Complex masses for Dirac and Weyl spinors

I'm trying understand how to rotate Dirac fields to absorb complex phases in masses. I have a few related questions:   1. With Weyl spinors, I understand, $$ \mathcal{L} = \text{kinetic} + |M|e^{i\theta}\xi\chi + \textrm{h.c.} $$ The phase removed by separate left- and right-handed rotations, e.g. $\xi \to e^{-i\theta/2}\xi$ and $\chi \to e^{-i\theta/2}\chi$. These phases cancel in the kinetic terms. Is it correct that with Dirac spinors, $$ \mathcal{L} = \bar\psi |M|e^{i\theta\gamma_5} \psi = \text{Re}(M)\bar\psi\psi +i\text{Im}(M)\bar\psi\gamma_5\psi $$ and the phase is removed by $\psi \to e^{-i\theta\gamma_5/2}\psi$? The appearance of the $\gamma_5$ in the phase troubles me a little - I suppose this is telling us that Weyl spinors are a more suitable basis than Dirac spinors?   2. If the field is Majorana, $\xi = \chi$, and the field can still absorb a phase? I think I must be making a trivial mistake.  For example, Majorana neutrino fields cannot absorb phases, leading to extra CP violation. And in SUSY, the gaugino Majorana soft-breaking masses are e.g. $M_1e^{i\theta}$. Can their phases be re-absorbed via a field redefinition? I don't think they can. So I must have a mistake.


Is particle entanglement a binary property?

Is the particle entanglement a boolean property? That is, when we consider two given particles, is the answer to the question "are they entangled" always either "yes" or "no" (or, of course, "we are not sure if it's yes or no")? Is there such thing as partial entanglement?


Is there a general systematic approach how to calculate the individual terms in an operator product expansion?

Is there a general systematic procedure or approach to obtain the analytic functions $c_{ijk}$ as well as the corresponding operators $$A_i(z)$$ that appear in the operator product expansion (OPE) $$ A_i(z)A_j(w) = \sum\limits_k c_{ijk}(z-w)A_k(w) $$ for any given two operators $A_i(z)$ and $A_i(z)$? I have followd (a limited number of) examples of OPE calculations that involved things like expanding until some correlators for which the expression is known appear and power series expansions of derivatives of operators, but from this I was not able to discern the general way to go.


Hollow stone columns provide more support?

In history class in elementary school I remember learning that the Greeks would build their stone columns hollow because they thought this provided more support. Is it true that a hollow column is stronger? Thanks!


Why don't you see multiple images of an object?

Consider the ray model of light. Let's say an object such as a pencil is illuminated, and consider one point on that pencil. Since there could be many rays of light bouncing off the same point on the pencil, one ray could hit the left side of your eye and another ray from the same point could hit the right side of that same eye. Why don't you see that point in 2 different places then? Thanks!


Explanation that air drag is proportional to speed or square speed?

A falling object with no initial velocity with mass $m$ is influenced by a gravitational force $g$ and the drag (air resistance) which is proportional to the object's speed. By Newton´s laws this can be written as:   1. $mg-kv=ma$ (for low speeds)    2. $mg-kv^2=ma$ (for high speeds).  I assume that $k$ is a positive constant that depends on the geometry of the object and the viscosity. But how can one explain that the air resistance is proportional to the velocity? And to the velocity squared in the second equation?


Resisting force depends on velocity?

Why does resisting force depend on velocity? I think there is no relation between resisting force and velocity of object. Please speak about it logically.


Confused about the theoretical origin of quadratic air drag

Though the mathematical concepts underlying quadratic air drag are quite straightforward (a single variable differential, just like the linear drag equation), my text book (and online text books) completely ignore the reasoning behind why the "drag" at high speeds on a trajectory through a fluid is proportional to the velocity squared. I know that this could not just be an empirical result. Is there an intuitive reason for this (perhaps like momentum transfer or kinetic energy?). If possible, I would also like to ask for an intuitive reason behind the units $m^2/s^2.$


Does the mass point move?

There is a question regarding basic physical understanding. Assume you have a mass point (or just a ball if you like) that is constrained on a line. You know that at $t=0$ its position is $0$, i.e., $x(t=0)=0$, same for its velocity, i.e., $\dot{x}(t=0)=0$, its acceleration, $\ddot{x}(t=0)=0$, its rate of change of acceleration, $\dddot{x}(t=0)=0$, and so on. Mathematically, for the trajectory of the mass point one has \begin{equation} \left. \frac{d^{n}x}{dt^n}\right|_{t=0} = 0 \textrm{ for } n \in \mathbb{N}_0\mbox{.} \end{equation} My physical intuition is that the mass point is not going to move because at the initial time it had no velocity, acceleration, rate of change of acceleration, and so on. But the mass point not moving means that $x(t) \equiv 0$ since its initial position is also zero. However, it could be that the trajectory of the mass point is given by $x(t) = \exp(-1/t^2)$. This function, together with all its derivatives, is $0$ at $t=0$ but is not equivalent to zero. I know that this function is just not analytical at $t=0$. My question is about the physical understanding: How could it be that at a certain moment of time the mass point has neither velocity, nor acceleration, nor rate of change of acceleration, nor anything else but still moves?


Does General Relativity encompass Special Relativity?

Can all of the predictions made in Special Relativity (SR) also be made in General Relativity (GR)?


The example of relativity of simultaneity given by Einstein

I understand (supposedly) the mathematics concerning the relativity of simultaneity in Special Relativity, but I have a nagging question regarding the original example given by Einstein supporting it (I'm only disagreeing with this specific example, not the concept). It is normally given as a person on an embankment and a person on a train. There is a relative speed between them (usually presented as the train passing the embankment). Now, when both people are at the same x-position (x=0), there is a flash of light at x = +dx and x = -dx. The argument as I keep seeing it is that the person on the embankment will say that both flashes reach him at the same time, whereas the person in the train will say that the flash in front of him reaches him before the other because he was moving toward it, and thus the observers will disagree on the simultaneity of the flashes. But given that the flashes occurred at the same distance from each of them, the speed of light is constant in both frames, and either one can claim to be at rest, then won't they, according to SR, necessarily see the flashes as simultaneous (both flashes have to travel the same distance in both frames since at the time of emission, the sources of both flashes were equidistant from both observers). I agree that the person on the embankment will say that the person on the train shouldn't see them as simultaneous (and vice versa) since either observer will see the other moving relative to the sources, but in each of there own frames, they must see the flashes as being simultaneous shouldn't they? Am I just misunderstanding the example? Thanks.


How to solve fixed-fixed beam with finite difference method?

What equations to use on this system to form a matrix $A$ with dimensions $[n,n]$ and load vector $q$ with dimension $[n]$ ? I am trying to get vertical displacement $w$. $$w = A^{-1}\times q$$ Boundary conditions are as follows: $$w(o) = 0 $$ $$w(L) = 0 $$ $$\phi(o) = 0$$ $$\phi(L) = 0$$ It is becouse in any point of beam I can't make equation: $$d^2y/dx^2*E*I=M=0$$ so I can't get the exact values of displacement. The problem is that everywhere I look for solution it is done on a beam with continous load over entire beam or with at least one joint and I have only half of the beam covered with continous load and no joints. From $d^4y/dx^4*E*I=q=0$ again I have too many unknown values. ![Scheme](http://imgur.com/YkIdCAI.jpg)


Car tires wearing in dependence on breaking system

Lets say I have two cars.   They are identical in every way, except that Car A has a normal breaking system, where most of the breaking power is inflicted on the front wheels, and some on the back, and Car B has a breaking system where all of the power is inflicted on the front wheels and none on the back. In the long run, in which car will the tires on the back wheels experience more wearing?   (For the sake of comparison, let's say we measure wear by the weight lost from the tires.)


Calculating energy U from $\partial U/\partial q$

Imagine $N$ oscillators with only two possible energies, $\epsilon_0$ and $ \epsilon_1$, with $\epsilon_1 > \epsilon_0$. Taking $\epsilon_0 = 0$ for now I showed $\Omega(q\epsilon_1) = \frac{N!}{(N-q)!q!}$ and then $$\frac{\partial S}{\partial q} = k \log(N/q - 1) $$ How can I use the above equation to show that $$U = N\epsilon_1\frac{e^{-\epsilon_1/(kT)}}{1+e^{-\epsilon_1/(kT)}} $$ I tried moving the $\partial q$ over to the right, and then have $dS = dU/T$, but i wasn't getting anything meaningful.


In the diode equation, why the exponential $\exp$ and the ideality factor $n$ are there? What do they represent & what is their significance?

In the Shockley diode equation, why the exponential $\exp$ and the ideality factor $n$ are there? What do they represent & what is their significance? I have to work on Solar Photovoltaics, and I need to understand the Shockley diode equation clearly.


Planetary model of atom still valid?

When I was in school, I learned (from Democritus) that an atom was similar to a solar system, with the nucleus being the sun, and the electrons being the planets. Of course, there are some differences:   * The "sun" isn't a single entity, but a collection of protons and nuetrons.    * Two planets can share an orbit (which might be possible in a solar system too, but it doesn't happen in our solar system).  Is this model still valid? Here are my problems with it:   * In "Surely You're Joking, Mr Feynman", Richard Feynman implies that electrons are more a theoretical concept than real objects.    * I have trouble understanding atomic bonds (ionic and covalent) in this model.    * I also have trouble understanding electron "orbit jumping" in this model, as well as several other things.  Is there a better model for someone learning this for the first time?


Reason for considering the positive root

In eqn. (3.11) of Srednicki's QFT book only the positive root is considered; i.e., $ \omega = + \sqrt{(k^2 + m^2 )} $ Why the negative root is not considered? And what is the $\omega$?


Why does the power in an inductor equal what it does?

I understand that power is that rate at which work is done and that because of this the power in an inductor is equal to $$P=\frac{d}{dt} \left(\frac12Li^2\right).$$ I also understand that the power is also equal to $$P=Li\frac{di}{dt}$$ since $L\,\frac{di}{dt}=V$ and $Vi$ is power. I understand that since the power is equal to both of these equations that they are equal to each other. The part that I don't get is mathematically how to get from one to the other.


Glueball mass in non-abelian Yang Mills theory

How can the glueball mass be calculated in Yang Mills theory?


Is there precision experimental evidence for Furry's theorem -- that only even degree VEVs are non-zero?

Is there precision experimental evidence for or contradicting Furry's theorem -- that only even degree VEVs are non-zero, specifically for the EM field?


Acceleration of particles in a pipe with variable cross section

I am currently making an exercise from Paterson,1983, IV.3.ii. It goes as follows > Water flows steadily and incompressibly along a pipe whose area cross > section $A(x)$ varies slowly with the coordinate $x$ along the pipe. Use the > conservation of mass and/or incompressibility to calculate the mean velocity > along the pipe at $x$, and calculate the acceleration of a moving particle > moving with this mean velocity. Take the mean velocity to be along the pipe > and depending only on $x$. I would solve the first part of the question as follows. Let $A(0)$ and $v(0)$ be the cross section and the velocity at the begin of the pipe respectively. Since the medium is incompressible, it holds that $A(0) v(0) = A (x) v(x)$ for every $x \in [0, L]$ where $L$ is the length of the pipe. The local velocity in the x-direction at any point $x$ therefore is $$v(x)=\frac{A(0) v(0)}{A(x)}$$ So the average velocity of a fluid particle therefore is $$\langle v(x)\rangle=\frac{1}{L}\int_0^L \frac{A(0) v(0)}{A(x)} dx $$ So far so good. It's the second part of the question I don't get: How can I know where the particle attains its average velocity and what the local acceleration is. I only know that it exists (intermediate value theorem for differentiable functions).


