title: Accounting for metric tensor derivatives in Einstein-Hilbert action

text: I'm puzzling over the canonical derivation of GR from the Einstein-Hilbert action; getting the derivation to gel with an explicit treatment of the functional derivative isn't working out. So the derivation (drawn here from wikipedia, though other literature is similar) begins, $$ I = \int{\sqrt{-g}d^4x} \left[\frac{1}{2 \kappa}R + \mathcal{L}\right] $$ and immediately proceeds to $$ \delta I = 0 =\int d^4x \delta g^{\mu \nu}\left[\frac{1}{2\kappa}\frac{\delta\left(\sqrt{-g}R\right)}{\delta g^{\mu \nu}}+\frac{\delta\left(\sqrt{-g}\mathcal{L}\right)}{\delta g^{\mu \nu}}\right]. $$ But the Ricci scalar depends on the first and second derivatives of the metric tensor, so why do we not have factors $$ \delta g^{\mu \nu, \alpha}, \delta g^{\mu \nu,\alpha \beta} $$ against which we vary as well? Maybe there is some identity that in this case causes these terms to vanish, but I don't see it.


title: Real world examples for projectile thrown upwards or downwards

text: I am preparing a physics course for high school about projectile motions. If a projectile moves with initial velocity $v_0$ in the gravitational field of the earth, the equation $$ s(t) = 1/2 g t^2 + v_0 t $$ holds, where $s(t)$ is the travelled distance at time $t$. Now I am looking for real world applications of this equation (or the corresponding equation for the velocity). More specifically I don't want any problems where the equation is somehow artificially embedded into a real world situation, for example "you fire the projectile of a signal pistol with initial velocity $v_0=...$, at which height is the projectile at time $t=...$, what is the maximal height..." projectile signal pistol In that example it is not clear **why** you know the initial velocity or why you want to calculate the maximal height (indeed, I think in most cases the manufacturer writes the maximal height into the manual, but then **why** should you be interested in calculating the initial velocity?) So the point is, in the problems/examples I am looking for, it should be clear, **why** one has the input data and **why** one wants to calculate other things using the equation of motion. Please stick to one problem/example per answer. Further references for the context of the example would be nice. _Edit_ : I should make clear, that I want examples where $v_0 \neq 0$. I am only intersted in the upward - downward -case. One example I was thinking of, was that of a vulcano that ejects stones. However I don't know much about vulcanos, so I don't know which initial data are known, and what people want so calculate...


title: Help w/ speed of sound experiment report question

text: Experiment was done by using an oscilloscope and a piezoelectric transducer to generate ultrasonic sound waves. We had to move the transmitter whilst receiver remains constant on a angled 1 meter track or vice versa. ![enter image description here](http://i.stack.imgur.com/FJ08d.jpg) > As you change the separation between the transmitter and the receiver, one > signal changes in size and the other remains the same. Which one of these > signals changes? What causes this change in amplitude? Why does one signal > remain the same? And that is the question I don't understand. I believe signal generated with a piezoelectric transducer was fed to the oscilloscope with the signal collected by the sensor (receiver). And the one collected with the sensor was the one that changed. (I don't know why. Maybe because of energy lost?) Also I have used this formula A = D/f to calculated the amplitude. 5.0X10^-6 m (5.0X10^-6 = 0.18 / 35950) Does this sound reasonable? With this amplitude I believe its enough for the signal to bounce off the receiver and transmitter and cause constructive and destructive interference. (Approx. receiver radius of 1.5 cm, transmitter 1.5 cm+)


title: A ball is thrown up, and a second ball is dropped- what is the maximum initial velocity of the first ball so that

text: This question only exists in one dimension, the y axis. A ball is thrown upwards with an initial velocity $V_o$ from a roof with height $h$. One second later, a second ball is dropped from the same roof. If $V_o$ of the first ball is greater than some value $V_{max}$,a value of $h$ does not exist that allows both balls to hit the ground at the same fime. Solve for $v_{max}$ This is my attempt at the solution, note: it is wrong. They will meet at value $\Delta h_1 = \Delta h_2$ $\Delta h_1 = V_o*t + \frac{1}{2}*a*t^2$ $\Delta h_2 = \frac{1}{2}*a*(t-1)^2$ Let $n = \frac{1}{2}a$ $\Delta h_1 = V_o*t + n*t^2$ $\Delta h_2 = n*(t-1)^2$ $V_o*t + n*t^2 = n*(t-1)^2$ When I simplify further, I now have $(V_o + 2n)*t = n$ This expression will give me the time it takes for the two values to be equal to each other. So now if I substitute for t to get the time it takes the second ball to hit the ground, $t = \sqrt{\frac{\Delta h}{n}} + 1$ I now get $(V_o + 2n)*(\sqrt{\frac{\Delta h}{n}} + 1) = n$ So an impossible height for $h$ is anything greater than or equal to 0. If I set it to 0, $V_0 = -n$ Solving this, I get $V_0 = 4.9 \frac{m}{s}$ However, this is incorrect. The answer is supposed to be $9.8 \frac{m}{s}$ How am I supposed to do this question?


title: What are hot electrons?

text: What are they? How are they created? And what do they have to do with plasmons? I searched the web, but I would like more reliable and straightforward sources.


title: Is reflection just a particular case of scattering?

text: I was talking to a colleague about optical scattering from a metallic nanoparticle, and we had a very simple doubt. If you have a particle that's small compared to the illuminated area, you can use Mie theory to calculate the scattering you'll get. However, if the particle is very big, you have a mirrored ball and light will be reflected from it. We don't call that scattering. Do we call the phenomenon "reflection" when we have "scattering" from a surface that's bigger than the illuminated area? Or maybe it has to do with the polar intensity diagram: if the scattered distribution is narrow we call it reflection. However, the term "diffuse reflection" contradicts this idea. NOTE: I understand that all these cases are just the solution to Maxwell's equations, and that we assign different names for different kinds of solutions. But my question is about how we name this processes.


title: Why do bass tones travel through walls?

text: I was in the shower while my roommate was listening to music and got to thinking about the fact that I could only hear the bass and lower drums through the walls. Why is this? The two possibilities I could think up were:   1. For some reason, sound waves at low frequencies (if I remember this means a lower pitch) are better at traveling through solids.   2. The bass sound waves are slightly vibrating the walls themselves, and this is somehow producing sound waves in the air on the other side of the wall. I have no clue if either are on the right track, and I'd really love to know!


title: Derivatives of delta function and equation of continuity for a single charge...

text: For a single charge $e$ with position vector $\textbf R$, the charge density $\rho$ and and current density $\textbf{j}$ are fiven by: \begin{equation} \rho(\textbf{r},t)= e\,\delta(r-\textbf{R}(t))\end{equation} \begin{equation} \textbf{j}(\textbf{r},t)=e\frac{d\textbf{R}}{dt}\delta(\textbf{r}-\textbf{R}(t))\end{equation} Suppose we want to check the equation of continuity \begin{equation}\frac{\partial \rho}{\partial t}+\nabla \cdot\textbf{j}=0 \end{equation} How to do it? How to deal with the derivatives of a delta function? Thanks


title: On Parallel Transport

text: Let's consider the issue of parallel transport in relation to the figure on the following Wikipedia link: http://en.wikipedia.org/wiki/Parallel_transport With reference to the Figure on the link: Instead of parallel transporting the vector from A to N lets (parallel) transport it from A to N' along the meridian where N' is a point just below N [say latitude=89.9999 degrees]. Now we move the vector parallel to itself along the line of latitude passing through N'to reach the corresponding point on the meridian NB.The vector is now almost parallel to the meridian NB[Since the concerned line of latitude is not a geodesic I have used the word almost]. The vector is moved down and then moved back to A along the equator. It turns by a very small amount.The exclusion of a tiny[you could make it microscopic] spherical triangle is causing so much of difference.Why?


title: What are the properties of the Electromagnetic wave $E=E_0e^{-i\omega t}$

text: My question is, whether this definition $E=E_0e^{-i\omega t}$ includes that it is a plane wave, since I am confused by the fact that we do not have any dependence on the position. So about what kind of electromagnetic waves are we talking if we use $E=E_0e^{-i\omega t}$?


title: How to prove equations for energy of a wave

text: In my textbook, it says that the energy (I) of a wave, determined by the power of a wave (P) divided by area (A), is determined by the following formula: $$I = \frac{1}{2}\rho v\omega^2A^2 = 2\pi^2 \rho vf^2A^2$$ where $\rho$ is density, $v$ is propagation speed, $\omega$ is angular speed ($\omega = 2\pi f$ and $A$ amplitude. Can anyone show how it is like this?


title: Hamiltonian Flow Map

text: I'm reading this article and am struggling with some of the terminology. What is the **flow map** for a Hamiltonian system? I'm looking for a rigorous definition really! Many thanks in advance.


title: Is there a star catalog for download?

text: Is there a downloadable star catalog which not only includes positions, but also spectral class, size, etc.? A CSV file would be great for the sake of easiness.


title: Quantum computers and algorithm performance

text: I have a question. Gradually quantum computers will emerge someday. So, nowadays making algorithm efficient is important; I mean, making it optimal to run as fast as possible. But once quantum PC emerges does the algorithm performance improvements are still important? Cheers


title: The energy of an electromagnetic wave

text: The intensity of an electromagnetic wave is only related to its amplitude $E^2$ and not its frequency. A photon has the same wavelength as the wave that's carrying it, and its energy is $h f$. So if a laser wave is kept at the same amplitude and the wave length is reduced, why does its intensity remain the same even though its photons now carry less energy? Why are the intensities of electromagnetic waves so different to sound waves (and other waves travelling through a medium) which are related to $f^2 E^2$?


title: Relation between shm and circular motion

text: I recently read in a book that combination of two simple harmonic motions of equal amplitude in perpendicular directions differing in phase by pi/2 is circular motion. I don't seem to understand this because I am not able to figure which two forces in circular motion are acting to cause two different simple harmonic motions. Any explanation would be thoroughly appreciated


title: Why don't I feel pressure on my body when swimming under water?

text: If I put a couple of lead bricks on my foot, there would be a definite sensation of a heavy, perhaps even painful, force. Calculating the pressure for $20 kg$ of lead over a $100 cm^2$ area of my foot, I get about $20 kPa$. Now I stand underwater in a pool. If my foot is $2 m$ below the surface, then the hydrostatic pressure would be $\text{density} \times \text{depth} \times \text{gravity}$, or $20 kPa$. Same as above. But in this case I wouldn't feel any pain or high pressure on my foot. Why the difference in sensation when under water, even though the pressure is the same in both cases?


title: Newton's second-law word problems with coefficient friction

text: Here is an example problem: > A sled of mass 50kg is pulled along a snow-covered, flat ground. The static > coefficient of friction is 0.30, and the kinetic coefficient friction is > 0.100. What force is needed to start the sled moving?   1. What force is needed to start the sled moving?   2. What force is needed to keep the sled moving at a constant velocity?   3. Once the sled begins to move, what force should be applied to accelerate it at a constant rate of $2.58 m/s^{2}$ ? I appreciate your help. I am not able to understand how to solve them, so an explanation on how to do it would help a lot.


title: What are the 't Hooft papers about classical models underlying QM?

