Yang-Mills Coulomb Gauge

My Question is how to explicitly move into the "Coulomb gauge" in Yang-Mills theory. Using the answer provided by QMechanic, one can move into the "temporal gauge" for Yang-Mills fields: Gauge fixing choice for the gauge field $A_0$ . Once in this gauge, to move into the "Coulomb gauge," does one take an explicitly _time independent_ $g$, and demand that it satisfy: $$ i\partial^\mu (g^{-1}(\partial_\mu - i A_{\mu})g) = 0 $$ Thus giving $\partial ^{\mu} A'_{\mu} = 0$? Are we guaranteed that such a $g$ exists? Thanks.


How to formally write down the Boltzmann equation?

Can someone write down the Boltzmann equation, not neglecting any of the variables of the involved functions and integrals? Specifically, how to concisely capture the "primed" variables in a sensible manner?


Photons-Wave/particle duality

I know that photons and electrons and such are said to have a wave particle duality, but what does that mean for a photon? When light strikes an object, are many photons emitted, enough to draw infinitely many rays, is only one emitted, or something in between? In particular, I'm having trouble with thin film interference: ![enter image description here](http://i.stack.imgur.com/E53nS.gif) The two resulting rays are said to constructively interfere, which is confusing to me. The two rays are clearly parallel, but not coinciding, so how do the two interfere at all? I think my problem is that I imagine light to be a single ray, with a linear oscillating magnetic field- what is the proper way to address these rays? Are they photons? Or are they small instances of a wave front? I've heard Huygens' Principle, but in this case we present single rays at the end, so I'm led to believe they really ARE rays, in which case they would be photons, and the interference problem would be a result of the wave/particle duality. The only other thought I've had with regards to the interference is that, as opposed to looking at the rays as one dimensional rays, they could be some kind of representation of a wave 'centered' around that vector, but that doesn't make sense either. I know it's a heavy question, but it's really confusing me.


pressure exerted by fluid

If I had a flexible tube sealed at both ends and I submerged it in water (held vertical) Would the bottom half of the tube compress and the top half expand? What would the pressure in the tube be? Say its a 2" length of tube with the bottom being in 12" of water


Propagation of light in transparent media: absorption and reemission or scattering?

In the two Phys.SE questions What is the mechanism behind the slowdown of light/photons in a transparent medium? and Why glass is transparent? transparent media were discussed. But I'd like to clarify one detail: is a photon absorbed (and delayed) by the medium and then reemitted, or scattered instantly? Is e.g. a laser beam still coherent after passing glass? As medium molecules are disordered, this should distort the phase of photons taking different paths.


Electrons in CRT

In a CRT, where do the ejected electrons go after they cause fluorescence on the screen, have they lost most of their energy, or do they actually go through the glass?


Angular momentum of particle rolling around inside of sphere

I have a hemispherical bowl in which I roll a small particle around the edge, starting from the top at point _A_ with a velocity $v_o$. It travels halfway around the sphere and reaches point _B_ , which is a vertical distance _h_ below _A_ , with a velocity $v_f$. Point _A_ is a radial distance of $r_o$ from the vertical centerline and point _B_ is a radial distance of $r$ from the vertical centerline. There is no friction. The goal is to solve for the angle, $\theta$, between the horizontal and the velocity $v_f$. Here is a diagram of the problem scenario: ![diagram of sphere](http://i.imgur.com/57qgEHI.png) My solution relies on the assumption that angular momentum only relies on the velocities in the plane perpendicular to the vertical centerline. Is that a safe assumption? Also, when dealing with energies, is rotational KE and linear KE the same? Should I be taking RKE into account? * * * $$ L_o=L_f $$ $$ mr_ov_o=mrv_f\cos \theta $$ $$ \theta = \arccos(\dfrac {mr_ov_o}{mrv_f}) = \arccos(\dfrac {r_ov_o}{rv_f}) $$ * * * $$ KE_o + PE_o = KE_f $$ $$ \frac 12 mv_o^2 + mgh = \frac 12 mv_f^2 $$ $$ v_o^2 + 2gh = v_f^2 $$ $$ \sqrt {v_o^2 + 2gh} = v_f $$ * * * $$ \theta = \arccos(\dfrac {r_ov_o}{rv_f}) = \arccos(\dfrac {r_ov_o}{r\sqrt {v_o^2 + 2gh}}) $$


Question regarding FTL information transfer between two inertial frames

Why can’t information travel faster than the speed of light, if the two endpoints to and from which the information is being sent are moving relatively to each other, as long as the information travels slower than the inverse of the velocity which the planets are moving apart at. According to the velocity addition formula: w=(u−v)/(1−uv/c2), as long as u < 1/v, then w is positive and the laws of causality are not violated. Is there a reason then that information cannot travel faster than the speed of light between two points moving relatively to each other. Also, couldn’t information travel an infinite number times the speed of light, if two points were not moving relatively to each other.


What force is acting on the charge in the dielectric?

For example I have a dielectric solid with a small charged ball in it. And I have external electric field $E$. So what force is acting on this ball? The field in dielectric is $\frac{E}{\epsilon}$, so the force should be $\frac{Eq}{\epsilon}$. On the other hand. If I remove a small piece of dielectric then the field in the hole will be $E$. Now I put charge in this hole and the force is $Eq$. The hole is just for charged ball and there is no free space. So what is the force?


Does sound propagate further in freezing weather?

A few days ago I went for a walk in the evening. We're having winter with a little snow and freezing temperatures. We're in a quiet, shallow valley with a train station about 1km from us. I heard a train coming so I wanted to wait for it to watch it arriving to the station. To my surprise, although I was hearing the sound coming from behind a hill, the train wasn't appearing. After several minutes, I gave up, and went back, and finally I saw the train arriving after another a few minutes. The train must have been several km away when I first heard it. I watched this phenomenon later and I realized that also I could hear and understand people talking on much larger distances than usual. This has not happened before, and my only idea is that it's because of the cold weather. I have two ideas how to explain it:   1. Cold air propagates sound better for some reason.   2. We have a few cm of snow covered by ice crust, as we had freezing rain a few days ago. I guess this can mean sounds aren't absorbed by earth and are reflected instead, which makes them propagate further. (I'd say this is more probable than 1.) Is any of this reasonable, or is there another explanation? (I'm not a native speaker so please feel free to correct any language errors.)


Why does a salt solution conduct electrical current?

How does e.g. sodium chloride (aq) conduct electricity? By accepting electrons (unlikely since they already have a full outer shell)? But they can't be hopping around themselves, can they? I mean, if I have two poles made of metals inserted into a beaker with this solution, and I try to let a current go through from the pole through the solution to the other pole, the ion itself moving can't be conducting electricity, right? The metal pole won't accept such an ion instead of an electron...


Did force of gravity cause evolution of large scale structures?

Did big bang create gravity? What role gravity is assumed to have played in the formation (starting from the big bang) of large structures of our universe and what other important physical mechanisms and processes probably led to the structure we observe today?


I have difficulty understanding exercise 24 in this document: > Two parallel wires I and II that are near each other carry currents i and 3i > both in the same direction. Compare the forces that the two wires exert on > each other. > > (a) Wire I exerts a stronger force on wire II than II exerts on I. > > (b) Wire II exerts a stronger force on wire I than I exerts on II. > > (c) The wires exert equal magnitude attractive forces on each other. > > (d) The wires exert equal magnitude repulsive forces on each other. > > (e) The wires exert no forces on each other. I think - if you use $F_m=IlB\sin \alpha$, which is Force on electric wire in uniform magnetic field - that $F_{II}=IlB\sin \alpha = 3IlB\sin \alpha > IlB\sin \alpha =F_{I}$ So answer would be b), but how is it possible because you have Newton's third law( the forces should be equal, but does it apply here) and there is not any magnetic field here. So do I use this Lorentz's law or which law do I use?


How quark electric charge directly have been measured?

How quarks electric charge directly have been measured when quarks never directly observed in isolation? (Due to a phenomenon known as color confinement.)


Hexadecapole potential using point particles?

We can get monopole $1/r$, dipole $1/r^2$, quadrupole $1/r^3$ and octupole $1/r^4$ potential falloff by placing opposite point charges at the corners of a point, line, square and cube, respectively. My book cryptically says "and so on", how does one get a $1/r^5$ and higher potential falloff with a finite number of point-charges in 3 dimensions?


How to derive the expression for Bose-Einstein distribution variance?

![enter image description here](http://i.stack.imgur.com/Oo2mp.png) Can anyone point me to a derivation of this expression? $n_s$ is the number of bosons in a state.


I am looking for 'Programming+Electronics+ Physics' field

I wish to pursue a career that somehow involves programming, electronics, and physics. What are such careers? Also, I have heard of some 1 year post-graduate diplomas/courses for specialization in Physics. Which universities/ colleges offer them?


Finding out Energy value

A Lagrangian is given by, $$L= \left(\frac{\pi}{2}\right)^2 R^d \left[\frac{1}{2}\dot A^2 - V(A_{max})\right]$$ $$E=\left(\frac{\pi}{2}\right)^2R^d V(A_{max}) $$ where V (A) now includes nonlinear terms and E is the energy which is found by taking the appropriate Legendre transform of the Lagrangian and evaluating it at the upper turning point of an oscillation, $A_{max}$. Now using the potential $V= \phi^2-\phi^3+\frac{\phi^4}{4}$, and $\phi=A(t)e^\frac{-r^2}{R^2}$we can write, $$V(A)= (1+\frac{d}{2R^2})A^2-\left(\frac{2}{3}\right)^\frac{d}{2} A^3+ \frac{A^4}{2^\frac{d+4}{2}}$$ $$V''(A)= (2+\frac{d}{R^2})-6\left(\frac{2}{3}\right)^\frac{d}{2} A+ 3\frac{A^2}{2^\frac{d}{2}}$$ > For $d=2$, they got $E_{\infty}=4.44$ and $d=3$ they found the value > $E_{\infty}=39.69$, but how? **Why do we write here $E_{\infty}$?** For more > information please check equations 13 and 14 in the link


Isotropic neutrino-lepton scattering

I'm a physics student and I'm attending an introductory course of particle physics. My professor stated that, in center of mass frame, the $\nu_\mu e^- \to \nu_\mu e^-$ elastic scattering has an isotropic angular distribution, while the $\bar{\nu}_\mu e^- \to \bar{\nu}_\mu e^-$ scattering has not. I can't figure why this should be true. Any help would be appreciated.


