A sphere rolling down a rough wedge which lying on a smooth surface

A sphere of mass $m$ and radius $r$ rolls down from rest on an inclined (making an angle $\phi$ with the horizontal ) and rough surface of a wedge of mass $M$ which stays on a smooth horizontal floor. Find the acceleration of the wedge and the acceleration of the sphere relative to the wedge. I have written out the equations of motion for the sphere and the wedge and also the equation of motion for the rotational motion of the sphere. But then I get stuck. I don't know how I should relate the angular accleration $\alpha$ of the sphere with the acclerations $a_{\perp},a_{\parallel}$ appearing in its equations of motion. Can anyone please help me out? Thanks!


How does a liquid crystal on Silicon affect the phase of incident light?

When a polarized light beam is directed to a phase-only LCoS system how is the phase modulated ? What is the physical effect behind ? Does voltage modulation has an impact on birefringence ?


Does the vacuum energy problem of quantum field theory only occur in the Hamiltonian approach, or also in the path integral approach and in AQFT?

In a standard QFT class, you're being indoctrinated that there is the "infinite vacuum energy density problem". (This is sometimes paraphrased as the "cosmological constant problem", which is in my opinion a misnomer since the calculation that calculates a finite value for the vacuum density and relates it to the cosmological constant is ill-founded and dubious.) The argument goes as follows: When defining the Hamiltonian $H$ from the Lagrangian density $\mathcal{L}$, we first define the Lagrangian function $L := \int \operatorname{d}^3 x \mathcal{L}$ and the momenta $\pi(x) := \frac{\partial L}{\partial \dot \phi(x)}$ and finally the Hamiltonian $H := \int \operatorname{d}^3 x \pi \dot \phi - L$. We then promote all the quantities that occured to operators and calculate the decomposition of $H$ into creation and annihilation operators $a^\dagger$, $a$. Then, the lecturer makes a big fuss (prototypical example) about the Hamiltonian being of the form $\int \operatorname{d}^3 p E_p \frac{1}{2} \left( a^\dagger a + a a^\dagger \right)$ = $\int \operatorname{d}^3 p E_p \left( a^\dagger a + \delta(0) \right)$ and the $\delta(0)$ being an infinite contribution to the vacuum energy density. (I always found the argument fishy because we can't know beforehand if some quantisation strategy will succeed or not, so we should rather start with a quantised Hamiltonian and ask whether its classical limit is the theory we started with, which can be answered with yes.) My question now is: Does a similar problem also occur in the Feynman path integral approach, or in any approach from algebraic quantum field theory?


interpretation of Green function

Is there a physical interpretation of the existence of poles for a Green function? In particular how can we interpret the fact that a pole is purely real or purely imaginary? It's a general question but I would be interested in the interpretation in quantum systems.


Is there a method which quantizes non-abelian gauge theories without path integrals formalism?

In the most QFT books there is a method of quantization of non-abelian theories through path integral methods. But I want to learn also the other methods without using of this formalism. Does anyone know these methods?


Unit Conversion

I need to convert this expression $ 81752 \left( \frac{BTU}{in^2 h ºR} \right )$ to $ \left( \frac{Kg}{s^3 ºC} \right )$ Is my draft correct? $ 81752 \left( \frac{BTU}{in^2 h ºR} \right ) \rightarrow \frac{1 BTU}{in^2 h} = \frac{454,3 W}{m^2} = \frac{1 ºR}{0,5555 K} = \frac{66,846533 W}{m^2 k} = \frac{66,896533 kg}{s^3 K} $ $ = \frac{1 K}{-272,2 ºC} = \frac{-2456 Kg}{s^3ºC} $


Interference - the shortest way from the point of constructive one to the point of destructive one

So this is a problem from Polish maturity exam. ![2 speakers](http://i.imgur.com/TtzkFnS.png) The image shows 2 speakers (G1, G2) and point B. The wavelength of sound coming from both speakers is 0.155 m, and the wave coming from both speakers is in phase. So point B is where the constuctive interference appears (you can calculate this quite easily). And the problem - "Draw an arrow in the direction of the shortest way possible from point B, where the sound intensity is high, to point A, where the sound intensity is low". Essentialy what they're asking is in which direction should we turn from point of constructive interference in order to, by going in a straight line, achieve point of destructive interference in the shortest way possible. So any thoughts on how to calculate, solve this?


Does light have mass/weight?

If I plug in my memory stick/USB into my laptop and load it up with documents etc. will its mass/weight increase?


General physics question involving Heisenberg Uncertainty Principle

Question: > An unstable particle produced in a high-energy collision is measured to have > an energy of $483\ \mathrm{MeV}$ and an uncertainty in energy of $84\ > \mathrm{keV}$. Use the Heisenberg uncertainty principle to estimate the > lifetime of this particle. Attempt at answer: I thought $$ (\Delta E)(\Delta t) = \hbar = 1.055 \times 10^{-34}\ \mathrm{J}\cdot\mathrm{s} = 6.591 \times 10^{-16} \mathrm{eV}. $$ So I tried solving for $\Delta t$, my time, with $\Delta E = 84\ \mathrm{keV}$. However, this answer is wrong (I get something like $7.85 \times 10^{-21}\ \mathrm{s}$, which is incorrect). I assume I'm supposed to incorporate $483\ \mathrm{MeV}$ somehow, but am not sure how. Any suggestions?


Comparing Graphene and Ni3 HITP2

A recently announced advance in Metallized Organic Films at MIT: http://pubs.acs.org/doi/abs/10.1021/ja502765n discusses a self-organizing material: Ni3HITP2 = Ni3(2,3,6,7,10,11-hexaiminotriphenylene)2 with a film conductivity of 40 Scm^-1 The wikipedia article talks about resistance of graphene of 10^-6 ohm-cm but mentions a sheet resistance of 1ohm/square under certain circumstances at the edge. I'm confused by the different metrics. How do the two materials compare?


Equation for a galaxy's overall gravitational pull?

Can the Newton's formula be used to express $g$ between two Galaxies in proximity? Gravity on Earth uses two concepts, one is the radius (R) of earth and the other is the distance $h$ from the surface of the Earth. Can you use similar concepts for galaxies? If that is the case, how do you measure $R$ and $h$, considering that: 1 - a galaxy does not have a spherical shape. 2 - a galaxy does not have much density. 3 - a galaxy does not appear to have a surface as would a star or a planet. What then would be the equation that I would use to calculate the pull from the Andromeda galaxy?


How fast is the Cosmic Microwave Background Radiation (CMBR) changing?

I know that the Cosmic Microwave Background Radiation (CMBR) is the leftover radiation from the "surface of last scattering". However, at every instant the surface is changing (at the rate of flow of time). So how constant is the CMBR?


Relativistic Time dilation and travel to exoplanets

NASA recently discovered the first habitable exoplanet with the Kepler space telescope. Recently I was on this site: http://www.cthreepo.com/lab/math1/ With the Long Relativistic Journeys calculator on that page, I calculated the amount of time it would take according to the traveler to travel 500 light years with half of the journey at a constant acceleration of `2g` and the second half with equal deceleration. The result was under 7 years. Does this require going near the speed of light? Or does it only require acceleration? If, say, we were to accelerate the space ship up to a certain speed and then decelerate periodically, could the same result be achieved? Or would no significant time dilation occur?


What exactly is Electromagnetic Radiation?

I don't understand how and why the electric and magnetic fields oscillate in the electromagnetic radiation wave, and any way where do these fields originate from, for there are no charged particles in the wave (there are no particles in it at all). Till now I've taken this for granted that the electric and magnetic fields oscillate at 90° to each other but recently I'm trying to study quantum physics but due to this confusion I couldn't understand it.


computer vs heater coil comparison

If I have a computer and a heater coil that consume exactly the same amount of energy, which would be more efficient at heating my room? This is assuming that they both have the same fans and heat dispersion characteristics. And if the answer is that they are the same, then why don't we change all of our personal household heaters into computers?


Renormalization Group and Ising with d=1 and D=1

I have a question about the results of RG on Ising model. I know it's possible to obtain two couple of relations   1. $K'(K)$, $q(K')$   2. $K(K')$, $q(K)$ between the coupling costants. My problem arise when i try to draw the flow diagram of the coupling constants. I know that this model dont allow phase transition except the trivial case $T=0$, but if we reiterate the relation (1) or (2) we increase or decrease the coupling costant. In one case i obtain $$K=0,T=\infty>>>>\cdots>>>>K=\infty,T=0$$ but in the other? **Boolean Ising Model** with $d=1$ dimension of lattice, $D=1$ dimension of vector space of the spins on lattice. The energy with zero external field is $$H=-J\sum_{<ij>}S_iS_j$$ note that there are overcounting. Then the partition function can be put in the following form $$Z=\sum_{\\{S\\}}\prod_ie^{KS_iS_{i+1}}$$ With a partial summation on even spins it became $$Z'=\sum_{S}\prod_ie^{K(S_i+S_{i+1})}+e^{-K(S_i+S_{i+1})}=\sum_{S}\prod_if(K)e^{K'S_iS_{i+1}},$$ where in the last i used the scaling proprieties   $$Z(N,K)=f(K)^{N/2}Z(N/2,K').$$ The relations for $f(K)$ and $K'(K)$ are: $$f(K)=2\cosh^{1/2}2K,$$ $$K'=\frac{1}{2}\log\cosh{2K}.$$ The extensivity of free energy states $-\beta F=\log Z=Nq(K)=\frac{N}{2}\log f(K)+\frac{N}{2}q(K')$.


Triple axis accelerometer direction-finding

Can I detect the direction of the disturbance source which propagated in the medium by waves, using only triple axis accelerometer? I guess yes, because I already have projections of all three axes (i.e. three accel detectors) on first arrived wavefront. Or maybe I'm wrong? What mathematics could be involved?


Binary Star system with one star stationary?

Can a Binary Star system be possible where in one star is stationary and the other star revolves around it? (Just like a planet revolving a star. i.e planets in the system and the star revolving around their own center of masses to balance the system).


Number of zero-modes on the sphere

Is it true that a field of conformal dimension $h$ (integer or half integer) has $1-2h$ zero-modes on the sphere, if $1-2h \geq0$. This seems to be right for different ghost fields :   * $c$ has weight $-1$ and 3 zero-modes   * $\gamma$ has weight $-1/2$ and 2 zero-modes   * $\xi$ has weight $0$ and 1 zero-modes   * $\eta$ has weight $1$ and 0 zero-modes I'm not sure how to compute the number of zero-modes on the sphere... Is it the argument given at p.167 in volume 1 of Polchinski's book ?


Is Newton's third law always correct?

Newton's third law states that every force has an equal and opposite reaction. But this doesn't seem like the case in the following scenario: For example, a person punches a wall and the wall breaks. The wall wasn't able to withstand the force, nor provide equal force in opposite direction to stop the punch. If the force was indeed equal, wouldn't the punch not break the wall? I.e., like punching concrete, you'll just hurt your hand. Doesn't this mean Newton's third law is wrong in these cases?


Sound level of organ pipe driven by helium

The sound pressure level of an organ pipe is a function of the gas flow rate delivered to the pipe. Source Would the sound level of an organ pipe driven by helium be lower than that of a pipe driven by air at the same blowing pressure (same volume flow rate) due to the lesser mass flow rate of the helium (helium being of lesser density than air)?


Trace operation on dynamic equation: physical meaning?

Suppose we have Heisenberg equation of motion for some observable $A$, $$ i\hbar\frac{dA}{dt}= -[H,A] $$ since the trace of any finite dimensional commutator structure vanish(not something like $[x,p]=i\hbar$ ), $$ Tr\left(\frac{dA}{dt}\right)=\frac{i}\hbar~Tr([H,A])=0 $$ so my question is, what does $Tr(\frac{dA}{dt})=0$ indicate?


Is lattice enthalpy positive or negative?

I've learnt that the lattice enthalpy (defined as the energy change from a solid ionic lattice to separate gaseous ions) is always positive, obviously. However, I've seen it explained as the opposite other places, so it's negative. What is correct?


