the massless limit) in classial field theory to mean that the difference between right handed and left handed electrons is conserved in QED. $$j^{\mu 5}~:=~\overline{\psi}\gamma^{\mu}\gamma^5\psi~=~j^{\mu}_R-j^{\mu}_L,\qquad j^{\mu}_{R,L}~:=~ \overline{\psi}_{R,L}\gamma^{\mu}\psi_{R,L}, $$ How does it relate to spontaneous photon emission ? When an excited electron emits a photon, its spin flips. This would mean that the axial charge for fermions changes. Is this process supressed for massless electrons ?


title: Conventions for Klein factors in bosonization of Quantum Hall edge states

text: I am not having much experience in the field of bosonization, hence the following question: In some papers (such as http://arxiv.org/pdf/cond-mat/9501007.pdf Eq. (6) ) a Quantum Hall edge is bosonized by introducing the equal time commutation relations $ \left[ \Phi_{i}(x) , \Phi_{j}(x')\right] = i \pi \left( K_{ij} sgn(x-x') + L_{ij}\right) $ with $K_{ij}$ being the $n\times n$ K-matrix describing the Quantum Hall phase and $L_{ij}=sgn(i-j)\left(K_{ij}-Q_{i}Q_{j}\right)$ being the Klein factors and $Q_{i}$ being some n-dimensinal charge vector. Still other papers (such as http://arxiv.org/pdf/1106.3989v3.pdf Eq.(A2) ) use $ \left[ \Phi_{i}(x) , \Phi_{j}(x')\right] = -i \pi \left( K_{ij}^{-1} sgn(x-x') + \theta_{ij}\right) $ with $\theta_{ij}=K_{ik}^{-1}L_{kl}K_{lj}^{-1}$ and $L_{ij}$ as before. Now there seems to be both a change in an overall minus-sign as well as a "conjugation" of the Klein factors by the inverse K-matrix. I am wondering why these two descriptions are equivalent?


title: What if a probably non wave equation, satisfies the wave equation?

text: What does it mean for a function to satisfy the wave equation? Does it necessarily mean that it is a wave equation? Take for instance the function $f(t,z)=(at+b z)^2$. This function satisfies the wave equation. Does this mean that it could describe a wave?


title: What's the difference between energy and temperature in field theory?

text: I'm familiar with the formalisms for both zero temperature and finite temperature field theory, but (somewhat embarrassingly) I don't actually have a good physical intuition for when physical scenarios call for one set of methods vs. the other. For instance, at say the LHC it's my impression that despite the huge energies involved during collisions, zero temperature field theory is appropriate for all relevant calculations. One the other hand (I could certainly be wrong here, again), it's my impression that heavy ion collisions such as those performed at RHIC are suited to finite temperature calculations instead. So my questions are as follows: 1) Generically, how does one know when to use finite vs. zero temperature field theory? 2) Can one a priori determine whether finite temperature corrections will be important? 3) If my impression are correct about the LHC and RHIC, is there a quick, back-of-the-envelope calculation which demonstrates that zero temperature and finite temperature field theory should be used in these two scenarios, respectively?


title: Freeze and "break apart" an object. How?

text: Is it possible to freeze an object, say an apple, and then drop it so that is will break apart into pieces? And if so, how can it be achieved?


title: Why does the monitor make a cracking noise?

text: After switching off the monitor, there's a single crack after a while. I wonder where exactly it comes from. I know that this is normal and not an indicator of being defect. This sound is also not related to a specific type of monitor. I've heard this sound from my CRT as well as from my TFT. I guess it possibly has to do with cooling down. I found a lot of statements that second this thought but there's no reliable source confirming this theory though. However, an explanation that only says that it has to do with the cooling process is not sufficient at all. I'm interested in what exactly makes this sound? Is it, for example, a capacitor which has fully unloaded (if this is the proper term). I really haven't the slightest inkling what exactly can make this crack noise.


title: Basic question about probability and measurements

text: Say I have a Galton box, i.e. a ball dropping on a row of solid bodies. Now I want to calculate the probability distribution of the movement of the ball based on the properties of the body (case A). For instance if I change the position of the ball the distribution might change (case B). I want to know the distribution of the ball after it has hit the body, as a function of various properties. Does it even make sense to speak of a probability distribution here? Basically what I had in mind before, is that quantum mechanics is probabilistic and that classical mechanics is deterministic. Does this mean one can actually calculate where each ball will end up, if the measurements are precise enough? ![mech](http://i.stack.imgur.com/8xL3M.png)


title: How intense a magnetic field would it take to keep an hypothetical iron-made moon orbiting around it?

text: The intention of the question is to provide an example of the weakness of gravity. I imagine a horseshoe magnet located at the Earth's centre (remove the Earth), and a ferromagnetic moon. How intense would the magnetic field need to be to keep such a moon in orbit at the same distance that the moon is from the Earth?


title: Electric engine transmission

text: If we compare a petrol engine and an electric engine car, it is sufficient for the electric engine car (e.g _Tesla Roadster_ ) just to have 1 gear (forward), whereas petrol engine can have up to 7. Why is it sufficient for the electric engine to have a single gear? Is it because it can deliver the required torque always?


title: Can long strings always snap?

text: In quantum chromodynamics, long flux tubes will always snap because a quark- antiquark pair gets created from the vacuum, and hadronization results with a quark attached to each new end. In string theory, if we try to stretch a single string out a long distance, will it also snap? If open strings with Neumann boundary conditions are allowed, this can always happen. Suppose this is not the case. If D0-branes exist, we can create a pair of them from the vacuum, and the string will still snap. Even if no D0-branes don't exist, or space filling branes either, as long as we have some Dp-brane for some p in between, the Dp-brane can be wrapped up as finite closed bubble, and we can create a pair of such bubbles. The string will still snap into "tadpoles". In some string theories, strings carry charges and are BPS states. Can they snap? I doubt it.