Mass eigenstate of neutrinos

Isn't mass a fixed and an intrinsic property of a particle? How can we talk about eigenstates of the mass in the context of neutrinos?


What are mass eigenstates?

According to Wikipedia > _Neutrino oscillation arises from a mixture between the flavor and mass > eigenstates of neutrinos. That is, the three neutrino states that interact > with the charged leptons in weak interactions are each a different > superposition of the three neutrino states of definite mass. Neutrinos are > created in weak decays and reactions in their flavor eigenstates. As a > neutrino propagates through space, the quantum mechanical phases of the > three mass states advance at slightly different rates due to the slight > differences in the neutrino masses. This results in a changing mixture of > mass states as the neutrino travels, but a different mixture of mass states > corresponds to a different mixture of flavor states. So a neutrino born as, > say, an electron neutrino will be some mixture of electron, mu, and tau > neutrino after traveling some distance. Since the quantum mechanical phase > advances in a periodic fashion, after some distance the state will nearly > return to the original mixture, and the neutrino will be again mostly > electron neutrino. The electron flavor content of the neutrino will then > continue to oscillate as long as the quantum mechanical state maintains > coherence._ What are these mass eigenstates?


What are the largest thermal gradients achievable in a lab environment?

I am looking for a system capable of creating a gradient of $100\, \mathrm{K}/\mathrm{\mu \textrm{m}}$ on a $30\, \mathrm{\mu}\textrm{m}$ spacing of a system mounted on a Si-N membrane. My so-called nanoheater is not up to the task.


Approximating mean daily and hourly temperature beyond Fourier series

Summary: What "well-known" and short parametrized mathematical function describes daily and hourly temperature for a given location? If you look at the mean daily temperature graph for a given location, it looks like a sine wave with a period of one year. Similarly, the hourly temperature for a given day for a given location also looks like a sine wave with a period of one day. However, closer inspection (Fourier analysis) shows that they're not. There are fairly strong components of frequency 2/year and 3/year for the daily temperature, and the hourly temperature also has strong non-single-period terms. Is there a parametrized function that reasonably describes the daily mean temperature and (a separate function) the hourly mean temperature? The parameters would be location-based. I realize I can keep taking more Fourier terms to increase accuracy, but I was hoping for something more elegant. For example, maybe the graph is a parametrized version of sin^2(x) or some other "well-known" function.


Math for Thermodynamics Basics

I am studying Statistical Mechanics and Thermodynamics from a book that i am not sure who has written it, because of its cover is not present. There is a section that i can not understand: ${Fj|j=1,..,N}$ $S= \sum_{j=1}^{N} F_{j}$ $<S>=< \sum_{j=1}^{N} F_{j}> = \sum_{j=1}^{N} <F_{j}>$ $\sigma^{2}_{S} =<S^{2}>-<S>^{2}$ line a: $=\sum_{j=1}^{N}\sum_{k=1}^{N} <F_{j}F_{k}> \- \sum_{j=1}^{N} <F_{j}>\sum_{k=1}^{N}<F_{k}>$ line b: $=\sum_{j=1}^{N}\sum_{k=1(k\neq j))}^{N} <F_{j}><F_{k}> +\sum_{j=1}^{N} <F_{j}^{2}> \- \sum_{j=1}^{N} <F_ {j}>\sum_{k=1}^{N}<F_{k}>$ line c: $=\sum_{j=1}^{N} (<F_{j}^{2}>-<F_{j}>^{2})$ $=\sum_{j=1}^{N} \sigma_{j}^{2}$ My question is what happened after line a to line b and after that to line c? My other question is, i have a little math, what should i study to understand such thermodynamics root math studies, calculus 1 or 2 or what else, can you specify a math topic? Thanks


Value of Ramanujan Summation In Quantum Mechanics

In mathematics, sum of all natural number is infinity. but Ramanujan suggests whole new definition of summation. "The sum of $n$ is $-1/12$" what so called Ramanujan Summation. First he find the sum, only Hardy recognized the value of the summation. And also in quantum mechanics(I know), Ramanujan summation is very important. **Question.** What is the value of Ramanujan summation in quantum mechanics?


What practical issues remain for the adoption of Thorium reactors?

From what I've read on thorium reactors, there's enormous benefit to them. Their fuel is abundant enough to power human civilization for centuries, their fission products are relatively short-lived, they're far less prone to catastrophic failure, and they don't produce anything you could feasibly use as a source of material for nuclear weapons. So what technical issues need to be resolved so that Thorium reactors become practical and put into wide use? Is it purely engineering issues that need to be overcome? Or are there problems of physics as well? If so, what are the technical problems and what research is occurring to overcome them? If none of the problems that face thorium reactors are insurmountable, then why aren't they the focus of research and development that nuclear fusion is? Are there real environmental issues? (If so what are they?)


Why, in EXAFS spectrum, does the absorption coefficient monotonically decrease with increasing photon energy?

In atomic physics, it is common knowledge that following the absorption edge, where the photon energy equals the binding energy of a core electron, a monotonic decrease in the absorption coefficient with increasing photon energy is observed. Obviously it is so common, that everyone mentions it, but nobody cares to explain. For example: ![absorbtion spectra graph](http://i.stack.imgur.com/ns5sv.gif) As I understand, in EXAFS (Extended X-Ray Absorption Fine Structure) case, a core electron is excited to the conduction band, and photons with higher energy should just excite it to the higher energy level in the conduction band. So why does absorption decrease?


Impervious nature of solid matter due to quantum degeneracy pressure

On Wikipedia the following statement is made without reference: > Freeman Dyson showed that the imperviousness of solid matter is due to > quantum degeneracy pressure rather than electrostatic repulsion as had been > previously assumed. Can anyone find the appropriate reference(s)?


Simulating Gravity in 3D Game?

Alright, I am writing a space simulator for a 3D game and I would like to implement gravity of objects into it. Is there a nice way to find a velocity vector which can be added to my engine output vector to create the effect of gravity. In addition, how would I be able to find the velocities required to get object A into orbit around object B at a certain distance?


How to keep the clock of a spaceship synchronised to the clock of an observer?

I read that the clocks of GPS satellites seem to run slower than the clock of stationary observer, because of their speed (special relativity) and seem to run faster than the clock of stationary observer because of their altitude (general relativity) . http://osg.informatik.tu- chemnitz.de/lehre/old/ws0809/sem/online/GPS.pdf From this, can you model the speed $\upsilon(t)$ and the altitude $r(t)$ of a **spaceship** launched from the ground vertically at $t=0$ such that the clock of the **spaceship** will always seem to be synchronised to the clock of the observer on the ground (from the observer's frame of reference) ?


Helium-4 superfluidity and gauge symmetry breaking

Is there an accessible account of superfluidity in Helium-4 as a manifestation of "global gauge symmetry" breaking? And what is meant by "global gauge symmetry"? I was taught that gauge symmetries were by definition local. Is it just a different terminology in condensed matter?


How powerful would an explosion have to be for its effects to be felt worldwide?

If an explosion were to occur at any point on earth, how powerful would that explosion have to be for it to be audible or otherwise detectable by every person on the planet? Detection could mean either seeing or hearing the blast or feeling the tremors created by the shock wave. Bonus question: is any such explosion possible without it destroying the planet, the atmosphere or wiping out all life on earth? * * * A rough estimate puts the average distance between most antipodes on land at just shy of 20 000 kilometres. The largest nuclear bomb ever detonated, Tsar Bomba, had a yield of 50-58 megatons of TNT and was detectable almost a 1000 km away, according to Wikipedia: > The heat from the explosion could have caused third-degree burns 100 km (62 > mi) away from ground zero. A shock wave was observed in the air at Dikson > settlement 700 kilometres (430 mi) away; windowpanes were partially broken > to distances of 900 kilometres (560 mi). Atmospheric focusing caused blast > damage at even greater distances, breaking windows in Norway and Finland. > The seismic shock created by the detonation was measurable even on its third > passage around the Earth. The most powerful volcanic eruption known was that of Mount Tambora in 1815. Classified as Volcanic Explosivity Index 7 (note that it goes up to 8) with an estimated yield of 800 Mt, it was heard about 2 600 km away. > On 5 April 1815, a moderate-sized eruption occurred, followed by thunderous > detonation sounds, heard in Makassar on Sulawesi 380 km (240 mi) away, > Batavia (now Jakarta) on Java 1,260 km (780 mi) away, and Ternate on the > Maluku Islands 1,400 km (870 mi) away. On the morning of 6 April, volcanic > ash began to fall in East Java with faint detonation sounds lasting until 10 > April. What was first thought to be the sound of firing guns was heard on 10 > April on Sumatra island more than 2,600 km (1,600 mi) away. The Wikipedia page on TNT equivalents list some interesting events but branches off into seismic and cosmic events after the entry for the Tsar Bomba. So, is it possible to go bigger?


Lagrangian Systems

Given a manifold $M$, Arnold's "Mathematical Methods of Classical Mechanics" defines a Lagrangian system as a pair $(M,L)$ where $L$ is some smooth function on the tangent bundle $TM$. The function $L$ is called the Lagrangian. In the case when $M$ is a Riemannian manifold and a particle in $M$ is moving under some conservative force field, taking the Lagrangian to be the kinetic minus the potential energy we recover Newton's second law. I know that one of the main advantages to the Euler-Lagrange equations over Newtons is the way in which they simplify constrained systems. I know another is the coordinate independence of the equations. However, in all applications I've seen the manifold is always Riemannian and the Lagrangian is always $K-U$. My questions are:   1. Why do we have this abstract definition of a Lagrangian system and of an abstract Lagrangian?    2. What are some of the cases in which $L$ is not $K-U$ that gives interesting results?    3. Or cases in which the manifold is not Riemannian?


How is a Hamiltonian constructed from a Lagrangian with a Legendre transform

many textbooks tell me that Hamiltonians are constructed from Lagrangians like $$L=L(q,\dot{q})$$ with a Legendre transformation to obtain the Hamiltonian as $$H=\dot{q}\frac{\partial L}{\partial \dot{q}}-L$$ but none of the textbooks explain how this is done. My specific problem is that I have Lagrangians that do not depend on $\dot{q}$ and therefore should have $\frac{\partial L}{\partial \dot{q}}=0$, hence $H=-L$. But my impression from the clues I have is that it is not that simple. Let's say the Lagrangian is $$L(q)=\ln(q)-(2q-10)\lambda$$ Now as far as I know the Legendre transformation should give a function $f^*(p)=\sup(pq-L(q))$ (this implies $p=\frac{\partial L}{\partial q}$) which is obtained by substituting the stationary point $q_s$ of $\sup(pq-L(q))$ into $pq-L(q)$ thus getting $f^*(p)=pq_s-L(q_s)$ (for instance wikipedia's Legendre Transformation page explains this). Doing this for the example above: $$\frac{\partial (pq-L(q))}{\partial q}=\frac{\partial (pq-\ln(q)+(2q-10)\lambda)}{\partial q}=p-\frac{1}{q}+2\lambda$$ must be 0 for a stationary point, thus $q_s=1/(p+2\lambda)$. And hence the transformation should be $$f^*(p)=p\frac{1}{p+2\lambda}-\ln(\frac{1}{p+2\lambda})+(2\frac{1}{p+2\lambda}-10)\lambda$$ and this should be the Hamiltonian. But this equation does obviously have nothing to do with the textbook Hamiltonian. Rather, an answer to another question has in a similar case treated $L(q)$ as being dependent on $\dot{q}$ implicitly (Writing $\dot{q}$ in terms of $p$ in the Hamiltonian formulation ... answer by Qmechanic). It mentions using the Dirac-Bergmann method for obtaining the Legendre transform. Trying something along the lines of this other question the above example seems to give $$p=\frac{\partial L}{\partial \dot{q}}=0$$ and $$p \approx 0$$ (an equality modulo constraint as the answer to the question linked above says). And then $H=\dot{q}p-L$. The difference seems to be that the Legendre transform is done with respect to two different variables, $q$ and $\dot{q}$ - but it was my understanding that it had to be done with respect to all variables the Lagrangian depends on. So how does the $qp_q$ term vanish if we have only the $\dot{q}p_{\dot{q}}$ term left? Thanks. edit: changed ln to \ln as Plane Waves suggested, and sup to \sup. And yes, sup is the supremum over all q, as Vibert said, sorry for forgetting to mention that.