text: Gerard 't Hooft states on his webpage: > _I have mathematically sound equations that show how classical models > generate quantum mechanics._ Also, there are some interesting discussions here on Physics SE about the question, see for example _Discreteness and determinism in superstrings_ and _Deterministic quantum mechanics_ [or searching for "+hooft +determini*"] and the links therein. Which of the 't Hooft papers in arXiv should I read in order to grasp the question? Could anybody provide an ordered list? I would like to restrict myself for the moment to those strictly related to quantum mechanics, to grasp the ideas within a known framework. (I have seen that the question extends to the realm of string theory, where I am for the moment nearly ignorant.)


title: Are the principles of space-time homogeneity and Isotropy independent of one another?

text: Einstein in deriving the Lorentz transformations, used the principles of space-time homogeneity and Isotropy. Does space-time isotropy follow from space-time homogeneity or are they completely independent of one another?


title: Calculating the required amount of acid/base to reach a pH

text: Bit stuck on working this out - or if it is even possible. I have a reservoir of solution (water/plant feed/dead flies) that is pH 6 and I want to get it to pH 6.25. Here is the information I have:   * Size of reservoir: 50 liters   * pH of reservoir: 6   * Required pH: 6.25   * Alkali/Base Stuff: Potassium Hydroxide at 50% Strength Is there any way I can work this out to say, 15 or 20%? Many Thanks


title: What are some creative illustrations of the nature of dissipative forces?

text: I'm teaching a conceptual introduction to physics for American 13-15 year old students this summer. One of the main ideas I want to hit on is the relationship between energy conservation, equilibrium, and dissipative forces. (e.g. When a box sliding over the floor comes to rest, its kinetic energy mostly goes into heating the floor. We expect this because there there are many degrees of freedom in the floor, while the bulk motion of the box is at most six degrees of freedom.) I'm looking for experiments and demonstrations of this effect. We can look at examples of turning mechanical work into heat (rub your hands together, hammer a nail, start a fire with friction), but this doesn't quite get across the idea of what thermal energy is. We might be able to observe Brownian motion, but since molecules are too small to see this has limited intuitive appeal for this age range. Ideally, I'd like to find systems where you can actually see the "microscopic" degrees of freedom alongside the "macroscopic" degrees of freedom. This could be actual physics demonstrations, or artificial scenarios in the form of games the students play out on a field (perhaps following certain rules about the field's layout as individual decision makers, but inevitably creating a certain distribution of students in different "zones" on the field) or simulations on a computer. All suggestions welcome, and if I implement it in the course next month, I'll report back on how it went. (Mods, could you please mark this community wiki?)


title: length contraction question

text: we know from eintein's theory of relativity that lets say, a ruler is travelling to a speed if light, then we can say that the ruler (from our view as observers) has shorten. but why, lets say we have a 15 km runway, and we let an electron run through it, in electron's perspective, the length of runway is now only around, very short lets say in centimeters. why is that so? i can grasp the idea if we can see actual objects flying, but when i think of the path as being shorten, i dont get the idea why its shorten also. thanks


title: Electromagnetic-Radiation

text: 1. describe how radiation from a resonant LC circuit is increased by opening out the capacitor plates?   2. Explain that a straight wire provides reactive elements of a LC circuit.


title: Pressure Rise in Closed Tank with 2 Inputs

text: Supposing you had a cylindrical closed tank inside the body of a submarine; the top of the tank is at 200m depth. The tank has height 2.5m and internal volume 1.8m^3. A section of pipework is connected between the top surface of the submersible (level with the top of the tank) and to the tank (assume inlet is at the bottom of the tank); an orifice plate will control the incoming flow rate. A separate pipe connects to the side of the tank, through which air will be supplied at a constant pressure of 30 bar; with volumetric flow rate 5.7m^3/min (if required). Assuming the initial pressure inside the closed tank is normal atmospheric (1 bar), how would you calculate the time for the pressure of the air bubble in the top of the tank to equalise with the pressure external to the top of the tank at 200m? Ignore frictional losses in the pipeworks. ![enter image description here](http://i.stack.imgur.com/3mgeQ.jpg)


title: Freshman year forces problem

text: I have been sitting with this problem for about 2 hours now. I know it involves simultaneous equations of some sort... I am trying to figure out maybe if there was not enough information given .. maybe the community could help me out :) Question: _It Is Question 2_ ![Image of question 2](http://i.stack.imgur.com/UDhJC.png) My Approach   ![enter image description here](http://i.stack.imgur.com/Q6QNS.png)


title: Must a reversible engine be a carnot engine?

text: I have this homework question: "Show that any reversible engine operating between T1 and T2 is a carnot engine." I think I have a solution, but it feels very hand-wavy. We know that any process that can be represented as a loop in the PV plane is reversible as the net entropy change will be zero. We must operate between two specifice temperatures, so the loop must comprise of two isotherms a T1 and T2. So the question is what curves join the isotherms. As a heat engine comprises of energy input at constant temperature, there will be no energy change between the isotherms. So the curves connecting the isotherms must be adiabatic curves. So we have a carnot cycle. Is this sufficient? I don't know why, but I doubt it.


title: Does sound absorption depends upon the amplitude of sound wave?

text: I can understand the mechanism of frequency dependant sound absorption by most materials but does the sound attenuation also depends upon the AMPLITUDE(sound pressure or rather loudness/sound intensity)??? I tried some googling but unfortunately could not find any reliable source but my self thinking and intuition tells me that the answer is yes it does does depend on amplitude of the sound wave. I think one possible mechanism for this amplitude dependent sound absorption may be like this- Most materials (including air) do not follow Hooke's law fully. That is in in simpler terms the restoring force is not proportional to the strain. And as such we can consider sound wave as a strain wave in a medium, the frequency of sound will not be amplitude independent and so hence the sound absorption, as it depends on the frequency and the frequency in turn depending on amplitude, owing to anharmonicity of the medium. Please correct me if am wrong fully/partially on this as I don't have full self confidence on what I'm thinking is true b'coz sound physics is not my field of expertise like most of you answering here will be experts, I suppose. THANKS FOR YOUR KIND ATTENTION. Any citation attached will be most appreciated. FOR ANHARMONICITY CHECK OUT LINK ATTACHED HEREBY- www.answers.com/topic/anharmonicity


title: How will water affect the electric field?

text: How will 2 charged bodies act when the separation medium between them is water ?


title: Local decoherence and entropy

text: Consider a quantum system consisting of two subsystems, $A$ and $B$. Let $\rho$ be the density matrix of the whole system $A\cup B$. Let $|\alpha\rangle$, $\alpha = 1,2\cdots d_B$, be the states of the subsystem $B$. Then $\rho$ can be written as the following: $$ \rho = \sum^{d_B}_{\alpha=1}\sum^{d_B}_{\beta=1}\sigma_{\alpha\beta}\otimes|\alpha\rangle\langle\beta|, $$ where $\sigma_{\alpha\beta}$ are sub-density-matrices for subsystem $A$ of size $d_A\times d_A$. Here $d_A$ is the dimension of the Hilbert space of the subsystem $A$. The reduced density matrix of subsystem $A$ is given by $$ \rho_A = \sum^{d_B}_{\alpha=1}\sigma_{\alpha\alpha}, $$ and the reduced density matrix of subsystem $B$ is given by $$ \rho_B = \sum^{d_B}_{\alpha=1}\sum^{d_B}_{\beta=1}\mathrm{tr}(\sigma_{\alpha\beta})\otimes|\alpha\rangle\langle\beta|. $$ Let us consider a process after which the quantum coherence of subsystem $B$ is lost. The density matrix then becomes: $$ \rho' = \sum^{d_B}_{\alpha=1}\sigma_{\alpha\alpha}\otimes|\alpha\rangle\langle\alpha|. $$ I am interested to know whether it is possible to relate the Renyi entropy of the new density matrix $\rho'$, defined as $$ S_\alpha(\rho')=\frac{\ln\mathrm{tr}(\rho'^\alpha)}{1-\alpha}, $$ to the Renyi entropy of density matrices $\rho$, $\rho_A$, $\rho_B$, or similar quantities. If the quick answer is no, I hope someone could point me to useful references.


title: Beam splitters- Direction of use

text: ![There are two cases I'm asking about The square in the middle is a cube beam splitter in the same orientation for both cases](http://i.stack.imgur.com/tl5sI.png) There are two cases I'm asking about. The square in the middle is a cube beam splitter in the same orientation for both cases. We are looking at the beam splitter from the top. In the first case, the beam comes in from the left and half gets transmitted and half gets reflected downwards. So, now changing only the beam direction in case 2, with the beam splitter in the same orientation, will the beam split like I've drawn? **Does the beam splitter work if the laser comes from opposite directions?** **What would happen if the Beam came from the top direction? The bottom?** Does the beam splitter only work one way?


title: I have a question about how sound is produced

text: How is sound produced looking from the atomic level? My thought process goes like this: Atoms are not perfect circles/solid spheres which we use to describe many macroscopic/classical phenomena.They have only electrical boundaries.There is also empty space between them. So, if atoms collide/vibrate how would it produce what we perceive as sound?


title: fiber optic second order PMD as an operator on the tensor product Hilbert space

text: Second order polarization mode dispersion (SOPMD) is a coupling mechanism between polarization and frequency. Take our photon to be the following tensor product: $\psi = \int \gamma_{\omega} | \omega \rangle \otimes (\alpha |H \rangle + \beta | V \rangle )d \omega$ then, can we model SOPMD as an entangling operator, $O_{SO}$, on this joint space? Furthermore, can we model certain kinds of noise in fibre transmission systems as this operator, $O_{SO}$, followed by the tracing out of one or both of polarization or frequency? $\rho^{\prime} = Tr_{Pol} (O_{SO} \rho O_{SO}^{\dagger})$


title: Questions on beta-decay

text: According to my textbook, for all the beta decays, it is required that the mass of the original atom to be heavier than the mass of the final atom. Is this due to the fact that all the beta decays involve the production of neutrinos and these neutrinos take away the energy of the atoms? It is further required that for beta-plus decay the mass of the original atom must be at least 2 electron mass heavier than the mass of the final atom. Why?


title: Definition of isotropy of radiation and its consequences

text: I have encountered different notions of isotropy of radiation and I would like to know if they are the same and what the exact definition of isotropy is, if one exists. Let's take black body radiation inside a cavity at thermal equilibrium for an example. It's a fact, that in this case the radiation is isotropic, but what does that mean exactly? Let's consider a point in the cavity and two small surfaces $d\sigma_1$ and $d\sigma_2$ located at said point, but with different orientations in space. Let $L_1(\theta_1,\phi_1,\nu)$ be the spectral radiance of $d\sigma_1$ where $\theta_1$ and $\phi_1$ are the angles of the solid angle $d\Omega_1$ and $\nu$ is a certain frequency and $L_2(\theta_2,\phi_2,\nu)$ be the corresponding spectral radiance of $d\sigma_2$. The angles $\theta_i$, $\phi_i$ are measured with respect to the normal to the surface $d\sigma_i$. The first notion of isotropy that I encountered is: $L_1(\theta_1,\phi_1,\nu)$ = $L_1(\tilde{\theta}_1,\tilde\phi_1,\nu)$ for all possible angles $\theta_1$, $\phi_1$, $\tilde\theta_1$, $\tilde\phi_1$. That means, if the surface is chosen, it does not matter in which direction you look. The second notion is: Let $\theta := \theta_1=\theta_2$ and $\phi := \phi_1 = \phi_2$. Then $L_1(\theta,\phi,\nu) = L_2(\theta,\phi,\nu)$ (Planck used this in his book "Vorlesungen über die Theorie der Wärmestrahlung" in the derivation of equation (21)). That means that the orientation does not matter. So which one is the definition of isotropy of radiation? Maybe none or both?


