title: Explanation for the next steps of chaplygin dipole

text: this post is the Chaplygin dipole, it's an interesting issue. **Can someone explain me these steps in other words please? any Explanation of any step will help me, I hope that together I will understand all! please explain me the steps in other words because I didn't understand what is written** In 1903 Chaplygin published another remarkable paper (Chaplygin 1903) devoted to the motion associated with a compact vorticity distribution in a two- dimensional unbounded inviscid flow. In the introduction of that paper he gave a precise formulation of the problem: Consider an unbounded mass of incompressible fluid in which the motion is parallel to the OXY plane; let the motion outside some circular cylinder be irrotational, the velocity being equal to zero at infinity. The question is to find a distribution of vortex lines inside the cylinder that gives rise to a uniformly translating vortex column with a continuous velocity distribution and with a positive pressure all around. As a first example of the solution Chaplygin considered in detail a case of rectilinear motion of a circular vortex of radius $a$ with a constant translation velocity $v_0$. By superimposing on the whole fluid fluid a uniform velocity $-v_0$ he obtained a stationary problem of a steady vortex cylinder placed in a potential flow with uniform velocity at infinity. By choosing the polar coordinate system $(r,\theta)$ , with the origin at the centre of the cylinder, the stream function $\psi_1$ for the potential flow around the cylinder is written as: ![enter image description here](http://i.stack.imgur.com/qpjbO.jpg)


title: How to put spin-1/2 Fermi sea into real space representation?

text: For $N$ spin-1/2 free fermions, the ground state is given by the Fermi sea, $$|{\rm FS}\rangle = \prod_{|{\bf k}|<k_F} c_{{\bf k}, \uparrow}^\dagger c_{{\bf k}, \downarrow}^\dagger |0\rangle $$ while we can take Fourier transform of the fermion creation operator $$c^\dagger_{{\bf k},\sigma } = \sum_{{\bf r}_{\sigma}} {\rm e}^{-i{\bf k\cdot r}_\sigma} c^\dagger _{{\bf r}_\sigma,\sigma}$$ But how to put it in a compact form of creation operators of real space?


title: Are lens spaces classified via a Weinberg angle?

text: I am thinking about Kaluza Klein theory in the 3 dimensional lens spaces. These have an isometry group SU(2)xU(1), generically, and in some way interpolate between the extreme cases of manifolds $S^2 \times S^1$ and $S^3$ (or, arguably depending of your parametrisation, across $S^2 \times S^1 ... S^3 ... S^1 \times S^2$). My question is, simply, if the application of Kaluza Klein construction forces different coupling constants to the two groups $SU(2)$ and $U(1)$, and then a kind of discrete Weinberg angle, or if they have always the same coupling. Note that when you build a lens space bundle over CP2 (which is equivalent to a $S^1$ bundle over $CP^2$ x $CP^1$, if I read correctly a remark of Atiyah reported by Kreck and Stolz), you get an isometry group of SU(3) times SU(2) times (1), so the same question could be asked here, in this more widely known setup. Of course that odd dimensional spaces do not carry chirality, but my question is just about having different couplings, that is all. It could be interesting, as a plus, to compare with coupling relationships coming from Non Commutative Geometry...


title: Force applied in a body moving at high speed

text: Consider a rod of length $l$ and uniform density is moving at high speed. I want to deflect the rod where should I need to apply the minimum force, so that the rod is deflected..?


title: String landscape in different dimensions

text: For D = 11 large (uncompactified) spacetime dimensions, the only "string theory" vacuum is M-theory For D = 10, there are 5 vacua. Or maybe it's more correct to say 4, since type I is S-dual to $Spin(32)/\mathbb{Z}_2$ heterotic? For D = 4 there is a monstrous number of vacua and there's no full classification > For which values of D a full classification exists, and what is it? My guess is that the classification exists for $D \geq D_{min}$ but what is the value of $D_{min}$?


title: On the Coulomb branch of N=2 supersymmetric gauge theory

text: The chiral ring of the Coulomb branch of a 4D $\mathcal N=2$ supersymmetric gauge theory is given by the Casimirs of the vector multiplet scalars, and they don't have non-trivial relations; the Casimirs are always independent. Also in Gaiotto's class of $\mathcal N=2$ non-Lagrangian theories, the chiral ring of the Coulomb branch doesn't (seem to) have relations. Is it a general fact? If so, how can we deduce it from the $\mathcal N=2$ supersymmetric algebras? * * * I was asked to clarify the definition of the Coulomb branch in non-Lagrangian theories; let's define them for $\mathcal N=2$ SCFT by the fact that $SU(2)_R$ symmetry acts on the Coulomb branch operators trivially.


title: Limits on stable charged particles

text: Are there generic model-independent limits on massive stable charged particles (say 10-500GeV)? I mean SU(2)/color singlets with just a hypercharge. For example the LEP search (CERN-EP/99-075) was only for masses greater than 45GeV. Recent LHC searches seem to make assumptions about lifetimes, R-Hadron formation etc. Are there generic limits from precision electroweak? References would be appreciated.


title: What happens when a ball stops bouncing?

text: If I were to drop a bouncy ball onto a surface, each successive bounce will be lower in height as energy is dissipated. Eventually, however, the ball will cease to bounce and will remain in contact with the ground. _What happens during that small moment of time that is the transition period?_ Here's my line of thought so far: Let's say, for example, that the ball could be modeled as an "ideal" bouncy ball. Each time it hits the floor, it rebounds with one-half of the impact velocity, the other half of which is absorbed and dissipated by various things. Will this ideal bouncy ball ever stop bouncing? The answer, as far as I can tell, is yes (well, _maybe_ ). Since each bounce has half of the initial velocity of the previous bounce, the time between each bounce also halves. The first bounce might take $1$ second, the second takes $0.5$ seconds, the third $0.25$ seconds, etc. This infinite series converges to a finite amount of time, but with an infinite number of bounces. Since it's obviously not possible for an actual ball to bounce an infinite number of times in finite time, it is clear that the above model doesn't work. Eventually, I decided that the problem must be in the fact that the model doesn't take into account the fact that ball spends time compressing and decompressing while in contact with the floor. Even once the ball touches the ground, the center of gravity continues to move downward as the ball compresses. Soon, the ball expands, and the center of gravity moves upward. Lets $h_1$ be the height of the center of gravity when the ball is uncompressed, and $h_2$ the height of the center of gravity of the ball when compressed, and $D = h_1 - h_2$. Eventually, the height of the bounce will be smaller than $D$, so that center of gravity simply vibrates between those two locations, and the ball never leaves the ground.


title: Maximum depth for bacteria on rogue planets

text: If microscopic life evolves on a planet which is subsequently ejected from orbiting its star, then the planet would cool down. How deep into the interior of the planet could organisms reach, assuming the mantle had cooled down enough? For example, on Earth there are rock-boring bacteria that have tunneled their way 3 km into the crust. The limiting factor for depth of bacteria on Earth seems to be temperature, but presumably if the mantle had cooled down then at some depth other factors would come into play such as pressure or rock hardness or composition.


title: Creating/Downloading a large Galaxy Dataset

text: I was wondering where I can get a more or less complete set of a galaxy to test an n-body simulation (preferably two colliding galaxies with approx 300k to 1M elements). Is it possible to extract this data from a public source like Nasa? Or does it make more sense to just generate the data? Note: I Found the SDSS Skyserver website, but I don't think it's possible process this data to get useful initial conditions and a more or less complete galaxy.


title: If a hot Jupiter was not tidally locked, then are there any specific cases where its wind speeds would be milder than those found on Jupiter?

text: If a hot Jupiter was not tidally locked, are there any specific cases where its wind speeds would be milder than those found on Jupiter? After controlling for the age of the hot Jupiter, of course. The one thing with giant planets is that they are so warm on the inside that the temperature gradient between the interior and the surface is smaller than that on a cold gas giant like Jupiter (though maybe the temperature gradient between the interior and surface is so large that our surface temperature may not matter as much). Now, what are the causes of wind? Wikipedia explains it below. > Wind is caused by differences in pressure. When a difference in pressure > exists, the air is accelerated from higher to lower pressure. On a rotating > planet, the air will be deflected by the Coriolis effect, except exactly on > the equator. Globally, the two major driving factors of large-scale winds > (the atmospheric circulation) are the **differential heating between the > equator and the poles (difference in absorption of solar energy leading to > buoyancy forces) and the rotation of the planet**. Outside the tropics and > aloft from frictional effects of the surface, the large-scale winds tend to > approach geostrophic balance. Near the Earth's surface, friction causes the > wind to be slower than it would be otherwise. Surface friction also causes > winds to blow more inward into low pressure areas. > > Winds defined by an equilibrium of physical forces are used in the > decomposition and analysis of wind profiles. They are useful for simplifying > the atmospheric equations of motion and for making qualitative arguments > about the horizontal and vertical distribution of winds. The geostrophic > wind component is the result of the balance between Coriolis force and > pressure gradient force. It flows parallel to isobars and approximates the > flow above the atmospheric boundary layer in the midlatitudes.[2] **The > thermal wind is the difference in the geostrophic wind between two levels in > the atmosphere. It exists only in an atmosphere with horizontal temperature > gradients**.3 The ageostrophic wind component is the difference between > actual and geostrophic wind, which is responsible for air "filling up" > cyclones over time.[4] The gradient wind is similar to the geostrophic wind > but also includes centrifugal force (or centripetal acceleration).[5] Now - here's the thing: if a hot Jupiter was not tidally locked, then the initial temperature difference between the day side and the night side would not be so great (although it is true that the difference is not that big in tidally locked hot Jupiters - but then that low night-day temperature contrast is irrelevant to our question as it is caused by the wind). One might initially believe that tidally locked planets do not rotate - **but that is not true** \- they do rotate, and their rotation speeds are not negligible because their orbital periods are so slow. So theoretically, one could consider the case of a non-rotating non-tidally locked hot Jupier (which would only be possible in very young hot Jupiters due to the timescale of tidal locking at small distances - but those hot Jupiters are still possible). Hot Jupiters that rotate at a very fast speed, though, would probably have a significant equator-pole temperature gradient that would drive a very fast wind speed. So - back to my original question. If a hot Jupiter was not tidally locked, **are there any specific cases** where its wind speeds would be milder than those found on Jupiter?


title: Could many widely separated space telescopes be combined for VLBI on IR/visible wavelengths?

text: I have read about ground-based Very Long Baseline Interferometry telescope arrays able to achieve huge resolution at IR/visible wavelengths. There are also space-ground VLBI configurations in operation today that work at radio wavelengths.   1. Is it feasible to employ many space-based optical/IR telescopes to achieve a high angular resolution visible light image of something the size of Jupiter? (Maybe during a transit or occultation, since it would be faint on its own.)   2. What would the separation need to be for a space telescope array of that size?   3. Would it be overly difficult to get precise enough timing, or to synchronize the relative motions of the orbiters or to deconvolve that motion from the sources?   4. Could it be made of many considerably smaller widely separated mirrors, or would it be better to use a fewer number of large mirrors?


