title: Integrals of Power

text: This question has two parts: a question about a specific expression and a question about a family of expressions. _In what follows I will omit all constants of integration_. If we integrate force with respect to time we get momentum: $\int F \, \operatorname{d}\\!t = mv$. If an object accelerates along a line then we can find its acceleration at any given point and write force as a function of distance. Doing so, and integrating gives kinetic energy (or work done): $\int F \, \operatorname{d}\\!x = \frac{1}{2}mv^2$. I decided to try the same thing with power using the definition $P = Fv$. When I integrated with respect to time I got, as expected, $\frac{1}{2}mv^2$ which is work done. If an object accelerates along a line then we can measure the power at regular distances and write power as a function of distance. Then we may integrate power with respect to distance: $$\int P \, \operatorname{d}\\!x = \int mav \, \operatorname{d}\\!x = m\int\frac{\operatorname{d}\\!v}{\operatorname{d}\\!t}v\, \operatorname{d}\\!x = m\int\frac{\operatorname{d}\\!x}{\operatorname{d}\\!t}\frac{\operatorname{d}\\!v}{\operatorname{d}\\!x}v\,\operatorname{d}\\!x = m\int v^2\operatorname{d}\\!v = \tfrac{1}{3}mv^3$$ **Here is the first question** : _What is $\frac{1}{3}mv^3$? Does it have a name? How can I understand it conceptually?_ **Here is the second question** : Momentum is $mv$, kinetic energy is $\frac{1}{2}mv^2$, this new quantity $\frac{1}{3}mv^3$ has turned up. It seems that $\frac{1}{k}mv^k$ for positive whole numbers $k$ plays a role. These can all be combined into one power series. Provided $v$ is very small, we have $$mv + \tfrac{1}{2}mv^2 + \tfrac{1}{3}mv^3 + \cdots + \tfrac{1}{k}mv^k + \cdots \equiv m\ln\left(\frac{1}{1-v}\right)$$ _Do the expressions $\frac{1}{k}mv^k$ have any general meaning; and does this logarithmic expression have any use?_ (I realise that the units in the sum are all different.)


title: Surface gravity of Kerr black hole

text: I'm going through Kerr metric, and following the 'Relativist's toolkit' derivation of the surface gravity, I've come to a part that I don't understand. Firstly, the metric is given by $$\mathrm{d}s^2=\left(\frac{\Sigma}{\rho^2}\sin^2\theta\omega^2-\frac{\rho^2\Delta}{\Sigma}\right)\mathrm{d}t^2-2\frac{\Sigma}{\rho^2}\sin^2\theta\omega \mathrm{d}\phi \mathrm{d}t+\frac{\Sigma}{\rho^2}\sin^2\theta \mathrm{d}\phi^2+\frac{\rho^2}{\Delta}\mathrm{d}r^2+\rho^2 \mathrm{d}\theta^2$$ With $$\rho^2=r^2+a^2\cos^2\theta,\quad \Delta=r^2-2Mr+a^2,$$ $$\Sigma=(r^2+a^2)^2-a^2\Delta\sin^2\theta,\quad \omega=\frac{2Mar}{\Sigma}$$ The Killing vector that is null at the event horizon is $$\chi^\mu=\partial_t+\Omega_H\partial_\phi$$ where $\Omega_H$ is angular velocity at the horizon. Now I got the same norm of the Killing vector $$\chi^\mu\chi_\mu=g_{\mu\nu}\chi^\mu\chi^\nu=\frac{\Sigma}{\rho^2}\sin^2\theta(\Omega_H-\omega)^2-\frac{\rho^2\Delta}{\Sigma}$$ And now I should use this equation $$\nabla_\nu(-\chi^\mu\chi_\mu)=2\kappa\chi_\nu$$ And I need to look at the horizon. Now, on the horizon $\omega=\Omega_H$ so my first term in the norm is zero, but, on the horizon $\Delta=0$ too, so how are they deriving that side, and how did they get $$\nabla_\nu(-\chi^\mu\chi_\mu)=\frac{\rho^2}{\Sigma}\nabla_\nu\Delta$$ if the $\Delta=0$ on the horizon? Since $\rho$ and $\Sigma$ both depend on $r$, and even if I evaluate them at $r_+=M+\sqrt{M^2-a^2}$ they don't cancel each other. How do they get to the end result of $\kappa$?


title: Is a white object always white?

text: Is a object that’s white with the lights on, still white although it looks different under a different light or in total darkness?


title: Proof Human Designs for Fusion Could Work

text: Are there convincing proofs on paper that fusion is possible the various ways humans are trying to design machines to do it or is the only way absolutely proven to work in theory also the only way actually working reactors (i.e. stars) work, that is, getting a huge quantity of mass and letting gravity do its work?


title: Knowing the mass and force acting on a particle, how do we derive the relativistic function for velocity with respect to time?

text: Use this scenario: An electron gains speed in the Stanford Linear Accelerator (SLA) across 3000 meters, reaching a final velocity of 0.95c due to a constant force pushing on the electron. Given the mass of the electron and the constant force pushing on the electron, what is the function for velocity in terms of time? How do we derive this?


title: Matrix representing the quantity - why can some matrices not be physical quantity?

text: In Heisenberg picture, my textbook says that the following matrix $A = \frac{5}{3}\Sigma_1 + i\frac{4}{3}\Sigma_2$ cannot represent physical quantity. the book says this is because $\frac{5}{3}\Sigma_1$ and $\frac{4}{3}\Sigma_2$ cannot have definite values. At here, $\Sigma_1$ and $\Sigma_2$ represent Pauli matrices. I am curious why this is; how can $\frac{5}{3}\Sigma_1$ and $\frac{4}{3}\Sigma_2$ cannot have definite values?


title: How is charge measured and mass of the electron at the same time?

text: 1. There are few constants that usually come together, $e/m$ also $h/e$. How are they decoupled?    2. If the speed of light is "derived" as Wikipedia states how meter defined and time?


How can the properties of the Earth be determined? And how is the gravitational constant then determined from them? A satellite can determine the G.M.. But, how they are individually determined nowadays ?


title: Does a microwave resonantly excite the rotational levels when cooking?

text: Wikipedia states there is no resonance absorption, but says at the same time that the molecules are oscillating like dipoles, which is kind of the same if you are exciting the rotational levels ? The spacing in the rotational levels will be almost independent of wavelength in the $\mathrm{MHz}\, -\, \mathrm{GHz}$ range?


title: How the bond angle of a water molecule is measured?

text: How the angle of a water molecule is measured? The bond angle for water was first determined by vibrational and rotational analysis of its infrared absorption spectra. How is bond angle extracted?


title: What is the physical meaning of enhanced gauge symmetries in string compactifications?

text: For some very special cases of string compactification backgrounds, we get mysterious enhanced gauge symmetries. What is their physical meaning, and why do they arise? Is there any geometrical interpretation for the enhanced gauge symmetries?


title: Extracting acceleration vector from rotated aircraft

text: Suppose we have an aircraft with accelerometer measuring accelerations along each axis. It is mounted in a way so it is perpendicular to the plane in all axes (that should be obvious). We also have gyrometer sensor mounted on the plane - it can tell us how is the plane currently aligned (rotated) in air. **1st case:** Suppose we have an aircraft on the ground. Then we can get our desired acceleration component just by reading the accelerometer's Z axis reading (Z axis is pointing from sky to the center of the earth). **2nd case:** The plane is in the air, doing some loops for example - now to get the desired reading as previously (acceleration towards earth) we would need to compensate the Z axis acceleration with current rotation and/or accelerations. So, I have all the data but I don't know what physical/mathematical formulas I should use to extract the acceleration component pointing towards the earth. I would also be glad to see some tips about what literature I should read to be able to do this.


title: Solar neutrino predictions

text: The solar neutrino problem has been "solved" by discovering that neutrinos have mass and they oscillate. So how accurate are _now_ our predictions about the number and types of solar neutrinos that reach the earth?


title: Characteristic x-ray in energy spectrum

text: Context: Monte Carlo simulation of a linear accelerator photon beam. The energy spectrum for photons as calculated from the phase space files found in here has a peak somewhere near $500\,\text{keV}$. The characteristic x-rays on the other hand for tungsten fall roughly in the range $60-70\,\text{keV}$. These photons represent a beam just after the flattening filter and just before the secondary collimators. **Question 1** : Does anyone have any idea why the peak is so off? Or perhaps it represents something entirely different? **Question 2** : Since we are here, one more clarification: The term "characteristic x-rays" is used in _electrons_ energy spectrum, right ? Isn't it wrong to use it the way I do here ? ![enter image description here](http://i.stack.imgur.com/Nx3kd.png)


title: What is the place of an electromagnetic field in the electromagnetic spectrum?

text: What place should I give an electromagnetic field (produced by a current conducting coil) in the electromagnetic spectrum? What will its wavelength and frequency be?


title: After normalizing a wavefunction I don't know how to calculate probability on an interval (-0.1 + 0.1)

text: This is quite large homework where I 1st had to normalize the wavefunction $\psi = Axe^{-x^2/2a}$ and I got a constant $A=\sqrt{2/(a\sqrt{\pi a})}$. How do I calculate the probability now for the interval (-0.1nm , + 0.1nm) if $a=1nm^2$? * * * I first tried to calculate the normalisation factor but the units are weird... is this possible? Is most likely my normalisation factor wrong? $$A=\sqrt{\frac{2}{1nm^2 \cdot \sqrt{\pi\cdot1nm^2}}}=1.06\frac{1}{nm\sqrt{nm}}$$ * * * **EDIT:** After thinking again I decided to include the derivation of the normalisation factor $A$: \begin{align} &\phantom{=}\int\limits_{-\infty}^\infty A^2x^2 \exp\left[\tfrac{-x^2}{a}\right]dx = 1 = A^2\int\limits_{-\infty}^\infty \underbrace{x}_f\cdot \underbrace{x\exp\left[\tfrac{-x^2}{a}\right]}_{dg/dx} dx= \\\ &=A^2\left[\smash{\underbrace{\left.x\left(-\tfrac{a}{2}\right)\exp\left[\tfrac{-x^2}{a}\right]\right|_{-\infty}^{\infty}}_0}-\int\limits_{-\infty}^\infty\left(-\tfrac{a}{2}\right)\exp\left[\tfrac{-x^2}{a}\right]dx\right]=\\\ &=\tfrac{A^2a}{2}\int\limits_{-\infty}^{\infty} \exp\left[\tfrac{-x^2}{a}\right]dx = \tfrac{A^2a}{2}\int\limits_{-\infty}^{\infty} e^{-t^2}\sqrt{a}\,dt = \tfrac{A^2a\sqrt{a}}{2}\int\limits_{-\infty}^{\infty} e^{-t^2}dt = \frac{A^2a\sqrt{a}}{2}\sqrt{\pi} \end{align} Which means that: \begin{align} \frac{A^2a\sqrt{a}}{2}\sqrt{\pi} &= 1\\\ A&=\sqrt{\frac{2}{a\sqrt{a\pi}}} \end{align}


title: Chiral anomalies à la Fujikawa: Why don't we just take another measure?