title: Plane Polar coordinates for simple pendulum in moving lift(elevator)

text: Can any one help me with the following. A simple pendulum is suspended from the ceiling of a lift. It is moving upward with acceleration $a$. The string of the effective length of the pendulum makes angle $\alpha$ with the vertical. Choose the point of suspension as origin, $x$-axis along the horizontal and $y$-axis vertically downwards. Express $x$ and $y$ in terms of plane polar coordinates at time $t$ after the lift takes off Now I understand it will be $x = l \sin\alpha$ and $y = l \cos\alpha$ where $l$ is the length of the pendulum string but it seems correct answer is $y = l \cos\alpha - \frac{1}{2}at^2$ can any one please explain why and how the other factor is included in the equation and why it is not affecting the $x$ and affecting only $y$


title: Can we describe mathematics using filters and matrices?

text: Can mathematical filters or ultrafilters be used to predict quantum physics 'events' as accurately as using matrices like Schrodinger did? Is there a way to explain some of the predictive power of quantum mechanics with matrices on an amateur level?


title: Conflict between Lippmann–Schwinger equation and Gell-Mann and Low theorem about energy

text: Lippmann–Schwinger equation states that scattering state will have the same energy as free state, while Gell-Mann Low theorem says that they have different enery. Lippmann–Schwinger equation says: That is, if $|f\rangle $ is the eigenstate of interacting picture's free Hamiltonian $\hat{H}_0^{(I)}$ with eigenvalue $E$, then $$|F \rangle := \hat{U}(0,-\infty)|f\rangle$$ is the eigenstale of the full Hamiltonian $\hat{H}^{(I)}(0)$ at $t=0$ with the same eigenvalue $E$, where $\hat{U}(t,t_0)$ is the evolution operator in interacting picture. On the other hand, Gell-Mann and Low theorem states that : Let $|f\rangle$ be an eigenstate of $H_0$ with energy $E_0$ and let the 'interacting' Hamiltonian be $H=H_0 + gV$, where $g$ is a coupling constant and $V$ the interaction term. We define a Hamiltonian $H_\epsilon=H_0 + e^{-\epsilon |t|}gV$ which effectively interpolates between $H$ and $H_0$ in the limit $\epsilon \rightarrow 0^+$ and $|t|\rightarrow\infty$. Let $U_{\epsilon I}$ denote the evolution operator in the interaction picture. The Gell-Mann and Low theorem asserts that if the limit as $\epsilon\rightarrow 0^+$ of $|F_\epsilon \rangle = \frac{ U_{\epsilon I} (0,-\infty) |f\rangle}{\langle f | U_{\epsilon I}(0,-\infty)|f\rangle}$ exists, then $|F_\epsilon \rangle$ are eigenstates of $H$. But the energy $E$ will not be same as $E_0$ and will have a shift. Here is a proof of my first state that energy will be same ![enter image description here](http://i.stack.imgur.com/rhih5.png) ![enter image description here](http://i.stack.imgur.com/rH5wk.png) ![enter image description here](http://i.stack.imgur.com/onN6Q.png) ![enter image description here](http://i.stack.imgur.com/zbIlB.png) ![enter image description here](http://i.stack.imgur.com/QZ3RB.png) ![enter image description here](http://i.stack.imgur.com/ArOis.png) ![enter image description here](http://i.stack.imgur.com/9kv4n.png) ![enter image description here](http://i.stack.imgur.com/7TjRL.png) ![enter image description here](http://i.stack.imgur.com/GkYAw.png)


title: “Relativistic Baseball”

text: On Randall Munroe’s new blog “what if”, he answers the question: > “What would happen if you tried to hit a baseball pitched at 90% the speed > of light?” http://what-if.xkcd.com/1/ He concludes: > … the air molecules in front of this ball don’t have time to be jostled out > of the way. The ball smacks into them hard that the atoms in the air > molecules actually fuse with the atoms in the ball’s surface. Each collision > releases a burst of gamma rays and scattered particles. > > These gamma rays and debris expand outward in a bubble centered on the > pitcher’s mound. They start to tear apart the molecules in the air, ripping > the electrons from the nuclei and turning the air in the stadium into an > expanding bubble of incandescent plasma. > > … Suppose you’re watching from a hilltop outside the city. The first thing > you see is a blinding light, far outshining the sun. This gradually fades > over the course of a few seconds, and a growing fireball rises into a > mushroom cloud. Then, with a great roar, the blast wave arrives, tearing up > trees and shredding houses. > > Everything within roughly a mile of the park is leveled, and a firestorm > engulfs the surrounding city. The baseball diamond is now a sizable crater, > centered a few hundred feet behind the former location of the backstop. Is that all actually reasonable, or is it all just hyperbole?


title: Electric Magnetic potential and Lorentz transform

text: I have heard that the scalar potential and the magnetic vector potential in the electromagnetic four potential become the four vector by the Lorentz transform. Thereafter, the Lorentz transform leads to gauge theory.   1. How do the four potentials become the four vector?    2. Is the Lorentz transform related to gauge theory?    3. Is gauge theory related to the development of the four vector?


title: Capillary tube of insufficient length

text: I was wondering if we have a very thin glass tube placed in a tub of liquid and the portion of the tube outside the liquid is lesser than the height to which the liquid can rise because of capillarity, what will happen? Will the liquid overflow?


title: Normal reaction on wheels when car is taking a turn

text: I recently read in a book that for a car taking a turn on a horizontal surface, the normal reaction of the road on the outer wheels is always greater than the normal reaction on the inner wheels. However, I feel it should be the other way round. For example, in real life situations the outer wheels are lifted off the ground when taking a really sharp turn at high speed. This clearly shows that inner reaction is greater. Any help would be appreciated.


title: Is Athene's Theory of Everything a respectable theory?

text: Athene's Theory of Everything is a _very_ popular youtube video proposing a "theory of everything": ![screenshot of video on youtube](http://i.stack.imgur.com/peKpZ.png) **How respectable is this video?** Is it complete hogwash, or is there an element of truth to it? Can someone with the necessary experience in physics please debunk or validate the contents of this video once and for all. Millions of people are waiting. I have - at best - an enthusiastic layman's understanding of quantum and modern physics. From my perspective, the video didn't immediately strike me as bullsh*t. It is suspicious as it is uploaded by a gamer on youtube, but it seems intelligent and well thought out. * * * **Note:** Asking here for a review of a theory that has not been published in an academic journal would usually be closed. But this video has **over 2.5 million views** and I think this needs to be addressed by those who would otherwise not have the time to review a non published theory.


title: Why normalizing spin state is different? Why no need dxdydz?

text: An electron in a spin state: $$X=A\quad\begin{pmatrix} 2i\\\2\end{pmatrix}\quad$$ In order to get A, I have to normalise it, my question is how should write? Since I have to normalise why no need integral? Like $$\int\limits_{-\infty}^{\infty}X{X^*}dx dy=A^2\int\limits_{-\infty}^{\infty}[(-2i){\bar x}+2{\bar y}][(2i){\bar x}+2{\bar y}]?$$ I don't know how to write,then I find book,it says $X{X^*}=A^2|4+4|$,so how should I write? $X{X^*}=A^2 \quad\begin{pmatrix} 2i\\\2\end{pmatrix}\quad \quad\begin{pmatrix} -2i\\\2\end{pmatrix}\quad$ Or $X{X^*}=A^2[(-2i){\bar x}+2{\bar y}][(2i){\bar x}+2{\bar y}]$


title: The delay between neutrinos and gammas in a supernova, and the absolute mass scale of neutrinos

text: In a supernova explosion (of some type), there is a huge amount of neutrinos and gamma rays produced by a runaway nuclear reaction at the stellar core. In a recent comment, dmckee noted that the neutrinos actually make it out of the supernova earlier than the gamma rays because the latter do scatter in trying to get out, and that the precise details of this delay, if measured, could shed light on just how small the (nonzero) neutrino mass is. Is there some simple way of explaining the different models at stake, or what the essential physics is behind this? Is there some accessible reference about this?


title: How does the height of a jump change if the body is scaled down?

text: This is a physics question that appeared in the movie 'The Internship.' > If you are reduced to the size of a coin and put into a blender what would > you do? Apparently the answer was that with a reduced size you could jump higher and jump out the blender. I would think that if you are smaller your strength will also be smaller so you could only jump a few centimeters and could not escape from the blender. So how does it work?


title: beta decay equation balance

text: Quark doesn't constitutes more fundamental particle and proton and neutron consist of quarks. Now come to beta decay. > $n \rightarrow p + e^{-} + \bar{\nu}_e $ How can an electron emit from an neutron whereas electron is a fundamental particle? Is the electron created from gluon?


title: Wire forces in an inductor

text: For an inductor, the electromotive force comes from Faraday's law, as I understand it. The governing equation includes the number of turns, the magnetic flux (field and area), and a time differential. $$ \text{emf} = n \frac{ d \phi }{ dt } = n \frac{ d ( B A) }{ dt } $$ The idea is that $B$ changes, because that comes directly from the current value. So does $A$ want to change too? Since increasing area would decrease the flux, I picture a natural tendency to close or open the area in the same way as the inductor does for the current, because they're contained in the same derivative. Does that imply that the wire experiences an outward or inward force? Inductors you can buy tend to be fairly well secured, but if they weren't, is there are combination of $I$ and $dI/dt$ that would make the loop balloon out, and also contract in? I ask because I'm curious if this is used as artificial muscle fibers. Make a connected series of loops like a chain, and then alter the current to contract and expand it. Other examples of such artificial muscle fibers I've seen use less direct material properties. But I've not seen the idea mentioned before, and in fact, I've never seen it mentioned that inductors experience wire forces. I'm not really sure how to formalize that in the first place.


title: Adiabatic expansion in van der Waals gas

text: Given a Van der Waals gas with state equation: $$\left( P+\frac{N^2 a}{V^2}\right)\left( V-Nb \right)=NkT,$$ show that the equation of an adiabatic process is: $$\left( V-Nb\right)T^{C_V}=\text{constant}.$$ I began by setting $dQ=0$ in $$dU=dQ+dW,$$ one then gets $$0=dU+PdV.$$ Now given $U=\frac{3}{2}NkT-\frac{N^2 a}{V},$, I plugged it's derivatives into $$dU=\left( \frac{\partial U}{\partial T}\right)_V dT+\left( \frac{\partial U}{\partial V} \right)_T dV,$$ from which I obtained $$0=C_V dt+\left(P+\frac{N^2 a}{V^2} \right)TdV=C_V dT + \frac{NkT}{V-Nb}dV,$$ using VdW's equation. Dividing by $T$ and integrating gives $$C=\log{T^{C_V}}+\log{(V-Nb)^{Nk}},$$, which is equivalent to $$C'=(V-Nb)^{Nk}T^{C_V},$$ for $C$ and $C'$ constants. Now the expression so obtained seems very similar to what I was looking for, but I can't seem to get rid of the $Nk$ exponent. Anyone got a different approach to this problem, or a way to get the desired formula? Thanks in advance.