Not working brakes: just another energy conservation problem

> A car is driving down a mountain ($v=90 km/h=25 m/s$, when the driver > realizes that brakes aren't working. He try to lose velocity going up an > inclined ($20°$) plane, with a friction coefficient of $k=0.60$. How many > meters will it take to halt? I've tried as following ($s$ is the request): $$K=\frac{mv^2}{2}$$ At the end, the potential energy gained is: $$U=mgh=mg\cdot s\cdot sin \alpha$$ In the mainwhile the energy lost due to the friction is: $$L_f=F \cdot s=mg \cdot cos(\alpha) \cdot s$$ But the work done by non conservative forces (friction) is also: $$L_f=U-K$$ And I have: $$mg \cdot cos(\alpha) \cdot s=mg\cdot s\cdot sin \alpha-\frac{mv^2}{2}$$ $$g \cdot cos(\alpha) \cdot s=g\cdot s\cdot sin \alpha-\frac{v^2}{2}$$ $$9.22s=3.35s-312.5$$ But I get a negative time. What's wrong? I'm sure that there is a stupid error, but I can't find it. The correct result (reported on the textbook) is 120 m.


As I remembered, at the 2 poles of a battery, positive or negative electric charges are gathered. So there'll be electric field existing within the battery. This filed is neutralized by the chemical power of the battery so the electric charges will stay at the poles. Since there are electric charges at both poles, there must also be electric fields outside the battery. What happens when we connect a metal wire between the 2 poles of a battery? I vaguely remembered that the wire has the ability to restrain and reshape the electric field and make it within the wire, maybe like a electric field tube. But is that true?


What is the mechanism of conduction of electric current in a conductor?

I am just very much confused about what mechanism is followed when we connect the - terminal of battery to the + terminal through a conductor? > What happens inside the conductors that makes the electrons flow? I proposed a conduction mechanism that when we connect a wire to the - terminal, we are actually touching a neutral thing and a charged thing, electrons flow in the section in conductor just next to the - terminal and that section gets charged to equal potential but simultaneously there appears a potential difference between this and the next section in the conductor because the next section is still neutral so now to equalize the potential the electrons flow to this next section and this part goes on till the end of conductor and current flows. I just want to know whether I am thinking in the right direction.


Bystander effect: last works?

Do you know the last works about Bystander Effect on cells caused by radiations? I have found research papers until 2004.


Some Dirac notation explanations

Equation for an expectation value $\langle x \rangle$ is known to me: \begin{align} \langle x \rangle = \int\limits_{-\infty}^{\infty} \overline{\psi}x\psi\, d x \end{align} By the definition we say that expectation value is a sandwich: $\langle \psi|\hat{x}|\psi\rangle$. So: \begin{align} \langle \psi|\hat{x}|\psi \rangle = \int\limits_{-\infty}^{\infty} \overline{\psi}x\psi\, d x \end{align} * * * Can you first confirm that these three lines are correct (I am not sure if I understand Dirac's bra-ket notation right). If they are wrong please explain: \begin{align} \text{1st:}& & \langle \psi | \hat{x} | \psi \rangle &= | \psi\rangle \cdot \hat{x}|\psi \rangle\\\ \text{2nd:}& & \langle \psi | \hat{x} | \psi \rangle &= {\langle \psi|}^\dagger \cdot \hat{x}|\psi \rangle\\\ \text{3rd:}& & \langle \psi | \hat{x} | \psi \rangle &= {\langle \psi|}^\dagger \cdot \hat{x} \langle\psi |^\dagger\\\ \end{align} How do i derive relations $\langle\psi|\hat{x}|\psi\rangle = \langle \psi |\hat{x}\psi\rangle$ and $\langle\psi|\hat{x}|\psi\rangle = \langle \hat{x}^\dagger\psi |\psi\rangle$?


Intrinsic angular momentum in classical mechanics

Please note, I am only interested in classical mechanics discussion on this. Please do not involve quantum mechanics. Inspired by this question: Is Angular Momentum truly fundamental? My question is: Can there be a concept of angular momentum separate from "orbital" angular momentum in classical mechanics? For example, can there be a thing such as "intrinsic" angular momentum in a classical theory that could be distinguished from the limit of shrinking a spinning ball to size zero?    It seemed obvious to me that the answer should be no. -- No such distinction could be made in a classical theory. However searching on related topics and reading more brought me to this article in wikipedia:   http://en.wikipedia.org/wiki/Einstein%E2%80%93Cartan_theory   While I don't follow the math details, the overview comments seem to claim there is a classical notion of intrinsic spin, but GR cannot handle intrinsic angular momentum. And a different classical theory is presented which does, and differs from GR. For instance, some important tensors can be non-symmetric now, which can't happen in GR no matter the size we make a spinning ball. Therefore there seems to be a real distinction between "intrinsic" angular momentum, and the limit of shrinking a spinning ball to size zero, even in classical mechanics. This blows my mind. So if the answer to the above questions are YES!, can someone help explain what this classical "intrinsic" angular momentum is?


Is there an intuitive reason the brachistochrone and the tautochrone are the same curve?

The brachistochrone problem asks what shape a hill should be so a ball slides down in the least time. The tautochrone problem asks what shape yields an oscillation frequency that is independent of amplitude. The answer to both problems is a cycloid. Is there an intuitive reason why these problems have the same answer? Proposed operational definition of "intuitive": Imagine modifying the problem slightly, either to the brachistochrone tunnel problem (tunnel through Earth), or by taking account of the finite radius of the ball. If the answer to the original question is intuitive, we should easily be able to tell whether these modified situations continue to have the same brachistochrone and tautochrone curves.


Transparent boundary condition

I am interested in the finite-difference beam propagation method and its applications. I try to solve the Helmholtz equation. At first, i would like to solve numerically it for the easiest case, without nonlinearities. Just to make sure I'm on the right way. But i really don't understand how to wright the boundary condition. I chose the transparent boundary condition and i need to write it properly to solve numerically the equation. So, for a linear, homogeneous and instantaneous medium the Helmholtz equation is writen (in 3D case, z is the propagation direction) $$ \frac{\partial^{2} E(x,y,z)}{\partial x^{2}} + \frac{\partial^{2} E(x,y,z)}{\partial y^{2}} + \frac{\partial^{2} E(x,y,z)}{\partial z^{2}} = - (k_{0} n )^{2} E(x,y,z) $$ It can be solved if the initial condition is known, $E(x,y,0)$. Introducing operator $\hat{S}$ $$ \hat{S} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + (k_{0} n )^{2} $$ The equation can be written in the following form $$ \frac{\partial^{2} E(x,y,z)}{\partial z^{2}} = -\hat{S} \ E(x,y,z) $$ The solution of this equation is $$ E(x,y,z) = \exp \left [ - i \sqrt{\hat{S}} z \right ] E^{+}(x,y,0) + \exp \left [ i \sqrt{\hat{S}} z \right ] E^{-}(x,y,0) $$ Considering only the forward propagating component and introducing the propagation operator $\hat{P}^{+}$ the electric field at $z=\Delta z$ can be written through the value of the field at $z=0$ (initial condition written earlier) and so on. $$ E(x,y,\Delta z) = \hat{P}^{+}(\Delta z) \ E(x,y,0) $$ where $$ \hat{P}^{+}(\Delta z) = \sum \limits_{n=0}^{\infty} \frac{1}{n!}\left[- i \sqrt{\hat{S}} \right]^{n} \Delta z^{n} $$ Obtained expression can be adopted to the Crank-Nicholson scheme. But it is also necessary to write the boundary condition. How to write the boundary condition if the medium is confined in the transparent walls ?


Penetration of armor plate

Is there a simple mathematical expression for the stopping power of a given thickness of armor, given the thickness of armor plate, the radius of a cannon ball, the density of the cannonball and the armor, the tensile strength and/or toughness of the armor, and the speed of the cannonball? For simplicity assume the cannonball is a solid metal sphere and that the armor plate is homogeneous. I realize that in modern warfare the projectiles are pointed and armor plate isn't a homogeneous slab, but I want to understand the simple case. (My question is inspired in part from reading about Civil War ironclads, but I also saw the question about chain mail and thought that if that question was legitimate this one should be more so.) In case my question isn't clear, what I'm asking is something like the following. Suppose it took 10 cm of iron armor to stop a 20 cm diameter cannonball moving at 300 meters/sec. How thick would the armor have to be to stop a 40 cm cannonball moving at the same speed? Or what if you doubled the speed? Or what if you doubled the tensile strength of the armor? Etc...


Speed of light, observed speed while travelling at the speed of light

I was watching Discovery channel the other night, they were telling that time slows down when you travel at a higher speed. This means there is a difference between the actual speed you travel at, and the perceived speed. Does anybody know what the perceived speed is, the speed it seems you're travelling at? What does this imply? Does it have any meaning, in some theories perhaps, that the perceived speed is lower? Does this mean that you can never actually travel at the speed of light? (All this is if we don't account for the fact that's impossible to reach the speed of light.) I'm also not into physics, so try to keep it simple...


Deriving Newton's Third Law from homogeneity of Space

I am following the first volume of the course of theoretical physics by Landau. So, whatever I say below mainly talks regarding the first 2 chapters of Landau and the approach of deriving Newton's laws from Lagrangian principle supposing Hamilton's principle of extremum action. Please keep this view in mind while reading and answering my queries and kindly neglect the systems to which Action Principle is not applicable: If we use homogeneity of space in Euler-Lagrange equations, we obtain a remarkable result i.e. the conservation of momentum for a closed system. Now, this result, using the form of Lagrange for a closed system of particles, transforms into $ \Sigma F = 0 $ . Now, how from this can we conclude that the internal forces that particles exert come in equal and opposite pairs? Is it because for 2 particles this comes out as $ F_{1} + F_{2} = 0 $ and we take the forces exerted by particles on one other to be independent of other particles (i.e. Superposition Principle) as an experimental fact? I doubt it as whole of Newtonian Mechanics is derivable from Lagrangian Mechanics and supposed Symmetries. So, according to me, a fact like Newton's Third Law should be derivable from it without using an additional experimental fact. I have an idea to prove it rigorously. Consider two particles $i$ and $j$. Let the force on $i$ by $j$ be $F_{ij}$ and on $j$ by $i$ be $k_{ij}F_{ij}$. Now the condition becomes $\Sigma (1+k_{ij})F_{ij}=0$ where the terms to be included and rejected in summation understood. As this must be true for any value of $F_{ij}$, we get $k_{ij}=-1$. I don't know if this argument or refinement of such an argument holds or not. I can see many questions arising in this argument and it's not very convincing to me. I would like to hear from you people as to if it is an experimental result used or not? If not, then is the method given above right or wrong? If wrong, how can we prove it? **Addendum** My method of proof uses the fact of superposition of forces itself, so it is flawed. I have assumed that the coefficients $k_{ij}$ are constants and don't change in the influence of all other particles which is exactly what superposition principle says. As the superposition of Forces can be derived by superposition of potential energies at a point in space and potential energy being more fundamental in Lagrangian Mechanics, I restate my question as follows: _Is the principle of superposition of potential energies by different sources at a point in space derivable from the inside of Lagrangian Mechanics or is it an experimental fact used in Lagrangian Mechanics?_ I, now, doubt this to be derivable as the fundamental assumption about potential energy is only that it is a function of coordinates of particles and this function may or may not respect superposition.


