title: Representation of SU(3) generators

text: Let's discuss about $SU(3)$. I understand that the most important representations (relevant to physics) are the defining and the adjoint. In the defining representation of $SU(3)$; namely $\mathbf{3}$, the Gell-Mann matrices are used to represent the generators $$ \left[T^{A}\right]_{ij} = \dfrac{1}{2}\lambda^{A}, $$ where $T^A$ are the generators and $\lambda^A$ the Gell-Mann matrices. In adjoint representation, on the other hand, an $\mathbf{8}$, the generators are represented by matrices according to $$ \left[ T_{i} \right]_{jk} = -if_{ijk}, $$ where $f_{ijk}$ are the structure constants. My question is this, how can one represent the generators in the $\mathbf{10}$ of $SU(3)$, which corresponds to a symmetric tensor with 3 upper or lower indices (or for that matter how to represent the $\mathbf{6}$ with two symmetric indices). What is the general procedure to represent the generators in an arbitrary representation?


title: Justifying order of magnitude reasoning

text: So in the context of a set of notes I am reading about acoustics I get to equation (23) in this paper. Basically it comes down to showing that ( **note the dots above the a's meaning time derivative!)** $$ f(t) = -a_0c_0\int_{-\infty}^t \dot{a}(t'+\frac{a_0}{c_0})e^{-\frac{c_0}{a_0}(t-t')} dt'\ = -a_0^2\dot{a}(t),. $$ under the assumption that $\frac{c_oT}{a_0} >>1$. $T$ can be thought of as a characteristic time scale for this problem. For those interested in more than just the math, please see the paper. Now I show this in the following way: Integrating by parts: $$ \begin{align} f(t) = & \\\ =& -a_0^2\dot{a}(t+\frac{a_0}{c_0})+a_0c_0\int_{-\infty}^t \ddot{a}(t'+\frac{a_0}{c_0})\frac{a_0}{c_0}e^{-\frac{c_0}{a_0}(t-t')} dt'\\\\\ & \end{align} $$ By expanding $\dot{a}$ as a taylor series we get: $$ \dot{a}(t+\frac{a_0}{c_0}) = \dot{a}(t) +\frac{a_0}{c_0}\ddot{a}(t)+O\left(\frac{a_0^2}{c_0^2}\right) $$ Now if we make the order of magnitude estimate: $$ \frac{a_0}{c_0}\ddot{a}(t) \simeq \frac{a_0}{c_0T}\dot{a}(t) $$ then I hope it is clear by applying this reasoning to the equation for $f(t)$ that we have shown $f(t) = -a_0^2\dot{a}(t)$ for $\frac{c_oT}{a_0} >>1$. **Now here comes my question:** How can I justify the time derivative operator behaving as $1/T$, where $T$ was just given as the length scale of the problem (which seems so arbitrary)? Is my reasoning above correct? I was hoping someone could put my mind at ease about the "hand wavy" nature of these order of magnitude approximations.


title: Shielding magnetic field without mu-metal

text: I need to shield my device from magnetic interference, including earth magnetic field (if you move device around, it might be enough to cause slight currents i guess) and magnetic field caused by power nets, wires with large currents e.t.c . I know this could be achieved by making case out of mu-metal, but it seems that I can't find it anywhere (especially in small quantities). So, are the other, easier ways to shield magnetic field? Will multiple sheets of steel work for example?


title: How do permanent magnets manage to focus field on one side?

text: The actuator of a hard drive head consists of two very strong neodymium magnets, with an electromagnetic coil between them. If you take that apart, the magnets attract each other very strongly. There's no doubt the field between them is very strong. But if you flip them back to back, there is no repulsion force - there is pretty much nothing. While the magnets are very strong on one side, they seem completely inert on the other. I understand how a U-shape of a shaped magnet allows it to focus field near the two exposed poles, but how does a flat rectangular plate manage to keep the field over one flat, wide surface and nearly none near the other? [sorry about the heavy edit but it seems the question got totally muddled with irrelevant discussion about unipolar magnets and possibility or impossibility to remove magnetic field from one side. Only heavy rephrasing may help.] Edit: some photos: 1: The magnets stuck together. They hold really hard, I'd be unable to just pull them apart but I can slide one against the other to separate them. ![1](http://i.stack.imgur.com/pTjpC.jpg) The magnets in "inert position" - they don't act on each other at all, just like two inert pieces of metal. ![2](http://i.stack.imgur.com/0HTae.jpg) The magnets seem to have two poles located on the surface off-center. Here, a normal magnet placed against the hard disk magnet, centering itself on one side, then flipped - on another. ![3](http://i.stack.imgur.com/AvckM.jpg) ![4](http://i.stack.imgur.com/nG9EA.jpg) ![5](http://i.stack.imgur.com/UBtIH.jpg) The metal shield seems to act like completely unmagnetized ferromagnetic. I can stick the "normal magnet" any way I like to it, and it doesn't act on another piece of ferromagnetic (a needle) at all. ![6](http://i.stack.imgur.com/9iqcP.jpg)![7](http://i.stack.imgur.com/Mn1EC.jpg) When I apply a small magnet to it, it becomes magnetic like any normal "soft" ferromagnetic - attracts the needle weakly. It behaves as if the (very powerful) neodymium magnet glued to the other side wasn't there at all. ![8](http://i.stack.imgur.com/yrW8S.jpg) Unfortunately the neodymium magnets are glued very well and so fragile I was unable to separate any without snapping it, and then the "special properties" seem to be gone.


title: Repulsive gravity

text: IANAP, so feel free to berate me for thinking apocryphal thoughts! Just as magnetism has two charges, in which particles of like-charge repulse and particles of dissimilar charge attract, might gravity have two charges in which particles of like-charge attract and particles of dissimilar charge repulse? In practice, the state of magnetism means that there is no system composed of many particles in which all particles attract. Rather, there is a net 0 charge if there are equal numbers of each particle type. My silly theory regarding gravity would mean in practice that there would be two (or more) "clumps" (or universes) in existence, which are racing away from each other. So in our clump (universe) we see only attracting particles, because all the opposing particles have long since separated out and are racing away beyond the boundary of the observable universe. Just like the alien who lands in China and assumes that all humans have slanted eyes, we only observe the attracting particles (or "charge") and disregard the other, unobservable, "charge". Is there any way to disprove this idea, or like string theory can I go one believing it as it can never be disproved? Thanks.


title: Attractiveness of spin 2 gauge theories

text: > **Possible Duplicate:**   >  Why is gravitation force always attractive? I have heard that the attractiveness of gravitation is due to the fact that it is a spin 2 gauge theory. Why is this so? I find this very interesting and would love to understand this fact.


title: Gravitational Force (conceptual)

text: 1. Why is gravitational force always an attractive force?    2. And is the Newtonian formula of gravitational force applicable for very small particles like electrons and protons etc.? From **Formula of Gravitational Force** , I'm referring to: $$F_G = \cfrac{GMm}{R^2}, $$ where $M$ and $m$ are the masses of objects. A logical explanation will be much appreciated.


title: Why gravity is an attractive force?

text: Why gravity is an attractive force? * * * One may say that it is because of space time curvature but General Relativity is built on this law: $\displaystyle G \frac{m_1 \times m_2}{r^2}$ (To be more precise, it is derived from it's potentiel form known as Gauss's law for gravity that can be written like this: $\nabla \cdot g = - 4\pi Gp $). So GR can't explain why gravity is attractive. General Relativity only explains how gravity occurs in terms of space time. * * * Another may stick with the Quantum theory stating that gravitons are responsible for the force of gravity, but why a massless particle with spin 2 will exchange positive momentum?


title: What happens when two smooth surfaces touch?

text: I am wondering what will be the physics to explain how two neutral, chemically nonreactive objects stick. I know that using van der Waals formalism, we can treat neutral body electrodynamic forces and arrive with attractive forces that pull the objects together. Now, once the objects touch (say a mechanical cantiliver in a MEMS sensor like the one used in an iPhone), what happens to the forces? A quantitative answer or some estimate on how strong the attractive force is for simple cases will be very appreciated. in response to anna's comment : Let us consider what happens in vacuum for ultra smooth surfaces, with no residual electrical charge and fully chemically stablized surfaces (example, silicon crystals with stabilised surface bonds).


title: generalizing spin rotations

text: I have a question about the relation: $\exp(-i \vec{\sigma} \cdot \hat{n}\phi/2) = \cos(\phi/2) - i \vec{\sigma} \cdot \hat{n} \sin(\phi/2)$. In my texts, I see $\phi\hat{n}$ always as c-numbers. My question is whether or not this relation can be generalized for $\hat{n}$ being an operator? If so how exactly would the expression be different? Thanks.


title: Did Einstein know about Michelson-Morley experiment?

text: The second postulate of special relativity deals with constancy of light in inertial reference frames. But, how did Einstein came to this conclusion? Did he knew about the Michelson-Morley experiment?


title: vander waals and casimir forces

text: Does one need to invoke quantum mechanics to explain casimir force or vander waals force. I see that textbooks show derivation of vander waal force with no QM but casimir force is typically described with QM. Is there a difference between vanderwaal and casimir forces ? Are there distinct examples of these two forces in real life. Is there a way to prove a given force is vanderwaal and not casimir or vice versa.


title: Electrons - What is Waving?

text: If an electron is a wave, what is waving? So many answers on the internet say "the probability that a particle will be at a particular location"... so... the electron is a physical manifestation of probability? That doesn't sound right. This page http://mwolff.tripod.com/see.html seems to suggest that the spherical wave pair is what gives the electron the properties of a particle. I could be misinterpreting, but this answer makes more sense to me - if only I understood what was waving!


title: $\lambda=\frac{2h}{p}$?

text: I am studying quantum physics and there is something I don't understand: I know that for any particle $E=hf$ (Einstein relation) and $v=\lambda f$ ($v$ is the speed of the particle). I also know that the kinetic energy is $E_k=\frac{mv^2}{2}$. Solving those 3 equations for $\lambda$: $$h\frac{v}{\lambda}=\frac{mv^2}{2}$$ I finally find $\lambda=\frac{2h}{mv}=\frac{2h}{p}$ which is not consistent with the De Broglie relation $\lambda=\frac{h}{p}$. Where am I wrong in my development ?


title: Wave Packets, Group velocity, and Phase velocity

text: Mathematically, you find that the wave function of a particle $\Psi (x,t)$ moves with the same velocity as the velocity of the particle ($v_{particle} = v_{group}$). Is there a reason why the particle velocity should not equal the phase velocity?


title: Group Velocity and Phase Velocity of Matter Wave?

text: In quantum mechanics, what is the difference between group velocity and phase velocity of matter wave? How can it also be that phase velocity of matter wave always exceeds the speed of light?


title: Speed of a particle in quantum mechanics: phase velocity vs. group velocity

text: Given that one usually defines two different velocities for a wave, these being the phase velocity and the group velocity, I was asking their meaning for the associated particle in quantum mechanics. And is one of them more representative for a particle?