title: Axion related questions

text: I have several question regarding axion. Could anyone give me some brief introduction to what is a axion string, axion field and how is this related to fermion zero mode and chiral zero mode?


title: Is there a mathematical way to describe how a flame flickers?

text: I love the way candles flicker. It's a great effect and I almost want to see it replicated in an actual lightbulb. I was curious if there is any way to express that mathematically? I'm not that familiar with physics but I go in assuming that most things can be expressed in math, just very complex in some cases!


title: Possible mechanics based on the known symmetries in the nature (investigating rumor)

text: Somewhere I've heard about a relative new mathematical result regarding mechanics. Specifically, there is a list of the known symmetries of mechanics (both Newtonian and relativistic), i.e. different invariances such as rotation and translation symmetries. These symmetries specify the possible laws of mechanics. There is a relative new (maybe as young as 10-20 years) result, which shows, that based on the group of the currently known symmetries of Nature, only two possible mechanics can exist: the Newtonian and relativistic. As I understand, it is only a rumor. Is it true? Where can I read more about this?


title: 2D CFTs and permutation orbifolds

text: Suppose we have 2 systems with the same partition function, does this mean the 2 systems are the same? For example, in 2D CFTs, would the equality of two partition functions imply that the underlying theories are the same (in the CFT sense, I mean same central charge, same OPE, etc). Suppose we take the $\text{N}^{th}$ symmetric product of a mother CFT with a partition function $Z(\tau,\bar{\tau})$ and then I orbifold by the permutation group $S_N$ or any cyclic subgroup $\mathbb{Z}_N$ to get the partition function of the permutation orbifold $\mathbf{Z}$. Now suppose we find another system with the same partition function $\mathbf{Z}$, does this mean that this system should be equivalent to the permutation orbifold? Please feel free to edit or correct my question. Thank you all in advance!


title: How irreversible processes are possible?

text: Susskind says that all laws of mechanics are reversible and any valid mechanic law most be reversible: you can always determine the previous state of any physically valid system. However, the simplest operations in computation, _reset_ as well as the binary AND and OR operators, are irreversible. You may say that "this is only a virtual simulation" and everything is reversible in reality. However, because of the dissipation, almost all processes in the world are irreversible. So, despite irreversibility is completely forbidden, everything seems to be irreversible and entropy only increases. I want to understand how this contradiction is resolved (at least in one case). I also do not understand why do you represent a system configuration space by a box. You say that the state is a point in that box and its trajectory represents the state evolution. However, reversibility requirement demands that the trajectories do not intersect. This means there must be only one trajectory. So, you should rearrange all the dots of the configuration box into a line (it might be a segmented into unconnected circles). However, that might be another question.


title: Are there moments in particle collision experiments where the particle beam is in open air?

text: I keep encountering the story of Anatoli Bugorski, who apparently got hit in the head by a proton beam at the U-70 PS in Protvino, Russia. I find it difficult to believe that this is actually possible, because I've always been told that particle accelerators (from CRT's to the LHC) require a vacuum to prevent beam loss and defocussing and whatnot. How do you get your head in the path of a particle beam if it travels in a vacuum tube? Or, is a vacuum always needed if you want to accelerate particles and do scattering experiments? More concrete, is this incident possible? Was the U-70 operating without vacuum?


title: Do objects in free-fall contribute to earth's weight as a whole still?

text: If everything in earth's atmosphere contributes to earth's overall weight, including air etc. Does an objects weight in free-fall effect this overall weight in anyway. Just say the weight of earth was 10kg, and consisted of 8kg earth mass, 1kg air and 1kg as an object. If that 1kg was in free-fall, what effect will this have on the 10kg weight of the planet. I understand that this can not be replicated in any other environment as as soon as the object stops exerting downward force on the object it's inside, it no longer contributes to the containing objects weight. But how does this work with our entire planet, which creates the gravitational pull that defines that very rule? Or is this much simpler than I am imagining?


title: Deformation of the cylinder

text: There is given a cylinder with fixed side surface. On one base is being rendered constant pressure $p$, so cylinder cambers ![enter image description here](http://i.stack.imgur.com/ipfx9.png) I'm trying to calculate the form of cambering. The problem is that height of the cylinder can't be neglected, so one can't use results for mambranes from elasticy theory. My approach was in proposing that this form will be close to the form of cambered parallelepiped in simillar conditions rotated around z axis: ![enter image description here](http://i.stack.imgur.com/aZzAU.png) Even if this proposition is right, I didn't succeed in calculations for parallelepiped, so I'm asking help with this problem.


title: How is this classical "paradox" resolved in electromagnetism?

text: A magnet and a coil move relative to each other. In the frame of reference of the magnet, there is a magnetic field and consequently a force acting on the charges in the coil according to the Lorentz force $F=qv\times B$ but there is no net electric field. In the frame of reference of the coil, there is a magnetic field and also an electric field, induced by the magnet, $E'$ that moves the charges in the coil, producing a current. But, in the first case no work is done on the charges, since the force is perpendicular to the velocity. In the second case, the force $qE'$ does work on the charges. How is this "paradox" resolved in classical electromagnetism?


title: Is $f=ma$ an identity?