The relation between Hamiltonian and Energy

I know Hamiltonian can be energy and be a constant of motion if and only if:   1. Lagrangian be time-independent,    2. potential be independent of velocity,    3. coordinate be time independent. Otherwise $$H\neq E\neq {\rm const},$$ or $$H=E\neq {\rm const},$$ or $$H\neq E={\rm const}.$$ I am looking for examples of these three situation.


Definition of Lorentz transformations as transformations of the universe?

Following Arnold's [1] definition of the universe as an affine space $ A ^4$ with the group $\mathbb R ^4$ acting on it, we may define a galilean transformation as _an affine map $g:A^4 \to A^4$ which preserves the galilean structure_ , i.e. which preserves time intervals and spatial distance beetween simultaneous events. “Time” is an application $t:\mathbb R ^4 \to \mathbb R $ and $P,Q\in A^4$ are simultaneous if $t(P-Q)=0$? Furthermore, to say that a coordinate system $\varphi _1$ (read: a bijection $\varphi _1 :A^4 \to \mathbb R ^4$) is in uniform motion with respect to another $\varphi _2$ means that $\varphi_2 ^{-1}\circ \varphi _1 $ is a galilean transformation of $\mathbb R \times \mathbb R ^3 $ (to give a galilean structure to this space we take $t$ to be the projection on the first coordinate). On the other hand, a Lorentz transformation beetween two coordinate systems in relative motion is often defined as a _linear_ transformation _of the space of coordinates_ : $\Lambda :\mathbb R \times \mathbb R ^3 \to \mathbb R \times \mathbb R ^3 $. Question is: is it possible in special relativity to characterize a) the universe as an affine space $A^4$ (with the group $\mathbb R \times \mathbb R ^3$ acting on it) with some additional structure in a way similar to the galilean universe? I think the answer is yes, the structure is given by the pseudo metric on $\mathbb R \times \mathbb R ^3$, $|x|^2 =c^2t^2 - x_1 ^2 -x _2 ^2 -x_3 ^2$. Is this sufficient to fully characterize the universe $A^4$. b) Lorentz transformation as transformations of $A^4$ that preserve its structure. Again, is it sufficient to say that Lorentz transformations are the linear ones which preserve the space-time distance given by $|.|^2$? And why linear and not affine?


Fine tuning and parametric modelings

When I perform parametric modeling, if there is significant multicollinearity between variables I think should be independent, but in fact are not, I run into the case where one or more of the coefficients becomes exceeding small (or large) relative to the others. How is that different than what occurs in fine tuning problems of the standard model?


Negative temperature and Absolute hot

This video explains that heat at negative temperatures flows from the negative object to the normal object. If the temperature of the normal object is absolute hot, what happens with the heat? The heat can't be transferred to the absolute hot object, and it apparently does not flow in the other direction, so what happens to it?


Which experiment would be able to detect change in the speed of light?

1. Since the unit of distance is defined in terms of speed of light in vacuum, if the speed of light in vacuum were to change by "magic", what experiment would be able to detect that?   2. Is there a theory which says that the speed of light is not a function of time (in the same medium)? edit. people voting for close, please see my comment


Can infrared light be sent to long distance?

I need to send infrared light from its emitter to a distance of about 10-12 feet. Is this possible?


Equivilence of One Flux Quantum and Zero Flux

In Ady Stern's review of the Quantum Hall effect, he says of a quantum hall system "The spectrum at $\Phi = \Phi_0$ is the same as the spectrum at $\Phi = 0$..." Can someone explain why this is? It seems like the applied magnetic field certainly changes the hamiltonian, and thus the spectrum, but apparently not when the flux is a single quantum. Also I apologize for any newbie mistakes or if this is answered elsewhere, I'm pretty new to stackexchange. Thanks.


What are some ways to (approximately) symbolically diagonalize Hamiltonian operator?

Specifically the Hamiltonian takes the form of $$\hat H = \frac{\Delta }{2}{\hat \sigma _z} + {\omega _1}\hat a_1^\dagger {\hat a_1} + {\omega _2}\hat a_2^\dagger {\hat a_2} + {g_1}\left( {{{\hat a}_1}{{\hat \sigma }_ + } + \hat a_1^\dagger {{\hat \sigma }_ - }} \right) + {g_2}\left( {{{\hat a}_2}{{\hat \sigma }_ + } + \hat a_2^\dagger {{\hat \sigma }_ - }} \right),$$ a three body version of Jaynes-Cummings model. I'm currently trying to diagonalize this Hamiltonian, a first step in our application of quantum Zeno effect to a three-body system. I guess this Hamiltonian simply has no close-form diagonalization, just like in classical physics there is no closed-form general solution for a three-body system. So my question is: what are several symbolic approximation techniques to diagonalize an Hermitian operator? Better if that techniques particularly suits this Hamiltonian. The values of $\Delta, \omega_1, \omega_2, g_1, g_2$ need not be general; they can be set, say, all equal in order to simplify calculation.


Global symmetry in string theory

It is often stated that in quantum gravity only charges coupled to gauge fields can be conserved. This is because of the no hair theorem. If a charge is coupled to a gauge field then when it falls into a black hole the black hole acquires a corresponding field. However if it is not then the black hole doesn't "remember" it. Apparently it implies we can't have exact global symmetries, only gauge symmetries. How is this expectation realized in string theory? Is it true string vacuum sectors cannot posses global symmetry? Can we prove it?


Poincare Symmetry in QFT

Given that spacetime is not affine Minkowskispace, it does of course not possess Poincare symmetry. It is still sensible to speak of rotations and translations (parallel transport), but instead of $$[P_\mu, P_\nu] = 0$$ translations along a small parallelogram will differ by the curvature. I have not thought carefully about rotations and translations, but basically you could look at the induced connection on the frame bundle, to figure out what happens. This is all to say that spacetime has obviously not _exact_ Poincare symmetry, although the corrections are ordinarily very small. Most QFT textbooks seem to ignore this. Of course it is possible to formulate lagrangians of the standard theories in curved space and develop perturbation theory, too. But since there is no translation invariance, one can not invoke fourier transform. My questions are:   * Why is it save to ignore that there is no exact poincare symmetry? Especially the rampant use of fourier transforms bothers me, since they do require exact translation invariance.    * How does one treat energy momentum conservation? Presumably one has to (at least) demonstrate that the covariant derivative of the energy momentum tensor is zero. Any references that discuss those issues in more detail are of course appreciated.


Space between particles

I am a high school student, and I am just wondering what is the space between each particle, like what is the gap around each atom? I have found no text book cover this topic. Is it a vacuum?


What is in the space between a nucleus of an atom and its electrons?

There is a common analogy about the structure of an atom, such as the nucleus is a fly in the centre of a sports stadium and the electrons are tiny tiny gnats circling the stadium (tip of the hat to 'The Greatest Show on Earth') but what is in the space between the 'fly' and circling 'gnats'?


If empty space is not really empty, what does the space between an atomic nucleus and its electrons consist of?

I'm currently reading Brian Greene's **The Elegant Universe** and have also picked up his other book - **The Fabric of the Cosmos**. In **The elegant universe** Brian makes up an interesting point that empty space is not really empty. This is according to string theory. Now I may be completely wrong, but lets consider the **assumption** that empty space is not really empty. Now there are several analogies to explain the composition of an atom. One favorite analogy says that the atomic nucleus can be considered to be the size of a football placed in the center of a football field, with the electrons moving just outside this field. _That's a lot of space!_ **Assuming** that empty space is not really empty, what does this atomic space consist of?


Does vacuum (empty space) exist?

Added: **5 times down vote for now! Down voter is this religion or physics, please try to explain your decision.** I'm confused about this. In physics we know for a vacuum, but I think that there is a contradiction in this term. The quantum fluctuations are the phenomena that contradicts the vacuum existence because, according to them, the vacuum isn't the empty space. In vacuum the creation of particle-antiparticle pairs is allowed for small times and this is also proven in practice. From the other side, the relativistic theory says about space-time that interactions like gravitation bend the vacuum (empty space). There seems to be a contradiction to me. If vacuum is an empty space then we can't bend it, because we can't bend something nonexistent. In other words: we can't bend 'nothing'. **Can empty space really exist in physics?** **EDIT1:** quote: Luboš Motl > "By definition, the space without energy is the space whose total value of > energy is equal to 0." But this space is nonexistent, so it is abstract and should exist only in our mind... Why is so? My second statement of this post says that in physics we cant bend nothing because such a bending is only thinking and not physics! Another possibility is that space is unknown kind of energy, but this is contradiction in modern physics! **EDIT2:** Can any physics believe that nothing exists? By mathematical logic no! And mathematic is elementary tool in physics. Anything other than that is religion! **Nothing (empty space without energy) is only a logical state!** **EDIT3:** quote: Roy Simpson: > The General Relativity Vacuum is a space-time model region without matter. and Luboš Motl says: "By definition, the space without energy is the space whose total value of energy is equal to 0." Agree... But this is only mathematical Euclidean space + time so this is only a mathematic and not physics! With other words: this is only a method of mathematical mapping. But, in real (not theoretic) physics we can't mapping empty things. Empty is only a logical state! **EDIT4:** Roy Simpsons argumentation seems to me acceptable. quote: Roy Simpson: > Einstein struggled with this too, and the problem has come to be known as > the "Hole argument" within GR. You have to decide whether you are just > interested in GR's vacuum (empty space) or the full physical vacuum which > includes quantum aspects as well. Thanks


What would be the effects if Jupiter was to blown up?

So, in one science fiction story, that tries to be realistic as possible apart from few space magics, humanity has contingency plan to blow up Jupiter. As in, totally destroy it in one massive nuclear explosion. I'd like to know the effects of such event. Would it totally wreck the solar system or would the whole plan be non-issue?


A simple example of symmetry setting the properties of a Physical System

Does anybody know of an example were one could derive some important properties of a physical system from a symmetry of said system. I´m specially looking for simple classical examples, which could serve to illustrate the importance of finding symmetries of a system to non-physicists (high school students or first year undergrads)


How far can water rise above the edge of a glass?