Effect of slits and a lens

We have a screen with two slits (Young style) separated by a distance $d$, one of them receives a planar wave of $600nm$, the other receives a planar wave of $400nm$. Behind the slits there's a screen we observe. Only with this, as the waves have different wavelengths, I guess there can't be any interference, we will only see the difraction pattern, the two functions of the form $\sin^2(x)/x^2$, with the principal maximums separated a distance $d$. Am I right here? Now, we put a convex lens behind the slits so the screen in which we observe is in the focal plane. Ok, I have three configurations: 1.- In the first one, the system is configured such that the slits are far away from the lens. Here, we can approximate the wave that arrives as a planar wave, and therefore the lens will perform the Fourier transform in the focal plane of the screen. The diffraction of the slits also performs the Fourier transform, so this configuration should lead to having only two bars of light in the screen, centered in the focus. Am I right? 2.- The slits are in the focal plane on the lens, such that the lens is in the middle of slits-screen. Here, the same thing should happen, right? As the light comes from the focal plane, the lens must do the fourier transform with no extra things, and we should get the two bars, again both of them in the same line (center of the screen). Am I right here? 3.- The last one, I can't see... the lens is just behind the slits so the distance between slits and lens is $\approx 0$. My guess here is that the two centers of the intesity distributions $\sin^2(x)/x^2$ will go to the center of the screen (the focus) because they go perpendicular to the lens, but the rest of the pattern will just be compressed a little. Again the wave don't interfere, so the intensities just sum up, and the resulting will be: $$I=I_1\frac{\sin^2(\alpha x)}{(\alpha x)^2}+I_2\frac{\sin^2(\beta x')}{(\beta x')^2}$$ Being $\alpha$ and $\beta$ the some factor of compression due to the lens, that could actually be a function of $x$. Am I right here? Am I completely wrong? What would happen?


$\hat{\imath}$ component of force exerted on an electron by a magnetic field?

> The magnetic field over a certain range is given by $\vec{B} = > B_x\hat{\imath} + B_y\hat{\jmath}$, where $B_x= 4\: \mathrm{T}$ and $B_y= > 2\: \mathrm{T}$. An electron moves into the field with a velocity $\vec{v} = > v_x\hat{\imath}+v_y\hat{\jmath}+v_z\hat{k}$, where $v_x= 5\: \mathrm{m/s}$, > $v_y= 8\: \mathrm{m/s}$ and $v_z= 9\: \mathrm{m/s}$. The charge on the > electron is $-1.602 \times 10^{-19}\: \mathrm{C}$. What is the > $\hat{\imath}$ component of the force exerted on the electron by the > magnetic field? Answer in units of $\mathrm{N}$. I know that $\vec{F}=q\vec{v} \times \vec{B}$, so plugging in I have: $$\vec{F}=(-1.602 \times 10^{-19})<5,8,9> \times <4,2,0>$$ My confusion is to whether or not multiply my velocity vector components by charge (the scalar) or if I take the cross product between $\vec{B}$ and $\vec{v}$ first? I'd also like to know why whichever operation comes first does in fact come first.


Is there a general physics simulator for learning purposes?

Is there a complete physics simulator that I can use to do general simulations for learning purposes? For example:   1. Create a sandbox.   2. Fill with a gas.   3. Load a 3d solid model like this (but 3d).   4. Fill it with a dense liquid.   5. Load gravity.   6. Watch, measure and understand how a barymeter works. It doesn't need to be precise, just usable, so I guess it is not impossible. The point would be to simulate and visualize any kind of exercise you would find in your physics book. It would be the mother of the learning tools. If it doesn't exist, is anybody interested in programming it?


Can the vanishing of the Riemann tensor be determined from causal relations?

Given a _Lorentzian manifold and metric tensor_ , "$( M, g )$", the corresponding _causal relations_ between its elements ( _events_ ) may be derived; i.e. for every pair (in general) of distinct events in set $M$ an assignment is obtained whether it is _timelike_ separated, or _lightlike_ separated, or neither ( _spacelike_ separated). In turn, I'd like to better understand whether causal separation relations, given abstractly as "$( M, s )$", allow to characterize the corresponding Lorentzian manifold/metric. As an exemplary and surely relevant characteristic (cmp. answer here) let's consider whether _the Riemann curvature tensor vanishes_ , or not, at each event of the whole set $M$ (or perhaps suitable subsets of $M$). Are there particular causal separation relations which would be indicative, or counter-indicative, of the Riemann curvature tensor vanishing at all events of set $M$ (or if this may simplify considerations: at all events of a _chart_ of the manifold); or on some subset of $M$? To put my question still more concretely, consider as possible illustration of "counter-indication": (a)   Can ~~any _chart of a 3+1 dimensional Lorentzian manifold with everywhere vanishing Riemann curvature tensor_ (or, at least, a whole such manifold) contain ~~ [Edit in consideration of 1st comment (by twistor59): -- _the Riemann curvature tensor vanish at least in one event of a 3+1 dimensional Lorentzian manifold_ if each of its _charts_ contains -- ]   * fifteen events (conveniently organized as five triples): $A, B, C$; $\,\,\,\, F, G, H$; $\,\,\,\, J, K, L$; $\,\,\,\, N, P, Q\,\,\,\,$, and $\,\,\,\, U, V, W$,   * where (to specify the causal separation relations among all corresponding one-hundred-and-five event pairs): $s[ A, B ]$ and $s[ A, C ]$ and $s[ B, C ]$ are _timelike_ ,   $s[ F, G ]$ and $s[ F, H ]$ and $s[ G, H ]$ are _timelike_ ,   $s[ J, K ]$ and $s[ J, L ]$ and $s[ K, L ]$ are _timelike_ ,   $s[ N, P ]$ and $s[ N, Q ]$ and $s[ P, Q ]$ are _timelike_ ,   $s[ U, V ]$ and $s[ U, W ]$ and $s[ V, W ]$ are _timelike_ , $s[ A, G ]$ and $s[ G, C ]$ and $s[ A, K ]$ and $s[ K, C ]$ and   $s[ A, P ]$ and $s[ P, C ]$ and $s[ A, V ]$ and $s[ V, C ]$ are _lightlike_ , $s[ F, B ]$ and $s[ B, H ]$ and $s[ F, K ]$ and $s[ K, H ]$ and   $s[ F, P ]$ and $s[ P, H ]$ and $s[ F, V ]$ and $s[ V, H ]$ are _lightlike_ , $s[ J, B ]$ and $s[ B, L ]$ and $s[ J, G ]$ and $s[ G, L ]$ and   $s[ J, P ]$ and $s[ P, L ]$ and $s[ J, V ]$ and $s[ V, L ]$ are _lightlike_ , $s[ N, B ]$ and $s[ B, Q ]$ and $s[ N, G ]$ and $s[ G, Q ]$ and   $s[ N, K ]$ and $s[ K, Q ]$ and $s[ N, V ]$ and $s[ V, Q ]$ are _lightlike_ , $s[ U, B ]$ and $s[ B, W ]$ and $s[ U, G ]$ and $s[ G, W ]$ and   $s[ U, K ]$ and $s[ K, W ]$ and $s[ U, P ]$ and $s[ P, W ]$ are _lightlike_ , the separations of all ten pairs among the events $A, F, J, N, U$ are _spacelike_ ,   the separations of all ten pairs among the events $B, G, K, P, V$ are _spacelike_ ,   the separations of all ten pairs among the events $C, H, L, Q, W$ are _spacelike_ , and finally the separations of all twenty remaining event pairs are _timelike_   ? Conversely, consider as possible illustration of "indication": (b)   Is there a _3+1 dimensional Lorentzian manifold with ~~everywhere vanishing Riemann curvature tensor_ (or, at least, one of its _charts_ ) which doesn't ~~ [Edit in consideration of 1st comment (by twistor59): -- _nowhere vanishing Riemann curvature tensor_ such that all of its _charts_ \-- ] contain   * twenty-four events, conveniently organized as four triples ($A, B, C$; $\,\,\,\, F, G, H$; $\,\,\,\, J, K, L$; $\,\,\,\, N, P, Q$) and six pairs ($D, E$; $\,\,\,\, S, T$; $\,\,\,\, U, V$; $\,\,\,\, W, X$; $\,\,\,\, Y, Z$; $\,\,\,\, {\it\unicode{xA3}}, {\it\unicode{x20AC}\,}$),   * where (again explicitly, please bear with me$\, \\!^*$): the sixty-six separations among the twelve events belonging to the four triples are exactly as in question part (a), each of the six pairs is _timelike_ separated, the separations of all fifteen pairs among the events $D, S, U, W, Y, {\it\unicode{xA3}}$ are _spacelike_ ,   the separations of all fifteen pairs among the events $E, T, V, X, Z, {\it\unicode{x20AC}\,}$ are _spacelike_ , $s[ D, {\it\unicode{x20AC}\,} ]$ and $s[ S, Z ]$ and $s[ U, X ]$ are _spacelike_ ,   $s[ E, {\it\unicode{xA3}} ]$ and $s[ T, Y ]$ and $s[ V, W ]$ are _spacelike_ , $s[ A, {\it\unicode{xA3}} ]$ and $s[ A, Y ]$ and $s[ A, W ]$ are _spacelike_ ,   $s[ A, {\it\unicode{x20AC}\,} ]$ and $s[ A, Z ]$ and $s[ A, X ]$ are _timelike_ ,   $s[ A, E ]$ and $s[ A, T ]$ and $s[ A, V ]$ are _timelike_ , $s[ C, {\it\unicode{x20AC}\,} ]$ and $s[ C, Z ]$ and $s[ C, X ]$ are _spacelike_ ,   $s[ C, {\it\unicode{xA3}} ]$ and $s[ C, Y ]$ and $s[ C, W ]$ are _timelike_ ,   $s[ C, D ]$ and $s[ C, S ]$ and $s[ C, U ]$ are _timelike_ , $s[ F, {\it\unicode{xA3}} ]$ and $s[ F, D ]$ and $s[ F, S ]$ are _spacelike_ ,   $s[ F, {\it\unicode{x20AC}\,} ]$ and $s[ F, E ]$ and $s[ F, T ]$ are _timelike_ ,   $s[ F, V ]$ and $s[ F, X ]$ and $s[ F, Z ]$ are _timelike_ , $s[ H, {\it\unicode{x20AC}\,} ]$ and $s[ H, E ]$ and $s[ H, T ]$ are _spacelike_ ,   $s[ H, {\it\unicode{xA3}} ]$ and $s[ H, D ]$ and $s[ H, S ]$ are _timelike_ ,   $s[ H, U ]$ and $s[ H, W ]$ and $s[ H, Y ]$ are _timelike_ , $s[ J, D ]$ and $s[ J, U ]$ and $s[ J, Y ]$ are _spacelike_ ,   $s[ J, E ]$ and $s[ J, V ]$ and $s[ J, Z ]$ are _timelike_ ,   $s[ J, T ]$ and $s[ J, X ]$ and $s[ J, {\it\unicode{x20AC}\,} ]$ are _timelike_ , $s[ L, E ]$ and $s[ L, V ]$ and $s[ L, Z ]$ are _spacelike_ ,   $s[ L, D ]$ and $s[ L, U ]$ and $s[ L, Y ]$ are _timelike_ ,   $s[ L, S ]$ and $s[ L, W ]$ and $s[ L, {\it\unicode{xA3}} ]$ are _timelike_ , $s[ N, D ]$ and $s[ N, S ]$ and $s[ N, W ]$ are _spacelike_ ,   $s[ N, E ]$ and $s[ N, T ]$ and $s[ N, X ]$ are _timelike_ ,   $s[ N, V ]$ and $s[ N, Z ]$ and $s[ N, {\it\unicode{x20AC}\,} ]$ are _timelike_ , $s[ Q, E ]$ and $s[ Q, T ]$ and $s[ Q, X ]$ are _spacelike_ ,   $s[ Q, D ]$ and $s[ Q, S ]$ and $s[ Q, W ]$ are _timelike_ ,   $s[ Q, U ]$ and $s[ Q, Y ]$ and $s[ Q, {\it\unicode{xA3}} ]$ are _timelike_ , and finally the separations of all ninety-six remaining event pairs are _lightlike_   ? (*: The two sets of causal separation relations stated explicitly in question part (a) and part (b) are of course not arbitrary, but have motivations that are somewhat outside the immediate scope of my question -- considering Lorentzian manifolds -- itself. It may nevertheless be helpful, if not overly suggestive, to attribute the relations of part (a) to "five participants, each finding coincident pings from the four others", and the relations of part (b) to "ten participants -- four as vertices of a regular tetrahedron and six as middles between these vertices -- pinging among each other".)