Some questions about calculation central charge in a CFT in $d$ spacetime dimensions

This is based on this paper, http://arxiv.org/abs/hep-th/0212138   * For a CFT on a $S^d$ spacetime (of radius R) it seems to be claimed that the central charge is given by, $ c = \langle \int_{S^d_R} d^dx \sqrt{g} T_\mu ^\mu \rangle $    * Their equation 23 (on page 6) seems to indicate that if $W = -log Z$ is the free energy of the theory then it further follows that, $c = \frac{1}{d}R \frac{\partial }{\partial R} W$ (...I believe that the derivative is being evaluated at the value of the radius of the sphere..)    * Just below equation 26 it is claimed that, "...the central charge can be read off from the coefficient of log R in an expansion of W[R]..."  I would like to know the proof/derivation of three methods that have been spelt out to calculate the central charge of a CFT.


Work is force multiplied by distance What is this distance?

![Work](http://i.stack.imgur.com/8XUua.gif) What is the distance which the force is multiplied by it to get the work done? a) The distance of the body motion while the force was touching it? b) The total distance of the body motion?


Why my simple electromagnetic experiment does not work?

I make a setup (see attached picture).   * ONE coil (1" diameter x ~120 turns), which is separated with a gap, is mount on two sides. Each side has an iron rod (1" diameter x 6" long) to increase the focus of the magnetic wave.The wire for the core is 14 gauge motor wire (5.9 amp power rating max).   * Two super strong magnets (1" diameter x 0.5" thick each) are mount at the middle block (plastic) under Duck tape. Each of them is rated ~40 lbs holding force. Each side of the block has one magnet.   * I use 6 power resistors in series to create 50 ohm resistance. I connect the circuit to wall power 120VAC. Since the impedance of the coil is small, I think I give 2.4 amp @ 60 HZ to the circuit.    * Everything for holding is by very smooth plastic (Acetal Resin).    * Before I connect the power, I put the system at rest. The magnet is so strong that, they make the middle piece to go to either one of the iron.  I thought the MIDDLE PIECE would swing to the left and right with 60Hz. However, the system does NOT move at all. Any idea? ![Experiment setup](http://i.stack.imgur.com/PoK8O.jpg) ![enter image description here](http://i.stack.imgur.com/qBCen.jpg)


Which charge value to use and Why

$1\mu C=10^{-6}C$ vs $e=1.60\times10^{-19}C$. I am having trouble understanding when to use for $q$, $10^{-6}C$ or $1.60\times10^{-19}C$. I had a problem where $a=Eq/m$. I don't know which value of a charge go into the formula. I need to understand why.


Double slit experiment and representation of light waves

Consider the following image from Wikipedia and based on it I have a doubt. ![Image Credits: Wikipedia](http://i.stack.imgur.com/X0QtM.gif) I do not understand why are the light waves represented like the waves in water. Shouldn't the waves be like sine waves. Why is the slit part the starting of a new wave. Secondly, Why does this lead to to destructive interference rather than constructive because the waves are emitted at the same time. And if anyone can tell me how is double slit possible even when there is a single photon?


Slinky base does not immediately fall due to gravity

Why does the base of this slinky not fall immediately to gravity. My guess is tension in the springs is a force > mass*gravity but even then it is dumbfounding. ![](http://i.imgur.com/qr3dIsC.gif)


Potential and Kinetic Energy

In engineering school you learn the basic swing problem. Essentially that there is a transfer of kinetic energy (as seen in the velocity at the bottom of as swing) to potential energy at the top of the swing's arc. My question: What is the nature of KE and PE and how do they seamlessly "transfer" between the two? How is this explained by classical physics?


Space-time hospitality

Are there any reasons to assume that organic processes cannot arise in relatively flat spacetime environments?.. An argument for the existence of a critial value or window ..


Noether's theorem and gauge symmetry

I'm confused about Noether's theorem applied to gauge symmetry. Say we have $$\mathcal L=-\frac14F_{ab}F^{ab}.$$ Then it's invariant under $A_a\rightarrow A_a+\partial_a\Lambda.$ But can I say that the conserved current here is $$J^a=\frac{\partial\mathcal L}{\partial(\partial_aA_b)}\delta A_b=-\frac12 F^{ab}\partial_b\Lambda~?$$ Why do I never see such a current written? If Noether's theorem doesn't apply here, then is space-time translation symmetry the only candidate to produce Noether currents for this Lagrangian?


Taylor expansion of the metric

Consider a coordinate change $$ x^a\mapsto \tilde x^a=x^a+\epsilon y^a $$ In the note I am reading, the author calculate the change of metric by $$ g_{ab}(x) = \tilde g_{ab}(\tilde x)=\tilde g_{ab}(x^a+\epsilon y^a)=\tilde g_{ab}(x^a)+\epsilon\mathcal{L}_Y\tilde g_{ab}(x^a)+\cdots $$ My question is, why do we use $\mathcal{L}_Y\tilde g$ rather than $\nabla_Y\tilde g$ in the first order term? * * * Update Following is my newly try. As Prahar pointed out, I should write $$ g_{ab}(x)\mapsto\tilde g_{ab}(\tilde x)=\frac{\partial x^c}{\partial {\tilde x}^a}\frac{\partial x^d}{\partial {\tilde x}^b} g_{cd}(x) $$ Then we have $$ g_{ab}(x)=\tilde g_{ab}(x^a+\epsilon y^a)=\tilde g_{ab}(x^a)+\epsilon\nabla_Y\tilde g_{ab}(x^a)+\cdots $$ And I have to show $$ \nabla_Y\tilde g_{ab}(x^a)=\mathcal{L}_Yg_{ab}(x^a) $$ We have $$ \begin{align} \tilde g_{ab}(x^a)&=\frac{\partial x^c}{\partial {\tilde x}^a}\frac{\partial x^d}{\partial {\tilde x}^b} g_{cd}(x^a)\\\ &=\left(\delta_a^c-\epsilon\frac{y^c}{\tilde x^a}\right)\left(\delta_b^d-\epsilon\frac{y^d}{\tilde x^b}\right)\left.\right|_{x^a-\epsilon y^a}g_{cd}(x^a-\epsilon y^a)\\\ &=g_{ab}(x^a-\epsilon y^a)-\epsilon\left[\frac{y^d}{\tilde x^b}g_{ad}+\frac{y^c}{\tilde x^a}g_{cd}\right]\left.\right|_{x^a-\epsilon y^a}+O(\epsilon^2)\\\ &=g_{ab}(x^a) - \epsilon\nabla_Yg_{ab}(x^a)-\epsilon\left[\frac{y^d}{\tilde x^b}g_{ad}+\frac{y^c}{\tilde x^a}g_{cd}\right]\left.\right|_{x^a} +O(\epsilon^2) \end{align} $$ So we get $$ \nabla_Y\tilde g_{ab}(x^a)=\nabla_Yg_{ab}(x^a)-\epsilon\nabla_Y\left(\nabla_Yg_{ab}+\frac{y^d}{\tilde x^b}g_{ad}+\frac{y^c}{\tilde x^a}g_{cd}\right)\left.\right|_{x^a} + O(\epsilon^2) $$ On the other hand, we have $$ \mathcal{L}_Yg_{ab}(x^a)=\nabla_Yg_ab{x^a}+g_{ac}\nabla_bY^c+g_{cb}\nabla_aY^c $$ I cannot see these two are identical. Is there anything wrong in my deduction?


Stability theory

I'm studying stability theory recently and met a lot of phrases like linear stability and nonlinear instability. After searching on Google, I became more confused. Thus I wonder if there is any approperiate or famous books discussing these issues of dynamical systems.


How fast would a truck have to go to pull a pedestrian onto the road?

Let's suppose a pedestrian **P** is walking or standing next to a highway.   Suppose a truck **T** will drive by the said pedestrian at speed **V** leaving distance **L** between the two. Assuming **L** is a reasonably small finite size how would one calculate the force acting upon the pedestrian from both the general differences in pressure and turbulence? Shortly put, how fast would the truck have to go _in practice_ so that the air flow left in its wake would pull the pedestrian onto the road? ![Sample image](http://i.stack.imgur.com/Q8R1P.png)


Is Initial Coherence a Problem in Experiments?

In experiments, or optical setups, is the initial coherence of lasers something to be concerned about? You can calculate interference patterns, but those generally rely on the phase shift between the two sources being zero, right? In a laboratory, how do you know two lasers will be in phase?


Tricks at manipulating creation/annihilation operators

Manipulation of terms in algebras different from the standard one (e.g. boolean algebra) can be a bit unnatural but there are always shortcuts that can help you. I was wondering if there is a list with the standard tricks for manipulating creation and annihilation operators of bosons and fermions, instead of using intensively their commutation relations and having to rediscover the wheel every time. This would prevent me (and many others) from losing a big amount of time at blind guess-checking... Even a good set of exercises can count as an answer.


POVM positive density matrix decomposition

I have to prove that given a density matrix $\rho$ in a finite d-dimensional Hilbert space $\mathcal{H}_d$, it always exists at least one informationally complete POVM measure $\\{E_i\\}_{i=1}^{d^2}$ such that $$ \rho=\sum_{i=1}^{d^2} \alpha_i E_i, $$ with $\alpha_i\geq 0$. Without the last constraint on the $\alpha_i$ it is obvious, how can I prove it imposing the constraint $\alpha_i\geq 0$? Any idea?


Having trouble understanding the concept of current and Ohms law

Circuit in series: $10\,\text{V}$ power supply, 2 resistors connected. Each resistor is $10\,\Omega$, total of $20\,\Omega$ resistance. If I put an ammeter at the end of the circuit, will the current be calculated using 20 Ohms or 10? Electrons flow through the first 10 Ohm resistor and are slowed down, yes? Then they flow through the second 10 Ohm resistor. They have passed through a total of 20 Ohms BUT at 10 Ohms per resistor. Will the current be the same as if it had passed through 1 20 Ohm resistor? That violates everything I understand.


Are there any known (closed form even if approximate) solutions to problems in relativistic elasticity?

There are several useful known solutions to the EFE with relatively simple / trivial stress-energy-momentum tensor, such as the Schwarzschild solution. Despite the idealizations made therein they are enormously important and insightful. I am curious to know 1) Do any such solutions exist for the case of relativistic elastic solids? 2) If not, what are the principal obstacles in constructing solutions, even approximate (say linearized) ones? 3) If I understand correctly, various competing versions of a "theory" of relativistic elasticity have been around for several decades. Is there a reason why there are multiple theories? 4) Also, IIUC, matters of existence and uniqueness for the Cauchy problems arising from such theories have also been dealt with. I came across this paper by Beig and Schmidt, in which the authors deal with questions of local existence and uniqueness for a theory they describe as a "relativistic version of classical elasticity". Is this a generally accepted state of affairs or are there caveats? 5) There seem to be several disparate sources of information about this topic, is there a nice "unified" resource to get started? Thanks.


One problem on Classical Mechanics

![enter image description here](http://i.stack.imgur.com/L1h39.jpg)![I'm wondering why in this problem I can't apply newton second law? The mass m of the moving part will be the mass solved in the problem.](http://i.stack.imgur.com/HXxjS.jpg) I'm wondering why in this problem I can't apply newton second law? The mass $m$ of the moving part will be the mass solved in the problem.


About photons and mirrors

If a photon hits a 'perfect' mirror (with no environment interference) would the mirror move a bit?


Non Classical mechanic answer to : length of time a thrown object spends in rest before falling down?

When an object is thrown upwards, when it eventually comes to rest and starts falling, for how long is it stationary? What about an particle in electric field having an initial velocity towards it's same charge? That too would come to rest and reverse velocity, the question is for how long is it at full stop with rest? Classical physics gives a time of 0, but is that correct? Is it really at rest for 0 seconds? That answer seems a bit counter intuitive.


How to liquefy Hydrogen?

I have got a science project and my teacher has recommended me to do "Liquefying Hydrogen". I have been continuously thinking about that but I have not come to a solution. Can anyone please tell me how to liquefy and store hydrogen??