text: When deriving the chiral anomaly in the non perturbative approach for a theory of massless Dirac fermions, you start by showing that the path-integral measure is not invariant unter the chiral transformation: $\psi \rightarrow U(x) \psi = e^{i\alpha(x)\gamma^5}\psi\\\\\overline{\psi} \rightarrow \overline{\psi} \,\overline{U}(x) = \overline{\psi} \, \gamma^0 e^{-i\alpha(x)\gamma^5}\gamma^0 = \overline{\psi} \, U(x)$ and the measure transforms as: $d\overline{\psi}d\psi \rightarrow \det[U \overline{U}]^{-1}\,d\overline{\psi}d\psi = \det[U]^{-2}\, d\overline{\psi}d\psi $ The you proceed with evaluating the determinant to find the anomaly term. (It is done like this for example in Weinberg chap. 22.2) But now imagine we write the massles Dirac Lagrangian in its chiral components $\psi_L$ and $\psi_R$: $\mathcal{L} = \overline{\psi} i \gamma^\mu D_\mu \psi= i \psi_L^\dagger \bar{\sigma}^\mu D_\mu\psi_L + i \psi_R^\dagger {\sigma}^\mu D_\mu\psi_R $ This is now a Lagrangian of two independent Weyl spinors, one left handed one right handed. The chiral transformation from a above now reads $\psi_L \rightarrow e^{-\alpha(x)} \psi_L \,\,, \hspace{1cm} \psi_L^\dagger \rightarrow e^{\alpha(x)} \psi_L^\dagger\\\ \psi_R \rightarrow e^{\alpha(x)} \psi_R \,\,, \hspace{1cm} \psi_R^\dagger \rightarrow e^{-\alpha(x)} \psi_R^\dagger$ We write the measure for the the path integral now as $d\psi_L^\dagger d\psi_L d\psi_R^\dagger d\psi_R $ which is lorentz and gauge invariant. But it is invariant under the chiral transformation! So what happend, where is our Anomaly gone? In rewriting the measure $d\overline{\psi}d\psi$ as two measures of Weyl fileds the $\gamma^0$ got dropped, which before was responsible for changing the sign in the exponent of the transformation once more and made the measure non-invariant. But i don't see anything wrong in rewriting the measure in this way. Even for the massive case this should be a legible procedure, or am i wrong? Probably somebody can give me a lesson in constructing measures for Weyl spinors ot tell me else what is going on here. Cheers!


title: Why do people rule out local hidden variables?

text: I bet the automatic response to my question would be "Bell's theorem" and of course I am not disputing Bell's proof. I am however uncertain of one of his assumptions. The so called "no conspiracy" assumption states that we somehow posses this magical thing called free will which makes us able to break free from the causal chain of events that make us measure the $y$-axis instead of the $x$-axis and so on in experiments. Obviously this makes little sense once reviewed under the light of logic. What this naturally implies is that, if a theory is fully deterministic, it is instantly super-deterministic. Simon Kochen and John Conway published a theorem in 2009, "The Strong Free Will Theorem", which simply-put states: either everything is deterministic (super-deterministic) or every particle has free will. Among the few who accept this to the fullest extent is Gerard 't Hooft. He has proposed that what we call sub-atomic particles are really just templates and that there is a realm beneath the quantum where the true fabric of reality is "hidden". By accepting determinism he escapes Bell's theorem (due to not having to accept the "no conspiracy" assumption) and can construct a local, deterministic and realist hypothesis of reality. In Gerard 't Hooft's model there is cellular automata somewhere near the Planck scale which gives rise to the "emergent quantum mechanics". This means that there exist no superpositions in objective reality, the cat is always either dead or alive, no collapse of wave functions, no branching universes, non-locality or retro- causality. The big question for me is basically: Why do people have such a hard time accepting determinism? Some will object and say "oh but it's SUPER- determinism", but that makes no sense whatsoever, either everything is determined (determinism) or only somethings are (quasi-determinism). By accepting it we escape Bell's theorem. Bell himself was well aware of this and mentioned it a few times in interviews in the 80's. I know some are concerned that if we accept complete determinism we can no longer do science because it's all a HUGE CONSPIRACY. Gerard 't Hooft has answered critics who bring up this here: arXiv:1112.1811, section 6. Additionally, another author who has taken a deeper look into the worries of "conspiracies" in QM interpretations is Peter J. Lewis in his paper "Conspiracy Theories of Quantum Mechanics" (also here). So to all of you who dismiss local hidden variables I ask: Why?


title: Calculate power $P=V \times I$

text: I have an adapter which mentions like this...               Input: 100-240V~1.8A            50-60hZ      So how much power does it really consume? I just knew `Power = Volts X Current`. But not sure how to apply the value since the volts is mentioned as **100-240V** the value. Sorry for a very basic question....


title: Does quantum mechanics violate the equivalence principle?

text: I have a question about equivalence principle in quantum mechanics. Consider a Schroedinger equation under gravitional field $$\left[ - \frac{1}{2m_I} \nabla^2 + m_g \Phi_{\mathrm{grav}} \right]\psi = i \partial_t \psi \tag{1} $$ where $m_I$ and $m_g$ are the inertia and gravitational masses, respectively. $\hbar=1$ unit is adopted. To the contrary as the classical mechanics $$ m_I \frac{ d^2 x}{ dt^2} = m_g g \tag{2}$$ we can choose a transformation $x'=x-\frac{1}{2} g t^2$ to "switch off" the gravity. But it seems the transformation will not switch off gravity in quantum mechanics, Eq. (1). Does it mean quantum mechanics break the equivalence principle? (I can think about relativistic Hamiltonian, but it will not resolve the problem, as far as I can see)


title: Explanation request for flashes of light

text: I have observed what I believe to be the same phenomenon on two occasions and wanted to ask for possible explanations. Both times, I saw flashes of whitish light, near the ceiling of a room. They lasted only a small fraction of a second. They were not that bright: like a lamp turning on and off. They were in a part of the ceiling where there are no sources of light. My guess: some unequal electrical charges in the air discharging. I haven't heard of this happening just randomly, powerfully enough that it can be seen. (Potentially) Relevant details: 1) The first time, my girlfriend and I both saw and remarked on the flash at the same time. This is my only evidence that it wasn't just physiological. 2) Both flashes took place in Michigan. The second was in the bathroom of a heated university building during cold weather. I don't remember the circumstances of the first exactly, but I think it was in the living room of an air-conditioned house during the summer. Thanks for any possible explanations. I'm not a physics student or anything, this is really just a curiosity for me. Also when I try to google this I only run into "paranormal phenomena" websites. :) Also I'm not sure what tag to use, so people should feel free to change tags if there are more relevant ones.


title: String-net models on non-trivalent lattices

text: I have just started reading about string net models. The following aspect wasn't entirely clear to me: String net models are most naturally defined on trivalent networks, that is to say networks where we attach 3 "legs" to each "point" of the lattice. The model is then fully specified by defining the string types, branching rules (the "$N_{ij}^{k}$") and the orientation of the strings. While this approach appears natural (especially based on what I have read on Tensor categories) most of the "simplest" String net models (such as the $Z_2$ Kitaevs Toric Code or more generically the $Z_N$ Wen Plaquette Model) can also be defined on square lattices where we seem to have 4 "legs" attached to each "point". I was wondering how and whether one can always "reduce" a String net model on an arbitrary N-valent lattice (N "legs" attached to each "point") to a String Net Model on a trivalent lattice. Similarly: Can one construct arbitrary String Net Models on N-Valent lattices by understanding the model on some trivalent lattice? But in order not to complicate things: What is the "recipe" to reduce the Z2 Kitaev Toric Code on a planar square lattice to some "trivalent lattice"? I am looking forward to your responses!


title: What happens after a rocket is fired with velocity equal to or greater than escape velocity?

text: When a rocket is fired a speed greater than or equal to escape velocity, then it does not have any gravity effect of the earth. So, suppose if a rocket is having that much energy which will only last for 1 sec. Then it will move with 11.2km/s and it escapes from the earth. My question is what happens after that?


title: Is it possible for an electrical device to generate more heat than another using less power or the same?

text: People are claiming that a Plasma TV uses more power which explains why it uses more heat. We refer to power as wattage in this regard. Countering this, is it possible that a device can draw the same wattage or less and produce more heat? Basically, can something create more heat than something else with the same or less drawn power?


title: Do we know exactly how fast we are going relative to the center of the galaxy

text: I mean total inertial on my body siting here at my computer "California US." Ok so the earth is rotating on its axis and in turn around the sun and the sun around the galaxy. The object the question how much kinetic energy is there in a litter of water "at rest" on my desk.


title: separation between slip rings of DC generator

text: ![enter image description here](http://i.stack.imgur.com/9lj41.gif) this is a DC generator which converts mechanical energy to Direct Current. The commutators in a DC generator are separated (as you can see in the image). It is explained in our book that it prevents the change in direction of current. But i don't understand, even if we don't keep them separated, how will current change direction? In short, what is the function of the gap between the two split rings?


title: What are wimpy particles?

text: Can anyone give a brief overview of what a wimpy particle is?


title: Does quantum Zeno effect play role in astrophysics?

text: For example, do two galaxies situated in proximity reduce the atom decay rate in each other? What happens with decay quanta escaped to infinity? Does the radius of apparent horizon effect the lifetime of the atoms in the universe? Do atoms placed on a planet's surface decay slower than those placed at a distance from the planet? Is this decay slowing equivalent to the time slowing due to general relativity? Is it affected only be the mass of the nearby object or by dielectric constant of the material?


title: Simple mechanics problem

text: I have no idea how to approach this problem I drew the diagram but that hasn't helped that much. ![enter image description here](http://i.stack.imgur.com/CaRP1.jpg)


title: Does ABS cause a force that resists turning a car into a corner?

text: ABS systems work by allowing a tire to continue to rotate rather than "locking-up" (stop turning) due to loss of traction with the road surface. A rotating tire can influence the direction of the car, a "locked-up" tire has no directional influence. This also gives the tire more traction than a fully "locked-up" tire. As a car enters a turn, angular momentum causes weight to shift to the tires on the outside of the turn, reducing weight, and thus traction, on the inside front tire. My question is: does ABS's effect of increasing traction on the inside front tire cause a force on the car that causes it to go more straight (resist turning) compared to a car that has no ABS and whose inside front tire is allowed to stop (or mostly stop) turning?


title: Withstanding charlatans: How to protect friends from buying "water structuring facilities"?

text: In my country it is a big business now, to sell "water structuring facilities". Promoters telling that after passing through them the water obtaining fantastic abilities basing on the fact that "water can keep information coded in molecular domains, chained by weak hydrogen bonds". This "structured" water is a panacea for all diseases. I guess it is all started from this guy, but now they have a bunch of popular documentary films and many official academics supporting these ideas. Which arguments should I use to convince my friends they're charlatans? I don't have a grade in physics, but from my basic university course I believe it is nonsense. I think there must be some popular and simple facts that can be convincing enough for non-professionals who can understand physics at the high school level.