title: Moment of inertia of a disc with masses attached at the rim

text: Does the moment of inertia of a disc with some masses attached at the rim be the same as one without the attached masses? Or is it necessary to use parallel axis theorem to incorporate the moment of inertia of the attached masses about the same axis of rotation?


title: Showing the thermal expansion coefficient is the sum of the linear expansion coefficients

text: Given that the thermal expansion coefficient is defined as: $$ \beta=\frac{\frac{\Delta V}{V}}{\Delta T} $$ and the linear thermal expansion coefficient is defined as: $$ \alpha = \frac{\frac{\Delta L}{L}}{\Delta T} $$ Show that $\beta $ is equal to the sum of its linear expansion coefficients in all three directions: $$ \beta = \alpha_x+\alpha_y+\alpha_z $$ I first began as writing: $$ \alpha_i \Delta T= \frac{\Delta L_i}{L_i} $$ Where i denotes a Cartesian coordinate. We then have that: $$ (\alpha_x+\alpha_y+\alpha_z) \Delta T= \frac{\Delta L_x}{L_x}+\frac{\Delta L_y}{L_y}+\frac{\Delta L_z}{L_z} $$ Simplifying to: $$ (\alpha_x+\alpha_y+\alpha_z) \Delta T=\frac{\Delta L_x*L_y L_z+\Delta L_y*L_x L_z+\Delta L_z*L_y L_x}{L_x L_y L_z} $$ Implying that the numerator is the change in volume of our object. Upon further thinking, I realized that if we had a cube that started off with a one meter edge, and expanded to a cube with a two meter edge, we should have a volume change of 7, but the numerator of my equation gives 3. This leads me to conclude something is wrong with my formulation, but I do not see it.


title: Total Vs Partial in Lagrange density?

text: I have a question regarding the red term below. This is the integration by parts during the derivation of the Euler-Lagrange equation for continuous systems. Why is this not the time derivative $d/dt$.... why do we use the partial? I have seen the equation expressed in both ways and wondered what the physicality's are. \begin{equation} \int^{t_2}_{t_1}dt\frac{\partial \mathcal L}{\partial (\partial _t\phi)}\frac{\partial }{\partial t}\delta \phi =\frac{\partial \mathcal L}{\partial (\partial _t\phi)}\delta \phi \bigg|^{t_2}_{t_1}-\int^{t_2}_{t_1}\color{red}{\frac{\partial }{\partial t}}\bigg(\frac{\partial\mathcal L}{\partial (\partial _t\phi)}\bigg)\delta \phi dt \end{equation} This particular Lagrange density is a function of $\mathcal L=\mathcal L(\phi,\dot \phi ,\partial _x\phi)$


title: Is thermal noise "quantum random"?

text: Is the randomness that can be extracted from thermal noise "as random" (that is, even theoretically inaccessible to measurement according to our knowledge of quantum mechanics, and not just random for practical purposes) as the one that comes from "true" quantum phenomena, like radioactive decay of nuclei? Due to the uncertainty principle, we can't predict both a particles position and speed infinitely accurate, and therefore we shouldn't be able to predict thermal noise, so it would seem that there is at least some level of quantum mechanics involved, but is that the same randomness that dictates the decay of nuclei, tunneling and other quantum phenomena? So, is there a true distinction between thermal and "quantic" noise on a theoretical level, or are they basically the same? In other words, is the thermal movement of atoms and molecules fundamentally deterministic or indeterministic?


title: Degeneracy in quater-wave stack

text: Consider a 1D photonic crystal, the quarter-wave stack, and its band structure. A famous conclusion is that there's no gap at Brillouin zone's center. In other words, successive bands are degenerate at k = 0. It's this degeneracy I try to understand. My current understanding is that, at k = 0, the quarter-wave stack supports standing waves and since each layer is exactly a quarter-wavelength in thickness, the energy densities of a certain pair of successive bands correspond to the same energy level. Is there any alternative explanation of this fact? Can one predict this degeneracy from group theory?


title: What is the physical meaning of the norm of the electromagnetic four-potential?

text: In SI units this would be $\frac{1}{c^2}\phi^2 - A_x^2 - A_y^2 - A_z^2$. Is this just not a physically meaningful quantity at all because it's not gauge invariant?


title: Does space curvature automatically imply extra dimensions?

text: Total newbie with basically no physics knowledge here :) I would welcome any correction to the steps of my reasoning that lead to my question, which could easily turn out to be invalid :) My current understanding is that General Relativity is definitely accepted by the vast majority of scientists, and that according to General Relativity spacetime is curved by the presence of mass. I also seem to understand that not as many scientists (although probably still the majority) agree that extra dimensions of space exist. But if we agree that spacetime can "curve", aren't we automatically saying that extra dimensions of space exist? I mean, if I grab a stick and I bend it, it becomes curved with respect to our good old 3D space. The stick needs to be in a 3D space, with respect to which it can be straight or curved. So, if the thing that gets curved is not an object in space but space itself, doesn't space need to be in "another" space in order to be curved? I apologize in advance if I used inaccurate terms here, which I most likely did :)


title: Can a thermometer really measure the temperature of a substance?

text: When we measure the temperature of a substance by using a thermometer and waiting until the two come into thermal equilibrium, the thermometer will not display the original temperature of the substance before the thermometer was introduced - the measured temperature will be skewed a little depending on the thermometer's starting temperature. (For this same reason we cannot lower the temperature of a substance to absolute zero by repeated thermal contact with a substance already at absolute zero). For most practical purposes this tiny inaccuracy due to the thermometer may not be of significance, but I am wondering if it is correct to say that a thermometer cannot actually measure the temperature of a substance?


title: Polarization effects of second-harmonic generation (SHG)

text: In one of my labs, we were using an Nd:YAG at 1064 nm and we put it through a KTP nonlinear crystal to perform SHG. We noticed that before the crystal, the 1064 light was unpolarized, but the 532 light was polarized. How does SHG polarize the light?


title: Noncommutative Field Quantization

text: I'm studying noncommutative (quantum) field theory, and I have confusion need to be clear. I'm reading Szabo's and Douglas's .pdf of noncommutative QFT. As I understand, in the book they just construct Field Theories in noncommutative spacetime, which have action functional (of Weyl operator or of normal function with star-product). What is the relation between this Field Theory in NC spacetime and Field Theory in normal spacetime? Can we quantize the field theory in NC spacetime (as we quantize the field theory in normal spacetime)? (Field theory in NC spacetime has the quantized spacetime, but not quantized field, doesn't it?) I cannot find materials about the Quantized Field theory in NC spacetime. (just field theory in NC spacetime.)


title: Ghosts in theories of gravity and holographic theories

text: I want to understand when a theory leads to ghosts in gravity. Is there any relation between ghosts and non-linear higher order theories? Ghost is a clasical or quantum field concept?


title: Quantum Hall Effect Dark Matter Detector?

text: Has anyone used a Quantum Hall effect detector to detect dark matter? I was looking at the following animation on wikipedia: http://en.wikipedia.org/wiki/Quantum_Hall_effect#Integer_quantum_Hall_effect_.E2.80.93_Landau_levels One could tune the magnetic field so that the topmost Landau level is arbitarily close to the Fermi energy. If dark matter particles even slightly interacted with the electrons in that Landau level then they would be pushed up to the Fermi level. The Hall voltage would then jump from one Hall plateau to another.


title: Gauge field with flat connection

text: Consider a gauge field $A_z^a$ with a flat connection $$F_{z{\bar z}}^a = \partial_z A_{\bar z} ^a - \partial_{\bar z} A_z^a + f_{bc}{}^a A_z^b A_{\bar z}^c = 0$$ where $f_{bc}{}^a$ is the structure constant satisfying $[T_a, T_b] = f_{ab}{}^c T_c$. We define the gauge covariant derivative $$ (D_z)^a{}_c = \delta^a_c \partial_z + f_{bc}{}^a A_z^b $$ This satisfies $$ (D_z)^a{}_c (D_{\bar z})^c{}_b - (D_{\bar z})^a{}_c (D_z)^c{}_b = f_{cb}{}^a F_{z{\bar z}}^c = 0 $$ Now, consider the following equation > $$ (D_z)^a{}_b B^b_{\bar z} - (D_{\bar z})^a{}_b B^b_z = 0 $$ **What is the > most general solution to the above equation?** Clearly, one solution is (owing to the flatness of the connection) $$ B_z^a = (D_z)^a{}_c \phi^c $$ for any $\phi^c$. Are there more solutions? PS - $(z,{\bar z})$ are coordinates on $S^2$.


title: Grand partition function of gas of non-interacting spin-1 bosons in magnetic field

text: Consider a gas of non-interacting spin 1 bosons in a uniform B field, each subject to a Hamiltonian of the form: $ H(\vec{p},s_z) = \frac{p^2}{2m} - \mu_0 s_z B$ where $s_z$ can take the three possible values of -1, 0 and 1. Assume further the orbital effect $\vec{p} \rightarrow \vec{p} - e\vec{A}$.   * First, how would you go about determining the grand partition function of the system, and the average occupation number for the three possible spin orientation? I know how to do this with no magnetic field present. I'm tempted to simply add the magnetic interaction term, but given the way the grand partition function is normally defined (in terms of distribution set), I find it a bit confusing here.   * Second, what is meant here by orbital effect?


title: Resolution of the Eye-Light Rays and Diffraction

text: Please look at the problem. The problem is solved for me. However, I don't understand why the angle alpha is considered to stay constant when the rays go through the lens. The ray along the distance s DOES NOT get refracted, which I agree. But the ray along the top point of the circle DOES get refracted once it gets to the liquid inside the eye. Can someone clarify why the textbook does not take that into account? http://imgur.com/0USADP7


title: Wavefunctions and Quark Confinement

text: While I have a decent knowledge of general relativity (and, of course, classical mechanics), I am quite a novice when it comes to quantum mechanics, so I apologize if this is a rather basic question. In a proton, neutron, or other hadron, the constituent quarks are bound together, thanks to color confinement. The strong nuclear force gets stronger as quarks are moved further apart, so it is thought to be impossible to "see" an isolated quark. My question is this: **A quark's wavefunction extends throughout all of space (prior to any de- coherence), so how can it be considered "bound" to the other quark(s) with which it constitutes a hadron?** My logic is that the quark could be anywhere, not simply tightly bound to the others; this flies in the face of quark confinement. Where am I wrong?


title: When the magnetic force is perpendicular to the velocity what will be affected on a charge?

text: I guess it can also be stated as a two part question. Is the velocity of a charge always perpendicular to the magnetic force, and when the magnetic force is perpendicular to the velocity what is affected on a charge?


title: Newton's rings: What causes the other rings?

text: ![enter image description here](http://i.stack.imgur.com/BAaQx.jpg) This is from an experiment we did in physics class. We shone a sodium light at a convex lens on top of a sheet of glass - and this image was captured by a USB microscope. I know what causes the main rings - but what about the surrounding ones, and why are they different colours?


title: electrostatics particle charging

text: When ash particles are charged negatively ( corona charging, ionization, tribo electric ) and then collected on to the fabric; particles arrange themselves in porous dendritic structure whereas without charging particles dont form the porous structure. What is the physics behind this sort of arrangement of particles?


title: What happens when a charged (negative or positive) object touches an insulator?

