title: Integrating a Dirac delta function with the argument dependent of a parameter

text: How can I handle the integral $$ \int_{t_1}^{t_2} \delta(D - x(t)) dt, $$ with $D$ a constant. I want to do a change of variables to perform the integral over $x$ but I am not sure how to proceed.


title: Doppler effect problem with moving mirror

text: This was the given question: > A light beam of intensity $I$ and frequency $f$, directed along the positive > $z$-axis, is reflected perpendicularly from a perfect mirror which itself is > moving along the positive $z$-axis with a constant velocity $v$. Find the > reflected intensity $I'$ and the frequency $f'$ of the reflected light. If a light beam is directed towards a mirror which is moving towards us with a constant velocity $v$, the apparent frequency of the reflected beam would be higher. However, I would like to know if there is any change in the observed intensity of the light beam. Also, just rechecking, does the intensity of a light beam depend only on the amplitude? Thanks in advance


title: Why does the specific thrust of an ideal turbojet drop with increasing compressor pressure ratio?

text: As the pressure ratio increases in an ideal turbojet (fixed flight Mach number), the specific thrust, $\frac{F}{\dot{m_{air}}}$ rises, reaches a peak for small pressure ratios and then starts to decrease with increasing pressure ratios in the compressor. It seems counter-intuitive to assume that the specific thrust should decrease with increasing compressor pressure ratio since we're generating a greater pressure gradient, therefore should be able to generate more thrust.


title: Maximum permissible speed while going down a ramp

text: So, I was playing hill climb racing and I noticed that if we move with high speeds towards a ramp going down we just jump it off. While lower speeds, help us to stay in contact with the ramp. ![enter image description here](http://i.stack.imgur.com/DYQew.png) Sticks : ![enter image description here](http://i.stack.imgur.com/hVjYJ.png) Off it goes : ![enter image description here](http://i.stack.imgur.com/EoIGL.png) So, for fun, I decided to calculate maximum permissible speed which allows us to stay in contact with the ramp. Here's my working : Let's consider the case of a cylinder of mass $M$ and radius $R$. Consider sufficient friction so as to not allow any slipping. ![enter image description here](http://i.stack.imgur.com/bdOPk.png) Lets start.... $u = w_0 R$ $v = w R$ $I = \frac 1 2 MR^2$ As there is no slipping, Friction does no work. Initial Energy : Taking initial level to be U=0 for gravitational potential energy : $\frac 1 2 M u^2 + \frac 1 2 I w_0^2 +MgR =\frac 3 4 M u^2 + MgR$ Final Energy : $ \frac 1 2 M v^2 + \frac 1 2 I w^2 + MgRcos\theta =\frac 3 4 M v^2 + MgRcos\theta $ As Energy is conserved, $ v^2 = \frac 4 3 (gR(1-cos\theta) + u^2)$ Now we have to find N in terms of v and put it $\geqslant 0$ But I cannot see how to find N? Please help.


title: Why are we living in the $q$ part of the phase space?

text: In Hamilton mechanics and quantum mechanics, $p$ and $q$ are almost symmetric. But in the real world, the $p$ space isn't as intuitive as the $q$ space. For example, We can uniquely identify a person by its position, but not its momentum. Two fermions can easily have the same momentum while they cannot hold the same position. Are there particles that cannot have the same momentum while can hold the same position? What causes the breaking of the symmetry of $p$ and $q$?


title: Waveguides Transmission Mode Determination

text: How do I know if I have TE, TM, or TEM rectangular conductive waveguide? For instance, I am doing a lab where we want maximum magnetic field in the waveguide, does that mean we want the TE because $\vec{E}=0$ near boundary? Is it the material of waveguide that tells you, frequency of signal, or something else? Thank you! BTW I am using microwave radiation.


title: Griffiths phase

text: What are Griffiths effects in the context of condensed matter physics? From a cursory examination of the literature I've gathered the following: it seems that ordered systems have a "clean" critical point, at which the system makes a sharp phase transition, and that disordered systems have a "dirty" critical point which is perturbed from the clean critical point due to the randomness of the system. A Griffiths phase then occurs when the control parameter is between the clean and dirty critical points, and is characterized by rare occurrences of local order in an otherwise disordered phase. Is this correct? What does "rare" really mean? Is this a recently discovered phenomenon?


title: How to modulate the coherence of light field？

text: I want to use a Ronchi grating to modulate the coherence of incoming light field. The grating is placed at the object plane of an optical system under test. So can anybody answer my question about > How to modulate the coherence of light field?


title: Feynman rule 4-point vertex WW -> ZZ

text: I am looking for the Feynman rule of the 4-point gauge boson interaction of W+ W- -> Z Z. I am guessing it looks like the Yang Mills 4-point vertex for gluons, but with helicity included. Equation 8.81 in my professor's lecture notes seems to imply polarization, or am I misinterpreting?


title: Thermal expansion is an expression of which conservation laws?

text: Many objects get larger as they heat up and contract as they cool down. Which conservation laws are applied to describe this phenomenon? How do they interact with each other to produce this effect?


title: Can composite field consists of two fermions cause cosmic inflation rather one scalar field?

text: Inflation is triggered by one scalar field, can a field composite of fermions do the same in the early universe


title: How does Newton's 2nd law correspond to GR in the weak field limit?

text: I can only perform the demonstration from the much simpler $E = mc^2$. Take as given the Einstein field equation: $G_{\mu\nu} = 8 \pi \, T_{\mu\nu}$ ... can it be proved that Newton's formulation of gravitational and mechanical force (e.g. $F = ma$) corresponds to Einstein's in the limit when masses are small and speeds are relatively slow?


title: Amplitude and phase in vector wave field

text: Is it possible to make some separation of amplitudes and phase for a general vector-wave field? For example, like a paraxial approximation of a complex scalar field of the form $$\Phi(x,y,z) = A(x,y,z)e^{i\theta(x,y,z)}.$$ If this is possible, what are some common methods?


title: How can electrons be confined in Quantum dots?

text: Atoms are in the range of $1$ Angstrom while Quantum dots are in the range of $2$-$10$ nm. In any atom, $99.9$% is unoccupied. So if I have a Quantum dots of size $3$ nm and suppose in my Quantum dot, I have $20$ atoms, however when I do the math, I find that electrons can never be confined $3$ dimensionally in that quantum dot. So how can electrons be confined in Quantum dots?


title: Applying a force on a rigid body on a certain point

text: I have a rigid body with an origin point (at the center of mass). I want to apply a force on a certain point. So what is the force applied to the origin of this rigid body? Description image: ![description image](http://s7.postimage.org/7vmrbtarv/drawing.png) Note: I don't want the moment of torque. More explanation: Assume $p$ is a point in the space of a rigid body. So if we had a force $\vec F_{p}$ at a point $p$, how can we get the $\vec F_{center}$?


title: Why doesn't water come out of tap/faucet at high pressure when I turn it on?

text: (tap=faucet) When I turn a tap on full and then put my thumb over the spout covering, say, 90% of it, then the water spurts out. If I turn it on to, say 10%, then the water dribbles out. What's the essential difference between the way that my thumb covers most of the hole and the way the tap valve opens that makes the water come out at a different pressure in either case?


title: "Weight" of moving object in a car collision

text: From time to time I see safety warning about keeping loose items in your car. The last warning used a 2kg object, and claimed that if a collision occurred at $50{km\over h}$ it would have a weight equal to 80kg. At $90{km\over h}$ it would have a weight of 256kg. How are these "new weights" calculated?


title: Movement of a nudged block on a high friction surface

text: If I apply a single force to an object ‘floating in free space’, then it will either translate (if the force is in line with the Cof G ) or more generally it‘ll rotate about the C of G due to the couple =Force x perpendicular distance. I’m happy so far…. BUT… ... this of course never actually happens (except in space…) and more particularly what if my arbitrary shaped object is sitting on a table (as most blocks do!) Then experience and experimentation says that any offset force will actually cause a momentary rotation about a point other than the ‘centre’ of the object. Presumably the position of this real rotation centre is such as to balance out the effect of friction resisting the force, and to balance turning moments in some way? Any suggestions or hints on how to find this real rotation centre for a given frictional coefficient, shape (rectangle/ circle / arbitrary) and pushing force direction?


title: Math needed for undergrad Statistical Mechanics/Thermal Physics

text: A professor recommended me to take a course on Statistical Physics as preparation for agent-based computing in social sciences. Now I have no experience in physics beyond basic highschool, and mathematics I have taken several discrete math courses (for computer science) and no calculus yet. I have had basic differentiation/trigonometry/integration in high school as well. What math/physics do i need to be able to understand undergrad stat mechanics? Thanks a lot for your answer! (btw. I am already planning to take classical mechanics, because that's what the university states as prerequisite)


title: On the singularity $r=0$ of the Schwarzschild metric

text: I faced following sentences: > _Unlike the co-ordinate singularity at $r = 2M$, the origin of the > Schwarzschild metric $r = 0$ has a true curvature singularity. It was first > believed that this singularity was an artifact of spherical symmetry and > that a generic collapse would evade the singularity. However, work by > Hawking and Penrose showed that this was not so and that singularities were > generic rather than special. The ubiquity of singularities is guaranteed by > thesingularity theorems by Hawking and Penrose._ **Question1:** What is the meaning of "generic collapse"? **Question2:** When is a singularity an artifact of spherical symmetry?


How would one perform an $N$-body simulation in General Relativity (GR) for something like galaxy formation or galactic dynamics? Suppose one wants to simulate the rotation curve $v(r)$ for galaxies with an $N$-body simulation, and wants to do it in the framework of GR / the Einstein equations.


title: Elastic Ball Collision and contact angle

text: For a project I need to make a good simulation of balls moving around in a space. The project will be without any air drag and without losing energy in bounces. Since there is no way to lose energy the energy should be the same before and after a bounce. However, this is not the case. The formula I'm using to calculate the new speed after a bounce are the bottom 4 on this page: http://williamecraver.wix.com/elastic-equations The problem right now is that the balls lose quite a bit of speed whenever they bounce. However there is some similarity. Every single time they do bounce the energy down in a half parabola looking like this: http://imgur.com/Rlj3yni (the white line on the right is the total energy). I was wondering what I did wrong and I think it may be the contact angle. How would you calculate the right contact angle? And if I'm not doing that wrong, what is wrong then? The code I'm using:               double dbotsx = Math.abs(b2.x - x);     double dbotsy = Math.abs(b2.y - y);          double difdir = Math.atan2(dbotsy, dbotsx);     //difdir = Math.toDegrees(difdir);          double dir1 = Math.atan2(dy, dx);     //dir1 = Math.toDegrees(dir1);     double dir2 = Math.atan2(b2.dy, b2.dx);     //dir2 = Math.toDegrees(dir2);          double v1 = Math.sqrt(Math.pow(vx, 2) + Math.pow(vy, 2));     double v2 = Math.sqrt(Math.pow(b2.vx, 2) + Math.pow(b2.vy, 2));          double m2 = b2.m;          double vxe1 = ((((v1 * Math.cos(dir1 - difdir)) * (m - m2) + 2 * m2 * v2 * Math.cos(dir2 - difdir))) / (m + m2)) * Math.cos(difdir) + (v1 * Math.sin(dir1 - difdir) * Math.cos(difdir + (Math.PI / 2)));     double vye1 = ((((v1 * Math.cos(dir1 - difdir)) * (m - m2) + 2 * m2 * v2 * Math.cos(dir2 - difdir))) / (m + m2)) * Math.sin(difdir) + (v1 * Math.sin(dir1 - difdir) * Math.sin(difdir + (Math.PI / 2)));          double vxe2 = ((((v2 * Math.cos(dir2 - difdir)) * (m2 - m) + 2 * m * v1 * Math.cos(dir1 - difdir))) / (m + m2)) * Math.cos(difdir) + (v2 * Math.sin(dir2 - difdir) * Math.cos(difdir + (Math.PI / 2)));     double vye2 = ((((v2 * Math.cos(dir2 - difdir)) * (m2 - m) + 2 * m * v1 * Math.cos(dir1 - difdir))) / (m + m2)) * Math.sin(difdir) + (v2 * Math.sin(dir2 - difdir) * Math.sin(difdir + (Math.PI / 2)));


title: Gravitational field of a particle in SR

text: According to special relativity, what is the gravitational field due to a particle moving with a constant velocity v? Would it be correct to assume that the particle has a stronger gravitational field because of the relation $M=\gamma m$, where m is the rest mass and M is the relativistic mass? How is the spacetime affected?