text: In his The Principles of Natural Knowledge, Alfred North Whitehead writes that famous $f=ma$ is an identity: > It has been popular to define force as the product of mass and acceleration. > The difficulty to be faced with this definition is that the familiar > equation of elementary dynamics, namely, > > $mf = P$ > > now becomes > > $mf = mf$ > > It is not easy to understand how an important science can issue from such > premises. How does he obtain the identity?


title: What were the maximal collision energy of the LHC if it were rewired to $e^{-}-e^{+}$ collisions?

text: Yes, I know, that it weren't a large _hadron_ collider then. No, I don't think, that the creators of the LHC didn't know what they do. I am just curious. * * * How could it be calculated? A such rewiring of the LHC (also, making it capable to work with electron-positron pairs) weren't a little bit cheaper as the now planned next linear collider?


title: Could strings be geons?

text: * Is it possible that string theory strings are geons? This may be an overly speculative or naive question, but is there an obvious reason why not? Both strings and geons seem to have roughly the same length scale (comparing the rq from here to thePlanck length).


title: A Spin up particle in QFT

text: This appears like a question that is rarely addressed in field theory pedagogy (perhaps because the answer is obvious): **how does one describe a particle of definite spin in quantum field theory?** For example, given some state in a theory of spinors (say a single particle for simplicity): $|\psi\rangle = \int \psi(p) a^\dagger(p)|p\rangle$, where $a$ is the ladder operator you obtain by canonically quantizing, say a Dirac field. Since this is a Dirac field, it describes a particle of spin 1/2. **How does one extract information about the spin of this particle?**. For example, what would the creation operator for a single particle of definite spin up look like?


title: Why does the earth have to feel a pull when something falls?

text: Heck, I'm not even worried about the speed of a reaction. But remember that if I fall towards the earth, with a force, the earth has the same force exerted upon it in the opposite direction. I was thinking, after reading a comment on a previous gravity question, that gravity should be propagated by particles (this would explain why you instantly feel gravity, its like entering a stream of particles). This would also explain why it would take time to feel a change in the gravitational field. If the change propagates at c, that would explain how similar gravitons are to light. Anyway, if I absorb gravitons, why does the planet care? It acts as though it absorbed equal and opposite gravitons in terms of its resultant force experience. Do we state that a disturbance in the field is sensed, rather like a pressure buildup? I imagine in analogy a water buildup when I stand in the shower propagating backwards up the shower head. It seems as though disturbances in the field would explain it, implying virtual gravitons in an analogy to virtual photons perhaps. Feel free to ignore that if I'm mistreating the idea of virtual photons or whatever-I have heard them described as field disturbances though. To further clarify, an analogy: suppose I am a rotating sprinkler at high frequency. As I throw down water all about me, a post in the garden "absorbs" the water I throw at it. I, however, do not react at all, just the water. However, according to what we observe, a planet reacts to a person's weight, and this in my perspective can only be answered if the planet "senses" when its gravitons have been absorbed and tries to fill in the gap, throwing in more gravitons than before. Since more gravitons are emitted forward, this would explain why the earth feels a forward push, given that gravitons carry negative energy. (Of course they have to have negative energy, otherwise when you get hit by one it would push you away). This seems more like a fluid force if the fluid moves from regions of low to high density, filling in the gap from absorbant things, in this case all mass- containing objects. Does this sound ok, or am I totally wrong on a point of importance?


title: Lorentz group representations in QFT: what's the vector space?

text: In QFT, a representation of the Lorentz group is specified as follows: $$ U^\dagger(\Lambda)\phi(x) U(\Lambda)= R(\Lambda)~\phi(\Lambda^{-1}x) $$ Where $\Lambda$ is an element of the Lorentz group, $\phi(x)$ is a quantum field with possibly many components, $U$ is unitary, and $R$ an element in a representation of the Lorentz group. We know that a representation is a map from a Lie group on to the group of linear operators on some vector space. **My question is, for the representation specified as above, what is the vector space that the representation acts on?** Naively it may look like this representation act on the set of field operators, for $R$ maps some operator $\phi(x)$ to some other field operator $\phi(\Lambda^{-1}x)$, and if we loosely define field operators as things you get from canonically quantizing classical fields, we can possibly convince ourselves that this is indeed a vector space. But then we recall that the dimension of a representation is simply the dimension of the space that it acts on. This means if we take $R$ to be in the $(1,1)$ singlet representation, this is a rep of dimension 1, hence its target space is of dimension 1. Then if we take the target space to the space of fields, this means $\phi(x)$ and $\phi(\Lambda x)$ are related by linear factors, which I am certainly not convinced of. _EDIT: This can work if we view the set of all $\phi$ as a field over which we define the vector space, see the added section below._ I guess another way to state the question is the following: we all know that scalar fields and vector fields in QFT get their names from the fact that under Lorentz transformations, scalars transform as scalars, and vectors transform like vectors. **I would like to state the statement "a scalar field transforms like a scalar" by precisely describing the target vector space of a scalar representation of the Lorentz group, how can this be done?** _ADDED SECTION_ : Let me give an explicit example of what I'm trying to get at: Let's take the left handed spinor representation, $(2,1)$. This is a 2 dimensional representation. We know that acts on things like $(\phi_1,\phi_2)$. Let's call the space consisting of things of the form $(\phi_1,\phi_2)$ $V$. Is $V$ 2 dimensional? Viewed as a classical field theory, yes, because each $\phi_i$ is just a scalar. As a quantum field theory, each $\phi_i$ is an operator. We see that in order for $V$ space to be 2 dimensional after quantization, we need to able to view the scalar quantum fields as scalar multipliers of vectors in $V$. i.e. we need to view $V$ as a vector space defined over a (mathematical) field of (quantum) fields. We therefore have to check whether the set of (quantum) fields satisfy (mathematical) field axioms. Can someone check this? Commutativity seem to hold if we, as in quantum mechanics, take fields and their complex conjugates to live in adjoint vector spaces, rather than the same one. Checking for closure under multiplication would require some axiomatic definition of what a quantum field is.