Analogy between magnetic bottle and Van Allen's radiation belt

A magnetic bottle is an arrangement that permits to confine charged particles. Here you can find a review for charged particle rotating in a magnetic field and at the bottom of the page a description of the magnetic bottle: Motion of a charged particle in magnetic field Let's use this image since it's more clear ![enter image description here](http://i.stack.imgur.com/y45wc.jpg) What results from this picture is that a particle inside a magnetic bottle makes a circular motion following magnetic field lines which have a sort of cylindrical symmetry as it's expressed by this simulation: Magnetic Bottle 1 So you can imagine to put the earth in the middle of the picture: in this way the two rings are to the two magnetic poles. In the case of a Van Allen's radiation belt motion simulation (Radiation Belt as a Magnetic Bottle), particles move around a magnetic field lines instead of moving around the earth. But from this simulation: Magnetic Bottle 2 you can see that a motion around a magnetic line is possible but i can't understand heuristically why. So the questions are:   1. how a particle can move around a magnetic line if there is closed to it another one? so how can a particle choose around which line move (losing in this way the cylindrical symmetry)?   2. why the two type of motions are in disagreement? Is the motion in the first picture just an approximation perhaps?


Wind Turbine Impact

OK, I'm not sure this is not a silly question, but here goes. I've often wondered what the impact is of wind turbines on the weather. My question is, since wind turbines transform wind energy into electricity, this means that the winds lose in power. I'm well aware that for the effect to be noticeable, you'd need a pretty large wind turbine park. But I wondered how large? How would the effect scale with the size of the park? Bonus question: what about tidal turbines? Most closely related questions: Is it possible that windmills can consume the wind? What does the quantification of causes and effect look like for clouds in offshore wind turbine wakes?


Solving a Young Laplace equation for a meniscus against a flat plate

This is more of a math question and one, furthermore, that I know the final answer to. What I am asking is more of a "how do I get there" question as this question was generated during a self study situation. So, for a flat plate with a meniscus on it at some contact angle the form is: a y''$/(1+(y')^2)^{3/2}$ + b g y = 0 where a is surface tension, b is density, and g is the usual 9.8 m$/s^2$. Now, I know that when you "A first integration, together with the boundary condition dy/dx=y=0 as x goes to infinity yields: 1$/(1+y'^2)^{1/2}$=1-(bg/2a)$y^2$ " \--From de Gennes on Menisci My problem is with the integration. How do I integrate something like the 1st equation? I find myself running in circles. Worse, I know that this type of integration is something I've run across numerous times, but I can't seem to find the methodology in the current stack of books. So I beg, can someone either tell me where to look to find the appropriate methodology to attack this problem or take me through the process? I would be grateful for as long as it would be useful for you. Thanks, Sam


Why does Joule-Thomson expansion of a gas do no (external) work?

I understand one of the benefits of the Hampson-Linde cycle is that there are no cold-side moving parts, but isn't one losing an awful lot of energy in the throttling process? There must be something basic I'm missing here, because it looks almost like this cycle destroys energy: pressure drops while volume increases, and temperature drops at the same time. The expansion is isentropic (How does that differ from an adiabatic expansion? I thought the condition that dS = 0 constrained a gas to a unique trajectory - an adiabat - along which it does work. But unless I'm mistaken, a Joule-Thomson expansion does no external work, despite not being an expansion into an evacuated space. Where did the work "go"? Maybe an example might illustrate. If I let air out of my car's tyres, is that a Joule-Thomson expansion? If so, doesn't the expansion do work in lifting the atmosphere by 20 attometers? If not, what _is_ an everydayish example?


How fast can fire spread?

You can build a home-made flamethrower by using a spray and a lighter: ![enter image description here](http://i.stack.imgur.com/uxv6S.png) (from this YouTube video) But this can be dangerous if the fire gets in the can and the can explodes. So I guess the speed at which the particles come out of the box has to be high enough to prevent this. > **NOTE** : I don't want to do this at home, because I think it's too > dangerous. This is just a thought experiment! I guess the question " **how fast can fire spread?** " highly depends on the details (like: What kind of exothermic reaction is going on; how big is the pressure at which the spray exits the can; temperature around the can). I would like one theoretical answer (choose everything to make it as fast as possible / slow as possible) and a practical answer (make a reasonable assumption of what could go on with the lighter / spray stuff).


Trying to solve 2D Toda Lattice Equation with Lax Pair Approach

I am working on this Hamiltonian: $$ H = \frac{p_1^2 + p_2^2}{2m} + e^{q_2-q_1} + e^{q_2} + e^{-q_1} -3 $$ Thank you for the hint that it is a modification of the Toda Lattice Equation. Let me sketch what I tried until now and why it is not working: Analogous to the mentioned publications I introduced $$b_n:= \frac{1}{2}Exp{(\frac{q_n-q_{n+1}}{2})}\\\ a_n:= -\frac{p_n}{2} $$ where it follows directly with $\frac{\partial H}{\partial q_i}=-p_i$ and $\frac{\partial H}{\partial p_i}=q_i$: $$\dot{b_n} = (a_{n+1} - a_n)b_n \\\ \dot{a_n} = 2 (b_{n}^2 - b_{n-1}^2)$$ When now using the Lax Pair $L$,$B$: $$ L f_n = b_n f_{n+1} +b_{n-1} f_{n-1} + a_n f_{n}$$ $$ B f_n = b_n f_{n+1} - b_{n-1} f_{n-1} $$ it can be shown that $\partial_t L=[B,L]$. My problem arises in defining the border conditions of my couple $q_1$ and $q_2$ in the 2d lattice above, since one needs to shift to the 3d representation $\\{b_0,b_1,b_2\\}$ in order to satisfy the periodic conditions (One mutual coordinate $q_3 = 0$ coupled to the others). Since it can be shown easily that $\dot{\lambda} = 0$ (where $\lambda$ is an eigenvalue $Lv=\lambda v$) the constants of motion reduce to the calculation of the eigenvalues. But in this case the eigenvalues of $L$ dont seem to simplify, in fact it doesnt seem to be a solution, which was my inital goal. In general this approach seems to be at overkill for the 2d problem since it solves the n-dimensional Toda lattice.   1. Anyone knows of an easier approach to the 2d problem?   2. The Matrix $L$ seems to yield the wrong solution: $$ L = \begin{pmatrix} a_0 & b_0 & 0 \\\ b_0 & a_1 & b_1\\\ 0 & b_1 & a_2 \end{pmatrix}$$ The Matrix does neigher solve $\partial_t L = [B,L]$ (with $B=L_+ - L_-$) nor are eigenvalues constants of motion. Was has gone wrong?   3. Since the inverse scattering method can be applied here, I tried to get the scattering data, but actually I was not able to do the task. Any literature?


Why do we only have complete particle generations?

There are 3 generations of fermions in the standard model. I know that there is a theorem that states, that only complete generations are allowed. This means that there have to be quarks with three colors. With this, if we sum over all leptons and quarks in a generation the overall charge is zero. Why is it assumed that this must hold?


Two spinor tensors and Maxwell's equations

Let's have two symmetric (by the indices) spinor tensors $F_{ab}, F_{\dot {a}\dot {b}}$ and conditions $$ F_{ab}, \partial^{\dot {a} a}F_{ab} = 0, \quad F_{\dot {a}\dot {b}}, \partial^{\dot {a}a}F_{\dot {a}\dot {b}} = 0, \qquad (1) $$ $$ \partial^{a \dot {a}} = (\tilde {\sigma}_{\mu})^{\dot {a} a}\partial^{\mu}, \quad (\tilde {\sigma}_{\mu})^{\dot {a} a} = \varepsilon^{\dot {a}\dot {b}}\varepsilon^{a b}(\sigma_{\mu})_{b \dot {b}} = (\sigma_{0}, -\sigma_{i}). $$ by which these tensors may represent massless field with helicity $1, -1$ respectively. If our field is real, we must take the direct sum of these representations. It may be described through antisymmetrical 4-tensor rank 2. I know that there is the correspondence between an arbitrary antisymmetric tensor $F_{\mu \nu}$ and two symmetric spinor tensors $F_{ab} , F_{\dot {a} \dot {b}}$: $$ F_{\mu \nu} = \frac{1}{2}(\sigma_{\mu \nu})^{a b}F_{ab} - \frac{1}{2}(\tilde \sigma_{\mu \nu})^{\dot a \dot b}F_{\dot {a} \dot {b}}, $$ $$ F_{ab} = (\sigma^{\mu \nu})_{ab}F_{\mu \nu}, \quad F_{\dot {a} \dot {b}} = (\tilde {\sigma}^{\mu \nu})_{\dot {a}\dot {b}}F_{\mu \nu}. \qquad (2) $$ Then it obvious that I must substitute $(2)$ into $(1)$ and get Maxwell's equations. But I can't do it, because I have the problem with convolution of sigma-matrices. Can you help me?