title: Hydrogen transition and photon behavior

text: consider a transition for an electron in the Hydrogen atom from the ground state to the 1st excited state. Let's say this transition occurs through absorption of a photon of exactly the energy required to excite the electron. what happens to the photon right after it is absorbed? is it annhilated when it is absorbed? does its wavefunction cease to exist? I'm curious about this process thanks.


title: What accounts for the discrepancies in my calculations of year lengths?

text: A common exercise in many introductory astronomy texts is to use the lengths of various kinds days to calculate the approximate length of the corresponding year. For example, ratio $k$ of the length of the sidereal day to the mean solar day $D$, can be used to estimate the length of the tropical year, $a_t$ from1 $$ a_t \approx \left(k-1\right)^{-1}$$ and, similarly, the ration of the length of stellar day, $k'$, along with the established value for general precession, can be used to estimate the length of the sidereal year2 $$ a_s \approx \left(Pk'-1\right)^{-1}$$ where $P \approx 0.999999894$. But both of the values I arrive at using these calculations differ from established values. The value I get for $a_t$ is off by about a quarter of a second per year,3 while my value for $a_s$ is off by nearly 20 minutes.4 What accounts for these differences? Specifically,   1. Is my reasoning sound, or have a made an error in my thinking or calculations?   2. In particular, do my methods fail to incorporate some phenomena that should be accounted for, or fail to remove some that should be omitted?   3. If there are such omitted or wrongly included phenomena, do they correspond to the differences I'm seeing between my computed values and the established ones; for example, could I use those differences to determine some constant associated with the phenomena?   4. Are these differences just a result of the fact that each of these established values (for $a_t$, $a_s$, $k$, $k'$, and $D$) are determined independently, so that it is too much to expect them to "line up" any better than they do in these results?  * * * _Please forgive the extensive footnotes, but I want to be absolutely sure that my reasoning and calculations are, at least, sound._ 1: Given the average angular velocity, $\dot{\alpha }$, of the meridian with respect to the ecliptic coordinate system, it must be the case that in a mean solar day, the sun has passed through an angle in that system of$$\theta =\dot{\alpha} D-24^h$$so that it will take $$\frac{24^h}{\theta}=\left( \frac{\dot{\alpha }}{24^h}D-1\right)^{-1}$$days for the sun to complete the circuit of the ecliptic that determines the mean tropical year. And since by definition$$\dot{\alpha } = \frac{24^hk}{D}$$the length of the tropical year is$$\left(k-1\right)^{-1}$$ 2: Similarly, from the average angular velocity of the meridian with respect to fixed stars along the ecliptic, $\dot{\sigma }$, we can calculate the angular progress of the sun against fixed stars over the course of a mean solar day$$\phi =\text{ }\dot{\sigma} D- 360{}^{\circ}$$and since this progress exactly corresponds to the angular progress of the earth on its orbit with respect to fixed stars (i.e. excluding precession of the ecliptic) it will take$$\frac{360{}^{\circ}}{\phi }=\left( \frac{ \dot{\sigma }}{360{}^{\circ}}D-1\right)^{-1}$$mean solar days for a complete circuit of that orbit, which is the definition of the sidereal year; where $\dot{\sigma }$ consists of two components: the angular velocity of the earth on its axis which is by definition exactly $360{}^{\circ}$ per stellar day, $D/k'$, and the rate that that axis precesses relative to fixed stars$$\dot{\sigma } = \frac{360{}^{\circ}k'}{D}-\frac{50.28792\text{''}}{a_j}=\frac{360{}^{\circ}k'}{D} P$$where, where $a_j$ is a julian year and thus$$P\approx 0.999999894$$so that it will take$$\left(Pk'-1\right)^{-1}$$days for the sun to complete full circuit against the background stars, which is the definition of the duration of the sidereal year. 3: Using the IERS values for $D$ and $k$ I arrive at a value for $a_t$ which is too small by $0.265284\text{ s}$. 4: Using the IERS values for $D$ and $k'$ and the rate of precession, I arrive at a value for $a_s$ which is too large by $18.7156\text{ min}$, while, confusingly, ignoring the effect of precession (by setting $P=1$) produces a closer value, that comes up short by $1.6943\text{ min}$, which is more than an order of magnitude smaller ($4.1478\text{"}$) than the expected effect of omitting precession over the course of a sidereal year ($50.2888\text{"}$). Note that I'm pretty sure my discrepancies are not a result of daily variations (due to the obliquity of the ecliptic and the eccentricity of the Earth's orbit) in the values of ecliptic velocities, since these all average out over the course of a year to exactly the values used.   For example, the daily motion of the celestial sphere is around the celestial pole, not the ecliptic pole, so even if the velocity of the meridian around the pole is constant, the velocity with respect to the ecliptic will vary as the meridian sweeps through the ecliptic at varying declinations. Though the average ecliptic velocity over a complete circuit of the ecliptic will be $\dot{\alpha}$, the meridian completes less than a full circuit of the ecliptic in completing a full equatorial circuit (due to precession), and more than a complete circuit in a mean solar day (due to the sun's movement). However both of these omitted and added segments of the ecliptic follow the sun in the course of a year through the full range of declinations, so that over the course of a tropical year the average is $\dot{\alpha}$. So in the calculations for the tropical year, it is the fact that things average out for complete equatorial circuits that lets me use $24^h$ as a constant in the ecliptic system (as in the first equation in note 1), while it is the fact that things average out for mean solar days that let me apply $\dot{\alpha}$ to $D$.


title: What is three-photon interference?

text: Whilst reading this paper on a quantum processor that performs a type of matrix computation, I came across the concept of 'three-photon interference'. A quick Google search shows that this process is frequently referred to in quantum optical experiments but unfortunately, no wikipedia article exists for it, can anyone yield some insight on this? My own guess is that it is exactly what it says: the interference pattern that results when 3 photons are combined. Why is this distinction so important?, why 3 photons and not 4, for example?


title: Dispersion relation for TE and TM waves in general anisotropic medium

text: I want to calculate the dispersion relation (the relation between $\bf k$ and permittivity and permeability tensors and $\omega$) for a TE and a TM wave with wave vector $\mathbf k=k_x\mathbf {\hat x}+k_z\mathbf {\hat z}$ propagating in an anisotropic medium with general diagonal permeability and permittivity tensors $ \mu=\mu_0 \tilde \mu$ and $\epsilon=\epsilon_0 \tilde \epsilon$ with $\tilde \epsilon$ and $\tilde \mu$ given below. I write the Maxwell's curl equations for the fields with a space-time dependence of the form $\exp(-ik_0\bf k\cdot r+i\omega t)$ (with $k_0=\omega\sqrt{\epsilon_0\mu_0})$ in the $k$ domain as usual: $$\bf k\times \bf E=\bf {\tilde \mu} \bf B$$ $$\bf k\times \bf H=-\bf{ \tilde \epsilon} \bf E $$ where $\bf{ \tilde \epsilon}$ and $\bf {\tilde \mu}$ are dimensionless matrices: $$\begin{pmatrix} \epsilon_x & 0 &0 \\\ 0 & \epsilon_y & 0\\\ 0 &0 &\epsilon_z \end{pmatrix},\begin{pmatrix} \mu_x & 0 &0 \\\ 0 & \mu_y & 0\\\ 0 &0 &\mu_z \end{pmatrix}$$ writing the vector products $\bf k\times \bf H$ and $\bf k\times \bf E$ as matrix multiplication $\bar k \bf H$ with $$\bar k=\begin{pmatrix} 0 & -k_z &k_y \\\ k_z & 0 & -k_x\\\ -k_y &k_x &0 \end{pmatrix}$$ I rewrite the curl equations as: $$\cases{ \bar k \bf H=-\tilde \epsilon\bf E\\\\\bar k \bf E=\tilde \mu \bf H}\to\cases{(\bar k \tilde \mu^{-1}\bar k+\tilde \epsilon)\bf E=0 \\\\(\bar k \tilde \epsilon^{-1}\bar k+\tilde \mu)\bf H =0 }$$ Now, the determinant of the coefficient matrices $\bar k \tilde \epsilon^{-1}\bar k+\tilde \mu$ and $\bar k \tilde \mu^{-1}\bar k+\tilde \epsilon$ must be zero for the systems to have non-trivial solutions. My question is, where in this process I should consider the assumption of the waves being TM or TE? I know that the final dispersion relation depends on this.(being TE or TM)


title: Resources showing how to use differential forms in Physics

text: I've been learning for a while about multivectors and forms and how they simplify many things that in simple vector calculus seems to be complicated. The only problem until now is that differently from vector calculus, I'm not being able to grasp how to use those objects in Physics. For instance, I know that a $k$-blade represents an oriented piece of $k$-dimensional vector subspace from a vector space, I know that $k$-forms represents ways of "measurements" of those multivectors, and all of that stuff, but I simply don't know how to use them in practice like I've learned in the past with dot product, cross product, integration on the plane and space and so on. Is there any book ou there that show those things in a suitable way for a _physicist_? I know that there are many books good for mathematicians out there, but they mainly focus on proving consequences of the definitions, rather than applying them. I'm searching for something that shows how to really apply in Physics all that elements from calculus on manifolds. What I really mean is the following: most mathematics books on the topics tell how to prove theorems only. And many times I see the exposition and think "it's not possible to use this in practice" while there are tons of interpretations and usages in Physics. I just didn't find yet a book that show this. Just an example of what I say: in electrostatics we start with Coulomb's law that gives us the electric force between two charges. From that and the superposition principle we get an expression for the electric field. All of this is done with vectors, so that when we study those objects we use divergence, curl, and all those machinery from vector calculus. It seems straightforward to represent those things as vectors, but it doesn't seem obvious to do so with forms. Indeed, I don't even know how to write this in terms of forms without writing first with vectors and then transforming with a metric. **EDIT:** I'm mainly searching for resources that covers this topic in the lines that vector calculus is seem on Arfken's "Mathematical methods for physicists" book and the introductory chapters on Griffith's "Introduction to Electrodynamics" and Marion's "Classical Dynamics of Particles and Systems". I don't know if this kind of resource can help, but I thought this edit could make my question more specific.


title: Quantum Computing Power Advantages

text: Currently, the world's fastest supercomputer runs at 17.59 Petaflops, which consumes 9 megawatts of electricity. A qubit-based quantum computer has the potential to operate much more quickly for some operations than a classical computer would be able to (as noted in this question). In fact, classical computers face a major hurdle in computation speed due to the inherent capacitance of components. My question is: would a quantum computer able to compute things faster than a classical computer do so with less, about the same, or greater power draw for equivalent operations? Would that depend on its implementation?