title: Optimum magnet layout for tripping a coil sensor

text: If I had say a dozen or so small bar magnets and I wanted to used them to trip the traffic light sensors (which are a coil embedded in the road where a vehicle would stop for the lights), what would be the best layout? Would it be best to put them all end-to-end to create one long 'super' magnet? Or should I separate them into say a 3 by 4 grid with them all orientated the same way? Or is there some other optimal layout? By 'optimal layout' I mean, 'most likely to trip the sensor', which I'm guessing is the same as, 'would induct the most current in the sensor coil'.


title: Outcomes of earth slowing down spinning on its own axis

text: What are the possible outcomes if earth slows down spinning on its own axis? To be specific: Can the decrease in the internal centrifugal (or centripetal) force due to slowing down earth's spin:   a. change earth's structure given the fact that major part of earth is not solid and the earth's oblong shape may be in dynamic equilibrium of gravity vs this internal centrifugal force.   b. alter the sense of the "vertical" as indicated by the plumb line especially when we are off equator, given the fact that the direction of centrifugal force acting on a physical body has a non-zero component on the horizontal direction except on equator and poles? If this is true, will the altered horizontal component be sufficient enough to affect high rise buildings?


title: Sum of angular momentum of all electrons in a magnet

text: Can the sum of angular momentum of all rotating electrons in all the aligned atoms in a permanent magnet have a significant contribution to the macro angular momentum of the magnet? If yes, why does it not behave like a spinning top , like, resisting a change in its orientation?


title: Mass-Energy Equivalency

text: We call $E=mc^2$ the Mass-Energy Equivalency because it equates mass and energy together. But, by that same logic, shouldn't we call $E=\frac{1}{2}(mv^2)$, the equation of kinetic energy in Newtonian mechanics as the Mass-Energy Equivalence ?


title: Calculating the gravity gradient of a black hole

text: Given a black hole of, say, $10^8 \odot$ (solar masses), how can I calculate the gravity gradient ($\frac{\Delta g}{m}$?) at a given distance from it?


title: Gauss's Law problem

text: I've been thinking about this for a while, but I'm not sure how to proceed. I understand uniform charge-density problems, but the added non-uniform deal makes me uneasy:   * find the e. field inside a sphere (rad. = R) with nonuniform charge density $\rho = {\rho _0}\frac{r}{R}$ **Edit** I did this: ![](http://s8.postimg.org/qpv82s4xx/img.jpg). Can someone confirm if this is correct?


title: How can a singularity in a black hole rotate if it's just a point?

text: I guess nobody really knows the true nature of black holes, however, based on everything I know about black holes, there is a "singularity" at their center, which has finite mass but is infinitely small and thus infinitely dense. So a black hole is really just a point mass that dramatically warps the space-time around it so that an event horizon is achieved. What doesn't make sense to me is how a true mathematical "point" can rotate, or at least how it is even possible to determine if a point is rotating (or if it should even make a difference about whether or not it is rotating, for the same reason it would be impossible to tell if a perfect, featureless sphere were rotating (I know it would have higher rotational energy but lets assume you can't measure that because a true "point" can't attain rotational energy since it has no radius)). The only conceptual solution I have to describe a rotating black hole would be to compare it to a rotating whirlpool, where the surface of the water is the fabric of space-time... but this is different because there is no "singularity" in a whirlpool that _causes_ the water around it to rotate, whirlpools form from flow of water around a point (put into motion perhaps by some kid swirling water in a 2-liter bottle... it's due to an external cause), but the only thing that could cause rotation around a black hole would be the singularity itself, which doesn't make sense because, once again, how can we know if a true point is actually rotating? Aside from everything I've said, here is what I am actually asking: Why/how do some black holes rotate... what is the mechanism of their rotation (is analogous to a spinning point or a 3-d whirlpool, or something different)?


title: Do primary fields (in a CFT) satisfy the wave equation?

text: In a 2-dimensional CFT, do primary fields generally satisfy the wave equation? I know that if they have either purely holomorphic or purely anti-holomorphic, they do. But what about a general primary field of weights $(h,{\tilde h})$?


title: Splitting of degenerate states due to perturbation

text: In band structures we see avoided crossing when we have degenerate eigenstates (caused by perturbation due to potential energy). However along some direction in first Brillouin zone, even though the energies cross in dispersion relation, they do not split up. I vaguely learned that the reason for this is because of symmetry along those directions in Brillouin zone. Can anybody help me understand why we do not have splitting?


title: Hermitian conjugation in Radial Quantization

text: I'm a little confused about Hermitian conjugation in a radially quantized CFT. Now, in the Minkowski theory, Hermitian conjugation leaves the coordinates invariant, i.e. $t^\dagger = t$ and $x^\dagger = x$. We then Wick rotate the theory to Euclidean time $t \to - i \tau$ and therefore $\tau^\dagger = - \tau$. Next, we define the complex coordinates on the Euclidean plane $w = x + i \tau$ and ${\bar w} = x - i \tau$. The Hermitian conjugate of acts on this coordinate as $w^\dagger = w$ and ${\bar w}^\dagger = {\bar w}$. On the radial plane with $z = e^{- i w}$ we get $$ z^\dagger = e^{i w^\dagger} = e^{i w} = \frac{1}{z} $$ However, Francesco explicitly says (pg. 152, above equation (6.4)), that $z^\dagger = \frac{1}{z^*}$. Where am I going wrong? Can anyone explain this?


title: how do you work out the direction of an eddy current

text: I find it very difficult to work out the direction of the eddy current, can someone please explain the steps in working out the direction of it? there is a following question; I dont understand how part a is done? ![enter image description here](http://i.stack.imgur.com/DYVGq.png)


title: Why does Coulomb's law not hold for fast moving charges?

text: We all remember calculating the electric force of interaction between a stationary nucleus and a revolving electron using Coulomb's law. The electron in this case is moving. Here's what I think about the law's validity in this case. Considering two charges, if one is at rest and the other in motion, Coulomb's law still holds. Coulomb's law determines the electric force alone. Let us consider a Hydrogen atom at ground state. Considering the most probable states of an electron, the collection yields a circular orbit. In this case, the most probable distance is fixed. The revolving electron creates a magnetic field around it. However, the magnetic force acting on the nucleus is zero because the nucleus is stationary. And since the electron cannot exert a force on itself, the magnetic force acting on either is zero. This leaves us with the electric force alone that is given by Coulomb's law. But if both the charges are in motion, the total force cannot be given by Coulomb's law alone. Relativistic effects come into the picture. Here's a point I would like to make. In the case of moving charges( both moving slowly ), the total force acting on either, at some 'instant' can be given by the sum of the electric force( given by Coulombs law ) and the magnetic force. In the link given below, they have proceeded as I mentioned. Electric and Magnetic force between moving charges You see, Coulomb's law holds even in the case where charges move slowly. However a correction term( magnetic force ) was to be included. So, in what sense does Coulomb's law not hold in the case of fast moving charges? What relativistic effects show up in the case of fast moving charges thereby, leading to a breakdown in Coulomb's law?


title: Probability density in Hamiltonian Mechanics

text: I am currently studying Liouville's theorem compare wikipedia and there this mysterious probability density $\rho$ appears and I was wondering how one can determine this quantity analytically for a given problem. Let's think of the harmonic oscillator as a primitve example, how can I get an analytical representation of $\rho$?


title: Visual representations of black holes

text: Some black holes emit jets that are rapidly circling the poles of what appears to be a sphere at the center of the black hole, cf. Fig. 1. ![](http://i.stack.imgur.com/xFS7L.jpg) Is there really a sphere here and how far (relative to the event horizon) are people seeing these jets start?


title: Taylor expansion of translated fields

text: First of all, I would like to say that I am somewhat new to four-vector notation. I have a function of a four-vector that I want to expand. $$ A_\mu (\mathbf{x} + \mathbf{x}_0) = A_\mu (\mathbf{x}) + \mathbf{x}_0 \frac{\partial A_\mu (\mathbf{x})}{\partial \mathbf{x}} + \dots $$ where both $\mathbf{x}$ and $\mathbf{x}_0$ are four-vectors so that e.g. $\mathbf{x} = (ct,x,y,z)$ My question is then what this actually looks like? The second term, should I have an index on the derivative? Is the derivative term a sum of several terms? Could anyone help me write this out more explicitly, so that I am sure that I don't leave any terms behind?


title: Where are the L3, L4, and L5 of a hyperbolic orbit?

text: Do the L3, L4, L5 points exist in hyperbolic orbits? If yes, then where do they lie?


title: Kinetic energy and momentum conservation in an explosion?

text: My physics book says, _"A firecracker sliding on ice has the same total momentum before and after it explodes."_ I understand this part. This is because of Newton's 3rd law, and no external forces. This is what I really don't get. _"The same, however, is not true of a system's kinetic energy. Energetically, that firecracker is very different after it explodes; internal potential energy has become kinetic energy of fragments."_ It goes on to say, _"Nevertheless, the centre-of-mass concept remains useful in categorizing the kinetic energy associated with a system of particles._ " How is it that the kinetic energy increased but momentum stayed the same? My problem lies within the equation $K = \frac{1}{2}mv^2$ and $p= mv$. If kinetic energy increased doesn't that mean that the velocities increased as well? How else would $K$ become more positive? And since $K$ increased $v$ increased and thus $p=mv$ must increase?


title: Functional field integral in condensed matter field theory (Altland)

text: This is the action for the 1+1 dimensional interacting electron system; $$S_{cl}[\theta , \phi]= \frac{1}{2\pi} \int dxd\tau \left(g^{-1}v(\partial_x \theta)^2 + gv(\partial_x \phi)^2 + 2i\partial_{\tau} \theta \partial_x \phi \right).$$ I want to integrate out the Gaussian field $\phi$. This book says that it is just an "elementary" Gaussian integration. So, I tried some modification to this action; $$S_{cl}[\theta , \phi]= \frac{1}{2\pi} \int dxd\tau \left(g^{-1}v(\partial_x \theta)^2 + (\sqrt{gv}\partial_x \phi + \frac{i}{\sqrt{gv}}\partial_{\tau} \theta)^2 + \frac{1}{gv}(\partial_{\tau} \theta)^2 \right).$$ For this action, partition function is given by $$\int D\theta D\phi \exp[-S_{cl}].$$ Maybe, the second term in the action is related to Gaussian integral. But, I don't know how to calculate it. How can I calculate this?


title: What is a Killing vector field?

text: I recently read a post in physics.stackexchange that used the term "Killing vector". What is a Killing vector/Killing vector field?


title: Dimensional reduction and Schwarzschild solution

text: It is well-known that, given a Killing vector $\xi_\mu$ for a metric, this can be dimensionally reduced. This is exactly the case of the Schwarzschild solution due to the rotational symmetry. This procedure is quite general and one can always reformulate a general 2d (or 3d) model, as the Jackiw- Teitelboim model, of general relativity and obtain solutions for the 4d case with the given symmetry properties.A reference about this is this. The effect, as always happens in dimensional reduction, is to introduce another field into the gravity action. So far, I was not able to see the derivation of the reduced Schwarzschild metric from the dimensionally reduced Einstein equations. This is generally given for granted. So, I would appreciate to see such a derivation or to get some tutorial about this. Thanks.


title: Calculating car's acceleration from change in angle of hanging object?

text: The question essentially is based on a situation like this- A car has a small object hung from the cieling on a string (apparently at an angle of 0 degrees to the ceiling). The car is accelerating and the object is now hanging at a 30 degree angle (to the ceiling). How would I figure out how much the car is accelerating. PS - This is homework but Im stuck and would appreciate any advice. Thanks. Edit: changed angle from 45 to 30.


title: Capactiy of an orbtial to hold muons and electrons

text: In a normal atom, there is a limit of 2 electrons per obital due to the Pauli- Exclusion principle. I have seen people talking about replacing an electron with a muon, but since muons and electrons are distinguishable, would this then allow 2 electrons and 2 muons to occupy the same orbital, or would only 2 leptons be allowed to have the same set of quantum numbers? For example, would a Beryllium atom with 2 muons and 2 electrons use only the 1S orbital, or would the 2S be used as well? Spectroscopic methods should be able to distinguish between the two.