title: group and phase velocity of free particle

text: If Schrödinger wave equation is for matter waves then for a free particle Group velocity $V_g =2$ Phase velocity $V_p$ But matter waves satisfy the relation $V_g V_p = C^2$ where $V_p>C$ Does this contradiction tell free particle should not be dealt quantum mechanically?


title: Current Physical Applications of Elastic Fractals

text: Are there any known uses of modeling with elastic fractals in current physical applications? (Especially uses concerning with self-similarity)


title: It is possible to extract energy from black holes or..satellites?

text: When something falls down, potential energy is transformed into kinetic one. Furthermore, you can attach a pulley and extract some energy (like in an aljibe), for example with a dynamo... If the object fall in a deep well, we can extract more energy, till the object reach downhole.. But what if we could have a "bottomless pit"... Endless energy?.. Well we would need an infinite earth... Or like a "black hole aljibe"... hmmm that doesn't seem an option.. Oh, wait a minute, we already have one! A satellite is like always "falling down"... Then is exchanging "potential energy"(who knows from what reference system) into kinetic energy.. What about tied it to a pulley (a long rope I see) to a dynamo, or an earth winding with a magnet satellite, or a satellite winding with the earth magnet, or whatever.. The idea is to USE the "bottomless pit" energy.. Where is the mistake? (just in case there be any =P)


title: How fast will the projectile go the second time?

text: Say I have a linear motor [aka rail-gun] and use a x amount of electrical power. I fire the gun and the object exits at velocity v. I then reuse the same object as my projectile and fire the rail-gun a second time this time with 2x the electrical power. My lessons on momentum suggest it will go 2v. The Work-Energy Theorem says 1.414v. Which is right and why?


title: Will Gauss's law still hold in case of deviation from inverse square law?

text: Assume a spherical metallic shell over which a charge $Q$ is distributed uniformly. Applying Gauss's law $\displaystyle\oint\textbf{E}\cdot d\textbf{a}=\frac{Q_{\text{enc.}}}{\epsilon_0}$ by considering a gaussian spherical surface concentric with the shell but with smaller radius, one can easily show that the electric field inside the shell is 0 since there is no enclosed charge within the gaussian surface. What would change in the above analysis and result had the electrostatic interaction been different from the inverse square behavior (i.e., $1/r^{2+\epsilon}$ instead of $1/r^2$)?


title: How do ideas of leading singularities and Grassmanian help in curing infrared divergences when calculating N=4 scattering amplitudes?

text: Broadly speaking how do ideas of leading singularities and Grassmanian help in curing infrared divergences when calculating N=4 scattering amplitudes? My understanding is that one gets infra red divergences because the external gluon momenta becomes collinear with the the loop momenta. I am confused as to what Nima's and Freddie's collaboration are doing to avoid this? If people can clarify my confusion and direct me to the appropriate literature I would be grateful.Also, is there a clear way to understand this for theories like planar QCD?


title: One question about sextupole

text: In multipole expansion, we use monopole, dipole, quadrupole or octupole to describe an electromagnetic field. But I saw someone use sextupole to describe transition states. If we expand an electromagnetic field by spherical harmonics, $\ell=0,1,2,3$ represent monopole, dipole, quadrupole and octupole. Do we sextupole in electric field? For different $m$ in the expansion, are they also different modes?


I am writing this question here because I have a problem in understanding the Wigner Threshold law in Photodetachment and Photoionization. The Wigner Threshold Law is given by: $\sigma$=$E^{L+1/2}$. where $\sigma$ is the photodetachment cross section, E is the kinetic energy of the detached electron from the anion, and L is the detached electron angular momentum. I have the following questions: What is the threshold in this law? For this threshold law to be valid, should the energy be above or below the threshold? Where the literature says "near threshold", does this mean above or below the threshold - and how far? I found the threshold law for photodetachment but I couldn't find a threshold law for photoionization. Is there any threshold law for photoionization? It is known that the ejected electron and the neutral core (in the case of photodetachment) have an effective potential which is a sum of the interaction potential and a centrifugal potential: $Veff$=$V(r) + h/2mr^2 [l(l+1)]$. Is this centrifugal potential (second term) still valid in the case of photoionization? PS: If anyone can suggest a textbook or any other reference, having the derivation and details about the Wigner law on a fundamental level (a graduate level so that I can understand the basics very well), that would be good.


title: Energy band diagram of a metal-silicon-metal device, bias voltage applied

text: I have an intrinsic silicon layer sandwiched in between two aluminum contacts. I'm trying to figure out the band diagram of the entire device when a positive bias (much larger than the work function of the aluminium) is applied to one of the contacts. Intuitively, I would expect the i-layer part to be tilted (like in a p-i-n device), but I can't figure out what the connections between the i-layer and the metal contacts looks like.


title: Efficiency of the insulation of a house

text: I had an argument about the most cost-effective way to keep the energy bill low in the winter (here, temperature usually have an average of -20°C (-4°F)). He thinks that it's more effective to keep a temperature at a constant temperature at say 24°C (75.2°F), because It would cost as much or more to re- heat the house if you let the temperature drop. I simply think that the lower the temperature the better, because heat from the house will dissipate less and you need less energy to maintant the temperature. Am-I right? if so, which scientific theory could I look at to my make point?


title: Why do hydraulic jumps form?

text: I know that a hydraulic jump is formed when a zone of supercritical flow discharges into a zone of subcritical flow, but why exactly is a hydraulic jump formed? From a mechanical point of view, are the only forces acting in the diagram weight and friction? ![enter image description here](http://i.stack.imgur.com/3H1eU.gif) If someone could actually label the diagram with forces and explain them to me, I'd really appreciate it.


title: Are all solids mono-atomic? Is there not any di-atomic metals or compounds?

text: I want to know whether all solids are mono-atomic or not, and if there was diatomic solids or not,and if there was compounds of solids.


title: Do I need to focus the telescope?

text: I purchased a 90x refractor telescope with below configuration.   * Objective: 50mm diameter with 360mm focal length   * 2 Eye pieces: 6mm, 20mm (10-1/ 8" scope)   * 1.5x erecting eye piece It measures 13- 5/8” long x 2-1/2" diameter. I was trying to see a building half Km far from my balcony. But I can't see anything with any eyepiece other than white lighting circle. Moreover when I tried an eyepiece of toy binocular, I could see something, not much larger but at least visible. Is it something minimum distance of view for a telescope? Or do I need to focus it? (I don't know how)


title: Initial condition for Fourier transformed Schrödinger equation

text: I asked in this thread Time-dependet Schrödinger equation how to solve the Time-dependent Schrödinger equation. One of JamalS' recommendations was the Fourier transform, which is why I want to quote his example: * * * **Example** As an example, consider the case $V(x,t)=\delta(t)$, in which case the Schrödinger equation becomes, $$i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} + \delta(t)\psi$$ We can take the Fourier transform with respect to $t$, rather than $x$, to enter angular frequency space: $$-\hbar\omega \, \Psi(\omega,x)=-\frac{\hbar^2}{2m}\Psi''(\omega,x) + \psi(0,x)$$ which, if the initial conditions are known, is a potentially simple second order differential equation, which one can then apply the inverse Fourier transform to the solution. * * * Now, my question would be: What are meaningful initial conditions for this ODE? I mean, what you probably want to look at is how a wavefunction $\Psi(t=0,x)$ propagates in time? So how do you set up meaningful initial conditions for this Fourier-transformed Schrödinger equation? You don't need to refer to this particular ODE(with this potential). My question is rather: When you solve this ODE, what are appropriate initial/boundary conditions for this Fourier transformed ODE, cause this is were my imagination fails. If anything is unclear, please let me know.


title: Lift a weight with a pulley system

text: I have two weights, weighing one ten times the other. I should figure out a way to make a pulley system that can lift the heavier mass with the smaller one. I would like to use the least number of pulleys possible because of the friction, but I think that anyway the rope would suffer from the same problem. I found this: http://science.howstuffworks.com/transport/engines- equipment/pulley.htm But I cannot figure out how to continue. I would like to start from a system of 5\6 pulleys and make up a reasonable system. I believe that stress can be negligible. Can anyone show how to go through such problems starting from adding pulley and decreasing the needed force, pulley by pulley, and then reduce the number of them adding sheaves or more complicated shapes? Thanks. EDIT: Maybe adding some trusses among pulleys, can it be an advantage?


title: Is the Earth round?

text: Imagine you are a common man and want to prove that the Earth is Round, how would you prove it without any mathematical derivation or without the theory of the ships.


title: Why are most astronomy things spherical in the shape (like, the Sun, the Moon, the Earth, and other planets)?

text: What is the reason for all the astronomy things being spherical in the shape (like, the Sun, the Moon, the Earth, and other planets)?


title: Using the loop rule while charging circuits

text: We know that in an LC circuit the voltage across the capacitor is the same as the voltage across the inductor (the emf induced in it). This is a direct conclusion of KVL (Kirchoff's Loop Rule) applied to the circuit. But can we use KVL in such circuits, where we have things like capacitors and inductors which affect the voltages and current? For example, in a circuit with a battery and a capacitor, KVL cannot be applied. In fact, the voltages and current asymptotically approach the values predicted by KVL as the capacitor charges. Similarly, while deriving the time-dependence of current in an ac circuit with a capacitive load, the argument commonly presented is as follows: Voltage of source = Voltage across capacitor $$V_{cap}(t) = V_0\sin(\omega t)$$ $$Q_{cap} = CV_{cap}(t)$$ $$i = \frac{d(CV_{cap}(t))}{dt} = C\frac{d(V_0\sin(\omega t))}{dt} = CV_0\omega\cos(\omega t)$$ thus concluding that the current and emf are phase shifted by $\pi/2$. But isn't there a flaw if we use KVL in such circuits?


title: Degrees of freedom of particles in Lattice Boltzmann method

text: Is it true that in Lattice Boltzmann method particles have only one degree of freedom even in 3D case? Can someone explain that fact or provide a link? Thanks!


title: What constrains ghost fields $c^+$ and $b_{++}$ as $z$ going to infinity

text: In Green's superstring theory, Chapter 3, from equation 3.3.13 to 3.3.18, the author says that the singularity behavior of the ghost field $c^+$ must be no faster than $z^2$ as $z$ goes to infinity. And $b_{++}$ is required to approach $0$ as $z$ goes to infinity. May I ask what is the reason for such constraints?


title: Electric field and capacitance across a resistor

text: Using a simple lattice model of conduction, where electrons are accelerated by an electric field, and are slowed down by bumping into the lattice, you get the following equation for current density: $\vec{J}_n=nqμ_n\vec{E}$ Let's imagine an ideal DC voltage source connected with perfectly conducting wires to a resistor. Just from intuition, as the electrons reach the lattice of the resistor, you'd think that there'd be a pileup of electrons, since they don't have as much mobility in the resistor (almost like a traffic jam at tight roads). Do electrons or other charge carries collect at the end of resistors? If they do, is this what creates a voltage drop across resistors (or equivalently, an electric field across a resistor)? This idea of a collection of charge seems to imply a capacitance to the resistor. Do real resistors display any in-built capacitance?