title: Why point mass leaves it's own gravitational well?

text: I suppose that point mass has its own gravitational well. Why this point mass is still attracted by other gravitational waves. I expect that this point mass is surrounded by "gravitational hills". Why it is not blind to other weaker gravitational waves? If we think about gravity in terms of shape of spacetime. I mean that point mass (left chart) creates "sharp" gravity potential. Lowest gravity potential is exactly where point mass is located. Point stays in place (?). When mass is spread across spacetime (right chart) this is different. Lowest gravity potential is moved from the centre of the object. Object accelerates. Does point mass exist in real world? ![enter image description here](http://i.stack.imgur.com/A5RFM.png)


title: Compute and plot satellite position on 2D Earth map

text: I would like to create a simple satellite tracking software, that will compute the position of a satellite at any time, and show its location on a 2D map of Earth. Let's assume it is possible to find online position of a satellite in the Earth-centered inertial (ECI) coordinate frame ( in form of TLE for example ) and use some method to propagate its state to a moment in future. Then if I want to plot the position on Earth map, I will need to know orientation of Earth at a given moment in relation to the original ECI frame, probably with the help of ECEF (Earth-Centered, Earth-Fixed). And here come the questions, how do I :   * transform from ECI to ECEF    * take in account Earth precession and nutation   * do the proper time conversion, to also involve leap years, leap seconds, added seconds...  **EDIT :** During my online searches, I came across this article. It seems to be describing what I am looking for. However it uses a lot of terminology that I am unfamiliar with. It also doesn't tell a lot about precision of this technique and where the equations come from ( I need to be confident that I am implementing the right procedures ). I would still like to address all of the issues I listed in the question above.


title: Paths in the path integral

text: In the path integral approach one defines in some heuristic way the functional path integral \begin{equation} Z=\int{\cal{D}}\phi e^{iS(\phi)} \end{equation} and the one claims that one must integrate over all paths. My question is: Is the function space where one integrates the classical configuration space of the theory? If this is the case what is the formal definition of the configuration space of a field theory (so far I have seen it only defined as the space of all possible physical states)? Is it defined as the zero section of the phase space? Notice that to define the phase space we can define it through functional analysis. We can see the physical system as a PDE and define the phase space as the space of all initial data. EDIT: I am interested in the relevance or irrelevance of the function space for the configuration space. For example, in a globally hyperbolic spacetime with compact initial cauchy surface $\Sigma$ one can have well-posed problems for the scalar field, $\phi$ with initial data in $H^{1}(\Sigma)\times H^{0}(\Sigma)$. However one can also prove that the problem is well-posed for initial data in $H^{3}(\Sigma)\times H^{3}(\Sigma)$. TThese two spaces define two different sets of functions and two different phase spaces. Does the path integral give the same results for both cases or is the choose of a suitable function space at the classical level an ambiguity of the path integral approach?


title: What properties does the conductor making the plates for Casimir effect have?

text: Reading this explanation, I've understood that the divergence in computation of Casimir force on two parallel conducting plates is because of an unphysical model of ideal conductor, which makes EM field vanish on its boundaries. This looks much like unphysical model of rigid body in relativistic mechanics. Still, zeta regularization appears to somehow correctly remove the divergence and give sensible results. This must mean that effectively, it replaces our "too ideal" conductor with a "moderately ideal" one. So, the boundary conditions for EM field are no longer as simple as zero. So, my question is now: what boundary conditions does zeta-regularization implicitly impose on the EM field instead of the original ones? Does such change of boundary conditions in fact mean that the force will depend on which conductor the plates are made of?


title: What is the necessary condition on EM field for vanishing Poynting vector?

text: It is well known fact that both static and dynamic EM fields can give $E\times B \not=0$. What can we say about EM field if $E\times B =0$ ? Can previous equation still hold in case of dynamic EM field ? Thank you for help.


title: Is $4 \pi G$ the true most fundamental gravitational constant?

text: Newton's law of gravitation is: $$F = G m_1 m_2 \frac{1}{r^2}$$ It looks simple and natural. But that's only in 3 dimensions. Let's look what happens in $n$ dimensions: $$n=2 : F = 2 G m_1 m_2 \frac{1}{r}$$ $$n=4 : F = \frac{2}{\pi} G m_1 m_2 \frac{1}{r^3}$$ $$n=5 : F = \frac{3}{2 \pi^2} G m_1 m_2 \frac{1}{r^4}$$ $$n=6 : F = \frac{4}{\pi^2} G m_1 m_2 \frac{1}{r^5}$$ Oh no! Newton's force law becomes cluttered with unintuitive constants! But by defining $G^* = 4 \pi G$ Newton's law of gravitation can be reformulated as such: $$F = G^* m_1 m_2 \frac{1}{4 \pi r^2}$$ Immediately we recognize that $4 \pi r^2$ is simply the surface area of a sphere of radius $r$. But that's only in 3 dimensions. Let's look what happens in $n$ dimensions: $$n=2 : F = G^* m_1 m_2 \frac{1}{2 \pi r}$$ $$n=4 : F = G^* m_1 m_2 \frac{1}{2 \pi^2 r^3}$$ $$n=5 : F = G^* m_1 m_2 \frac{1}{\frac{8}{3} \pi^2 r^4}$$ $$n=6 : F = G^* m_1 m_2 \frac{1}{\pi^3 r^5}$$ $2 \pi r$ is the surface area of a 2 dimensional sphere of radius $r$. $2 \pi^2 r^3$ is the surface area of a 4 dimensional sphere of radius $r$. $\frac{8}{3} \pi^2 r^4$ is the surface area of a 5 dimensional sphere of radius $r$. $\pi^3 r^5$ is the surface area of a 6 dimensional sphere of radius $r$. Newton's law of gravitation in $n$ dimensions is: $$F = G^* m_1 m_2 \frac{1}{S_n}$$ Where $S_n$ is simply the surface area of a $n$ dimensional sphere of radius $r$. From this, it seems like $G^*$ would be a nicer definition for the gravitational constant.