title: Chirality when moving around legs in Feynman diagrams

text: Assuming one has the following term in a Lagrangian: $$ g (\overline{A_R} B_L)(\overline{C_R}D_L) $$ where A,B,C,D correspond to spin 1/2 Dirac particles and the subscripts $R$ and $L$ denote left- and right-handed particles. From what I remember does this correspond e.g. to the scattering process $$ \tilde{A_R} + B_L \rightarrow \tilde{C_R} + D_L $$ where $\tilde{A_R}$ denotes the anti-particle of $A_R$ etc. When 'moving around' the $B_L$ leg from the incoming to the outgoing part, one has to replace $B$ by its anti-particle $\tilde{B}$ if I remember correctly: $$ \tilde{A_R} \rightarrow \tilde{B} + \tilde{C_R} + D_L $$ However, I wonder what the chirality of B is ?


title: Plane waves in QFT

text: Suppose we work in the metric $(-1,+1)$. How do we describe an incoming particle with a plane wave; $\exp(-\mathrm ikx)$ or $\exp(+\mathrm ikx)$? What's the difference? Does it change if we work in the other flat metric $(+1,-1)$? It seems it's pretty arbitrary and not standard? More generally, how to determine direction of propagation of a plane wave written in exponential form?


title: Null vector fields given Bondi metric

text: I'm trying to understand how to compute the null future-directed vector fields if I have a given (Bondi) metric $g=-e^{2\nu}du^{2}-2e^{\nu+\lambda}dudr+r^{2}d\Omega$ with $d\Omega$-standard metric on the unit sphere. I was reading this paper http://www.jstor.org/stable/2118619 and there came the statement (no proof) that the null future directed vector fields, called $n$ and $l$, are given by $n=2e^{-\nu}\frac{\partial}{\partial u}-e^{-\lambda}\frac{\partial}{\partial r}$ $l=e^{-\lambda}\frac{\partial}{\partial r}$ with $g(n,l)=-2$. I do not understand where these expressions for the null vector fields came from, i.e. what kind of formulas did one use? I know that null vector fields are composed of null vectors, and a null vector $v$ is such that $g(v,v)=0$, but this doesn't really bring me a way to calculate $n$ and $l$. Attempt at the solution: Let $v=v_{1}\frac{\partial}{\partial u}+v_{2}\frac{\partial}{\partial r}$ be a null vector field. Then should hold: $g_{00}(v_{1})^{2}+2g_{01}v_{1}v_{2}=0$ this gives us two possible solutions, let call them $l$ and $r$, so as to be consistent with the paper. The first solution gives $v_{1}=0$ so one can write $l=\mathrm{something}\cdot\frac{\partial}{\partial r}$ and the second solution gives $v_{1}=-2e^{\lambda-\nu}v_{2}$, so that we can write $n=(-2e^{\lambda-\nu}\cdot\mathrm{somethingElse})\cdot\frac{\partial}{\partial u}+\mathrm{somethingElse}\cdot\frac{\partial}{\partial r}$. So I seem to get the general form of these null fields, but still have two open questions:   1. where do I see that they are "future-directed"?   2. how was the choice of "something" and "somethingElse" made? (in the paper, rather specific functions occur there without any motivation)


title: Validity of Bogoliubov transformation

text: In condensed matter physics, one often encounter a Hamiltonian of the form $$\mathcal{H}=\sum_{\bf{k}} \begin{pmatrix}a_{\bf{k}}^\dagger & a_{-\bf{k}}\end{pmatrix} \begin{pmatrix}A_{\bf{k}} & B_{\bf{k}}\\\B_{\bf{k}} & A_{\bf{k}}\end{pmatrix} \begin{pmatrix}a_{\bf{k}} \\\ a_{-\bf{k}}^\dagger\end{pmatrix} ,$$ where $a_{\bf{k}}$ is a bosonic operator. A Bogoliubov transformation $$ \begin{pmatrix}a_{\bf{k}} \\\ a_{-\bf{k}}^\dagger\end{pmatrix}= \begin{pmatrix}\cosh\theta_{\bf{k}} & -\sinh\theta_{\bf{k}}\\\\-\sinh\theta_{\bf{k}} & \cosh\theta_{\bf{k}}\end{pmatrix} \begin{pmatrix}\gamma_{\bf{k}} \\\ \gamma_{-\bf{k}}^\dagger\end{pmatrix}, $$ with $$\tanh 2\theta_{\bf{k}}=\frac{B_{\bf{k}}}{A_{\bf{k}}}$$ is often used to diagonalized such a Hamiltonian. However, this seems to assume that $|A_{\bf{k}}|> |B_{\bf{k}}|$. Is this true? If so, how else can the Hamiltonian be diagonalized?


title: Problem deriving kinetic energy from work

text: I'm currently reading a nice introductory book (german, could be translated as "Physics with a pencil"). The author works a lot with differential calculus and antiderivatives (integrals will be used later). I'm stuck at a kind of mathematical question inmidst the easier physics: So he takes the equation of force: $m \ddot{\vec{r}} = \vec{F}$ and dot-multiplies it with the speed $\dot{\vec{r}}$: $m \dot{\vec{r}} \cdot \ddot{\vec{r}} = \dot{\vec{r}} \cdot \vec{F}$ but I do not understand how he gets to the next step: $\delta_t \left(\frac{1}{2}m\dot{\vec{r}}^2\right) = \frac{d\vec{r} \cdot \vec{F}}{dt}\ \ \ $ (with $\frac{mv^2}{2} = E_{kin}$) because if I resolve the last equation again I find something different than what we began with, because the derivation of the left part leaves me no single $\dot{\vec{r}}$. $\frac{2}{2} m \ddot{\vec{r}} = \dot{\vec{r}} \cdot \vec{F}$ Maybe I misinterpret the application $\delta_t$ to both $\vec{r}$ or the $\frac{d\vec{x}}{dr}$-notation not applying to $\vec{F}$ here? (The author states before that he use the partial differentiation symbol $\delta_t$ in the same way as the total $\frac{d\vec{x}}{dr}$ because the partial states exactly what is to be derived.)


title: Amplitude at distance from source

text: So, there is a sound at $S$, whose intensity $I$ obeys the inverse square law ($I \sim \frac{1}{x^2}$). At point $P$, at a distance $r$ from $S$, the air molecules oscillate with an amplitude of $8μm$. Point $Q$ is at a distance of $2r$ from $S$. What is the amplitude of the air molecules at $Q$? What is the relationship between amplitude and distance?


title: Maxwells' equations and Coulomb's law

text: Coulomb's law and Maxwell's equations should be consistant as one can be derived from the other. Say we have a point charge with such a charge that $-kq=1$, meaning that at any point the electric field will have a magnitude of $$|E|=\frac{1}{r^2}$$ where $r$ is the distance from the origin (were we place our charge), and the vectors point towards the origin at all point. This would be equivalent to the following in cartesian co-ordinates: $$E=-\hat{x}\frac{x}{(x^2+y^2+z^2)^{3/2}}-\hat{y}\frac{y}{(x^2+y^2+z^2)^{3/2}}-\hat{z}\frac{z}{(x^2+y^2+z^2)^{3/2}}$$ We can verify that $$|E|=\frac{1}{\sqrt{x^2+y^2+z^2}}$$ Gauss's law in its differential form allows us to calculate a charge distribution that would give rise to such a electric field using the divergence operator: $$\nabla \cdot E=-\frac{1}{x^2+y^2+z^2}$$ from wolfram alpha: one, two Which absolutely doesn't make sense to me! Intuitively I would think it would be zero everywhere except (0,0,0). Or at least not to go of to infinity at any point. Could somebody please explain what's going on?


title: Online physics collaboration tools

text: I.e. online discussion with your friends. A forum is probably too overkill in this case. Yet so far nothing can beat direct communication. Important feature: the ability to archive discussions. We don't want to retell our story to people who just missed the "conference" (let the newcomers dig through the archives by themselves).


title: Neutron generators as Fusion reactor

text: One way of generating neutrons is by fusing D-D or D with T. In neutron generators that work by acceleration D-ions into a solid matrix of e.g. metal and T or D (metal hydrides). Such generators can produce up to 10^8 neutrons per second. I'm wondering, could they be tweaked to even higher levels to eventually produce a net amount of energy? The best result for the polywell was for example 10^9 fusions/sec, not far off from 10^8 fusions/sec.


title: Parallel transport of a vector along a closed curve in curvilinear coordinates

text: There is an expression indicating the change of the vector parallel translation along a closed infinitesimal curve in curvilinear coordinates (one way of introducing curvature tensor): $$ \Delta A_{k} = \oint \Gamma^{i}_{kl}A_{i}dx^{l}. $$ Then, before using Stokes' theorem, there need to require the uniqueness of the integrand. So, Christoffel symbol and vector's component expanded to a series near some point ${x^{\alpha}}^{0}$ on a curve: $$ \Gamma^{i}_{kl} = {\Gamma^{i}_{kl}}_{0} + (\partial_{\alpha}\Gamma^{i}_{kl})_{0}(x^{\alpha} - {x^{\alpha}}^{0}) + \frac{1}{2}(\partial_{p\alpha}\Gamma^{i}_{kl})_{0}(x^{\alpha} - {x^{\alpha}}^{0})(x^{p} - {x^{p}}^{0}) + ..., $$ $$ A_{i} = (A_{i})_{0} + (\partial_{\alpha}A_{i})_{0}(x^{\alpha} - {x^{\alpha}}^{0}) + \frac{1}{2}(\partial_{p \alpha}A_{i})_{0}(x^{\alpha} - {x^{\alpha}}^{0})(x^{p} - {x^{p}}^{0}) + ... $$ After that there are a words about the fact that the ambiguity of the operation of parallel transport is covered in quadratic terms. What are some reasons to "believe" that?


title: Ideal gas temperature and pressure gradients?

text: Consider an ideal gas in a $d\times d\times L$ box with the $L$ dimension in the $x$-direction. Suppose that the opposite $d\times d$ sides of the box are held at temperatures $T_1$ and $T_2$ with $T_2>T_2$ and that the system reaches a steady state. According to these notes, the thermal conductivity of an idea gas scales as the square root of temperature; $k=\alpha\sqrt{T}$ in which case by Fourier's Law one gets that the temperature gradient in the $x$-direction is $$ T(x) = \left[T_1^{3/2}+(T_2^{3/2}-T_1^{3/2})\frac{x}{L}\right]^{2/3} $$ What is the corresponding pressure gradient $P(x)$ in the steady state?


title: Virtual particles and S-matrix

text: One of methods of introducing of virtual particles is using perturbation theory. We say that scattering matrice amplitude $M_{in \to out}$ contains of $\delta(P_{out} - P_{in})$, which realizes energy-impulse conservation and relativistic connection $E^{2} - p^{2} = m^{2}$. When we leave only a few terms of the expansion this leads to unconservation energy at the interaction stage. But in final we obtain that energy-impulse is conserved. So my question is following: how exactly the dropping of terms of higher order by the coupling constant leads to energy-impulse non-conservation and how exactly do we obtain that in final energy-impulse is conserved? Also there is a question about delta-function. Whether it is administered solely by requiring the conservation of energy-momentum? Or it arises naturally from the very formulation of the problem?