Electrostatics with Yukawa mass

I seem to have forgotten how to solve Laplace's equation when there is a Yukawa mass $\mu$. I want to find the Yukawa potential due to a homogeneously charged sphere of radius $R$ and charge density $\rho$. **My attempt** I need to solve $(\nabla^2-\mu^2)\phi = -\rho$ inside and outside the sphere. Since there is spherical symmetry, I can convert to spherical coordinates, and assume the potential $\phi$ has no angular dependence: $$ \frac{1}{r}\frac{\partial^2}{\partial r^2} (r\phi) - \mu^2 \phi = -\rho $$ Outside the sphere, there is no charge, so I solve the homogenous equation giving the solution: $$\phi_\text{out}(r) = A \frac{e^{-\mu r}}{r}+B \frac{e^{\mu r}}{r}$$ and inside the sphere, where there is charge density, I solve the inhomogenous equation, giving the solution: $$\phi_\text{in}(r) = \frac{\rho}{\mu^2}+C \frac{e^{-\mu r}}{r}+D \frac{e^{\mu r}}{r}$$ **Problem** I have four constants of integration ($A$, $B$, $C$, $D$) but only three pieces of information to fix them.   1. The potential should not blow up as $r\rightarrow\infty$: I must have $B=0$   2. It should not be singular as $r\rightarrow 0$: I must have $D=-C$. And therefore, $$\phi_\text{in}(r) = \frac{\rho}{\mu^2} + 2 C \frac{\sinh( \mu r)}{r}$$   3. Also, I must have continuity at the surface of the sphere $r=R$: Then I have $$A=\big(\frac{\rho R}{\mu^2}+ 2 C \sinh \mu R\big)e^{\mu R}$$ My potential has an undetermined constant $C$; What is the final piece of information that will fix $C$?


How to charge a field?

In a previous post [ Noether theorem, gauge symmetry and conservation of charge ] we were discussing the different ways to demonstrate the current conservation: via the first Noether theorem applied to a global $U(1)$ gauge symmetry, or via the covariant (or minimal, or Weyl, or ...) substitution with a $U(1)$ local gauge symmetry and antisymmetry of the electro-magnetic field applied to the equations of motion. I nevertheless still get confused about something: what is a charge ? More precisely, how to charge a field ? Indeed, if one believes that only the global gauge construction is able to demonstrate the conservation of the charge, one has to admit that the charge is not defined that way ! The same demonstration applies for the conservation of the particle number. Indeed, I used in [ Noether theorem, gauge symmetry and conservation of charge ] a Lagrangien for _uncharged_ particles to show the conservation of a particular current through Noether theorem. **So the question becomes: how to charge this particle ?** In particular: how to charge this particle and still conserve the global gauge ? NB: In the local gauge construction, it seems to be easier to give a charge to the system, but then only the second Noether theorem applies, and people get annoyed by that. **To conclude, a** (possibly important) **remark about my point of view:** I would like to understand the problem of "charging the field" in a condensed matter perspective, when I believe we have no a priori idea of the charge of the quasi-particles appearing in our effective theories. Can we overcome this problem ? Of course, any point of view (especially the opinion of QFT physicists who might have already tackled this problem) is warmly welcome :-) And the question "how to charge a field ?" is just a warm-up for this more complicated one "how to define the charge of an effective, emergent field ?" which is entirely subsidiary question for the moment. **IMPORTANT EDIT** I just became aware of this question [ How does non-Abelian gauge symmetry imply the quantization of the corresponding charges? ] which is strongly related, and well answered. I nevertheless think one can continue to discuss here about the condensed matter problem. Indeed, there are some people who believe that a superconductor (among other emergent phases of matter) have charge neutral excitations [see e.g. http://dx.doi.org/10.1103/PhysRevB.41.11693 which is too old to be on arXiv, but this one: http://arxiv.org/abs/cond-mat/0404327 rephrases it in a different way.]. So the question follows: how can these excitations at low temperatures become again charged at higher temperature ? How can we measure charge current from an uncharged excitation ? How can we "charge" a field after all ?


Is a scaled-up aircraft carrier 'ski-ramp' a viable system to impart enough velocity to significantly assist a spacecraft to orbit?

Not a boost straight up to escape velocity, just sufficient added momentum to make a significant economic saving on fuel cost, mass and complexity of a standard launch. It seems ludicrous to me how much fuel in a standard launch appears to be wasted pushing thousands of tonnes of rocket fuel in a vertical direction from a standing start. The system I envisage would have a horizontal track of possibly 2-3km (distance to be debated), acceleration via external systems (electromag or JATO?), continue up a ramp of slowly increasing gradient, with craft's own engines now running, and exit the ramp at between Mach 1 or 2. Safety run-off track if craft's own engines fail to ignite properly. Winged style craft for stability from ramp & shuttle-style landing. Similar I know to the Phys.SE 'Rail gun' question here, but acceleration could be more survivable. Had a longer script but it appeared to over-run the site limit. Acknowledge AdamRedwine's railgun question -would have added my contribution to that but my Reputation today is only 1. Acknowledge Gerry Anderson's Fireball XL5 puppet show on British TV in the '60s - the launch concept has always stayed with me! Submitted for criticism.


What causes stress concentration (aka stress risers/raisers) at corners?

I've read a few explanations about why stress concentration occurs at sharp corners but I don't find the explanations intuitive. Can anyone explain it perhaps using an analogy such as atoms "holding hands" with neighbor atoms or a similar easy to understand analogy? Most explanations I've shown just show the result of a finite-element-analysis based stress analysis. This shows the emergent property but doesn't explain the underlying mechanism that causes the stress to concentrate at corners. Better understanding the cause of the crashes of the De Havilland Comet jet aircraft due to metal fatigue at the corners of their rectangular windows is the context for this question. See: http://www.cracked.com/article_19623_6-small-math-errors-that-caused-huge- disasters.html The answer I'm looking for should not contain any math.


Estimate quarter mile time

I need to estimate a drag race quarter mile time given the car's weight, bhp and preferably the drive (FWD, RWD, 4WD). I know $v(t) = ds/dt$ and $a(t) = dv/dt = d^2s/dt^2$, but how can I get the function $s(t)$ to calculate $v(t)$ and $a(t)$? I thought maybe Power to Weight ratio would help for solving the acceleration but I have no idea what to do with it and maybe the drive would just be a constant which is then subtracted from the calculated time. Can you help me?


Why does vibration loosen screws?

I am trying to figure out why vibrations (say, from an engine) loosen screws. It seems to me that there is evident symmetry between loosening and tightening a screw. I am wondering what breaks this symmetry.


Uniform constant magnetic field and traditional attractive force

Why uniform constant magnetic fields can not exert net force on a piece of iron whatever strong it might get?


Derivation of relativistic energy

The concept of relativistic energy comes from it's conservation in relativistic mechanics for an elastic collision. It seems to me that another possible derivation could equate the energy of a single particle before, with the kinetic energy after of N particles after the energy is distributed amongst them in the form of the Boltzman distribution, as well as individually being of the form $mv^2$. Are there any problems in doing this?


physically modelling the Saltatory nerve impulse transmission?

The nerve impulse transmission is specifically a biophysical process. Under a _resting_ stage, the membrane is already polarised (presence of charge on either side leading to a potential difference across it, due to its finite capacitance). Changes in this polarisations mediated by several agents, cause some part of this membrane to be depolarised (inversion of the charges in that localised area). This changes the potential difference across that region and starts ion flow from the depolarised region of the membrane to the adjacent polarised region, inducing the same depolarisation there. Myelinating a neuron is similar to lowering the capacitance of the neuron. Myelinating the neuron causes subsequent action potentials to be spatially separated, and the conduction of voltage between them occurs primarily through ion flow along the neuron. Here are some nerve conduction basics. > Now my actual question. **Modelling a neuron as a simple one dimensional > _membrane/cable_ , how can a lowered capacitance lead to a faster conduction > of the voltage perturbation i.e a change in voltage (depolarisation), which > is initially limited to a small localised region? This conduction of the > voltage change can occur through the ion flow along the inner side of the > membrane and also due to long distance changes in potential due to the > altered distribution of charges along the membrane**. A very good physical > modelling of the neuron is given here and here, but because of the quite > complicated mathematical nature of the modified telegrapher's equation which > appears as the final answer, I am unable to understand how lowered > capacitance increases the speed of voltage conduction? Tell me if this question is off-topic or too biological to be answered here.


Help with a solid sphere sliding and then rolling

Initially at time $t=0$, a solid sphere slides with velocity $v$ along a horizontal surface. The coefficient of friction is u. Find the required time for the sphere to stop sliding (the sphere will then be rolling). Ive tried using Newtons equations of motion but I'm not sure what the final velocity will be? At first I thought it may be zero because it will be rolling but obviously it will still have some translational velocity. I've also found the work done by the friction on the sphere and taking this figure away from the initial kinetic energy leaving (what I thought) would be the rotational kinetic energy that the ball has. However I can't get to the answer. I'm almost certain I'm just missing a small trick but I have no idea where to go with it.