title: Integrating out fields from classical systems

text: Has anyone ever heard of integrating out fields from classical Lagrangians if they are quadratic?


title: Particles from String theory

text: I understand that the strings in string theory are posited to be many, many orders of size smaller than say, a quark, electron or any other particle. But if this is so, how does the string "expand" to produce the gargantuan, by comparison, particle? Is this expansion caused by its vibrational pattern or is there another way by which the extremely small string manifests itself into a much larger object? I am, of course, under the impression that each "elementary" particle is generated by the vibrational pattern of only one individual string, i.e., one string produces one electron, etc.


title: Fourier Transform of E-Field with Decay Constant

text: Given an atomic transition with associated E-field $E(t) = E_{0}\cos(\omega_{0}t)e^{-t/\tau}$ where $\omega_{0}$ is the natural line frequency and $\tau$ is the decay constant of the simple harmonic oscillator. Find an expression for the line flux, that is $I(\omega)/I(\omega_{0})$. I'm trying to do the following Fourier transform: $$ f(\omega)=\int^{\infty}_{-\infty}E_{0}\cos(\omega_{0}t)e^{-t/\tau}e^{-i\omega t}dt $$ I'm not really sure how to do this integral. I tried calculating it from $-\tau/2$ to $\tau/2$ but I get an answer with the sine of a complex number in the numerator that will not go away when I square it to find Intensity. Any help would be appreciated. Thanks


title: Do atoms get created or are they recycled?

text: Basically, are the atoms that make up my body right now something that has existed since the big bang?


title: Temporal part of Quantum Wavefunction

text: I was hoping that someone could give me the more fundamental reason that we take as the temporal part of a quantum wavefunction the function $e^{-i\omega t}$ and not $e^{+i\omega t}$? Clearly $e^{-i\omega t}$ solves the time dependent Schrödinger equation whereas $e^{+i\omega t}$ does not. However, the Schrödinger equation, when it was first developed, was merely a hypothesis. It was new physics and, as such, could not be derived from previous work. Hence, why did Schrödinger and his contemporaries choose $e^{-i\omega t}$ and, thus, why does an antiparticle with wavefunction temporal dependence $e^{+i\omega t}$ correspond to backwards time travel or negative energy?


title: Electromagnetism: Conductors

text: Even though the thermal velocity of electron in a conductor is comparatively high, the thermal velocity is not responsible for current flow? Why is this the case?


title: Naive question about time-dependent perturbation theory

text: In time-dependent perturbation theory where $H=H_0+V$ and $V$ is considered small and has _no explicit time dependence_ , the standard text-book treatment of the leading order probability amplitude for the system to make a transition from $|i\rangle$ to $|f\rangle$ is $$ P_{f\leftarrow i}(\Delta t)=\big|\langle f|V|i \rangle\big|^2\frac{4\sin^2(\omega_{fi}\Delta t/2)}{\hbar^2\omega^2_{fi}} $$ If I consider transitions between two states of the same energy, I take the $\omega_{fi}\rightarrow0$ limit, giving me $$ P_{f\leftarrow i}(\Delta t)=\big|\langle f|V|i \rangle\big|^2(\Delta t)^2., $$ which grows without bound in time, as $\Delta t^2$. I would take this to mean that perturbation theory fails at long times. Why then am I allowed to take the large time limit to derive Fermi's Golden rule, without risking failure of perturbation theory?


title: Why is body frame angular velocity nonzero?

text: This question is relevant to Euler's angles and Euler's equations for a rigid body. Why aren't $\omega_1$, $\omega_2$ and $\omega_3 = 0$ in the body frame? How can we measure $\vec\omega$?


title: Does a wing in a potential flow have lift?

text: I have a hard time understanding whether or not a wing placed in a potential flow, assuming there is no viscosity and no friction with the wing, will produce a lift. I've seen several contradictory (to me) versions of what happens in this case, in particular:   * In Landau - Lifshitz, Vol 6, §11 it is shown that the force vector on any object in a potential flow is zero (i.e. there is neither lift nor drag). The mathematics behind this derivation are above my understanding, thus I can't really follow how this is shown and what assumptions are made along the way.   * The Wikipedia article on the d'Alambert's paradox states that a body in a potential flow does not experience drag. On the other hand the article does not say anything about lift forces and the German version explicitly states that it does not apply for lift forces.   * A common explanation for the lift produced by wings seems to be that its asymmetric shape causes a higher velocity of the flow above the wing and a lower below. Due to Bernoulli's equation that would result in a pressure-difference and thus a force. Honestly I don't quite get this explanation as I don't see where the different velocities are supposed to come from (without friction and thus without the circulation resulting from turbulences behind the wing). So now I'm wondering, do or do not wings have lift in irrotational, inviscid, incompressible fluids.


title: How much lift does the average latex helium filled party balloon produce?

text: How much lift does the average helium filled party balloon produce? (not including any extras like ribbon string)


title: How to show that two spin one half particles must form a anti-symmetric wavefunction?

text: I am stuck on how to connect the ideas that two spin half particles must form a anti-symmetric wavefunction. Is there a proof on how to show that two spin one half particles must form a anti-symmetric wavefunction? \begin{align} \Psi(x_1,x_2) = \Psi_1(x_2) \Psi_2(x_1) \- \Psi_2(x_1) \Psi_1(x_2) \end{align} This would indeed prove that spin half particles must obey the Pauli Exclusion Principle, because if two fermions were in the same state, then the wavefunction would equal zero.


title: Would wearing clothing that is black on the inside and white on the outside keep you cooler?

text: The Straight Dope ran an explanation of why nomads often wear black clothing - it absorbs heat better from the body. On the other hand, white clothing reflects sunlight better. Is it possible to get the best of both worlds and wear clothing that is black on the inside and white on the outside?


title: Perfectly vertical spinning top

text: Consider a non-spinning top. If a top is perfectly vertical, and the interface between its base and the ground is perfectly flat, it should stay in (unstable) equilibrium. I.e. it does fall. What about if the top, in addition to being perfectly vertical, is also spinning? Does this situation differ from the previous one?


title: What is the most counter-intuitive result in physics?

text: I think that relativity and quantum mechanics would provide some good examples.


title: Loschmidt's paradox - really a paradox?

text: Is Loschmidt's paradox a paradox even today? In other words, is the paradox resolved or not?


title: Repulsion of the pieces of a  broken magnet

text: > **Possible Duplicate:**   >  Why does it seem like a broken magnet's poles flip? I have experienced that if we break a bar magnet into two pieces and try to bring those broken faces together it gets repelled each other. Why is it so? consider a bar magnet, $ NN--------SS $ $NN$ for north pole and $SS$ for south pole. I broke that into two pieces, So one piece would be $ NN--SS'$ and another piece would be $'NN--SS$ .( $'$ indicates the broken part). Now if I bring those two together, $SS'$ of first should attract the $'NN$ of the other. But in the reality it is getting repelled. or is there any pole exchange occuring? $$NN--------SS$$ gives either $$NN---SS'$$ and $$'SS---NN$$ **OR** $$SS---NN'$$ and $$NN'\---SS$$ so it will repel each other. Please explain why is this happening? Edit: The cut has been made perpendicular to the polar axis.


title: What happens to the bar magnet field when it breaks?

text: References says alignment of a magnetic field of a bar magnet remains same even if we cut it into two or more pieces. According to this cut edges should attract with each other. When I tried it was working in the opposite way. why is it so ?


title: Broken bar magnet repels?

text: **![Why a bar magnet broken from between repel on joining again even if the poles are opposite, i.e. North and South.](http://i.stack.imgur.com/bM4tl.png)** **Why a bar magnet broken from between repel on joining again even if the poles are opposite, i.e. North and South.**


title: Length contraction as $v>c$

text: What would theoretically happen to an observed object's length (let it be a 5 meter line) moving at some speed greater than that of light in a straight path?


title: Non-renormalizable corrections to GUT unification

text: While writing these answers: Hypercharge for U(1) in SU(2)xU(1) model and Is there a concise-but-thorough statement of the Standard Model? , it occured to me that the unification prediction for Georgi-Glashow is relying on high-scale renormalizability, if you have a nonrenormalizable SU(5) coupling between the SU(5) Higgsing field and the gauge field, you can make the couplings of the three subgroups different. In general, after breaking, you can make different couplings for SU(2),SU(3) and U(1), depending on the Higgsing details, by allowing nonrenormalizable terms of sufficiently high order. If you Higgs in a natural way, using an SU(5) hermitian tensor field (spacetime scalar) with expectation (a,a,b,b,b), a term of the form $tr(\phi GG)$ where G is the SU(5) gauge field considered as an antihermitian SU(5) matrix, gives an SU(3) coupling correction which goes as (b-a)/M, where M is the higher string or planck scale. Considering that the string scale can be as low as $10^{17} GeV$, while the GUT scale can be as high as $10^{16} GeV$, depending on the details of low energy matter and high energy compactification, is it obvious that the failure of couplings can't be 10%? Question: Considering that the GUT scale is close enough to the Planck scale that renormalizability is not exact, what are the precise natural errors in coupling unification? ### EDIT: Clarification Lubos Motl gave an answer which misses the point, so I will give a more detailed description of the question. The question is: how bad can coupling constant unification fail _at the GUT scale_ just from lack of renormalizability, ignoring running. When you have a big gauge group break into a smaller gauge group, the field strength tensor for the big gauge group is multiplied by $1\over g^2$, and decomposes into the field strength tensors for the smaller gauge groups, which are then also multiplied by 1/g^2 so that the couplings are the same (at the unification scale--- forget about running--- this is a classical question). But you have a scalar VEV doing the breaking. Normally scalars can't interact with gauge fields to alter the coupling--- the coupling is nondynamical. But the reason for this is renormalizability--- you can't have a renormalizable scalar-scalar gauge-gauge derivative interaction. But ignore renormalizability, and suppose that there is an additional term in the Lagrangian of the form (the obvious space-time indices on F are suppressed) $$ A \mathrm{tr}(H F F) $$ Where F is the field strength tensor $F^{\mu\nu}$ for SU(5), A is a coupling constant of dimensions inverse mass, and H is an SU(5) adjoint scalar, which acts to Higgs SU(5), whose VEV is diag(a,a,b,b,b) in some gauge. The field H is invariant under SU(2) rotations of the top two components (because the H VEV is $\delta^a_b$ on the upper two-by-two block, and this is an invariant tensor of SU(2)), SU(3) rotations of the bottom 3 components (because the H VEV is $\delta$ on the bottom 3 by 3 block), and any diagonal phase rotations, including U(1) rotations that make up the Georgi-Glashow hypercharge. So this VEV breaks SU(5) to the standard model gauge group. Now, if you decompose the field strength F into the SU(2) field strength $W$ and the SU(3) field strength $G$, the nonrenormalizable term above decomposes into $$ A a \mathrm{tr}(W^2) + A b \mathrm{tr}(G^2) $$ So that the VEV of the scalar alters the _coupling constant_ of the two gauge groups in the decomposition by the Higgsing, and this happens at the GUT scale, ignoring any running. This scalar-gauge-gauge nonrenormalizable term is in addition to the ordinary kinetic term, so the full W and G action is $$ ({1\over 4g^2} + Aa) \mathrm{tr}(W^2) + ({1\over 4g^2} + A b) \mathrm{tr}(G^2) $$ The coupling constants do not match just from the breaking. The difference in their reciprocal-square coupling at the GUT scale is $$ 4A(a-b) $$ Now A is order ${1\over M_\mathrm{PL}}$, and $(a-b)$ is order $M_\mathrm{GUT}$, so you get an order $M_\mathrm{GUT}\over M_\mathrm{PL}$ difference between the coupling constants already at the GUT scale, before any running. I know that a term like this is ignored in running calculations, since these are RG flows in renormalizable low energy theories only. It is not included in higher loop corrections, or in any threshhold corrections, and to calculate it requires a full string background (and numerical techniques on the string background). The question is whether this term is typically small, or whether it is typically large enough to make the arguments against Georgi-Glashow from failure of unification moot. ### EDIT: more clarification Unification relies on the coupling constant of a subgroups being equal to the coupling constant of the big group. The reason is that the $FF$ kinetic term breaks up to the subgroup kinetic terms. But this depends on renormalizability. If you have _nonrenormalizable_ scalar FF interactions, you break the equality of subgroup couplings. These terms can be neglected at ordinary energies, but not at 10^16 GeV. This is for Lubos, who downvoted the question, I am not sure why. If you downvote this, you might also wish to downvote this: Does the ruling out of TeV scale SUSY breaking disfavor grand unification? But I urge you to think about it instead, as I am 100% sure that they are both saying correct things.