title: Do we have an idea about the amount of matter in the universe?

text: Do we consider the amount of matter in the universe to be "infinite"? Or do we have an idea about "how much" there is?


title: What entities create a gravitational field?

text: It is well known that masses create a gravitational field. Photons are affected by gravitation, but do they generate a gravitational field as well? What about the other gauge bosons? Do gravitons create a gravitational field?


title: Why do we consider the evolution (usually in time) of a wave function?

text: Why do we consider evolution of a wave function and why is the evolution parameter taken as time, in QM. If we look at a simple wave function $\psi(x,t) = e^{kx - \omega t}$, $x$ is a point in configuration space and $t$ is the evolution parameter, they both look the same in the equation, then why consider one as an evolution parameter and other as configuration of the system. My question is why should we even consider the evolution of the wave function in some parameter (it is usually time)?. Why can't we just deal with $\psi(\boldsymbol{x})$, where $\boldsymbol{x}$ is the configuration of the system and that $|\psi(\boldsymbol{x})|^2$ gives the probability of finding the system in the configuration $\boldsymbol{x}$? **(Added)** (I had drafted but missed while copy pasting) One may say, "How to deal with systems that vary with time?", and the answer could be, "consider time also as a part of the configuration space". I wonder why this could not be possible. **Clarification** (after answer by Alfred Centauri) My question is why consider the evolution at all (What ever the case may be and what ever the parameter may be, time or proper time or whatever). My motivation here is to study the nature of the theory of quantum mechanics as a statistical model. I am looking at it from that angle.


title: How do we deal with Gribov ambiguities when calculating in quantum gauge theories?

text: How do we deal with Gribov ambiguities when actually calculating in quantum gauge theories? Any literature references?


title: Is there anything special about the Sun's photosphere in terms of density?

text: The Earth has a definite boundary between rocky/ watery surface and gaseous atmosphere. The same cannot be said of the sun. Even though the photosphere gives an apparent "edge" to the sun via the human eye, the matter of the sun continues on significantly further. I have even heard a claim that from a certain perspective, all of the planets out to the KBOs are orbiting "inside" of the sun. My question concerns what happens when we graph the density of the sun versus distance from the center. I suspect the result will resemble exponential decay; is this correct? Does the graph have any unusual characteristics (such as a lack of smoothness) at any point along the distance axis, in particular at the photosphere?


title: Microscopic fields inside a conductor

text: In a neutral conductor if we assume electrons as point charges, the electric field in the space between them cannot be identically zero. This microscopic field may be very weak. What if we were very close to one of electrons? Shouldn't the electric field diverge?


title: Eigenvalues of a mean correlation matrix (integral over correlation matrices with arbitrary density)

text: Consider a stationary dynamic system with state $s(t)$ and correlation structure described by $C_{ij}(\tau)=\mathbb{E}[(s_i(t+\tau)-\bar{s_i})(s_j(t)-\bar{s_j})]$. Given an arbitrary density function $f(\tau)\ge0$ (that is, $\int f(\tau)d\tau=1$), are there any known results regarding the eigenvalues of the matrix $C_f=\int f(\tau)C(\tau) d\tau$? More specifically, I would like to derive conditions related to stability and thus I am interested in cases of all eigenvalues have negative real part. I consider here the general case and do not assume detailed balance (that is, $C(\tau)$ is not symmetric). Thank! EDITED: thinking about it, I'm likewise interested on conditions on $f$ given $C(\tau)$; that is, given such $C(\tau)$, what are the conditions on $f$ so that all eigenvalues have negative real part.


title: Why does Planck's law for black body radiation have that bell-like shape?

text: I'm trying to understand Planck's law for the black body radiation, and it states that a black body at a certain temperature will have a maximum intensity for the emission at a certain wavelength, and the intensity will drop steeply for shorter wavelengths. Contrarily, the classic theory expected an exponential increase. I'm trying to understand the reason behind that law, and I guess it might have to do with the vibration of the atoms of the black body and the energy that they can emit in the form of photons. Could you explain in qualitative terms what's the reason?


title: Dirac equation in curved space-time

text: I have seen the Dirac equation in curved space-time written as $$[i\bar{\gamma}^{\mu}\frac{\partial}{\partial x^{\mu}}-i\bar{\gamma}^{\mu}\Gamma_{\mu}-m]\psi=0 $$ This $-i\bar{\gamma}^{\mu}\Gamma_{\mu} $ is the spin connection: a) Why is it needed? b) Why is it minus could it be a plus (is it due to metric signature)? c) Is the a suitable method to calculate it for a given metric e.g FRW space- time?


title: Time and distance where velocity is a function of time

text: I'm creating a video game set in space in which spaceships travelling between places accelerate until they hit a maximum velocity. They then travel at that velocity until a time when they need to start decelerating so as to arrive at their destination and not overshoot. There is no relativity or friction or anything else 'fancy' involved For the algorithm I'm writing to do this, I need to know a) the time it will take a ship to reach the half way point in its journey (assuming it continues to accelerate forever), and b) the time it will take to reach maximum velocity, and c) the distance travelled at time t (assuming continuous acceleration). I can then use that to tell the ship at what time to start decelerating, depending on how far the whole journey is. However, to accelerate the ships, I multiply their current velocity by 1.5 every second (to decelerate, I divide by 1.5). i.e. v(t) = v(t-1) * 1.5 I can also assume Initial velocity (velocity at time 0) is a constant and always positive If anyone can point me in the right direction as to what formula to use (or even better, the pseudo-code to do it), I'd be most appreciative


title: study quantum mechanics without physics background

text: I am a first year PhD math student, and must decide: _should I study Quantum Mechanics, although I don't have undergrad background in Physics?_ Let me be more specific about my situation:   1. **Background:**   I'm a first year PhD math student with undergraduate background in Computer science. I switched from Computer Science to Math because I want to study Quantum Computing, in particular involving Quantum Mechanics.      I only learned "general physics" (for non-physicists) in my undergraduate studies, and in particular didn't learn anything about Lagrangians or Hamiltonians, and very little about Maxwell's or Schrodinger's equations; and that was some time ago now besides.      I also don't know anything about Partial Differential Equations, and am planning to review my Linear Algebra.   2. **Situation:**   My math department allows me to take one qualifying exam in Math and the other in another department (though the procedure is rather complicated.) I wish that I could take Quantum Physics as the second qualifying exam, but I should be extremely cautious about this decision. (To me, qual exams in my math department are really challenging, not to mention in other department). Now, I have to take some undergrad courses in math since I did not have math knowledge in undergrad, so if i take physics courses then the time to meet my math degree requirement has to last longer.   3. **Expectation:**   I want to study Quantum Information/Computing and in the long term to study Quantum Mechanics. I think the sooner I take the course Quantum Mechanics, the better I study Quantum Information/Computing, but I know everything is not as easy as I expect. Do I need to prepare more before taking graduate Quantum Mechanics? Your suggestion, experience will definitely help me to decide. Thank you.


title: Why is the observed ratio of baryon to photon density considered too small?

text: Many descriptions about matter-antimatter asymmetry starts with the statement that observed baryon to photon density (about 6.1x10^(-10)) is too small. What is the reason? What ratio is expected?


title: Lennard-Jones induced pseudo-molecules

text: It can be shown that the Lennard-Jones potential \- which describes the interaction between particles in non-ideal gases - gives rise to pseudo- molecules: after a triple "collision" of three (classical) particles under suitable conditions two of them can rotate around each other, forming something like a pseudo-molecule. This is similar to the process of asteroid capture under gravitation where also a third body is necessary to carry away the excess energy. Under realistic circumstances those pseudo-molecules won't be very stable and probably don't play an important role in the thermodynamical treatment of non- ideal gases. This might have to do with the fact that three-particle collisions are too rare and that accordingly only two-body interactions are considered. But assume that three-particle collisions that give rise to pseudo-molecules occur often enough and that the pseudo-molecules are sufficiently stable. The following questions came to my mind: >   1. Can ever greater clusters emerge? > >   2. What can be said statistically resp. thermodynamically about the > distribution of cluster sizes? > >   3. How would a given distribution (e.g. Poisson) affect the macroscopic > properties (especially: of **which** macroscopically observable properties) > of such a thermodynamical system? > > * * * Historical motivation: I try to empathize with (idealized) ancient atomists that knew nothing about chemical bonding and did believe only in elastic collision and contact interaction among atoms, i.e. some sort of glueing. An appropriate Lennard-Jones potential - with a sufficiently deep and narrow potential well - can model both. Which kinds of structures can emerge in such a system? The other way around: What would the atomists have to have known and considered to recognize that their model does not work?


What are the known relationships between rotation of planets/moons and their distance to Sun? Or any other known attributes? For example, the sidereal year for planets is directly related to their mass and their distance from the sun, but what about sidereal day? What are the known relationships for it?


title: Mass of a galaxy via luminosity

text: Is there a general formula for calculating the mass of a galaxy, or even a nebula from the luminosity? Or, is there a way of calculating the total mass of a galaxy from its energy output? Is there a Hertzsprung–Russell diagram equivalent for galaxies? I know about gravitational lensing or velocity dispersion via the virial equation, and the Schechter function, and using doppler spread to calculate a mass.


title: Finding angular acceleration from torque

text: We have to analyze this video Givens:   * An applied net torque due to the wind on the windmill is equal to 1500 N*m.   * Each (of the 3) propeller props weighs approximately 45 Kg and has a Moment of Inertial equal to about 65% of that of a rod of the same mass being spun about its end.   * This torque is applied for approximately 45 seconds before the explosion, prior to which the windmill was at rest. Question: What was the angular acceleration caused by the torque? So here's my attempt at it: T=Ia (a is alpha) T=ML^2/3 * a * .65 (due to the whole 65% thing. Actually not sure if I should put 3*M for each propeller) And so this is where I get stuck. I'm not given L, so I'm not sure how to work around this. I could also use $T=1/2MR^2\times a$, but then I dont know R.


title: What is needed to claim the discovery of the  Higgs boson?

text: As I understand the Higg's boson can be discovered by the LHC because the collisions are done at an energy that is high enough to produce it and because the luminosity will be high enough also. But what is needed to claim a real "discovery" ? I guess there is not one event saying "hey, that's an Higg's boson" ... I also guess that this was the same kind of situation when the top quark was discovered. How does it work ? **Edit:** There is a nice introduction to the subject on this page of the CMS experiment, and the various ways to detect it, for example through the following process. ![alt text](http://i.stack.imgur.com/hTjQC.jpg)


title: Why did the asteroid belt between Mars and Jupiter form as it did?

text: I'm curious about why the asteroid belt wasn't pulled by Mars's or Jupiter's gravity or formed into either moons or planets. Why did it form into an asteroid belt instead?