Force applied perpendicular to direction of motion, static or kinetic friction?

Let's assume we have a 1 kg cube on a flat uniform surface. The coefficients of friction between them are $\mu_k = 0.25$ and $\mu_s = 0.50$. This cube is moving at 1 m/s in the y direction, with whatever force is needed to overcome friction being applied. There is no force or motion in the x direction. Assume $g = 10\, {\rm m / \rm s^2}$ in the z direction. Now, if a force is applied in the x direction which type of friction applies, static or kinetic? Edit: To clarify, if the cube is moving forward, and gravity is acting down, the new force is being applied to the right. My rudimentary model of friction is that it acts like two pieces of sandpaper where the bumps can interlock when there is not motion, but when they are moving over top of each other they don't have time to settle in. In that model, I would think kinetic friction would apply, as the direction of movement shouldn't matter. However, I realize that in reality friction is a result of molecular attractive forces, and probably a lot of other more complicated things I'm unaware of. Because of this, I suspect the answer may lie somewhere between static and kinetic friction.


What made up light photons?

mass is energy per c square $m=E/c^2$ energy is made up of photons but what made up photon itself? what made up a single photon? * * * **Replay to comment:** but as we can see in history early phyisicists though that atom in non-fissionable (thats why they called it atom), now we call photon elementary but it may be a composite particle!, why not?


How do I intuit viscosity in a rotating fluid?

Suppose I have two plates with a viscous fluid in between. I slide them in the same direction (a direction in their own plane), one at $5 \,\text{m/s}$ and the other at $6 \,\text{m/s}$. Due to the relative velocity, there will be a viscous damping force tending to equalize the velocities of the plates. Now suppose I have two concentric cylindrical shells, the second with a radius $6/5$ that of the first. I put viscous fluid between them and rotate the entire system such that the cylinders have tangential velocities $5 \,\text{m/s}$ and $6 \, \text{m/s}$. Because the system is executing rigid rotation, there will be no viscous damping. Suppose I take a small cube of the viscous fluid in either scenario. In both cases, there's a similar velocity gradient across the cube, but only in one case does the cube dissipate energy. Why? I can work this out by writing the Navier-Stokes equation and transforming to cylindrical coordinates, but I am still having a hard time visualizing the kinematics such that this becomes clear to me.


Earth's rotation isn't that orbit?

If the earth is rotating at some $465~\text{m}/\text{s}$ at the equator and that's really fast.   1. Shouldn't we in that case be in orbit with the earth just not fast enough?    2. How fast do we need to move?   3. If the earth stopped rotating (gradually), shouldn't gravity increase significantly?   4. Do satellites need extra speed when they are released or does it just depend on their inertia from the earth's rotation?   5. If the satellite decelerated against the earth direction of rotation will it fall?    6. If the earth rotated extremely fast, shouldn't the earth's crust along with everything else get ejected away by the centrifugal force? I'm sorry if the questions cover many matters but I preferred to ask them all at once instead of asking each separately.


Maximizing Multiplicity of Einstein Solid == (Temperature = $\infty$)?

If I have a system consisting of 2 Einstein solids (`A` and `B`) is it equivalent to say that maximizing the multiplicity of the system is the same as setting the temperature to $\infty$? I have two reasons to support this Idea. 1) The total multiplicity of the system can be expressed as: $$\Omega_{tot} = \Omega_A(q_A,N_A) \Omega_B(q_B,N_B)$$ Which is the product of the individual multiplicities, and $q = q_A + q_B = $ Total quanta of energy shared between the system, and $N = N_A + N_B =$ total number of quantum harmonic oscillators. I wish to maximize the multiplicity with respect to oscillator `A`'s total internal energy ($U_A)$ $$\left( \frac{ \partial \Omega_{tot} }{ \partial U_A } \right) = 0 \implies \left( \frac{ \partial Ln( \Omega_{tot} )}{ \partial U_A } \right) = \frac{ \Omega_{tot}'}{ \Omega_{tot}} = 0$$ This is true because Ln(x) is a monotonic function. But it should be noted that the definition of temperature is such that: $$\frac{1}{kT} = \left( \frac{ \partial Ln( \Omega_{tot} )}{ \partial U_A } \right) = 0 \implies T= \infty$$ `k` here is the boltzman constant. 2) This makes some intuitive sense. On the Gaussian distribution as a function of Internal energy, the maximum multiplicity lies on the peak. The change in temperature gives a quantification of the direction of heat flow. If the maximum multiplicity is attained, then heat cannot flow any higher, it can only flow downward. This corresponds to infinite temperature. I hope my intuition here is correct.


Time-reversal symmery and topological insulators

I have a very naive question about the notion of time-reversal symmetry applied to topological insulators that are studied in experiments. If I understand correctly, the exsistance of time-reversal symmetry is crucial for the occurence of topologically protected surface states in topological insulators. At least this is what the theory of topological insulators tells us. But I have a problem with applying this theory to real materials. In real world there is always dissipation and therefore there is an "arrow of time" and time-reversal symmetry does not hold (right?). So how can we talk about time-reversal symmetry in real materials that we call topological insulators? Please help me find where I went wrong in my argument. I realize that there must be something wrong but can't figure out what exactly.


What happens to this potential energy?

Let's say I turn on a Van de Graaff which creates a large positive charge. Now let's say I have an object with a positive charge in my hand and I start walking toward the Van de Graaff from $x$ meters away. If I walk an arbitrary distance towards the Van de Graaff, I am doing work and the charge in my hand is gaining potential energy. Now somebody standing next to the Van de Graaff suddenly turns it off. What happens to that potential energy of the charge? By the conservation of energy it should be conserved somehow, but it appears not to be. Any help is appreciated.


Non-uniform lens flares

I assume -- and maybe wrong -- that lens flares, in photography, are caused by light getting "trapped" inside the curved lens and bouncing around (due to refraction and non-perfect total internal reflection) causing bright spots at periodic intervals, when they partially escape on the imaging side. This would give the classic circular(ish) spots when a camera pans up towards, say, the sun, where the shape of the spots mimics that of the lens/lenses and aperture. However, I was looking at the Hubble Deep Field image and noticed that a few of the brighter stars have _only_ horizontal and vertical flares. Similarly, stars (as in the light source) are often drawn as _n_ -pointed polygons, implying distinct flares at regular angles. What is it about the lens, in these cases, that causes flares to propagate seemingly non-uniformly or directionally?


Calculating vector components from other vectors

I'm busy studying for my Physics exam this evening and I've come across a vector problem that I cant quite solve. Any help would be appreciated! We are given the following vectors:               A = 5i - 6.5j     B = -3.5i + 7j      We are also told that a vector C lies in the xy-plane and is perpendicular to A. The scalar product of B and C is equal to 15. With this information we have to calculate the components of the vector C. I have tried calculating the magnitude of B and then trying to solve the equation:               |B||C|cos{theta} = 15      But I didn't get very far. I have a feeling I'm just not seeing something and making a careless error. Thanks in advance!


Kinematics with non constant acceleration

A particle experiences an acceleration described by $$ a=kx^{-2} $$ where x is the displacement from the origin and k is an arbitrary constant. To what value does the velocity v of the particle converge to as x approaches infinity if the particle starts at some point x0? If I approach this problem with energy, then $$ W = \int F \mathrm{d}x $$ $$ = \int_{x_0}^\infty mkx^{-2} \mathrm{d}x $$ $$ = mk(-\infty^{-1}+{x_0}^{-1}) $$ $$ W = K = mk{x_0}^{-1} $$ $$ \frac{1}{2}mv^2 = mk{x_0}^{-1} $$ $$ v = \sqrt{2k{x_0}^{-1}} $$ How would I solve this problem with pure kinematics? (there appears to be some sort of cyclical dependency where acceleration affects velocity, velocity affects displacement, and displacement affects acceleration) Likewise, two particles experience accelerations described by $$ a_1=k_1x^{-2} $$ $$and $$ $$ a_2=-k_2x^{-2} $$ where x is the distance between the two particles What two velocities do the particles reach as x approaches infinity if the two particles are initially separated by some x0?


How two photons interfere in a double slit experiment

I have a very basic question about photons in double-slit experiment. I am not good at math, and have some Quantum mechanics knowledge. Math free explanation would be very good to me. When we do double slit experiment with single photon at a time, it will be detected at some location of screen. Over the time period the accumulation of detected locations show interference pattern. The reason for the interference pattern is that photon's location in not well defined, the presence of photon in particular place can be determined by prbability and this probability is presented as a wave. that probability wave is splitted in to two while passing the two slits, collide with each other and causes interference pattern. Question 1: Is this reason for single photon interference correct? Question 2: How is the probability wave behaving while two-photons passing double-slit at a time? Are 4 probability waves (2 for each) interfering?


What will happen after escaping earth's gravitational field?

Suppose that I escaped the gravitational field of earth. Then: am I going to be pulled by Sun's gravity?


Entropy is constant. How to express this equation in terms of pressure and density?

In hydrodynamics of an ideal, non-compressive flow we use 5 variables: pressure $p$, density $\rho$ and velocity field $\mathbf{v}$. So we need 5 equations. Landau's "Hydrodynamics" states that the equations are:   1. The mass continuity equation $\frac{\partial \rho}{\partial t}+\nabla (\rho \mathbf{v}) = 0$   2. The Euler equation (3 components) $\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v}\cdot \nabla)\mathbf{v} = -\frac{1}{\rho}\nabla p$   3. A statement of the fact, that there is no dissipation of energy $\frac{d s}{d t} = 0$ ($s$ is entropy per unit mass) My question: how to express the last equation in terms of the actual variables we are using? We have to assume some form of the equation of state for the fluid in order to do it right? Landau in his typical fashion glance over it, assuming the readers' perfect understanding of thermodynamics, which is not my case unfortunately. PS. The question is a little related to a more general one I posted yesterday: Explicit form of the entropy production in hydrodynamics


What frame(s) of reference are used to measure the rotation of the Sun around the galaxy ?

I can find various speeds and estimated durations listed at numerous places but none specifically describe the frame of reference. Possible options as example of kind of answer I expect.   * Local Galactic cluster    * Distance quasars    * The cosmic background radiation? **\--------- UPDATE ---------** Thanks AIB and voithos. Lot of reading for me. Though technically, I still don't have an answer that meets the following criteria.   1. rotational velocity(average preferably) of sol around best estimate of center of Milky-way galaxy.    2. publicly available reference(I don't have immediate access to some of the books given)    3. frame of reference external to Milky-way galaxy. As I note below the only reference (wmap5basic_reprint.pdf) I can read that uses an external frame of reference doesn't specifically state the vector is rotational (despite wikipedia article assuming such). The topic is barely touched on in that paper. I realised the speeds are variable. What I had not realised is that the whole idea of a (relatively) clearly defined x orbiting y system doesn't really scale up well from the local solar system to the galactic scale. The galaxy is more like a whirlpool or tornado compared to the "clockwork" appearance of the solar system. Although both the solar system and galaxy are constantly(very slowly) changing "fluid" rotational systems, the galaxy is obviously far more fluid than the solar system. Also we have not yet been able to observe anything about it's center. Or in other words, we are not "orbiting" the galaxy, we are part of the galaxy. I suspect the topic is more in the realm of "fluid dynamics" than "orbital mechanics" I've accepted AIB's answer as the most enlightening to me personally. Also, it would appear I have wiki-sidebar blindness. Apologies for that. FYI   The paper referencing the speed relative to the CMB, as mentioned in wikipedia article can be found here http://cmbdata.gsfc.nasa.gov/product/map/dr3/pub_papers/fiveyear/basic_results/wmap5basic_reprint.pdf The relevant section appears to be 7.3.1. "... implies a Solar System peculiar velocity of 369.0 ± 0.9kms-1 with respect to the CMB rest frame." Although it's not obvious to me what vector that velocity is along. Though Dipole Anisotropy in the COBE DMR First-Year Sky Maps gives a specific velocity(including vector) for the local galactic group in relation to the CMB rest frame "implied velocity of the Local Group with respect to the CMB rest frame is 627 +/- 22 km/s toward (l,b) = (276 +/- 3 deg, 30 +/- 3 deg)." FYI   Other reference frames that are external to the local galaxy are "The Supergalactic coordinate system"


Proton gas density

As far as I know the lightest gas is hydrogen due to low mass of its nucleus, but what if we were to somehow strip hydrogen atoms of electrons and enclose protons in a container made of teflon (high electronegativity). Would such a proton-only gas be stable? And what would be its density under atmospheric pressure (assume teflon envelope around the gas)? Would electrostatic effects between positively charged protons contribute to lowering density of such gas?