title: Connection between momentum and energy

text: What is the connection between momentum and energy? > Which of the answers is the correct? > >   * A particle can have zero momentum but energy. >   * A particle can have zero energy but momentum. >   * Energy and momentum are always unambiguously connected. > Well, I zero momentum but energy could be potential energy, but that is about it. It must have zero velocity (or zero mass) and therefore have zero kinetic energy. If it has zero energy, it has zero velocity and therefore zero momentum. But momentum is a vector and depends on the direction, energy is a scalar and does not care for the direction. So the connection is not that unambiguous after all. What is the correct answer to this problem?


title: Can a single atom have angular momentum? At what angel does a single atom reflect a single photon?

text: Groups of atoms, say two of them, can have angular momentum as a group, but only because they individually have linear momentum and are bound together through a force that causes them to pull on each other so that the group as a whole appears to spin. Does it even make sense to talk about the angular momentum of a single atom? I was thinking about this when trying to understand how a photo's angle of reflection is determine, since there really is no such thing as a continuous surface. There are just groups of atoms. So how does the interaction differ when a photon hits a single atom vs. the situation where everything else is exactly the same but the atom is surrounded by 8 other atoms in a planar arrangement (i.e. a 3x3 grid)?


title: How can a block which is not receiving the direct force have a greater acceleration?

text: ![coefficient of friction between two blocks is 0.5 and lower surace is smooth.](http://i.stack.imgur.com/0KbTb.png) I solved it like this: $$F(\text{st max})=5\text{ N}$$ For the top block, $$\begin{align} 6\text{ N} - 5\text{ N} &= 1a \\\ a &= 1\ \mathrm{m/s^2} \end{align}$$ For the lower block, the driving force will be the frictional force, so $$\begin{align} 2a &= 5\text{ N} \\\ a &= \frac{5}{2} = 2.5\ \mathrm{m/s^2} \end{align}$$ I am confused as to how the lower block could have acceleration greater than the upper block, since the force is acting on the top block.


title: Why does a window become a mirror at night?

text: In day, when you look in the room through the window out, you can clearly see what happens outside. At night when it's dark outside but there's light inside you can look in the window but it becomes a mirror. Why?


title: Can Planck's constant be derived from Maxwell's equations?

text: Can mathematics (including statistics, dynamical systems,...) combined with classical electromagnetism (using only the constants appearing in chargefree Maxwell equations) be used to derive the Planck constant? Can it be proven that Planck's constant is truly a new physical constant?


title: Angular frequency. Wrong interpretation at Wikipedia?

text: This and this articles mention that the angular frequency is:               number of oscillations per unit of time      But this doesn't seem to be correct since the angular frequency is the _speed of wave rotation_ , and Wiki actually says about the _wave frequency_. Is this correct or I simply don't understand something?


title: Conceptual puzzles when using Coulomb's potential in solving the hydrogen atom in non relativistic QM

text: When solving the hydrogen atom in non relativistic quantum mechanics we use Coulomb's potential, $\sim1/r$, to describe the interaction between the proton and the electron. I am worried about two conceptual points: 1-Wouldn't this potential be modified as r becomes smaller and smaller? (because now the electron will feel different potential at very short distances because then it can see 3 quarks with different charges, not a point like proton) 2-Also very short distances means very strong electric field, strong enough for particle anti particle creation, hence we do not have non relativistic quantum mechanics anymore, specially when the number of particles is not conserved. Then what justifies our solution of the hydrogen atom without taking the previous two points into account?


title: Pedagogical projects in QFT for self-study purposes?

text: I'm reading Srednicki's QFT textbook. I'd like to do some projects that put the content learned into actual use. Is there any good project topic? I don't mind if it's already worked out some where. I just want to experience the process of actually putting QFT into use, or simply relating the textbook knowledge to more problem-oriented situation. The textbook problems are smaller drills.


title: Vacuums and free space

text: Do physicists use the terms "vacuum," "quantum vacuum," and "free space" synonymously? For example, I have read that based on conservation arguments, the spontaneous splitting of a photon into an electron-positron pair "in free space" is prohibited. And yet I have also seen this event termed "vacuum polarization," which would seem an odd usage if vacuum and free space are indeed synonymous. (My apologies if any of my premises are invalid; I am just a mathematician.)


title: The second resonance of string?

text: What is the relationship between "the second resonance " and string and the wavelength. Like in this question: if the length of the string is 2cm with second resonance, then what is wavelength?


title: Sound Propagation using Finite-Difference Time-Domain (FDTD) considering wind effects

text: I am trying to plot the propagation of sound from a fixed source in a 2D environment using Finite-Difference Time-Domain (FDTD) Method taking into account the effects of the wind velocity. I came across the following set of equations for solving the equation for pressure with respect to time at different grid points in the 2-D environment. $$ \frac{∂p}{∂t} = -( v_x\frac{∂}{∂x}+v_y\frac{∂}{∂y})p - \kappa(\frac{∂w_x}{∂x}+\frac{∂w_y}{∂y}) +\kappa Q $$ $$ \frac{∂w_x}{∂t} = -(w_x\frac{∂}{∂x}+w_y\frac{∂}{∂y})v_x- (v_x\frac{∂}{∂x}+v_y\frac{∂}{∂y})w_x-b\frac{∂p}{∂x}+bF_x $$ $$ \frac{∂w_y}{∂t} = -(w_x\frac{∂}{∂x}+w_y\frac{∂}{∂y})v_y- (v_x\frac{∂}{∂x}+v_y\frac{∂}{∂y})w_y-b\frac{∂p}{∂y}+bF_y $$ where, $ b = \frac{1}{\rho} $ is the mass bouyancy. But, I am unable to figure out what $\vec{F}$ and $Q$ are? They are mentioned somewhere as dipole and monopole pressure sources respectively. What does they actually mean?


title: Pair production intefering with gamma-ray laser black hole fabrication

text: A common "proposal" to make a micro black hole is to use on the order of 10^12 kg of gamma-ray lasing medium and focus all the light at a small point. However, intense light will interact with itself and produce particles such as electron and positrons, etc. Even visible light can do this, if it's strong enough (i.e. NIF pulsed laser intensity levels). Would this self-interaction create a cloud of hot plasma that "blurs" the focusing beams too much to make a black-hole?


title: The Preference for Low Energy States

text: The idea that systems will achieve the lowest energy state they can because they are more "stable" is clear enough. My question is, what causes this tendency? I've researched the question and been unable to find a clear answer, so I was hoping someone could explain what's going on behind the scenes here (or if nothing is and it's just an observed law that isn't explainable).


title: Highest naturally occuring binding energy of electrons

text: I was wondering which element has the highest binding energy of an electron. Is it simply the 1s electron of the heaviest stable element? If so, can somebody tell me where I can find a table of binding energies?


title: Same Plane Polarization of Light

text: When light impacts a surface, how does it know what direction to be polarized? I'd prefer a conceptual answer, there are randomly polarized photons impacting random molecules that can be randomly organized, and yet somehow only the light waves in the same plane are reflected. (Or the majority of reflected molecules are polarized, you can still SEE things, even with polarized glasses, so I'm not sure it's a totally exclusive reflection) In any case, why is it that polarization uniquely chooses the plane of the surface the light is coming from? I've read that it's the electrons freely moving in the plane, while the other orientations just get absorbed into heat, but I don't see why other orientations besides those moving parallel to the plane can't have free moving electrons.


title: Explanation for the notion that physical laws break down at the Big Bang

text: I've often heard the phrase "physical laws break down at the big bang". Why is this? Divide by zero? Please provide the mathematics.


title: Hard circuit problem

text: I want to find the p.d. between points A and B. I'm not sure how to approach this problem though, could I have a hint? ![enter image description here](http://i.stack.imgur.com/chmiO.jpg)


title: Is information about the speed of light hidden in its spectrum?

text: Can the speed of light in the vacuum (c) be inferred from the spectrum of light? If that is not the case is it possible to tell from lights spectrum that it has entered a different medium, e.g. can the correct fraction of c be inferred from the spectrum then?


title: Normal force at the bottom

text: > A solid sphere of mass $m$ is released from rest from the rim of a > hemispherical cup so that it rolls along the surface. If the rim of the the > hemisphere is kept horizontal, find the normal force exerted by the cup on > the ball when the ball reaches the bottom of the cup. I can't understand why the normal force will be $$ mg + mv^2/(R-r)$$ Can someone explain this with a free body diagram?


title: Physical intuition for higher order derivatives

text: Could somebody give me an intuitive physical interpretation of higher order derivatives (from 2 and so on), that is not related to position - velocity - acceleration - jerk - etc?


title: Quantum uncertainty of particle falling in black hole

text: A stationary observer at infinity sees a particle of mass m falling in a supermassive Schwarzschild black hole. He observes an increasing redshift and sees the particle ceasing to progress when it approaches the black hole's horizon. What happens to the positional uncertainty of this particle in the reference frame of the distant observer? A straightforward scaling argument (inserting the Hawking temperature into the equation for the thermal de Broglie wavelength for a particle of mass m) yields a thermal areal uncertainty scaling as the black hole circumference times the particle's Compton wavelength. Is this the correct limiting behavior?


title: Is there a name for the derivative of current with respect to time, or the second derivative of charge with respect to time?

text: This measurement comes up a lot in my E&M class, in regards to inductance and inductors. Is there really no conventional term for this? If not, is there some historical reason for this omission?


title: Degeneracy of Energy Levels for 2 identical particles in a One Dimensional box