title: Is the electromagnetic spectrum discrete?

text: I'm just starting to learn physics and I have a question (that is probably stupid.) I learned that energy levels that the bound electron can have are discrete. I also learned that when an electron transitions from one level to another, a photon with a specific wavelength (energy) is created and released into the wild. My question is this: are there holes in the electromagnetic spectrum, values of frequency that the created photon can't possibly have?


title: Why are hydrogen energy levels degenerate in $\ell$ and $m$?

text: Is there a good physical picture of why the energy levels in a hydrogen atom are independent of the angular momentum quantum number $\ell$ and $m$?


title: What Problems for 50 year space probe to Alpha Centauri?

text: I'm thinking speculatively. NASA is doing experiment with ion thrust tech. Assuming, that inside the next 5 years, we could create a probe which would take on the order of 30 to 50 years to reach Alpha Centauri: What problems would it have? ie: Is the biggest challenge going to be communications back to Earth? Maybe a small probe can't transmit a strong enough signal (especially since we'd be essentially looking straight at the star, from our point of view) ? Or Maybe the challenge in transmitting is really the targetting, to keep the Antenna pinpointed on Earth? Or would there be problems with powering the probe over a 50 year journey (at which point we'd really want it to have several years left of usable lifetime) ? Or is there problems with space probes, outgassing or other gradual losses which would cause the device to decay after 50 years of vaccum? Do the long-lasting Voyager probes help answer any of these issues? I know they are MUCH closer to earth than what I'm talking about, but at least they are still alive and transmitting... This is a speculative question, so I hope it is allowed by the forum rules. I'd like to think about how such problems can be overcome -- but first I really need to know what the problems might be. So I'm looking for thoughts or info... EDIT: perhaps I was optimistic. if a 50 yrs travel time is not practical, even with a long duration thrust like ion drive, then feel free to posit a longer flight time.


title: Why does the conductivity $\sigma$ decrease with the temperature $T$ in a semi-conductor?

text: We performed an undergrad experiment where we looked at the resistance $\rho$ and Hall constant $R_\text H$ of a doped **InAs** semiconductor with the **van der Pauw method**. Then we cooled it down to around 40 K and did temperature dependent measurements up to around 270 K. We were asked to create the following three plots from our measurements and interprete them. This is conductivity $\sigma = 1 / \rho$ versus the inverse Temperatur $T^{-1}$. I see that increasing the temperature (to the left) decreases the conductivity. I do understand that higher temperatures do that since the electrons (or holes) have more resistance due to phonon scattering. However, since higher temperatures mean a higher amount of free electrons, I would think that $\sigma$ should go up, not down. ![](http://chaos.stw-bonn.de/users/mu/uploads/2013-12-07/plot1.png) The densisty of holes $p = 1/(e R_\text H)$ does increase with the temperature, that is what I would expect: ![](http://chaos.stw-bonn.de/users/mu/uploads/2013-12-07/plot2.png) And the electron mobility $\mu = \sigma R_\text H$ decreases with the temperature as well: ![](http://chaos.stw-bonn.de/users/mu/uploads/2013-12-07/plot3.png) Now, I am little surprised that even though $p$ goes up with $T$, $\mu$ and $\sigma$ go down with $T$. Are the effects of phonon scattering and other things that increase the resistance that strong?


title: Is it correct to say "like poles attract, unlike poles repel" while two magnets are placed such that one is inside another?

text: As we know a solenoid is considered as a electromagnet(magnet) if there's a current flowing through it. if a soft iron core is placed inside the solenoid, the former get magnetised. Consider the solenoid as a hollow bar magnet, and the magnetised soft iron core inside as a bar magnet. The poles of the magnets next to each other is of same polarity. Is there attracting force between the two magnets? If so, does it mean "like poles attract, unlike poles repel" while two magnets are placed such that one is inside another?


title: How can such a high exponent arise in this physics equation?

text: This site http://what-if.xkcd.com/14/ states that during a helium flash, "the reaction rate is proportional to the 40th power of the temperature". Taking for granted that this is true, how can such a large exponent arise in a physic equation? It just makes no sense to me that some physical process could give rise to this phenomenon. An explanation of how it can arise for a helium flash would be satisfactory, but the question is really more general. How can a physical process accumulate enough factors to get up to x^40? Thermodynamic processes tend to be in the x^2 or x^3's and geometry can only add a few more x^3's. Note that we are not talking about the scale of things (obviously the 40th power is not a big deal in the length or time scales), we are talking about a physics /equation/.


title: How can the Air friction force be applied to an object?

text: Suppose we have an object and we throw it straight upward in the air. How do we apply the Air friction force to this object while moving upwards and after that downwards? Sorry if it's easy because I'm still a novice!