When you fill a glass with water, water forms a concave meniscus with constant contact angle $\theta$ (typically $\theta=20^\circ$ for tap water): ![enter image description here](http://i.stack.imgur.com/VV6ik.png) Once you reach the top of the glass, the water-air interface becomes convex and water rises up to a height $\Delta h$ above the edge of the glass, allowing you to fill the glass beyond the naive capacity $\pi r^2 h$: ![enter image description here](http://i.stack.imgur.com/C44BZ.png) So when getting myself a glass of water, I came to wonder exactly how much this increases the capacity of a glass, and what physical constants are involved. My intuition would be that for a very large glass, $\Delta h$ converges to a constant so that the effective water capacity of the glass grows like $\pi r^2 (h+\Delta h)$ (to make things simple I'm assuming that the glass is very thin: $\Delta r\ll r$). Perhaps such a constant depends on the precise shape of the rim of the glass. But if not, perhaps it is a constant multiple of the capillary length? So, what can we say about $\Delta h$, the "rim contact angle" $\alpha$, or the shape of the water-air interface when the glass is filled at maximum capacity?


How do electrons move?

Are they spinning around the nucleus? If so what makes them move? I just do not understand how they move or if they are "attracted" to something is it another particle that makes it move.


How to detect radiation on the metal (coin)?

I've got metal coin : http://www.worldpeacecoin.org/ Ruble/dollar, a coin of disarmament with certificate. But, I am very spleeny person, I fear of it's radiance level and I don't know if I can trust it or check it somehow. That could look weird but my fears feels realistic. I only know that Soviet R-12 (SS-4) nuclear missile is nuclear missile and I'm not sure how do they extract that metal from it? I think that any metal that could be extracted from nuclear missile should be dangerous for my health because of radiance level. Please tell me if there is a method to check the radiance level of it or why should not I fear it?


Poincare Patch covers half of the hyperboloid of AdS

We start with the general case of $AdS_{p+2}$ i.e AdS space in $p+2$ dimension. \begin{equation} X_{0}^{2}+X_{p+2}^{2}-\sum_{i=1}^{p+1}X_{i}^{2} = R^2 \end{equation} This space has an isometry $SO(2,p+1)$ and is homogeneous and isotropic. The Poincare Patch is given by \begin{equation} ds^2 = R^{2}\left(\frac{du^2}{u^2}+u^2(-dt^2 +d\mathbf{x}^{2})\right) \end{equation} According to Equation (2.27) of the article http://arxiv.org/abs/hep- th/9905111, The second metric covers only half of the hyperboloid. Firstly, how do I show this. Secondly, when I go to the asymptotic limit (small radial distance), should the topology of the two spaces be different?


Can the Earth's magnetic be used to generate electricity?

Since the Earth has a magnetic field, can it, in theory, be run through a conductive metal coil to create electricity?


Faraday Effect, Satellites, and Electromagnetic Atmosphere

If I launched a Satellite into orbit and dropped a silver wire caged in carbon fiber with aerodynamic wings (for support), would the Faraday effect of the wire traveling through earth's electromagnetic atmosphere (as the satellite orbits earth) generate a lot of electricity? Also, would the wire being in the atmosphere eventually pull the satellite back down to earth?


can we get electrical energy from gravitational energy?

Now I'm trying to give an answer of the question. Satellite rotates about earth by the gravitational force of earth. Now if we place the satellite in such an orbit so that it rotate in opposite to earth rotation and having a greater angular rotational speed. Now there is an existence of earth magnetic field. Though it is very poor but strength of magnetic field is not the major factor of energy conversion. So as in electrical machine rotor rotates and stator gives magnetic field and electricity generates is it not possible to extract gravitational energy by using satellite as rotor and geomagnetic field as stator? how much it is effective? Edited: OK eventually this question becomes duplicate. Now I am asking how can we store this energy. Can we store this energy? or not.


Satellite Power

Electromagnetic induction is the production of voltage across a conductor moving through a magnetic field. Since an orbiting satellite is passing through the Earth's magnetic field would a voltage be induced in a conductor inside the satellite? I would expect this voltage to be incredibly small but if true where is this energy "coming from"?


Could we obtain free electrical energy from the Earth's magnetic field?

If there exists a natural magnetic field due to the metallic core of the earth... could we then extract and get free energy from the earth core ??


Right topology for infinite dimensional "Hilbert" spaces with indefinite or semidefinite norm

For positive definite infinite dimensional Hilbert spaces, there is the standard Cauchy norm topology. What if this state space has an indefinite norm or a positive semidefinite one, as in gauge theories or Faddeev-Popov ghosts? Which infinite sums are valid, and which aren't? Similarly, for the algebra of operators, which norm topology do we choose? Not the W*-one? The C* one?


Superluminal neutrinos

![XKCD](http://imgs.xkcd.com/comics/neutrinos.png) I was quite surprised to read this all over the news today: > Elusive, nearly massive subatomic particles called neutrinos appear to > travel just faster than light, a team of physicists in Europe reports. If > so, the observation would wreck Einstein's theory of special relativity, > which demands that nothing can travel faster than light. --source Apparently a CERN/Gran Sasso team measured a faster-than-light speed for neutrinos.   * Is this even remotely possible?   * If so, would it be a real violation of Lorentz invariance or an "almost, but not quite" effect? The paper is on arXiv; a webcast is/was planned here. News conference video here


Can neutrinos really travel faster than the speed of light?

A few years ago scientists found that neutrinos were going faster than the speed of light. Then they found some problem with the calculations. I read an article that was made this year about faster than light neutrinos. I'm not sure if the journalist is ignorant or if they found that the neutrinos really were going faster.


What are the implications of the speed of light broken?

> **Possible Duplicate:**   >  What would be the effects on theoretical physics if neutrinos go faster > than light? I don't know if it's been asked before, but I couldn't find a thread about it. I guess the news have spread to your ears already, but the speed of light has been broken with neutrinos and I can't come up with any scenario on the implications of this breakthrough. I suppose the 4 constants of the universe don't hold anymore, but I'm no physicist and would like the see your thoughts on the subject. What would Einstein think?? Sorry if this has already been answered.


What are the implications of superliminal neutrinos?

> **Possible Duplicates:**   >  What would be the immediate effects if light does not go at the maximum > speed possible?   >  Superluminal neutrinos Do we re-write physics? Or can it be fitted in current theories? I always hear about Lorentz violation in Plank scale or somewhere there. But what about this?


Neutrinos and the Cosmic Speed Limit

> **Possible Duplicate:**   >  Superluminal neutrinos An article in the newspaper a few months ago said that Neutrinos travel faster than Light. Another article then said that this observation was flawed due to faults in the electrical systems. Is this true? And if not, will this lead to redefining the Cosmic Speed Limit ( _c_ ) as proposed by Einstein?


Concerning a previous possible tachyon observation?

A few years ago there was a story about the Large Hadron Collider where a possible tachyon was supposedly observed. It was later shown it didn't occur yet the incident made me think. If a large experiment using extreme amounts of energy trying to duplicate some processes or events of the 'distant' past , would the intense amount of energy in this 'early' Universe experiment create a situation where the speed of light was different?? If a particle was observed emanating from this experiment and it seemed superluminal could this be due to a different speed of light measurement due to this phenomenon.


What would be the effects on theoretical physics if neutrinos go faster than light?

Earlier today, I saw this link on Facebook about neutrinos going faster than the speed of light, and of course, re-posted. Since then, a couple of my friends have gotten into a discussion about what this means (mostly about time-travel), but I don't really know what this really implies. This made me wonder... What are the biggest and most immediate implications of this potential discovery? **Related:** Superluminal neutrinos


Can neutrinos travel faster than the speed of light?

> **Possible Duplicate:**   >  Superluminal neutrinos   >  What would be the immediate effects if light does not go at the maximum > speed possible? This is a hot topic right now, so I thought we should get a question going on it and hopefully keep it up to date with the latest evidence for or against this discovery. Sources: [1] (Associated Press), [2] (Guardian.co.uk), [3] (Original Publication - Cornell University) > Scientists around the world reacted with cautious shock on Friday to results > from an Italian laboratory that seemed to show that certain subatomic > particles can travel faster than light. > > The journey would take a beam of light around 2.4 milliseconds to complete, > but after running the Opera experiment for three years and timing the > arrival of 15,000 neutrinos, the scientists have calculated that the > particles arrived at Gran Sasso 60 billionths of a second earlier, with an > error margin of plus or minus 10 billionths of a second. The speed of light > in a vacuum is 299,792,458 metres per second, so the neutrinos were > apparently travelling at 299,798,454 metres per second. Ignoring the boilerplate media hype about the possibilities of time travel and alternate dimensions - I'm looking for academic sources that might suggest how this could be true, or alternatively, how this discrepancy could be accounted for. * * * I read the article published with their findings. It looks like they took an insane amount of care with their measurement of distance and time. One of the most common skepticism of people who know nothing about the experiment is stuff like: > You might worry about[...] have they correctly accounted for the time delay > of actually reading out the signals? Whatever you are using as a timing > signal, that has to travel down the cables to your computer and when you are > talking about nanoseconds, you have to know exactly how quickly the current > travels, and it is not instantaneous. [2] This experiment doesn't use that sort of 'stopwatch' timing mechanism though. There is no 'T=0', and no single firing of neutrinos. What is detected is watermark patterns in the steady stream of particles. The streams at the input and output are time stamped using the same satellites and any position along each stream has a precise time associated with it. By identifying identical patterns at input and output streams, they can identify how long it took particles to travel between the points. [1] ![Time](http://i.stack.imgur.com/MYJhd.png) As for distance, they use GPS readings to get the east, north, and altitude position along the path travelled to great precision. So much so that they even detect slow earth crust migration and millimetres of changes in distance between source and destination when something like an earthquake occurs. When your particles are travelling on the scale (730534.61 ± 0.20) metres, this is more than enough precision: ![Distance](http://i.stack.imgur.com/o4gUG.png) It's going to take a lot more than grassroots skepticism to think of what could have caused this discrepancy. I've seen suggestions such as the gravity of the earth being different along the path of the neutrinos, which warps space/time unevenly. The neutrino might not actually be travelling as far as they think if space/time is contracted at one or more points along the path where gravity varies. Anyways, I'll be interested in seeing how it pans out. Like most scientists, my guess is an unaccounted for systematic error (because they definitely have statistical significance and precision and on their side) that has yet to be pointed out, but it probably won't take too long with all the theoretical physicists that will be pouring through this experiment.


What would the impact be on the physics world if neutrinos DO travel faster than light?

> **Possible Duplicates:**   >  What would be the immediate effects if light does not go at the maximum > speed possible?   >  Superluminal neutrinos I was reading this article about a group of scientist thinking that they might have surpassed the speed of light with neutrinos, and are wanting other groups to check their work. If their research is proven true, what impact would this finding have on the world of physics?