text: i am studying for a physics exam and came across an exercise i cannot seem to crack. I have read through Feynmans lecture books & a lot of the internet but i am somewhat stuck. first the question, and then what i have got so far: (Note the problem is originally in German, I hope my translation does not change the problem) _**Problem Description:_** We look at 2 identical (indistinguishable) particles in a one-dimensional box with infinite potential energy at the bounds. The box' width is 2a. in the box the potential energy is 0 and on the bounds it is infinite. we take as granted that the particles DON'T interact. _**1)_** Find the 4 lowest Energies the system can have. _**2)_** How many states of the same energy does every of those 4 levels have if both particles have spin 1/2 (fermions)? _**3)_** How many states of the same energy does every of those 4 levels have if both particles have spin 1 (bosons)? _Ok, now lets see how far I've come_ _**1)_** We know, that the Energy levels of 1 particle in such a box of width 2a is given by $$E_n = \frac{n^2h^2}{32ma^2}$, \quad\quad n=(n_1+n_2)$$ This gives me the following 4 states: $$E_1 = \frac{h^2}{32ma^2},\quad E_2 = \frac{h^2}{8ma^2},\quad E_3= \frac{9h^2}{32ma^2},\quad E_4 = \frac{h^2}{2ma^2} $$ _**2)_** Here it gets tricky for me.Noting $\Phi(x)$ as the complete statefunction, $\Psi(x)$ for the Place and $\Xi(x)$ for the spin-state function and the subscripts $_s$ for symmetric and $_a$ for antisymmetric, we know that the complete function must be antisymmetrically for fermions. this is achieved by combining a symmetric place & antisymmetric spin or the other way around. Only looking at $\Psi(x)$ for now, we find the following n-states for our 2 body system. noting $n := (n_1,n_2) **n=1** (1,0) with $\Psi_a$ and $\Psi_s$ (2 degeneracies) **n=2** (1,1), $\Psi_s$ (1 degeneracie) (2,0) $\Psi_a$ and $\Psi_s$ (2 degeneracies) **n=3** (3,0) $\Psi_a$ and $\Psi_s$ (2 degeneracies) (2,1) $\Psi_a$ and $\Psi_s$ (2 degeneracies) **n=4** (4,0) $\Psi_a$ and $\Psi_s$ (2 degeneracies) (3,1) $\Psi_a$ and $\Psi_s$ (2 degeneracies) (2,2) $\Psi_s$ (1 degeneracie) Now i have a look at the spin-states $\Xi(x)$ for Spin = S = 1, using the formulas Number of antisymm spinstates # $\Xi_a(x) = S(2S+1) = 1$ Number of symmetric spinstates # $\Xi_s(x) = (S+1)(2S+1) = 3$ Now we Combine those to antisymmetric functions using $\Phi_a = \Psi_a+\Xi_s$ or $\Phi_a=\Psi_s+\Xi_a$ To count the states of the same energy i use this formula N states = $( N\Psi_a \cdot N\Xi_s)+( N\Psi_s \cdot N\Xi_a)$ (N denoting the number of states using the degeneracy of each n-combination) This finally results in the following degeneracies $N E_1= 2*1 + 2*3 = 8$ $N E_2= 2*1 + 3*3 = 11$ $N E_3= 4*1 + 4*3 = 16$ $N E_4= 4*1 + 5*3 = 19$ **3)** Now the same for Bosons (Spin S = 1). In this case the complete function $\Phi$ must be symmetrically, so we have to combine antisym spin and antisym place or both symm. The degeneracy of the place function stays the same as in 2), but for the spin we have to recalculate using: Number of antisymm spinstates # $\Xi_a(x) = S(2S+1) = 3$ Number of symmetric spinstates # $\Xi_s(x) = (S+1)(2S+1) = 6$ To achieve symmetric complete functions, we also need to adjust our N E formula which will be changed to : N states = $( N\Psi_a \cdot N\Xi_a)+( N\Psi_s \cdot N\Xi_s)$ (N denoting the number of states using the degeneracy of each n-combination) This finally results in $N E_1= 2*6 + 2*3 = 18$ $N E_2= 2*6 + 3*3 = 21$ $N E_3= 4*6 + 4*3 = 36$ $N E_4= 4*6 + 5*3 = 39$ _**Conclusion and MY question_** : Sadly this exercise does not have a solution, so I decided to post my attempt here and hope that some of you might find any errors I made and help me if they spot if and where my thoughts went rogue. Hoping for some good input! (Please add questions if something is not clear - it could be the translation).


title: Why do the trees` trunk assumes a circular/curvy shape but not rectangle or square?

text: Why does the trees grow circular? What are the forces that force a plant to grow with a circular trunk? If the stability of the tree were the reason then why does the trees bend in their natural growth? Silly question isnt it??


title: How do I calculate the heat lost or gained by surroundings?

text: How do I calculate the heat lost or gained by surroundings (Q surr) given mass ($m$), change in temp ($\Delta T$), and specific heat ($c$)? What equation would I use? How can I tell whether it's lost or gained?


title: Physics of a fixed wheel moving on a flat surface held at distance from a center point

text: Ok, so my boss is trying to make a car turntable. In essence, he has a two boards that sit atop a rotating ring. He wants to put two wheels at the end of each board (8 wheels total). He thinks that you can angle wheels properly (2d) such that they can be flat (axis is parallel to the floor) and the whole contraption move smoothly. Take a look at this picture (Edit: copied below - Mark) ![alt text](http://i.stack.imgur.com/YujYw.jpg) Ok, so the question, will this design move smoothly? My guess is "no." I reason that if the wheels are unbanked then they will move in a straight line when driven, and the ring will realign them periodically to its circular path (probably causing a skip at the wheel as the stress is released). He thinks it will just work.. Please use technical terms with very easy layman definitions so I can reference wikipedia and texts and learn more about this. Sorry, for the utter lack of useful physics terms.


title: Can you enter a timelike hypersurface?

text: As I understand it, a timelike hypersurface is one that has only spacelike normal vectors. But does this not imply that a the geodesic of a particle crossing it must be spacelike at that point? But geodesics of massive particles follow timelike geodesics...I suppose the same question arises for the event horizon of a black hole, which is a null surface, but in that case I just imagine that a particle doesn't have to have a timelike tangent vector at that point since it isn't locally flat.


title: Eye sensitivity & Danger signal

text: Why are danger signal in red, when the eye is most sensitive to yellow-green? You can check luminosity function for more details...


title: Forces and angles

text: "The little ball with the mass of 100g has gotten stuck in a chute as depicted in the picture. What forces, and how large are they, that are acting on the ball?" ![image-of-problem](http://i.stack.imgur.com/V647D.png) This is how I solve it:   1. I find the force of gravity acting on the ball: $0.1kg * 9.8m/s = 0.98N$   2. Next I construct a right triangle from the point of contact. Since the ball touches in two places the force of gravity is halved on each side. ![images-of-problem](http://i.stack.imgur.com/rq4Pd.png) Where $x$ is the normal force on the ball.   1. $x = 0.49N * sin(20^o) = 0.17N$   2. There's two of these normal forces, one on each side. As such we have a gravitational force of 0.98N and two normal forces, each of 0.17N. The answer given in the book is that there is a gravitational force of 0.98N, and the normal force is 1.4N. What am I doing wrong here?


title: Levi Civita Symbol and contravariance vs covariance

text: I have a question regarding the Levi-Civita symbol and contravariance vs covariance. Some of this was asked in a previous post, but I think I need more clarification. Consider the magnetic field: \begin{equation} B^k = \tilde{\varepsilon}^{ijk}\partial_i A_j \end{equation} Qn 1: In the formula I wrote it as the Levi-Civita symbol $ \tilde{\varepsilon}$, is it right or should it be the Levi-Civita tensor $\varepsilon$ instead? Or does that not make sense since the formula $\vec{B} = \nabla \times \vec{A} $ is true only in cartesian coordinates and so we cannot use a tensor? (hope that made sense) Qn 2: I wrote $B^k$ with an upper index to indicate that it is a contravariant vector - so that means under coordinate transformations $\tilde{\varepsilon}$ transforms too. Except that I'm confused as to how it transforms? Carroll (in his book spacetime & geometry) says that the Levi-Civita symbol is defined not to change under coordinate transformations (so its entries remain +1 -1 0), yet goes on to derive a transformation law $\tilde{\varepsilon}_{i'j'k'\cdots} = |g| \tilde{\varepsilon}_{ijk} \frac{\partial x^i}{\partial x^{i'}} \frac{\partial x^j}{\partial x^{j'}} \frac{\partial x^k}{\partial x^{k'}} \cdots$. So which is it? Qn 3: However in some books they write \begin{align} B_k = \epsilon_{ijk}\partial_i A_j. \end{align} In this case it seems like they are ignoring the tensorial nature of the object and just treating $\tilde{\varepsilon}$ as simply a symbol that gives the values +1, -1 or 0 in the summation. This occurs too, for example, in writing the commutation relations down for the spin operators: \begin{align} [J_i, J_j] = i \tilde{\varepsilon}_{ijk}J_k, \end{align} where I am pretty sure in this case the symbol is just thought of as just a number. (although $J_i$ are vectors of the Lie algebra...?). So, am I attributing too much meaning to the Levi-Civita symbol? Qn 4: In the course of my work I'm doing I have to contend with the angular momentum operator, which I wrote as \begin{align} L^k = \tilde{\varepsilon}^{ijk}g_{ia}r^ap_j. \end{align} In cartesian coordinates this reproduces $L^x, L^y, L^z$, since for example $L^z = x p_y - y p_x$. But is this formula right? Changing to spherical coordinates doesn't seem to give $L^z = \frac{\partial}{\partial \phi}$ and so on... Qn 5: Just a general question about the Levi-Civita symbol. Does it make sense to raise/lower indices in the Levi-Civita symbol? I know that you can do it for the tensor because the Hodge star operator uses it. sorry for the lengthy post and thanks in advance for the help!


title: Evolution of mass and velocity distributions under newtonian gravitation

text: Let $\rho(r,t)$ and $v(r,t)$ be mass and velocity distributions. Given $\rho(r,0)$ and $v(r,0)$ (initial conditions) what is the differential equation that describes the evolution of $\rho(r,t)$ and $v(r,t)$ in space and time, assuming Newtonian gravitation? You may assume $r$ to be one dimensional, if it makes writing the equations convenient. **EDIT** I think $v(r,t)$ should be replaced with $p(r,t) = \rho(r,t)v(r,t)$ and name it as momentum distribution to avoid some technical faults. **EDIT 2** It seems that to completely characterize an inviscid fluid flow, Euler equations are needed. The law of conservation of mass is already given in the answer. A component of pressure is missing in the equation related to conservation of momentum.(first equation). Lastly a third equation corresponding to law of conservation of energy should be introduced.(3rd one of the euler equations of inviscid fluid flow).For the third equation we assume that the internal energy per unit volume of the fluid is zero.


title: Theorems on instability of classical systems of charged particles?

text: Classically, a hydrogen atom should not be stable, since it should radiate away all its energy. I remember hearing from my favorite freshman physics prof ca. 1983 about a general theorem to the effect that all classical systems of charged particles were unstable, and I went so far as to contact him in 2009 and ask him if he remembered anything about it, but he didn't. There is something called Earnshaw's theorem (there's a Wikipedia article) that says that _static_ equilibrium is impossible. It certainly seems plausible that something similar holds for dynamical equilibrium, in some sense that I don't know how to define properly, much less prove rigorously. Intuitively, I would expect that any classical system of $m$ charged particles should end up having no "interesting structure," in the sense that the final state will consist of radiation plus $n \le m$ zero-size clusters whose charges $q_1,\ldots q_n$ all have the same sign (some possibly being zero); asymptotically, they all end up diverging radially from some central point. Does anyone know of any formal proof along these lines? Are there counterexamples, although possibly ones with initial conditions corresponding to zero volume in phase space? How would one go about stating the initial conditions on the radiation fields?


title: 3 current loops with non-conservation of momentum, when one specifically considers SIGNs

text: Thought experiment I. Consider two loops of wire, 2 small dipoles B and C , with a common axis z (facing each other) and (say) 30 cm apart B to C. At the speed of light, information (including a change in magnetic field) will require 1 nanosecond to travel from C to B.   1. Have the current on in loop B for some period of time at the start, so the B-fields at C is established in the +z-direction.   2. Turn loop B off rapidly (fall time < 0.1 ns, say) at the same time that a current in loop C is turned ON (rapidly, rise time <0.1ns, and opposite sense with respect to the previous current in loop B ).   3. In this way, as the current is turned on in loop C, it is immersed in the field from loop B and therefore both receives an impulse to the right, in the +z-direction. However, loop B will be "off" (and open so no effective eddy currents) when the "return" field from loop C arrives. Thus, loop C (which is free to move) will experience an impulse giving it momentum in the +z direction (to the right), whereas loop B will not experience an impulse to the left. I think this argument is sufficiently simple to sketch and to ponder. Thought Experiment II. However, If you argue that there is momentum to the left "in the magnetic field" from loop B, I will add a third loop to the left (call it A), and again, as B is opened rapidly (short fall time) -- at the same time that a current in A is turned ON (rapidly, and SAME sense with respect to the previous current in loop B ). **You see, while one may hand-wave (without equations and ignoring signs) that a magnetic field "carries momentum", the fact is that one can choose the DIRECTION of the current in a loop (or loops) which absorbs energy from that field, and * _thus one can choose the DIRECTION of the momentum in the sensing loop._ *** In this way, loops A and C (both free to move) as they turn on are immersed in the field from B while having currents in the opposite sense (A attracted by loop B whereas C is repelled) -- therefore BOTH loops receive an impulse to the right, in the +z-direction. Oh, and I will need to turn off the currents in loops B and C rather quickly, so that they both receive impulses in the +z direction without "feeling" the B fields from each other, for they will be "off" when those fields arrive. If you're concerned about fringe fields, I can add a rod of very high magnetic permeability down the z-axis, extending from A to C, so that essentially all the magnetic field is contained on the z-axis. Whew -- simple thought experiment, but one that could actually be done IMO. What will happen? Will there be detectable momentum imparted to the right (+z direction), but not to the left?