title: Accelerated charge inside sphere (again!)

text: Sorry to go on about this scenario again but I think something is going on here. Imagine a stationary charge $q$, with mass $m$, at the center of a stationary hollow spherical dielectric shell with radius $R$, mass $M$ and total charge $-Q$. I apply a force $\mathbf{F}$ to charge $q$ so that it accelerates: $$\mathbf{F} = m \mathbf{a}$$ The accelerating charge $q$ produces a retarded (forwards in time) radiation electric field at the sphere. When integrated over the sphere this field leads to a total force $\mathbf{f}$ on the sphere given by: $$\mathbf{f} = \frac{2}{3} \frac{qQ}{4\pi\epsilon_0c^2R}\mathbf{a}.$$ So I apply an external force $\mathbf{F}$ to the system (charge + sphere) but a total force $\mathbf{F}+\mathbf{f}$ operates on the system. Isn't there an inconsistency here? As the acceleration of charge $q$ is constant there is no radiation reaction force reacting back on it from its electromagnetic field - so that's not the answer. Instead maybe there is a reaction force back from the charged shell, $-\mathbf{f}$, to the charge $q$ so that the equation of motion for the charge is given by: $$\mathbf{F} - \mathbf{f} = m\mathbf{a}\ \ \ \ \ \ \ \ \ \ \ (1)$$ This reaction force might be mediated by an advanced electromagnetic interaction going backwards in time from the shell to the charge so that it acts at the moment the charge is accelerated. Now the total force acting on the system is the same as the force supplied: $$ \mathbf{F} - \mathbf{f} + \mathbf{f} = \mathbf{F}.$$ If one rearranges Equation (1) one gets: $$\mathbf{F} = (m + \frac{2}{3} \frac{qQ}{4\pi\epsilon_0c^2R}) \mathbf{a}$$ Thus the effective mass of the charge $q$ has increased.


title: Quantum randomness and brownian motion in biological systems, e.g., fertilization

text: I am looking for examples of physical indeterminacy impacting the macroscopic world. By physical indeterminacy, I mean physical sources of randomness such as quantum indeterminacy or brownian motion. One example is of particular interest here: whether such randomness influences which sperm cell fertilizes a particular ovum, or whether such biological systems are too large to be affected by random perturbations, either at the atomic level (quantum) or molecular level (brownian motion).


title: Is the curvature of space around mass independent of gravity?

text: Is the curvature of space caused by the local density of the energy in that area?Could gravity be a separate phenomenon only arising from the curvature of space? For instance if the density of energy in a particular area cause that area of space to ”curve” but the effect that we understand as gravity, (causing anything with mass to be attracted to each other) is only arising as a consequence of that space being curved. I guess it seems to me that things other than mass can cause the curvature of space (electromagnetic fields, an enormously high density of photons in a small area or at least I think so, but I'm not sure about the photons, and if a black hole rotating causes frame dragging (which I'm assuming means the surrounding physical metric of space is probably some mechanism, or thought experiment where you could ball up space tight enough to become a black hole even without any matter in it. I guess it's another question I Could ask.


title: What is a free fermion model?

text: Title says it all really.. Why is the XX spin chain a free fermion (non- interacting) model, and the XXZ chain not? Is it right that $\sum_l a_l^\dagger a_{l+1}$ isn't an interaction between fermions because it's creating a fermion on one site and destroying it on another? But why is $\sum_l a_l^\dagger a_l a_{l+1}^\dagger a_{l+1}$ an interaction term? Is something like \begin{equation} H_1 = -\sum_l (J+(-1)^lK) ( \sigma_l^x \sigma_{l+1}^x +\sigma_l^y \sigma_{l+1}^y) \end{equation} a free fermion model? If not, why not? **Edit** I don't have enough reputation to set a bounty, but if anyone could answer this question, I'd be very grateful! **Edit 2** Anyone?


title: Signs in proof of gravitation potential energy (GPE)

text: Proof of gravitational potential energy. Work done by gravity in bringing mass from infinity to a distance of $r$ between masses. When we use the integration formula and arrive at the answer we get $-GMm/r$ taking lower limit as infinity and upper as $r$. But this work should be positive as force and displacement are in same direction. Please explain. If my proof was wrong, then tell any other satisfying proof for GPE.


title: In truth, only atoms and the void

text: I have a question about this motto used by Sean Carroll in his blog: > _In truth, only atoms and the void._ Can you explain what this sentence means? My interpretation is that the sentence does not makes sense because in physics, "atom" has multiple meanings, the same goes for "void". These words have so many meanings that, used without qualification, they mean nothing. The word truth is also a notoriously loaded and undefined word. Does he mean the absolute truth (which has no place in physics) or is he using it colloquially, we don't know. Let's assume that he is using "atom" to mean "absolutely indivisible unit" and "void" to mean "absolutely empty space that contains nothing but atoms". This view is absurd because defines "void" to be "empty" and "not-empty" at the same time. But besides that, if in truth, there is nothing but atoms and the void, does Sean Carroll deny the existence of fields? Here are a few meanings that physicists give to "void" (a search of titles in arxiv containing "void"): rigid void, relativistic void, magnetic fields in voids, empty voids, nano void, dynamics of void and so on. So, in physics void can mean anything but "void". Then, what does "In truth, only atoms and the void" mean? Does it really have such a deep meaning to included as a motto of a blog?


title: Diffraction/Bragg's law : how does $2\theta$ come about?

text: I'm trying to get my head around a problem (I should have checked whether I had the answer in class, the exams are coming up now and I don't know if I'll get a lecturer response over the holidays) I can't get figure out the relationship between $\theta$ and $2\theta$ in the diagrams supposedly making clear how a crystal lattice diffracts. From what I've read just now my understanding is that Bragg diffraction is actually transmission part-way into the crystal, then reflection off of an atom inside, hence the angle is twice the incident angle. The textbook we've been referred to for further reading makes no note of a $2\theta$ (it can be read here). The example used in the lecture is below ![enter image description here](http://i.stack.imgur.com/VmKym.png) I obtained an answer by using a right-angled triangle made by the X-ray detector at the middle arrow, as this is the only perpendicular angle I can see. I really don't get how this would give the angle at the crystal as $2\theta$ (and I'm not at all confident that I should halve this angle to use in calculation of $\theta$ Can someone explain why I need to do so (as despite my misgivings, this is clearly indicated as the correct procedure). I've obtained an answer and am confident with the theory, maths etc., I just don't see the derivation of the $2\theta$ as opposed to $\theta$.


title: Dynamics of circular motion

text: If there is a disc rotating about its centre, let the surface be frictionless and if a coin is placed anywhere on the disc (not at the center) why doesn't it fly off even though there is a centrifugal force acting on it with respect to the disc's frame?


title: Why do 3D holograms need curved lines?

text: I'm making a scratched hologram and I'm interested in we have to make curved lines to get a 3D image.


title: Magnification multiplication using telescope arrays?

text: If we have an array of telescopes attached one after another, would the resultant magnification be multiplied?Also would such a contraption be feasible to make telescopes with amazing magnification?


title: Non-Linear Density Shell Problem

text: I'm trying to understand Newton's Shell Theorem (Third) http://en.wikipedia.org/wiki/Shell_theorem However this applies to a sphere of _constant density_. How is this formulated for sphere of _varying density_ , e.g., a ball of gas bound together by gravity? Actually, this requires another question: how does the density of ball of gas bound by gravity diminish with radius? **EDIT** as DMCKEE pointed out "You can't answer the problem ignoring thermodynamics", so I've removed that proviso.


title: How is the complexification of spacetime justified?

text: As always the caveat is that I am a mathematician with very little knowledge of physics. I've started my quest for knowledge in this field, but am very very far from having a good grasp. General relativity assumes that spacetime is a $4$-dimensional manifold. Ultimately, I know that physicists deal with Calabi-Yau manifolds (in string theory, I imagine). These are holomorphic complex manifolds. How, and when, is this change of hypotheses justified? Is one way to view the general relativistic space time that spacetime is a variety of dimension $2$ over $\mathbb{C}$? Or is this change done later? Why is it ever done?


title: Do we expect that the universe is simply-connected?

text: I heard recently that the universe is expected to be essentially flat. If this is true, I believe this means (by the 3d Poincare conjecture) that the universe _cannot_ be simply-connected, since the 3-sphere isn't flat (i.e. doesn't admit a flat metric). Is any/all of this true? If so (or even if not), what sorts of unexpected things might follow from the non-simply-connectedness of our universe?