Do the particles that were found to break the speed of light really break Einstein's theory of relativity?

> **Possible Duplicate:**   >  What would be the effects on theoretical physics if neutrinos go faster > than light? Update: Loose cable caused faulty results Apparently, researchers at CERN have found definitive evidence of particles traveling faster than the speed of light. The article writes: > If confirmed, the discovery would undermine Albert Einstein's 1905 theory of > special relativity, which says that the speed of light is a "cosmic > constant" and that nothing in the universe can travel faster. Is this just a publicity stunt, or is it possible that this is legit? (I would like to see the paper for these findings, I love how they conveniently leave that out). In other words, if this is found to be true, **would Einstein's theory of relativity _really_ be challenged?** There's some debate in the comments which suggest this doesn't do anything to Einstein's theory. E.g.: > The reporting here is incorrect. Einstein’s theory _DOES NOT HOLD_ that > nothing can travel faster than light. That is a very common misconception. > Einstein’s theory starts with one axiom (a self evident truth) and one > postulate (a statement deemed to be true without further argument). The > axiom from Einstein’s theory can be stated: the laws of physics should not > depend on the frame of reference of the observer. This is a self evident > truth. The postulate can be stated: light will be measured to travel at the > same speed by all observers regardless of reference frame. This postulate > was based on experimental evidence available in 1905 and still available > today. When the axiom and postulate are applied to observers traveling at > constant speed relative to one another, you get the special theory of > relativity, published as part of Einsteins 1905 paper. One of the > conclusions is that “the speed of light is constant and absolute in free > space”. It falls out of the mathematics. This conclusion is part of the > special theory of relativity. It has to date not been dis-proven. If it is, > then the postulate must be incorrect. It would mean that light _can_ be > measured to travel at a different speed depending on the frame of reference > of the observer. I don’t get from the article that this is what has > occurred. When the axiom and postulate are applied to gravitational and > accelerating frames of reference, you the general theory of relativity, > published in 1916. The general relativity mathematics bring forth strange > things like black holes, worm holes, time warps, time travel etc. in the so > called “fabric of space and time.” The stuff science fiction authors and > buff’s are so enamored with. Including me. Most of it has not been > practically realized. Only the more mundane stuff like gravity lenses, time > dilation, length contraction have been observed. If any experiment can be > performed that is in contradiction to the conclusions, then we would merely > say, as would Einstein, that the postulate of the constancy of the speed of > light, regardless of reference frame, must be incorrect. That’s not such a > big deal, really. It would change a lot of physics. It would be very > exciting. But it would just mean that the one postulate, one that none of us > have ever been able to intuit anyway, is incorrect. > > This is not the first time that experiments have been performed that have > particles traveling faster than light _in a medium other than free space_. > In this case neutrinos travel through, air, water and apparently rock faster > than light does. That does not violate the fundamental postulate that the > speed of light is constant regardless of the frame of reference of the > observer. > > Get it right. > > Dr. Karl Hudnut, UCAR – COSMIC. ## Who is correct?


Dirac adjoint of a matrix

The Dirac adjoint for Dirac spinors is defined as, $$ \bar{u} = u^{\dagger} \gamma^{0} \, . $$ However I have come across this, $$ \overline{\gamma^{\mu}} = \gamma^{\mu} \, , \tag{1} $$ (where $\gamma^{\mu}$ are the $4\times4$ gamma matrices). Naively applying the same rules as for the Dirac spinor clearly does not get us anywhere, $$ \overline{\gamma^{\mu}} = \gamma^{\mu \dagger} \gamma^{0} = \gamma^{0} \gamma^{\mu} \gamma^{0} \gamma^{0} = \gamma^{0} \gamma^{\mu} \neq \gamma^{\mu} \, . $$ So it seems that the Dirac adjoint for a matrix is defined differently, so in trying to figure this out I make the following reasoning, let $A$ be a $4 \times 4$ matrix and $u$ a Dirac spinor so that $Au$ is again a Dirac spinor. Taking the Dirac conjugate (which is defined) gives, $$ \overline{A u} = (A u)^{\dagger} \gamma^{0} = u^{\dagger} A^{\dagger} \gamma^{0} = u^{\dagger} \gamma^{0} \gamma^{0} A^{\dagger} \gamma^{0} = \bar{u} \;\underbrace{\gamma^{0} A^{\dagger} \gamma^{0}}_{ = \bar{A} ? } \, . $$ So my guess is that $\bar{A} = \gamma^{0} A^{\dagger} \gamma^{0}$. If this is the case it is straightforward to show that $\overline{\gamma^{\mu}} = \gamma^{\mu} $. **My question** is the following, is the above statement correct? Is it so that the Dirac adjoint is actually only defined for Dirac spinors but it can be sort of extended to $4 \times 4$ matrices as above (allowing one to write $\overline{A u} = \bar{u} \bar{A}$)? * * * Link where I found eq. (1) (page 93, eq. 3.249) Link where I found eq. (1) and the claim $ \overline{X} = \gamma^{0} X \gamma^{0} $ which appears to be missing a "$^{\dagger}$"? (page 9, eq. 5.54)


Triangular lattice arrangement of vortices in a superfluid

In a simply connected container containing a superfluid and rotating, there is a net circulation of superfluid. This is found due to the vortices formed, around which the superfluid rotates. These vortices have been found to generally form a triangular lattice arrangement. Why is the triangular lattice arrangement preferred by the vortices formed ?


Use of Principle of Equivalence

_Let $x^\mu$ be the coordinates of a reference frame, $K$, where all bodies feel the same constant and uniform acceleration $\textbf{a}=\textbf{g}=-\nabla\varphi$; let $\xi^\mu$ be the coordinates of a Locally Inertial Frame, $LIF$. Using the **Principle of Equivalence** , show that the linear part in $\textbf{g}$ of the interval for $K$ is_ $$ ds^2=(1+2\varphi/c^2)c^2dt^2-d\textbf{x}^2+o(1/c^2). $$ My attempt: P.E states the metric tensor, $g_{\mu\nu}$, respect to a generic system of coordinates its related to metric tensor of a flat Minkowskian $\eta_{\alpha\beta}=\text{diag(1,-1,-1,-1)}$ via the Jacobian of the diffeomorphism $x\mapsto\xi$. Obviously such a diffeo can be of the form $$\xi^0=ct,\xi^i=x^i-1/2g^it^2,$$ and the P.E states that $$ g_{\mu\nu}=\frac{\partial\xi^\alpha}{\partial x^\mu}\frac{\partial\xi^\beta}{\partial x^\nu}\eta_{\alpha\beta}, $$ so the 00 component is $$ g_{00}=\frac{\partial\xi^\alpha}{\partial x^0}\frac{\partial\xi^\beta}{\partial x^0}\eta_{\alpha\beta}=\frac{\partial\xi^0}{\partial x^0}\frac{\partial\xi^0}{\partial x^0}-\frac{\partial\xi^i}{\partial x^0}\frac{\partial\xi^j}{\partial x^0}\delta_{ij}=1-\Big(\frac{v^i-g^it}{c}\Big)^2\simeq1-2\varphi/c^2++o(1/c^2,\textbf{g}^2), $$ here i take $2v^itg^i/c^2=2x^ig^i/c^2=-2\varphi/c^2.$ Other problem arises becouse my $g_{0j}\neq0$. What i wrong?


relativistic acceleration equation

A Starship is going to accelerate from 0 to some final four-velocity, but it cannot accelerate faster than $g_M$, otherwise it will crush the astronauts. what is the appropiate equation to constraint the movement so the astronauts never feel a gravity higher than $g_M$? for a moment i thought the appropiate relationship was $$ \left\lvert \frac{d u}{d \tau}\right\rvert \le g_M $$ where the absolute value is of the spatial component of the four-acceleration But going down this route i get the following: $$ \lvert u_F \rvert = \int_0^{\tau_F} \left\lvert \frac{d u}{d \tau} \right\rvert\,d \tau \le g_M \int_0^{\tau_F} d \tau = g_M \tau_F $$ where $u_F$ is the spatial component of the final velocity, and $\tau_F$ is the proper time it takes to reach the final velocity. The above gives me: $$ \tau_F = \frac{ \lvert u_F \rvert }{ g_M } $$ i'm doing some silly mistake, because there are no gamma factors, and i'm getting a finite proper time to reach $\lvert u_F \rvert = c$


How much has the Milky Way moved since it's forming?

What i really want to ask how much has the Milky Way moved, relative to where it was "at the big bang" or the soonest time that makes sense (since i doubt "at the big bang" makes much sense in this question). I suppose the galaxies have non-zero impulse, otherwise we wouldn't see things like galaxy collisions. So, relative to where our galaxy or whatever was there (dust cloud?) "in the beginning", how much did we move?


Dependance of temperature on color of metals

I asked this question because I supposedly did last year, Stanfor Klein which belongs to the Solar Energy Laboratory of the University of Wisconsin says that "the color of a car does not affect its internal temperature". I wonder why Metals with different colors perhaps do not absorb different doses of temperatures? and as a consequence, when different metals of different colors are exposed to strong radiation it is not so differently warmed?


the sounds of an exploding star

We know that space cannot spread a sound wave as there is no "air" or a medium that would support the spread of a sound wave. However if we put ourselves in the vicinity of an exploding star, would it be possible to hear something? The question arises from the idea that within the explosion of a star (first few seconds or less) you may hear a noise due to the explosion of the star...


Residual Resistivity in alloys and metals

## Residual Resistivity * * * I saw that the graph of resistivity to temperature of alloys like nichrome is like so![enter image description here](http://i.stack.imgur.com/KkiDU.png) Meaning that even at 0 K it has some resistivity just like copper : ![enter image description here](http://i.stack.imgur.com/GnQ1v.png) I read some where "It is the residual resistivity due to defect scattering" Is this related to the defects that i studied in solid state chemistry about lattice defects.Can some body elaborate?   * An alloy is a mixture of metals and a temperature coefficient of resistivity comparable to metals then why is its graph more linear than metals.    * Is this because of the lattice structure of an alloy?


Discrete movement vs wave function collapse

I remember once, as a child, thinking that objects do not really "move," but that at a very small scale they would have to "disappear" and then "appear" again at their newly shifted position, just the way computers render moving particles based on refresh rates. This relates to Zeno's paradox which is solved by infinite sums. Then I heard about quantum wave function collapse and the double slit experiment, and then thought: oh, maybe nature solved the problem by turning anything that wants to move into a wave instead of making a single particle "appear" and "disappear" in new positions as it moves. Waves is by the way a very elegant solution in comparison. My question is: was my thinking correct? are waves (and wave collapse) nature's way to make particles move around?


Can a wormhole be created in space?