Noether's Theorem For Functionals of Several Variables

My question is on using a form of the single variable Noether's theorem to remember the multiple variable version. Does Noether's theorem, for functionals of a single independent variable, just say that, because $\mathcal{L}$ is invariant, we have $$\mathcal{L}(x,y_i,y_i')dx = \sum_{j=1}^n p_i d y_j - \mathcal{H}dx = \mathcal{L}(x^*,y_i^*,y_i'^*)dx^* = \sum_{i=1}^n p_i d y_i^* - \mathcal{H}dx^*?$$ It is usually stated by saying that $$\sum_{i=1}^n \frac{\partial \mathcal{L}}{\partial ( \tfrac{d y_i}{dx})} \frac{\partial y_i^*}{\partial \varepsilon} - \left[\sum_{j=1}^n \frac{\partial \mathcal{L}}{\partial ( \tfrac{d y_j}{dx})} \tfrac{\partial y_j }{\partial x} - \mathcal{L}\right]\frac{\partial x^*}{\partial \varepsilon}$$ is conserved, so is this equivalent to what I've written above (I've hopefully justified all of this below)?. (If there is any ambiguity in this question, I've offered a hopefully unnecessary explanation of the details below) The only problem with it I see is that it works fine for conservation of linear momentum, but it seems to show that conservation of energy and angular momentum are always zero, since $dx^* = 0$, unless I add a constant to the above, i.e. $$\mathcal{L}(x,y_i,y_i')dx = \mathcal{L}(x^*,y_i^*,y_i'^*)dx^* + C$$ which seems fine as adding constants to Lagrangians do not affect the EOM, so that's the only problem I see with it. I like the above expression, it's great for remembering Noether's theorem, it's more general than what almost every physics book presents as Noether's theorem (it's not just conservation of energy!), explains what Landau's additivity of integrals of motion comment means mathematically etc... **Can we generalize it to Lagrangian densities?** The statement of the multivariable Noether I know is that, for $$\mathcal{L} = \mathcal{L}(x_i,u_j,\frac{\partial u_j}{\partial x_i})$$ we have that $$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left[\sum_{j=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_j}{\partial x_i})}\left(\frac{\partial u_j^*}{\partial \varepsilon_k} - \sum_{i=1}^n\frac{\partial u_j}{\partial x_i}\frac{\partial x_i^*}{\partial \varepsilon _k}\right) + \mathcal{L}\frac{\partial x_i^*}{\partial \varepsilon _k}\right] = 0$$ I can hardly remember this, and as I've indexed it I can't turn it into anything involving what I _think_ is the Hamiltonian for a functional of several independent variables $$\mathcal{H} = \sum_{j=1}^np_{ij}\frac{\partial u_j}{\partial x_i} - \mathcal{L} = \sum_{j=1}^n \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_j}{\partial x_i})}\frac{\partial u_j}{\partial x_i} - \mathcal{L}$$ (I have no idea why Jackson adds a $g^{\alpha \beta}$) Can this be turned into something similar to my main equation, perhaps using $\delta_{ij}$s or $g_{\mu \nu}$s or something? An attempt: $$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left[\sum_{j=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_j}{\partial x_i})}\left(\frac{\partial u_j^*}{\partial \varepsilon_k} - \sum_{i=1}^n\frac{\partial u_j}{\partial x_i}\frac{\partial x_i^*}{\partial \varepsilon _k}\right) + \mathcal{L}\frac{\partial x_i^*}{\partial \varepsilon _k}\right] = 0$$ $$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left[\sum_{j=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_j}{\partial x_i})}\left(d u_j^* - \sum_{i=1}^n\frac{\partial u_j}{\partial x_i}d x_i^* \right) + \mathcal{L}d x_i^* \right] = 0$$ $$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left[\sum_{j=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_j}{\partial x_i})}d u_j^* - \sum_{k=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_k}{\partial x_i})}\sum_{l=1}^n\frac{\partial u_k}{\partial x_l}d x_l^* + \mathcal{L}d x_i^* \right] = 0$$ $$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left[\sum_{j=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_j}{\partial x_i})}d u_j^* - \sum_{k=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_k}{\partial x_i})}\sum_{l=1}^n\frac{\partial u_k}{\partial x_l}d x_l^* + \sum_{l=1}^n\delta^l_i \mathcal{L}d x_l^* \right] = 0$$ $$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left[\sum_{j=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_j}{\partial x_i})}d u_j^* - \sum_{l=1}^n (\sum_{k=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_k}{\partial x_i})}\frac{\partial u_k}{\partial x_l} - \delta^l_i \mathcal{L})d x_l^* \right] = 0$$ $$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left[\sum_{j=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_j}{\partial x_i})}d u_j^* - \sum_{l=1}^n \mathcal{H}d x_l^* \right] = 0$$ $$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left[\sum_{j=1}^m p_{ij}d u_j^* - \sum_{l=1}^n \mathcal{H}d x_l^* \right] = 0.$$ I don't know if this is right. References:   1. Gelfand, Calculus of Variations - Ch. 3, 4 & 7   2. Jackson, Classical Electrodynamics - Ch. 12 Sec 10 * * * Noether's theorem for functionals of a single independent variables states that, given a continuous one-parameter family of infinitesimal transformations of the form $$T(x,y_i,\varepsilon) = (x^*(x,y_i,\varepsilon),y_i^* (x,y_i,\varepsilon))= (x^*,y_i^*)= (x+\frac{\partial x^*}{\partial \varepsilon}\varepsilon,y_i+\frac{\partial y_i^*}{\partial \varepsilon}\varepsilon),$$ if a Lagrangian $$\mathcal{L} = \mathcal{L}(x,y_i,\frac{dy_i}{dx})$$ is invariant $$\mathcal{L}(x,y_i,\frac{dy_i}{dx}) = \mathcal{L}(x^*,y_i^*,\frac{dy_i^*}{dx})$$ under $T$ then the scary-looking quantity $$\sum_{i=1}^n \frac{\partial \mathcal{L}}{\partial ( \tfrac{d y_i}{dx})} \frac{\partial y_i^*}{\partial \varepsilon} - [\sum_{j=1}^n \frac{\partial \mathcal{L}}{\partial ( \tfrac{d y_j}{dx})} \tfrac{\partial y_j }{\partial x} - \mathcal{L}]\frac{\partial x^*}{\partial \varepsilon}$$ is conserved (more precisely, you do it in terms of the functional but that is irrelevant for this discussion). Examining this quantity for two seconds let's us notice we can write this conserved quantity as $$\sum_{i=1}^n \frac{\partial \mathcal{L}}{\partial ( \tfrac{d y_i}{dx})} \frac{\partial y_i^*}{\partial x} - \mathcal{H}\frac{\partial x^*}{\partial \varepsilon}$$ in terms of the Hamiltonian $$\mathcal{H} = \sum_{j=1}^n \frac{\partial \mathcal{L}}{\partial ( \tfrac{d y_j}{dx})} \tfrac{\partial y_j }{\partial x} - \mathcal{L} = \sum_{j=1}^n p_i \tfrac{\partial y_j }{\partial x} - \mathcal{L} $$ or, taking away the parameter, as $$\sum_{i=1}^n p_i d y_i^* - \mathcal{H}dx^*$$ Notice the similarity between the quantity in Noether's theorem and this: $$\mathcal{H} = \sum_{j=1}^n p_i \tfrac{\partial y_j }{\partial x} - \mathcal{L} \rightarrow \mathcal{L} dx = \sum_{j=1}^n p_i d y_j - \mathcal{H}dx$$ Does Noether's theorem just say that, because $\mathcal{L}$ is invariant we have, solving for $\mathcal{L}$, that $$\mathcal{L}(x,y_i,y_i')dx = \sum_{j=1}^n p_i d y_j - \mathcal{H}dx = \mathcal{L}(x^*,y_i^*,y_i'^*)dx^* = \sum_{i=1}^n p_i d y_i^* - \mathcal{H}dx^*?$$


A rigid rotating rod that breaks in two pieces

Suppose we have a rigid rod of lenght $L$ and homegenous mass density. One of its extreme points, say $P$, is fixed so that the rod can rotate around the axis passing in it. Initially the rod is held horizontally, then it is left and when it reaches the vertical position it breaks, without generating any impulsive force, in two equal pieces. I'm wondering about how to find the angular velocity of the two fragments immediately after the rod breaks. I tried but without succes, here is what I did. I suppose that energy is conserved, that is: $$ E_{rod}= E_{1} + E_{2} $$ where $1$ labels the upper fragment and $2$ the lower fragment of the rod. We have: $$ E_{rod}=\frac{1}{2}I_{rod}\omega_{rod}^{2} $$ $$ E_{1}= \frac{1}{2}I_{1}\omega_{1}^{2} $$ $$ E_{2}=\frac{1}{2}I_{2}\omega_{2}^{2} + \frac{1}{2}\frac{m}{2}V^{2}_{2} $$ where $V_{2}$ is the velocity of the center of mass of the lower fragment and the moment of inertia with respect to P are: $$ I_{rod}=\frac{mL^{2}}{3} $$ $$ I_{1}=\frac{mL^{2}}{24} $$ $$ I_{2}=\frac{7mL^{2}}{24} $$ **EDIT** According to V.Moretti's comments we know that energy and angular momentum are conserved, thus we have the system: $$ \left\\{\begin{array}{c} E_{rod}= E_{1} + E_{2} \\\ \vec{L}_{rod}=\vec{L}_{1} + \vec{L}_{2}\end{array}\right. $$ I think it is: $$ \vec{L}_{tot}=I_{rod}\vec{\omega}_{rod} $$ $$ \vec{L}_{1}=I_{1}\vec{\omega}_{1} $$ $$ \vec{L}_{2}=I_{2}\vec{\omega}_{2} + \frac{m}{2}\vec{r}\wedge\vec{V}_{2} $$ Moreover I think that $\vec{V}_{2}$ can be simply calculated considering the center of mass of the lower fragment as a point of the rod and then use the well-known relation between angular velocity and velocity: $$ \vec{V}_{2}=\vec{\omega}_{rod}\wedge \vec{r} $$ and thus, in a right-handed coordinate system with the z-axis exiting the screen, it is: $$ \vec{V}_{2}=(-\omega_{rod}\hat{z})\wedge(-\frac{3}{4}L\hat{y})=-\frac{3}{4}L\omega_{rod}\hat{x} $$ and: $$ \vec{r}\wedge\vec{V}_{2}=(-\frac{3}{4}L\hat{y})\wedge(-\frac{3}{4}L\omega_{rod}\hat{x})=-\frac{9}{16}L^{2}\omega_{rod}\hat{z} $$ Now there is nothing that indicates motion in the x-z and y-z planes, thus the conservation of the angular momentum should be a scalar equation, namely: $$ L_{tot}=I_{rod}\omega_{rod}=I_{1}\omega_{1} + I_{2}\omega_{2} + \frac{m}{2}\vec{r}\wedge\vec{V}_{2}= $$ $$ =I_{1}\omega_{1} + I_{2}\omega_{2} -\frac{9}{32}mL^{2}\omega_{rod}=L_{1} + L_{2} $$ Now I am trying to solve the system: $$ \left\\{\begin{array}{c} \frac{1}{2}I_{rod}\omega_{rod}^{2}= \frac{1}{2}I_{1}\omega_{1}^{2} + \frac{1}{2}I_{2}\omega_{2}^{2} + \frac{9mL^{2}\omega_{rod}^{2}}{64}\\\ I_{rod}\omega_{rod}=I_{1}\omega_{1} + I_{2}\omega_{2} -\frac{9}{32}mL^{2}\omega_{rod}\end{array}\right. $$ but, quite surprisingly, I am not having good results. What am I doing wrong?


Tensor product decomposition of SU(2)

I have a rather trivial question. I am looking for the decomposition of $1/2\otimes 1/2\otimes 1/2$. It should give, $0,1/2$ and $3/2$. I thought one must get as the overall dimension of this space 8, but counting, I just get 7. Does one have 2 singlets?


Where is the flaw in deriving Gauss's law in its differential form?

From the divergence theorem for any vector field E, $\displaystyle\oint E\cdot da=\int (\nabla\cdot E) ~d\tau$ and from Gauss's law $\displaystyle\oint E\cdot da=\frac{Q_{enclosed}}{\epsilon_0}=\int \frac{\rho}{\epsilon_0}~d\tau$ Hence, $\displaystyle\int\frac{\rho}{\epsilon_0}d\tau=\int (\nabla\cdot E)~d\tau$ Textbooks conclude from the last equation that $\displaystyle \nabla\cdot E=\frac{\rho}{\epsilon_0}$ My question is how can we conclude that the integrands are the same? Because I can think of the following counter example, assume $\displaystyle \int_{-a}^a f(x)~dx=\displaystyle \int_{-a}^a [f(x)+g(x)]~dx$ where $g(x)$ is an odd function. Obviously the 2 integrals are equal but we cannot conclude that $f(x)$ is equal to $f(x)+g(x)$ so where is the flaw?


The Double Slit Experiment and the changing of electron behaviour

As you will all know, when one tries to detect which slit an electron has gone through with close up observation, it changes from behaving like a wave and producing an interference pattern to behaving like a particle and producing two lines on the screen. Is there a full explanation as to why the electron changes its behaviour when observed?


Coordinate transformations of the metric tensor

Let's have metric (it describes the space-time of uniformly accelerating observer in Minkowski space-time) $$ ds^2 = v^2du^2 - dv^2. \qquad (.0) $$ I need to find expressions for $u = f(x, t), v = g(x, t)$, which leads to $$ ds^2 = dt^2 - dx^2, \quad u = f(x, t), \quad v = g(x, t). $$ How to get $f, g$? My attempt. I substituted $u = f(x, t), v = g(x, t)$ in $ds^2 = u^2dv^2 - du^2$ and then, by equating it to $dt^2 - dx^2$, got system of PDE: $$ g^2 (\partial_{x}f)(\partial_{t}f) - (\partial_{x}g) (\partial_{t}g) = 0, \quad g^{2}(\partial_{x}f)^{2} - (\partial_{x}g)^{2} = -1 , \quad g^{2}(\partial_{t}f)^{2} - (\partial_{t}g)^{2} = 1 . $$ It may be "simplified" to $$ (\partial_{x}g)^{2} - (\partial_{t}g)^{2} = 1 , \quad g^{2}\left((\partial_{t}f)^{2} - (\partial_{x}f)^{2}\right) = 1. \qquad (.1) $$ How to solve it? Or maybe there is more simple method to get an expressions for $f, g$? **Edit**. I got an equation $$ \partial_{tt}f - \partial_{xx}f = 0 . \qquad (.2) $$ But I can't get some conditions, which can help me to choose some partial solution. By the other words, I can't use $(.1)$ in the right way. So, what conditions may I use to solve $(.2)$? **Edit**. I got the solution by getting a solution for transition from Minkowski space-time to original space-time which metric is given by $(.0)$. I used method of separation of variables.


Fundamental invariants of the electromagnetic field

It is a standard exercise in relativistic electrodynamics to show that the electromagnetic field tensor $F_{\mu\nu}$, whose components equal the electric $E^i=cF^{i0}$ and magnetic $B_i=-\frac12\epsilon_{ijk}F^{jk}$ fields in the taken frame of reference, has two Lorentz invariant quantities, $$\frac12F^{\mu\nu}F_{\mu\nu}=\mathbf{B}^2-\mathbf{E}^2$$ and $$\frac14F_{\mu\nu}\ {}^\ast F^{\mu\nu}=\frac14\epsilon^{\mu\nu\alpha\beta }F_{\mu\nu}F_{\alpha\beta}=\mathbf{B}\cdot\mathbf{E}.$$ There is, however, a further Wikipedia article which states that these two quantities are _fundamental_ , in the sense that any other invariant of this tensor must be a function of these two. While I find this plausible, I have never seen a proof of this fact, and it is absent from e.g. Jackson. **Is there a simple proof of this fact?** I'm particularly interested in higher- order invariants, but I would also like answers to include a proof that these are the only two bilinears. To be more precise, I would like to see a proof that > Any function $I:F_{\mu\nu}\mapsto I(F)\in\mathbb{R}$ that takes > electromagnetic field tensors to real scalars and which is Lorentz invariant > (i.e. $I(\Lambda_{\mu}^\alpha \Lambda_{\nu}^\beta F_{\alpha\beta})= > I(F_{\mu\nu})$ for all Lorentz transformations) must be a function > $I(F)=I'(F^{\mu\nu}F_{\mu\nu},F_{\mu\nu}\ {}^\ast F^{\mu\nu})$ of the two > fundamental invariants described above. If there are multiple ways to arrive at this result, I would also appreciate comments on how they relate to each other.