Would gravity on the surface of a planet which is being consumed by a black hole change?

Assuming that the black hole starts grow in the exact center of the planet and that the general structure of the planet does not degrade as it is eaten from the inside, would the gravity on the surface of the planet be affected by the black hole growing. My uneducated guess is that as the matter becomes more compacted the gravity on the surface would actually decrease due to the fact that the distance to the mass, with respect to the surface, is increasing. Also would the evaporation of the black hole actually decrease the total amount of mass in the center of the planet?


Why do we need 2. Quantization of the Dirac Equation

As a Mathematician reading about the Dirac equation on the internet, leaves me with a great deal of confusion, about it. So let me start with its definition: The **Dirac equation** , is given by $ i \hbar \gamma^\mu \partial_\mu \psi = m c\cdot \psi $ where the Dirac matrices $\gamma^\mu$ are defined by $\gamma^\mu\gamma^\nu + \gamma^\nu\gamma^\mu = \eta^{\mu\nu}$ and where $\psi$ is a "solution". Ok, the first deal of confusion already starts with the $\psi$'s. It seems that people freely see them as spinor valued functios or as "operator fields". But if I understand this correctly, seeing them as operators, is not part of the original picture, but was later added as the so called **second quantization**. Right? Now my question is the following: Why do we need this second quantization of the Dirac equation? What experiments can not be described by the original Dirac equation? Maybe there is a list somewhere or such.


Rotational symmetry in integration

Can someone please tell me why $$4\int d^4x \, x^\mu x^\nu ~=~\int d^4x \, g^{\mu\nu}x^2 $$ by some rotational symmetry argument?


Purcell and the Magnetic Field

When showing that the magnetic field derived from the curl being proportional to current and the divergence being zero is unique, Purcell says that at some sufficiently remote enclosing boundary the difference of two magnetic fields (two possible fields from the curl), D, would take a constant value. Why is this true?


If the quarks in a neutron are (up,down,down), why isn't it negatively charged? Excuse the silly question, just wondering.


spread of fock state distribution and infinite revival time of rabi oscillation in spontaneous emission

In cavity QED for a 2-level atom, the revival time for oscillation b/w the states $\left|\ e\ 0\right\rangle$ and $\left|\ g\ 1\right\rangle$ (absorbing the same photon that is emitted) is said to be infinity because in general, Rabi oscillation revival time is proportional to the square root of the mean of the occupancies of Fock states. Why should the spread of the Fock states distribution go to large values or close to infinity in the spontaneous emission case, to make the revival time infinity ? Refer :- http://prl.aps.org/abstract/PRL/v94/i1/e010401 for a description of Rabi Oscillations Revival


Lecture Notes confusion: Constructing the Einstein Equation

This question is on the construction of the Einstein Field Equation. In my notes, it is said that > The most general form of the Ricci tensor $R_{ab}$ is > $$R_{ab}=AT_{ab}+Bg_{ab}+CRg_{ab}$$ where $R$ is the Ricci scalar. Why is this the most general form (involving up to the second derivative of the metric --- by definition of $R_{ab}$)? I suppose there are symmetry and degrees of freedom arguments. But I can only see why the LHS is a _possible_ form. I can't se why it is the most general form... > Taking the covariant derivative $\nabla_a$ of the expression above gives > $$C=\frac{1}{2}$$. This I understand. > Compare the resulting expression with the Poisson equation gives > $$A=\frac{8\pi G}{c^4}$$. This I _don't_ understand --- perhaps I am being silly again... but still. I assume the "Poisson equation" referred to here is $$E^i{}_i=4\pi\rho G$$ where $E^i{}_i$ is the tidal tensor and may be expressed as $$E^i{}_i=R^i{}_{aib}T^aT^b$$ where $T^a$ is the tangent vector of the geodesic. So contracting $$R_{ab}=AT_{ab}+Bg_{ab}+CRg_{ab}$$ with $T^aT^b$ gives $$4\pi\rho G=AT_{ab}T^aT^b$$. But then what? Or perhaps I have already made a mistake? I only know the energy-momentum tensor $T_{ab}$ to be of the form $$\begin{pmatrix}H&\pi_i\\\\\frac{s_i}{c}&T_{ij}\end{pmatrix}$$ where $H$ is the energy density, $\pi_i$ is the momentum density, $s_i$ is the energy flux. But I don't understand how it leads to $$A=\frac{8\pi G}{c^4}$$.


Hollow gold bar

A scammer got a hollow gold bar and fills it with a combination of lead and air, with the same average density as gold. What's the simplest way of discovering the fraud? I know that x-rays will see it, but are there simple means for analyzing it? (without destroying it)


Conservation of momentum leading to damage

What would be an intuitive way to damage objects in a physics game using impulses? Since momentum is conserved, so is impulse (the change in momentum for any two time periods) in a closed system. So if I have the impulse of a collision (that is, the change in momentum during a time interval for one of the two bodies) what would be the intuitive way to damage both the objects. Should each object receive the same damage based on the impulse of the overall system, or should the damage incurred be split unevenly based on the relative mass of each object? My intuition is that objects with larger mass would have higher hit-points, and so would be less affected if the same damage was incurred to both the objects in the collision.


Equation that tells me the rpm and mass of a spinning disk needed to keep a second large mass stable using gyroscopic effects

I am trying to figure out how large of a mass and how quickly I need to spin said mass to keep a two-wheeled robot stable. Ideally, I am looking for a formula that relates _m_ 1=mass of robot, _m_ 2=mass of spinning disk, _v_ =rotational speed of disk (rpm), and some sort of stability factor _s_ (or amount of mass _m_ 1 that it can counteract). Is there any such formula, and what would it be? Thanks!


Making an equal amount of positive and negative energy?

I was wondering if it would be possible to create an amount of positive energy out of a vacuum, in addition to an equal amount of negative energy, thus not violating the first law of thermodynamics, since the total amount of energy in the universe would remain the same. It could be like an "energy loan," with the positive energy being like the "loan money," and the negative energy being like the "debt." Is this theoretically (or even practically) possible? For example, I am asking if you could create 200 units of positive energy out of a vacuum, as long as you also created 200 units of negative energy too, so as not to violate the second law of thermodynamics. By negative energy, I mean energy that is below 0 degrees Kelvin. I came across the idea as I was reading about the zero-energy universe theory. It suggested that this mechanism was what made the universe, and that it could have been started by an unusually large quantum fluctuation. Does this idea conflict with the second law of thermodynamics in any way? Would entropy have to be considered, since negative energy reduces entropy, violating the second law? If so, is there a way of changing the idea a bit so there as to avoid the problem? Could "quantum interest" account for the missing entropy? Do any restrictions apply on the negative and positive energy, such as quantum inequalities? Can the positive and negative energy be separated? Is there a way to get around the restrictions, if there are any? Finally, if all of this is possible, could it be done? Would a possible method be to recreate an unusually large quantum fluctuation, as described in the second paragraph? Are there other ways of making the "energy loan"?


Is Ehrenfest theorem equivalent to Bohr's Correspondence Principle?

Ehrenfest theorem is usually dubbed as the quantum mechanical equivalent of Newton's law and Griffiths states, in the first chapter of his textbook, that Ehrenfests theorem enables us to work with expectation values in quantum mechanics as _Ehrenfest theorem tells us that expectation values obey classical laws._ On the other hand, the correspondence principle states that _the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers._ Are the two logically equivalent? References: http://farside.ph.utexas.edu/teaching/qm/lectures/node31.html


Functional integral aproach for Feynman rules

I am familiar with the basic ideas of quantum field theory but I feel uncomfortable when I have to derive Feynman rules by myself for a given action (for example in non-linear sigma models or electrodynamics). Could you provide a list of good books, articles, lecture notes and/or any other sources (except Peskin), where Wick's theorem and the Feynman rules are derived from the functional integral **in detail** , preferably with examples?


Gravity force and dark energy

If gravity is a fundamental force which bends spacetime and dark energy is energy which stretches spacetime, what is the difference between the terms force and energy?


dual variables for lattice fermions

I am quite familiar with duality transformations for lattice spin systems (i.e. systems with global $O(n)$ symmetry) and pure gauge systems (i.e. local $SU(n)$). However, after searching for a bit, I can't seem to find literature on dual variables for lattice fermions (systems with inner products of grassmann fields). For the spin and gauge cases we do harmonic analysis and fourier expanded the exponential of the action (or a character expansion for the gauge case). Is there an analogous process for lattice actions with grassmann fields? Thanks.


Why does Joule heating not occur when no current flows through a conductor?

Joule heating happens every time when the conduction electrons transfer kinetic energy to the conductor's atoms through collisions, causing these conductor's atoms to increase their kinetic and vibrational energy which manifests as heat. Then, why wouldn't it happen when no current is flowing through the conductor? When there is no current, the electrons are still moving randomly at a speed of $\mathrm{~10^5\ m/s}$, but at a zero average velocity. Then, why don't these electrons collide with the atomic ions making up the system and transfer energy to them causing them to heat up even when no net current is flowing?


String vibration

In many textbooks an illustration of a vibrating string at a fundamental mode shown (and wikipedia) shown like this http://en.wikipedia.org/wiki/File:Standing_waves_on_a_string.gif). However if you watch youtube videos of string vibrating in slow motion you see something like this (http://projects.kmi.open.ac.uk/role/moodle/pluginfile.php/1239/mod_page/content/1/ta212_2_008i.jpg) So which one is this? Is the first one just a schematic version of the second? Also the notion of a wave propagation in a string sort of relevant only to the second pictire only? Or am I wrong? Please clarify, I think I am missing something fundamental here. Thanks


Focusing laser & off-axis illumination & diffraction limit

I've thought I had a good understanding how resolution enhancement tricks works for projection lithography, until I tried to understand if it's possible to get sub-diffraction performance for focused laser beam: Let's assume we have a laser with nice and shiny perfect aspherical focusing lens (with performance greatly exceeding diffraction limit for it's NA). This system moves around and draw stuff. The question is - can we get enhanced resolution if we apply usual lithography trick of annular/quadruple illumination? Obviously, we won't reduce aberrations thanks to off-axis illumination, as there are none.