title: snails moving on a sphere

text: I am trying a variation of snails moving in an equilateral triangle. Suppose that you have $6$ snails arranged on the the surface of a sphere such that they are equidistant from each other. All the snails are numbered from $1$ to $4$. The snail numbered $1$ moves toward snail numbered $2$, and $2$ moves towards $3$, $3$ moves towards $4$, $4$ moves towards $5$, $5$ moves towards $6$, $6$ moves towards $1$, If they move at a certain speed $v$ when and where would all the snails meet.


title: Does the Standard Model require neutrinos to be massless?

text: I am an undergraduate student in Physics, I have a basic understanding of Particle Physics and Quantum Mechanics but none whatsoever of Quantum Field Theory. I know that Neutrino mixing requires neutrinos to be massive (but why? Physically, couldn't neutrinos mix if they were massless?), and that their mass is usually estimated to be lower than an upper threshold. But mathematically, does the Standard Model _actually_ predict an upper limit on the neutrino mass, or does it just say that they are massless? In the former case, what is it stopping it from predicting a lower limit? In the latter case, so is it wrong?


title: Chern-Simons Energy-Momentum Tensor

text: I'm assuming the following statement is true. I'm not finding any reference which shows that explicitly. > **Statement:** Chern-Simons term is a topological one and does not > contribute to the Energy-Momentum tensor. My problem is that I'm not finding this! I'm making some mistake and I don't know where! I'm taking the Chern-Simons term $${\cal L}_{CS} = \frac{\kappa}{2} A_{\sigma} \varepsilon^{\sigma \mu \nu} \partial_{\mu} A_{\nu}$$ Via Noether theorem this term contributes to the Stress-Energy-Momentum tensor through $$T^{\mu \nu}_{CS} = \frac{\partial {\cal L}_{CS}}{\partial \partial_\mu A_\alpha} \partial^\nu A_\alpha - \eta^{\mu \nu} {\cal L}_{CS}$$ Clearly $$\frac{\partial {\cal L}_{CS}}{\partial \partial_\mu A_\alpha} = \frac{1}{2} \kappa A_{\sigma} \varepsilon^{\sigma \mu \alpha}$$ So that, $$T^{\mu \nu}_{CS} = \frac{\kappa}{2}A_{\sigma} \varepsilon^{\sigma \mu \alpha} \partial^\nu A_\alpha - \eta^{\mu \nu} \frac{\kappa}{2} A_{\sigma} \varepsilon^{\sigma \lambda \gamma} \partial_{\lambda} A_{\gamma}$$ And it's not zero. * * * I tried other thing... may be what is zero is the contribution to the energy- momentum conservation... So I compute the following $$\partial_\nu T^{0 \nu}_{CS} = \frac{\kappa}{2}\partial_\nu \left[A_\sigma \varepsilon^{\sigma 0 \alpha} \partial^\nu A_\alpha - \eta^{0 \nu} A_\sigma \varepsilon^{\sigma \beta \alpha}\partial_\beta A_\alpha\right]$$ opening it $$\frac{2}{\kappa}\partial_\nu T^{0 \nu}_{CS} = \partial_0 \left[A_\sigma \varepsilon^{\sigma 0 \alpha} \partial^0 A_\alpha\right] + \partial_i \left[A_\sigma \varepsilon^{\sigma 0 \alpha} \partial^i A_\alpha\right] - \partial^0 \left[A_\sigma \varepsilon^{\sigma 0 \alpha}\partial_0 A_\alpha\right] - \partial^0 \left[ A_\sigma \varepsilon^{\sigma i \alpha}\partial_i A_\alpha\right] \\\= + \partial_i \left[A_\sigma \varepsilon^{\sigma 0 \alpha} \partial^i A_\alpha\right] - \partial^0 \left[ A_\sigma \varepsilon^{\sigma i \alpha}\partial_i A_\alpha\right]$$ Closing it back $$\frac{2}{\kappa}\partial_\nu T^{0 \nu}_{CS} = \left[\varepsilon^{\sigma 0 \alpha}\partial^i - \varepsilon^{\sigma i \alpha} \partial^0 \right] (A_\sigma \partial_i A_\alpha)\\\ \frac{2}{\kappa}\partial_\nu T^{0 \nu}_{CS} = \left[\varepsilon^{i \sigma \alpha} \partial^0 -\varepsilon^{0 \sigma \alpha}\partial^i \right] (A_\sigma \partial_i A_\alpha)$$ which is not something trivially zero. Where is my mistake? * * * Let me follow now ( _based on suggestions of Jamals in the answers_ ) the idea that we have to fix the gauge to notice there is no contribution to the Stress-Energy Tensor. We can compute that the form I've found to $T_{CS}^{\mu \nu}$ is not gauge invariant, in fact, doing $A^\mu \rightarrow A^\mu + \partial^\mu \chi$ we have $$T_{CS}^{\mu \nu} \rightarrow T_{CS}^{\mu \nu} - \kappa (\varepsilon^{\mu \sigma \alpha} \partial_\sigma A_\alpha)\partial^{\nu}\chi $$ If we look specifically to $T^{00}$ we have $$T_{CS}^{00} = - \kappa A_0 \varepsilon^{ij} \partial_i A_j$$ So that with a gauge transformation, $$T_{CS}^{00} \rightarrow T_{CS}^{'00} = - \kappa A_0 \varepsilon^{ij} \partial_i A_j - \kappa (\varepsilon^{0 i j} \partial_i A_j)\partial^{0}\chi $$ We can choose a gauge where $\partial^{0}\chi = - A^0$ and then $T_{CS}^{'00} = 0$, as we where looking for. Anyway, for $T_{CS}^{'\mu \nu}$ $$T^{'\mu \nu}_{CS} = \frac{\kappa}{2}A_{\sigma} \varepsilon^{\sigma \mu \alpha} \partial^\nu A_\alpha - \eta^{\mu \nu} \frac{\kappa}{2} A_{\sigma} \varepsilon^{\sigma \lambda \gamma} \partial_{\lambda} A_{\gamma} - \kappa (\varepsilon^{\mu \sigma \alpha} \partial_\sigma A_\alpha)\partial^{\nu}\chi $$ Things don't look fine. Let's take a look on the momentum vector, $$T^{'i 0}_{CS} = \frac{\kappa}{2}A_{\sigma} \varepsilon^{\sigma i\alpha} \partial^0 A_\alpha + \kappa (\varepsilon^{i\sigma \alpha} \partial_\sigma A_\alpha)A^{0} $$ It's pure gauge! Every term involve $A^0$ and we fixed that $A^0 = \partial^0 \chi$


title: Simulating a 1-dimensional wave on (a segment of) an infinite line

text: I'm trying to numerically simulate a 1-dimensional with a chain of linked harmonic oscillators as described here (the result can be seen here). The simulation behaves like a wave on a finite line segment with fixed or free boundaries (depending on the boundary conditions you set): when the wave reaches the border it bounces back (as it should by by conservation of energy). What I would like to do is to make my queue of oscillator behave like a segment of an infinite queue of oscillators, that means there must be no bounces and the energy must flow away throught the boundaries. I tried some strategies which turned out not to work such as adding invisible extensions of the queue and making it more dampened (actually it turned out to be equivalent to not adding the extension at all!). Any Idea?


title: Isn't $\varepsilon_0\int_{\text{all space}}\vec{E}_1\cdot\vec{E}_2 \,{\rm d}v$ just the potential energy?

text: I have two metallic spheres each with a charge of $q_1$ and $q_2$ respectively. What is the value of $$\varepsilon_0\int_{\text{all space}} \vec{E}_1\cdot\vec{E}_2 \,{\rm d}v$$ where $\vec{E}_1$ and $\vec{E}_2$ are the electric fields due to the two spheres and $\varepsilon_0$ is the permittivity of free space. The radius of the two spheres are $a$ and $b$ and the distance between their centers is $R$ with $R>>>a,b$ (As far as I know), $\frac{1}{2}\varepsilon_0\vec{E}_1\cdot \vec{E}_2$ would be the electric field energy density at any point, so this integral should be double of the net electrostatic interaction potential energy (as opposed to self-potential energy) between the two bodies. So I thought that it would be $$\frac{2q_1q_2}{4\pi\varepsilon_0R},$$ but according to the answer key in my test it is actually $$\frac{q_1q_2}{4\pi\varepsilon_0R}.$$ How? Where am I wrong?


title: What is the magnetic field inside hollow ball of magnets?

text: Setup: we have a large number of thin magnets shaped such that we can place them side by side and eventually form a hollow ball. The ball we construct will have the north poles of all of the magnets pointing toward the center of the ball, and the south poles pointing away from the center. The magnets in this case are physically formed such that in this hollow ball arrangement they are space filling and there are no gaps between them. Is such as construction possible? If so, what is the magnetic field (B-field) inside and outside the ball?


title: Observing photons

text: Your replies to my question about being able to see a photon, from the side (answer, unanimous, “no”) have raised in me some additional questions. Would it reasonable to think that in consequence, the photon, from any angle but directionally “in front”, is a black body? I ask because I am accustomed to think of the photon as a “light” particle, when perhaps it might be more correct to think of it as a “black” particle. When an observer sees light from a photon, isn’t it true that in fact, what is actually being seen is the photon itself? When the photon enters the eye of the observer, what happens (3.1) to the observer? (3.2)to the photon? 3.1 The photon has no mass, so the observer/photon combination remains at original observer mass. However, the photon has energy, what impact on the observer done this energy input have?   3.2 Upon entering the observer, does the photon disappear? If not what else can it do?


title: Cherenkov radiation in nuclear bomb

text: Would Cherenkov radiation occur at the explosion of a nuclear bomb? Suppose it would not be occluded by smoke or anything else for that matter.


title: List of experimental band gap

text: does anyone know where one can find a list, database, book of experimental bandgap values of semiconductors? Is there such a collection? Or do I need to scour papers one at a time to get the values?


title: Motivation for preservation of spacetime volume by Lorentz transformation?