Eigenvalues of the square of Pauli-Lubanski operator

Let's have Pauli-Lunanski 4-operator: $$ \hat {W}^{\nu} = \frac{1}{2}\varepsilon^{\nu \alpha \beta \gamma}\hat {J}_{\alpha \beta}\hat {P}_{\gamma}, $$ which easy transforms to $$ \hat {W}^{\nu} = \frac{1}{2}\varepsilon^{\nu \alpha \beta \gamma}\hat {S}_{\alpha \beta}\hat {P}_{\gamma}, $$ where $\hat {S}_{\alpha \beta }$ is the spin-tensor. For the square of this operator it's easy to get $$ \hat {W}_{\nu}\hat {W}^{\nu} = \hat {S}_{\alpha \beta}\hat {P}^{\beta}\hat {S}^{\alpha \gamma}\hat {P}_{\gamma} - \frac{1}{2}\hat {P}_{\delta}\hat {P}^{\delta}\hat {S}_{\rho \varepsilon}\hat {S}^{\rho \varepsilon}. \qquad (.1) $$ In the case of the rest frame $\hat {P}_{\gamma}\psi = 0$, so $$ \hat {W}_{\nu}\hat {W}^{\nu} \psi = -\frac{1}{2}\hat {P}_{\delta}\hat {P}^{\delta}\hat {S}_{\rho \varepsilon}\hat {S}^{\rho \varepsilon}\psi = -\frac{m^{2}}{2}\hat {S}_{\rho \varepsilon}\hat {S}^{\rho \varepsilon}\psi . $$ In the special case of 4-tensor I know, that $$ \hat {S}_{\alpha \beta}\hat {S}^{\alpha \beta} = 2(\hat {S}_{1}^{2} - \hat {S}_{2}^{2}), $$ where $\hat {S}_{1}$ is the spin-vector 3-operator. But what is $\hat {S}_{2}$? And what it do with eigenvalue? Without it $\hat {W}_{\nu}\hat {W}^{\nu} \psi = -m^{2}s(s + 1) \psi $, as it must be. **Addition**. Maybe, I understand my mistake. I ignored the first term in $(.1)$. At the rest frame in a case of diagonal operators of $\hat {P}_{i}$ it has structure $\hat {P}_{i} = (\hat {P}_{0}, 0, 0, 0) = (m, 0, 0, 0)$, so $$ \hat {W}_{\nu}\hat {W}^{\nu}\psi_{\mathbf p = 0} = \left(\hat {S}_{\alpha \beta}\hat {P}^{\beta}\hat {S}^{\alpha \gamma}\hat {P}_{\gamma} - \frac{1}{2}\hat {P}_{\delta}\hat {P}^{\delta}\hat {S}_{\rho \varepsilon}\hat {S}^{\rho \varepsilon}\right)\psi = \left(\hat {S}_{\alpha 0}\hat {P}^{0}\hat {S}^{\alpha 0}\hat {P}_{0} - \frac{1}{2}\hat {P}_{0}\hat {P}^{0}\hat {S}_{\rho \varepsilon}\hat {S}^{\rho \varepsilon}\right) \psi_{\mathbf p = 0} = $$ $$ =m^{2}\left( - \hat {\mathbf S}_{2}^{2} - \frac{1}{2}2 \left( \hat {\mathbf S}_{1}^{2} - \hat {\mathbf S}_{2}^{2}\right)\right) \psi_{\mathbf p = 0} = -m^{2}s(s + 1)\psi_{\mathbf p = 0 }. $$ But, anyway, what is $\hat {\mathbf S}_{2}$?


Fermion vs. Bosons and particle vs. wave: is there a link?

I'm puzzled since several years on this basic aspect of quantum mechanics. Quantum theory is supposed to describe particle-wave symmetry of our world. It also describes our universe in term of bosons and fermions. Now the experimental facts:   * It seems rather easy to observe wave particle of bosons (think about photon of course) whereas they particle behaviour is pretty complicated to observe. You need stringent experimental condition to observe the particle behaviour of photons for instance.   * It seems rather easy to observe particle behaviour of fermions (think about electron of course) whereas they wave behaviour is pretty complicated to observe. You need stringent experimental condition to observe the wave behaviour of electrons for instance. I was just wondering if some people figure out the wave-particle behaviour in terms of fermion vs. boson classification, or if the above experimental facts are just pure coincidence. I admit it would require a less stringent dichotomy boson/fermion, perhaps like in supersymmetry theories, that I do not know enough. Any comment are welcome.


Connection between $\Delta x \Delta p \geq \frac{\hbar}{2}$ and $\Delta E \Delta t \geq \frac{\hbar}{2}$

Is there a way to derive second equation from the first one? I mean is there a connection between those two uncertainty relations? \begin{align} \Delta x \Delta p &\geq \frac{\hbar}{2}\\\ \Delta E \Delta t &\geq \frac{\hbar}{2} \end{align}


How did the inflaton field "add" energy to the universe?

How did inflation add energy to the universe? What mechanism did this occur by? In other words, where did that energy come from? Was it due to the quantum fluctuation (or that scalar field rolling down a potential energy hill) to a lower, more stable energy state? This energy then manifested as exponential expansion, and then switched from expansion into heating (by filling up the universe with hot quark-gluon plasma)? Do we know the mechanism for this expansion->reheating switch?


Kronecker delta in inertial tensor

I feel confused in (11.9) how does the book prove the following identity: $$\sum\limits_{i} w_{i}x_{\alpha,i} \sum\limits_{j} w_{j}x_{\alpha,j} = \sum\limits_{i,j}w_{i}w_{j}x_{\alpha,i}x_{\alpha,j}$$ ![enter image description here](http://i.stack.imgur.com/w8u1I.jpg) I tried to derive it myself: $$\sum\limits_{i} w_{i}x_{\alpha,i} \sum\limits_{j} w_{j}x_{\alpha,j} = \sum\limits_{i}\sum\limits_{k}w_{k}\delta_{ik}x_{\alpha,i} \sum\limits_{j}\sum\limits_{m}w_{m}\delta_{jm}x_{\alpha,j} $$ Exactly from here, I got stuck, but I'm still not very much convincing about the result that the book got, becasuse it does not contain kroncker delta function for the part I tried to derive. Can anyone give me a mathematical proof?


Does the Earth rotate vertically too?

We know that the earth has seen several Ice Ages. And we know the Earth's magnetic poles seem to flip once in a while. Could all of this be explained by a once every 250 Million Years or so the Earth actually rotates head over heels, so to speak?


How can the reduction postulate be removed with the other postulates of QM still leading to correct predictions?

In the axiomatic presentation of QM, I've seen it stated many times that the reduction postulate is not needed and/or incorrect, and could be gotten rid of. However, without the reduction postulate, wouldn't tests of QM simply yield wrong results? If I measure a system that is in some linear superposition $\psi=\sum c_i |e_i\rangle$ of eigenstates of my observable $O$, the measurement postulate says I will get one of the eigenvalues as a result, with probability $|c_i|^2$. But now, since I no longer have the reduction postulate, I will calculate that as a result of the interaction with my measurement apparatus $\psi$ evolved linearly and unitarily into $\psi'$, and therefore still consists of (possibly a different) linear superposition $\sum c_i' |e_i\rangle$ of eigenstates of $O$. Therefore, again applying the measurement postulate, I calculate that I will now get $|e_i\rangle$'s eigenvalue with probability $|c_i'|^2$. After a sequence of $n$ such measurements, close together in time, the probability that I will get some particular $|e_j\rangle$'s eigenvalue in all the $n$ measurements is $(|c_j|^2 * |c_j'|^2 * ... * |c_j^{(n)}|^2) << 1$. However, we know from experience that making $n$ close in time measurements of the same observable will yield the same value with a probability of $1$ (or very close to $1$). So I don't see how the reduction postulate can just be dropped without either QM giving incorrect predictions, or without modifying some other postulates somehow?


Higgs Boson and its relation to the vacuum

The Higgs boson and the electroweak theory used symmetry breaking from condensed matter physics as its inspiration. The BCS theory of superconductivity is one such condensed matter symmetry breaking theory. I believe this theory uses some kind of model for the underlying atomic structure of the metal. In a similar manner the vacuum works in the same way with particles popping in and out with charges all the time. Is there a relation between the vacuum and the atomic structure of condensed matter. If so, could one define the vacuum with particular properties to match the Higgs potential? I know the Higgs potential is a vaccuum, but can properties be assigned to space itself and the fluctuating charged particles that pop in and out that are analogous to the charged atoms in a condensed matter metal and produce the Higgs fields this way? Maybe the vacuum is discrete like a metal lattice instead of continuous.


Quantum Hall effect for dummies

In the past few days I've become increasingly intrigued by the QHE, mainly thanks to very interesting questions and answers that have appeared here. Unfortunately, I am as of yet very confused by all the (seemingly disparate) stuff I learned. First, here are some random points that I've been able to gather   1. I(nteger)QHE occurs due to the presence of Landau levels   2. IQHE is an embodiment of topological order and the states are characterized by the Chern number that tells us about topologically inequivalent Hamiltonians defined on the Brillouin zone   3. IQHE requires negligible electron-electron interations and so is dependent on the presence of impurities that shield from Coulomb force   4. F(ractional)QHE occurs because of formation of anyons. In this case Coulomb interaction can't be neglected but it turns out an effective non-interacting description emerges with particles obeying parastatistics and having fractional charge   5. FQHE has again something to do with topology, TQFT, Chern-Simons theory, braiding groups and lots of other stuff   6. FQHE has something to do with hierarchy states So, here are the questions >   * Most importantly, do these points make sense? Please correct any > mistakes I made and/or fill in other important observations >   * How do explanations 1. and 2. of IQHE come together? Landau quantization > only talks about electron states while topological picture doesn't mention > them at all (they should be replaced by global topological states that are > stable w.r.t. perturbations) >   * How do explanations 4., 5. and 6. relate together >   * Is there any accessible introductory literature into these matters? >   * Do IQHE and FQHE have anything (besides last three letters) in common so > that e.g. IQHE can be treated as a special case? My understanding (based on > 3.) is that this is not the case but several points hint into opposite > direction. That's also why I ask about both QHE in a single question >


Is Newtonian physics invariant under fast-forward?

I am aware of how the laws of physics are invariant with a time reversal transformation, so that (ignoring entropy) a film of billiard balls running backwards cannot be differentiated from one running forwards. What about a film of billiard balls running in fast-forward motion? Ignoring relativistic effects, I think that there would be no way to tell the playback speed. OK, then imagine that one of the billiards were made out of delicate, thin glass, and that in the regular play-back film the motions are slow enough that it does not break. Playing the film in fast-forward motion, the delicate ball still would not break despite being hit by very fast-moving ones. Does this mean that Newtonian physics is not invariant under fast-forwarding? Is there a way of expressing this mathematically (analogous to saying t goes to -t for time reversal)? And finally, since the glass ball does not break, is this violation of intuition an illusion having to do with entropy, like with running a film in reverse, or is there some kind of modification that needs be made to the forces of the molecular bonds under fast-forward motion to prevent the glass from shattering?