title: Symmetrical twin paradox

text: Take the following gedankenexperiment in which two astronauts meet each other again and again in a perfectly symmetrical setting - a hyperspherical (3-manifold) universe in which the 3 dimensions are curved into the 4. dimension so that they can travel without acceleration in straight opposite directions and yet meet each other time after time. On the one hand this situation is perfectly symmetrical - even in terms of homotopy and winding number. On the other hand the Lorentz invariance should break down according to GRT, so that one frame is preferred - but which one? **So the question is: Who will be older? And why?** And even if there is one prefered inertial frame - the frame of the other astronaut should be identical with respect to all relevant parameters so that both get older at the same rate. Which again seems to be a violation of SRT in which the other twin seems to be getting older faster/slower... How should one find out what the preferred frame is when everything is symmetrical - even in terms of GRT? And if we are back to a situation with a preferred frame: what is the difference to the classical Galilean transform? Don't we get all the problems back that seemed to be solved by RT - e.g. the speed limit of light, because if there was a preferred frame you should be allowed to classically add velocities and therefore also get speeds bigger than c ?!? (I know SRT is only a local theory but I don't understand why the global preferred frame should not 'override' the local one). Could anyone please enlighten me (please in a not too technical way because otherwise I wouldn't understand!) **EDIT**   Because there are still things that are unclear to me I posted a follow-up question: Here


title: General relativity and global aspects

text: The theory of general relativity tells me something about the global structure of space-time, eg simply connected ?


title: Our Universe Can't be Looped?

text: With reference to the Twin-Paradox (I am new with this), now information of who has actually aged comes from the fact that one of the twins felt some acceleration. So if universe was like a loop, and the actually travelling twin again reached earth after completing the loop, then no such information would have biased the actually travelling twin, and the paradox will still remain (?). And universe thus can't be a loop, it must have an end point?


title: What is known about the topological structure of spacetime?

text: General relativity says that spacetime is a Lorentzian 4-manifold $M$ whose metric satisfies Einstein's field equations. I have two questions:   1. What topological restrictions do Einstein's equations put on the manifold. For instance, the existence of a Lorentz metric implies some topological things, like the euler characteristic vanishing.   2. Are there any experiments being done or even any hypothetical experiments that can give information on the topology? E.g. is there a group of graduate students out there trying to contract loops to discover the fundamental group of the universe?


title: Potential energy sign conventions

text: Almost every book on physics that I read have some weird and non-clear explanations regarding the potential energy. Ok, I do understand that if we integrate a force over some path, we'll get a difference in some origin- function values ($\int_{A}^{B} Fdx = U(B) - U(A)$). This function is the potential energy. Of course, whether we can define this term or not depends on the force. Now, here's an example of explanation (to be more precise - lack of explanation) regarding the GPE from one of the books: > "...When a body moves from some point A to point B, gravity is doing work: > $U_A-U_B=W_{A \to B}$. The magnitude can be calculated using an integral: > $W_{A \to B}=\int_{r_A}^{r_B} F(r) dr = \int_{r_A}^{r_B} > \left(-\frac{GMm}{r^2} \right)dr=(-\frac{GMm}{r_A})-(-\frac{GMm}{r_B})=U_A- > U_B$ > > ... > > Thus, when $r_A>r_B$, the magnitude is positive and therefore $U_A>U_B$. In > other words: when the distance between the bodies is being increased - the > gravitational potential energy of the system is also being increased. Giving absolutely no explanation on why all of a sudden they put a minus sign into the integral. From another book: > Work done by Coulomb's force: $W_{el}=\int_{r_1}^{r_2}\frac{q_1q_2dr}{4 \pi > \epsilon_0 r^2}=\frac{q_1q_2}{4 \pi \epsilon_0 r_1}-\frac{q_1q_2}{4 \pi > \epsilon_0 r_2}$ > > ... Calculating the work gravity is doing is no different from the > calculation of the work done by an electric field, with two exceptions - > instead of $q_1q_2/4 \pi \epsilon_0$ we should plug $G M m$, and we also > should change the sign, because the gravitational force is **always a force > of attraction**. Now, this is not satisfying at all. So what if it is an attraction force? How this should influence our calculations, if the work is defined as $|F| |\Delta x| \cos \theta$, so the sign only depends on the angle between the path vector and the force vector? Why they put a minus sign? Is is just a convention or a must thing to do? Some say the sign is important, others say the opposite. Some explain this as a consequence of that we bring the body from infinity to some point, while others say it is a consequence of an attractive nature of the gravitational force. All of that is really confusing me. Also, in some of the questions like "what work is required to bring something from point $A$ to point $B$ in the field of gravitational/electric force", the books sometimes confuse $U_A-U_B$ and $U_B-U_A$ - as I understand it - the work that **I** must do is always $U_B-U_A$. However the work that **the force** that is being created by the field do is always $U_A-U_B$, am I correct?


title: Is it possible for information to be transmitted faster than light by using a rigid pole?

text: Is it possible for information (like 1 and 0s) to be transmitted faster than light? For instance, take a rigid pole of several AU in length. Now say you have a person on each end, and one of them starts pulling and pushing on his/her end. The person on the opposite end should receive the pushes and pulls instantaneously as no particle is making the full journey. Would this actually work?


title: light travels a maximum speed... /?

text: > **Possible Duplicate:**   >  Is it possible for information to be transmitted faster than light? we know that speed of light is an unconquerable term in physics..light takes about 1 year to travel 9500000000000000 metres that is light year....suppose i have a needle of length 95000000000000000 metres long....suppose that one end of needle is at my hand and other end is at distance 95000000000000000 metres apart. . now if i push gently the end of needle at my hand then at the same time the other end at a distance of 95000000000000000 metres will move forward.....now light has taken 1 year to travel a distance of 95000000000000000 metres..and i have made a mechanical disturbance to travel the same distance with in no time from one end of the needle to other end placed at a distance of 9500000000000000 metres apart. . in other words i have made a mechanical disturbance to travel faster than light.. . .


title: Send a signal from Earth to a planet billion light years away instantly

text: If light was infinitely fast, we could just send a light signal from Earth to the planet. But I was wondering, if we made a perfectly non-elastic rope as long as the distance between earth and the far-away planet. Could somebody pulling the rope from the earth send an instant signal to a person holding the end of the rope on the far-away planet? Will the rope side on the other planet move at the exact same moment as the rope side on the Earth? Of course, we should ignore the obvious factors making this means of communication impossible -_-


title: Gravitational Effect Versus the Speed of Light

text: If, for some reason, the sun were to suddenly disappear altogether, I would like to know the following:   * would we "feel" it first (i.e. being thrown into outer space due to no longer having anything to orbit);   * would we see the sun disappear at precisely the same moment that we "feel" it?; or   * would we see it before we feel it (shortly before inevitably being subsequently thrown into outer space)? For reference: light from the sun takes ~8.5 minutes to reach us here on Earth. I have read that gravitational waves travel at the speed of light, however I personally would imagine that we feel it first for this reason: how could we possibly "feel" something **8.5 minutes after it has happened** if it is something as major as planet Earth being thrown into outer space? Surely we wouldn't be hurtling through space for 8.5 minutes before noticing, would we? But then that would lead us into the realm of the Universal Speed Limit and how, in theory, nothing should be able to exceed it (except, perhaps, subatomic particles being able to communicate instantly (MUCH faster than the speed of light), but that is a whole other question and not for discussion here). Any help would be great. Mods: if you feel that this question belongs in a theoretical Stack site, then please kindly move it to the appropriate one. Many thanks.


title: Solar Catastrophe

text: Consider all of sudden the sun vanishes. What would happen to planetary motion. Will it continue to move in elliptical path or move in a tangential to the orbit immediately after sun vanishes or move in elliptical orbit for some time after the vanishing of sun or any other cases? If so, please explain...


title: 'Push' in a rigid rod travel at speed of sound or speed of light

text: Two person, $A$ and $B$, each holding one end of a long solid rod. Now person $A$ pushes the rod on one end. **Question** : Is it correct that the information that the rod has been pushed will travel to the other end at the speed of light whereas the actual 'push' will travel at the speed of sound in the rod? i.e. If the rod has length $ ct $ , then will person $B$ feel the push in time $t$ or $ct/v$? ($c$ is speed at which EM waves propagates in this experiment, and $v$ is the speed of sound in the rod.)


title: How fast gravity force propagates?

text: > **Possible Duplicate:**   >  The speed of gravity Lets say for example that a pertubance occured on the Sun that changes its distance to the Earth. How fast we are going to notice the gravity changes? Is it immediately? Light speed?


title: Do the changes of gravity travel across distances instantly or limited to the speed of light?

text: > **Possible Duplicate:**   >  The speed of gravity It takes a long time for a radio signal to travel to the planet Mars. What if we made a special type of radio that could detect small changes in gravity from a fall away object. For example, a satellite in orbit around Earth. That satellite had a device that allowed it to make small changes in it's gravity field. The detector orbiting Mars could detect these changes, and convert it into a communications signal. Would these two devices be able to communicate with each other instantly using gravity? There by providing a real-time communications link between Earth and Mars?


title: If you make a steel rod from here to Alpha Centauri and move it, will the movement appear there instantly?

text: > **Possible Duplicate:**   >  Is it possible for information to be transmitted faster than light? imagine this theoretical situation:      You have make an extremely long piece of steel (or anything really) one end is placed on earth the other is all the way at Alpha Centauri, 4ly away.   Here on earth the end is hanging loosely, on Alpha Centauri there is a bell at the end of the rod. Now if you move our end straight in the direction of AC will the bell there ring instantly ? Or will it be slower/same as speed of light.      If I take make it small scale, for example moving a steel rod with a bell on the end, I can't imagine that there would be a delay, but of course the speed of light is the limit, so what could cause the delay ? Thanks for explanation.   My very sketch: http://imageshack.us/photo/my-images/221/steelrod.png/


title: Steel rod to Mars vs the speed of sound, how is supersonic travel possible?

text: I remember reading this passage in the "Feynman Lectures", where Dr. Feynman describes an experiment in which a theoretical metal rod of length equal to the distance between Mars and Earth is arranged between mars and earth. Then the "rod handler" on earth gives the rod a push upwards. So the question posed is, will the rod instantaneously move backwards on Mars? He then proceeds to say that actually the movement will be felt roughly after the time it would take for sound to propagate from Earth to Mars i.e. at the speed of sound. This is because the atoms in the rod actually propagate their position at the speed of sound. See also this Phys.SE post and links therein. **My question is:** If this were true, how is supersonic travel possible? Won't there be a scenario where the atoms in the components of the aircraft won't "catch up" because the propagation of the new positions is slower than the actual speed? Won't the aircraft disintegrate?


title: Does gravity spread instantly?

text: > **Possible Duplicate:**   >  The speed of gravity I am real noob in physics, so sorry if this question is really stupid. Today in a casual conversation I claimed that if the sun were to instantly disappear, we would have felt it in any way nos ooner than after about 8 minutes(which is the distance in light minutes from Earth to Sun). My friend said that light-wise I am right, but we would instantly have felt the gravitation change. I argued that gravitation doesn't spread faster than light. Am I right or is he right? Thanks in advance and sorry again for the noob Q :)


title: Does gravity act instantaneously?