title: Examples where an ill-behaved function leads to surprising results?

text: In mathematical derivations of physical identities, it is often more or less implicitly assumed that functions are well behaved. One example are the Maxwell identities in thermodynamics which assume that the order of partial derivatives of the thermodynamic potentials is irrelevant so one can write. Also, it is often assumed that all interesting functions can be expanded in a Taylor series, which is important when one wants to define the function of an operator, for example $$e^{\hat A} = \sum_{n=0}^\infty \frac{(\hat A)^n}{n!}.$$ Are there some prominent examples where such assumptions of mathematically good behavior lead to wrong and surprising results? Such as... an operator $f(\hat A)$ where $f$ cannot be expanded in a power series?


title: Is light particle of wave?

text: We know that Young's double slit experiment shows that light is a wave. On the other hand photoelectric effect shows that light is made up of photons. How can light be both at the same time?


title: Why is my tupperware sealed after heating it up in the microwave?

text: I had a vegetable soup in a crystal tupperware and I put it in the microwave to heat it up. The funny thing is that when I took it out, it was impossible to open it up. The lid is bent towards the inside, as if the air has escaped the tupperware somehow. Can anyone tell me why this might have happened? Thanks


title: What dark matter can AMS currently find (or exclude)?

text: The rumor mill is running again, this time it's about the AMS experiment (Alpha Magnetic Spectrometer) that's going to make a major announcement soon. I suppose they are looking for peaks in gamma and antiproton spectra due to dark matter annihilation. To get a better idea of what kind of results (positive or negative) we can expect, I'm wondering what kind of models they can probe and what their current reach is. For example, assuming neutralino dark matter, can they strictly exclude $\tilde\chi^0_1$ masses up to $x$ GeV (like LEP)? Or only under such-and-such conditions (e.g. $\tan \beta > \ldots)$? What parameter regions of cMSSM-like SUSY models are within reach, which axion masses, etc.?


title: Separation of solids from liquids using magnetic waves

text: Using magnetic forces, I want to separate solids from liquids in a solution as a centrifuge would do. Is there a way to hit a volume of liquid and get it to separate liquids and solids with heavy pulses of magnetic waves?


title: How to make DIY flight detector for double slit experiment?

text: I want to reproduce double slit experiment. So, is it possible to build flight detector (situated near one slit) at home? Is it possible to buy it somewhere?


title: After using annihilation operator on vacuum state, why it is $0$ instead of vacuum?

text: For bosonic systems, why $a|0\rangle=0$ and not $a|0\rangle=|0\rangle$?


title: Root Mean Square Speed of Gas

text: The RMS speed of particles in a gas is $v_{rms} = \sqrt{\frac{3RT}{M}}$ where $M$ = molar mass; according to this Wiki entry: http://en.wikipedia.org/wiki/Root-mean-square_speed * * *   1. The gas laws state that $pV = nRT$ where $n$ = the number of moles of gas.   2. Further more, the kinetic theory of gases gives the following equation $pV = \frac{1}{3}Nm(v_{rms})^2$ where $N$ = number of particles and $m$ = mass of gas sample. Combining **1** and **2** gives: $nRT = \frac{1}{3}Nm(v_{rms})^2$ which simplifies to: $v_{rms} = \sqrt{\frac{3nRT}{Nm}}$ As $n = \frac{N}{N_{A}}$: $v_{rms} = \sqrt{\frac{3RT}{N_{A}m}}$ Also $m = Mn = \frac{MN}{N_{A}}$. Therefore, $N_{A}m = MN$ Substituting this in: $v_{rms} = \sqrt{\frac{3RT}{MN}}$ However the RMS equation from Wikipedia contains no $N$ or reference to the number of particles. Why does this happen?


title: What does the notion of basis sets for photons in number of particle picture?

text: Since $$|n.\rangle=|u_{k_1}\rangle\otimes...\otimes|u_{k_1}\rangle\otimes|u_{k_2}\rangle\otimes...\otimes|u_{k_2}\rangle\otimes...\otimes|u_{k_m}\rangle\otimes...\otimes|u_{k_m}\rangle$$(There are $n_{k_i}$ terms in $|u_{k_i}\rangle\otimes...\otimes|u_{k_i}\rangle$.) means in each modes there is a fixed number of photons. Does $$|\\{n\\}\rangle$$ means the set of all the possible candidates of $|n.\rangle$ ?


title: Looking for specific Relativity example

text: Many years ago (in the '70s I think) I read an explanation of the meaninglessness of simultaneity at large distances. The example had to do with two people walking along a sidewalk in opposite directions, and an alien race on a planet millions of light-years away planning an invasion of the Solar System. The example showed that in one walker's reference frame the invasion fleet had departed, but in the other reference frame the fleet had not. At the time, the explanation made perfect sense, but I have forgotten the details and have never run across this example again. Does anybody know where this was, or have the text of the explanation?


title: Characteristic length, characteristic time and complexity of the process

text: Different physical processed (starting from elementary particles or even below to the universe itself) have different length scales $L$ and different characteristic times $T$. Larger processed tend to live at slower pace (very roughly, for electron orbitals it is $(10^{-10}\text{m},10^{-15}\text{s})$ while for stars - $(10^{11}\text{m},10^{12}\text{s})$). But going into details, objects of the same size have different characteristic times. For example, a human time scale (from 'blink of an eye' (100ms) to the lifespan (70ys)) is much shorter that for a rock of the same size. The question is if such relation (i.e. $T$ vs. $L$) is related to complexity of a process (either in terms of precision/robustness of it, long range correlations, or structure of interactions between its components)? Any 2D plot of "time scale" vs "length scale" of different processed would be appreciated.


title: Is there a unit system, where all physical laws looks nice?

text: And by nice looking law I mean with no constants. I mean, what would we need to set, so all laws would without those nasty constants in front of them? (I mean all of them, also $\pi$!) What would it mean? What I am trying to say with this question is: Is there a universal language, do particles and gravity live in the same space? If gravity and quantum world should go together, there should be one basis, where you don't need to use any constants.


title: Does dark energy have gravity?

text: So I keep running into this idea on the Internet that only mass has gravity, but isn't mass simply contained energy? Energy bounces around, or travels at the speed of light, because it is not in the form of an atom, it is in the form of light, photons, quarks, neutrinos, and things we haven't discovered yet. Wouldn't it make sense to say that if mass can warp space by its existence, then everything that exists must also have that effect as well, but to a much smaller degree, a square of the speed of light smaller, and we could only detect that gravity it had, if it wasn't flying around at the speed of light? The reason this is important is that light cannot escape a black hole, and I believe that nothing can escape a black hole over time, whether it be light, dark energy, photons, neutrinos, or anything that exists at all, meaning that the fate of the universe changes. If dark energy gets sucked into black holes because it has gravity by its existence, then the pushing effect it has turns into a pulling effect over time, and then the only logical conclusion is the universe is eternal, moving in an endless cycle of big bang, big crunch, big bang, which also explains why the Big Bang happened, and continues to happen, forever. If you say that dark energy remains at constant density despite the expansion of the universe, explain why it does this, and why you think that, and why it doesn't have gravity. When was this proven, and what caused the Big Bang?


title: Can an LC oscillator be used to generate visible light?

text: The LC oscillator is most commonly used to generate radio waves for practical use and the frequency $\omega$ of the LC oscillator equals that of the electromagnetic wave so produced. So, can they in principle be used to emit _visible light?_ The frequency of visible light is on the order of a few hundred terahertz, and the frequency of an LC oscillator is $$\omega = \frac{1}{\sqrt{LC}}$$ I admit, the product $LC$ does become very small (on the order of $10^{-30}$) when the numbers are plugged in, but making an inductor and a capacitor with small values isn't difficult, is it?


title: Deriving Lorentz transformation - Serway

text: in Serway, the derivation of Lorentz transformation started from these two equations $$x′=\gamma(x−ut),$$ $$x=\gamma(x′+ut′).$$ The book derived the first equation and then deduced the second. Is there a proof that if the first is right then the second is also right? To be specific I don't understand:   1. How $t$ was converted to $t′$ in the second equation.   2. Why the position of $\gamma$ didn't change in the second equation.  Note: I understand the first equation well but I don't understand how to reach the second from the first.


title: How quickly should a fluid come to hydrostatic equilibrium?

text: Let's say I'm holding a one-liter water bottle, full of water, which I then drop. Before dropping the water bottle, the equilibrium is for there to be a pressure gradient in the water canceling the gravitational force on the water. While the bottle is in free fall, the new equilibrium is constant pressure everywhere. Should I expect the water to come to this new equilibrium in the few tenths of a second it takes the water bottle to fall? I expect the answer is basically yes, because density changes (and therefore pressure changes) should propagate at around the speed of sound, and p-waves might bounce around a few times while exponentially dying away (depending on boundary conditions created by the material of the bottle?), at the end of which we have equilibrium. So for a 30-cm bottle with sound speed 1500 m/s, I might guess the time is a few times .02s, which is longer than the ~.5s it takes for the bottle to fall from my hand to the ground. Does this sort of reasoning make any sense? How can I justify it in a less handwavy manner?


title: Why proper acceleration is $du/dt$ and not $du/d\tau$?

text: Wikipedia says: > In relativity theory, proper acceleration[1] is the physical acceleration > (i.e., measurable acceleration as by an accelerometer) experienced by an > object. and says: > In the standard inertial coordinates of special relativity, for > unidirectional motion, proper acceleration is the rate of change of proper > velocity with respect to _coordinate time_. Why coordinate time? As far as I know the moving observer itself measures the proper things. It's clock measures proper time, it's speedometer measures proper velocity. So if the observer accelerates with $10m/s^2$ (and feels that proper acceleration), isn't the speed shown by the local speedometer changes $10m/s$ per second on the local clock?


title: Decoherence when no one is looking?

text: I understand that in the single-electron-at-a-time double slit experiment, if a detector is placed before the slit, the interference pattern vanishes. Suppose I left the detector on, but put a bag over its screen (I can't tell what state the electron is before it passes the slits), does the interference pattern come back? If so, does that mean the electron "knows" I'm not "looking" and proceeds to interfere with itself? Edit: I had photons passing through the slit initially. I've updated it to electrons to reflect a more common setup.


title: Lifetime of battery

text: If I directly connect two terminals of 3V battery (negative to positive) using copper wire, would it lose all its charge faster compared to another 3V battery that is used to lighten a 1.5V bulb?


title: Kinetic energy and temperature

text: I've randomly been thinking about smoothies and internal energy all weekend. If we have an assortment of fruit in the solid phase and then proceed to blend it all so that it ends up being in the liquid phase, am I right in saying that work has been done on the solid fruit such that its internal energy is higher, therefore it remains in the liquid phase at the same temperature as the room (once thermal equilibrium is reached). I'm having a bit of difficulty in understanding how another identical assortment of solid fruit and the smoothie liquid can both be at equilibrium if their kinetic energies are different (liquid has more than a solid right?). I thought that kinetic energy determines temperature??