From where comes the raindrop

First off all, I've no knowledge of how all this works. I've only got some common sense of "how things work"... But I've been wondering this for quite a long time now, so maybe you guys can help me. The question is rather simple: Where does the raindrop come from? And with that I mean: The raindrop that just hit the ground, where did that come from? From the clouds that are right above me, in front of me, or back of me? So let's say that we've got these 3 possibilities, which one is the most likely one? The first one, where the rain is traveling with the same speed horizontal as the cloud it came from: (so it came from the cloud directly above me) ![Raindrop 1](http://i.stack.imgur.com/FLkOU.jpg) The second one, where the rain is traveling faster than the clouds (due to wind?). So that the rain came from the cloud that yet has to pass me. ![Raindrop 2](http://i.stack.imgur.com/oHvYk.jpg) The third one, (And i do think that this is the right one) is that the rain is falling slower (horizontally seen) than the cloud (because the cloud is lighter). So that the rain that hits the ground came from the cloud that already past me. ![Raindrop 3](http://i.stack.imgur.com/0U8uE.jpg) An another possibility is that the rain is going left and right, (keeping it 2D for simplicity) because of the different wind regions (in the high clouds, the wind goes from left to right, but lower it goes from right to left). Is there even an "right" answer for this? Or are there too many variables that could change everything? And could you please explain it as simple as possible? I've got zero knowledge in this area... Thank you in advantage! -Also, i don't know if this is the right stackexchange site to ask...


How long does it take to scan a typical scanning electron microscope image?

I suppose the answer depends on resolution and imaging area, but can you provide some ball park measures of imaging times with an SEM?


Is $w=mg$ the right way to calculate mass?

How can you find the mass of an object by using a weight and a gravitational acceleration from another planet then earth? I was given $5.8\text{ N}$ and an acceleration of $4.0\ \mathrm{m/s^2}$. The answer I keep getting is $1.45\text{ kg}$, but the real answer is supposed to be $.58\text{ kg}$. I'm using the formula $w=mg$. Am I doing something wrong?


Factor of two differences for free field Green's functions in conformal field theory

I have a question about the expressions for free field Green's functions in conformal field theory. It comes from three origins 1) In Polchinski's string theory volume I p36, it is given > $$ \frac{1}{\pi \alpha'} \partial_z \partial_{\bar{z}} X^{\mu}(z,\bar{z}) > X^{\nu} (z',\bar{z}') = -\eta^{\mu \nu} \delta^2 (z-\bar{z}', > \bar{z}-\bar{z}') \tag{2.1.20} $$ 2) In David Tong's string theory lecture note, p 186 (in the number at the bottom) or p 193 (counting by acrobat reader), > $$\langle Y^a(z,\bar{z}) Y^b(\omega,\bar{\omega}) \rangle = G^{ab} > (z,\bar{z}; \omega,\bar{\omega} )$$ $$ \partial \bar{\partial } G^{ab} > (z,\bar{z}) = - 2 \pi \delta^{ab} \delta (z,\bar{z}) \tag{7.33} $$ here $\alpha'$ has been scaled by $X^a = \bar{x}^a + \sqrt{\alpha'} Y^a$ in p 185/192. Why there is a factor of 2 difference in the RHS of (2.1.20) (Polchinski) and (7.33) (Tong)? In page 77, it is given > $$ \langle \partial^2 X(\sigma) X(\sigma') \rangle = - 2\pi \alpha' > \delta(\sigma-\sigma') \tag{4.20}$$ I am not sure what is the definition of $\partial^2$. Since $\partial \bar{\partial}=\frac{1}{4} (\partial_1^2 + \partial_2^2) $ and $\delta^2(z,\bar{z})=\frac{1}{2} \delta(\sigma^1) \delta(\sigma^2)$ (Polchinski p33),the factor at RHS of Eq. (4.20) in David Tong's lecture note seems to agree with Polchinski (2.1.20), but not Tong's (7.33) In addition, David Tong's lecture is related to > We'll be fairly explicit here, but if you want to see more details then the > best place to look is the original paper by Abouelsaood, Callan, Nappi and > Yost, \Open Strings in Background Gauge Fields", Nucl. Phys. B280 (1987) 599 3) In the original paper, it is given > $$\frac{1}{2\pi\alpha'} \square G(z,z')= - \delta(z-z') \tag{2.7}$$ where $\square=\partial_{\tau}^2 + \partial_{\sigma}^2$. Again, a factor of two difference between (2.7) and (2.1.20) of Polchinski. In Eq. (2.1) of that paper states > $$ S= \frac{1}{2\pi \alpha'} \left[ \frac{1}{2} \int_{M^2} d^2 z \partial^a > X_{\mu} \partial_a X^{\mu} + i \int_{\partial M} d \tau A_{\mu} > \partial_{\tau} X^{\nu} \right] $$ In Polchinski's string theory volume I, p32, it is given > $$ S = \frac{1}{4\pi \alpha'} \int d^2 \sigma \left( \partial_1 X^{\mu} > \partial_1 X_{\mu} + \partial_2 X^{\mu} \partial_2 X_{\mu} \right) > \tag{2.1.1}$$ Since $d^2 z = 2 d \sigma^1 d \sigma^2 \tag{2.1.7}$ , is that the origin of factor of 2 difference? Because combininig Eqs.(2.1.1) and (2.1.7) leads to $$ S = \frac{1}{2\pi \alpha'} \int d^2 z \left( \partial_1 X^{\mu} \partial_1 X_{\mu} + \partial_2 X^{\mu} \partial_2 X_{\mu} \right) $$


Space translation of operators, states, and particle densities

In Sidney Coleman's Lectures he talked about space translations such that $$\tag{1} e^{ia P}\rho(x) e^{-ia P} ~=~ \rho(x-a),$$ but when I expanded the exponentials and used the commutation relation of $P=-i\frac{d}{dx}$ and $x$, I got $$\tag{2} e^{ia P}\rho(x) e^{-ia P} ~=~ e^{ia [P,~\cdot~]}\rho(x) ~=~e^{a\frac{d}{dx}}\rho(x)~=~\rho(x+a)$$ with a plus instead of $\rho(x-a)$ with a minus as Coleman (1). This was on Coleman's lectures on QFT (1975-76), Lecture 3, about 12:20 minutes into the lecture. Does anyone see where I may have gone wrong? References:   1. Coleman's lectures on QFT (1975-76), Lecture 3, about 12:20 minutes into the lecture.   2. S. Coleman, _Notes from Sidney Coleman's Physics 253a,_ arXiv:1110.5013, p. 23.


Why does the amount of dark matter increase the further away from the galaxy's center?

Why isn't the dark matter just randomly distributed? How does it know where to go?


Derivative chain rule in a triangle, confusing but interesting problem

I asked the question in math.stackexchange. But I think it is better to ask here again. I am new to these sites. Please forgive me if it is not polite. http://math.stackexchange.com/q/921001 You can reply any poster. I paste it again here. ![variables in triangle](http://i.stack.imgur.com/RmuHH.png) Refer to the above figure. Assuming the length of the 3 edges of triangle are $r_0,z_0,\xi_0$. And we have $\xi=\sqrt{r^2+z^2}$ (Eqn.1)and $\xi_0=\sqrt{r_0^2+z_0^2}$. The normal vector on the hypotenuse is $\hat{n}$, and the components are $n_r=\frac{z_0}{\xi_0}$ and $n_z=-\frac{r_0}{\xi_0}$ Meanwile$\frac{\xi}{r}=\frac{\xi_0}{r_0}=\frac{1}{\cos{\theta}}$(Eqn.2) and $\frac{\xi}{z}=\frac{\xi_0}{z_0}=\frac{1}{\sin{\theta}}$(Eqn.3). Now I have a funtion $T=T(r,z)$ on the hypotenuse, and I need to calculate $f=\frac{\partial{T}}{\partial{z}}n_r-\frac{\partial{T}}{\partial{r}}n_z=\frac{\partial{T}}{\partial{\xi}}\frac{\partial{\xi}}{\partial{z}}n_r-\frac{\partial{T}}{\partial{\xi}}\frac{\partial{\xi}}{\partial{r}}n_z$ (Eqn.4) If using Eqn.1, then $\frac{\partial{\xi}}{\partial{z}}=\frac{z}{\sqrt{r^2+z^2}}$ (Eqn.1*a)and $\frac{\partial{\xi}}{\partial{r}}=\frac{r}{\sqrt{r^2+z^2}}$(Eqn.1*b) and $f=\frac{\partial{T}}{\partial{\xi}}(\frac{z_0z}{\xi_0\sqrt{r^2+z^2}}+\frac{r_0r}{\xi_0\sqrt{r^2+z^2}})$. (Eqn.5) If using Eqn.2 and Eqn.3, $\frac{\partial{\xi}}{\partial{z}}=\frac{\xi_0}{z_0}$ (Eqn.6)and$\frac{\partial{\xi}}{\partial{r}}=\frac{\xi_0}{r_0}$(Eqn.7) and $f=\frac{\partial{T}}{\partial{\xi}}(\frac{\xi_0z_0}{z_0\xi_0}+\frac{\xi_0r_0}{r_0\xi_0})=2\frac{\partial{T}}{\partial{\xi}}$(Eqn.8) It looks to me that Eqn.6 and Eqn.7 are right. but Eqn.8 is wrong. because when $\theta$ goes to 90degree,it should be $f=\frac{\partial{T}}{\partial{z}}=\frac{\partial{T}}{\partial{\xi}}$ So what is the right expression for $f$, and why? ($f$ is azimuthal component of the $\nabla\times({T\hat{n}})$) as somebody asked.


Stresses in asteroid during close flyby

The acceleration of an asteroid (such as 2012DA14) as it approaches earth is proportional to the reciprocal of distance $r$ from earth center, squared. the derivative of the acceleration, or jerk, is then proportional to the cube of the reciprocal of distance $r$, and the speed $v$. Thus the jerk is rapidly increasing as the asteroid approaches. Will the asteroid experience stresses from the jerk that could possibly cause it to break up?


Why does a large train cause the ground to shake?

I work in a 4 story building that is approx. 150 feet away from a set of train tracks. When a large (40+ car) freight train goes by, the shaking in the building is perceptible. As I've watched the train go by, there does not appear to be any side to side movement and the speed is constant. What exactly is causing the vibrations in the ground? Is it simply the train's traversal over each segment of track? Surely the train itself is not "bouncing" along the track with enough energy transfer to shake the ground, right?


Can stimulated emitted photons be absorbed?

Typically a stimulated photon will be one of a pair with its stimulating photon. If the leading photon is absorbed by a particle in the ground state, will it then be re-emited by the stimulated emission caused by the second photon? If it causes a stimulated emission from an excited particle, will there then be a train of three photons, or will the new pair leave the excited particle excited? I am interested in the case where there is not a population inversion, but presumably the same rules would apply to lasers.


Is the fuel burn for a satellite launch affected by the position of the moon relative to the launch site?

The gross mass of a satellite rocket is tiny compared to that of Earth, and Luna. Between them, however, the two bodies set up tides in bodies of water which itself is again considerable mass. At the point of launch a satellite rocket might not be affected by the Moon's gravitation attraction. As it goes higher/ more distant from earth/closer to escape/ target orbit do it's fuel burn statistics change depending upon the position of the moon vis-a-vis launch location/payload release location?


Dynamics of counter-rotating flywheels

I've wondered about this for ages. If we create a pair of flywheels that rotate in the opposite direction with the same angular momentum, but are co- located and have the same mass and inertial moment (one can imagine various ways to accomplish this, at least approximately) -- it is clear to me that there cannot be any precession forces, but, if we try to rotate the entire assembly around an axis perpendicular to the axis of flywheel rotation, will the force needed to produce this secondary rotation be the same as if the flywheels were stationary, or will it require a proportionally greater force to rotate in this way, as it does with a single flywheel?


Reflectance of Titanium as Function of Thin Film Thickness

As far as I know, transmittance equals $e^{-\alpha x}$, where $\alpha$ is absorption coefficient and $x$ is thin film thickness($100-300\,nm$). My team and I have engineered a way to find absorption. Transmittance, T= (Output intensity)/(initial intensity). And, absorption A=(initial intensity-output intensity)/(initial intensity). After simplifying the equation, one should get $$A= 1-T$$ The problem I am facing is I do not know any equation, which will give me reflectance of Ti as a function of thin film thickness at $808\,nm$ wavelength. If someone gives me an equation of transmittance, please include the effects of absorption so that I can calculate the reflectance afterwards.