text: I know that insulators do not conduct electricity because they do not allow the free movement of electrons. Let's assume the object that's going to touch the insulator is negatively charged. Does some charge get transferred to the the insulator or does no charge get transferred at all? I want to understand this more conceptually. I don't really get why insulators are used in electrostatic experiments.


title: Torque, and the Law of the Lever

text: How fundamental is the Law of the Lever? It seems that we simply _define_ torque as being $r \times F$, if that's the case, then torque isn't a _derived_ quantity, is it? Something like the Law of the Lever is distinct from Newton's Laws, then?


title: How would one expect a massive graviton to behave?

text: Typically, adding a mass $m$ to a gauge boson causes the boson to only be able to travel over a finite distance, $L\sim m^{-1}$, limiting the range of the associated force. For example, photons become massive in superconductors and hence magnetic fields cannot penetrate very deep into superconductors. Should one expect the same behavior for a massive graviton? In the literature there are examples of massive gravity theories, such as the de Rham-Gabadadze-Tolly model (dRGT), which can give rise to self-accelerating universes due to a condensate of the graviton field (see here, for example). How does this phenomenon mesh with the usual reasoning that a mass limits the range of a gauge field?


title: Volume charge density of H-atom

text: I have a problem where I am supposed to calculate the volume charge density of a neutral hydrogen atom. The potential is given to be $$ \Phi = k \frac{e^{-ar}}{r} \left(1 + \frac{ar}{2}\right) $$ Now I tried to use the Poisson-Equation stating $$ \Delta \Phi = \frac{\rho}{\varepsilon_0} $$ which leads me to $$ \rho = \Delta \left( \underbrace{\frac{q}{4 \pi \epsilon_0}}_{= k} \frac{e^{-\alpha r}}{r} \left( 1 + \frac{\alpha r}{2} \right) \right) = k \Delta \Big( \frac{e^{-\alpha r}}{r} + \frac{\alpha e^{-\alpha r}}{2} \Big) = k \left( \Delta \left( \frac{e^{-\alpha r}}{r} \right) + \frac{\alpha}{2} \Delta \left( e^{-\alpha r} \right) \right) $$ Now I define $f = e^{-\alpha r}$ and $g = \frac{1}{r}$. The Laplacian of the product $fg$ is then $$ \Delta(fg) = g \Delta(f) + f \Delta(g) + \nabla (f) \cdot \nabla (g) $$ and the derivatives are $$ \nabla(f) = - \alpha e^{-\alpha r} \hat{ \mathbf r} \qquad \qquad \Delta(f) = \alpha^2 e^{-\alpha r} $$ $$ \nabla(g) = - \frac{1}{r^2} \hat{ \mathbf r} \qquad \qquad \Delta(g) = - 4 \pi \delta(r) $$ $$ \implies \nabla(f) \cdot \nabla(g) = \frac{\alpha e^{-\alpha r}}{r^2} $$ Inserting this back into the original equation yields $$ \rho = k e^{-\alpha r}\Big( \frac{\alpha^2}{r} - 4 \pi \delta(r) + \frac{\alpha}{r^2} + \frac{\alpha^3}{2} \Big) $$ However, this seems somewhat wrong to me since I would have expected the expression to be increasing from the origin and then decreasing after some $r=R$ since the potential of the electron hull should take over. Can anybody either confirm that this is correct or show me where I made the mistake? Apart from taking the derivatives like in cartesian coordinates, I have tried calculating the Laplacian by calculation in spherical coordinates as well using the spherical Laplacian $$ \Delta \Phi = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial \Phi}{\partial r}\right) $$ but still got the same result.


title: A question about the coupling between string and gauge field $A_{\mu}$

text: I have a question about deriving the coupling term of string and the gauge field on brane. According to David Tong's lecture note p184/(191 in acrobat), the coupling is given by $$ S_{\mathrm{end-point}}=\int_{\partial M} d \tau A_{a}(X) \frac{d X^a}{d \tau} \tag{1} $$ It is said that the coupling is obtained by exponentiating the vertex operator, "as described at the beginning of Section 7", $$ V_{\mathrm{photon}} \sim \int_{\partial M} d \tau \zeta_a \partial^{\tau} X^a e^{ i p \cdot x} \tag{2} $$ My question is about the logic of exponentiating the vertex operator. In the beginning of section 7 of the lecture note, in order to obtain the coupling between string and the gauge fields, the Polyakov action is extended in curved space $$ S= \frac{1}{4 \pi \alpha' } \int d^2 \sigma \sqrt{g} g^{\alpha \beta} \partial_{\alpha} X^{\mu} \partial_{\beta} X^{\nu} G_{\mu\nu}(X) \tag{7.1} $$ The coupling comes from the bending of spacetime, e.g. $$G_{\mu\nu} (X) = \delta_{\mu\nu} + h_{\mu\nu} (X) $$ $$ Z= \int \mathcal{D} X \mathcal{D} g e^{-S_{\mathrm{Poly}} -V} = \int \mathcal{D} X \mathcal{D} g e^{-S_{\mathrm{Poly}} } (1-V +\frac{1}{2} V^2 + \dots ) $$   $$ V= \frac{1}{4 \pi \alpha'} \int d^2 \sigma \sqrt{g} g^{\alpha \beta} \partial_{\alpha} X^{\mu} \partial_{\beta} X^{\nu} h_{\mu\nu}(X) \tag{7.2} $$ In order to obtain Eq. (1) from (2), where is the bending of metric?


title: About Pauli-Lubanski 4-vector

text: I saw you answered someone else about the Pauli-Lubanski 4-vector properties. I was doing some simple analysis of polarized leptons, started with the canonical convention of the spin operator wich is just $s=(0,\vec{s})$ and made the simple boost. However I am confused because I have found in the literature (such as Greiner's Gauge theory of Weak Interactions) a different boosted vector which is just what you mentioned in your answer, $$\mathbf{w} = m \mathbf{s} + \frac{ \mathbf{p}\cdot\mathbf{s}}{p_0+m}\mathbf{p}$$ If this is an approximation of a simple Lorentz Boost I cant seem to obtain it, could you show me explicitly where does this former expression comes from?


title: Is an alpha particle's curvature in a magnetic field visible with a homemade cloud chamber?

text: I'm trying to do some calculations to see just how strong a magnet you'd have to have, in order for curvature to be noticeable in a rudimentary cloud chamber, with lead-210 as an alpha particle source. I'm guessing that this would have to be huge, but I'm not getting through my back of the envelope calculations successfully. Starting from the tensor expression: $$\frac{d^2 x^\mu}{d\tau^2}=\frac{q}{m}F^\mu_\nu \frac{dx^\nu}{d\tau}$$ and expanding using the relation $d\tau=\frac{dt}{\gamma}$, I find: $$\frac{q}{m}F^\mu_\nu \frac{dx^\nu}{dt}=\gamma\ddot{x}^\mu+\dot{x}^\mu \frac{d}{dt}\gamma$$ Assuming the electric field is zero, no work is done, so $\frac{d\gamma}{dt}$=0, this gives in vector form (zero $ct$ component): $$\gamma^{-1} \frac{q}{m}\dot{\vec{x}}\times \vec{B}=\ddot{\vec{x}}$$ Plugging numbers in, with the mass being about four times that of a proton, and the charge being twice that of a proton, and the energy released by lead 210 in alpha decay $3792 \text{KeV}$, I find that the emitted particles have a velocity around $\frac{1}{20}c$, and I get an acceleration on the order of $10^{15}$ in a 1 tesla field, and calculating $v^2/a$ for the radius of curvature gives me a radius of .28 meters. I'm skeptical of my result because it seems great, but I haven't seen any images or stories of this being tried with a homemade cloud chamber and a neodymium magnet. Is this result correct? Is there something I'm not taking into account? It would be disappointing to set the experiment up and see no result! (I can post the Mathematica code used to query these values and calculate everything, on request.)


title: Object with friction in circular motion caused by a string

text: A physics problem in my textbook reads: > A 0.40kg ball, attached to the end of a horizontal cord, is rotated in a > circle of radius 1.3m on a frictionless horizontal surface. If the cord will > break when the tension in it exceeds 60N, what is the maximum speed the ball > can have? How would your answer be affected if there were friction? Obviously the first question is easy to calculate. But the second one gave me some trouble. The book answer states that friction would not affect the problem, however I believe it would. In order for a limp cord to accelerate the ball in uniform circular motion, the force moving the cord would have to go in a circle of its own if there were friction. Below I drew a picture of my idea of the problem. (The text in the middle says center of rotation). ![Visualization I created of my scenario](http://i.stack.imgur.com/OJlaT.png) You can see that the net force has to be greater than it would otherwise have been because of friction. Assuming that the book answer is wrong though, I have another question about the diagram I drew. Would the force of friction act tangent to the circle as I have indicated below? Or would it behave differently?


title: Trying to link Eulerian and Lagrangian perturbations

text: I'm trying to make sense of the link between a (linear) Eulerian (i.e., at a given point) and Lagrangian (following a fluid element) perturbation. I will here express not only where I'm stuck, but also my current understanding of the topic, so I beg that if you find any misconception, please point it out! **>> Background** So far, I (think I) know that in the **Eulerian** perturbation, if $h(\boldsymbol{r},t)$ is the perturbed quantity, where $\boldsymbol{r}=\boldsymbol{r}_0+\boldsymbol{\delta r}$ is the new position vector and $\boldsymbol{\delta r}$ is a small displacement around $\boldsymbol{r}_0$, then $$h(\boldsymbol{r},t)=h_0(\boldsymbol{r})+h'(\boldsymbol{r},t),\ \ \ \ \ \ \ \ (1)$$ where $h_0(\boldsymbol{r})$ is the quantity at an equilibrium state evaluated at $\boldsymbol{r}$ and $h'(\boldsymbol{r},t)$ is a small perturbation. On the other hand, the **Lagrangian** perturbation $\delta h(\boldsymbol{r})$ can be written as $$\delta h(\boldsymbol{r})=h(\boldsymbol{r}_0+\boldsymbol{\delta r})-h_0(\boldsymbol{r}_0)=\left[h(\boldsymbol{r}_0)+\boldsymbol{\delta r}\cdot \nabla h_0(\boldsymbol{r}_0)\right]-h_0(\boldsymbol{r})$$ or, by using the relation found for the Eulerian perturbation, eq. $(1)$, evaluated at $\boldsymbol{r}_0$, $$\delta h(\boldsymbol{r})=h'(\boldsymbol{r}_0)+\boldsymbol{\delta r}\cdot \nabla h_0(\boldsymbol{r}_0).$$ **>> The question** What I have trouble relating is $\delta h(\boldsymbol{r})$ and $h'(\boldsymbol{r},t)$. Both are small perturbations but, in this case, there is a explicit dependance on $\boldsymbol{r}_0$ by $\delta h(\boldsymbol{r})$. However, if I fix $\boldsymbol{r}_0$, both represent a perturbation about that point so...why are they different? My intuition (probably messed up by know; thanks QM) tells me that they are equal, but the notation is different in almost every book I've read on the topic. Can anyone shine some light here?


title: Is there any physics that cannot be expressed in terms of Lagrange equations?