Question: If air was significantly compressed, and then the force compressing it suddenly disappeared (no container), what would happen? Side Question: I'm imagining significantly compressed, but what kind of pressure would be necessary for that? Thoughts: It has been a while since I took chemistry and physics, but what I think is that it would rapidly expand and exert similar pressure to whatever the container was exerting on it in all directions? So similar to what an explosion from heating gas or what not would do?


title: Can a Celestial body have a mountain rise all the way to Synchronous Orbit?

text: On Earth, the tallest mountain is Mauna Loa. In the solar system, to date, the tallest mountain is Mons Olympus. Both fall far short of GSO. Mars gravity is 0.376 of Earth gravity. If Mauna Loa were on Mars it would have an altitude of ~80000 ft. If Mons Olympus were on Earth it would similarly be no more than ~27000 ft. Yet these are almost comparable altitudes. Is it possible for a mountain to rise as high as GSO? Are all known tallest peaks on planets in the Solar System similarly limited?


title: What are the forces used in this weightlifting example?

text: You have a person doing a bench press. Their hands are equidistant apart and they are centered on the bar - each hand about 1.5 feet from center of the bar. The bar weighs 45 pounds. In example 1 there are two 45 pound weights on each side at 2.5-3 feet each from the center (lets say each weight is 3 inches wide). In example 2 there are nine 10 pound weights on each side starting at 2.5 feet again and then ending 27 inches later. What are the forces that cause example to to feel heavier and is there any way to calculate it.


Is there any mathematical structure modelling parallel universes (multiverse)? If we established such structure, is it science or science-fiction?


title: returning to physics research?

text: What the the most realistic and practical ways for a PhD in physics who has left the fold to return to a career of research in physics? The traditional employer of choice, universities won't hire such people. What alternative career choices are there for someone without an impressive track record so far? Are there alternative forms of support, or part time careers earning just enough money to get by while not interfering with research?


title: Philosophical take on Schrodinger's cat paradox

text: I'm a mathematics undergrad currently doing a module on quantum mechanics. I am interested to delve into the more philosophical aspect of the paradox. Could anyone please give me some introductory ideas to introduce me the philosophical issues that arise?


title: What's the difference between gas and plasma?

text: A friend stated that stars are hot balls of gas, however we know that technically stars are plasma. Is his statement entirely incorrect? Can a plasma be considered a form of gas or should be referred to as a distinct form of matter?


What method should I use to solve for the final acceleration of a projectile being launched from the earth's surface? The question I am working on is: > A projectile is launched vertically from the surface of the earth with an > initial speed $v$. What is the magnitude of its acceleration $a$ > ($\mathrm{m}/\mathrm{s}^2$) $d$ km above the surface of the earth? I cannot use a formula for an object under constant acceleration because the force between the object and the earth changes at large distances. My textbook has formulas for the universal law of gravitation and also for gravitational potential energy. I am thinking that I may need to solve this problem using the energy formulas. The formula given for potential energy is: $$U(r) = - {GM_Em}/{r}$$ Using this I can solve for the final potential energy. How do I derive the final acceleration from this?


title: Slope of a channel

text: I have a trapezoidal channel like this: ![Trapezoidal channel](http://i.stack.imgur.com/WvOye.gif) > The question is to get the slope of the channel such that we have a Chézy > coefficient of 64 $m^{1/2}/s$. > > Given: $B= 76\: \mathrm{m}$, $y = 10\: \mathrm{m}$, $\tan(\gamma) = 5/12$ > where $\gamma$ is the angle between the horizontal line and the oblique > line. $q= 1200\: \mathrm{m^3/s}$. Here is how I solved it: we know that $$ v = C \sqrt{ R\space i}$$ so $$ i = \frac{v^2}{C^2 \times R}$$ $$v = \frac{q}{S} = \frac {1200}{ y \times(B + \frac{y}{\tan(\gamma)})} = 1.2\: \mathrm{m/s}$$ $$R = \frac{1000}{76 + \frac {20}{5/12}}=7.8\: \mathrm{m}$$ and then we have: $$i = \frac{1.2^2}{64^2 \times 7.8} = 4.5 \times 10^{-5}$$ # Question: Is this an expected value of a slope? shouldn't this be something like: $4.5$ or $45$?


title: Possible career paths

text: I currently hold a Bachelors in Computer Science and a Minor in Physics. As far as that goes, I've taken courses in the basic things (University Physics 1 and 2), Thermodynamics, Circuitry and Modern Optics. I also read about various topics online in my spare time. I'm also planning on going for a Masters in Computer Engineering at NYU Poly. My question is what are some possible Physics related career paths for someone with my background? I just graduated this passed May, so I'm fresh out of the academic world. Are there places in National Labs for people like myself? I live probably no more than an hour or so from the Brookhaven National Lab on Long Island and would like to somehow get involved there. Is such a thing even feasible?


title: Origin of interaction in inelastic neutron scatting

text: In solid state physics, inelastic neutron scattering is a commonly-used experimental technique for probing the energy spectrum of phonon and magnon excitations. This technique relies on the interaction between neutrons and the excitations. What is the origin of the interaction in these cases? Is there a similar interaction that would allow the determination of electronic band structures using the technique?


Why does quantum cryptography give us uncrackable codes? What makes it 'uncrackable'? Articles in for example pop science magazines always claim QC produces uncrackable coded, however I highly doubt these claims. p.s. - Just type in 'quantum cryptography uncrackable codes' and you'll find a ton of hits.


title: Some sort of conservation equation

text: As far as I know, in General Relativity, an expression of the kind $\nabla_{\mu} X = 0$ states that, associated to $X$, there exist a charge which is conserved. The first example that comes to mind is the conservation of energy-momentum $\nabla_{\mu} T^{\mu\nu} = 0$, as a consequence of the contracted Bianchi identities and assuming Einstein's field equations. My question regards the meaning of the following expression: $\nabla_{\mu} \left( R^{\mu\nu} K_{\nu} \right) = 0$ where $R^{\mu\nu}$ is the covariant Ricci tensor and $K_{\nu}$ is a **generic** vector field. Does this equation mean something as it stands? Or do I have to impose a metric (and hence a form of the Ricci tensor) in order to find more informations? The fact is that I don't understand what kind of conserved quantity could be related to the vector $R^{\mu\nu} K_{\nu}$. The equation comes from a rather peculiar way to obtain a modified theory of gravity. The reference is here, but the original source is there I admit that the procedure is not entirely clear to me, but basically it seems to appeal to a "gauge theory"-like approach in order to obtain modified gravitational field equations. It's kind of lengthy, but ultimately they consider commutators of covariant derivatives of successive orders to obtain relevant quantities, such as the Riemann tensor (first-order commutator) and matter current (second-order commutator). For the latter, the analogy with gauge theories is necessary. Finally they get the modified gravitational field equations, eq.(16) in the first reference. The very nature of the approach makes it necessary to use a vector field $K_{\nu}$ in order to carry on the calculations, but the character of such a vector field is never specified (there's no need to, in fact). Only after eq.(16) they comment that Einstein's case is recovered if $\nabla_{\mu}K_{\nu}=0$. Going back to the equation that I posted... in order to have some hints about the meaning of it all, starting from eq.(16) in the paper, I just impose the vacuum condition $T^{\mu\nu}=0$ and see what happens: $\left(\nabla_{\mu}R^{\mu\nu}\right) K_{\nu} + R^{\mu\nu} \left( \nabla_{\mu} K_{\nu} \right) = 0$ and that's it. I'm not at all convinced about the comments they make about the vector field (they call it "substratum") but the overall procedure doesn't seem entirely bad.


title: Diagonal matrix in k-space

text: I'm having some trouble with an integration I hope you guys can help me with. I have that: ${{\mathbf{v}}_{i}}\left( \mathbf{k} \right)=\frac{\hbar {{\mathbf{k}}_{i}}}{m}$ and ${{\mathbf{v}}_{j}}\left( \mathbf{k} \right)=\frac{\hbar {{\mathbf{k}}_{j}}}{m}$ Now according to my teacher when I multiply these two, and integrate the angular part ($\,d\theta \,d\varphi$) over $d{\mathbf{k}}$, with $d\mathbf{k}={{k}^{2}}\,dk\sin \left( \theta \right)\,d\theta \,d\varphi$, I should end up with: ${{\mathbf{v}}_{ij}}{{\left( \mathbf{k} \right)}^{2}}=\frac{{4\pi{\hbar }^{2}}}{3{{m}^{2}}}{{k}^{2}}$ My question is why?I see that when there is no angle dependency I get the $4\pi$, and nothing touches $\hbar$, $m$ and $k$. But where to the $1/3$ come from? That's probably my main concern. So anyone with a hint or something ? Thanks in advance.


title: Why are black mosquito nets so much less visible than white ones?

text: I have just been installing a couple of mosquito nets for windows in our apartment. The material is polyester, fibers are approximately 0.2mm thick, and the hexagonal "holes" apprixmately 1.5mm in diameter. Just like the packaging says ("buy black ones if you want to have a better view"), the black ones are almost invisible from a little distance (~2 meters, its a little bit darker, almost in effect like an ND photo filter). From the same distance, the white ones are clearly visible, with good eyes you can also spot the "holes" (the effect looks almost like if you put the black point in photoshop into a much whiter area, making the contrast much less). Why is that so? Is there a physical effect that objectively reduces the "image information" that reaches the human eye? I would appreciate an answer that -- if applicable -- contains a short average human explanation, and additional one that goes a bit more in-depth into the "hobbyist physics" level


title: Conjugated-Phase Matching

text: I am working on imaging applications of phase-conjugation in optics, in particular phase-conjugation imaging from recorded holograms. I have found several useful sources, but I cannot find a book that allows me to familiarize with the derivations from scratch. I would like to find a book where I can follow the derivation from a recorded hologram to the construction of an image by phase conjugation so as to estimate cross-resolution, depth-degradation, etc. Do you know a book (on optics, radars, math-physics) where I can find such an expostion? Thanks in advance.


title: Voltmeter forming a closed circuit

text: A battery is connected to a 10Ω resistor as shown in Figure 2. The emf (electromotive force) of the battery is 6.0 V. ![Circuit diagram](http://i.stack.imgur.com/UXIQe.png) When the switch is open the voltmeter reads 6.0 V and when it is closed it reads 5.8V. Explain why the readings are different. The answer is: when switch is closed a current flows (through the battery)   hence a pd/lost volts develops across the internal resistance But why does the voltmeter not form a closed loop with the circuit and hence cause energy to be dissipated by internal resistance, without having the other loop's switch closed?