Is wormhole a practical concept? If not what did the scientists do, to theorize it? Does it have any limitations pertaining to speed of light?


Layman explanation and demonstration of positive lightning

Inspired by the thunderstorm overhead, and after reading the question and answers Voltage and current of positive lightning \- what is an effective means to explain this phenomenon to a layman? Also, as demonstrations are more often than not a better way of explanation, how could positive lightning be safely and meaningfully demonstrated?


What is the reason behind band gap narrowing in semiconductors

I want to know why some semiconductors band gap decreases after doping with elements. **_Burstein-Moss band-filling effect_** can be useful to explain band gap widing in a semiconductor materials but i was unable to find any logical explanation for band narrowing effect. Can you please explain the mechanism of band gap narrowing. This below quotes were taken from a research article. > There is general agreement that two competing phenomena are dominant in > affecting the absorption edge in heavily doped semiconductors. First, the > well-known Burstein-Moss band-filling effect which shifts positively the > measured band-edge energy with increasing carrier concentration. In this > case the measured optical gap $E_{m}$ is the sum of the optical gap of the > lightly doped material $E_{0}$, plus that due to filling of the conduction > band due to Is $\Delta E_{BM}$, I. E. $E_{m}=E_{0}+E_{BM}$. Thc second > phenomenon which affects the optical absorption edge with increasing donor > density is due to a change in the nature and strength of the interaction > potentials between donors and the host crystal. This latter effect gives > rise to a band-gap shrinkage and to some increased tailing of the absorption > edge. In this case the measured optical gap is $E_{m}=E_{0}+E_{BM}-\Delta > E_{g}$., where $\Delta E_{g}$ is the gap shrinkage. I didnt understand the explanation given in second phenomenon. What is author meant by due **_to a change in the nature and strength of the interaction potentials between donors and the host crystal_**. Advance thanks for your help


Triple-right triangle experiment: what's the minimum distance?

Among the other ways, one way to prove the Earth is round is the **triple- right triangle**. The idea is simple:   1. Starting from point A you move in a straight line for a certain distance.   2. At point B, turn right 90° degrees, move along the line for the same distance.   3. At point C, turn again to the right and do the same.   4. We'll eventually get back at the starting point: point A and C are the same location, thus we just created a triangle with 90° degrees. This proves that that Earth has a spherical shape (not a perfect sphere), since these movements would only create a square with three sides if we were to do it on a flat surface. However, the "problem" of this experiment is that it's not really doable on a small scale. The distance must be so much that the curve of our planet can be taken into consideration. Walking 1 meter, then one meter and then another meter won't create a triangle, since the curve of the planet is not that strong. So my question is: what's the minimum distance we'd need to travel for this experiment to work?


Why do two bodies of different masses fall at the same rate (in the absence of air resistance)?

I'm far from being a physics expert and figured this would be a good place to ask a beginner question that has been confusing me for some time. According to Galileo, two bodies of different masses, dropped from the same height, will touch the floor at the same time in the absence of air resistance. BUT Newton's second law states that $a = F/m$, with $a$ the acceleration of a particle, $m$ its mass and $F$ the sum of forces applied to it. I understand that acceleration represents a variation of velocity and velocity represents a variation of position. I don't comprehend why the mass, which is seemingly affecting the acceleration, does not affect the "time of impact". Can someone explain this to me? I feel pretty dumb right now :)


why do all objects of varying mass acelerate the same?

> **Possible Duplicate:**   >  Confused about the role of mass why is it that two object of varying mass will fall at the same speed in a frictionless enviorment like the moon? Is it because the object needs to overcome more momentum or what?


Why do objects with different masses fall at the same rate?

> **Possible Duplicate:**   >  Confused about the role of mass Today we were in our Literature class talking about the Renaissance and the Enlightement and our teacher also said that scientific experiments were being conducted, and she gave as an example the experiment in which they dropped objects with different mass from a tower to see which object would land first. She then said that she herself didn't know the outcome and since I'm known in my class as the #1, they asked me and I said that they landed at the same time (to my shock many classmates even disagreed with me about this fact). The teacher asked me to explain why and we haven't had anything about inertia in our physics class, so I was forced to use my own self-thought knowledge of physics: I said gravity does pull harder on heavier objects, but heavier objects have more resistance to move, in other words, they have more inertia. The entire class except for the teacher disagreed with me even though they usually don't. One fairly annoying kid asked me to explain why a feather falls way slower than a bowling ball, and I explained why and I also made the claim that in a vacuum they would fall at the same rate. Was I right or my class? Of course I know my explanation is lacking, but it has truth to it, right?


Acceleration of two masses by gravitational force

> **Possible Duplicate:**   >  Confused about the role of mass I know that two different masses fall at the same rate in the same gravitational field because the greater gravitational force of the heavier one is exactly offset by its greater inertial resistance. What I don't understand is why the larger mass wouldn't fall more slowly at first and then once the inertial resistance is overcome, it would then accelerate faster than the less massive one. It seems that the rate of acceleration (not velocity, which obviously does) would vary according to the distance of the fall


Theoretical Physics - How to?

Although I doubt somewhat whether this question is really appropriate for this site, I hope it gets answered anyways. I guess, what I'm wondering is:   1. How does one get to work as a theoretical physicist and - probably more importantly - what do theoretical physicist actually do all day long?   2. How are theoretical physicists distinguishable from mathematicians? Does a physicists day look very different from that of a mathematician?   3. I have a great interest in physics, but I'm not really much interested in doing experiments: Would it be advisable to do my bachelor in mathematics and try to get into theoretical physics later on?   4. Is there a real chance of getting into research afterwards? (not that any kind of answer to this question would ever stop me from trying...) Well, I hope this question is acceptable. I think 1) might for example be answered by giving a link to a blog of a working theoretical physicist, who gives some insight into his or her everyday life, or some kind of an essay on the topic. Of course any other kind of answer is greatly appreciated. Thanks in advance! Kind regards, Sam **Edit after several answers:** Thanks a lot for all the responses! I found it very interesting and helpful to to get some input from you guys. Although the opinions seem to differ a bit, one can definitely see many overlaps, too. I do still have some time do decide, and will definitely look at some books suggested here, visit some lectures and try to get a feel for what it would be like to do either physics or maths. Thanks again for your effort! :-)


Non-Locality of Space - QFT (Srednicki's book)

I was going through Mark Srednicki's book on QFT. It says in the relativistic limit the Schrodinger equation becomes something like : $$ i\hbar\frac{\partial}{\partial t} \psi(\vec x,t) = \sqrt{-\hbar^2c^2\nabla^2+m^2c^4}\psi(\vec x,t) $$ Now he says that if I expand the square root (say binomially) it will have infinite no. of spatial derivatives acting on $\psi(x,t)$; this implies that equation is not local in space. What exactly does it mean to say the equation is not local in space?


Discrete Values for Observables vs Average Values (Quantum Mechanics)

When considering observables and their corresponding operators, would it be correct to believe that discerning discrete values for an observable is possible ONLY when $\psi$ is an eigenfunction of the operator? Alternatively, would it also be correct to believe that the average value of an observable is ALWAYS obtainable regardless if $\psi$ is an eigenfunction of the operator? Thanks for your help.


Derivation of $a_{j}$ coefficients in the quantum harmonic oscillator

In Griffiths' book page 53, when we derive the solution of the quantum harmonic oscillator by using the power series way, we have: $$a_{j+2} = \frac{2j+1-K}{(j+1)(j+2)}\, a_{j} .$$ And for large $j$, we have: $$a_{j+2}\approx\frac{2}{j}\,a_j.$$ Up to this point I totally agree (one just takes the limit). However, the subsequent derivation of solution $a_{j}$ and $h(\xi)$ I attached from the Griffiths' textbook are very confusing.   * How did it go from $a_{j+2}\approx\frac{2}{j}a_j$ to the solution of $a_{j}$?    * Also, how do the second and third approximations work in $h(\xi)$? My questions are mainly mathematical. I very much hope someone can provide a derivation or refer a link where these questions may already be answered. ![enter image description here](http://i.stack.imgur.com/qklld.jpg) ![enter image description here](http://i.stack.imgur.com/wBTkg.jpg)


Linearity of Quantum Mechanics?

The proof of the No-Cloning Theorem states "By the linearity of quantum mechanics, ..." -- Could someone please give me a rough sketch/outline of what this means. Does it have to do with the Hilbert Space that wave functions live in? I apologize if this question isn't specific enough, I just wanted to fully understand this concept.


What's the meaning of the general solution and the particular solution in differential equations?

Can anybody cast some physical insight into this? I've been studying differential equations on my own and don't understand how you can have a whole host of general solutions. It seems like a rather curious situation which we don't come across in other areas of mathematics. Is there anything more to the discussion that I'm missing?


Learn QM algebraic formulations and interpretations

I have a good undergrad knowledge of quantum mechanics, and I'm interesting in reading up more about interpretation and in particular things related to how QM emerges algebraically from some reasonable real world assumptions. However I want to avoid the meticulous maths style and rather read something more meant for physicists (where rigorous proofs aren't needed and things are well- behaved ;) ) I.e. I'd prefer more intuitive resources as opposed to the rigorous texts. Can you recommend some reading to get started?


Books on foundations of QM

I am seeking for books on foundations of Quantum Mechanics with subjects like the EPR experiment, Bell's theorem, the problem of measurement, entanglement, decoherence, nonlocality, interpretations, QM without observer, different formalisms etc. I prefer modern books, post graduate oriented, with history, formalisms and different approaches for a young researcher specialize himself on the subject. EDIT: the other question marked as "duplicate" do not fit my needs. I prefer books with higher modern mathematics, deep arguments and well-defined positions within the philosophy of science, post graduate oriented, if those exists. I need a book to get plenty knowledge as to do research in that academic area, not an intuitive approach.


Good quantum physics textbooks

> **Possible Duplicate:**   >  Learn algebra and interpretation of QM I'm a physics student it's my last year and I find my self struggling with usage of quantum mechanics. Now I'm starting to learn nuclear physics, solid state physics and soon quantum field theory and I see that I have holes in my knowledge. It's not that I'm completely ignorant, I've passed the exam in quantum physics I understand the concepts, but it doesn't feel right, I don't have the intuition like I do for example, classical mechanics or electrodynamics.   For example, of course, there are problems that I can't solve in classical mechanics or electrodynamics but I always know where to start I have an intuition about the problem and in most scenarios I have an ideas what solutions should look like. But in quantum physics, usually even for simple problems I can get stuck or don't know where to start.   I want to rebuild my knowledge from a new source.   The literature I've used so far is:   Introduction to Quantum Mechanics by David J.Griffiths,   Quantum physics by Leonard Schiff,   Modern Quantum Mechanics (Revised Edition) by J.J.Sakurai.      First one is an easy read, good introduction, but it doesn't use bra/ket notation.   Second on I just didn't like, it was hard to read, short on examples.   Third one I really like, but I'd like to see more examples and I can't check solutions for problems at the end of the chapters anywhere.      Could you please recommend me couple of more choices? I prefer books with lot of examples of solving problems. Also, while it's fine that book starts with wave functions, solving Schrodinger equations etc, I'd like a book that uses bra/ket notation.


