title: Measure absolute speed

text: Currently I'm 17 years old, going to secondary school. So, my ideas might be totally wrong...   I know that everything is relative. In the example of speed, the earth moves, and the galaxy moves, etc. My physics teacher told me that the speed of light is absolute, which means that the speed of the light source doesn't influence the speed of light in space. So, I was thinking that that fact could helps us to measure the absolute speed of our planet in space. Not relative to the sun, or the galaxy. The way of measuring it was following: `A` is the light emitter.   `B` is the light sensor, in combination with a very very precise timer.   `D` is the signal broadcasting point.   `~~>` is light, going from `A` to `B`.                A ~~~~~>~~~~~~~>~~~~>~~~~~>~~~~~~~~>~~~~~~>~~~~~~~~~>  B       \__________________________D_________________________/      So, how it works — in my head — is that you send a signal from `D` to both `A` and `B`. The distances from `A` to `D` and from `B` to `D` are equal, so this should get the signal to both A and B in the same time. The distance between A and B is constant, say K. As soon as the signal reaches B and A at the same moment, B starts the very precise timer and waits for the light coming from A, at the same time, A starts emitting light. According to my knowledge, you should be able to calculate the speed of the whole situation along the axis A,B. Why? Because if our absolute speed is along with the light direction, it should take longer for the light to reach B, because the distance is bigger. Otherwise, we are moving in the opposite direction of the emitted light, so we are going towards the light, so, we make the distance for the light to travel shorter, because we are concede towards the light. Compare it with a car (`C`) driving on the highway, next to a high speed train (`T`). The train goes faster than the car. Compare both situations: 1) Train and car moving in the same direction, train starts behind the car.               T  ---------------->                               C ------>      2) Train and car moving in opposite direction towards each other.               T  ---------------->                     <-------  C      In Situation 1, it will take longer for the car and train to meet.   In Situation 2, it will be pretty quick that they meet, because they going towards each other. It is that difference in time that can be used to calculate our absolute speed, I think. To define our absolute velocity vector, we can do this measurements three times, each test perpendicular to the two others, so we can apply Pythagoras to get our absolute speed as a scalar. My teacher could hardly believe that it would work, so he thought that something should be wrong to my theory. What do you think, assuming that we have very precise measuring tools?


title: Hypothetical very massive particles

text: I'm looking for a table or compilation of hypothetical very massive ($m\gtrsim 1$ TeV) particles and their expected masses (or bounds on them or relation with other scales). All I know is (please, correct me if I'm wrong):   1. Supersymmetric particles: $M_{GUT}\gtrsim m\gtrsim 0.1$ TeV.   2. Massive neutrinos related to see-saw mechanism: $m\sim M_{GUT}$.   3. Magnetic monopoles: $M_P\gtrsim m\gtrsim 0.1$ TeV.   4. Dilaton: $M_P\gtrsim m\gtrsim 0.1$ TeV.   5. X bosons in GUT: $m\sim M_{GUT}$.


title: Could Dark Matter be a manifestation of tachyons tunnelling between black holes?

text: Among the many Dark Matter candidates, I wondered if there are any along the lines of the title. The inner horizon of a black hole, with its mass inflation, and/or the high spacetime curvature, seems a violent enough place for the strangest things to happen that can anywhere, and tachyons (if these exist) seem to have enough counterintuitive properties that they might perceive far greater space volumes accessible inside black holes than outside, so that any created inside and straying out would readily tend to find their way back into the same or another black hole, with some energy change while in transit that would manifest itself outside the hole(s) as "dark mass" (assuming this was a fairly steady state process). Piling speculation on speculation, one might further envisage that a tachyon which exited a black hole and lost enough energy outside (and hence gained speed) for long enough might greatly increase its chance of staying outside and thus manifest itself as dark energy. The latter would also explain why the expansion of the universe is accelerating: If the average distance between black holes (between galaxies for example) is increasing with cosmic expansion, then on the preceding assumption slightly more tachyons can be expected to remain outside black holes, and so on. It seems all this would make most sense (if at all) if tachyons appeared somehow "spread out" in normal space, like a field, as opposed to conventional sub-luminal particles. Apologies if this is all too vague and speculative for some peoples' liking. I am not axe-grinding, quite the opposite, just curious to hear any informed comments on an idle fancy which is most likely untenable.


title: Collision time of Brownian particles

text: Let's assume two spherical particles $p_1$ and $p_2$ of finite radius $r_1$ and $r_2$, which are at locations $(\pm\frac{d}{2},0,0)$ a distance $d$ apart at initial time $t$. These particles diffuse with coefficients $D_1$ and $D_2$, respectively. How can I obtain the probability distribution of the collision time (that is, if the time at which they collide is $t + \Delta t$, I would like to know the probability density function of the random variable $\Delta t$), or at least some of its moments, in these two cases:   1. when the domain is infinite, e.g., $\mathbb{R}^3$, and   2. when it is bounded (inside a sphere with surface area $A$ and the origin $(0,0,0)$ as its center)? I tried to look up for an answer in Van Kampen's book, but I couldn't really come up with an answer.


title: Can a large thermonuclear bomb be used for planetary defence?

text: How effective would a thermonuclear device of 47 megatons or higher like the zsar bomba be in space? Could we use as planetary defence on somthing like an asteroid as long as it is detected in time and what effect if any would it have on our planet and our satellites in orbit?


title: Electron model under Maxwell's theory

text: I was not able to recall my memories, so: What is the formula that states the frequency of electrons revolving around nucleus is equal to the frequency of light (or photon) emitted (or radiated)? (I am of course talking of Maxwell's theory; in reality, we know that this is not true.)