text: My favorite way of deriving the Lorentz transformation is to start from symmetry principles (an approach originated in Ignatowsky 1911; cf. Pal 2003), and one of my steps is to prove a lemma stating that the Lorentz transformation has to preserve volume in spacetime, i.e., in fancy language, it has to have a Jacobian determinant of 1. My visual proof of the lemma (for 1+1 dimensions) is given at the link above, in a figure and its caption. Intuitively, the idea behind the proof is that it would be goofy if a boost could, say, double the area, since then what would a boost in the opposite direction do? If it's going to undo the first boost, it has to halve the area, but then we'd be violating parity symmetry. The actual proof is actually a little more complicated than this, though. If a rigorous proof could be this simple, I'd actually be satisfied and say the result was so simple and obvious that we should call it a day. But in fact if you want to get it right it becomes a little more involved, as shown at the link where I give the actual proof. One way to see that the intuitive argument I've given here above doesn't quite suffice is that it seems to require that area in the $x$-$y$ plane be preserved under a boost in the $x$ direction, and that's not true. This is different from the approach based on Einstein's 1905 axiomatization. In that approach, you derive the Lorentz transformation and then prove the unit Jacobian as an afterthought. Although my proof of the unit Jacobian from symmetry principles works, I've always felt that there must be some deeper reason or better physical interpretation of this fact. Is there? Takeuchi 2010 says on p. 92: > This conservation of spacetime area maintains the symmetry between [...] > frames, since each is moving at the exact same speed when observed from the > other frame, and ensures that the correspondence between the points on the > two diagrams is one-to-one." This seems clearly wrong to me, since you can have one-to-one functions that don't preserve area. (Takeuchi is writing for an audience of liberal arts students.) Mermin has a very geometrical pedagogy that he's honed over the years for a similar audience, and he interprets space-time intervals as areas of "light rectangles" (Mermin 1998). Since he uses the 1905 Einstein axiomatization, conservation of area, which is equivalent to conservation of spacetime intervals, comes as an afterthought. One thing that bugs me is that the area-preserving property holds for Galilean relativity (and my proof of it holds in the Galilean case without modification). So any interpretation, such as Mermin's light rectangles, that appeals specifically to something about SR seems unsatisfying. The notion of area here is really just the affine one, not the metrical one. (In Galilean relativity we don't even have a metric.) Another way of getting at this is that if you start with a square in the $x$-$t$ plane and apply a Lorentz transformation, it becomes a parallelogram, and the factors by which the two diagonals change are the forward and backward Doppler shifts. Conservation of area then follows from the fact that these Doppler shifts must be inverses of one another. Laurent 2012 is an unusual coordinate-free presentation of SR. He interprets $\epsilon_{abcd}U^aB^bC^cD^d$ as a 3-volume measured by observer whose normalized velocity vector is $U$. Restricting to 1+1 dimensions, $\epsilon_{ab}U^aB^b$ is the length of vector $B$ according to this observer. He gives the example of associating affine volume with the number of radioactive decays in that volume. The implications of this are vague to me. What is the best way of interpreting the preservation of spacetime volume by Lorentz transformation? Please do not reply with answers that start from the known form of the Lorentz transformation and calculate the Jacobian determinant to be 1. I know how to do that, and it's not what I'm interested in. W.v. Ignatowsky, Phys. Zeits. 11 (1911) 972 Bertel Laurent, Introduction to spacetime: a first course on relativity Mermin, "Space-time intervals as light rectangles," Amer. J. Phys. 66 (1998), no. 12, 1077; the ideas can be found at links from http://people.ccmr.cornell.edu/~mermin/homepage/ndm.html , esp. http://www.ccmr.cornell.edu/~mermin/homepage/minkowski.pdf Palash B. Pal, "Nothing but Relativity," http://arxiv.org/abs/physics/0302045v1 Takeuchi, An Illustrated Guide to Relativity


title: Question on dimensions of CFT operators (ref: MAGOO, hep-th/9905111)

text: Right now I am having this silly difficulty from the following:   1. BTW, Conformal dimension/scaling dimension is -ve of mass dimension ..right?   2. In p-63 of Magoo, after 3.15 eq, they said a.) $\phi$ is dimensionless..why? b.) They, after getting length dimension of boundary field as $\Delta d$, referred to 3.13 to comment that O operator has conformal dimension $\Delta$. I know its right..But as in the lhs of 3.13, the exponent must be dimension less, it seems O should have length or conformal dimension to be $-\Delta$.. I know I am making a silly mistake here..Thanks for answering.


title: Particle current operator in general vs Particle current operator for tight binding Hamiltonian

text: I am referring Mahan Many-Particle Physics. There are 2 particle current operators -one in general and one for the tight binding Hamiltonian. How do we go from the general current operator (1.195 in Mahan) $ j_{i}(\mathbf{r})=\frac{1}{2mi}\left(\psi^{\dagger}(\mathbf{r})\nabla\psi(\mathbf{r})-\psi(\mathbf{r})\nabla\psi^{\dagger}(\mathbf{r})\right) $   [whose Fourier transform is $\int d^{3}r\,\,\exp(-i\mathbf{q}.\mathbf{r})\,\, j_{i}(\mathbf{r}) = \frac{1}{2mi}\sum_{\sigma\beta}c_{\sigma}^{\dagger}c_{\beta}\int d^{3}r\,\,\exp(-i\mathbf{q}.\mathbf{r})\,\left(\phi_{\alpha}^{\star}(\mathbf{r})\nabla\phi_{\beta}(\mathbf{r})-\phi_{\beta}(\mathbf{r})\nabla\phi_{\alpha}^{\star}(\mathbf{r})\right)$ giving $j_{l}(q) = \frac{1}{m}\sum_{k\alpha}\left(k+\frac{q}{2}\right)c_{k+q,\sigma}^{\dagger}c_{k,\sigma} $] to (1.204) current operator $ j=-iw\sum_{j\delta\sigma}\delta c_{j+\delta,\sigma}^{\dagger}c_{j\sigma}$ for the Tight Binding Hamiltonian $H=w\sum_{j\delta\sigma}c_{j+\delta,\sigma}^{\dagger}c_{j,\sigma}+\frac{1}{2}\sum_{ij,ss^{\prime}}n_{is}n_{js^{\prime}}V_{ij}$? (here $\phi(r)$ are plane waves.) I mean we use an alternative definition of current in terms of polarization to get to the form of current operator in real space for the tight binding Hamiltonian but on fourier transforming this operator, it should return us a form similar to the current operator generally defined in Fourier space-- I don't see how we will get the mass m of $j_{l}(q)$ on taking the fourier transform of $ j=-iw\sum_{j\delta\sigma}\delta c_{j+\delta,\sigma}^{\dagger}c_{j\sigma}$


title: LHC vacuum tube diameter

text: What's the diameter of the tube where the protons fly around? I dont mean the overall diameter (~1-10 m), instead the inner most ultra-high vacuum tube where the protons are. The inner pipe, the capillary, the channel.. I would guess maybe 1 mm?


title: Condensed matter application of AdS/CFT

text: I'm going to be teaching a course on gauge/gravity duality (aka AdS/CFT) in the winter. The focus will be on applications in particle theory including $N=4$ SYM, the viscosity/entropy bound, and aspects of large $N$ QCD. I would like to include at least one application to condensed matter physics. There has been a lot of work on this over the last couple of years but I haven't followed it very closely. Can anybody suggest a nice application that is simple enough to work through in a few lectures for students who know QFT and GR and will have had some exposure to AdS/CFT?


title: Rolling stone on a frictional surface

text: Consider a spherical rigid stone rotating with angular velocity $\omega$ being dropped vertically onto a horizontal rigid surface with the coefficient of friction $\mu$. Can the stone roll on the surface? If it can, what type of the motion is it? Constant velocity, constant accelaration, or varying acceleration? Ignore air resistance. In reality if we drop a rotating ball, it will roll on the surface. But I am not sure whether it is because of the deformation of the ball at the surface of contact.


title: Definition of a 'tunneling lifetime'

text: I'm given a one-dimensional potential with two wells, one local minimum at some higher energy and one deep global minimum next to it, separated by a barrier of own shape and height (phase qubit). I used a WKB approx. in conjunction with a harmonic approximation of the barrier to find the tunneling probability, i.e. the transmission coefficient T. Now, I'm asked to estimate the tunneling lifetime of the metastable levels in the elevated well. How do you relate the tunneling probability to the lifetime?


title: Where does the extra kinetic energy come from in a gravitational slingshot?

text: I read in this answer in this site that the KE a free-falling ball acquires is _not originated_ by the attracting body but that energy _was actually stored_ in the ball when it had been lifted to the height it dropped from. In this way, it was said, gravity is subject to the **conservation of energy** principle and cannot change the total energy of an object. Consider now the maneuver known as a _gravitational slingshot_ (also _gravity assist_ ) used by space probes such as the Voyager 2. A space probe approaches a planet with velocity $v$, slingshots around, and ends up with velocity $v+2U$, where $U$ is the velocity of the planet. Consider the energy of the probe. Before, it was $E_i=\frac{1}{2} mv^2$ and after it is $E_f=\frac{1}{2} m(v+2U)^2$. It looks like $E_f$ is much bigger than $E_i$ - but where did the additional energy come from? Is this not a violation of the _conservation of energy_ principle?


title: Sharpness of starlight

text: I am interested in argument that the observed lack of blurriness in starlight refutes the "tired light" theories. I have not been able to find this argument expanded in detail. There are two ways that I can think of that we observe the lack of blurriness:   1. In absorption spectra, the lines are sharp, indicating a sharpness in light frequency.   2. In the apparent size of a distant object: if the mechanism that caused the light to become tired was the result of some collision, then the direction of travel of the photons would presumably deviate slightly making the apparent size larger and the object appear blurry. Can upper limits be placed on any blurriness from experimental data in both cases? What would it mean if there is a difference between the two, for instance if we could detect a small blurriness in direction, but none in frequency?


title: Calculating the number of quanta emitted as Cherenkov radiation in a wavelength interval dl?

text: This website gives an equation for it http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/cherenkov.html It appears as plaintext but this is what I believe it is parsed: $n = \dfrac{dl 2πα L \sin^2θ}{l^2}$ And this is the text surrounding and about the equation: > For most media blue light predominates over longer wavelengths of light, > because the number of quanta emitted as Cherenkov radiation in a wavelength > interval dl at wavelength l over a path length L is given by: > > where α is the fine structure constant, equal to about 1/137. Notice that > the refractive index, and therefore the angle θ, changes with wavelength l > as demonstrated when a prism produces a spectrum from white light. This > suppresses the rate at the small wavelengths of ultraviolet and beyond. Other websites list this formula but appear to have copied and pasted the surrounding text around it. It's not an equation in the given form but I assume from the text around it that an $n = $ could be added for the number of quanta. Three questions: Have I interpreted this equation correctly? If so, why didn't they cancel the common factor of $l$. And can someone link or show a derivation of this equation or whatever the correct equation is?


title: How to interpret tensor form PDE in terms of matrix algebra

text: From this mathwork page "c for system", the usual second order PDE is written in tensor form: $$ -\nabla\cdot(\mathbf{c} \otimes \nabla \mathbf{u})+\mathbf{a}\mathbf{u}=\mathbf{f} $$ and $\mathbf{c}$ is claimed to be a $N$-by-$N$-by-$2$-by-$2$ tensor. Each side of the equation should correspond to a $N$-by-$1$ vector in this case, and now I am confused... I want to express this equation in matrix form, since $\mathbf{c}$ is a 4th rank tensor while $\nabla{u}$ is 2nd order tensor($2$-by-$N$ matrix?), how to define the tensor operation $\otimes$ then? Seems wiki doesn't give the answer. Cand anybody elaborate the rule here? (Also how to cast the gradient operator in this case?) Any physical insight is honestly appreciated!


title: Can I levitate an object without using an electromagnet?