title: Why can you "suck in" cooked spaghetti?

text: We all know that there is no "sucking", only pushing. So how are cooked spaghetti pushed into your mouth? The air pressure applies orthogonal on the spaghetti surface. Where does the component directed into your mouth come from?


title: Calculating the heat capacity of a system

text: I have started reading _Statistical Physics_ by F. Mandl and I would appreciate some help with the following exercise > A system consists of $N$ weakly interacting subsystems. Each subsystem > possesses only two energy levels $E_1$ and $E_2$, each of them non- > degenerate. > > Obtain an exact expression for the heat capacity of the system. Alright, I haven't got a clue. Let's start with definitions. The heat capacity is given by $$C = \frac{\partial E}{\partial T}$$ with $E$ the mean energy, $T$ the temperature. Then we also have $$\frac1T = \frac{\partial S}{\partial E}$$ with the entropy $S = k\ \ln \Omega(E)$. Now, for a given energy $E$, there is a (unique) number $n \le N$ with $E=n E_1 + (N-n)E_2$. And with this $n$ we get $$ \Omega(E) = {N \choose n} = \frac{N!}{(N-n)!n!}$$ And then I thought about writing the heat capacity as $\frac{\partial E}{\partial n} \frac{\partial n}{\partial S} \frac{\partial S}{\partial T}$ or something like that. But in everything I have tried, I'm having trouble to differentiate with respect to $T$. I don't really get a grip on how the other things depend on $T$. Thanks for your help!


title: What are insightful/impressive experiments for adults

text: Similar to this question: what experiements do you think are impressive for layman adults? I for example like how you can simulate an Einstein Ring with the bottom of a wine glass.


title: Content of the N=2 Graviton Multiplet in 6D

text: I've been looking into extended Supersymmetry in higher dimensions recently. What I keep wondering about are some components of the gravity multiplet, that seem to appear from the construction of Supergravity from superconformal methods. The appearance of the graviphoton is clear from the N=2 structure. But what puzzles me is the anti-selfdual two-tensor that I usually find as a part of the gravity multiplet. How does the anti-selfdual tensor get introduced into the gravity multiplet? I would be most grateful for an answer that can at least shed some light on this without referring explicitly to superconformal tensor calculus.


title: Error in books of conformal field theory?

text: If you look at the book Conformal Field Theory (by Philippe Francesco, Pierre Mathieu and David Senechal) or the lecture notes Applied Conformal Field Theory (by Paul Ginsparg), and many other places: The conformal transformation is defined as the subset of coordinate transformations under which the metric changes as follows, $$ g_{\mu \nu}'(x') = \Omega(x) g_{\mu \nu}(x) $$ under a coordinate change $x'=x'(x)$ where, $$ g_{\mu \nu}'(x')=\frac{\partial x^{\alpha}}{\partial x'^{\mu}} \frac{\partial x^{\beta}}{\partial x'^{\nu}} g_{\alpha \beta}(x). $$ But I think the first equation should be: $$ g_{\mu \nu}'(x') = \Omega(x') g_{\mu \nu}(x'). $$ Although this looks like a typo, but it is consistant in many lectures/books. So I am little confused. Am I thinking it wrong? Books/Lectures are probably thinking that in the end of the transformation, they are relabeling $x'$ as $x$. But then that should be explicitly mentioned. **Derivation of conformal killing vector equation using book's version of the equation:** Consider an infinitesimal change, $$ x'^{\mu} = x^{\mu} + \epsilon \xi^{\mu}(x) $$ and, $$ \Omega(x) = 1+\epsilon \omega(x) $$ Then the equation according to the books: $$ g_{\mu \nu}'(x') = \Omega(x) g_{\mu \nu}(x) $$ gives, $$ g_{\mu \nu}'(x') = g_{\mu \nu}(x) + \epsilon \omega(x) g_{\mu \nu}(x) $$ Also we can expand, $$ g_{\mu \nu}(x') = g_{\mu \nu}(x) + \epsilon \xi^{\alpha} \partial_{\alpha} g_{\mu \nu}(x) $$ The Lie derivative of the metric, $$ \mathcal{L}g=\lim_{\epsilon \to 0} \left( {g'_{\mu\nu}(x')-g_{\mu\nu}(x') \over \epsilon} \right) = \omega(x) g_{\mu \nu}(x) -\xi^{\alpha} \partial_{\alpha} g_{\mu \nu}(x) $$ Which implies, $$ -(\xi_{\mu;\nu}+\xi_{\nu;\mu})=\omega(x) g_{\mu \nu}(x) -\xi^{\alpha} \partial_{\alpha} g_{\mu \nu}(x) $$ as the killing vector equation. Where as if you consider the equation: $$ g_{\mu \nu}'(x') = \Omega(x') g_{\mu \nu}(x') $$ Then you get following similar steps, $$ -(\xi_{\mu;\nu}+\xi_{\nu;\mu})=\omega(x) g_{\mu \nu}(x) $$ Then which Killing vector equation is correct?


title: real world applications of Mathematics which use functions with singularities, not just as a matter of mathematical taste but for conceptual reasons

text: **EDIT** [for aptness of this question to this site, read 'real world applications' as 'applications in Physics'] The concept of function (of the form $f : \mathbb{R} \to \mathbb{R}$ ) has been used in various real world applications of Mathematics. In almost all of the applications the general consideration such as approximating a given function with another function is viewed through a lens of mean square error or one of several other such very similar metrics. My question is concerning the use of functions which have singularities. I've heard from some Physicists that the use of such functions is just a matter of mathematical taste as such functions could be easily approximated with smooth functions and hence the choice of using such functions is just a matter of mathematical taste and it wouldn't make any conceptual difference. I tend not to agree with such an argument because it all depends on the lens of approximation that is being used. If we do not use the metrics like the ones mentioned above then it could be different. A function which contains singularities is fundamentally conceptually different from a smooth function which approximates it no matter how good the approximation is. So my question is, are there any applications of Mathematics in which functions with singularities are used not just as a matter of mathematical taste but for conceptual reasons ?