title: Symmetry groups

text: I am quite new to this subject. I am just repeating in a few words, what I have learned so far: There are 4 fundamental forces of nature: strong, weak, electromagnetism and gravity. Physicists are trying to re-generate a condition, a very high temperature during the Big Bang, to find out which force actually works during that time. The Weinberg-Glashow and Abdus Salam model unified weak and electromagnetism to electro-weak force. Gravity still lays undefined in the Standard Model. At Planck's scale, physicists observe that the 4 forces of nature unify to one single force. We have not yet been able to define that force as yet but the quest is on. Now the U(1), SU(2), SU(3)........What are these? Can anyone please help me understand, sequentially how the theories evolved? The Grand Unified theory is one which is SU(5)? Does SU stand for special unitary group?


title: Why is the Horizontal Force Constant in Deriving the One Dimensional Wave Equation

text: My textbook in deriving the wave equation for a one dimensional elastic string stated that the horizontal direction force is constant.I understand that the horizontal components of the tensions on either side of the element have to be equal since it is assumed it moves only in the vertical plane but why does it have to be constant? Don't they just have to be equal?


title: Finite potential well - transcendent equation for even solutions

text: I have a finite square well like the one on the picture below: ![](http://i.stack.imgur.com/3PGLK.png) I have done some calculations on it and got a _transcendental equation_ for even solutions which is like this: $$ \boxed{\dfrac{\mathcal{K}}{\mathcal{L}} = \tan \left(\mathcal{L \dfrac{d}{2}}\right)} $$ where: \begin{align} &\mathcal{K}\equiv \sqrt{\frac{2m (W - W_p)}{\hbar^2}} &&\mathcal{L} \equiv \sqrt{\frac{2mW}{\hbar^2}} \end{align} I have been looking in Griffith's book where he says that _transcendental equation_ can be solved graphically so i did try to write it down so i could draw it (i inserted $\mathcal K$ and $\mathcal L$): $$ \sqrt{1 - \dfrac{W_p}{W}} = \tan\left( \frac{\sqrt{2mW}}{\hbar} \frac{d}{2} \right) $$ I noticed that i don't even know what am i looking for. Is it the total energy $W$? If so i must know potential $W_p$ outside the well, width $d$, and a mass of a particle $m$. But still how could i graphically solve this? * * * EDIT: I set all the constants to be $W_p=d=m=\hbar=1$ and ploted a graph for an even solutions: ![enter image description here](http://i.stack.imgur.com/25DUN.png) I allso plotted a graph for the odd solutions which for which _transcendental equation is_ ($\mathcal K$ and $\mathcal L$ stay the same as before): $$ \boxed{-\dfrac{\mathcal{L}}{\mathcal{k}} = \tan \left(\mathcal{L \dfrac{d}{2}}\right)} $$ $$ -\sqrt{\frac{1}{1 - W_p/W}} = \tan\left( \frac{\sqrt{2mW}}{\hbar} \frac{d}{2} \right) $$ ![enter image description here](http://i.stack.imgur.com/jDq1I.png) Could anyone please confirm that i got the right graph for transcendental equation?


title: How deep can my knowledge of particle physics go without the maths?

text: By no means do I have the mathematical background to understand most of the math used in elementary particle physics. My current knowledge is of all the elementary particles and how they interact and build upon each other to build the macroscopic world. Will I be able to continue forward and understand more in particle physics or should I stop now and start learning probability theory and whatever else I might need? Is there a place where I can start to learn the maths? Is there a place where I can learn all the maths but on a novice level? The coolest thing I have seen/understood is how the electromagnetic force works: Photons bounce back and fourth between particles in atoms creating the force field. So the thing that keeps my hand from going through the keys as I type is the electromagnetic force, specifically photons (please correct me if I am wrong about any of this).


title: Probabilistic vs Statistical interpretation of Double Slit experiment

text: Why is it assumed that the results seen in the double slit experiment are probabilistic and not just a statistical result of some unknown variable or set of variables within the system.


title: The temperature in space

text: 1. We know that cosmic microwave background temperature is about 2.7K. But what temperature we will measure in space using a simple Kelvin thermometer in the shadow? Can it be lower than 2.7K?   2. Suppose a space ship is flying in our solar system with a speed of 0.99c relative to earth. Will it measure a different temperature? Much higher? And if so, will the intense heat cause the ship itself to warm up?


title: Using eigenvalues to determine the stability/behaviour of the system

text: first time I've been on physics.se but have used the math and cs before... Anyway, here's my question: If we have a damped pendulum described by the equation $$y'' + ay' + b = 0 , a>0$$ Using the conversion $x_0=y$ and $x_1=y'$ we can convert this into a set of first-order ODEs as follows: $$ x_0'=x_1$$ $$x_1'=-ax_1+b$$ or in matrix form : $$\mathbf{x'}=\begin{bmatrix}0 & 1\\\0 & -a\end{bmatrix}\mathbf{x}+\begin{bmatrix}0 \\\ b\end{bmatrix}$$ Now, the exercise where I'm doing this now says we can use the eigenvalues to determine the stability of the system the eigenvalues of the system. The eigenvalues are $\lambda = 0$ and $\lambda = -a$... I know that an eigenvalue of 0 means a neutral equilibrium such that a small change in the variable means it remains at the changed variable and that a negative eigenvalue means a small change will return to its original value, but I'm struggling to see how this relates to the physical situation of the damped oscillator, and what it means about the behaviour of the system.. Any thoughts would be appreciated


title: Distance at which a 300m asteroid is visible

text: Using the best available telescopes, and assuming a typical albedo of 10%, at what distance can a 300m asteroid be seen (well enough to start plotting its course) in 1) visible and 2) infrared wavelengths?


title: Is it possible to create a head-mounted optical device that allows the user to see the wind?

text: Is it possible to create a head-mounted optical device that allows the user to see the wind, by means of thermal imaging / infrared detection, variations in the light's refractive index, or something similar?


title: Help: 3D visualization of magnetic field around moving point charge

text: The diagrams I'm able to find online only show the concentric field lines in the particle's plane, perpendicular to the motion, which of course generalizes to the cylindrical shape around a conductor. Can someone share a nice representation of the three-dimensional magnetic field all around a moving point charge?


title: How does a steady wind flow generate sound?

text: When a wind blow through sharp edge, say, edge of a paper, you can see the vibration of the paper and hear the sound. For this type of oscillation, it should be a damped oscillation with external driving force $\mathcal{F}(x)$: $\ddot{x}+2\lambda\dot{x}+\omega_{0}^{2}x=\mathcal{F}(x,t)$ with the driving force in the form of sinusoidal $\mathcal{F}(x,t)\sim F\sin(\omega t)$ (It should be a 3D version though). However, if the air flow is really steady, the external force should be constant so there is no oscillation at all from this equation. A simple guess is that the motion of the paper influence the streamline of air flow so that the pressure gradient provides the exact sinusoidal driving force. This mechanism is reasonable because the propagation of sound wave has the oscillation relationship between the displacement and the pressure. A question here is how the air flow sustain the energy loss due to the damping. It is also not easy to think of the initial condition as well as external force for these oscillation. So 1) What exactly is the physical mechanism to generate sound when a 'steady' air flowing through a sharp edge? 2) Another similar question is how the driving force act in the resonance in musical instrumental such as pipe? A 'steady' flow of air is also provided somewhere to the pipe, but air oscillates inside the pipe. **Edit** : From the answers, people suggest few mechanism for different situations. Few more questions: 3) For static object, Kármán vortex street can form behind the object. So is the sound frequency the same as frequency of vortex generation, or the same as the expansion of the vortex? Sound is a spherical wave propagating outward, so identify the sound frequency should locate the point of generation. 4) Where is sound generated in the situation (1)? Is the sound frequency the same as the frequency of the vibrating paper? Fluid mechanics is a difficult topics, but there are approximation for special cases trying to explain the underlying mechanism, such as the flag fluttering cited by j.c.


title: Linearity of quantum mechanics and nonlinearity of macroscopic physics

text: We live in a world where almost all macroscopic physical phenomena are non- linear, while the description of microscopic phenomena is based on quantum mechanics which is linear by definition. What are the physics points of connection between the two descriptions?


title: Could teleportation ever really happen?

text: Could any form of matter teleportation every really happen? If so, what form would it take? Some science-fiction talks about literally _sending_ the atoms from one location to another; others talk of creating a duplicate and destroying the original. Question originally added here. **Update:** there are a couple of comments asking me to specify what I mean by **teleportation**. I'm talking about stuff of Star Trek - taking an object or person and transporting them almost instantly across great distances (thousands of KMs) by:   1. breaking them up at the atomic (or smaller) level   2. transporting those atomic components (or the _information_ , "recipe" to recreate) and   3. (re-)assembling at the new location.


title: How to determine the order of indications of a clock?

text: Given the description of a clock $\mathcal A$, as   * (1) a set $A$ of all (more than 2) distinct indications of this clock, **in no particular order**   (where the individual indications contained in set $A$ are denoted below as "$A_J$", "$A_Q$", "$A_V$" etc.), together with   * (2) a function $t_{\mathcal A} : A \rightarrow \mathbb R$   (such that we may speak of set $A$ and function $t_{\mathcal A}$ together as "a clock, $\mathcal A$" at all), and, in line with Einstein's assertion that:   _"All our well-substantiated space-time propositions amount to the determination of space-time coincidences {such as} encounters between two or more {... participants}"_,   given   * (3) a sufficient suitable account, **in no particular order** , of participants (such as $A$, $J$, or $Q$) and their encounters (where $A_J$ is $A$'s indication at the encounter of $A$ and $J$, $A_Q$ is $A$'s indication at the encounter of $A$ and $Q$, and so on) -- How can **the order** of indications of clock $\mathcal A$ be derived? Or to consider the simplest case:   How can be determined whether   $A_J$ was between $A_Q$ and $A_V$, or   $A_Q$ was between $A_V$ and $A_J$, or   $A_V$ was between $A_J$ and $A_Q$   ? **Note:**   It should not be assumed that clock $\mathcal A$ was "monotonous";   in other words: it is not an acceptable answer to argue   "if $(t_{\mathcal A}[ A_V ] - t_{\mathcal A}[ A_Q ]) \times (t_{\mathcal A}[ A_Q ] - t_{\mathcal A}[ A_J ]) > 0$ then $A_Q$ was between $A_V$ and $A_J$". **Another note in response toa comment by Jim:**   The encounters (coincidence events) in which the participant took part whose set of indications at those encounters is called $A$ (and who is therefore him/her/itself conveniently called $A$ as well) may be denoted as   "$\mathscr E_{AJ}$", "$\mathscr E_{AQ}$", "$\mathscr E_{AV}$",   among others;   corresponding to the indications "$A_J$", "$A_Q$", "$A_V$" of $A$ (and, together with function $t_{\mathcal A}$, therefore indications of clock $\mathcal A$) which had been named explicitly above. As far as a Lorentzian manifold may be assigned to those events in which $A$ took part (together with additional events, as may be required) and as far as the corresponding Causal structure would be determined, those events in which $A$ took part were all elements of one chronological (or timelike) Curve. My question is consequently, in other words, whether and how elements of such a chronological (or timelike) curve may be ordered, given their parametrization only by some arbitrary function $t_{\mathcal A} : A \rightarrow \mathbb R$,   together, of course, with a sufficient suitable account (3, above) of all participants involved, and their encounters, if any, in no particular order.