text: I know that it's possible to make an object levitate using an electromagnet to hold it up. But is it also possible to do this with regular magnets? Is there a special kind of magnet I need in order to have one powerful enough to hold an object up? I'm asking because I have this idea in mind where I want to make a decorative item that is levitating an inch or so above its container. ![alt text](http://i.stack.imgur.com/30uUp.png) **A** is the object I want to levitate.   **B** is the container.   **C** shows the magnetic field that is repelling **A** and **B** from each other achieving the levitation. The size of this would be as small as a food plate, maybe even smaller. **B** is barely a pound or two in weight. Is that possible?


title: Theoretical Stability of "AB-matter"

text: Alexander Bolonkin has proposed the possibility of manipulating nucleons to produce stable, macroscopic structures of nuclear matter at zero pressure (which he calls "AB-matter"), by analogy with the nanotech ideas of directly manipulating atoms to build high-tech materials. The basic claim is that an unbounded number of alternating protons and neutrons can be arranged in a fiber held together by residual nuclear force and a small contribution from magnetism due to the nucleon magnetic moments, and prevented from collapsing and held rigid by electrostatic repulsion. Superstrong macroscopic structures can then be built by combining these basic nuclear matter needles. Bolonkin is a legitimate scientist (PhD in aerospace engineering), but not a nuclear physicist, and has gotten papers on this stuff published, but not in physics journals (for example, "Femtotechnology: Nuclear AB-Matter with Fantastic Properties" in _American Journal of Engineering and Applied Sciences_ and "Femtotechnology: Design of the Strongest AB Matter for Aerospace" in _Journal of Aerospace Engineering_ ). Furthermore, nobody else seems to have published anything on this topic, all of which makes me rather skeptical of his claims. So, ignoring the issue of how you'd construct it in the first place (assume we find some helpful Cheela to do it for us or something), could a linear arrangement of alternating protons and neutrons at zero pressure (i.e., not confined in a neutron star or something) remain stable and not collapse into one big nucleus, or segment itself into a bunch of individual nucleii? And does it make any difference if the fiber is kept under tension by some external means?


title: Polarizing Beam Splitter Interferometer

text: In "Dance of the Photons" by Anton Zeilinger (pgs. 82-84), Zeilinger has a polarising beam splitter interferometer as such- http://img.photobucket.com/albums/v608/steviet/IMG_20140905_0001_zpsc8add996.jpg (assume the length of paths are the same) The input photon is polarised 45 degrees. The output is that a 45 degree polarised photon is found in the lower output. (1) Would the output be the same if a photon of no definite polarisation (say, as part of an entangled pair) is inputted into the interferometer? (2) What path would the output be if the input photon was polarised in the Vertical direction? Would, at the 1st PBS, it be transmitted, and if this diagram is correct: http://physics.illinois.edu/people/kwiat/Interaction-Free- Measurements_files/current.gif (bottom PBS, the alignment of the PBS is the same as my set-up), a |V> polarized photon coming in from the top towards the 2nd PBS (which it would if it is transmitted at the 1st PBS in my set-up) would the photon end up to the right of the PBS, not below it? Many thanks Stevie


title: Determine the effective radius of a gear?

text: I'm trying to determine the gear radius using a ruler by $cm$ but I just don't know what to measure: 1- **the length from gear center point to the end of the gear tooth ?** ![enter image description here](http://i.stack.imgur.com/JM2VH.jpg) Or 2- **the length from gear center point to the start of the gear tooth ?** ![enter image description here](http://i.stack.imgur.com/cC1d6.jpg)


title: From Berry's phase to artificial Gauge potential

text: How a nonzero geometric phase in a loop is used to generate artificial gauge potentials? If possible, can you also tell how to generate the non-abelian artificial gauge potentials.


title: Dipole moment induced in a spherical particle

text: Consider a spherical metal particle made out of gold. If there is an external charge somewhere near the gold particle, is there a way to calculate the resulting dipole moment that is induced by the external charge? Or is there a way to calculate the induced dipole moment that results from an external dipole moment instead of an actual charge? Would be glad if you could give me some ideas or equations to tackle these two problems.


title: Conditions of applicability of potential flow about an airfoil

text: In many cases the flow about an airfoil is calculated by solving the Laplace equation, (for example in the Hess-Smith panel method). If the velocity field is irrotational and its divergence its zero, that field will satisfy the Laplace equation. My question is: how do you tell when these two conditions (zero curl and zero divergence) are still a good approximation of the actual flow in the reality? If the flow is incompressible, its divergence will be zero, right?. But in what conditions you can treat air as a incompressible fluid? And how can I tell if the flow can be treated as irrotational? I thought that this was related to viscosity, but I've read that in some cases there can be irrotational flows with viscous fluids and also rotational flows with inviscid fluids. And the Kutta condition, that is fundamental in the Hess-Smith method, is based on the fact that the fluid has viscosity. And I've also read that it's important, to apply the Hess-Smith method, that the airspeed is subsonic everywhere. What are the reasons of this condition? I'd be happy to receive explanations or also useful links. Thank you.


title: Distance of image from concave mirror

text: A concave mirror of focal length $8cm$ forms an inverted image of an object placed at a certain distance. If the image is twice as large as the object, what is the distance of the object and the image from the pole of the mirror? I started off with the relation: $\frac{h_o}{h_i} = \frac{d_o}{d_i}$ since the image is inverted, $h_i = -2h_o$, and $\frac{h_o}{h_i} = \frac{1}{-2}$. $\implies \frac{d_o}{d_i} = \frac{1}{-2} \implies -2d_o = d_i$. Using the mirror formula $\frac1f = \frac{1}{d_i} + \frac{1}{d_o}$ and substituting $d_i$ for $-2d_o$, $\frac1f = \frac{1}{-2d_o} + \frac{1}{d_o}$ $\implies \frac1f = \frac{1 + -2}{-2d_o} = \frac{1}{2d_o} \implies d_o = \frac{f}{2}$ Since $f = -8cm$ (using Cartesian sign convention), $d_o = -4cm$, but how can a real image be formed when the object is placed at less than the focal length?


title: What are the different ways to measure the spatial curvature of the universe?

text: Just what the question asks. Assuming the Friedmann-Rovertson-Walker (FRW) metric, what measurements can be performed to determine the spatial curvature of the universe.


title: Negative energy levels in the diagram for a hydrogen atom

text: ![enter image description here](http://i.stack.imgur.com/6raba.jpg) The higher the number of the shell (n), the higher is the energy level of the electron. However, why was it necessary to have negative values. So for example, when $n=1$, the energy could be $5 eV$ and for $n=2$, $6 eV$... having positive values could also have supported the idea that as $n$ increases, energy of electron increases. What is the point of having negative numbers, does it somehow aid calculations?


title: Work done on stationary rocket

text: Suppose I have a rocket thats exerts a force (with negligible loses in mass), which cancels out the downward force of gravity. Clearly my rocket could be moving at a constant velocity (ignore air resistance) or not at all. Nothing seems fishy yet, but what about this: Clearly, no work is being done, but energy is still being expended to keep the rocket in place. So how can the energy expended be found (please no answers of the form: bond enthalpies of fuel etc., I am sure there is an explanation for this in Newtonian kinematics/energetics)? Because if we try to use the formula $\mathrm{Work} = F \cdot d$, then $d$ could be zero or anything at all, which doesn't tell us anything useful. Another similar scenario would be that of two rocket exerting forces in opposite directions, clearly energy is being expended but no work so what to do?


title: How to write the single electron spin-orbit coupling under an external magnetic field?

text: As we know, without the external magnetic field, the single electron spin- orbit coupling(SOC) has the form $\boldsymbol{\sigma}\cdot(\boldsymbol{\nabla} V\times \mathbf{p})$ up to a coefficient, where $V$ is the static electric potential. Now if we add an static external magnetic field $\mathbf{B}=\boldsymbol{\nabla}\times \mathbf{A}$, does the SOC preserve the above form $\boldsymbol{\sigma}\cdot(\boldsymbol{\nabla} V\times \mathbf{p})$ or change into $\boldsymbol{\sigma}\cdot(\boldsymbol{\nabla} V\times (\mathbf{p}-e\mathbf{A}/c))$? The notation $\mathbf{p}$ always represents $-i\hbar(\partial_x,\partial_y,\partial_z)$ in all of our formulas. This reminds me another question, may be worthless: Does this expression $\mathbf{r}\times(\mathbf{p}-e\mathbf{A}/c)$ make sense? How to write the orbital angular momentum under an external magnetic field? Thank you very much.


title: Is the reduction map completely positive?

text: I am struggling with proving the complete positivity of a general map ( granted it is CP ). The reduction map is defined as $$ \rho \rightarrow \mathrm{Tr}(\rho)I - \rho $$ It is a trivial job to show the positivity of this map. I don't know whether this map is CP at first place. If it is , how should I proceed ? I would like a general algorithm (if it exists). Otherwise I would like someone to help with a counter-example. I tried with bell states and failed.


title: Gravity of very distant objects

text: As far as I know stars emit a finite number of photons in all directions in a given period of time and as an observer goes further away he experiences less and less photons to the point where the photon sphere is so spread out that sometimes he will not experience any photons making the star appear to be blinking. So my question is: Does the same apply to gravity and is it also "blinking" at very long distances? And if so, then could theoretically an observer (by chance) experience a long enough period between such blinks as to drift away as if he wasn't being influenced by gravity of this object?


title: Impact of Ag nano-particle size on OPV efficiency : simulation

text: I would like to simulate the impact of gold nano particles size on the efficiency of Organic photovoltaic devices(inverted ofcourse) looking for any simulation plateform where this idea could be done any sugguestions ?


title: State of constant motion

text: Why does an object remains in its state of constant motion if there are no forces acting on that object? My understanding is that all the energy of the motion will be kept inside and a change in the speed needs a change in the amount of the energy stored by that object. But how is that energy stored in a form of speed? Does anything changes inside the object's structure if it is moving?


title: Is it really proper to say Ward identity is a consequence of gauge invariance?

text: Many (if not all) of the materials I've read claim Ward identity is a consequence of gauge invariance of the theory, while actually their derivations only make use of current conservation $\partial_\mu J^\mu=0$(which is only equivalent to a global phase symmetry). I'm aware of the fact that a gauge field has to be coupled to a conserved current to keep gauge invariance, but a non-gauge field can also be(though not must be) coupled to a conserved current and in that case Ward identity should still hold. So do you think it is at least misleading, if not wrong, to claim Ward identity is a consequence of gauge invariance?


title: How much iron ware to make a Faraday cage

text: In a thunderstorm I was thinking the following: suppose I am rowing in a lake during a thunderstorm. How big a Faraday cage do I need to make to protect myself? If lightning strikes the cage, will it dissipate away through the water?


title: Where does the light go when I turn off the light switch?

text: As above... Surely it must go somewhere? My child asked me this question and it was a difficult one to answer.


title: Why is the string theory graviton spin-2?

text: In string theory, the first excited level of the bosonic string can be decomposed into irreducible representations of the transverse rotation group, $SO(D-2)$. We then claim that the symmetric traceless part (i.e. the **35** rep) is the spin-2 graviton - but isn't the label "spin-2" intrinsically 3+1 dimensional? I.e. it labels the representation under the little group $SU(2)$?