Casimir Forces and its associated Feynman Propagator

This is a continuation to my previous question, in which I began an attempt solve the Casimir Force problem using path integrals. As one of the answers there suggest I solve the Feynman propagator subject to the boundary conditions $x=0$ and $x=L$ at the plate boundaries. The equation for Feynman propagator is $$ (\Box^2+m^2)\Delta_F(x-x') = -\delta(x-x') $$ The solution to the free field is $$ \Delta_F(x-x') = \lim_{\epsilon\rightarrow0}\int \frac{d^4p}{(2\pi)^4}\frac{e^{ip_{\mu}(x^{\mu}-x'^{\mu})}}{p^2-m^2+i\epsilon} $$ What would be the boundary conditions that I have to exactly impose ? Imposing a boundary condition would mean, I think we might have to introduce the a new function (I don't if am right, but this is in general true for Green's function I guess) $$ \Delta_F(x-x') \rightarrow \Delta_F(x-x') + F(x-x') $$ where $F(x-x')$ is such that it satisfies the Boundary condition. Now my question is in case I have boundary condition (like below) how do I solve the differential equation for the boundary conditions like, (take plates to be at $z=0$ and $z=L$) $$ \Delta_F(x-x')\bigg|_{z=0} = \Delta_F(x-x')\bigg|_{z=L} = 0 $$ EDIT 1: It just occurred to me that there might be short route to this problem with some conceptual reasoning, I gave this a try.. Considering the region between the plates, I know the momentum is quantised in the z-direction, so I have (which is some sense imposed by the boundary condtions) $$ p_z = \frac{n\pi}{L} $$ Now using the Feynman propagator in momentum representation, which is $$\widetilde\Delta_F(p) = \frac{1}{(p^0)^2-(\textbf p^2+m^2)+i\epsilon}$$ In this I can substitute for, $p_z$, which will give me $$ \widetilde\Delta_F(p) = \frac{1}{(p^0)^2-(p_x^2+p_y^2+\Big(\frac{n\pi}{L}\Big)^2)+m^2)+i\epsilon} $$ Now can I get back to position representation, but with integral on $p_z$ replaced by a sum over $n$. Am I right in doing this procedure ? EDIT 2 : Following the procedure that I have mentioned, for a simple (1+1) case of the Feynman propagator in position representation, I have $$ \Delta_F(x-x') = \sum_{n=1}^\infty\int\frac{dp_0}{(2\pi)^2}\frac{e^{ip_0(x^0-x'^0)}e^{i\frac{n\pi}{L}(z-z')}}{(p^0)^2-\big(\big(\frac{n\pi}{L}\big)^2+m^2\big)} $$ EDIT 3 : $$ \text{Tr}\log{\Delta} = - \sum_n \int dp_0 \log{\bigg(p_0^2 - \bigg(\frac{n\pi}{L}\bigg)^2 + m^2\bigg)} $$ But this term seems to diverge, how does one obtain a cutoff in the context of this problem. (A cutoff for $p_0$ integral is also needed I guess).


Who is right? Force exerted on a body. Me or my friend?

> Three masses $m_1 = 3\text{ kg}$, $m_2 = 9\text{ kg}$ and $m_3 = 6\text{ > kg}$ hang from three identical springs in a motionless elevator. The > elevator is moving downward with a velocity of $v = -2\text{ m/s}$ but > accelerating upward with an acceleration of $a = 5\ \mathrm{m/s^2}$. (Note: > an upward acceleration when the elevator is moving down means the elevator > is slowing down.) What is the force the bottom spring exerts on the bottom > mass? Take $g = 10\ \mathrm{m/s^2}$. **My argument** The elevator is steadily slowing down and it get an upward acceleration. So as it goes down the net force is $$\begin{gather}F_s - mg = ma \\\ F_s = 6(10 + 5) = 6(15) = 90\text{ N}\end{gather}$$ But this is the force the spring has. It is the same as the force EXERTED on the body attached? Is there a Newton's third law involved? **Friend's argument** He says that BECAUSE it has an upward acceleration and it is going down that we need to be concern with $$Fs = mg - ma = 90 - 60 = 30\text{ N}$$


Using Quantum Teleportation in a way to have the effect of matter teleportation

So, I understand that quantum teleportation is the transfer of a quantum state from one particle or system of particles and its correlations to another receiving system consisting of particle(s). Can't this be used in a way to achieve teleportation as depicted in sci-fi? You start out with a translationally entangled particle pair (position and momentum are correlated) as described by this paper: http://pra.aps.org/abstract/PRA/v61/i5/e052104 Now according to these papers, you can theoretically teleport the position and momentum information of particle. Atomic teleportation of the external degrees of freedom, (Their position and momentum) http://pra.aps.org/abstract/PRA/v49/i2/p1473_1 http://prl.aps.org/abstract/PRL/v86/i14/p3180_1 http://iopscience.iop.org/0295-5075/75/6/847 Firstly, I need someone to help me understand **exactly what is being teleported in regards to the papers I've listed**. I understand completely the discrete case of quantum teleportation with spin or polarization, I'm not asking for that. This is what I know: 1) Start out with a translationally entangled pair. 2) Interact an "input" particle with one of the pairs. 3) Make a measurement of the input particle with one of the pairs' position and momentum. (This is the step I don't really understand) But I know that this is analogous to the Bell state measurement in discrete quantum teleportation that is widely described everywhere. 4) This result is communicated to the other laboratory where appropriate "shifts" of position and momenta are done to the entangled pair just like in the discrete case again. So, if position and momentum information is teleported, then does this mean that if the input particle was propagating in the X-direction like a wave- packet, then after teleportation, the receiving particle will now move in the X-direction relative to its original position? This is quite confusing to me. I want to understand this because I want to ask the trillion dollar question, "What if you are to replace the input particle with a input MOLECULE?" For a diatomic molecule, its simply two atoms of the same type. According to this wikipedia article: http://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals I know there are vibrational states in molecules which ARE essentially position and momenta information right? If we are to set up TWO translationally entangled pairs of atoms of the same type... And have them simultaneously interact with the molecule in step 3), _then can't we do the appropriate SHIFTS of the entangled pairs of atoms to make them turn into the input molecule?!_ So, the overall effect is that if you start out with H2 in Lab A, and had a translationally entangled pair for each atom, then you'll end up with H2 in Lab B after teleportation which is exactly what we want if we want to stick close to the sci-fi sense. If what I am saying makes sense, then can't this be in principle be extrapolated to larger molecules and eventually cells, and organs, to an entire human? I know that this is a large jump... However, what I'm trying to say is that according to this kinda scheme I'm asking about, its sticking to real physics, and its essentially accomplishing teleportation in the sense people know about. Do I make sense?


A few questions related to frame dragging

I am trying to get my head around a few concepts related to frame dragging and related physics. In regards to black holes that have no charge and all their mass is tied up in rotational kinetic energy then the ergosphere is a maximum size. By definition, as I see it defined, the outer boundary of the ergosphere is the point at which a photon becomes stationary with respect to outside observation if it is **orbiting** retrograde. The radius at the **equator** of this maximal case is $2GM$ making it twice the inner event horizon radius if I am understanding correctly. The innermost stable orbit for massive particles is at GM in a pro-grade direction. If any of this is incorrect please let me know. So my questions are: >   1. Pretty much by definition it is impossible for a massive particle to go > slower than the speed of light relative to outside observers if it is within > the ergosphere - true? What is the limit of this speed relative to outside > observers? I'm looking for the limit before it passes into the inner > horizon. > >   2. Since the frame dragging extends outward (presumably to infinity with > extremely negligible effect) what would happen in the case where you had a > ring of orbiting black holes equally spaced with spin aligned such that the > axis of rotation was tangential to the circle? Without touching event > horizons it would appear to be able to pull massive particles through the > center of the circle in a straight line rather than a curved path as above - > correct? And if the holes were close enough would that allow the ergospheres > to merge (temporarily as I'd assume everything would merge eventually) would > that allow faster than light travel for massive particles in a straight > line? I'm neglecting that the orbits of the holes are unstable and would > radiate gravitational waves and that they are massive enough so tidal forces > are 'small' compared to the particle or mass of particles. > >   3. If the holes above are arranged as stated would this extend the > ergosphere inward toward the center if they are closely spaced at some > distance? I've seen cases for general non-maximal spin cases of two holes > and am unable to determine how this would play out. > > Thanks for taking the time to read and if I put too much in a single question I apologize but didn't want to spam several closely related questions.


Where is the Higgs Field?

Where is the Higgs Field? I get the idea of how it works but I couldn't find anything explaining where it is.


what kind of system respects $SU(N)$ symmetry?

I read this post, Is the symmetry group of two spin 1/2 particles $SU(2) \times SU(2)$ or $SU(4)$? If the picked answer is correct, can I believe that an $N$-degenerate system respects $SU(N)$ symmetry? But if so, I am confused in this way: can we understand that a spin $j$ particle contains $2j+1$ internal degenerate states? Then the symmetry of this particle is $SU(2j+1)$, right? If $j>2$, then it is larger than $SU(2)$! I must be wrong somehow since I know little about group theory. So, basically, what kind of system respects $SU(N)$? * * * **Update 1** Since I have not enough reputation, I could not add comment on the linked post, which I am still confused, so I may also post here(related to this issue): There are $M\geq 2$ particles, if each particle respects $SU(N)$ symmetry, and if   * A. there is no interaction between them, then what is the symmetry group, if   *     * A1.they are non-identical particles?     *     * A2. they are identical particles?     * B. there is pairwise interaction between them, and the interaction is $SU(N)$ invariant,   *     * B0. what do I mean by $SU(N)$ invariant?     *     * B1. symmetry group for non-identical case?     *     * B2. for identical case?


$t_1$, $t_2$, $t_3$ Hermitian generators of $SU(2)$

What is the exact $SU(2)$ representation to which these Hermitian generators belong? \begin{equation} t_a=\\{t_1,t_2,t_3\\}=\left\\{\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\\ 1 & 0 & 1 \\\ 0 & 1 & 0 \end{pmatrix},\,\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -i & 0 \\\ i & 0 & -i \\\ 0 & i & 0 \end{pmatrix},\,\begin{pmatrix} 1 & 0 & 0 \\\ 0 & 0 & 0 \\\ 0 & 0 & -1 \end{pmatrix} \right\\} \end{equation} I'm a bit puzzled with this, these are not the generators of the triplet representation of $SU(2)$ (the triplet in $SU(2)$ is the adjoint rep which is real and constructed with the structure constants), nevertheless they are used as if they were in much literature. What are really these generators? They seem like the extension to 3 dimensions of the Pauli matrices. The 3 dimensional fundamental rep (this does not make sense to me)? some kind of non-irreducible representation?


Help with calculating resistance at a given temperature

> The resistance of a bulb filament is $100\Omega$ at a temperature of > $100^\circ \text{C}$. If its temperature coefficient of resistance be $0.005 > \space \text{per} ^\circ \text{C}$, its resistance will become $200\Omega$ > at a temperature of > >   1. $300^\circ\text{C}$ >   2. $400^\circ\text{C}$ >   3. $500^\circ\text{C}$ >   4. $200^\circ\text{C}$ > Now the linear approximation is $R_t=R_0(1+\alpha T)$ Therefore \begin{align*} 100\Omega &= R_0(1+0.005\times 100) \\\ \therefore R_0 &= \frac{100}{1.5} \\\ \text{Now } 200 &= \frac{100}{1.5}(1+0.005\times t_2) \\\ 0.005\times t_2 &= 2 \\\ t_2 &=400^\circ\text{C} \end{align*} But there is also this formula $\alpha=\frac{R_2-R_1}{R_1(t_2-t_1)}$ \begin{align*} 0.005 &=\frac{200-100}{100(t_2-100)} \\\ t_2 &= 1/0.005 + 100 \\\ t_2 &= 300^\circ\text{C} \end{align*} Why are these answers differing?