title: Have general relativistic effects of the sun's rotation been measured?

text: I was wondering if general relativistic effects of the sun's rotation have also been measured, like gravity probes A and B measured GR effects from the earth.


title: Photon on null geodesic

text: If given an FRW metric $ds^2 = -dt^2 + a^2(t)[dx^2+dy^2+dz^2]$ and for the trajectory followed by a photon (null geodesic; $ds^2=0$) with affine parameter $\lambda$, know that $g_{\mu\nu}\,\frac{dx^{\mu}}{d\lambda}\,\frac{dx^{\nu}}{d\lambda} = 0.$ How does one find $\frac{dt}{d\lambda} = \frac{w0}{a(t)}$? Already found the nonzero Christoffel coeffs and the remaining geodesic equations, $ 0 = \frac{d^2x^i}{d\lambda^2} + \Gamma^i_{ti}\,\frac{dt}{d\lambda}\,\frac{dx^i}{d\lambda} $ and $ 0 = \frac{d^2t}{d\lambda^2} + \Gamma^t_{ii}\,\frac{dx^i}{d\lambda}\,\frac{dx^i}{d\lambda} $ with $ \Gamma^i_{ti} = \frac{1}{a(t)}\,\frac{da(t)}{dt} $ and $ \Gamma^t_{ii} = a(t)\,\frac{da(t)}{dt} $ I expect it to be pretty simple (yet not seeing it) and that the null geodesic in the $x$-direction come from something like (?): $ ds^2 = -(dt^2)+a^2(t)\,dx^2 = 0 \rightarrow \frac{dt}{a(t)} = \pm dx $ I also believe that $dx/d\lambda = c$ if the affine parameter is related by $\lambda = t$. This is where I am confused and need it to move on to look at the cosmological redshift and derive the ratio of emitted and observed energies of a photon at $t1$ and $t2$. Anything to straighten me out would be great!


title: Does quantum computing rely on particular interpretations of quantum mechanics?

text: It is my understanding that quantum computing relies on quantum superposition and entanglement to work--qbits must exist in all states simultaneously before giving a particular result when observed. Would this mean that quantum computing is impossible in interpretations of quantum mechanics in which qbits are not in reality existing in all states simultaneously until observed? Thus, wouldn't quantum computing be incompatible with non-local hidden variable interpretations (deBroglie-Bohm, for example), or with other interpretations in which the underlying reality is deterministic such as 't Hooft's?


title: Derivation of $ E=h\nu$

text: Is it possible to derive the relation $ E=h\nu$ from Schrodinger equation or the basic principles of quantum mechanics or is it something which is considered to be an axiom with no explanation?


title: Product of $\gamma^5 \sigma^{\mu\nu}$

text: I'm trying to prove that $\gamma^5 \sigma^{\mu\nu}=\frac{i}{2}\epsilon^{\mu\nu\alpha\beta}\sigma_{\alpha\beta}$ I started with the left hand side and expanded the $\gamma^5$ to $\gamma^5=\frac{i}{4!}\epsilon^{\mu\nu\alpha\beta}\gamma_\mu\gamma_\nu\gamma_\alpha\gamma_\beta$ After opening the $\sigma^{\mu\nu}$ matrix and multiplying it by the $\gamma^5$ from left I got $\frac{i}{2}\left\lbrace \frac{i}{4!}\epsilon^{\mu\nu\alpha\beta} \gamma_\mu\gamma_\nu\gamma_\alpha\gamma_\beta\gamma^\mu\gamma^\nu-\frac{i}{4!}\epsilon^{\mu\nu\alpha\beta} \gamma_\mu\gamma_\nu\gamma_\alpha\gamma_\beta\gamma^\nu\gamma^\mu\right\rbrace$ But after using anticommutation of the matrices and the fact that $\gamma^\mu\gamma_\mu=4$ I get $\frac{i}{2}\frac{2i}{3}\left[\gamma_\alpha,\gamma_\beta \right]$ which would be good if the $\frac{2i}{3}$ would be $\frac{i}{2}$ I tried several times but always got the same result. What am I missing here?


title: Observed composition of UHE cosmic rays

text: How much is known about the composition of ultra high energy cosmic rays (say $E>10^{20}\text{ eV}$)? I get the impression that the particles are often assumed to be protons or other heavier nuclei, but what _empirical_ basis do we have for this? Is there experimental evidence that excludes photons or leptons as candidates? Do we know if the UHE's are matter or antimatter? I understand that there is quite a bit of literature examining the possible mechanisms of _production_ of UHE's as nuclei of various masses. For example, it could be easier for Iron nuclei to get up to high energies in a magnetic field since they would have to be accelerated for less time than protons. However, I don't want to put the cart before the horse; the fact is that we don't know _where they come from_ , and so we should not base our knowledge of _what they are made of_ on these theories.


title: Does rotation cause gravity?

text: I want to ask that what causes gravity. My opinion is that it is the rotation that causes gravity, when we see a whirlpool in water it takes anything that comes near to it in to the center of it. I have a doubt that is even gravity caused due to rotation of earth. When earth rotates, it takes the atmosphere with it and that pressure is what we experience it as gravity, as it takes towards the center of the earth we assume it as a downward force. Is my assumption correct?


title: How does Spacetime Curvature increase the velocity of particles falling towards the earth?

text: ![enter image description here](http://i.stack.imgur.com/jbG4x.jpg) Two particles fall side by side, towards the earth. The horizontal distance between them is 10m. As they advance nearer and nearer to the earth's surface, the horizontal distance decreases, from 10m to say 8m and the particles move closer and closer towards each other. I understand that happens because spacetime curves inwards and so the particles move towards each other. What I dont understand is how the spacetime curvature effect the particle's velocity. How does spactime curvature induce acceleration? A non-mathematical answer would be preferred.


title: Commutators in bra-ket notation

text: 2-d Hilbert space, with 2 (orthogonal) kets $|a\rangle$ and $|b\rangle$ Operator $A=|a\rangle\langle b| + |b\rangle\langle a| $ Operator $B=-i|a\rangle\langle b| +i|b\rangle\langle a| $ Commutator $[A,B]=AB-BA$ When I try to compute the commutator I end up getting expressions like $|a\rangle\langle a| $ , i.e a bra multiplied by a ket. How am I meant to calculate this?


title: Relationship between frequency and amplitude of mechanical waves

text: Can two mechanical waves carry same energy whose frequencies are different?(assuming same medium but having different amplitudes


title: Average Velocity of a body moving in a circle with constant speed $v$

text: > A Body is moving with constant speed $v$ along a circle of radius $R$. Find > the average velocity of the body from time $t = 0 $ to $t= \frac{R}{3V}$. My attempt at the question: Let distance traveled along the circle be $d$. $$ d= \left(\frac{R}{3v}\right)\cdot v \Rightarrow d= \frac{R}{3}$$ Let angle subtended by arc be $\theta$. Then $$ \theta = \frac13 rad. $$ By sine rule I can find out the displacement $S$:   $$S = 2R\sin(1/6) $$ Hence average velocity : $$\text{average velocity} = \frac{S}{t} \Rightarrow \text{average velocity}= 6v\sin(1/6) $$ But the correct answer is $\dfrac{3v}{\pi}$.   I need some help here as to how to solve it.


title: Static geodesics in GR

text: Can we find static geodesics of the type $$x^{\nu}=x_0^{\nu}+\delta_0^{\nu}\tau$$ in some space-time other than Minkowski's?


title: Do the effects of special relativity occur instantaneously?

text: Imagine a second earth one lightyear away from our earth. Now imagine a bunch of scientists with a plane and some atomic clocks who are living on this second earth and who perform the Hafele-Keating Experiment, which is the experiment that was done in 1971 on our earth to prove the relativistic effects of kinematic and gravitational time dilation: > Joseph C. Hafele, a physicist, and Richard E. Keating, an astronomer, took > four cesium-beam atomic clocks aboard commercial airliners. They flew twice > around the world, first eastward, then westward, and compared the clocks > against others that remained at the United States Naval Observatory. When > reunited, the three sets of clocks were found to disagree with one another, > and their differences were consistent with the predictions of special and > general relativity. _(Wikipedia)_ Now lets suppose both earths are not moving or at least that they are not moving in relation to each other. Ignoring the gravitational time dilation, we can tell already from the experiment of 1971 that the atomic clocks on the fast moving plane will run a bit slower than the static atomic clocks on earth-two due to the kinematic time dilation of special relativity. Because earth-one is not moving in relation to earth-two we can expect that if there was another atomic clock on earth-one it would run just as fast as the clocks on earth-two, hence also faster than the ones on the plane. Now my question: Does this asynchronity between the time on earth-one and the time on the highspeed-plane flying around earth-two start the moment the plane starts (ignoring the short time of acceleration the plane needs) or does it start much later, when the light (information) of the flying plane arrives at earth-one (which would be one year delayed because of the distance of one lightyear between the two earths)? In other words: Do the effects of special relativity occur instantaneously, independed of the distance between the observing and the observed frame of reference?


title: Correspondence between wave function and state vector

text: I am confused with connection between state $| \psi \rangle$ of a quantum system and corresponding wave function $\psi(x)$ (at a given time). I have been told that for every state $| \psi \rangle$ we can define $\psi(x) \equiv \langle x | \psi \rangle$, where $|x\rangle$ are eigenkets of position operator. So basicaly wave function is continuous coefficient in the expansion of $| \psi \rangle$ in the basis eigenkets $| x \rangle$, right? But this can be true for any state of the system only if the state space of the system is a subspace of the space spanned by position eigenkets $| x \rangle$, right? Moreover this has to be true for every quantum system. How is this possible?


title: Are covariant derivatives of Killing vector fields symmetric?

text: I'm reading the Lecture Notes on General Relativity by Matthias Blau, and in section 9.1 (point 1) he writes: > Let $K^\mu$ be a Killing vector field, and ${x^\mu(\tau)}$ be a geodesic. > Then the quantity $$Q_K=K_\mu\dot{x}^\mu$$ is constant along the geodesic. Now, in equation (9.2) he writes $$\nabla_\nu{K}_\mu\dot{x}^\nu\dot{x}^\mu=\frac{1}{2}(\nabla_\nu{K}_\mu+\nabla_\mu{K}_\nu)\dot{x}^\mu\dot{x}^\nu$$ which vanishes due to Killing's equation. The thing I don't really understand is why ${\nabla_\nu{K}_\mu-\nabla_\mu{K}_\nu=0}$. I can see that because ${\mu,\nu}$ are dummy indices, then \begin{align}\nabla_\nu{K}_\mu\dot{x}^\mu\dot{x}^\nu&=\nabla_\mu{K}_\nu\dot{x}^\nu\dot{x}^\mu\\\&\Longrightarrow\,\nabla_\nu{K}_\mu-\nabla_\mu{K}_\nu=0 \tag{1}\end{align} Is this argument valid? If so, does $(1)$ hold in general for Killing vectors? As I'm almost sure it doesn't, despite the title of my question (all Killing vector fields would be gradient fields), does it happens only along the geodesic? If so, what _interpretation_ would this condition have?


title: Permittivity of an object

text: Is it possible that an object is a conductor and its permittivity is equal to the permittivity of vacuum? Is there any relationship between the permittivity and the conductivity?