title: Returning Light

text: What happen if someone is just close enough to a black hole, and he emits light tangentially out, and the light got bent and circles around, what will he see, can he see the light source from his back?                            +---------->                 /            \        light   /              \               ^                \     observer  +   O black hole /     sees light\               /                \             /                  <----------- light path


title: Newton's second law, gravity and buoyancy

text: > A body of mass $m$ and volume $V$ is immersed into the sea. The body moves > under the action of two forces: gravity and buoyancy (Archimedes). > > Gravity force has a magnitude $mg$, where $g$ is a constant acceleration due > to gravity. Buoyancy force has a magnitude $\rho V g$, where $\rho$ is the > water density, and acts in the direction opposite to the gravity force. > > Part 1. > > Assume that the water density is a constant, $\rho V >m$ always holds, and > at the initial time moment the body is released from rest at the depth of > $h>0$ metres below the sea surface. Assume that the origin is at the sea > surface and the positive direction of motion is towards the centre of the > Earth. > > (a) Complete the diagram by showing all forces acting on the body, and the > initial displacement of the body. > > (b) Write down Newton's second law of motion. Solve this equation, i.e. > write down the expressions for the velocity and the displacement as > functions of time. > > (c) Show that the body will move towards the sea surface, and when it > reaches the surface it has the velocity $v= -\sqrt{\frac{2gh(\rho > V-m)}{m}}$. 1a) I can do 1b) I know that Newton's second law is $mx=F$ with $F$ being the forces. I have written the formula as $mx=\rho Vg - mg$ but I am not sure if this is correct. Then I've divided both sides by $m$ to get $x$ on its own and integrated with respect to t (time) to get $x$ as a function of time. Giving me $x=-gt+\frac{\rho Vgt}{m} + c$ 1c) I'm not sure where the $h$ has come from, or how I could input it into my previous equation


title: What defines the thickness of a Nomarski prism?

text: Lets say I want to design a Nomarski prism that would split the ordinary and extraordinary beam by an angle of 0.32 mrad. I used a raytracer to find the internal angle between the quartz wedges. However I couldn't decide how thick the quartz wedges should be. The original patent mentions the thickness but I don't understand the sentence: http://www.google.co.uk/patents?id=QI5kAAAAEBAJ&zoom=4&dq=nomarski%20patent%202924142&pg=PA6#v=onepage&q=Fig.%204&f=false ![Picture of Nomarski Patent](http://i.stack.imgur.com/0Trmk.png) The picture shows the Nomarski prism from the side. The optical axis of the system is horizontal. The internal angle between the wedges (or blades) is $\alpha$. The _optic_ axis (note the difference between optical axis of the system and optic axis of the crystal) of the first prism Q2 is perpendicular to the screen. The wedge Q1 has its axis inclined in-plane. Q2 is 0.5mm thick at the center. Q1 is 1mm thick at the center. **Both compensate their birefringence.** This last expression isn't clear to me. I have the feeling that I understand this only vaguely.   1. Is the birefringence in Q2 stronger because of the orientation of the optic axis? (Probably yes)   2. Why does the birefringence need to be compensated? (must have something to do with the adjusting the bias by shifting the Nomarski prism transversal to the optical axis in the microscope)   3. How is the total thickness of the prism (here 1.5mm) chosen? Can a thinner Nomarski prism be used with higher angles?


title: Velocity of the photon

text: As far as I know photon travels in a wave pattern, that means it has $x$ and $y$ axis velocity components. If photon travels along $x$ axis, then what is it's maximum $y$ axis velocity?


title: Equation of motion of a photon in a given metric

text: I have this metric: $$ds^2=-dt^2+e^tdx^2$$ and I want to find the equation of motion (of x). for that i thought I have two options:   1. using E.L. with the Lagrangian: $L=-\dot t ^2+e^t\dot x ^2 $.   2. using the fact that for a photon $ds^2=0$ to get: $0=-dt^2+e^tdx^2$ and then: $dt=\pm e^{t/2} dx$. The problem is that (1) gives me $x=ae^{-t}+b$ and (2) gives me $x=ae^{-t/2} +b$.


title: Astronomical measurements and data

text: For the sub-nuclear physics there is a database of the Particle Data Group, I was wondering if there was a similar on-line collection for astrophysics, such as energy absorption and emission spectra for different sources.


title: Energy of liquid and gas at same pressure is different?

text: Air compresses (change in volume) and it creates pressure. It's the internal energy (momentum of each molecules) creates that pressure. But in liquids as it can't be compressed (may be by .3% if I am sure about water) it can't be pressurised, right? This is only possible by an external force on water itself. Does it means that work done to generate same pressure on liquid is lower than gas? In this case pressurized water holds less energy that pressured gas of same pressure. This is my thought, could someone tell me my thought is right or wrong?