How to calculate error of parallax and sextant based navigation?

1. First of all, why wasn't the sextant ever used for land navigation? The horizon is easier to see at sea, but land based sextants could be used in conjunction with artificial horizons (as at sea when horizon is hidden by fog).   2. Parallax has been used by both the US army and navy to measure distance to targets. The devices that used this principle were called coincidence rangefinders. It seems this system was still used after the introduction of radar. Why was this system eventually phased out?   3. Finally my main question: How can one calculate the error for measurements made by sextants and parallax devices?


Help explore a self-feedback camera-monitor chaotic system

We are trying to emulate the chaotic system Jim Al-Khalili demonstrate (3 min video). In our chaos lab, we are trying to research the chaotic system shown in the video. We are using just a webcam and a regular PC monitor. Our goal is to build a bifurcation tree for this system and to show how by changing the parameters of the system (location and distance of the camera from the monitor, time delay (between capturing and showing what is captured on the screen)) we can see a transition from "order" to chaos. The problem is we are not sure what exactly is the mathematical representation of the system, and what is the thing that is "doubling" and going to chaos (the y axis of a bifurcation tree - like voltage peaks on the diode in a chaotic RLD system). How do we approach this subject (if it's even possible)?


What is evidence for an irreversible change?

Knowing some about thermodynamics and reactions, I do understand how it can be shown that a change is reversible. But irreversible? Why can't it be that a change that was deemed irreversible thousands of years ago via new changes perhaps developed by physicists or changes, processes and reactions from some other part of the world or space, a change that was deemed irreversible in the future can be shown to be reversible? I think that for instance diseases that were deemed irreversible as science progressed, we could make changes that were priorly said to be irreversible, in fact reversible.


Should entropy have units and temperature in terms of energy?

I've been thinking about entropy for a while and why it is a confusing concept and many references are filled with varying descriptions of something that is a statistical probability (arrows of time, disorder, etc.). Could this confusion be in the nature of it's units and how it is scattered around in different equations? When Maxwell published his paper on the molecular distribution of velocities in 1859. This has led to the identification of temperature with the mean kinetic energy of atoms or molecules in the gas. At that point we could have redefined temperature to units with energy which make sense instead of K. This would make the new $T_a=kT$ where $T_a$ is the new absolute temperature in the units with energy. Once temperature is in the units of energy this makes entropy unitless (makes more sense to me): $S = ln(W)$ In addition this makes other equations more clear: Maxwell's gas law identity has the form (for atomic particles of mass m) $\frac{3kT}{2} = \frac{m<ν^2>}{2}$ where T is the absolute temperature and $<ν^2>$ , the average of the squared velocity of the atoms, and k Boltzmann’s constant. But with $T_a$, the relation $\frac{3kT}{2} = \frac{m<ν^2>}{2}$ will become simpler $\frac{3T_a}{2} = \frac{m<ν^2>}{2}$ And the gas constant R in the equation of state for ideal gases would be changed into Avogadro number $N_{AV} = 6.022 × 10^{23}$ and the equation state of one mole of an ideal gas will read: $PV = N_{AV}T_a$, instead of $PV = RT$ **Why bother?** This would make entropy identical conceptually and formally to information by redefining temperature in terms of units of energy. This creates a strong association between entropy and probability and makes the second law (which isn't absolute anyway) less mysterious. Would it make sense to have temperature in units of energy at other levels of physics and entropy unitless say in the case of black holes entropy?


Correspondence principle and quantum computers

I just read this article at https://medium.com/the-physics-arxiv- blog/7ef5eea6fd7a about the work of a physicist called Bolotin, that states that P!=NP (from computer science) implies that large quantum mechanical objects are not possible. The author starts by explaining the Schrodingers cat thought experiment. He then says "Nobody knows why we don’t observe these kinds of strange superpositions in the macroscopic world", which I find strange because you cannot observe a superposition, you either see that the cat is alive or dead. But then he writes "For some reason, quantum mechanics just doesn’t work on that scale. And therein lies the mystery, one of the greatest in science." I thought that the correspondence principle actually explains nicely why quantum mechanics works very well on a large scale? The main point of the article is that if you can have a large quantum mechanical system, you would get P=NP, which is believed to be not true by most computer scientists. But isn't that exactly what they try to do with quantum computers?


Why does string theory have such a huge landscape?

I was browsing through _Foundations of Space and Time,_ a compilation of essays on various theories of quantum gravity. The following passage in the introduction intrigued me: > _Each compactification leads to a different vacuum state.... at least one > state should describe our Universe in its entirety.... the enormous number > (~10^500 at last count) of solutions, with no perturbative mechanism to > select mechanism to select among them, leads some critics to question the > predictive power of the theory..Even more worrying is that, while the theory > is perturbatively finite order by order, the perturbation series does not > seem to converge._ I don't know anything about string theory and so I could not make head or tails this. All I know is that ~$10^{500}$ is a very large number.   1. What exactly is a 'solution' in string theory? Is it a spacetime metric of some sort or the terms of a S-matrix of some sort?    2. Why are there so many 'solutions'?    3. I thought string theory was supposed to be finite, why do perturbative series still diverge?   4. Is there any experimental technique to limit the number of 'solutions'?    5. Will experimental techniques be able to pinpoint a solution within present day string theorists' lifetimes too? If not, how long will it take before we can experimentally probe these things?    6. Are string theorists completely relaxed about these issues? Or are they in anguish?


Phases of the moon video

I am an educator, and I am looking for a specific video. In the video, they ask some middle school students and some college graduates about why the moon has phases. Most of the students in both the groups get the answer wrong, saying that the phases of the moon happen because of the shadow of the Earth. I remember that they also interviewed the teacher of the middle school students, and she was really perplexed that her students didn't know the right answer. Another thing I remember from the video was that the middle school kids were not sure of their answers, but the college graduates (who are at their graduation ceremony) were really sure. I want to use that video for educational purposes, but I can't find it. Can you please help?


Need help buying binoculars online for astronomy (10x50)

After reading much online I've decided to by a 10x50 porro prism Binoculars. The one I have in mind http://www.letsbuy.com/celestron-upclose-10x50-p-34652 has BK7 prisms. The one I would like to buy is this: Olympus 10x50 DPS I but it is never available and a tad bit more expensive. Is buying binoculars or telescopes online a bad thing as I've read you need to inspect them carefully before buying? Also will the prism difference (BK7 vs BAK4) make much of a difference for the 10x50 range?


detailed balance in the context of the ising model

I am having a very basic problem understanding the idea of detailed balance, particularly in the context of the Ising model. Most references I have found contain the following phrase: "In equilibrium, each elementary process must be equilibrated by its reverse process". What does it mean for one process to equilibrate another? In particular, I am trying to understand the section here: http://en.wikipedia.org/wiki/Ising_model#Algorithm_Specification in which they state that the given form of selection probabilities is required by detailed balance, but I suspect that that will follow once I understand what detailed balance actually means. Thanks!


Grounded conductor inside a uniform electric field

I am working on a textbook problem of a grounded conductor inside a uniform electric field. The textbook states that "grounded" means potential = 0. In my opinion, "grounded" should mean "same potential as infinity". But in this case we can't set potential at infinity equal zero. So my question is, what is actually the meaning of "grounded". Am I right that it means "equal potential with infinity"? What does it mean by "grounded" in this question when the potential at infinity can't be set to zero?


If the velocity of particle $A$ exceeds that of $B$, is the acceleration of $A$ greater than $B$?

Two particles $A, B$ are travelling along parallel straight paths. At some point, the velocity of $A$ exceeds that of $B$. Does this _necessarily_ mean that the acceleration of $A$ is greater than the acceleration of $B$? If you look at the $v - t$ graph of the two particles, the lines would intersect. Probably, starting off, the velocity of $B$ would be greater, but since the slope of the velocity of $A$ would be greater it would intersect with the graph of $B$ and exceed it. I couldn't think of any other situation. So, my conclusion was that the acceleration has to be greater. But my textbook says otherwise. How come? EDIT: This is question 13 from chapter 2 in Resnick halliday physics. To clarify: the problem does NOT assume that initally A's velocity was lower than B's. (See comments)


Superpositions with two observers

This is a bit of an odd question. I'm not a physicist, so bear with me if I say something wrong. Lets say you have some sort of quantum event where matter is in a superposition. Standing next to you is another scientist waiting to observe the results (and, in theory, collapse the suposition). You go to get a cup of coffee while your fellow scientist stays in the room. Your fellow scientist observes the result of the experiment while you are out of the room but does not tell you what the result was. There are three possible options here for when you return to the room:   1. The matter is still in a superposition for both you and the other scientist.   2. The matter is no longer in a superposition for either of you (even though you have not observed the event and have no knowledge of what happened).   3. The matter is in a superposition only to you. So which option is it? If the answer is option 2, would that mean that we as humans have some kind of "superpower" forcing quantum superpositions to collapse as soon as we see them?


Why aren't there more than three generations of the leptons and quarks?

There are three generations of electrons, neutrinos, and quarks. The second and third generations of electrons and quarks are unstable and decay into lighter particles. Why are there exactly three generations? Is it possible that there are more generations of increasingly massive and unstable particles that we aren't able to discover yet (for instance, at higher energy levels that we are capable of), or is there some known reason why it has to be exactly (and can't be more than) three? edit: in addition to the link above, I also found this question helpful: What Do We Get From Having Higher Generations of Particles?


Is it safe to use any wireless device during a lightning storm?

I need "educated" reasons whether it is safe to use any wireless device during a lightning storm. Most people said don't use it but they cannot explain why.


Radio waves and frequency of photon

Is 89MHZ station emitting photons of 89MHZ frequency? (I mean $\nu$ in $E=h\nu$).