title: What happens when some asteroid comes in the way of warp drive?

text: Recently Harold white, NASA scientist shared his knowledge and the things he is working on. That is Warp drive for interstellar navigation. This spaceship compresses the space in front of it and expands the space behind it. So the ship is able to move faster than light. In stead it moving it just warps the space around it. According to Einsteins General theory of relativity its possible. Now I am thinking what will happen when some asteroid or big mass come in front of it, when its in motion. I cant think of anything. Anybody has any idea?


title: Pendulum with changing length over time. What's wrong?

text: I tried to find the equation of this pendulum, but I think I did something wrong. I know I have to get the Bessel's equation but I can't see it. It's a simple 2-D pendulum, without any dissipation. The length is $l$. \begin{gather} x(t)= l(t) \sin\theta(t), \\\ y(t)=l(t) \cos\theta(t), \\\ \dot x(t)= \dot l(t) \sin\theta(t) + l(t) \dot \theta(t) \cos\theta(t) \\\ \dot y(t)= \dot l(t) \cos\theta(t) - l(t) \dot \theta(t) \sin\theta \end{gather} and: \begin{equation} \dot x^2 + \dot y^2 = \dot l^2 + l^2 \dot \theta^2 \end{equation} The potential and kinetic energy: \begin{gather} V=-mgy= -mgl(t) \cos\theta(t), \\\ T=\frac{m}{2} (\dot l^2(t)+ l^2(t) \dot \theta^2(t)) \end{gather} So the Langrangian is: \begin{equation} L=T-V= \frac{m}{2}(\dot l^2 + l^2 \dot \theta^2) + mgl \cos\theta \end{equation} After this, we get: \begin{gather} \frac{ \partial L}{\partial \theta} = -mgl \sin\theta, \\\ \frac{ d}{dt} \frac{\partial L}{\partial \dot \theta}= m2l \dot l \dot \theta +ml^2 \ddot \theta. \end{gather} From thism we get the equation: \begin{equation} \ddot \theta + 2 \dot l \dot \theta = g \sin\theta. \end{equation} And for the radial part: \begin{gather} \frac{\partial L}{\partial l} = ml \dot \theta^2 + mg \cos\theta , \\\ \frac{d}{dt} \frac{\partial L}{\partial \dot l}= m \ddot l. \end{gather} The equation is: \begin{equation} \dot l = l \dot \theta^2 + g \cos\theta \end{equation} My question is how can I get to the Bessel's equation from here?


title: Gauss' law and an external charge

text: Gauss' law states that the net outward normal electric flux through a closed surface is equal to $q_{total, inside}/\epsilon_0$. However, I'm a bit confused of why the presence of an external charge doesn't influence the solution. I always hear as a response - "any field lines entering the enclosed surface, will also leave it", which is fine. However, the field is inversely proportional to $r^2$, therefore, it will "exit" with a different magnitude and thus, with a different flux. So how they possibly cancel each other out? And if possible, I really look forward to a simple explanation, without involving too much math. I just want to get the concept. I'm sorry for such a newbie question.


title: Curiosity Rover (MSL): current coordinates

text: I'm looking for information on the **current coordinates** of Mars Science Laboratory's Curiosity Rover. I've only found the landing site coordinates `4.5895°S` `137.4417°E` (Bradbury Landing). ![Author:NASA/JPL-Caltech/Univ. of Arizona](http://i.stack.imgur.com/YyKl6.jpg)


title: How the pressure of a system changes in terms of density?

text: I am modeling a closed natural circulation loop, filled with water. Some parts of the loop are heated, some are cooled and other are assumed adiabatic. As an effect of heating and cooling the density of water changes and so does the total pressure in the loop. My question is as follows: > Is there a way to calculate the total pressure of the system in terms of, > for example, mean density, mean temperature and total volume of the system? > For air, the ideal gas equation should be a nice approximation, but it is > not applicable for liquids. The problem becomes more complex when the water > starts to boil at some point, then it is a two-phase fluid. I have seen answer to this question What equation of state is needed for liquid states? but it does not help in my case.


title: Positivity in the Pauli/Bloch/coherence vector representation

text: Suppose $\rho$ is an $n$-qubit state and $\vec{x}$ is a vector of coefficients in the Pauli representation (also called the Bloch or coherence vector). That is $$ x_k = {\rm Tr}(\rho \sigma_k), $$ where $$ \sigma_k = \sigma_{k_0}\otimes\sigma_{k_1}\otimes\cdots\otimes\sigma_{k_{n-1}}, $$ and each $\sigma_{k_j}\in\\{\sigma_0=I,\sigma_1=X,\sigma_2=Y,\sigma_3=Z\\}$ is a single qubit Pauli operator. Now the question is, given a vector $\vec{x}$, what is the most computationally efficient method to determine if the state $$ \rho = \frac1{2^n} \sum_k x_k \sigma_k $$ is a valid density matrix.


title: Is this 2D structure triclinic?

text: ![enter image description here](http://i.stack.imgur.com/v8Uhd.jpg) The only rotation axis obvious to me is rotation by 360 degrees, the identity. Vertical mirror planes I've been dicing and cutting it through several planes and I still see none. Yet, the structure looks fairly symmetric and I feel like I'm missing something. No center of inversion either. Not too terribly sure about horizontal mirror planes. I suppose that one bisecting the plane horizontally would be there but I am not entirely sure if this is appropriate. Right now I'm saying this structure belongs to the C1 point group.


title: Calculate electricity generation

text: A friend of mine stays near a water stream and wanted help to use its water. The height up to which the water could be raised using the water flow is 12ft. How much electricity can it produce? Is it at least enough to power half the house?


title: Do particles have different spins in different frames of reference?

text: Let's say we have two photons, whose momentum vectors point to opposite directions. Also spin angular momentum vectors of the photons point to opposite directions. (Sum of spins is zero) Now we change into a frame where the momentum vectors point into almost same direction. Do the spin vectors point into almost same direction in this frame? (Is the sum of spins not zero in this frame?) (Photons were far away from us, moving to the left and to the right, then we accelerated towards the photons) (Momentum and spin vectors are either parallel or anti-parallel in massless particles, according quantum-mechanics) I mean: Can acceleration of an observer change spins of particles that the observer observes?


title: How the pipes of water don't freeze in -40 celsius?

text: Does anyone have a good answer how comes that the water in the pipes in our houses doesn't freeze even if it is -40 celsius for a whole week? of course assuming we don't make the water to fluid in it


title: Asynchronous generator run in vacuum chamber

text: what will be happen if we put asynchronous generator in vacuum chamber & run it above its synchronous speed. After reaching its over synchronous speed we will cut off electrical supply. Can it run continuously? Can we generate electricity from it?


title: True or false? Particle physics

text: 1. It is not possible to prove the point of origin of a photon   2. It is not possible to prove the point of origin of a free electron   3. It is not possible to prove that protons or neutrons exist inside a nucleus Can anyone confirm?


title: What's the meaning of the coupling change after a renormalization (in the 1-dim Ising Model)?

text: What does it mean that after the theory (1-dim Ising model here, but the question is general) is renormalized one time and $g_i\rightarrow g_i'$, that the couplings are _weaker_ , even if the theory is still the same theory, just reformulated? It is said that it can now be viewed at a theory at high temperatures $-$ the theory flowing with the renormalization group towards a critical point. But what does that mean that it's actually still the same theory? The unrenormalized theory itself gives different results at different temperatures, right? So how does this fit together?


title: Gas Circulation Using Pressure Difference

text: Dear all, see attached picture Please, is it possible to have the gas recirculated from the gas phase to the liquid as described in the diagram assuming the gas is not soluble in the water. These are the conditions. 1\. The valve remained closed, 2\. The liquid level didn’t change 3\. P1 increases. Could P1 get to a level where it would force itself through the pipe and go through the liquid. If P1 increases, what happens to P2. ![enter image description here](http://i.stack.imgur.com/CHf0S.jpg)


title: How to get $t$ from $a(v)$?

text: I read what if we have acceleration given as a function of velocity we can calculate time as $$t(v) = t_0 + \int_{v_0}^{v} \frac{dv}{a(v)}.$$ Why?


title: Explain to me how this water cooler works?

text: Goodday all, I was recently reading up on a few projects that might be of interest to me when I found "CPU Bong water coolers", there isn't much online on them so I figure I would ask y'all. If you do a quick google search you can see what this thing actually is. They get a length of PVC and orient it vertially, at they bottom they attach a 45 split with the 45 open end facing up. The very bottom of the pipe sets in a bucket with water. Water from the bucket is pumped through a CPU waterblock and collects heat. The water continues to a shower head situated at the top of the PVC pipe. The water flows through the shower head and strait down the tube into the bucket. While the water is falling through the tube air is forced over it via a fan over the open end of the 45 split. I think the idea is that the air gonig over the small droplets of water will strip off molecules that are close to evaporation. When they evaporate they pull heat from the last of the water in that drop, cooling the drop. All the drops are cooled this way. Is this how this thing works? How could I get a rough estimate the size I would need to remove lets say 300 watts from the incoming water? (I know I will get people who quote "But what if the butterfly flaps its wings in Europe, that thing will then only cool 299.4553456 watts; I just need something generic. Should I get a 10foot PVC pipe or a 6in pipe?) Does it scale? If a 1ft pipe can effectively remove 10 watts, does a 10ft pipe remove 100 watts? If you think it depends on to many factors to tell me, what are the factors? Humidty, ambient, CFM of the fans, how small the drops are from the shower head, etc and which should be maximized and which should be minimized.


title: Do primary first class constraints change the electric field in the Hamiltonian form of Maxwell's theory?

text: In my understanding of Dirac's theory of constrained Hamiltonians, the primary (and also the secondary) first class constraints are generators of canonical transformations that do not change the physical state: the electric field is part of the physical state so it has zero response to a primary first class constraint. However, a paper http://arxiv.org/abs/1310.2756 recently appeared which says that the primary first class constraints change the physical state. The paper gives a direct calculation which I'll reproduce below. Using the notation in Dirac's Lectures on Quantum Mechanics, the p's are $B^{\mu}$ and the q's are the electromagnetic potentials $A_{\mu}$. The primary first class constraints are $B^{0}\approx 0$. So, the generator of the primary first class constraints is, $$ G=\int d^{3}x \xi(x) B^{0}(x) $$ The response of the electromagnetic field is given by the PB, $$ \frac{dA_{\mu}}{d\epsilon}=[A_{\mu},G]=\delta^{0}_{\mu}\xi(x)\ . $$ The paper defines the electric field as, $$ E_{r}=A_{r,0}-A_{0,r} $$ and denies any relation between $E_{r}$ and the canonical momenta $B^{r}$ until the dynamical equation $\dot{q}=[q,H]$ has been used. The paper gets the response of the electric field to the primary first class constraint as, $$ \frac{dE_{r}}{d\epsilon}=\frac{\partial}{\partial t}\frac{dA_{r}}{d\epsilon}-\frac{\partial}{\partial x^{r}}\frac{dA_{0}}{d\epsilon}=-\xi_{,r} $$ and this is troubling me because the response should be zero. I thought I understood constrained Hamiltonians but now I'm not sure, please help.