text: A lot of physics, such as classical mechanics, General Relativity, Quantum Mechanics etc can be expressed in terms of Lagrangian Mechanics and Hamiltonian Principles. But sometimes I just can't help wonder whether is it ever _possible_ (in the future, maybe) to discover a physical law that can't be expressed in terms of Lagrangian Equations? Or to put it in other words, can we list down for all the physical laws that can be expressed in terms of Lagrangian equations, what are the mathematical characteristics of them( such as it must not contain derivatives higher than 2, all the solutions must be linear etc)?


title: Why is there a deep mysterious relation between string theory and number theory, elliptic curves, $E_8$ and the Monster group?

text: Why is there a deep mysterious relation between string theory and number theory (Langlands program), elliptic curves, modular functions, the exceptional group $E_8$, and the Monster group as in Monstrous Moonshine? Surely it's not just a coincidence in the Platonic world of mathematics. Granted this may not be fully answerable given the current state of knowledge, but are there any hints/plausibility arguments that might illuminate the connections?


title: Superconformal algebra

text: I had earlier also asked a question about super conformal theories and I am continuing with that, now with more specific examples. I am quite puzzled with it given that I see no book explaining even the basics of this. I am merely picking this up from stray notes and papers and mostly discussions. So for ${\cal N} = 2$ superconformal algebra in $2+1$ dimensions the symmetry group was $SO(3,2)\times SO(2)$ and possibly the primary states of this algebra are labelled by the tuple $(\Delta, j,h)$ where $\Delta$ is the scaling dimension and $j$ is the spin and $h$ is its $R$ charge (or whatever it means to call it the $R$ charge highest weight) I guess that in this context by “primary state” one would mean those which are annihilated by the special conformal operators (K) as well as the $S$ operators of the algebra. Though confusingly enough the terminology seems to be used for operators as well.   * I would like to know where can I find the complete superconformal algebra to be listed which can enable me to find the $(\Delta, j,h)$ of the states which are gotten by acting other operators n the primaries. I guess its only the momentum operators and the supersymmetry that is left to create the tower of the “descendants”  I guess that once one knows the above algebra I can understand these mysterious tables that I see listing the “states” corresponding to the supersymmetry operators. If for the identity operator the tuple is $(0,0,0)$ then I would like to understand how for the $Q$ the possible labels are apparently $(0.5,\pm 0.5,\pm 1)$ and for $Q^2$ the labels are $(1,0,\pm 2)$, $(1,0,0)$ and $(1,1,0)$. For $Q^3$ the labels are $(1.5,0.5,\pm 1)$ and for $Q^4$ the labels are $(2,0,0)$. I would like to know how the above labels for the operators are derived. It gets harder for me to read this since for the operators the R symmetry as well as the spinorial index are both being suppressed.   * Now consider a primary labelled by $(\Delta, j,h)$ such that it is in the long representation and hence $\Delta >j+\vert h\vert +1$. Then I see people listing something called the “conformal content” of this representation labelled by the above state. For the above case the conformal content apparently consists of the following states, $(\Delta, j,h)$, $(\Delta+0.5, j\pm 0.5,h\pm 1)$, $(\Delta + 1 , j,h \pm 2)$, $(\Delta +1 , j+1,h)$, twice $(\Delta + 1, j,h)$, $(\Delta + 1, j-1,h)$, $(\Delta + 1.5, j\pm 0.5,h \pm 1)$ and $(\Delta + 2, j,h)$ I would like to know what exactly is the definition of “conformal content” and how are lists like the above computed. Similar lists can be constructed for various kinds of short representations like those labelled by $(j+h+1,j,h)$ ($j, h \neq 0$), by $(j+1,j,0)$, by $(h,0,h)$, by $(0.5,0,\pm 0.5)$, by $(h+1,0,h)$ and $(1,0,0)$. Its not completely clear to me a priori as to why some of these states had to be taken out separately from the general case, but I guess if I am explained the above two queries I would be able to understand the complete construction.


title: Why can't massless particle exceed speed of light?

text: Why massless particle can't exceed speed of light?


title: How accelerometers sense constant velocity movements

text: There is something I don't understand about accelerometers. I know it's possible to make an Inertial Measurement Unit (IMU) by using "three" accelerometers. So with that, I could calculate the $x$, $y$, $z$ coordinates of something in 3-D space. The point that I don't understand is, how could it sense constant-speed movements? When the velocity is constant, acceleration is zero. So how does it work?


title: Glass melting at near absolute zero?

text: I read this report and summarise here but my question is - if quantum mechanics will make glass melt at temperatures near absolute zero and it is near absolute zero then wouldn't this be a huge issue for satellites and space shuttles? http://www.sciencedaily.com/releases/2011/02/110202102748.htm Scientists Use Quantum Mechanics to Show That Glass Will Melt Near Absolute Zero ScienceDaily (Feb. 4, 2011) Prof. Eran Rabani of Tel Aviv University's School of Chemistry and his colleagues at Columbia University have discovered a new quantum mechanical effect with glass-forming liquids. They've determined that it's possible to melt glass -- not by heating it, but by cooling it to a temperature near absolute zero. This new basic science research, to be published in Nature Physics, has limited practical application so far, says Prof. Rabani. But knowing why materials behave as they do paves the way for breakthroughs of the future. "The interesting story here," says Prof. Rabani, "is that by quantum effect, we can melt glass by cooling it. Normally, we melt glasses with heat."


title: Steady State Temperature of an Object Orbiting the Earth

text: This may be irrelevant or stupid to ask but I couldn't come up with a good answer. At least, we could not agree on with my friend the other day. I would like an estimate of the temperature of a human body orbiting the earth. I know this may seem strange and a bit funny.. From what I know or read (or maybe 'guess' would be a better word), any body of mass would cool down eventually because it radiates (i.e. most of the objects in space are cool). OK. But imagine that a body is accidentally thrown from a space shuttle orbiting the earth, what would happen? I guess first of all it depends on how it receives the sunlight.. Anyways, thanks for your answers! **A FOLLOW UP QUESTION** According to @zephyr's answer, the steady state temperature would be around $271 K$, so in which distance to the sun one body should be to have a steady state temperature of (say) $295 K$? Or doesn't the distance make a difference?


title: Lagrangian for Relativistic Dust derivation questions

text: In most GR textbooks, one derives the stress energy tensor for relativistic dust: $$ T_{\mu\nu} = \rho v_\mu v_\nu $$ And then one puts this on the right hand side of the Einstein's equations. I would like to derive this from some action. I have read through all standard GR textbooks, and the only one which talks about this is General Theory of Relativity by P. Dirac. If you don't have the book, let me quickly reproduce the derivation here. The action for the relativistic dust is: $$ S_M = -\int \rho c \sqrt{v_\mu v^\mu} \sqrt{ |\det g| } d^4 x = -\int c \sqrt{p_\mu p^\mu} d^4 x $$ where $p^\mu=\rho v^\mu \sqrt{ |\det g| }$ is the 4-momentum _density_. Now we vary with respect to $g^{\mu\nu}$ as follows: $$ \delta S_M = - \delta \int c \sqrt{{p}_\mu {p}^\mu} d^4 x = $$ $$ = - \int c {\delta(g^{\mu\nu} {p}_\mu {p}_\nu) \over 2\sqrt{{p}_\alpha {p}^\alpha}} d^4 x = $$ $$ = - \int c { {p}_\mu {p}_\nu \over 2\sqrt{{p}_\alpha {p}^\alpha}} \delta(g^{\mu\nu}) d^4 x = $$ $$ = - \int c { \rho v_\mu \rho v_\nu \sqrt{ |\det g| }^2 \over 2 \rho c \sqrt{ |\det g| } } \delta(g^{\mu\nu}) d^4 x = $$ $$ = - \int {1\over2} \rho v_\mu v_\nu \delta(g^{\mu\nu}) \sqrt{ |\det g| } d^4 x $$ From which we calculate the stress energy tensor using the standard GR formula for it: $$ T_{\mu\nu} = - {2\over\sqrt{ |\det g| }}{\delta S_M\over\delta g^{\mu\nu}} = $$ $$ = - {2\over\sqrt{ |\det g| }} \left( -{1\over2} \rho v_\mu v_\nu \sqrt{ |\det g| } \right)= $$ $$ = \rho v_\mu v_\nu $$ If we vary with respect to $x^\mu$, we obtain the geodesic equation (the calculation is lengthy, see for example here, or the Dirac book). I'll be happy to clarify any of the derivations above if needed. Now my questions: 1) Why isn't this in every GR textbook? Is there some problem with the derivation? 2) The geodesic equation follows from this action $S_M$. Standard way to derive the geodesic equation is to maximize the proper time $\tau = \int d \tau = \int \sqrt{-{1\over c^2} d s^2} = \int \sqrt{-{1\over c^2} g_{\mu\nu} d x^\mu d x^\nu} $. Is there some relation between this $\tau$ and $S_M$ since both give us the same geodesic equation? 3) Is it correct to simply say, that all GR for relativistic dust (without electromagnetism) follows from this action: $$ S = {c^4\over 16\pi G} \int R \sqrt{ |\det g_{\mu\nu}| } d^4 x -\int c \sqrt{p_\mu p^\mu} d^4 x $$ when varied with respect to $g^{\mu\nu}$ it gives the Einstein's equations with the $T_{\mu\nu} = \rho v_\mu v_\nu$ tensor on the right hand side, when varied with respect to $x^\mu$ it gives us the geodesic equation (for each particle of the dust). 4) Why do we need to hide the $\sqrt{ |\det g_{\mu\nu}| }$ in the 4-momentum _density_ and do not vary it? Dirac says that it is because $\rho$ and $v^\mu$ are not independent quantities when varying, but I don't understand the argument. 5) The standard way is to use the Hilbert action, the action for the elmag field, the stress energy tensor for dust and derive the geodesic equation as a conservation of the stress energy tensor, that follows from the conservation of the Einstein's tensor. How is this approach related to the above? Isn't it physically better to simply postulate the total action and derive everything from it? Note: Dirac shows how to incorporate electromagnetism by simply using the standard action for it and the procedure above then gives the correct elmag. tensor on the right hand side of Einstein's equations and the Lorentz force at the right hand side of the geodesic equation, as well as Maxwell's equations for the elmag field.


title: Dimensional Analysis: Buckingham Pi Theorem

text: I am studying for a fluids quiz and I am having a few problems relating to dimensional analysis but for the time being fundamentally I have a problem selecting the repeating variables. Like does anyone know an effective way of determining them or an approximate list one should start with? In particular if one looks here ![Problem 7.5R](http://i.stack.imgur.com/C5Tkk.png) Source: http://instructional1.calstatela.edu/ckhachi/CE303/Ch7%20ReviewProb_Solutions.pdf I don't understand two things:   1. How do you choose what repeating variables to use - my assumption based on what I've read in the text and online is you want something that is not dimesionless, that is the variable must not be and when you put it all together it cannot be dimessionless.   2. Now assuming I understand that correctly, do you always just set it equal to $F^0L^0T^0$? Sidenote - so one thing I couldn't have done is chosen D and l as repeating variables, because they are the same thing basically. Apart from that I could have done, any combination of $ \mu, V, D, or \sigma $ correct?


title: What does an electron's wavevector mean inside of a crystal?

text: With a plane wave, I always took the direction of the wavevector, $k$, as the direction of propogation (magnitude proportional to the inverse wavelength). Alternatively, it could represent the momentum (minus a factor $\hbar$) of a particle. However inside a crystal, the electron wavevector and the electron velocity are not necessarily in the same direction. I'm thinking here of a 2D material with a cylindrical Fermi surface where the momentum may have a z component, but the Fermi velocity does not. In everyday cases you would expect momentum and velocity to be in the same direction, moreover I considered the propogation of the wave to be in the same direction as its particle analogue. I realise that inside a crystal the electrons are no longer simple plane waves, but what then does the $k$ vector mean?