Harmonic Oscillator Expectation Value

In Calculating the expectation value of the quantum harmonic oscillator, I've come across a problem for finding $\left \langle x \right \rangle$ for the coherent state $\left| \alpha \right \rangle$ $$x = \sqrt{\frac{\hbar}{2m\omega}}(a_+ + a_- )$$ $$\sqrt{\frac{\hbar}{2m\omega}}\left \langle \alpha \right| (a_+ + a_- ) \left| \alpha \right \rangle =\sqrt{\frac{\hbar}{2m\omega}} \alpha ^* (a_+ \alpha + a_- \alpha )$$ This doesn't agree with the result my book has given, which is $\sqrt{\frac{\hbar}{2m\omega}} (\alpha + \alpha ^* )$. I'm not used to working in bra-ket notation, should it's extremely likely that I'm missing something small here. The general principle is that $\left \langle \beta \right| A \left| \beta \right \rangle = \beta ^* (A\beta)$, right?


Is the Schwarzschild black hole unphysical?

To obtain the Schwarzschild metric from Einstein equations of general relativity, we suppose that the energy density is a distribution : $$ \rho (\vec{r}) = M \delta(\vec{r})$$ The Schwarzschild radius, corresponding to the horizon, is $R_s = 2 M$, in units $G=c=1$. A physical interpretation of this, is that you cannot put more energy than $M$ in a sphere of radius $R_s$. The way I understand this, is because the total energy of the black hole, sum of its mass energy (positive) and its auto-gravitational energy (negative), should remain non-negative. However, for the Schwarzschild black hole, for each radius $r$ between $0$ and $R_s$, the total mass in the sphere of radius r, is : $$ m(r) = \int_0^r \rho(\vec{u}) d^3u =\int_0^r \delta(\vec{u}) d^3u = M$$ So it seems to violate the principles cited above, and so the Schwarzschild black hole should be unphysical. If it makes sense, a physical black hole (static, with spherical symmetry) should have a mass/energy density $\rho (r)$, such that, whatever the value of $r$ is, we have the inequality (in units $G=c=1$): $$m(r) = \int_0^r \rho (u) 4\pi u^2du \le \frac{r}{2}$$


Is there a symmetry associated to the conservation of information?

Conservation of information seems to be a deep physical principle. For instance, Unitarity is a key concept in Quantum Mechanics and Quantum Field Theory. We may wonder if there is an underlying symmetry, in some space, which may explain this conservation of information.


two electron spins line up in opposite direction and the Ising model

What cofuses me with the Ising Model is the claim that these nearby little magnets want to face the same direction. facing the same direction, they have lower energy. This seems to conflict with the fact that north poles of magnets repel each other. So little magnets act different from big everyday magnets? But then, it's said that if you bring two electrons near enough that they can feel each other's magnetic field, then if you wait enough, a photon will be emitted and they will get into an entangled state such that their spins are facing opposite directions. This seems to conflict with the claim that little magnets facing the same direction have lower energy. I am confused. It just looks like there are two kinds of little magnets and one kind's north poles repel each other, and another kind's north poles attract each other. What is really going on?


Why does a "speed of sound" exist?

I've recently read that wind cannot be faster than the speed of sound (german source). But why is there a speed of sound? I understand (well, mostly accept to be honest) that the speed of light in vacuum is a maximal speed for all matter. And I understand that you need more energy the more you accelerate particles. But why can't you make wind faster?


Ratio of radiation sources in the uranium cycle

Sorry, I know y'all are probably getting a lot of questions re: Fukushima, but I had a very specific one and no-one has been able to answer it. I am specialised in medical radiation, and have been explaining the risks to workers in this accident, but I need to know what are the rough proportions of alpha, beta (+/-) and gamma radiation flying around. If I have this right, most of the decays in the uranium cycle are alpha or beta+ and beta-. Some isomeric transitions (and obviously the beta+ decays) will release gammas. Obviously there will be a small amount of xrays too, from Bremsstrahlungs and such. But what are the percentages of this in general uranium fuel material (I guess 'spent' fuel is the real problem here, not fuel undergoing chain reaction)? Is 50% of the radiation alphas? Is 80% gammas? It makes a huge difference from a biophysics perspective as alphas are easy to protect against (for the workers) but gammas are unshieldable. Beta- decays are in between, pretty easy to deal with. Anyone know the uranium fuel cycle well enough to give a rough estimate here? EDIT: Added some thoughts in my response to Bars, who helped a lot. Any other thoughts that follow on from those insights?


Depends river stream on river level?

If we have referral particle in the middle of the river, will it's velocity increase with higher river level?


Effect of introducing magnetic charge on use of vector potential

It is well known that Maxwell equations can be made symmetric w.r.t. $E$ and $B$ by introducing non-zero magnetic charge density/flux. In this case we have $div B = \rho_m$, where $\rho_m$ is a magnetic charge density. But this means that $B$ cannot be expressed as the curl of vector potential anymore. Does it mean that it is impossible to develop electrodynamics of non-zero electric and magnetic charges in terms of vector and scalar potentials? What happens to $U(1)$ gauge invariance then? P.S.: I know that for the point-like magnetic charge vector potential can still be introduced, but not in the entire space. My question is related to non point-like magnetic charge densities.


how does a lizard walk on vertical walls?

One day I was observing a lizard walking on walls as easily as we walk on horizontal surfaces. I got curious to know the physics behind this but I couldn't get to any useful conclusion. I tried to Google it but all the hypothesis like lizards create vacuum while they walk or they secrete some sort of chemical (and few others) were rejected due to some or the other reason, so I thought maybe here I could get some help on the subject.


What's the most efficient strategy to shake a bottle to create a global flow?

Consider a bottle of orange juice with pieces of the fruit gathered at the bottom of the bottle. You need to shake the bottle to evenly distribute the pieces in the bottle. The first part of the process seems straight forward, just giving the bottle a few sharp shakes to create turbulence and destroy the conglomerate of fruity pieces. However, what's the most energy-efficient strategy to create a global flow to distribute the stuff? Since at the start of the process, the interesting parts are gathered at the bottom of the bottle, a strong axial component would be neccesary, for energy reasons a laminar flow would be interesting (assuming the dispersal has allready happened). Edit to add: Efficient means the least energy input from me. If your mixing method involves throwing the bottle off a cliff, the energy for cyrrying it up there counts.   Where I live, juice comes in round bottles or in boxy containers.


Rate of evaporation of water

In our everyday life, does the temperature or humidity have a greater effect on how fast things dry? Are there any theories/models that explain this quantitatively?


Exchange operator in terms of rotation operator

I have studied about exchange operators and rotation operators and I know that an exchange between 2 particles in a combined state is the same as rotating each particle 180 degrees (according to http://en.wikipedia.org/wiki/Exchange_symmetry). Could someone explain the mathematical equation behind this? I have tried to search but to no avail.


Dark-energy sources and respective implications

Firstly, I want this to be a very layman/simple level. According to Sean Carroll, we have certain possibilities for where dark-energy comes from: http://www.youtube.com/watch?v=v6o2bUPdxV0 Including   * A modification of general relativity   * Vacuum energy   * A modification of vacuum energy And he goes on to say that **if vacuum energy is the right answer the universe will continue to expand**. What would be the implications if the right answer is a modification of general relativity? Are there any other theories that explain dark energy that make predictions about the future like this?


Why doesn't matter pass through other matter if atoms are 99.999% empty space?

The ghostly passage of one body through another is obviously out of the question if the continuum assumption were valid, but we know that at the micro, nano, pico levels (and beyond) this is not even remotely the case. My understanding is that the volume of the average atom actually occupied by matter is a vanishingly small fraction of the atom's volume as a whole. If this is the case, why can't matter simply pass through other matter? Are the atom's electrons so nearly omnipresent that they can simultaneously prevent collisions/intersections from all possible directions?


How many state configurations are possible for $N$ particles in completely different states? I cannot remember if the total number of state configurations for $N$ particles in completely different states is $n^n$ or $n!$. Also, if $n$ particles are in completely different states, does this represent the maximum possible entropy of this system?


Was Titanic's captain's decision correct?

I just saw a documentary on Titanic. Someone in that documentary told Captain Smith > "Shall we close the water tight doors, sir?" and the Captain replied > "No, let it be open. Let the water be distributed uniformly throughout ship, > so that we could lower the life boats." Note: The quotes are not exact but very near to exact. * * * ## My Question If and if, the watertight doors were shut, was there a chance of Titanic to escape from its tragic fate? And what exactly is a water tight container? I presume that this is related to physics :)


Carrying water on person, or on the frame when bicycling

So, the question is as follows: What is the difference in work exerted by the rider in the two following scenarios? a) Rider + bike. Water carried in a holder on the frame   b) Rider + bike. Water carried in a backpack (for simplicity, assumption that weight of holder = weight of backpack and backpack is perfectly aerodynamic, not increasing drag) I would assume that in case a), as the weight is closer to the bike's centre of mass, it would require the least amount of extra work, compared to b), where the weight is more distant from the centre of mass. Thanks in advance.


What's the difference between exchange spin wave and magnetostatic spin wave?

So far I've heard of three kinds of spin waves   * Magnetostatic spin waves (MSW)   * Dipole-exchange spin waves (DESW)   * Exchange spin waves (ESW) What's the difference?


Electron revolving in an atom

When an electron revolves around the nucleus in P or d-orbitals why does not it collide with the nucleus.   I mean to say that the shape of the orbital narrows near the nucleus , so shouldn't it collide with the nucleus?


Twisted supermultiplets

What is a twisted supermultiplet, in a generic supersymmetric theory? Which ordinary fields belong to one of such twisted supermultiplets? I am confident with the idea of a supermultiplet or a superfield, in $\mathcal{N}=1,2,4$. However, I don't know what the term _twisted_ stands for. Also, why is it useful to consider such things?


Complementarity between the laws of physics?

Is this following proposal plausible, worth considering, or dismissable as lunatic fringe science? What if the universe isn't really what we think it is but some universal quantum computer where we read off one output at a time? This universal quantum computer is initialized to a superposition of program instructions -- which are identified with the laws of physics -- and after reading off one output after another, one finds there are in general two different kinds of questions we can ask; general relativity type questions, where we get GR type answers, and quantum mechanical type questions, where we get QM type answers. But we have to choose between one or the other, and whichever we choose, we get the answer appropriate for that domain. The place where both are united is the Planck scale, but unfortunately, the Planck scale as way too inaccessible to us, and is hence unobservable. Can this explain the confusion about UV/IR mixings? String compactifications are computer programs determining the parameters of the effective field theory describing the domain of answerable experiments. A complementarity between the laws of physics themselves. GR/QM complementarity. Is there any flaw with my proposal?


Is it a total or an explicite time derivative in the Schrödinger equation?

I am always dubious when I need write Schrödinger equation: do I write $\partial / \partial t$ or $d/dt$ ? I suppose it depends on the space in which it is considered. How?


Perturbed stress-energy tensor in a cosmological context?

In the theory of cosmological pertubations, we can write the metric of a null- curvature expanding Universe as : $ds^2 = -c^2\left(1+2\frac{\psi}{c^2}\right)dt^2 + a^2 \left(1-2\frac{\phi}{c^2}\right)\left(dx^2+dy^2+dz^2\right)$ From that metric we can compute the perturbed Einstein tensor $G_{\mu\nu}$. Now, I search what is the form of the stress energy tensor associated to this in order to write the Einstein equation : $G_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}$. So, starting from the fact that we consider a perturbed fluid : $\rho=\bar{\rho}+\delta\rho$ $P=\bar{P}+\delta P$ what is the $T_{\mu\nu}$ associated to that fluid ? * * * Remark : Here are some indications (see 4.4.1) : http://www.damtp.cam.ac.uk/user/db275/Cosmology/Chapter4.pdf but I do not understand very well why the author write the $T_{\nu}^{\mu}$ instead of the $T_{\mu\nu}$ (so if someone knows why, please explain it to me).


Switching from an accelerated frame of reference to a locally inertial reference system

_Using the equivalence principle, show that the interval for an accelerated observer ($\textbf{g}$ uniform and constant) has the form_ $$ ds^2|_{\text{first order in }\textbf{g}}=(1+2\varphi/c^2)c^2dt^2-d\textbf{x}^2+o(1/c^2), $$ where $\varphi=-\textbf{x}\cdot\textbf{g}$.


Why does the wave description say that probability oscillates, while the phase interpretation says constant amplitude?