I have read in wikipedia this statement "The energy carried from one winding of a transformer to another, in quantum terms **is carried by virtual photons, not real photons** " (wikipedia src: virtual particle) Of course anything in wikipedia could be solved "changing it" =), but I wonder, someone put it for a reason, then in the case it have sense, _I don't know what is the limit for that, what are the frequencies for virtual or real photon interactions._ (if is the frequency what makes the difference, or what ?) Perhaps it's related with the kind of interaction, I mean photoelectric effect and the induction are different mechanism , but I don't see clearly how "real photons" disappear from the picture, if they are the lesser energy that can be transfered anyway. I have put that into a comment for this answer Low frequency electromagnetic waves but I think is better to open an specific question.


title: Equivalence of canonical quantization and path integral quantization

text: Consider the real scalar field $\phi(x,t)$ on 1+1 dimensional space-time with some action, for instance $$ S[\phi] = \frac{1}{4\pi\nu} \int dx\,dt\, (v(\partial_x \phi)^2 - \partial_x\phi\partial_t \phi) $$ where $v$ is some constant and $1/\nu\in \mathbb Z$. (This example describes massless edge excitations in the fractional quantum Hall effect.) To obtain the quantum mechanics of this field, there are two possibilities:   1. Perform canonical quantization, i.e. promote the field $\phi$ to an operator and set $[\phi,\Pi] = i\hbar$ where $\Pi$ is the canonically conjugate momentum from the Lagrangian.   2. Use the Feynman path integral to calculate all expectation values of interest, like $\langle \phi(x,t)\phi(0,0) \rangle$, and forget about operators altogether. My question is > Are the approaches 1. and 2. always equivalent? It appears to me that the Feynman path integral is a sure way to formulate a quantum field theory from an action, while canonical quantization can sometimes fail. For instance, the commutation relations for the field $\phi$ in the example above look really weird; it is conjugate to its own derivative $\Pi(x,t) = -\frac{1}{2\pi\nu}\partial_x\phi(x,t)$. The prefactor is already a little off. For this to make sense, we have to switch to Fourier transformation and regard the negative field modes as conjugate momenta, $\Pi_k=\frac{1}{2\pi\nu}(-ik)\phi_{-k}$. A more serious example: it seems to me that the Feynman integral easily provides a quantum theory of the electromagnetic gauge field $A_\mu$ whereas in canonical quantization, we must first choose an appropriate gauge and hope that the quantization does not depend on our choice. > Could you give a short argument why 2. gives the right quantum theory of the > electromagnetic field? (standard action $-\frac1{16\pi} > F^{\mu\nu}F_{\mu\nu})$


title: Exactly how is the constant measured velocity of light deduced from Maxwell's equation?

text: For electromagnetic radiation the velocity of propagation is $c = 1/\sqrt{\mu_0 \epsilon_0}$. Since both $\mu_0$ and $\epsilon_0$ do not vary in any inertial frame, then $c$ must be constant in any inertial frame. Now apply the argument to sound. The velocity of sound in a rod clearly doesn't change regardless of the velocity of the rod, but it's obvious that if the rod moves past me at the velocity of a sound wave in that rod, the sound wave will be standing still as I view it. In other words, there appears to be something wrong with the argument in the first paragraph since the same argument fails for sound, which is also a wave with a propagation velocity. So how exactly does light differ from sound here? I'm looking for an explanation that would be understandable to a budding scientist who is a non- physicist. (I'm trying to prepare to explain special relativity to my daughter (a geologist who's taking a course in cosmology), and want to be able to explain special relativity from first principles, in such a way that special relativity becomes obvious and simple).


title: Statistical count

text: I am reading the book"Heat and Thermodynamics" by Mark Waldo Zemansky and Richard Dittman. In the chapter "Statistical Mechanics" it says if I have $N_{i}$ distinguishable particles in any of $g_{i}$ degenerate states associated with energy $\epsilon_{i}$ and if there is no restriction on multiple occupancy then there will ${(g_i)}^{N_i}$ number of ways in which those particles can be occupied. Then it says in case of indistinguishable particles the number of ways become $\dfrac{{(g_i)}^{N_i}}{N_{i}!}$. when I do it in tabular method with $g_{i}$=3 and $N_{i}$=3 with the indistinguishable particles it gives 10 ways, not in agreement with what's written. Please help!


title: Acceleration vector - deceleration vs direction

text: If acceleration of something $= - 10 \text{ m s}^{-2}$ And forwards is define as north. Does that mean the object is getting slower (decelerating) or accelerating in the reverse direction (south) How can you tell the difference?


title: How do I find the force an object exerts on adjacent objects when I push on it?

text: Suppose that I have a situation like the figure below, with several circular objects in contact. I know:   * $\vec{A},\vec{B},\vec{C}$: the positions of the points of contact between circles   * $\vec{O}$: the center of one circle   * $\vec{F}$: the force applied to that circle ![Image of circles](http://i.minus.com/jbggsUzWrXFbzS.png) I need to find the forces $\vec{N}_1,\vec{N}_2,\vec{N}_3$ which the main circle exerts on the other circles. How can I do that?


title: What happens when we cut objects?

text: What is the role of the molecular bonds in the process of cutting something? What is the role of the Pauli exclusion principle, responsible for the "hardness" of matter? Moreover, is all the energy produced by the break of bonds transformed into heat?


title: How to compute the speed necessary for an airplane to fly?

text: I give some physics lessons to a friend. She asked me a question that I am unable to answer. Could you help me ? A plane has a weight of $2\times10^6$kg. The surface of the wing is $1200 \text{m}^2$. We assume an air density equal to $1\text{kg}/\text{m}^3$. The speed of air under the wing is $100\text{m}/\text{s}$. What should be the speed of air over the wing so that the plane remains in the air ? I searched for expressions of lift (as for instance on Wikipedia http://en.wikipedia.org/wiki/Lift_(force)) and tried to link it with the speed of fluid over and under the wing but didn't find anything relevant with the only pieces of information I have in the exercise. In my opinion, the speed of air will depend on the shape of the wing... And if we don't know anything about the shape, we are unable to answer the question. Am I wrong ? Any idea ?


title: Bound states of the Schrödinger equation on $S^1\times \mathbb{R}^3$

text: I wana study the $\textbf{bound states}$ of a particle in the space $\mathbb{R}^3\times S^1$ which is under the effect of a time independent potential $V(\vec{r})$ but is free on $S^1$. So, I begin by writing the Schrödinger equation $i\hbar\frac{\partial}{\partial{}t}\psi=\left(\frac{-\hbar^2}{2m}[\nabla^2+\frac{\partial}{R^2\partial\theta^2}]+V(\vec{r}\right)\psi $ and proceed to solve by separation of variables (the coordinate $\theta$ is an angle used to parameterize a point of $S^1$, and $R$ is the radius of the space) $\psi(\vec{r},\theta;t)=\phi(\vec{r})\beta(\theta)N(t)$ This way I get three equations, the first is just the time dependent one, just the same we get solving the Schrödinger equation in $\mathbb{R}^3$ $i{}N'(t)=\omega{}N(t) $ whose solution is $N(t)=C_1e^{-i\omega{}t} $ The second equation is the one corresponding to $\theta$ $-\frac{\hbar}{2mR^2}\beta''(\theta)=\omega_1\beta(\theta)$ where $\omega_1$ is just a constant that appears after separating the variables. The solutions are $\beta(\theta)=C_2e^{i\theta{}R\sqrt{\frac{2m\omega_1}{\hbar}}}+C_3e^{-i\theta{}R\sqrt{\frac{2m\omega_1}{\hbar}}} $ The second equation is analogous to the typical time independent Schrödinger equation but this time the $\omega$ is not the same as in $N(t)$ because of the $\theta$ part. $\frac{-\hbar^2}{2m}\nabla^2\phi+V(\vec{r})\phi=\hbar(\omega-\omega_1)\phi $ Having arrived here I am unsure how to proceed. I want to study the $\textbf{bound states}$, so, do I have to consider the solutions with $\omega$ negative or the ones with $(\omega-\omega_1)$ negative, since this is the constant appearing in the equation who has the potential? Furthermore, what boundary conditions should I apply to $\beta(\theta)$? Writing the derivation in detail would have taken too much space so I fear that some things I have written may not be too clear, let me know. I would appreciate any hint. Thanks!


title: Energy of a black hole : a quizz question

text: Consider a standard eternal Schwarzschild black hole of mass $M$. One wants to speak about the energy $E$ of a black hole, so let 's present this as a quizz. 1) This is a non-sense to speak about extended conserved quantities in General Relativity, including total energy. 2) $E=0$ 3) $E=M$ 4) Other results $(E = \frac{M}{2}$, etc...) * * * Well, while $(1)$ seems correct, I would vote for $(2)$, the argument would be that negative gravitational energy is compensating positive non-gravitational energy (mass energy)


title: Book on optics in curved space-time

text: As evidenced from my earlier questions on vision and curved space, I am struggling a little bit with visual perception in curved space-time. I would like a book recommendation on optics and vision in curved space-time.   1. I find books on Gravitational lensing to be inadequate as Gravitational lensing itself deals mostly with specific cases in astronomy. It does not have the breadth and scope I require.   2. I want the book to discuss in detail how space (of low curvature) could affect visual perception in everyday life scenarios. Very high mathematical rigor is desired.   3. I would also really like the book to have pretty colors and pictures for aesthetic value. If you know any book that come close to satisfying my conditions, please do tell me. I will be very grateful if you can. Also, as per the books policy, it is also OK to provide an explanation for any subdiscipline of this topic instead, especially if that subdiscipline is not covered by most resources. But please do not make such an answer Community Wiki.