Matter-antimatter asymmetry problem

As we know that matter-antimatter asymmetry is one of unsolved problems in physics. One possible solution to this problem is given as baryogenesis which produce asymmetry in rate of creation between matter and antimatter particles. But doesn't alternate solution like "different regions of space with different type of particles " holds more ground.. Means in one region matter particle dominates (the region we live ) and another region anti matter particle will dominate. When particle-antiparticle pair created from energy (at that energy is so much high that particle- antiparticle pair could be created from energy according to $E=mc^2$) before they meet and annihilate with each other they also has equal chance of meeting with same type of particle in neighborhood, because after some time of big bang the four force were united and gravity was as powerful as remaining three forces. So now in 50% of all pairs destroyed by annihilation and other 50% clumps together with same type of particle means matter with matter and antimatter with antimatter and then inflationary epoch throw these clumps from each other at very large distance so that they can not meet each other and have chance of annihilation. And we now can not see these regions because of accelerated expansion of universe because of dark energy. Doesn't this hypothesis is more or at least equally valid with baryogeneis hypothesis.


Equation to estimate droplet volume falling from a needle of a known orifice diameter at a given flow rate

I am doing some research to see if there is a simple equation (or an equation that can be simplified) to estimate the volume of a drop of liquid falling due to gravity based upon the approximate flow rate, and perhaps the viscosity/surface tension of the fluid. Basically this is a drip chamber fluid set for delivering medications. Clinicians routinely calculate fluid flow based upon the number of drops per minute assuming a fixed drop size. However since drop size varies with flow rate. Using a fixed volume size has some amount of inaccuracy. I would like to see if I can be more accurate in determining actual flow rate.


The source of gravitation in a spacetime without matter

In a discussion concerning: Physical meaning of non-trivial solutions of vacuum Einstein's field equations there were a number of answers claiming that the flatness of the Ricci space (Rµv=0) does not necessarily entail the flatness of the Riemannian spacetime. This is explained by the fact that besides the Ricci tensor, the Riemann curvature tensor depends also on the Weyl tensor said to govern the propagation of gravitational radiation. And although Rµv=0, the Weyl tensor can still be non-zero. That being so, it seems that there are some gravitational forces not included in Einstein's field equations, as the Ricci tensor (Rµv) and the Einstein's tensor (Gµv) are said to be trace-reversed (meaning that when one of them vanishes, the other does too). So, if Rµv=0 does not exclude the presence of gravitation - gravitational radiation can still be found through Weyl tensor - then (since Gµv=0) Einstein's gravitational field equations must be incomplete? Yet there immediately arises another - and in my opinion, more important - question: **What is the source of this gravitational radiation, since Rµv=0 means the whole universe is void of matter?**


Intuitive explanation of the inverse square power $\frac{1}{r^2}$ in Newton's law of gravity

Is there an _intuitive_ explanation why it is _plausible_ that the gravitational force which acts between two point masses is proportional to the _inverse square_ of the distance $r$ between the masses (and not only to the inverse of $r$)?


Gravity force strength in 1D, 2D, 3D and higher spatial dimensions

Let's say that we want to measure the gravity force in 1D, 2D, 3D and higher spatial dimensions. Will we get the same force strength in the first 3 dimensions and then it will go up? How about if we do this with Electromagnetic force? I've include the Electromagnetic force just to see if I can find an analogy to the gravity force behavior.


Newtonian gravity equation in a 2 dimensional world

I am wondering if my line of thought is correct - and thus the resulting answer to the problem above would be correct. As we know the gravitational force (of two point masses) is given by $$F = G\frac{m_1m_2}{r^2}.$$ So the gravitational force/vector field reduces with the distance squared. Now this is the formula in 3 spatial dimensions - and I always picture it as a point with gravitational field lines moving outward. Then the "strength" of the field would be the density of the lines. And hence the density drops with the distance squared (as it is inversely proportional to the area of the sphere at that distance). Now taking this line of thought to other situations we can think of course about a hypothetical 2 dimensional world. Here gravity would also be. And here we can also see the density of the "gravitational field lines". However as they propagate only in 2 spatial dimensions the density would be inversely proportional to the circumference of the circle at a distance $r$. And hence the formula would lose the square and become like: $$F = G\frac{m_1m_2}{r}$$ (With change $G$, and obviously we can't talk about mass in 2d). Is this line of thought correct?


Asymptotics of the Wigner 6j Symbol

So, in doing some numerical computations in QFT, I've run into the following Wigner 6j-Symbol: $ \left\\{ \begin{array}{ccc} x & J_1 & J_2 \\\ \frac{N}{2} & \frac{N}{2} & \frac{N}{2} \\\ \end{array} \right\\} $ In the regime where $x \ll J_1,J_2,N$ and $J_1 \approx J_2 \approx N$, and $N$ is large. I would like to know if there is an asymptotic formula for such a symbol, or if one can be derived. Using symmetries we can get $ \left\\{ \begin{array}{ccc} x & \frac{1}{2} \left(J_1+J_2\right) & \frac{1}{2} \left(J_1+J_2\right) \\\ \frac{N}{2} & \frac{1}{2} \left(N+J_1-J_2\right) & \frac{1}{2} \left(N-J_1+J_2\right) \\\ \end{array} \right\\} $ Perhaps this could help, I'm really not sure.


Solving the equation of relativistic motion

How does one solve the tensor differential equation for the relativistic motion of a partilcle of charge $e$ and mass $m$, with 4-momentum $p^a$ and electromagnetic field tensor $F_{ab}$ of a constant magetic field $\vec B$ perpendicular to the plane of motion. $$\frac{dp^a}{d\tau}=\frac{e}{m}F^a{}_bp^b$$ ? Let the the initial condition be $$p^a=(E_0 ,\vec 0)$$ I can see that the differential equation resembles that of a SHM equation or a cosh, sinh one if it's a scalar equation. However, I don't know how to deal with a tensor equation. Could anyone please explain? Thank you.


Redshifted from what?

We need to know two of the following three to calculate the third: redshifted color, baseline color, and velocity. The velocity is related to the difference between the redshifted color and the objects baseline color. How do we know the baseline color of distant objects to know the amount of redshifting?


can gapped systems have gravitational anomalies?

The question is in the title. If it is possible, what are some examples of gapped systems--either quantum field theories or condensed matter systems--which exhibit some kind of anomaly when coupled to a metric with curvature or placed on a spacetime with non- trivial topology?


Neutral shell(with a charge inside) in an electric field

> A positive point charge $Q$ is kept eccentrically inside a neutral > conducting shell. An external uniform field E is applied. Then: > > a) Force on Q due to E is zero > > b) Net force on Q is zero > > c) Net force acting on Q and conducting shell considered as a system is zero > > d) Net force on the shell due to E is zero My try: Since the shell is neutral, thus net force on it by E must also be zero. Also since Q is electrostatically shielded(by placing it inside a shell), thus net force on it must also be zero. But the answer I'm getting(as you might have guessed) is wrong(that's why I've asked this question). Please point out the flaw in my reasoning.


Is a Perfect/Lossless Mirror possible?

In traditional mirrors, some of the input light is absorbed by atoms in the mirrors surface and are 'lost' as heat, degrading the quality of the reflected image. Could this loss be compensated by an array of "powered elements" arranged to reflect nearby photons repelling them with some sort of electo-magnetic force, to completely prevent loss? _This would realize theperfect mirror, reflecting all light back._ ![enter image description here](http://i.stack.imgur.com/OD5FI.png)


What is the Levi--Civita connection of a Wick rotated metric?

A Wick rotation is a transformation that allows to change from a Lorentzian manifold to a Riemaniann manifold. In the cases when this is possible, is the Levi-Civita connection of the Riemaniann manifold the same connection as the initial Lorentzian manifold?


Angular momentum of a rotating black hole

Is there an upper limit to the angular momentum of a rotating (Kerr) black hole?


How does reflection work?

In Newton's model of light as being composed of particles, it's easy to imagine reflection as being the rebounding of individual corpuscles off a surface. However, since light can also behave like a wave, it poses a challenge in visualizing reflection. How does a wave reflect off of a surface, whether it be specular reflection or diffuse reflection? Must the wave be first absorbed, and then re-emitted? Or is there a different mechanism?


How do mirrors work?

Apparently, light is just a certain wavelength, or "the visible spectrum" of electromagnetic waves. If I recall correctly, my physics teacher explained to me that electromagnetic waves are basically consisted of two, interchanging parts. The "electric" and the "magnetic" parts of the wave are somehow manifesting each other in synchronized intervals, or they are (more probably) causing each other and canceling themselves out in the process. I checked wikipedia and the transition seems to be somehow interpolated (but then again, many things in nature are). So, which part of the mirror actually reflects the wave? Which of these two parts? Both? How come the wave doesn't get heavily distorted in the process? I guess the actual electrons of atoms of silver play a role, but why isn't every material reflective, then? Because is isn't "perfectly" flat? If I lined up atoms of a non-metal element in a perfect plane (maybe several rows, actually), would it reflect light just as mirrors do?


What exactly is the connection between the Jacobi and Bianchi identities

While reviewing some basic field theory, I once again encountered the Bianchi identity (in the context of electromagnetism). It can be written as $$\partial_{[\lambda}\partial_{[\mu}A_{\nu]]}=0$$ Here, $A_\nu$ is of course the electromagnetic potential. This formula is immediately reminiscent of the Jacobi identity: $$[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0$$ This is even clearer in general relativity, where we have $$\nabla_{[\lambda}R_{\rho\sigma]\mu\nu}$$ which we can rewrite, remembering the definition of the Riemann tensor in terms of the commutator of covariant derivatives, as $$[[\nabla_\lambda,\nabla_\rho],\nabla_\sigma]+[[\nabla_\rho,\nabla_\sigma],\nabla_\lambda]+[[\nabla_\sigma,\nabla_\lambda],\nabla_\rho]=0 $$ This all looks like there should be some profound connection here, but I'm incapable of pinpointing it. Maybe one of the experts here can make this more precise? I'd love to get to know more about this. Any comments are much appreciated. I'd also be grateful if someone could suggest (more) appropriate tags to use.


Heat flow in a hollow cylinder

Consider a hollow cylinder of different outer radius and inner radius and two different temperatures are maintained at the outer and inner surfaces such that inner temperature is higher. Because of this heat will flow radially outward. Now I don't understand why we need to use integration to find rate of heat flow that is, why isn't the temperature gradient constant? Why can't we directly apply heat conduction formula?


Constant of motion

An exercise from Goldstein (9.31-3rd Ed) asks to show that for a one- dimensional harmonic oscillator $u(q,p,t)$ is a constant of motion where $$ u(q,p,t)=\ln(p+im\omega q)-i\omega t $$ and $\omega=(k/m)^{1/2}$. The demonstration is easy but the physical significance of the constant of motion is not so clear to me. Indeed I can show that $u$ can be rewritten like: $$ u(q,p,t)=i\phi+\ln(m\omega A) $$ where $\phi$ is the phase and $A$ the amplitude of the vibration of the oscillator. I can also demonstrate that $m\omega A=\sqrt{2mE}$, where $E$ is the total energy of the oscillator. But there is any further significance of $u$ that I'm missing?