title: Is period of rotation relative?

text: My question is inspired by the following answer by voix to another problem: "There is a real object with relativistic speed of surface - millisecond pulsar. The swiftest spinning pulsar currently known, spinning 716 times a second. Surface speed of such pulsar with radius $16 \mathrm{km}$ is about $7\times10^{7} \mathrm{m/s}$ or $24\%$ speed of light." This period of the pulsar is that measured here on Earth, that is by remote observer. However, pulsar has both enormous surface rotational speed and quite strong gravitational field. All processes at relativistic speed or strong gravity are time dilated. Is the period affected as well? What is difference of periods of rotation measured by a distant observer and an observer on the pulsar surface?


title: Capacitor charging and discharging

text: I have a conceptual question. The circuit is shown below. Let us assume that the capacitor shown in the figure is of 2.2 micro-farads having rated level of 5 volts and we decide to charge it up to 2 volts so that we don't smoke it. Circuit - ![enter link description here](http://i.imgur.com/3nfxU.png) If we connect $R_1$ and $R_2$ in the circuit for charging and discharging respectively, how will the circuit work? The red loop shows the charging circuit and the green loop shows the discharging circuit. The circuit is as shown until the capacitor is charged and then the switch S is closed. Will the circuit discharge instantaneously? If yes, when will the capacitor be re-charged again? Will be it charged after the long exponential charge decay of the capacitor or will it start to charge itself after it has a certain amount of charge left? Will this process of charging and discharging be continuous if the switch S remains in closed position? Let me start with some basics. The red loop circuit will charge with $R_1\times C$ time constant. So the charging will be quick because of low $R_1$. It will be 22 microseconds. After the capacitor has been charged to 2V (max given by power supply), we close switch S and then the discharge process will start. As expected it will also be quick and similar to charging time constant. Will this capacitor charge again? If yes, when? And can it be controlled?


title: Forcing quadrupole moments to vanish for a neutral system

text: For a system of electric charges $q_i$, at positions $\mathbf{r}_i$, with a nonzero net charge $Q=\sum_i q_i$, one can define a "centre of charge" in the obvious way as $$ \mathbf{r}_c=\frac{1}{Q}\sum_i q_i\mathbf{r}_i. $$ This concept is definitely not as useful as one might naively hope, but it does have a physical significance as the position of the origin that sets the system's dipole moment (and therefore the dipolar term in a multipolar expansion) to zero. That means, then, that the monopole approximation is far better in the far field than normally: the electrostatic potential goes as $$ \Phi(\mathbf{r})=\frac{Q}{|\mathbf{r}-\mathbf{r}_c|}+O\left(\frac{1}{|\mathbf{r}-\mathbf{r}_c|^{3}}\right) $$ instead of $1/r^2$ as usual. * * * For a neutral system with $Q=0$, however, the concept of centre of charge is meaningless and there is no monopole term in the expansion. The relevant concept is then the dipole moment, $$ \mathbf{d}=\sum_i q_i(\mathbf{r}_i-\mathbf{r}_0), $$ which is independent of the position $\mathbf{r}_0$ of the origin. However, this means that the relative importance of the subleading term in the multipole expansion is higher than above: $$ \Phi(\mathbf{r})=\frac{\mathbf{d}\cdot\mathbf{r}}{|\mathbf{r}-\mathbf{r}_0|^2}+O\left(\frac{1}{|\mathbf{r}-\mathbf{r}_0|^{3}}\right) $$ **My question is:** for a neutral system, is it possible to find a suitable position $\mathbf{r}_0$ for the origin that will set the subleading, quadrupole term to zero? Since this entails a system of five equations (linear in $\mathbf{r}_0$ when $Q=0$), I suspect this is impossible in general geometries. If this is the case, which geometries allow for vanishing quadrupole moments and which ones don't? In the cases where one can do this, does this position have a special name? More generally, if all multipole moments up to some $l\geq0$ are zero, (when) can the subleading term be made to vanish?


title: Definition of pressure in Bernoulli's principle

text: I have a difficulty to understand Bernoulli's Principle which states increased fluid speed decreases the pressure. But in a wind tunnel when we measure the pressure with a pitot tube, both the pressure and the velocity of the wind is higher at the narrow section. Does the pitot tube measure another kind of pressure than the one in Bernoulli's principle? What is P1 and P2 in the figure at this link: http://eidein.edublogs.org/2008/07/29/36-bernoullis-principle/


title: Velocity of satellites greater than required velocity

text: If $v=\sqrt{\frac{GM}{r}}$ where and the $v$ is perpendicular to the gravitational force, the object will go in a perfectly circular orbit.   I want to discuss a few more possibilities and please be so kind to roughly draw or describe the shape of orbit and change in $r$.   1. If the $v$ was decreasing constantly as a result of air drag constantly.    2. If the $v$ was intentionally increased by a certain amount.    3. If an object were to given a velocity, $v\gt\sqrt{\frac{GM}{r}}$ perpendicular to $g$. Can anyone show an animation/diagram of how this would hold true in Einstein's way of looking at gravity(space-time curvature).


title: Phonon-polariton literature resources?

text: What is a good resource for studying phonon-polaritons?


title: Electric field inside and outside a metallic hollow sphere

text: 1) It is known that inside a metallic hollow sphere it will not experience outside electric field because of the charge separation of electrons and holes at the surface of sphere and creating an equal and opposite field. So it is better to stay inside the car during lightning. Why stay in the car during thunderstorms? But isn't there a maximum value of electric field the metallic surface can resist because at some point (higher value) of electric field the movement of electrons will get stopped because it can't generate more electron holes pairs and which will limit the further change in increment of electric field to zero. Since lighting produce such a high value of electric field is it safe to stay inside a car during lightning? _assuming no heat generated during lightning_ 2) Similar case about charge inside a earthed metallic hollow sphere. If I keep some charge inside the metallic hollow sphere, outside field is zero and inside field is dependent upon the charge. If I increase the charge inside the metallic hollow sphere to such a level which is more than the total electron charge on the surface of sphere then the earthing has no effect because it has already neutralized the maximum amount of holes (+ve) by accepting the electrons from earth. So the field will escape outside.


title: Defining electric potential energy

text: When we say that the electric potential energy at a point is X is that assuming we are using a positive test charge?


title: What are the fields in this problem?

text: In problem 3 of chapter 2 of Landau Lifshitz "Mechanics," I don't understand the meaning of the fields as defined in the following statement: > Which components of momentum and angular momentum are conserved in motion in > the following fields? > > (a) the field of an infinite homogeneous plane,   >  (b) that of an infinite homogeneous cylinder,   >  (c) that of an infinite homogeneous prism,   >  (d) that of two points   >  etc. I just don't get what he's trying to say. Does he mean an electric field produced by a uniformly charged infinite plane, or an uniformly charged cylinder, etc.?


title: Moving along friction surfaces

text: If a particle moves along a one dimensional surface with constant friction. As the particle moves from point $A$ to point $B$ it loses an amount of energy equals $E(A,B)$. Consider that the particle returns back to point $A$. It loses another $E(A,B)$ units of energy so it loses total $2*E(A,B)$ of energy but doing integration to get the change of kinetic energy, We add integral of the force of friction from $A$ to $B$ to the same integral from $B$ to $A$ so we get zero. How can this happen?


title: Clarification on a Goldstein formula steps (classical mechanics)

text: At page 20 of Classical Mechanics' Goldstein (Third edition), there are these two steps given between eqs. (1.51) and (1.52): $$\sum_i m_i \ddot {\bf r}_i \cdot \frac{\partial {\bf r_i}}{ \partial q_j}= \sum_i [\frac {d}{dt}(m_i {\bf v}_i \cdot \frac{\partial {\bf v}_i}{\partial \dot q_j})-m_i {\bf v}_i \cdot \frac{\partial {\bf v}_i}{\partial q_j}]$$ and $$\sum_j \\{ \frac{d}{dt}[ \frac{\partial}{\partial \dot q_j}(\sum_i \frac{1}{2}m_i v^2_i)] - \frac{\partial}{\partial q_j}(\sum_i \frac{1}{2}m_i v^2_i)-Q_j \\}\delta q_j .$$ Why does "$ \frac {1}{2}$" appear in the second formula?


title: Simple heat transfer question

text: > You add an unknown volume of milk of $5.2 ^\circ C$ to a cup of coffee ($40 > mL$ of water, temperature: $80.3 ^\circ C$). After a while of stirring the > temperature reaches $73.2 ^\circ C$. The properties of water are the same as > those of coffeewater. Neglect the warmth capacity of the cup: > >   * _Calculate the warmth that has left the coffee._ > >   * _Calculate the added milk in mL._ > > $$Q = cm\Delta T$$ $$ c = 4.18\times10^3$$ $$m = 0.04 $$ (I believe so, 1L of water is 1kg, so 40ml=0,04 kg) $$\Delta T = -7.1$$ $$Q=0.04\times4.18\times10^3\times-7.1 \approx -1.2\times10^3\text{ J}$$ so the answer is approx. $1.2\times10^3\text{ J}$ Is this correct, or am I missing something? The next question: $$m = \dfrac{Q}{c\Delta T}$$ $$m = \dfrac{1.2\times10^3}{4.18\times10^3\times68} \approx 4\text{ mL}$$ This also doesn't seem to be correct. What am I doing wrongly here?


title: Why does the local inertial compass coincide with the stellar compass?

text: I found this physics paper via a non-duality site and I wished that I could understand it. Could someone please either read it and explain it to me or else point me to pages that would help me understand the necessary concepts? Somehow I don't get the concept of inertial compass. But even if I did understand the terms, I would not be qualified to judge this thesis... http://www.worldnpa.org/pdf/abstracts/abstracts_21.pdf Thanks :)


title: What are the forces of constraint if there are multiple equivalent constraints?

text: Suppose a large (rigid) block is sitting on top of two smaller blocks of equal height $1$, both of which rest on the ground. We wish to find the position of the block (easy) and the forces of constraint on the block coming from the two smaller blocks. If we write the Lagrangian along with the equations of constraint $z=1$, we can find the motion of the object using Lagrange multipliers, and we can also find the force of constraint. If there were constraints $\\{G_n\\}$, then the force of constraint in the $z$ direction would be $F_{n} = \lambda_n \frac{\partial G_n}{\partial z}$ on the constraint $G_n$. However, this tells us nothing about how much force each individual block is exerting. We know from experience that the upward force from the smaller blocks must equal $mg$, where $m$ is the mass of the larger block, and furthermore, that the force is evenly distributed, so that each of the two smaller blocks exerts an equal amount of force. It seems natural to indicate the existence of two blocks by writing down the constraint $z=1$ twice and then trying to solve the resulting system of equations. However, this doesn't seem to get us very far. So, how is it possible to detect the fact that the force on the two smaller blocks is distributed equally?