title: Will the earth never cool, but only get hotter?

text: Since the earth is in a vacuum and therefore there is no thermal transfer of heat to anything else, how can it even cool down? It seems like its average temperature would always be constant, ignoring outside sources of heat. However, if you then consider that there is constant radiant heat transfer from the sun to the earth, now you have a net gain in energy/heat. I also read somewhere that the solar wind actually blows a bit of our atmosphere into space, but that seems like a miniscule amount of heat loss compared to how much heat is gained from the sun. Will the earth ever cool at all?


title: Spontaneous pair production?

text: So I've been looking into particle-antiparticle pair production from a gamma ray and don't understand one thing. Let's say I have a 1,1 MeV photon and it hits a nucleus - electron-positron pair with some momentum will be created and the nucleus will probably get some momentum as well because of the impact. But why does the photon need the nucleus at all? Why can't it just fly through space and suddenly, with some probability, change into a electron-positron pair with momentum? I see that the momentum of the system wouldn't be conserved but I don't really understand how the nucleus helps it. Thank you a lot!


title: Correlation Function of ground state; Physical Meaning

text: I was asked to find the correlation function of the ground state of the QHM: $$\langle0|\hat x(t)\hat x(t-\tau)|0\rangle$$ I found that this evaluated to $\frac{\hbar}{2m\omega}e^{i\omega \tau}$. I was then asked to take the fourier transform of this (w.r.t $\tau$), which evaluates to $ \delta(x-\omega)$ (times constants). I am curious as to what the interpretation of this would be? The correlation function shows that the number states of the harmonic oscillator oscillate, even though expectation of x and p vanish. But the fourier spectrum of x would suggest that it only oscillates at a given frequency?


title: Information Retrieval

text: This question is motivated by the issue of information retrieval from black holes, but it is essentially a question about quantum information. It is widely believed (in certain circles) that the information about the history of black hole formation is continuously leaked out as the black hole is evaporated, and is encoded by the Hawking radiation quanta. This way one does not lose information when the black hole is completely evaporated, and furthermore one does not need to postulate some implausible ways in which macroscopic amount of information is stored in an increasingly microscopic object, only to be somehow released at the very last stage of the evaporation process. To make this more quantitative, I am wondering if similar questions were asked in the quantum information community. Suppose you have an encrypted message of a certain length which is revealed to you over time, and you are trying to decode it. Obviously the more time passes the more information you can have about the message, and when you reach a time which scale with the size of the message you are expected to gain access to the full information. I'm looking for some more quatitative knowledge about this question. For example: For a generic decoding, is it known how much time you need in order to gain access to a finite fraction of the information? Are there some universal results about the asymptotics of such process (in the spirit of "critical exponents")? Are there bounds on optimal encoding, whose purpose would be to delay a release of a finite fraction of the information to later and later times? In particular, can there be encoding which releases a finite fraction of the information only at the very last stages of the decoding? I realize this is probably an hopelessly pedestrian question, pointing me towards some review literature might be the way to go in such case. Edit: Thanks everyone for your answers, they all were useful in different ways. The question was vague enough for it not to have a single correct answer, but I'll choose Peter's because he somehow managed to read my mind (though there was no way to tell from the way the question was phrased). Hope to ask more precise questions on related topics in the future.


title: Relative motion and how to calculate magnitude and velocity of objects relative to different given objects

text: A canoe has a velocity of 0.550m/s southeast relative to the earth. The canoe is on a river that is flowing at 0.540m/s east relative to the earth. ![enter image description here](http://i.stack.imgur.com/KtcSL.jpg) Alright so I attempted many different things an still do not understand the nature of the question nor do I understand how to get an answer. Find the magnitude of Vc/r of the canoe relative to the river now the next question Find the direction of the velocity of the canoe relative to river expressed as an angle measured south of west. Now I figured that the angle would be equal to theta if you laid our the vectors tail to tip with -Vr/e \+ Vc/e however I have no earthly idea on how to calculate this other than the equation: arcsin(Ay/A with the resulting vector of -Vr/e \+ Vc/e being defined as A and Ay being the y component of A but im not sure how to calculate the y component of A. If this question doesn't make much sense I apologize I am struggling with the concept as a whole.


title: Derivation of cylindrical line heat source problem?

text: I have a line heat source embedded inside a cylinder and I am trying to find the temperature distribution T(r,t). By using the similarity variable, the solution to the differential equation is $$\theta(\eta)=C_{1}\int_0^\eta \\! \frac{e^{-u}}{u} \, \mathrm{d}u + C_{2}$$ and my boundary conditions are $-4\pi kh\frac{\partial\theta}{\partial \eta}|_{\eta=0}=q_{0}$ and $\theta\to0$ as $\eta\to\infty$. When I apply the first boundary condition using the derivative I get $1/0$ which is indeterminate so what do I do there? Also, the second boundary condition is supposed to give the exponential integral function, but I am not sure where that comes from. I guess I am stuck on applying the boundary conditions to this. The derivation says the final answer should be $$\theta(\eta)=\frac{-q_{0}}{4\pi h k}\int_\eta^\infty \\! \frac{e^{-u}}{u} \, \mathrm{d}u$$


title: Calculating the velocity needed to make a distance in the horizontal with a given height and no upward motion

text: In 1780, in what is now referred to as "Brady's Leap," Captain Sam Brady of the U.S. Continental Army escaped certain death from his enemies by running over the edge of the cliff above Ohio's Cuyahoga River, which is confined at that spot to a gorge. He landed safely on the far side of the river. It was reported that he leapt 22ft across while falling 20ft ![enter image description here](http://i.stack.imgur.com/7MH00.jpg) Now how do I calculate the minimum speed needed to make the "jump"?


title: Time difference in transmission of AC and DC

text: Is there any time difference while considering the transmission of AC and DC ? Is either one of them faster than the other one in terms of transmission of energy/power from one point to another? I mean to say that suppose there are two power stations very very far apart, if one station turns on the switch for the transmission to start, would it matter that DC has constant field, and AC has alternating field for the time taken in transferring energy from one station to another ? Addendum : to elaborate the question i would add that as current in DC is related to drift velocity of electrons which is not invloved in the transmission during AC, my question is rather related to the flow of electrons. To clarify a little more, do the electrons present right at the tip of the other point of transmission cover same distance while vibrating(travelling distance of amplitude) or drifting in same time ?


title: Meaning of probability in a multiverse/a many-world interpretation?

text: Consider me tossing a coin and I got tail as a result on observing it. Then, what would be the result of the 'parallel me' in another universe? If the 'parallel me' gets head as a result then, considering all the parallel universes the probability of occurrence of any event would be 1! In that case the whole concept of probability (considering all the parallel universes) goes wrong. What's your opinion about this or am I thinking in the wrong way? (I actually asked this question while participating in 'I'm a Scientist get Me Out of Here' but I would also like the views of Physics SE members.)


title: Education : Students devising and solving their own problems

text: I apologize in advance for the nature of this question. I have been thinking lately of methods to increase the quality of my self- study. I have been meaning, but never actually got to, in addition to solving numerous problems in the textbooks, to construct and solve my own problems, based on the material taught in each chapter. Would that be helpful, educationally, or would it be a waste of time? Is this practice carried out in general?


title: Laplace operator's interpretation

text: What is your interpretation of Laplace operator? When evaluating Laplacian of some scalar field at a given point one can get a value. What does this value tell us about the field or it's behaviour in the given spot? I can grasp the meaning of gradient and divergence. But viewing Laplace operator as divergence of gradient gives me interpretation "sources of gradient" which to be honest doesn't make sense to me. It seems a bit easier to interpret Laplacian in certain physical situations or to interpret Laplace's equation, that might be a good place to start. Or misleading. I seek an interpretation that would be as universal as gradients interpretation seems to me - applicable, correct and understandable on any scalar field.


title: Can a photon be emitted with a wavelength > 299,792,458 meters, and would this violate c?

text: Just curious if the possibility exists (not necessarily spontaneously) for a photon with a wavelength greater than the distance component of c to be emitted, and would this inherently violate the scalar c?


title: Finding the mass of bed/Patient

text: One of my friends has a problem and we don't know how to get this done, We want to to know the mass of a patient who is laying on a bed. One scales was put under the foot of a bed and weighed 232kg and the other under the head of the bed and weighed 220kg. The centre mass of the patient is 0.8 meters from the foot of the bed. What is the mass of the patient?


title: The commutator of Killing vectors

text: I'm going over an assignment for my general relativity course. My solution to the question below strikes me as too short, considering that it appeared in the "longer questions" section of the assignment. > **Question:** > > A Killing vector $V^\mu$ is one that satisfies the equation $\nabla_{\mu} > V_{\nu}+\nabla_{\nu} V_{\mu}=0$. The commutator of two vectors $U^\mu$ and > $V^\mu$ is defined as $[U,V]^\mu=U^\nu \nabla_\nu V^\mu-U^\nu \nabla_\nu > U^\mu$. Show that the commutator of two Killing vectors is itself a Killing > vector. My solution could be (and is likely to be) incorrect, but I'm just using the definitions that I've seen so far. **My solution:** To get from $[U,V]^\mu$ to $[U,V]_\mu$ we have to use the metric i.e. $[U,V]_\nu=g_{\nu \lambda}[U,V]^\lambda$. Now I just plugged this in to the Killing equation to get (for the first term) $\nabla_{\mu}[U,V]_\nu=\nabla_{\mu}g_{\nu \lambda}[U,V]^\lambda$. Using that the metric is covariantly constant, we have $\nabla_{\mu}g_{\nu \lambda}=0$, hence the whole term is equal to zero. The same thing happens with the second term, so that the whole expression is zero. I can't see that I made a mistake, but it seems way too short a solution. When I looked at the solutions provided in the course, they are about a page long, and begin $$\nabla_{\mu}[U,V]_\nu + \nabla_\nu[U,V]_\mu=\nabla_\mu (U^\tau \nabla_\tau V_\nu-V^\tau \nabla_\tau U_\nu ) + \nabla_\nu (U^\tau \nabla_\tau V_\mu-V^\tau \nabla_\tau U_\mu )=\dots$$ i.e. they do things explicitly. Actually, having written this out, I can see that what I've done can't be correct otherwise every vector would be a killing vector just by writing $V_\mu$ as $V_\mu=g_{\mu \nu}V^\nu$. However I don't know what I've done wrong, so any help would be very much appreciated!


title: Vanishing of the Ricci tensor in higher spacetime dimensions

text: I understand how, if the Riemann tensor is 0 in all its components, since we construct the Ricci tensor by contracting the Riemann, Ricci tensor would be 0 in all components as well. I've read that vanishing of the Ricci tensor in 3 spacetime dimensions implies the vanishing of the Riemann curvature tensor, but that in higher dimensions that does not hold. Can somebody explain why is that so? Is it because we have more independent components of the Riemann tensor in 4 spacetime dimension, than in 3 (20 vs 6)? Also if the number of independent components of Riemann tensor in $n$ spacetime dimensions is $$N(n)=\frac{n^2(n^2-1)}{12}$$ and since we know that the Riemann tensor has 256 components, does that limit the spacetime dimensions of it's usage? Or does that mean, that for example in 10 spacetime dimensions, there won't be any independent components of Riemann tensor?