The wave description of a particle illustrates an oscillating probability of the particle being found in any point in space. When a particle travels, it carries along with it a phase that oscillates in the complex plane. When the particle travels, the amplitude never changes although the phase does, which doesn't matter. The only property that matters is the amplitude. Why does the wave description say that probability oscillates and the phase interpretation says constant amplitude? What am I missing?


Kinetic energy of a cylinder

It is a long cylinder (you can approx $R=0$), and it has a fixed point in one os its ending points, it's rotating on a plane and I have to calculate the kinetic energy from reference systems situated in the center of mass, and both extremes. If I calculate it in the fixed point, it has only rotational energy: $$\frac{1}{6}ML^2\dot{\varphi}^2$$ If I choose the center of mass, the moment of inertia changes but I have to add to that the energy of that moving point $1/2Mv_{CM}^2$, and the result is the same. My problem is that when I choose the moving extreme, the result changes: The moment of inertia is the same as with the fixed point: $1/3ML^2$, and the speed at that point is $1/2ML^2\dot\varphi ^2$, so adding them gives you something different from the $1/6$ result, and they should be the same, ¿What am I doing wrong?


Higgs maths video Brian Greene

What do you think of the video below? ==> http://www.youtube.com/watch?v=KWj00MCqSxs Is the explanation in this video largely correct, or missing something? Is the mathematical description in the video of the higgs mechanism right? He said that phi is re-written as a m + h. What does it mean? Is the m in this video the vacuum expectation value? Thanks for all answers!


How close are humans to creating gold from other elements?

Gold is made when dying stars collide. How close are humans to doing that? Or is it even possible to do that unless gravity is absurdly high?


Where does the extra equation come from to determine the forces from an object on a table?

I have a question about basic statics but somehow I cannot manage to find the answer on my own (btw, this is not a homework. It's been so many years since school for me...). The problem is very simple: we have an object with weight $D$ at a given location on table wit with four legs ($F_1$ to $F_4$). What is the force applied on each leg? (for simplicity, I'm just using the same labels $F$ and $D$ for both the location and the force) ![object on a table](http://i.stack.imgur.com/ZXhRB.png) $W$, $H$, $x$, $y$ and $D$ are given. To find the forces on each leg, as far as I remember, I have to consider two general equations: $\sum F=0$ and $\sum M=0$. So I have: $$ F_1 + F_2 + F_3 + F_4 - D = 0 $$ Also, considering the moments round the point $F_1$: $$ W(F_2+F_3) - xD = 0 $$ $$ H(F_3+F_4) - yD = 0 $$ But this just give me 3 equations! I missing one more equation and cannot figure it out.


Harmonics in an open organ pipe

How to solve the following question? > _An open organ pipe has two adjacent natural frequencies of 500 and 600 Hz. > Assume the speed of sound in the air 340m/s. The length of the organ pipe > is?_ What is **open organ pipe**? It means two open ends? Like this? Or one closed and one open? ![enter image description here](http://i.stack.imgur.com/DmRsb.jpg) Should I use this formula? $L=\frac{(2n+1)v}{4f}$


Convolution kernel of poisson equation by FFT

I'm trying to solve poisson equation using FFT. In genral it is a convolution of the charge density with potential well of point charge ( Green's function of laplace equation ) which is $1/r$ I'm not quite sure about fourier transform of this convolution kernel. the 2D fourier transform of $1/r$ is $ \frac{1}{\sqrt{k_x^2+k_y^2}}= 1/k_r$ acording to this: http://sepwww.stanford.edu/public/docs/sep103/jon3/paper_html/node3.html while in many papers about FFT solution of poisson equation is used kernel like this: $ \frac{1}{(4-2cos(2*\pi*k_x)-2cos(2*\pi*k_y))} $ For example here: http://www.physics.buffalo.edu/phy410-505-2004/Chapter6/ch6-lec2.pdf In one dimension the kernel is $1/k^2$ **Basicaly, I would like to make clear what is the kernel in 1D, 2D and 3D case both in real space and Fourier space**


Solutions of damped oscillator differential equation

I am reading about damped harmonic motion in my Physics book (Gerthsen Physik) and there are two things that irritate me: # Stokes friction It says that Stokes friction would be $$F = -m \gamma \dot{x}.$$ To me, it does not make any sense why it depends on the mass of the object. It means that the deceleration does not depend on the mass. Or is it just since it is proportional to the volume and the volume usually depends on the mass. # linear combination of solutions Then the differential equation is set up, I fully understand it, except the confusion about the friction. $$ m \ddot{x} + \gamma \dot{x} + \omega_0^2 = 0 $$ With $$\delta = \frac{\gamma}{2}$$ and $$\omega = \sqrt{\omega_0^2 - \delta^2}$$ the solution to the $$x(t) = \exp(\lambda t)$$ approach are $$\lambda = -\delta \pm i \omega t$$. One of the plus or minus variants are a valid solution, but the linear combination of both are all solutions. I understand that. The book then says that the general solution is the combination of real and imaginary parts. Alternatively, one can add a phase angle and have one "compact" solution like this: $$ x(t) = x_0 \exp(-\delta t) \exp(i ( \omega t + \phi ) ) $$ The $\exp(i \phi)$ certainly makes up a linear combination of real and imaginary parts, but how is it a combination like $$c_1 \exp(i\omega t) + c_2 \exp(-i\omega t) ?$$


Fluid properties

I cannot understand the following sentence, > Without flow , however, shearing stresses cannot exist, and compression > stress or pressure is the only stress to be considered. (from _A Physical Introduction to Fluid Mechanics_ , by A. J. Smits, John Wiley & Sons, 2000) Could someone explain this to me?


about satellite is it possible to get signal from two satellite at a time?

1. Is it possible to get signal from two satellites at the same time?   2. If we go far away from a certain country to another, and be able to continue to recieve the same signal?


Are Black Holes set to take over the Harmonic Oscillator in the 21st century?

A few years ago I attended a talk given by Andy Strominger entitled Black Holes- The Harmonic Oscillators of the 21st Century. This talk, http://media.physics.harvard.edu/video/?id=COLLOQ_STROMINGER_091310 was quite formidable and has had a lasting impression on my physics studies since. I'd consider the talk transformative in my approach. Now, imagine hearing the power of the harmonic oscillator model for the first time before taking an EM or QM class and simply having a spring mass system in mind... Fast forward a few years and consider learning the eigenfunctions of the harmonic oscillator Hamiltonian ${\Omega_n}$ are an orthonormal basis set for $\mathcal{H}=L_2(\mathbb{R^d},d\mu)$. A statement equivalent to: 1) Schwartz Space $\mathscr{S}$ being Fourier invariant 2) The Fourier inversion theorem holding Now, the power of the harmonic oscillator is coming to fruition in what I can solve in physics. Admittedly, there is still much in the field that is beyond me at this point that is well represented by the harmonic oscillator model. This leads me to first ask what are the best known fits for the harmonic oscillator model? However, this question is not the main focus of this post as I am also deeply interested in models extending beyond the harmonic oscillator paradigm. Namely, what are examples of objects currently being modeled as Black holes? It's interesting (for me, hopefully more) to ask this question in the "firewall" era, given that I attended this talk before the now famous AMPS paper. So, in summary I am asking for examples of physical objects modeled by harmonic oscillators, or black holes, relevant in the field today. It would be interesting to compare the usage of the models in light of the theme of Andy Strominger's talk.


Modelling the movement and jumps of a chalk while drawing a dashed line on a blackboard

You probably know that if you try to draw a line using a piece of chalk on a blackboard , under some conditions (for example, $\alpha<\frac{\pi}{2}$ in the picture below) you will have a dashed line pattern instead of a continuous line. My question is : (how) Can you model this special movement of the chalk ,and specially , **find the length of line segments and the distance between them** (which are the _visible_ characteristics of the motion)? Note:gravity is present. (I think it affects the solution, at least in some models) ![the chalk](http://i.stack.imgur.com/jy0C5.png) The pattern: ![enter image description here](http://i.stack.imgur.com/8awdr.png)


Mathematical Modelling for Efficiency of PV thermal technology

As the efficiency of PV thermal or wind energy decreases with time(from the installment). Is there any mathematical model that gives its relation with time. i.e., whether it is a linear decrease or some other function of time. I need to use the result of this in my research on energy markets.


WKB approximation to loop diagrams

I'm a bit confused with the terminology here. This paper claimed to use WKB method to calculate the usual loop diagrams. Notice that the vertex is approximated by expanding around the classical (on- shell) solution of the massive particle propagator. However this method doesn't look like the usual WKB I know. AFAIK in WKB (eikonal) approximation we are supposed to expand the propagator around the particle limit of the solution, but NOT the classical field limit. So, how do you describe the WKB method in QFT? Is it practical enough to give us quick intuitive calculation on our system? (probably in this sense, the WKB+1loop is the crudest semiclassical approximation we can do -> first order in $\hbar$ and coupling constant)


Conservation of energy in transformers

How does energy remain conserved in a transformer if emf is increasing, or decreasing? Does the current decreases to accomodate? Does Ohm's law still hold here? Although we know, Ohm's law is not universal.


What is the smoking gun signature of string theory? Suppose we have a complete and consistent model of quantum gravity with a zero or negative cosmological constant, but all we are given is its complete S-matrix. Using only this information, how do we figure out whether or not this model corresponds to a string theory compactification? If the cosmological constant is positive, what can we use in place of the S-matrix?


Rayleigh-Taylor Instability dependence on acceleration direction

I'm trying to bolster my understanding of the Rayleigh-Taylor instability, and I've gotten stuck on the point of which fluid (more or less dense) is being accelerated into the other. Cases of uniform acceleration make sense (e.g. plane-parallel fluids under gravity; or being accelerated by a piston), but my understanding falters in more exotic configurations. Consider the following: A wind tunnel(-like) setup---without gravity---in which one fluid (density $\rho_1$) is at rest, while another ($\rho_2$) is blown into the first with a constant velocity. Because the two substances are colliding --- they'll both feel an acceleration. But because there is no acceleration between the **reference frames** of $\rho_1$ and $\rho_2$, how do you tell which configuration is stable, and which is not?


Transferring CFT correlations from $\mathbb{R}^3$ to $S^3$

There seems to be a simple method to transfer a CFT's correlations from $\mathbb{R}^3$ to $S^3$ but I am not understanding why it is supposed to work. The idea is that somehow because, $ds^2_{S^3} = \frac{4}{(1 + \vert \vec{r} \vert^2 )^2}ds^2_{\mathbb{R}^3 }$ it follows that all that one needs to do is that replace all occurences of $\vec{r} - \vec{r}'$ by $\frac{2(\vec{r} - \vec{r}') }{\sqrt{ (1 + \vert \vec{r} \vert ^2 )(1 + \vert \vec{r}' \vert ^2 ) } } $   * Why is this supposed to work?    * What is the necessary relationship between the two CFTs for this to work? (like if one uses this on the free scalar field theory then it will turn out that w.r.t the Laplacians on $\mathbb{R}^3$ or $S^3$ the two 2-point functions satisfy _different_ P.D.Es respectively and hence they come from two different Lagrangians - since we anyway knew that the Lagrangian for the conformally coupled scalar on the flat space is not the same as the Lagrangian for the conformally coupled scalar on $S^3$)   * Is it part of some general idea that would work between other pairs of manifolds?


Quantum computing problem

Suppose that a qubit is in the state $|\varphi\rangle=a|0\rangle+\sqrt{1-a^2}|1\rangle$, where $a\in[-1,1]$. If we first perform a standard basis measurement on this qubit and then perform a $\\{|u\rangle,|u^\perp\rangle\\}$-basis measurement, where $$|u\rangle=b|0\rangle+\sqrt{1-b^2}|1\rangle \text{ for some }b\in[-1,1],$$ what is the probability that the outcome of the second measurement is $|u\rangle$, in terms of $a$ and $b$? Edit: I've found a solution, it should be $ab+\sqrt{(1−a^2)(1-b^2)}$.


Translator Operator

In _Modern Quantum Mechanics_ by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $$\left| x+dx\right\rangle \simeq \left| x \right\rangle.$$ But for every $\epsilon > 0$ $$\langle x+ \epsilon \left| x \right\rangle = 0.$$ Therefore this limiting process $$\lim_{\epsilon \rightarrow 0} \left| x+ \epsilon \right\rangle = \left| x \right\rangle$$ does not make sense for me. I couldn't derive commutator relation without using these.