title: Trace of stress tensor vanishes ==> Weyl invariant

text: You often see in textbooks the statement that ${T^\mu}_\mu = 0$ implies Weyl invariance or conformal invariance. The proof goes like $\delta S \sim \int \sqrt{g} T^{\mu\nu} \delta g_{\mu\nu} \sim \int \sqrt{g} {T^\mu}_\mu$ where I have varied the action with respect to the metric and assumed $\delta g_{\mu\nu} \propto g_{\mu\nu}$ (i.e. a Weyl transformation). This does not seem to be completely general because I can imagine a Lagrangian containing matter fields with non-trivial conformal weights. Then the full variation under Weyl tranformation contains a term proprotional to the matter equation of motion. So I would conclude that the correct statement is more like ${T^\mu}_\mu = 0, \quad \& \quad \frac{\delta S}{\delta \phi} = 0\implies \textrm{Weyl invariant}$ Is it true that Weyl invariance only holds when the matter fields are on-shell or am I missing something? Thanks.


title: How to get "complex exponential" form of wave equation out of "sinusoidal form"?

text: I am a novice on QM and until now i have allways been using sinusoidal form of wave equation: $$A = A_0 \sin(kx - \omega t)$$ Well in QM everyone uses complex exponential form of wave equation: $$A = A_0\, e^{i(kx - \omega t)}$$ **QUESTION:** How do i mathematically derive exponential equation out of sinusoidal one? Are there any caches? I did read Wikipedia article where there is no derivation.


title: Do you regret being a Physicist?

text: I'm a high school student and physics is my passion and of course I want to major in physics, but I get a lot of negative responds from my parents, friends and some teachers when I say I want to be a theoratical physisct. I'm sure some of you struggled with this since this site obviously has a lot of researchers and physics students. The most common questions I recieve: 1- What do physicsts do if every thing is already discovered? (This question is annoying and stupid) 2- What is the average annual salary for a physicist? 3- Why is physics even important? 4- So will you end up eventually as a high school physics?


title: Beta particle energy

text: what does it mean that a beta particle has a 1.6 MeV of energy , does it mean that it can ionize 120000 hydrogen particle with ionization energy of 13.6 MeV ?


title: Optimal velocity in rain

text: **_Problem statement:_**   > A sphere is rolling in rain from point A to point B. The vertical speed of > rain is V and horizontal speed of rain is v, as shown in the picture. Angle > between horizontal component of rain's velocity and sphere's velocity is > $\varphi$. What is the optimal speed of sphere so that it would be as dry as > possible?   >   > ![Problem picture](http://i.stack.imgur.com/sljvY.png) So here's what I tried to do:   The velocity of rain in respect to the sphere is $\vec{v} = \vec{v_s} - \vec{v_r}; v=\sqrt{v_s^2 + v_r^2} = \sqrt{v_s^2+(\sqrt{V^2+v^2})^2}=\sqrt{v_s^2+V^2+v^2}$. I thought that maybe if I found the derivative of this function in respect to $v_s$ and equated it to 0, that could be the answer. But I don't know how to find the derivative of such function and the solution doesn't seem right to me, though I couldn't come up with anything better.. Any help appreciated!


title: Can we have consistent histories inside a black hole?

text: A consistent history is a POVM set of observables corresponding to a time- ordered product of projection operators. For gauge theories, not any old operator will do, only gauge-invariant observables. In quantum gravity, there are no localized observables, meaning observables have to be defined with respect to the conformal boundary. Inside a black hole, this is none other than the black hole singularity, which is often spacelike. Spacelike singularities are enigmatic in string theory. This raises the very interesting question; can we have consistent histories inside black holes?


title: Effect of temperature on radioactivity?

text: I'm researching the effect of temperature on uranium radioactivity, however I can't find any solid empirical evidence to prove the notion that temperature does not affect radioactivity. Can anyone link to some solid data on the topic, or offer some advice on where to find such data?


title: How do I solve for the phase constant given the amplitude and the angular frequency?

text: > A piston (with mass M) in a car engine is in vertical simple harmonic motion > with amplitude A. The engine is running at a period T. Suppose a small piece > of metal with mass m were to break loose from the surface of the piston when > it (the piston) is at the lowest point z and velocity is v. At what position > in meters does the piece of metal lose contact with the piston? Also the maximum velocity and acceleration are given. The text book gives these equations: $x(t) = Acos(wt+φ)$ $v(t) = -wAsin(wt+φ)$ $a(t) = -w^2Acos(wt+φ)$ So I propose that the best way to figure this out would be to solve for the time at which acceleration is at its max, then plug that time into $x(t)$. But I must first solve for the phase constant. Is this correct? If so, how do I find the phase constant?


title: Voltage of open circuit

text: > A battery with emf $\varepsilon$ and internal resistance $r$ is connected > with a resistor $R$ in the following open circuit. What is the voltage > $V_{ab}=V_a-V_b$? > > ![enter image description here](http://i.stack.imgur.com/8RVNX.png) The answer is $- \varepsilon$. " _No current. There is no voltage change across R and r._ ". But I don't really understand why ... I was thinking intuitively it should be $0$? Then thinking of how to get 0, I was thinking ... $V_a = - \varepsilon$ since its on the negative terminal, the $V_b = + \varepsilon$ since its on the positive terminal. But $V_a - V_b = -2 \varepsilon$ ... how do I make sense of this?


title: Fluid Dynamics: Bernoulli Air Tunnel

text: ![Air Tunnel](http://i.stack.imgur.com/qzqzF.png) I have the following question as part of a lab intro to explain the more complicated stuff I'l actually have to submit in the lab. I put all the questions here just to show my thought pattern/what I'm supposed to be working through. Given:               Measurement 1 (fan off):     Micrometer reading = -0.010 inches     Transducer reading = 3.000 volts     Measurement 2 (fan on):     Micrometer reading = +0.500 inches     Transducer reading = 3.586 volts      (a) The relationship between pressure and voltage is linear, calculate an equation which relates the transducer voltage (in volts) to the pressure difference (in Pa) - _I understand this, as this is simply do a linear interpretation two points, etc._ (b) Port #8 in the test section is displayed as 3.450V on the meter. Use (a) equation to convert this value to a pressure differentialin Pascal. Calculate the air velocity at this point - _Plug and chug the answer from part (a)_ (c) The average velocity through the test section at Port #8 is equal to the point velocity in (b). What is the Mach number? Is this incompressible ﬂow? - **this I don't know how to find I know M = V/a though** (d) Apply continuity to determine theaverage velocity at Port #2. Based on the same assumption as (c) the average velocity through the test section at Port #8 is equal to the point velocity - **and since I don't understand (c) I don't understand this either**


title: Inserting metal into parallel plate capacitor

text: > The plates of an isolated parallel plate capacitor with a capacitance C > carry a charge Q. The plate separation is d. Initially, the space between > the plates contains only air. Then, an isolated metal sheet of thickness > 0.5d is inserted between, but not touching, the plates. How does the > potential difference between the plates change as a result of inserting the > metal sheet? The answer is: The potential difference will decrease. How do I get this result, it seems right but how do I reason it out? $$V = Ed$$ There was mention about " _E is the same outside the conductor. But the field inside is 0._ ". But in the formula, E is the field outside conductor right? If so it remains the same? Then $d = 0.5d$?


title: Expected value inequality

text: Why is $\langle p^2\rangle >0$ where $p=-i\hbar{d\over dx}$, (noting the **_strict_** inequality) for all normalized wavefunctions? I would have argued that because we can't have $\psi=$constant, but then I thought that we can normalize such a wavefunction by using periodic boundary conditions... So I don't how to argue that the inequality should be strict... Is it that otherwise it would be trivial? Added: It is clear that $\langle\psi|p^2|\psi\rangle\geq0$. i am just wondering why the inequality is _strict_. Thanks.


title: Where the angular momentum has gone?

text: I came across an interesting problem when I prepared for the preliminary exam on electromagnetism. Below is the problem in its original words: > A metallic sphere of mass, $m$, and radius, $a$, carries a net charge, $Q$, > and is magnetized with uniform magnetization, $M$, in the $z$-direction. > > (a) Determine the total angular momentum associated with the electromagnetic > field. > > (b) The magnetization of the sphere is now reduced to zero. Assuming that no > external mechanical forces act on it (PS: this is possible. For instance, > just heat up the sphere and the magnetization will be eliminated ), > determine the angular velocity and the sense of rotation of the sphere when > $M=0$. > > (c) Describe the origin of the torque that causes the sphere to rotate. For part (a), it is straight forward to calculate electric and magnetic field E, H inside and outside of the sphere. Generated by the net charge $Q$ uniformly distributed on the surface, the electric field $E$ is isotropic outside the sphere and vanishes inside. The magnetic field H is generated by magnetization $M$ and can be calculated using the "magnetic scalar potential" formalism since there is no free current. The result is that H is also uniform inside the sphere and takes the dipole form outside the sphere. The total angular momentum $J$ is obtained by integrating over the region where a local angular momentum density of the EM field exists, i.e. outside of the sphere. It is in the $z$-direction and I find it to be $$J= \frac{2 Q M a^{2}}{9 \epsilon_0 c^{2}}$$ For part (b) and (c), it is interesting to ask why the sphere would rotate after magnetization $M$ is erased. What I think up is that as $M$ decreases to zero, a electric field with non-zero curl is generated, as implied by Faraday's law. This electric field is in the azimuthal direction, and will act on the surface charge $Q$ to provide a torque. However, a straightforward calculation can show that the angular momentum this torque can transfer is $$J'= \frac{Q M a^{2}}{9 \epsilon_0 c^{2}}$$ Obviously a factor of 2 is missing, so this only accounts for half of the angular momentum stored originally in the EM field. So I am perplexed by where the other half has gone? I come up with 2 possibilities:   1. since the charged sphere begins to rotate, there is also non-zero angular momentum stored in EM field after magnetization is removed. But it could hardly account for the missing amount, since in that case the electric field $E$ is proportion to $Q$, the magnetic field $H$ to $Q^2$ ( one $Q$ from the charge and a second $Q$ from the angular velocity). Hence one expects the angular momentum stored is proportional to $Q^3$, which is very different from $J$'s linear dependence on $Q$.   2. maybe in the process of magnetization removal, EM radiation carries away some angular momentum. But this possibility has to be validated quantitatively before it can convince people. What I feel is that if the process of removal is infinitely slow, then the effect of radiation can be neglected. Then what is the correct answer? I provide quantitative result for $J$ and $J'$ in the hope that someone can check them and point out that I have made a wrong calculation lol. But I am really eager to hear from any explanation that sounds real ... Electromagnetism is as interesting as it was 3 years ago, but I just don't remember as much as I did then ...


title: What did Marie Curie do for atomic theory?

text: There appears to be a distinct lack of agreement in the physics community on what exactly Marie Curie did for atomic theory. Many journals state that Curie was responsible for shifting scientific opinion from the idea that the atom was solid and indivisible to an understanding of subatomic particles. However, JJ Thompson had already discovered the electron prior to Curie's work, and Rutherford proved the nature of atomic structure - so what did Marie Curie really do for atomic theory?