text: some one states that the earth would travel in it path for 8 min if the sun vanished. The equations of gravity show the Moon would crash into the Earth and the Earth would head for Mars the instant the Sun is gone. Can someone show how there is a delta Time and the relaxing of the bend in space-time fields is finite?


title: Here's a way to transmit data faster than the speed of light

text: > **Possible Duplicate:**   >  Is it possible for information to be transmitted faster than light by using > a rigid pole? Assume there is a long rod or a string connecting two points separated by a distance of several light years. It will take light several years to travel, but by a simple pull or push of the string or rod we can transmit date in binary format, what do you think?


title: what is the speed of gravitational interactions?

text: Example. Two bodies with masses m1 and m2 distance between them one year of light. What happens to the force of attraction between the two bodies when one body suddenly disappears? Is the force disappears at the same time or the force will act on second body for one year?


title: How much time does it take to affect?

text: We know gravitational is continuously acting on us. But let us assume that we are hanging in the space alone away from anything to affect our position. And suddenly a giant planet appear's a few million km from us then we will feel it's gravitational force on our body. But I wanted to know will it work instantly or it will take some time to take effect just like light take some time to reach us ?


title: Can Information Travel Faster Than The Speed Of Light?

text: Many believe that nothing can travel faster than speed of light, not even information. Personally, i think theoretically information can. Consider this following imaginary experiment: Imagine we are living on a planet that is big enough for a, let's say, 10-light-seconds-tall tower to erect. We hang a pendulum near the planet's surface using a long thin wire at the top of the tower. If someone at top of the tower cut the wire then the pendulum will instantly falling to the ground. In this case we can say that the information "someone cuts the wire" travels 10-light-seconds distance in no time. Since someone on the surface can only see the act of cutting the wire 10 seconds later, can we infer that the information travels faster than light?


title: Speed/direction of gravity for a moving source

text: Consider the Earth, and a bowling ball held 186,000 miles (1 light second) above it. When the ball is released, it will start to fall vertically downwards towards the Earth. Now consider the case if the Earth is moving sideways at 1000 miles/second. The bowling ball is released just as the Earth passes directly underneath. Does the ball fall a) vertically again, or b) does it fall towards where the Earth was 1 second before? Gravity propagating at the speed of light, would suggest answer b) but as most of the matter in the universe is travelling at very high speed, and planetary orbits are circularish, I think the answer is a)


title: If I move a long solid stick can I send message fastest than light?

text: > **Possible Duplicate:**   >  Is it possible for information to be transmitted faster than light? I mean by using a perfect solid stick long enough and moving it forward and backward can I send information fastest than light ? Can you imagine a solid stick long enough to reach the moon and using it to comunicate with the lunar base. Will be information faster than light? What are the theoretical reason other than technical reason for rules this as impossible? We should rule that a perfect unelastic solid exist ?


title: Does gravity travel at the speed of light?

text: Whenever I did calculations in high school physics involving gravity, it was either "a ball falling to the earth" type scenario, or a basic measurement of the gravitational attraction between two planetoids. I think I read somewhere recently that gravitational changes "aren't reflected instantly across the universe but instead propagate at the speed of light" somewhat like the ripples in a pond expanding. Is this true? Is there experimental evidence to confirm or deny this? What is the theoretical basis for it?


title: Gravity Concept Question

text: Thought experiment: Imagine the sun suddenly disappeared; lets say it some how transported to the edge of the observable universe.What will be the effects on the space-time? (1) What does General Relativity (specifically) have to say about the disappearance of mass from the portion of space-time being analyzed? I predict that the space will oscillate ,like a spring; interacting with it self (assuming the mass was spherical, and did not cause a force to act upon space in any direction), until it dampens out due to the interference, and initial "springiness" of space.But I do not know, I am no GR specialist. (2) Can someone also tell about what will happen if the opposite happened, (a mass suddenly appeared)?


title: How faster is the effect of force exerted by gravity?

text: We all know that Earth revolves around the Sun due to Gravitational force of the Sun. Also we know that it takes just over 8 minutes for the Sun's light to reach Earth. Now let us say, hypothetically if the sun were to suddenly extinguish (or disappear), would the Earth still not experience any effect of it until just over 8 minutes later (equivalent to time taken by light to reach from Sun to Earth) ? That is, Would the earth still continue to revolve (due to the centripetal force) for next 8 minutes after Sun disappearance and then thrown out of its revolving orbit (due to centrifugal) OR will it be immediately thrown out on the moment the Sun disappeared (i.e., not after 8 minutes) ?


title: Transfer of energy faster than the speed of light

text: In a vacuum: On my left I have a rod that is 558000 miles long (3 light seconds). A button is secured to its end which activates a laser beam sent back to me once I've pushed the rod in the direction of the button. On my right I have a laser aimed at a reflector that is 558000 miles away pointing straight back toward me. I push the rod on my left and activate the laser on my right at exactly the same time. Which laser beam will reach me first, the one on the right, or the one on the left?


title: sending information over a wire--mechanically

text: > **Possible Duplicate:**   >  Is it possible for information to be transmitted faster than light? I've thought about this since I was a little kid. I know it isn't exactly feasible, but it still bothers me. I hand you a really long wire, and we agree that "a long tug means 1, two short tugs mean 0", then you move off into the galaxy, a few light-years away from me. I proceed to give you information by tugging on the wire. With a really tight wire, couldn't I talk to you faster than the speed of light?


title: can we break the speed of light

text: I know this is impossible but I just want to know where I went wrong. here's my proposal: Let us imagine that we have a source of light and a switch that are far one light year from us. so it would take light one year to get to us. If we had a very long rod that connects my hand to the switch. When I move the rod a little distance I will reach the swich. So the time needed for me to close the switch is less than that made by light.right? Would I break the speed of light this way?


title: Spinning theoretical object moving faster than the speed of light

text: Let's say you have the Earth, or any celestial body, spinning as it is. What if you build a tower from the surface, and extended it out into space. If it was built far enough, could the furthest end exceed the speed of light?


title: Time taken for gravity of a distant object to interact with a newly created particle?

text: > **Possible Duplicate:**   >  The speed of gravity   >  Does gravity travel at the speed of light? Imagine there is a large mass $m_1$ (e.g. a star) 1 light-year away from us. It is stable, stationary relative to us and has been in place for a long time, much more than a year. A small mass $m_2$ (e.g. a proton) has just been created locally, 1 light-year away from $m_1$. How much time does it take for $m_2$ to feel the gravitational pull of $m_1$, and how can this be explained with the virtual-graviton theory of gravity? Some possible answers I can imagine: a) Immediately $m_2$ interacts with virtual-gravitons sent by $m_1$, a year ago. b) 1 year. It takes this long for freshly launched virtual-gravitons from $m_1$ to reach $m_2$ and vice-versa before any effect is felt on either mass c) 2 years. There needs to be an exchange of information / virtual-gravitons between $m_1$ and $m_2$ and this is the minimum time it could take. d) None of the above


title: Will we feel the gravity of a star 10 light years away for the next 10 years if, somehow, it vanishes today from its position?

text: I was watching a relativity video, and although I am not sure, I felt that it was trying to tell that the effect of gravitation of a body is instantaneous, in the sense that a sudden change in the mass of a body will instantaneously be felt by any object at a distance, even before light can be exchanged between them. So I want to make sure if its true. You don’t have to explain me why this is so if you don’t wish to. Also, please tell me if this has been experimentally established, or it is just a theory.


title: Why doesn't pushing balls in a tube propagate the movement faster than the SoL?

text: > **Possible Duplicate:**   >  Is it possible for information to be transmitted faster than light by using > a rigid pole? On one episode of QI they asked the question, "How fast do electrons move travelling around an electric current." The answer is (more or less) "very slowly", with the explanation that it's not the electrons that move fast, it's the force (I think). They likened it to pushing on one end of a long tube of touching marbles and observing how quickly one fell out the other end. This made me think about the tube. I realise that this tube cannot allow a force to propagate through the tube instantaneously because that would be faster than the speed of light, so my question is, why not? * * * **TL;DR;** Why doesn't a marble exit a long tube filled with touching marbles immediately when a force is applied to the marble at the other end?


title: How the effect travel's?

text: Let us assume that we placed lot's of ball touching each other in a hollow cylindrical tube, now if we push one ball at the end the ball at the other end move's instantly. So how do the information from one side of the tube travel's to the other side of the tube instantly. and at what speed at it travel's. If it travels at the speed of light so if make a long tube of the length of about $3*10^8$m it would take 1 second to see the effect at the other end assuming that nothing will bend or compress all thing's are ideal? I have read other related question but it didn't contained what I actually wanted to know. So it is not a duplicate.


title: Can a pushed plank beat light and break the laws of physics?

text: Imagine you are one lightyear away from a photon sensitive (light sensitive) switch. So it is obvious that light would take one year to reach to the switch. Now I have a one lightyear long plank. I simply point the plank towards the switch and press it. Now I just did work which light would take 1 year to do in a matter of seconds. Now the question is, did I break the laws of physics? Please do not give an answer like " it is impossible to make a plank this long ". Just think about it. Even I am thinking about this question right now.


title: What happens when I move a very long bar?

text: > **Possible Duplicate:**   >  Faster than light information Hey there, I just want to make clear that I'm new here and I don't know whether this is the right place to ask such a hypothetical question. So if it's not, I'd be glad if someone could tell me where to ask. :) Imagine you and a friend of yours were very far away from each other, let's say the distance is 1 light-year. Now you're holding a bar which measures exactly 1 light-year and currently touches your friend's shoulder. What happens when you move that bar (in order to poke your friend)? Will there be a delay? And what magnitude will that delay have? A year? Several years? Looking forward to your answers. :)


title: speed of gravitational waves

text: Do gravitational waves have a certain speed? Is it the speed of light or infinite, or am I misunderstanding what a gravitational wave is? I think it is a ripple in spacetime caused by interactions between objects, but i could be wrong.


title: Faster than light communication

text: > **Possible Duplicate:**   >  Is it possible for information to be transmitted faster than light? Suppose you have a very long rod, both ends are in space and the rod is 100 light years long. If you were to push the rod forward at one end, how long would it take for this be registered at the other end?. Is there theoretically a material the rod could be made of that would allow this?. Is this a possibility for faster than light communication across astronomical distances?. Also is "intermolecular" actually a word?.


title: Information faster than light?

text: Imagine that you have two typewriters on the other side of the galaxy. Their typebars are connected by ropes (stretched to the maximum), so anything you type will instantly appear at the second typewriter. Are the sentences information? If so, why not send information faster than light? If not, I'd at least ask if such a transfer is actually immediate. ![enter image description here](http://i.stack.imgur.com/8ft0S.png) Thank you very much.