title: How much more efficient is a road bike than a mountain bike?

text: What would be a good estimate on the difference of efficiency between a road bike and a mountain bike? A number of links cite all the usual reasons: thinner tires, better aerodynamics... But I'm not able to find one where they quantify this efficiency


title: Simultaneous conservation of linear and angular momentum

text: Suppose there is a ring in the space where there is no gravity. The width of the ring is $r$ which is negligible compared to its inner radius $R$. The ring is in horizontal position. Now imagine a human starts to move on this ring. Before the start of the motion angular and linear momentum are zero and system is isolated. When the human walks on the ring linear and angular momentum must remain zero. $$m_{human}*R^2*\omega_{human} = -M_{ring}*R^2*\omega_{ring} \rightarrow \omega_{human}=-\frac{M_{ring}}{m_{human}}*\omega_{ring}$$ There is no problem with this part. Now we consider the linear momentum. The linear momentum of the ring after the beginning of motion is zero because for every element of mass there is another element of mass moving in the opposite direction. However, the linear momentum of the human which can be considered a point mass is not zero. The magnitude of the linear momentum of the human is $ m_{human} \cdot \omega_{human} \times R$. So the total linear momentum is not zero despite the fact that the initial linear momentum was zero and the system is isolated. If there is not anything wrong I have to conclude that $\omega_{human} = 0$. What am I missing here?


title: Diffraction between two beams of right- and left-circularly polarized light

text: If we have an optical / microwave two-way relay, my understanding is that one strategy to avoid destructive interference-based signal degradation is to use orthogonal linear polarizations. Here, optical signals from one (transmitting) antenna are rotated 90 degrees (pi/2 radians) with respect to optical signals from the other (transmitting) antenna. However, what happens when two beams of circularly polarized light interact? Say we emit signals with left-circular polarization from one (transmitting) antenna, and either left-circular or right-circular polarization light from the other (transmitting) antenna? What happens with regards to destructive interference and signal degradation? I ask because it seems to me that circularly polarized light is more "robust" to atmospheric or environmental perturbations that could otherwise effect specific linear polarizations. So my guess is that there must be consequences for using circularly polarized light in a two-way optical relay (unless its just a matter of required power or antenna cost/complexity).


title: How does a cyclotron work?

text: Doing high school level physics and I got stuck at a question regarding a cyclotron. I need to know how does it work and the reason alternating current is used and the structure of the two semi circle of the cyclotron.


title: Dual Pairs in Four Dimensions

text: Following the conversation here, I am wondering if anyone knows of an example of dual pair with 4-dimensional N=1 SUSY which relates a non-Abelian gauge theory on one side to a theory with a Lagrangian description but no non- trivial gauge group. Cannot think of one off the top of my head, which doesn't mean it does not exist or even is well-known...


title: Finding the electric field for a shell of charge

text: Suppose we have charge density defined $$ \rho(x,y,z) = \begin{cases} 0 & 0 \leq r < a \\\ K & a \leq r\leq b\\\ 0 & b< r \end{cases} $$ For some constants $K,a,b$ How would we find the electric field for all points in space? Any help I would greatly appreciate! I imagine we argue that the field must be symmetric, but I was having difficulty proceeding from there.


title: How to calculate the kinetic energy of supernova ejecta?

text: I believed I could use $0.5\times M_{ej} V_{ej}^2$, with $M_{ej}$ being the ejected mass and $V_{ej}$ being the velocity of the ejected mass. But I noticed in this and this that the the mean velocity is somehow calculated as following : $V_{mean}=(3/5)^{0.5}\times V_{ej}$. And so the kinetic energy was calculated as : $(3/5)(0.5)\times M_{ej} V_{mean}^2 =(3/10)\times M_{ej} V_{mean}^2$. So is there any explanation for how this factor of $0.3$ or the mean velocity was calculated ?


title: Are there two aspects of Born's rule?

text: I am having some problem understanding Born's rule. I am getting a little bit confused. Here it goes; Let $f(x,t)$ be a solution of Schrodinger equation. Then Born's rule says that the square modulus of $f(x,t)$ gives the probability of locating a particle at position $x$ at time $t$. Since quantum mechanics also says that a quantum state in superposition collapses to one of its eigenvalues upon observation, then I understood $x$ as an eigenvalue corresponding to the position operator and that Dirac-delta function is its eigenstate. But then I do encounter another form of Born's rule which states that given that $g(x,t)$ is a linear combination of quantum states superimposed on each other, then the probability of $g(x,t)$ collapsing into one of its eigenfunctions, say $h(x,t)$ is the square modulus of the dot product of $g(x,t)$ and $h(x,t)$. Yes, I do understand how this dot product yields the coeficient of $h(x,t)$ but I don't understand well the connection between this latter rule and the former. For instance, is it correct to say that the solution to Schrodinger equation, $y(x,t)$ is a linear combination of Gaussian bell functions that tends to Dirac-delta functions as we increase the certainty of the positions? If yes, is the square modulus of $y(x,t)$ equals to the square modulus of the dot product of $y(x,t)$ at the Dirac-delta function with eigenvalue of $x$ at time $t$?


title: What is the definition of density as a function?

text: (Before I start, I don't know which tag is suitable for this post. Please retag my post if it bothers you.) Let's say there is a string on $[0,1]$ with a mass given by $m(x)$. ($m(x)$ means the mass contained in the interval $[0,x]$.) Then the inertia momentum of this string is given by, $$\int_0^1 x^2 dm(x).$$ (This is a Riemann Stieltjes Integral.) If $m$ is differentiable on $[0,1]$, this is exactly the same as $\int_0^1 x^2 m'(x) dx$. I thought the definition of 'density at a point' is given by 'derivative of mass', but I think this definition is not sufficient for many cases. Is there a precise definition of density at a point?


title: Why isn't the product rule used in the definition of mechanical work?

text: Mechanical power is normally defined as $P = \mathrm{d}W/\mathrm{d}t$, and work is normally defined as $W = \vec F \cdot \vec x$. Today an undergrad pointed out a confusion he had from Griffiths' E&M book where he has the line $$P = \frac{\mathrm{d}W}{\mathrm{d}t} = \vec F \cdot \vec v$$ Which is a definition of power I had seen before. But the student was confused because it seems like, if you have a time dependent force $\vec F(t)$, the math should work out like: $$P = \frac{\mathrm{d}W}{\mathrm{d}t} = \frac{\mathrm{d}\vec F(t)}{\mathrm{d}t} \cdot \vec x + \vec F \cdot \frac{\mathrm{d}\vec x(t)}{\mathrm{d}t}$$ (From the product rule.) However, I've never seen the above formula, so I'm guessing it's wrong. Also, it clashes with the $\vec F \cdot \vec v$ version that I'm pretty sure works completely fine with a time dependent force. I gave him a pretty weak answer and warned him that it's probably not correct: I told him that if you start with the differential form of work, $dW = \vec F \cdot d \vec x$, it seems to assume that the force stays the same in that tiny $d\vec x$, so it's constant there, and then dividing both by $dt$ gives you the equation we're looking for. Wikipedia seems to say something similar, basically using the step of $d\vec x = \vec v \ dt$ to get $\vec F \cdot \vec v$, which also assumes that $\vec v$ is constant over $d \vec x$. So, assuming one of those are correct, I see how they get $\vec F \cdot \vec v$. But what's the flaw in the product rule thing?


title: What was the muzzle velocity of a home made gun launch strait in the air if air time was 8.2sec?

text: I built a potato gun and wanted to calculate the muzzle velocity. I remember from physics that I could run the numbers by calculating time from launch until landing. After pointing strait into the air and launching the potato was air-born for 8.2sec. Without air resistance that comes out to a muzzle velocity of around 90mph which is probably consistent as we can't see the projectile leave the muzzle. I would like to get a more accurate calculation of what that would be with air resistance. Assuming a spherical potato that is 4ounces with a 2" diameter. Launched at 90 degrees with a 8.2sec air time.


title: Force on a potentially rolling ball initially at rest

text: I'm a graduate student in mechatronics, but I give myself headaches on this astonishingly simple 2D problem. Imagine a ball (perfectly rigid) on an horizontal plane (perfectly rigid), initially at rest. The only forces acting on it are its weight $P$ and the normal reaction of the plane. Now, at time $t = 0$, I apply an additional horizontal force to the ball $F$, big enough ($F > \mu \times P$) in its centre of mass. An opposite tangential friction force appears ($T = \mu \times P$). And the problem begins! Since the horizontal forces aren't equal, the ball will accelerate linearly. Moreover, the tangential force create a torque, and the ball will also rotate. But because we are still at time $t = 0_+$, both the linear and the angular velocity are nil. So the cinematic rolling-without-slipping condition $v = R \times \omega$ is fulfilled, so in fact, there is no friction and the ball is only slipping… That's pretty paradoxical. It's easy to know if the ball will move or stay immobile. But how can I determine if the ball will slipping and rolling, or just rolling without slipping?


title: Does GR provide a maximum electric field limit?

text: Does GR provide a limit to the maximum electric field? I've gotten conflicting information regarding this, and am quite confused. I will try to quote exactly when possible so as not to confuse things more with my paraphrasing. The author of the Motion Mountain physics textbook claims in his book there is a limit, and clarifies on his site that ( http://www.motionmountain.net/wiki/index.php?title=Dislike_Page ): > "electromagnetic fields are limited in magnitude. Now, every electromagnetic > field contains energy, and energy density is limited by general relativity: > if energy density is too high, a black hole appears. The smallest possible > black hole then leads to a field limit. If you deny an upper field limit, > you deny general relativity. However, general relativity has been confirmed > in every experiment so far." This sounds very obvious and intuitive to me. However one of my physics TA's got very upset when I used this in a thought experiment when discussing some limits in physics. When I told him the textbook I got that from, he looked it up, and commented on the Motion Mountain wiki website his argument: > "I'd like to add something here. Suppose for a moment the energy density > limit is correct, then if an object is sped up until it length contracts > enough such that the energy density is greater than this limit, does it turn > into a black hole? No, quoting John Baez "The answer is that a black hole > does not form"* > http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/black_fast.html > *The claim that "if energy density is too high, a black hole appears" is an > incorrect oversimplification. The issue is that in GR, gravity depends on > more than just the energy density (component T^00 of the stress energy > tensor). So we can't just look at the "relativistic mass" (E/c^2) to judge > whether a black hole forms. It actually isn't even enough to look at the > "invariant mass", for while the trace T of the stress energy tensor for a > particle is just its invariant mass, for an electromagnetic field it is > identically zero even though electromagnetic fields curve spacetime in GR. > So none of these concepts of mass are sufficient when discussing gravity > using GR (especially when considering electromagnetic fields), because > gravity couples to the entire stress energy tensor. I hope this was > helpful." This makes much less sense to me, and I don't understand how energy density tending to infinity could EVER avoid being a black hole. No offense to my TA, but I'm skeptical as he's disagreeing with a textbook author. Plus, the author's response was that my TA is another Einstein denier, and not worth responding to. So I'd like a third party's answer on this. Does GR provide a theoretical limit to the strength of an electric field? Is it best to just ignore my TA on this one?