Why do smaller objects become harder to break?

When grabbing a typical tree branch of at least two feet, it's so easy to snap with a less than one inch circumference that even a toddler can do it. However, after breaking it, the smaller halves with multiply from the center, going like this: 1 - BB (before break) 2 - AB (after break) 4 - AB (half of second break equals four) 8 - AB (continuing multiplication factors of two) And so on.... But what I want to know is why this is ... in quantum physics. I want to know why the more you break it, the harder it is to keep breaking it from the center of each half. This applies to tearing as well ... if you tear a cardboard box in half down the middle it's easy, but if you turn it over and continue on it keeps getting harder as the pieces get smaller.


Why is optical orbital angular momentum (OAM) called "topological charge"?

The terminology "topological charge" is frequent in lots of research papers related to optical vortex or optical OAM, it is used to represent the optical OAM. Why? How to comprehend it?


Why doesn't a bike/bicycle fall if going with a high speed?

Why does a bike/bicycle fall when its speed is very low or close to zero and is balanced when going with a high speed?


Why does a bicycle (without any support of stand) falls down being at rest, but not under motion?

I have always seen a bicycle not standing without any support, it either falls down to the right or to the left, may even to some other direction. But,the same two wheeler when under motion,moves balanced, without falling to either side,even it not falls, when the two wheeler is bent to left or right by the driver, like the one which we could see in the bike races. **So, what makes the bicycle at rest to fall? And what makes it not to fall when under motion?** _MY VIEW ON THE CONCEPT_   If the bicycle is at rest, it will be acted upon by downward force ($mg$,where $m$ is mass and $g$ is acceleration due to gravity). If it is affected just by this force, it should have stood without falling, because there is no force which pulls it to either side to make it fall. I think, it might fall, either due to the direction of wind, or due to the nonuniform distribution of mass, or due to other external forces. If we consider the same bicycle to be under motion, I don't know what makes it not to fall,even affected by nonuniform distribution of force,or direction of wind. I think there might be something related with vectors, angular momentum,and forces like centripetal and centrifugal. (What ever I said, is just my opinion about the concept, I don't claim it to be true. Correct me,if I am wrong anywhere.)


Why do we fall when we ride bicycle slowly but we don't when riding it fastly

> **Possible Duplicate:**   >  Why doesn't the bike fall if going with a high speed? Why is it that, when we ride bicycle at good speed, it is perfectly balanced and we don't fall. But we lose balance when we ride bicycle at lower speeds.


How do bikes stay upright?

> **Possible Duplicate:**   >  Why doesn't the bike fall if going with a high speed? How does a bike stay upright? I know there are various theories involving castor effects, and gyroscopic motion but as far as I'm aware no one knows for certain how a bike stays up right. What is the best theory right now?


Is Conformal Symmetry Local or Global?

I'm just brushing up on a bit of CFT, and I'm trying to understand whether conformal symmetry is local or global in the physics sense. Obviously when the metric is viewed as dynamical then the symmetry is local, because essentially then we're dealing with a change of variables under which the metric transforms with a local scale factor $\Omega = \Omega(x)$. Usually, however, we think of the metric as fixed. David Tong's excellent notes suggest that in this case the symmetry should be thought of as global. But I'm not sure I agree. Say we work in 2D and have general conformal transformation given by holomorphic function $f(z)$. Under a conformal transformation $z\to w$ say, viewed actively, a general field $\Phi$ will transform to $$(\frac{dw}{dz})^h \Phi$$ where the prefactor is clearly dependent on the spacetime point. This would suggest that the transformation is local in the physics sense. Perhaps the distinction he is trying to make is between physical and gauge transformations. But then again I might be wrong because I thought that only global transformations had nonzero conserved quantities, and there's definitely a conserved current for conformal symmetry. Could anyone help to clarify this for me?


What is $\langle \sigma_\mu \rangle$ $\langle \sigma_\mu \rangle$ for the Pauli Matrices?

What is \begin{align} \sum_{\mu=0}^{3} \langle \sigma_{\mu} \rangle^2 = ? \end{align} $\sigma_{\mu}$ are the Pauli matrices. The Bra-Ket notation is used in this question: \begin{align} \langle \sigma_{\mu} \rangle = \langle \Psi \lvert \sigma_{\mu} \lvert \Psi \rangle , \end{align} where $\Psi$ is the Pauli spinor of two complex components.


Derivation of normal shear stress

I am self-studying this note and I am stuck in the derivation of the normal shear stress. Specifically I can't see how the relations (23) and (24) come about. Specifically, what I don't understand is $$ \tau'_{xx} = \frac{\tau_{xx}+\tau_{yy}}{2}+\tau_{yx} \tag{23} $$ and $$ \tau'_{yy} = \frac{\tau_{xx}+\tau_{yy}}{2}-\tau_{yx}\tag{24} $$ Can someone elaborate on the note to make it clearer?


Are "confinement" and "asymptotic freedom" two sides of the same coin?

On Wikipedia it says that the two peculiar properties of quantum chromodynamics (QCD) are: confinement and asymptotic freedom. Asymptotic freedom is the idea that at low energies we _cannot_ use perturbation theory because the coupling becomes extremely strong. One the other hand, at sufficiently high energies, the coupling becomes very small and so we can use perturbation theory. This can all be determined by investigating the $\beta$-function. My question is: > Why is the asymptotic freedom of quarks not considered to be a sufficient > explanation for confinement? My logic: According to asymptotic freedom: at low energies the coupling becomes so strong that quarks cannot be isolated which explains confinement. If I am right (which I'm probably not), then why is confinement still considered to be such a mystery?


How to do linear stability analysis on this system of ODEs?

I was trying to do linear stability analysis of spring pendulum. I arrived at the differential equations which describe the system. But I am unable to proceed to linear stability analysis. Is it possible to do linear stability analysis on 2nd order differential equations by finding eigen values of Jacobian matrix? The equations are as below : $mr'' - m(L+r)(\theta')^2-mgcos(\theta)+kr' = 0$ and $(L+r)\theta''+2r'\theta'+gsin(\theta)=0$ * * * UPDATE : The fixed points are $(mg/k,0)$ and $(-mg/k,\pi)$


Why in the relativistic quantum mechanics $ \gamma_4$ name is not used instead of $ \gamma_5$?

I have seen in the in the Dirac equation $$\gamma_0,\gamma_1,\gamma_2,\gamma_3.$$ Then I have seen the definition of a new matrix $$\gamma_5=i\gamma_0\gamma_1\gamma_2\gamma_3.$$ Now my question is why the name of the new matrix has not been given as $$\gamma_4.$$ Is there any historical reason behind this or it is simply taken as $$\gamma_5$$ without any special reason skipping $$\gamma_4~?$$


Electron propagator in a strong background B field

Near a magnetar with $B>B_\rm{QED}$, the strong B field will suppress the Compton scattering cross section of photons with a specific polarization (E-mode). Some references I know deal this problem using time-dependent perturbation in quantum mechanics. My question is there an intuitive way in QED to see how this takes place and why polarization is important in this case. The electron propagator will be modified. How can we include the background field in the filed theory calculation and derive it?


How many oxygen molecules touch you in your lifetime?

Following up on a comment by BlueRaja to this beautiful answer of Ilmari Karonen, I would like to phrase this follow up question: How many air molecules hit your average human's skin during their lifetime? Actually, make that **how many oxygen molecules do you breathe in your lifetime?** (In case you want some motivation: Ilmari calculates that the chance for a random oxygen molecule in the atmosphere (though I note that local effects could actually play a role for this one) to have existed in that form since the time of _Homo erectus_ is about one in 1014. However, there's a lot of them molecules, so maybe you _do_ brush up with quite a lot of them often.) Any takers?


If a paper disc is cut into a spiral, does its moment of inertia change?

It is obvious that there is no change in the mass of it and its radius. But the shape of the object does change. Does it mean its moment of inertia will also change?


outward pressure in an confined electron gas

Suppose i have some electrons stored in a empty shell container with a negative ion layer in the inner surface so the electrons keep bouncing inside without being able to leave the inner cavity. I want to compute the electrostatic pressure that the electrons exert on the shell container. So i assume the electrons have a uniform volumetric charge density $\rho$, and the container has a radius $R_0$. My derivation is as follows: electric field at radius $r$ is $$\frac{1}{3 \epsilon_0} \frac{\rho r^3}{r^2} = \frac{1}{3 \epsilon_0} \rho r$$ force on an element of charge $dQ = \rho dr dA$ is $$F = p dA = \frac{1}{3 \epsilon_0} \rho^2 r dr dA $$ which means that the pressure satisfies $$ \frac{dp}{dr} = \frac{1}{3 \epsilon_0} \rho^2 r $$ which by simple integration leads to $$ p = \frac{1}{6 \epsilon_0} \rho^2 r^2 $$ evaluated at the shell this gives $$ p(R_0) = \frac{1}{6 \epsilon_0} \rho^2 R_0^2 $$ Which looks reasonable. But the part that i'm not sure is if this result is sensitive to the assumption that the charge density is uniform inside the volume. The relaxed density of the electrons will tend to be maximum near $R_0$. > How can i estimate the relaxed density radial function and/or how will this > change my result?


My mind has been busy recently by this question: What is physics of Collision between solid and liquid (or gas)? What is Conservation law's of such Collision? Why, when we drop a solid object in water, water shows an Strange behavior (infact what is Conservation law's of energy and momentum when we drop a solid object in water )? Figure 1: Strange behavior of water after impact ![Strange behavior of water after impact](http://wp.streetwise.co/wp- content/uploads/2012/05/water-drop.jpg)


Jupiter Core is Solid Hydrogen

I was watching a show on the science channel about gas giants; there is something I do not understand. I am not a scientist, so this may be obvious to some. I learned that there a three states of an given physical object; solid, liquid, and gas depending on how cold or hot the object is. An easy example is ice, water and steam from coldest to hottest. So the theory is that Jupiter has a super-heated solid and very dense core that is made up of hydrogen. How does a gas like hydrogen become a solid while being super-heated? Is it that the pressure is so much that the gas is compressed into a solid? If so how much pressure does it take to compress hydrogen into a solid? How does the heat play into the equation?


Elastic Collision

A cube has a side length of 20 cm. An atom in the gas moves around the cube as shown. It continually bounces off the four lateral walls of the cube. The atom has a mass of 6.6×10−27 kg. Because of the elastic collisions with the walls, the atom maintains its speed of 300 m/s as it moves around. The figure below shows the view from above looking down into the cube at the path of the atom. The repeated hits cause there to be a force applied to each of the four lateral faces. Find the force (in Newtons) that it applies to Face 1. Find the force it applies to Face 2.![enter image description here](http://i.stack.imgur.com/mrgk6.jpg) I have tried the thing as F=change in momentum w.r.t time for face 1 chnge inn momentum is 2mv sin(60') how to calculate change in time? or is there any other approach to the problem??