title: The scissor paradox: can we pass the information faster than light?

text: click to view the image   Before I start, I want to say that this is not a duplicate of "Is it possible for information to be transmitted faster than light by using a rigid pole?", Since point A is not a real object, it is possible for A to exceed the speed of light. **_Notice that I have already taken the bending effect into account and decided that if the widths of the bars are small enough, then it's safe to treat the scissor as if it weren't bending_**. In other words, when they are thin, then the motion will obey the equation in a small region around point O. Please read to the end and have a look on my analysis in the last part before you explain.    Suppose we have a "scissor", which is composed by two bars with a same width of length "l". Bring them closer to each other, during this process, we have the equation(#):$$v=-\frac l4\frac{cos\frac{\theta}2}{sin^2\frac{\theta}2}\omega$$, where v is the velocity of A, \theta is the angle between BC and BA as shown in the picture.It's also an arbitrary function of time t, or $\theta=\theta(t)$. $\omega$ is the angular velocity of $\theta$, thus $\omega=\frac d{dt}\theta(t)$.Here is the derivation:   1\. Draw a line vertical to L from A, intersecting with L at C.     1. Since $sin\theta=\frac{AC}{AB}$, $$AB=\frac{AC}{sin{\theta}}$$.   2. Since $cos\frac{\theta}2=\frac{AO}{AB}$, $$AO=AB*cos\frac{{{\theta}}}2$$   3. from 2 and 3 we have: $$AO=\frac{cos\frac{\theta}2}{sin{\theta}}*AC$$   4. Since $AC=l$, the equation in 4 becomes: $$AO=\frac{cos\frac{\theta}2}{sin\theta}*l$$or$$AO=\frac l{2sin\frac {\theta}2}$$   5. Differentiate each side with respect to t, and denote $\frac d{dt}AO$ as "v", getting the equation(#): $$v=-\frac l4\frac{cos\frac{\theta}2}{sin^2\frac{\theta}2}\omega$$    Again, we just look at a small area around point O where our equation is obeyed. Now, at any time $t_0$, $\theta$ and $\omega$ are independent of each other, thus $\omega$ is a free variable. At t= 0, let's set the value of $\omega$ really large and constant, so that the instantaneous speed of A is larger than even the speed of light. Later, since the factor $\frac{cos \frac{\theta}2}{sin^2 \frac {\theta}2}$ in equation(#) is quite large when $\theta$ is close to 0, and since large $\omega$ leads to small $\theta$ during a short period of time, the velocity of A will be increased by this factor. Thus overall, $\omega$ is set to be a large constant, l is a constant, the only changing factor is increasing. Therefore, the velocity of A will increase. Meanwhile, we emit a beam of light. A is in a superluminal motion. And A will arrive at a detector first, causing the bars to touch the detector, and sending the command to the machine. Then the light beam comes later, thus being detected later. Notice that here again, we place the detector close enough to point O, so that our equation is valid. Remark: Someone may argue that we are not sure if A could attain a speed higher to that of light. So suppose at t=0, the force has already affected the area around point O, and is now traveling further.Meanwhile, ${\theta}_0=\frac{\pi}3$ A is speeded up to 0.999c, then since our formula is valid at t=0 and later period of time, let's apply it and set $l=0.001m, c=3*10^8m/s,{\theta}_0=\frac{\pi}3$. Then plug them in, we get $$\omega=-3.461*10^{11}rad/s $$. Now we want to know its velocity at $t=1*10^{-12}s$, and we get $v=6.89062*10^8m/s$, faster than light.    According to relativity, this shall violate casualty, and therefore could never happen. With which kind of mechanism is this paradox solved? I have taken the bending effect into account, and decided that as long as the widths of the bars are small enough, then it's safe to treat the scissor as if it weren't bending.


title: The speed of gravity?

text: Sorry for the layman question, but it's not my field. Suppose this thought experiment is performed. Light takes 8 minutes to go from the surface of the Sun to Earth. Imagine the Sun is suddenly removed. Clearly, for the remaining 8 minutes, we won't see any difference. However, I am wondering about the gravitational effect of the Sun. If the propagation of the gravitational force travels with the speed of light, for 8 minutes the Earth will continue to follow an orbit around nothing. If however, gravity is due to a distortion of spacetime, this distortion will cease to exist as soon as the mass is removed, thus the Earth will leave through the orbit tangent. What is the state of the art of research for this thought experiment? I am pretty sure this is knowledge that can be inferred from observation.


title: Consequences of inverse square law with vast distances (Gravity); (in addition, is light speed broken)?

text: As is well known, the gravitational force between two masses is dependent on the spatial distance between them. Therefore, even at vast distances, the masses exert equal and opposite forces on one another. I know that our theories need for this to be the case (to derive things such as escape velocity, which calls the force at r = $\infty$ to be zero (by utilizing a limit where the distance approaches infinity)), yet the universe is so vast and complex (with an unimaginable number of masses). With so many masses in space, it seems as though this could make a huge difference (over a large distance, such as special relativity's involvement in magnetism) . Also, say there is a supernova 400 light years away from earth. The time independent theory states that we should immediately notice a difference in the net forces acting on us, yet that would mean information travelled faster than light, correct? My main point is this: Is it absolutely imperative that there is still a force infinitely far away? Would the universe still function if we were to cut it off at something like a light year? ***** (This is my main point) Does the infinite reach of the gravitational force have anything to do with the scaffolding of the universe as a whole ?


title: Relativistic canonical transformation

text: What is relativistic canonical transformation? I need every piece of information about it. Does anyone know a reference or an article about relativistic canonical transformation? For example, in classical mechanics, under one and only one condition, you can say that a transformation is canonical and that is: $J\cdot M\cdot J^T= M$   where $J$ and $M$ are two matrices which are represented in _Goldstein (3rd Edition) - Page 342_


title: is there a way to split a black hole?

text: Classically, black holes can merge, becoming a single black hole with an horizon area greater than the sum of both merged components. Is it thermodynamically / statistically possible to split a black hole in multiple black holes? If the sum of the areas of the product black holes would exceed the area of the original black hole, it seems to be a statistically favorable transition by the fact alone that would be a state with larger entropy than the initial state


title: Does Energy change sign when time is reversed?

text: In classical physics if one reverses time then energy does not change sign. For example in the formula for kinetic energy one has: $$E = \frac{1}{2}m v^2$$ If you reverse time the velocity $v$ changes to $-v$ but $E$ stays the same. What happens in quantum mechanics? In QM one has the formula: $$E = h \nu = \frac{h}{\Delta t} $$ If $\Delta t$ changes to $-\Delta t$ then in QM the energy should change sign if the time is reversed. Why is there a difference between classical and QM?


title: Precision of Coulomb's law

text: Up to which precision has the coulomb law proven to be true? I.e. if you have two electrons in a vacuum chamber, 5 meters appart, have the third order terms been ruled out? Are there any theoretical limits to measure the precision ( Planck's constant?). Obviously there are practical limitations ( imperfect vacuum, cosmic rays, vacuum fluctuation). Still, does anyone know what was the smallest amount ever correctly predicted by that law? * * * Edit : Summary On the high end of the energy spectrum a precision of 10^-16 has been shown ( 42 years ago ) For electron point charges at large distances the law might brake down due to practical reasons. For moving particles QED gives a correction to the law: http://arxiv.org/abs/1111.2303


title: Is this a photograph of Surface Plasmon Resonance?

text: Does this photograph depict surface plasmon resonance? ![PHOTO 1 - Ellipsometric style photograph produces blue-green and purple resonance waves encircle nanogold-like tubule.](http://i.stack.imgur.com/RRN4e.jpg) PHOTO 1 - Ellipsometric style photograph produces blue-green and purple resonance waves from nanogold-like tubule. PHOTO 1 was cropped from PHOTO 2... ![enter image description here](http://i.stack.imgur.com/0hV22.jpg) ...an accumulation of cellular debris contained in a 1976 TEM, gratuitously provided by FDA/NIH. The original TEM images were produced on silver-halide photographic gel paper and the darkness of the biological sample can be attributed to staining the biological with nanogold. (NOTE - The dark images in the TEMS are silver atoms precipitated by electron impact with the target biological into film and then and duplicated by the development process in the photographic gel. The TEM images are not ink...this technology preceded digital printers.) This photographing technique is not novel and a variation of the medium is used for nuclear emulsions according to 'Anna.v' who answered my first question as to whether or not PHOTO 3 depicted a particle annihilation. PHOTO 3 ![enter image description here](http://i.stack.imgur.com/qQeHk.jpg) Answer from Anna.v: > There exist pictures of positron annihilations and creation of electron > positron pairs. Here is one: (Sorry, unable to duplicate image as the > question about particle annihilation.. was closed, fortunately after this > insightful answer.) > > A positron in flight annihilates with an electron into two gammas, which are > invisible. One of them materializes at a certain distance from the track > stop, resulting in a new electron-positron pair (marked with green) > > These are taken in bubble chambers with a magnetic field perpendicular to > the plane. What you are photographing is much more like what is seen in > nuclear emulsions. > > Chemical and biological energies are of the order of eV. To create a > positron with a mass of ~500.keV is not possible. From what I see the TEMS > has at most keV energies in the electron beam You may be seeing muons from > the continuous muon background at sea level. The flux if I remember > correctly is 1 per centimeter square per second (all energies). These could > kick off electrons and even have enough energy to create electron positron > pairs but it would not be a repeatable phenomenon since the flux is random. > Also to see how elementary particle tracks would look in your material you > would need to calibrate it at some accelerator lab. The photographic gel emulsion that produced the original images has yet to be calibrated, however, the initial impression that PHOTO 1 depicted a blue-green purple resonance Googled me into awareness of SPRs which have been studied extensively and detected elaborately but lack photographic depictions, at least from an online search. I suspect the photographic gel may have been doped with gold vapors, but coming from a government lab, the paper lacks commercial markings. The predominant issue in my analysis arises from information that blue-green resonance changes the color of 50 nm gold to red. The color of colloidal gold depends on the size and shape of the nanoparticle, so possibly this shape generates an anomaly with blue-green purple resonance and gold coloration. The underlying paradigm may depend on whether or not the questioned SPRs were generated by the original electron beam from the TEM or arise from interaction with the possible presence of gold vapor in the gel when photographed. Any insight would be greatly appreciated. Thank you, Walter Kyle


title: Finding induced current in a loop at an instant

text: I'm working on this problem; however, I cannot seem to get anywhere. Given information: The rectangular loop in the figure has 2.1x10^-2 ohm resistance. What is the induced current in the loop at this instant? ![\[Picture of Problem\]\(http://session.masteringphysics.com/problemAsset/1075420/4/34.P38.jpg\)](http://i.stack.imgur.com/IAVF7.jpg) It asks for the answer of the current of the loop in terms of I of the rod. I attempted the problem: > I_loop = emf/R = vLBA / R > > emf = d(magnetic flux)/dt = d(B * A)/dt > > B = (mu_0 I)/(2 pi d) From here I'm not exactly sure what to do. Any ideas?