title: relationship between kinematics and area under curve

text: My teacher is giving us an honors pre-calc/calculus introduction to physics and the kinematics learning target he has given us is to understand the "relationship between area under a curve and the equation that calculates motion". I understand that the area under a curve (from the curve to the x-axis) will calculate the slope of the curve (which makes absolutely no mathematical sense to me), but I don't know how that relates to calculating motion. Please understand that my teacher is a very strict, high-standards, superfluous teacher who works at a university speed (he says so himself, and I'm only in HS) so there may be gaps in my understanding. I know what $\Delta$ means and I have a very, very loose understanding of the relationships between acceleration, velocity, and distance.


title: How can I put a permanent current into a superconducting loop?

text: I know that you can use induction to create a current in a superconducting loop, but this only works as long as the coil that induces the field has a current flowing through it. And obviously, this cannot work for the really big magnets that are used in MRI scanners or NMR machines. How does it work?


title: How to get a function for the voltage across a capacitor connected to an AC voltage source?

text: I am looking for the way of obtaining a solution for $V_{c}$ ,as a function of $t$ depending of $\omega$, of the following differential equation related to an electrical circuit involving a low-pass filter : $ \frac{ d V_{c}(t)}{d{t}} + \frac{V_{c}(t)}{\tau} = \frac{V_{s}(t)}{\tau} $. where, $V_{c}$ is the voltage across the capacitance, $V_{s}$ is the voltage given by the AC voltage source, $\tau$ is the time constant, considering that, $\tau = R C$ , $V_{s} = V_{in} \sin{( \omega t )}$ , $I_{R} = I_{C} = \frac{V_{s}(t) - V_{c}(t)}{R} = C \frac{dV_{c}(t)}{dt}$. I approached the problem by first solving the homogeneous part ( $ \frac{ d V_{c}(t)}{d{t}} + \frac{V_{c}(t)}{\tau} = 0$ ) and for which I get the following solution : $V_{c}(t) = K e^{\frac{-t}{\tau}}$ (where $K$ is a constant). I now need find the particular solution (to get the general solution : $S_{general} = S_{homogeneous} + S_{particular}$ ). I think the particular solution might be of the type : $V_{c}(t) = A \cos{(\omega t + \phi)}$. Edit : Once I replace the particular solution in the equation, I come to something depending on $\omega$ , $\phi$ and $\frac{A}{V_{in}}$ , but I don't see how to continue using the sum function sum formulas.


title: Single-mode fibers and ray-theory of light

text: I have read ray-theory cannot be applied to single-mode fibers. Why? I know rays are kind of an approximation to describe the real behavior of light, and electromagnetic theory and photonics are closer to nature, but are there no "rays" in single-mode fibers? Thank you.


title: How is a vacuum formed and measured by this pneumatics attachment?

text: Please excuse me if this question is misplaced and also for my narrow understanding of physics. Firstly, I'm asking simply because the robot I've now been set to work with uses an air pneumatics system. By way of a small vacuum it can move about objects from point A to point B. This vacuum is somehow created from a small attachment that is connected to a steady supply of pressurised air. This crude illustration is made on MS Paint shows how the vacuum is made. I'm still puzzled as to why the input of air doesn't simply flow out of **both** of the other openings. ![crude illustration](http://i.stack.imgur.com/CwyCl.png) **EDIT** I can intuit that the volume of air from the exhaust is the sum of the input and the additional air taken from the vacuum but **what is the force of the vacuum and how is it measured?** (Side note) > Does the surrounding environment (atmospheric presure) have any impact on > the vacuum? Any response is much appreciated.


title: Is WiFi safe for humans?

text: I have wifi on at my home almost all the time. I learned that the frequency of visible light is around several hundreds of THz, while the frequency of wifi is only several GHz, which is only 1:100,000. But I'm not sure how the amplitude of wifi wave compares to the amplitude of the visible light? Are the frequency and the amplitude the only concerns when we talk about the radiation? Can someone educate me to quantify the effect wifi radiation could have in our life? Thanks, Elgs


title: Divergence of conservative electric field

text: I have a little doubt about the following: according Gauss law in the form of Maxwell's equation, we know that: $$ {\rm div} (D)~=~ \rho(v) $$ This just tells us that the electric field has nonzero divergence if there is some nonzero charge distribution $\rho(v)$. But, for a point charge $$D=(qk\hat{r}/r^2)$$ while, in polar coordinates,   $$div(\hat{r}/r^2)=(1/r^2){(d/dr)(r^2/r^2)}= 0 $$ why does this apparent contradiction arise?


title: Paradox while Cooling & Heating in Air?

text: I was thinking about how asteroids get "burned" up in the upper atmosphere as they approach the earth surface due to the atmosphere of earth heating the asteroids immensely as a result of air compression which is nothing strange. This however seems opposite of things I observe day to day. While I am travelling in my car I open my window, and to my surprise the car became very chilly and I develop goosebumps (body response to cold), so I am inclined to think that the temperature of my body dropped slightly as a result of me travelling in our Air. This creates a sort of paradox, for a person travelling at asteroid-speeds s\he would say that as we speed up we **heat** the objects up, but for a person who travels at day-to-day speeds they would say as we increase in speed we **cool** things down, but both cannot be correct so **who is correct**? So my question is at what speed does temperature raise and not fall as we travel through air? And if so why is that?


title: Inscribed and Escribe Squares and Circle

text: Assume a circle(a) of diameter d. Inscribe a square A centred in the circle with its diagonal equal to the diameter of the circle. Now escribe a square B with the sides equal to the diameter of the circle. Show how to obtain the ratio of the area of square A to square to the area of square B.


title: Human eyes vs aberrations

text: There are no perfect lenses in nature. Aberrations of some kind will always be there. But why healthy human eyes circumvent this issue? Or, are there any aberrations we don't "see?"