A jet fighter pilot wishes to accelerate from rest at 5.00 g to reach Mach-3 (three times the speed of sound) as quickly as possible. Experimental tests reveal that he will black out if this acceleration lasts for more than 5.00s . Use 331m/s for the speed of sound. What is the greatest speed he can reach with an acceleration of 5.00 g before blacking out? What I have done is: Using the formula of acceleration, $a=(\Delta v)/(\Delta t) = (\Delta v)/5 sec$ Then, I plugged in 5 g into a. $5.00 g=(\Delta v)/ 5 sec$ I assume that the final velocity is $3(331 meters/sec)$ because the fighter pilot wants to reach Mach-3. So, I have $5.00 g = (3(331) - v_0)/5$. And this is where I am stuck. Can one tell me how to continue from here, or if I am completely wrong?


title: Image K-Space and SNR

text: 1. I am imaging a sample using CMOS camera. The pixel ratio is 6um x 6um and a total resolution {480, 752}. I understand that each pixel on the camera sensor is 6um x 6um in size (Have I understood this correctly ?).   2. If I have to find the signal-noise ratio of the 2D image to determine the image quality. In doing so I have to convert it into Fourier domain using FT. How can I know the sampling interval to calculate the FFT of the image data ?. Does the sampling interval depends on the pixel size ?   3. Suppose this is the Power spectral density of the image. How can I know the noise level and signal level from the figure 2?. I have assumed the sampling interval to be DelT =[(2*Pi)/PixelSize] Image : ![Image](http://i.stack.imgur.com/obzXE.png) PSD: ![PSD](http://i.stack.imgur.com/5ZVzo.png)


title: Why does a stack of transparencies appear white when a single sheet is clear?

text: My daughter and I found a box of old transparencies (clear plastic sheets) and something struck me as odd: even though each transparency is clear, the whole stack of transparencies looks whitish - why is that?


title: Curve object in liquid under pressure

text: I would like to know how red forces are compensate in this study. A black solid object is put in a liquid (helium or hydrogen for example). It's a curved solid. Solid don't move up or down, imagine it like same density than liquid. I study translation at right or at left. Gravity is perpendicular to the screen. Like study a liquid is difficult because forces are very small and there are a lot of problems: temperature, atomic effect, etc. I study with balls. If I look details of atoms of liquid, they place like a part of circle (like the curvature of the solid). But like atoms pressure atoms at left or at right (in all directions), a red force will appear. This force decrease force to right. How pressure equilibrate red forces ? ![enter image description here](http://i.stack.imgur.com/hbUGy.jpg) * * * Blue color is balls. Front: move up the top layer of balls and move down bottom layer of balls, balls let the black object move to the right. Rear: move down top layer of balls and move up bottom layer of balls. Like object is symmetric the energy to move balls is 0. Object has always the red force on it. ![enter image description here](http://i.stack.imgur.com/bJl4p.jpg) ![enter image description here](http://i.stack.imgur.com/LSEos.jpg) The container must block balls : ![enter image description here](http://i.stack.imgur.com/6HV5W.jpg) * * * maybe with torque it's easier to see the problem ? ![enter image description here](http://i.stack.imgur.com/C23e0.jpg) Not necessary balls are compressible. There is a net momentum on walls. Balls don't move in the shape.


title: Change of heat capacity fluid when you add solvent

text: I am considering a liquid for which I know $C_p$ or $C_v$. I am wondering how this changes when you add a (minor) amount of solvent to the liquid. Is there a general theory around describing how a random liquid's heat capacity changes when you add a random solvent? Or is it a very specific process for the particular liquid and solvent one considers? References to literature would be very much appreciated!


title: Does a Faraday cage block all magnetic field frequencies lower than x?

text: Where x is a function of the dimension of its holes. I mean, if I build a faraday cage able to shield magnetic field at 100kHz, will it we able to shield also all frequencies lower than 100kHz, such as 50Hz?


title: Causes of hexagonal shape of Saturn's jet stream

text: NASA has just shown a more detailed picture of the hexagonal vortex/storm on Saturn: > http://www.ibtimes.com/nasa-releases-images-saturns-hexagon-mega-storm-may- > have-been-swirling-centuries-1496218   >  https://en.wikipedia.org/wiki/Saturn_hexagon Is that theoretically understood what is the cause behind this eye-catching nontrivial, regular yet not circular, shape? If so, what is the cause? I expect some explanation in terms of "nonlinear equations" of "mathematical physics" and "solitons". P.S. (added a day after this question and the first answer was posted): On my blog where I posted the same question, people came up with some articles and phrases like "Rossby waves" and "resonance of latitude-dependence Coriolis frequency".


title: In QM, do we deal with basis or orthonormal sets?

text: Most textbooks say, that given a (countable) basis ${|\phi_n\rangle}$ of a Hilbert space, that every vector $|\psi\rangle$ of the space can be written as: $$\psi\rangle=\sum_{n=1}^\infty a_n|\phi_n\rangle$$ But this an infinite linear combination, and every vector should be expressed by a **finite** linear combination. So I think that they are an orthogonal set, which _generates_ the vector space as an infinite linear combination (the difference between infinite and finite LC means that orthonormal sets are usually _smaller_ than basis). But for the rest of the properties, ${|\phi_n\rangle}$ looks like a basis to me. **So, is ${|\phi_n\rangle}$ a basis or just an orthogonal set?**


title: Applications of analytic continuation to physics

text: I posted this on math.SE, but didn't get much response. It might fit better on this site. Holomorphic functions have the property that they can be uniquely analytically continued to (almost) the entire complex plane. So, just by knowing how the function behaves at a teenie-weenie open disc in $\mathbb{C}$, the behavior at a much larger scale is uniquely defined. Does this extraordinarily beautiful property have any direct applications in physics? For example, you know how some kind of field looks in a tiny area and you can somehow assume it is described by a holomorphic function and by analytic continuation you "extrapolate" the field to a much bigger domain.


title: Speed of Sound/Push in Supersonic Flight

text: I've learned from things like the Veritasium Slink Drop and the old "break the speed of light by pushing on the end of a very long pole" thought experiment that objects have a "speed of push" which is commonly referred to as the speed of sound. My understanding of this being that information about movement can't propagate through a medium faster than the speed of sound in that medium. So my question is: What happens to air when a supersonic jet flies through it? On the one hand it would seem that air can't move out of the way of the jet faster than the speed of sound in the air, however the jet obviously displaces the air at supersonic speeds anyway. So what's happening?


title: Chirality and helicity

text: I have massless Dirac equation and chirality and helicity operators which are given as $$ \hat {P}_{ch}\Psi = \gamma_{5}\Psi, \quad \hat {P}_{h}\Psi = \frac{(\hat {\mathbf S} \cdot \mathbf p)}{|\mathbf p|}\Psi . $$ I showed that both of them are lorentz invariant operators, they commutes with dirac free hamiltonian and their eigenvalues are $\hat {P}_{ch}\Psi (x) = \pm \Psi (x), \quad \hat {P}_{h}\Psi (x) = \pm \Psi (x)$. Is it enough for showing their identity? Or I also need to connect the given eigenvalue of $\hat {P}_{h}$ with corresponding eigenvalue of $\hat {P}_{ch}$?


title: Is there a wave function for anyons?

text: People talk about anyons a lot. But i have never seen an anyon wave function. I suspect that there is no such thing as a wave function for anyons. I mean, anyons are not generalizations of bosons or fermions. For bosons and fermions, one can have many-body hilbert spaces, but for anyons, there is no such thing.


title: Only Head On collision?

text: Let us consider two spheres A and B. Suppose they are interacting with each other (In broad sense one can say they are colliding). Let for the time being refer to "striking by coming in contact as collision". Now suppose the sphere B collided in such a way that its direction of motion is not along the line joining their centeres, in short it was not a head on collision. Then can this collision be an elastic one or is it so that there can be an elastic collision if and only if it is a head on collision? I hope the answer provider or commentator will justify his answer with reasons.


title: What really happens with Time Dilation?

text: I know if you move your time moves slower than someone who is stationary, by Lorentz's transformation. However, I don't get how this happens. What does it mean when time moves slower? How does it affect my watch for example when I move close to the speed of light? Does an electron for example feel time-dilation?


title: decomposing forces

text: I have a question about something I'm not sure about. I will first explain the story. Image If you calculate the force of a truck going downhill and applying its brakes:   * The mass is $50000\text{ kg}$   * The deceleration is $7\ \mathrm{m / s^2}$   * And the angle is 7 degrees. $F_1$ is the componenent of the gravity that is parallel to the to surface of the ground ( the F// or FII). $$F_1 = 50000 *9.81* \sin(7^\circ) = 59777\text{ N (force downwards)}$$ The resulting force is $50000 \times 7 = 350000\text{ N}$ (using the deceleration value) (force downwards). $$F_\text{brake} = 59777 + 350000 = 409777\text{ N (force upwards)}$$ If your truck goes up instead of down this time and then applies its brakes, do you need to do this? $$F_\text{resulting} = F_\text{brake} + F_1 = 409777+59777 = 469554\text{ N}$$ Is it right to count $F_1$ twice, the first time in the F included in the brake and the second time separately?


title: Extra help in Optics

text: I am in an optics class, and we are using the text "Introduction to Optics" third edition by Pedrotti. The book is completely useless in the course. The questions in the review section of the chapter cannot usually be done with some help from the internet. So I was wondering if anyone had a free text or online resource that would be of any help to some second year physics majors.


title: "The Schwarzschild Proton" paper - Is the science valid?

text: I'm a layman without a university background in physics / math. Since I don't have a background, reading a paper is more of an effort. Consequently, when I come across an interesting paper, I can't really just give it a glance, and see if it is science, or psuedoscience. This question is about this paper: > The Schwarzschild Proton. Nassin Haramein. Originally at The Resonance > Project; archived here by the Wayback Machine on 20/02/2012. [Edit by > bcrowell, 19/08/2013: this web page provides a very detailed discussion, and > offers heavy criticism.] If the advent of the internet has taught me anything, its that there is an inverse relationship between cost to publish and the need to vet what is published On the other hand, Lisi's E8 paper has taught me that a paper doesn't necessarily need to be correllated to an establishment like the Perimeter Institute or the Institute for Advanced Study (two examples off the top of my head) to be worth reading. Is the paper worth reading?


title: Force constant of metals - Kohn anomaly

text: In _Introduction to Solid State Physics_ (Kittel), it assumed the force constant between plane $s$ and $s+p$ $C_p=A\frac{\sin pk_0a}{pa}$ in metals to represent a Kohn anomaly. It says such a form is expected in metals. But I have no idea why. Thinking of typical potential energy-distance curve with one inflection point, shouldn't $C_p$ has only one node? Why only metals have such shape of $C_p$ and thus Kohn anomaly? I guess there are much complex quantum mechanical principles underlying the Kohn anomaly. Is it possible to explain the reason of that $C_p$ shape and Kohn anomaly in the classical manner?


title: Write a parabolic equation in kinematics

text: How might I go about writing a parabolic equation in standard form $ax^2 + bx + c$ given all of the following measurements: > $X_0, X_f, Y_o, Y_f$: the initial and final x and y positions.   >  $V_{x0}, V_{xf}, V_{y0}, V_{yf}$: the initial and final x and y velocities.   >  $t_x, t_y$: the time it takes for $x_0$ to transition to $x_f$; same for y. > These are equal in parabolic equations.   >  $a_x, a_y$: acceleration in x and y. I don't know how I would give the trajectory as a normal parabola from this data because I'm not sure which variables to substitute and where into $ax^2 + bx + c$.


title: Moving charge as a magnet, is the sign relative?