title: Canonical quantization in supersymmetric quantum mechanics

text: Suppose you have a theory of maps $\phi: {\cal T} \to M$ with $M$ some Riemannian manifold, Lagrangian $$L~=~ \frac12 g_{ij}\dot\phi^i\dot\phi^j + \frac{i}{2}g_{ij}(\overline{\psi}^i D_t\psi^j-D_t\overline{\psi}^i\psi^j) -\frac{1}{4}R_{ijkl}\psi^i\psi^j\overline{\psi}^k\overline{\psi}^l,$$ where $g_{ij}=g_{ij}(\phi)$ is the metric, $R$ its riemann tensor, and covariant derivative $$D_t\psi^i~=~ \partial_t \psi^i +\Gamma^i_{jk}\dot{\phi}^j\psi^k.$$ (Details taken from the book _Mirror Symmetry,_ written by Vafa et al., paragraph 10.4.1.) Taken for granted that the above Lagrangian is classically supersymmetric, with susy transformations given by $$ \begin{aligned}\delta\phi^i &= \epsilon \overline\psi^i-\overline\epsilon \psi^i \\\ \delta\psi^i &= i\epsilon (\dot\phi^i-\Gamma^i_{jk}\overline\psi^j \psi^k)\\\ \delta\overline\psi^i &= -i\epsilon (\dot\phi^i-\Gamma^i_{jk}\overline\psi^j \psi^k). \end{aligned}$$ How can I quantise the classical supercharges $$Q=i\overline\psi_i\dot\phi^i, \qquad \overline Q=-ig_{ij}\psi^i\dot\phi^j$$ in a way such that $$ \delta F=[\epsilon Q+\overline\epsilon\overline{Q},F]_{\pm}$$ where $F$ is either a fermionic or bosonic field and $\pm$ is used appropriately? The natural answer would be something like calculate conjugate momenta $$ p_i=\frac{\partial L}{\partial\dot\phi^i}, \qquad \pi_{i\psi}=\frac{\partial L}{\partial\dot\psi^i},$$ and impose canonical commutation relations $$ [\phi^i,p_j]=i\delta^i_j, \qquad \\{\psi^i,\pi_{\psi,j}\\}=\delta^i_j.$$ Since in doing this I face non-trivial ordering issues, which the book doesn't seem to be worried about, and moreover its quantized version of conjugate momentum to $\phi$ seems wrong to me, as well as its quantized $Q$ doesn't seem to reproduce the correct transformations for the fields, I ask if someone could clarify this. Moreover, looking in the paper _Constraints on supersymmetry breaking_ by Witten, in the neighbourhood of eqs. (90), (91), he seems to claim that the correct definition of conjugate momentum is derivative with respect to covariant derivative instead of time derivative, and this is another thing which leaves me with some doubts.


title: Apostol or Spivak for mathematical physics?

text: I came across many recommendations for both of these books, but I'm not sure which one should I use to study calculus... I know most of the methods used in calculus and I use them frequently, but I'm shooting for mathematical physics and I was wondering which of these books would be better for someone aiming for something which is both mathematically rigorous and written in a good style for a physicist. Also, if there is some other book that fits the description, I would appreciate the recommendation! :)


title: stopping distance

text: if it takes a 4 door sedan x amount of feet to go from 65 mph to 45 mph (and i don't know how many feet that is) on a dry straight road, how many feet would it take for a large Mack truck transporting a full load of fluid, which would be most likely twice as heavy, to slow down from 65 to 45? Also, as the truck slows, it is moving to the right. Because of the fluid in the tank pushing forward and to the right also, would this have an effect on how well the truck can travel straight or would it naturally pull to the right side?


title: Direction change of gyroscopic precession with change in moment of inertia

text: In the classic bicycle wheel gyroscopic precession, using the right hand coordinate system of xyz and right hand rule of rotation, Such as ![bicycle wheel](http://i.stack.imgur.com/0ERpO.jpg) X axis is in the longitudinal direction (spin axis) z axis pointing in the direction of gravitation acceleration (downwards) Using Euler's equation, ![Euler's equation](http://i.stack.imgur.com/xdWeg.png) Am I right to understand gyroscopic effect as: The holding force of the string on the wheel produces a torque about the wheel which cause the wheel to have a natural tendency to swing downward. However, when there is a spin of the wheel in the x-axis, that force torque which will cause the wheel to drop actually produces a resultant torque that in the Y-axis, resulting in the wheel rotating about the Y-axis. Due to this rotation in the Y-axis, coupled with the spin in the x-axis, it produces a secondary precession effect of torque which prevents the wheel from dropping down. Is my understanding of precession correct? The wheel produces this direction of precession because of the input spin (x-axis) of the wheel. If the spin is reversed, the precession direction will be reversed. Is this correct thinking too? For a bicycle wheel, the moment of inertia in the spin (x-axis) is larger than the moment of inertia in the y and z direction. Now, my question is, based on the Euler's equation, if the moment of inertia is the other way round, what happens? For instance, instead of a bicycle wheel, what happens if we use a long rod where the moment of inertia in y and z direction is larger than the x direction? Will the direction of precession be reversed too?


title: Expected number of quanta in harmonic oscillator states

text: I'm working my way through A Squeezed State Primer, filling in details along the way. Let $a$ and $a^\dagger$ be the usual annihilation and creation operators with $[a,a^\dagger]=1$ and $|n\rangle=\frac{1}{\sqrt{n}}(a^\dagger)^n|0\rangle$. With $\mu$ and $\nu$ complex numbers, define \begin{eqnarray*} b &=& \mu a+\nu a^\dagger \\\ b^\dagger &=& \mu^\ast a^\dagger+\nu^\ast a \end{eqnarray*} Choose $\mu$ and $\nu$ so that $b$ and $b^\dagger$ satisfy $[b,b^\dagger]=1$, ie. $|\mu|^2-|\nu|^2=1$. So $b$ and $b^\dagger$ give a set of states 'isomorphic' to the usual number eigenstates. Define generalised number states $|n'\rangle$ by \begin{eqnarray*} b|0'\rangle &=& 0\\\ |n'\rangle &=& \frac{1}{\sqrt{n}}({b^\dagger})^n|0'\rangle \end{eqnarray*} (So the prime as attached to the state, not the $n$.) With $N'={b^\dagger}b$ we have $\langle n'|N'|n'\rangle=n$. Inverting the relationship between $a$ and $b$: \begin{eqnarray*} \mu^\ast b &=& |\mu|^2a+\mu^\ast\nu{a^\dagger}\\\ \nu{b^\dagger} &=& \nu\mu^\ast{a^\dagger}+|\nu|^2a\\\ \end{eqnarray*} \begin{eqnarray*} a &=&(|\mu|^2-|\nu|^2)a &=& \mu^\ast b-\nu{b^\dagger} \\\ a^\dagger &=& (|\mu|^2-|\nu|^2){a^\dagger} &=& \mu{b^\dagger}-\nu^\ast b \\\ \end{eqnarray*} So the question is, what are the expected number of quanta in the $|n'\rangle$ states? I think I can compute this via: \begin{eqnarray*} \langle n'|N|n'\rangle &=& \langle n'|{a^\dagger} a|n'\rangle \\\ &=& \langle n'|(\mu{b^\dagger}-\nu^\ast b)(\mu^\ast b-\nu{b^\dagger})|n'\rangle \\\ &=& \langle n'||\mu|^2{b^\dagger} b+|\nu|^2b{b^\dagger}-\mu\nu({b^\dagger})^2-\mu^\ast\nu^\ast b^2|n'\rangle \\\ &=& \langle n'||\mu|^2{b^\dagger} b+|\nu|^2b{b^\dagger}|n'\rangle \\\ &=& \langle n'||\mu|^2N'+|\nu|^2(N'+1)|n'\rangle \\\ &=& n(|\mu|^2+|\nu|^2)+|\nu|^2 \\\ &=& n(2|\mu|^2-1)+|\mu|^2-1 \\\ \end{eqnarray*} Is that right? For $n=1$ I get $3|\mu|^2-2$. Page 323 of the paper appears to say it's $2\mu^2-1$ but I may be misunderstanding it. Where is my error?


title: Deriving or justifying fundamental constants

text: Is there a fundamental way to look at the universal constants ? can their orders of magnitude be explained from a general points of view like stability, causality, information theory, uncertainty? for example, what sets the relative magnitudes of Planck's constant compared to say charge of electron or is it just a matter of choice of units. Does the physics become more or less cumbersome, insightful if we set all fundamental constants to 1 in appropriate units.


title: Ashcroft Mermin Solid State Physics Eq. 2.60ff

text: I'm trying to follow the steps in Eq. 2.60 of said book. What I cant seem to figure out is how to change the integration variables from 'k' to 'E', as they state. The equation is $$\int \frac{d\textbf{k}}{4\pi^3} F(\epsilon(\textbf{k})) = \int_0^\infty \frac{k^2 dk}{\pi^2} F(\epsilon(k)) = \int_{-\infty}^\infty d\epsilon \, g(\epsilon) F(\epsilon)$$ I can follow the first transformation (why is $\textbf{k}$ suddendly $k$?), $$\int\frac{1}{4\pi^3} k^2 F(\epsilon(k)) \, dk \int_0^\pi \sin \theta \, d\theta \int_0^{2\pi} d\phi = \int_0^\infty \frac{k^2 dk}{\pi^2} F(\epsilon(k))$$ But what's happening in the second step is unclear to me. In the book it says, "one often exploits the fact that the integrand depends on $\textbf{k}$ only through the electronic energy $\epsilon = \hbar^2k^2/2m$,...", but I'm unsure how this is used. Could anybody point this out to me?


What defines the adiabatic flame temperature? In a case I have to solve, I need to describe the combustion of natural gas (Groningen natural gas, to be specific). However, I am having some problems understanding the adiabatic flame temperature, since on the internet I find some temperatures, but to me it seems that it would depend on the initial conditions, such as the start temperature and the fuel-oxidizer mixture. The combustion should take place inside a ceramic foam, and I am assuming that this would happen isobaric at atmospheric pressure, and the inserted gas mixture before combustion would be at room temperature. I need to describe a stationary situation, in which the flue gas will lose a certain amount of heat according to a given formula to foam (which will emit the same amount of energy through thermal radiation). Would the adiabatic flame temperature be equal to the temperature of the exhaust gas before the heat loss to the ceramic foam? $$ {\Delta}T=\frac{Q}{c_p}, $$ with ${\Delta}T$ the increase in temperature of the exhaust gas (relative to the temperature of the inserted fuel-oxidizer mixture) and $Q$ the net calorific value of the gas (taking in to account that the amount of moles change during the combustion). Or is the adiabatic flame temperature something completely different?