title: Differences In Potential Equations

text: Could someone please describe the differences between the uses each of these potential equations: > Potential due to a point charge: > > $V = \frac{k \cdot q}{r} - \frac{k \cdot q}{{r}_{\text{ref}}}$ * * * > Coulomb's potential (why are we missing the $r_{\text{ref}}$???): > > $V = \frac{k \cdot q}{r}$ * * * > Electrostatic potential energy of a two-charge system: > > $U = q_1 \cdot V = q_1 \cdot \frac{k \cdot q_2}{r} = \frac{k \cdot q_1 \cdot > q_2}{r}$


title: what happens to information when black holes merge

text: When a small black hole mergers with the larger black hole, Would an observer trapped in the smaller black hole be able to detect any difference? Does the observer still exists in the smaller black holes timeframe? If the larger black hole was built up from millions of smaller black holes do those event horizons merged together? Or do they pile up like BBs in a bowl?


title: Is the magnetic field a feature of our universe, or is it a consequence of the electric field?

text: Is the magnetic field a feature of our universe, or is it a side effect of the electric field? In other words: if we were to simulate in a computer a system of moving charged particles, taking in account only equations for the electric field, would we see magnetic effects?


title: Fields versus Photons

text: Does the field of a set of photons behave differently from a single photon. e.g. Suppose I have a group of photons with their Electric fields $\mathbb{E}_n$ all aligned. So $$\mathbb{E}_{\text{tot}} = \mathbb{E}_1 + \mathbb{E}_2+ \text{...} + \mathbb{E}_n$$ and upon this field a single photon with parallel $\mathbb{E}_i$ field is incident. Does $\mathbb{E}_i$ scatter? Or do I require such a large number of photons in $\mathbb{E}_{\text{tot}}$ that the field polarizes the underlying quantum vacuum to produce particle pair that scatter the $\mathbb{E}_i$ particle. More simply put, can an electric field by itself scatter a photon?


title: Superposition of Ricci scalars

text: Suppose I have two point/line singularities in spacetime (what is important to me is that they are localized). Also suppose I have some fields in spacetime and that the two singularities interact with each other via the fields (fields are non-minimally coupled to the curvatures). Now if I denote the Ricci scalar of the entire system by $R$ and for each of the singularities by $R_1$ and $R_2$, is $R=R_1+R_2$? I am not able to make up my mind. I don't think so and I think only to some order in perturbation this is true. But your help/comments will be greatly appreciated.


title: Free Body Diagram of Fluid Statics Problem

text: I need help determining the forces, and the direction of them in this fluid statics problem. A hollow cylinder with closed ends is 300mm in diameter and 450mm tall. It has a mass of 27kg and a tiny hole at the bottom. It is lowered slowly into water and then released. I need to calculate: a. The pressure of the air inside the cylinder. b. The height to which the water will rise. c. The depth to which the cylinder will sink. I was already given the answers to these problems for clarification but I think I'm going to start with the free body diagram (FBD) to part c. Now I know there is a weight from the cylinder going down and a buoyant force going up because of the water and the submerged volume of the cylinder. Now does the air trapped in the cylinder also create a buoyant force or some kind of force also pushing up? And how would I calculate this force if there is one? After that, I'm sure I can get the rest of the calculations.


title: Chance of Positron Escaping an Atom

text: Suppose a positron was created in the nucleus of the atom. Now, suppose we have 1,000,000 of these atoms all in their own vacuum. Statistically, would a large percent of positrons successfully leave these atom without being annihilated by an electron? Assume we have an element such as Sodium-22. Say we have billions of atoms surrounding each atom, are the chances still low or high for a positron to escape?


title: Newtons third law and Atwood Machines, confusion about tension

text: The classic way of solving these problems is like so: $T = mg - ma$ and $T = mg + ma$. However, according to Newton's third law, is this not wrong? I know how to solve these problems, but the concepts are confusing to me. First off, apparently both tensions in the string are equal to each other. But I don't understand this, and according to Newton's third law, this should be false? ![enter image description here](http://i.stack.imgur.com/gJylI.gif) Assuming $T_1$ acts on $m_1$ and $T_2$ acts on $m_2$ Newton's third law should mean that $T_1 = -m_1g$ because the force of gravity from the box $m_1$ acts on the rope, and the rope reacts with equal and opposite force. But according to this same logic, $T_2 = -m_2*g$ And since the tension in the rope must be equal at all points, and also since $m_1 \ne m_2$ this means that this is false. Does Newton's third law not apply to Atwood's machines? Or am I confused?


title: Transient Stress and Piezoelectric Generators

text: From what I can gather, the piezoelectric effect links mechanical stress to the potential difference across a material, with a two-way relationship between the two (motor or generator). I don't know much about piezoelectricity, but is an imposed stress all you need to produce a potential difference? Can a constant stress be applied or does it need to be alternating like the magnetic flux in Faraday's Law? If it can be constant, why not just build piezoelectric foundations under our buildings to get tons of free energy?


title: How can $J_1^2, J_2^2, J_{1z}, J_{2z}$ commute mutually?

text: I'm reading through J. J. Sakurai's Modern Quantum Mechanics book and currently looking at the "Angular-momentum addition" part. Here, it says you have two options and that one option is to construct simultaneous eigenket $\vert j_1j_2;m_1m_2\rangle$ of $J_1^2, J_2^2, J_{1z}, J_{2z}$ since the four operators commute with each other. I understand that $J_1^2$ and $J_{1z}$ commute, but I'm not sure how $J_1^2$ and $J_{2z}$ can commute intuitively. "commute" means that one can measure both at once right? But total angular momentum of spin 1 and angular momentum of spin 2 are independent. Where am I wrong here?


title: Does mass equal angular momentum?

text: At the wikipedia pages for angular momentum ($L$) and moment of inertia ($I$) we find the equations: $$L=I \omega$$ $$I=m r^2$$ where $m$ is mass and $r$ is the distance between said mass and barycenter and $\omega$ is angular velocity. Now rename the angular velocity as follows: $\omega=\frac{d\theta}{dt}$. Note that the area of an ellipse sector swept out from a focus is given by: $$A(\theta)=\frac{1}{2}\int_0^{\theta} r^2 \:\: ds$$ And so: $$\frac{dA}{d\theta}=\frac{r^2}{2}$$ Note that for a mass in an elliptical orbit this area is proportional to time. Therefore, we can write: $$\frac{d\theta}{dt}\sim\frac{d\theta}{dA}\sim\frac{1}{r^2}$$ Substituting this back into the first two equations at the top of this post gives: $$L\sim m$$


title: Equilibrium of a sphere in a water tank

text: A rigid sphere of radius $R_S$ made from a material with specific gravity $SG_s$ is completely submerged in a tank of water with radius $R_t$ and initial depth $L$ as shown in the figure ![enter image description here](http://i.stack.imgur.com/UZb7T.png) The sphere is placed over a small hole of radius a in the bottom of the tank that is open to the atmosphere. Find an inequality were the sphere will remain sitting on the bottom of the tank plugging the hole. You may assume that $a<< R_S$ How do I begin to solve this? Can someone point me in the right direction?


title: Faraday tensor, antisymmetric rank two

text: $F^{\mu \mathcal{V}}$ is defined in http://www.lecture-notes.co.uk/susskind/special- relativity/lecture-7/relativistic-lorentz-force/ How to show that $F^{\mu \mathcal{V}}$$F_{\mu \mathcal{V}}$, is a lorentz scalar and how to find its value in terms of vectors, Electric field (E), and magnetic field(B). This is mentioned in properties (point number 3) in wikipedia article https://en.wikipedia.org/wiki/Electromagnetic_tensor How to prove that?


title: Wave function of hydrogen atom including spin of nucleus

text: How do I write the wave function of hydrogen atom taking into consideration of nucleus spin? For example consider $2S_{\frac{1}{2}}$ state with nucleus spin $I$, then wave function $\psi=\langle2S_{\frac{1}{2}},F,F_{3}|$ where $F$ is the total angular momentum of hydrogen atom $F=J+I$ and $F_{3}$ is the projection of it along z axis. Now what will be the explicit form of the wave function? Thanks in advance.


title: About divergence of a vector field and geodesic sphere

text: I have a question. I want to know the difference between the sphere and the geodesic sphere. Another question: given a vector field, $Y$, on a manifold $M$ defined by: $Y(p)=p$ for every point $p \in M$, I tried to calculate the divergence of $Y$. I think that it is equal to $n$, where $n$ is the dimension of $M$, but I don't know exactly how I can compute it.


title: What did Hawking mean? 'Time started at the big bang'. Book suggestions please

text: **After writing down this question, I have come to realize that. What I really want is _reading materials_ on the questions below.** Before the big bang there was no such thing as 'time' (Steven Hawking on why the pope was wrong when he stated that physics and cristian beleves could go together as someone 'must have started the big bang'). Hawking stated that according to his theories time started at the big bang. Before the big bang, no time, hence no creator. Hawking actually said this in a speach, in memorial to Gallileo when the pope made official appologices on behalf of the church for banning Gallileo. I read this years ago so I might be wrong in some detail. **I would like something to read to understand what Hawking meant when he said that there could be no creator since time 'started' at big bang and did not exist before that**. I found this paper but I am not sure I understand what it means. Concentric circles in WMAP data may provide evidence of violent pre- Big-Bang activity. I think I have to read other things before actually digging in to that paper since I find it pretty hard. **What can I read as an introduction before reading this paper. Frankly, I don't understand it.** From a physics phd student I was informed that many physics researchers actually are unsure that the big bang really existed. Since they don't believe that everything could have existed in a singular point. Do they? (Might be a tricky question to answer). When you spread out time it turns out that time can not be spread out. I.e. it does not work like a rubberband. Or perhaps, When you compare time (maybe as a result of jouries at extreme speed where the two objects then have different ideas of how long time it has taken) does not follow form a bell curve statistically. Why not? The uncertainty evaluated at t = 0 also gives strange results but I am not sure it is related. _Perhaps it is perhaps not_. **I know a lot of calculus, a lot of linear algebra and basic quantum mechanics.**


title: Free Energy of N Spin 3/2 Particles

text: This question is from the book "Introductory Statistical Mechanics" by Bowley and Sanchez. The question is as follows: Calculate the free energy of a system with N particles, each with spin 3/2 with one particle per site, given that the levels associated with the four spin states have energies $\frac{3}{2}\epsilon$, $\frac{1}{2}\epsilon$, $\frac{-1}{2}\epsilon$, and $\frac{-3}{2}\epsilon$ and degeneracies 1, 3, 3, and 1 respectively. The answer given at the end of the book is $-3Nk_BT\ln(e^{\epsilon/2k_BT}+e^{-\epsilon/2k_BT})$ The way I proceeded to write the partition function is as follows: The degeneracy due to spin $g_n^{spin}=2s+1=4$ Hence the partition function is: $$Z=4e^{-3\epsilon/2k_BT}+12e^{-\epsilon/2k_BT}+12e^{\epsilon/2k_BT}+4e^{3\epsilon/2k_BT}$$ Note that I have also multiplied by the degeneracy indicated in the body of the question. However, since $F=-k_BT\ln(Z)$ the answer is not the same as quoted in the appendix of the book. I would be very grateful if you could provide insight on this issue.


title: Synchronising the Earth's rotation via mass redistribution

text: How much material would have to be moved per year from mountain-tops to valleys in order to keep the Earth's rotation synchronised with UTC, thus removing the need for leap seconds to be periodically added? Would it be a feasible project to undertake in order to resolve the current disagreement about the future of the leap second?


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

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