text: A thought experiment: Suppose we are moving in a spacecraft (A), carrying an **excess** of electrons on spaceship surface, but we can't go outside to measure it, and it's moving at constant speed $v$, with respect to other distant ship (B). When (A) pass in front of the ship (B), suddenly the two metal ships start to approach at increasing speeds, _it's because of the force from magnetic field generated by the relative speed of moving charges_ . We have to curb and redirect the ship, and of course solve the original problem, but we don't know about the excess of electrons on our surface, a scientist on board make hypothesis some minutes before the impact and thinks that the problem is a **lack** of electrons! (and uses some command to increment them, worsening the crash) **First question:** How a moving (and electrically charged) observer at constant speed could explain metals coming to him? Meanwhile on the (B) ship, technicians thinks their ship surface is not charged at all, and the force is because their metal structure itself, they thinks (A) ship is the charged one, and they don't know what to do because they are neutral, and action should be taken at the (A) ship to neutralize it because attraction would occurs regardless of the sign of electric charge. **Second issue:** As it has not sense to ask which ship is moving at speed v, because it's relative, does it have sense to ask about charge sign? Is there some way to tell which sign is the own charge? It's possible to know inside a system the sign of the outside visible charge? Help those ships captains... before they crash!. =)


title: Why can interactions be neglected for the Integer Quantum Hall effect?

text: Though the statement is made often, I've not seen any justification for neglecting electron-electron (Coulomb) interactions in the fully filled $\nu =1$ IQH state. I would highly appreciate if someone could provide an argument for the same. Also, I don't have a background in solid-state physics, maybe I'm missing something obvious?


title: Mathematical Elegance in a Cosmological Theory Considered Necessary?

text: In general, it seems cosmological theories that encompass more and more of the phenomena of the universe are expected to be more and more mathematically elegant, in conception if not in detail. Our experience seems to teach that it is a reasonable expectation to assume more fundamental theories will be more "elegant" or "beautiful" than previous theories. Two questions:   1. It seems like a similar expectation is what triggered Einstein to reject Quantum Mechanics. We know his experience led him astray. How do we know our experience isn't causing a similar result in our search for a Theory of Everything?   2. A correllation between "mathematical elegance" and explanatory power would seem to infer that "elegance" is more than just a human construct. How can that be? Why should there be a correllation between what we find pleasing, and how the Universe works?


title: Why does sweetness of coke change after freezing completely

text: I freeze my coke in a freezer completely to solid and then keep it out to melt and as it melts portion by portion I starts to drink, initially It will be very sweet and later it wont be sweet at all. why does this happen? We know that all sugar in coke will be dissolved (Solid solution) but how a major portion of this melts faster than water in the juice their by getting more sweet? How these sucrose can escape through the crystals formed by the ice and join in the water. I measured the sweetness (Brix) and found it varies.


title: Can entanglement be used to decrease entropy of two systems?

text: I know that entanglement can create correlation between systems. Would it be true to say that entanglement create constraints between systems? Can constraints be used to lower the total entropy of systems? In other words, say we have two separate systems with certain entropy values. If we create some kind of interaction between the two systems and did so without investing energy, would it be possible to create constraints between the two systems such that the total entropy decreases?


Why are the physical sciences described perfectly by mathematics?


title: How come this paper clip is " floating" on the subway floor?

text: What's the physics behind the paper clip floating? Technically this was filmed on a subway floor in motion. So I'm guessing it has some Newtonian mechanics involved here, and maybe some other stuff I don't know? Please explain. Thank you. http://www.youtube.com/watch?v=-qJoe8W-N98&feature=youtu.be Sorry if this might not be the proper place to ask this question. I do not know much of physics.


title: Does human body parts like hand or leg move with uniform acceleration?

text: So is this true. I mean does a movement by hand or leg make uniform or constant acceleration?


title: Heating effects of microwaves

text: I understand the heating effect(dielectric heating) of microwaves. This heating is caused by using a frequency of 2.45 GHz and this is the same frequency at which Bluetooth works and the L and S bands of communication. The intensity of microwaves used for communication is very less than used in microwave oven. **What is the numerical value of microwave heating as a function of intensity (for example, in the case of a mobile phone and a microwave oven). And how can I calculate it?** Note: I have gone throught this post Could cell-phone radiation cause cancer? and the related wikipedea article http://en.wikipedia.org/wiki/Mobile_phone_radiation_and_health. I know there are not any adverse effects of microwaves but I want to be sure that's why I want to know the numerical figures.


title: What does periodicity of $e^{-iHt/\hbar}$ mean in physical terms?

text: The unitary time evolution operator $U(t)=e^{-iHt/\hbar}$ has some distinct flavour of periodicity to it because of $e^{x+2\pi i}=e^x$. Is this periodicity reflected in any way in physical systems? Does it have any physical meaning? (Sorry for a rookie question. In case this is too simple for this site, tell me where to migrate it.)


title: Is there a way to quantify how similar a polycrystal should behave to a single crystal?

text: So in solid state classes we learned about phenomena like band structure and others arising from a periodic potential. Then we get to doing actual experiment and find out that materials being single crystals are actually fairly rare exceptions rather than the rule: more often, the material is a polycrystal or even amorphous. This kind of upsets my knowledge of what to expect, because all these things I learned were based on perfect crystals. But, then, I talk to people more knowledgeable than I about things like the band structure of an evaporated film, and they seem to give me the infuriatingly vague answer that we can still kind of expect to see the same properties of the band structure, but maybe not exactly. I don't even know where to begin theoretically -- it seems like if you have a polycrystal, it's now more like a bunch of quantum wells the width of the grain sizes mashed together, and they can tunnel into each other through their grain boundaries. I guess I could think of something using the knowledge of tunneling I know, but considering the grains are often oriented in random directions, I wouldn't know what to do. So my question is, is there a way to know what to expect of a polycrystal when you know the properties of the material in bulk? Or is it necessarily the type of thing you'd have to figure out from experiment?


title: In quantum mechanics(QM), can we define a high-dimensional "spin" angular momentum other than the ordinary 3D one?

text: Inspired by my previous question Questions about angular momentum and 3-dimensional(3D) space? and another relevant question How to define angular momentum in other than three dimensions? , now I get another question: In classical mechanics( **CM** ), for a $n$-dimensional space, the **orbital angular momentum( an antisymmetric tensor)** is defined as $L_{ij}=x_ip_j- x_jp_i$, where $i,j=1,2,...,n$. In **QM** , after canonical quantization, the **orbital angular momentum** $L_{ij}$ become some Hermitian operators and satisfy the following commutation relations, $[L_{ij},L_{kl}]=i\delta_{jk}L_{li}+i\delta_{li}L_{kj}+i\delta_{jl}L_{ik}+i\delta_{ik}L_{jl} \tag{1}.$ And as we know, in **QM** , a 3D angular momentum $S=(S_x,S_y,S_z)$ is called a **spin** only if $S^2_x+S^2_y+S^2_z=S(S+1) \mathbb i,$ **where $\mathbb i$ is the identity operator.** So in **QM** , more generally, can we define a $n$-dimensional **"spin"** $S_{ij}$ $(i,j=1,2,...,n)$, where $S_{ij}$ is an antisymmetric tensor and are Hermitian operators which satisfy eqn$(1)$, further more $\sum S_{ij}^2=$real number$\times \mathbb i$ ? **By the way:** More questions concerning the **definition of rotation groups** for angular momentum can be found here , who are interested in may have a look, thanks.


title: Do Maxwell’s Equations contain the Ideal Gas Law?

text: Maxwell’s Equations contain some other Laws as special cases. Does it possibly include the Ideal Gas Law as well? – the idea being that material substances can be modeled by appropriate subsets of the electromagnetic spectrum.


title: Earth-Centered Inertial (ECI) reference frame as approximate inertial frame of reference

text: In many practical applications, one can consider the Earth-Centered Inertial (ECI) reference frame approximately as an inertial reference system, though strictly speaking, it is non-inertial. Is there any quotable reference, where this claim is supported by a detailed estimation how small the effects are that one neglects if one considers the Earth's frame as inertial?


title: In Scully and Druhl delayed choice quantum eraser experiment, are there limits to the lengths of the delay?

text: I'm in humanities, not physics, so please bear with me. I am trying to understand this experiment and have a few unanswered questions. I have read the other posts on this site that discuss the experiment, but I'm still fuzzy on a couple of things. As I understand, there are two setups for this experiment, one involving splitting a photon into two after it passes through the double slits. One photon will reach a screen and create an interference pattern or not. The other has a 50% chance of indicating which-path information or not. This 50% chance, however, is not decided until after the first photon reaches the screen. In the second setup, the photon leaves a trace that could indicate which-path information but the trace is not observable to the experimenter. The experimenter then chooses whether to keep or erase this information. My major question pertains to the second setup. Must the experimenter decide whether to erase the which-path trace before the photon hits the screen? (It is still delayed choice because the decision has made after the photon has left the chamber where the trace was made.) What happens if no choice is made? Does the existence of the trace (unobserved but not yet erased) mean no interference pattern will show up? In the first set up, I have a similar question. I understand the photon that may indicate which-path information will go randomly to one of four detectors, two that can distinguish which path information and two that cannot. The interference pattern can only be observed after the data from these detectors has been collected and correlated, as only those that reached the which-path detectors will form an interference pattern. What I don't understand is what happens if photons are sent through the apparatus one at a time. What shows up on the screen in this case? The screen should be observable before the second photon's fate has been determined. Any insight would be greatly appreciated.


title: Is sound relative when travelling fast?

text: If I was listening to music through my headphones and travelling close to the speed of light and time is slowing down around me , would the music slow down as well?


title: Can a big microwave used to dry clothes?

text: I was wondering if it is possible for a microwave in principle to dry clothes (assuming of course it is big enough).


title: Why does a piece of paper travels up when inserted into a kite's thread?

text: ![enter image description here](http://i.stack.imgur.com/P9Yr4.png)When I was young, I would insert a piece of paper into a kite's thread. The paper would travel upward till it reaches the head of that kite. What makes it to go up against the gravity even if the kite's head is about hundreds of meters away from the ground?


title: What happens if we use an optical fibre for light at a different wavelength than the one specified on the fibre?

text: Say the fibre is for 700 nm, and we shine light at 1064 nm into it. Would it change the mode? The polarisation? The beam shape (i.e. introducing ellipticity in the beam)?


title: Why are the "coupling constants" constant?

text: The coupling constants (in the gauge theory) fix an inner product on the lie algebra of the gauge group and we use it to define strength of the fields. we are using ad-invariant inner products which are determined by some numbers. In other words, the set of all ad-invariant inner products form a space of more than one dimension and to fix a specific inner product we need to choose some numbers which are the coupling constants. This is the story which happens over each point of the space-time. Mathematically one can produce a theory in which these numbers (and so inner product) changes from a point to another (similar to Riemann metric on a general Riemannian manifold). In other words we can have coupling fields rather that constants. What is the physical reasoning which disallow us having such a theory (with coupling fields)? And what would be the physical implications if the nature follows such a model?