title: Fracture because of high-speed rotation

text: I was watching a rerun of an early MythBusters episode, where they look at whether CDs in high-speed drives can explode / fail simply because of being rotated too fast. The following are some calculations I did: Let the thickness of the CD be $t$, made of material with density $\rho$. Consider a small element of the CD at a distance $r$ from the center. The element subtends an angle $d\theta$ at the center, and is of length (along the radius) $dr$. The centripetal force needed to pull this in at angular velocity $\omega$ is $m r \omega^{2}=\rho \left(r d\theta dr t\right) r \omega^{2}$. The CD has a central spindle hole of radius $r=1.5\ cm$, and is itself of radius $R=12\ cm$. The force then exerted on an elemental area subtending angle $d\theta$ at the inner edge of the CD is (integrating the previous quantity from $r=r$ to $R$) $\frac{\rho \left(R^3 - r^3\right) d\theta t \omega^2}{3}$, which means the inside edge has pressure approximately $\frac{\rho R^3 \omega^2}{3r}$ (dividing the previous quantity by the area $t r d\theta$, and assuming $R^3$ dominates $r^3$). Fracture now happens when $\frac{\rho R^3 \omega^2}{3r} > T$, the fracture strength of polycarbonate plastic (which being brittle, is approximately its ultimate tensile strength). Substituting $\rho = 1.22\ g\ cm^{-3}$, and $T = 75\ MPa$ (all approximate values obtained from Wikipedia), I got $\omega > 17000\ rpm$. My question is, is the calculation correct? Or is there something I'm missing or misinterpreting? The value is the same order of magnitude as what MythBusters reported -- fracture at $\approx 25000\ rpm$. This is heartening, but it would be nicer to know that the number is off only because of incorrect inputs (strength of plastic, density etc.). As nice as MythBusters sometimes is, it would be nicer still if they did a little more hard science. * * * Correction: As akhmeteli points out below, the CD dimensions are incorrect. The central spindle hole has radius $7.5\ mm$, and the CD itself has radius $60\ mm$. There's also an arithmetic mistake in the above calculation. The updated answer is therefore $\approx 24000\ rpm$. That would ordinarily make me very happy, but as both user758556 and ja72 argue, this calculation would be useful as nothing more than a sanity check. What's really happening is a little too complicated for these arguments to properly explain.


title: At what scale does UV/IR mixing happen in string theory?

text: Imagine, if you will, a string background with an extremely tiny value for the string coupling. The Planck scale is many orders of magnitude smaller in size than the string scale. Is the UV/IR mixing scale the Planck scale, or the string scale? What happens at intermediate scales? Gravitons are string modes, and as such, their spatial extent is the string scale. They can never resolve anything smaller than that. On the other hand, D-branes are more sensitive.


title: The Higgs explains how particles acquire mass. Could it explain how much?

text: It's my understanding that nothing in the Standard Model predicts the mass values of the fundamental particles, so I guess that means we don't currently know how to make models of Higgs interactions predict the masses theoretically. My question is more about upcoming experiments when the LHC comes back online. As more data on the details of the mechanism come in, might a more accurate picture of Higgs (and the details the mass acquisition process with it) yield empirical models that specify how much mass a particle gets? Or are the reasons the Standard Model doesn't predict the masses too deep to be resolved by simple extensions of the theory?


title: Planck time & length

text: Why is Planck time the shortest possible duration ever? It's defined as the duration needed by light to travel Planck's length, but surely, give me any number, I can give you a lower number than that? So what's so special about Planck's time? Is the universe discrete, in the sense that time moves through discrete $N$ Planck units of time? In the same way, when you say Planck length is the smallest possible distance ever, does that mean every particle( or it's constituents) jumps from one point to another where distance between the points is Planck's length, kinda like strip lighting, where bulbs lit in a sequence give the illusion that the glowing section is moving along the strip?


title: How Felix Baumgartner has reached the speed of sound quickly

text: I have watched Felix Baumgartner freefall; but I wonder how Felix has reached the speed of sound quickly, in a matter of some seconds, then we had no idea of its speed? Any explanation please.


title: Toolbox for Complex Networks and Graphs

text: Is there a toolbox which helps in visual simulation and modeling of a network (say a mesh or ring) consisting of coupled synchronized system of nonlinear equations(ODE) which represent a system of lasers?


title: How to measure force of impact inside container?

text: I am in 7th grade and for my science fair project, I need a way to measure the force on a dropped object when it hits the ground. What I am trying to determine is which packing materials provide the best protection for an object in a collision. So I am planning on dropping containers filled with different packing materials surrounding some sort of force-measuring device in the middle. But I don't know how to either obtain or construct the force-measuring device. Because I am measuring the effectiveness of the packing material, I need to measure the force inside. One method I thought about was having a metal ball sitting on top of clay. After hitting the ground, the ball will dig into the clay. I can measure how deep the impact is and assume that the deeper the hole, the greater the force. But I am not sure if this will work. Does anyone have any suggestions on how to measure force (using either my idea or something else entirely)?


title: Is there something special in the visible part of electromagnetic spectrum?

text: I always wondered how much information we get from color. Things we see have different colors; edible products change color when began to spoil so we have a notion what color a fresh product should have. And the colors we see are very small part of electromagnetic spectrum. So is there something special in the visible spectrum or any part of electromagnetic spectrum of the same size is just as good?


title: What is a bilateral constraint?

text: In the realm of mechanics/rigid body dynamics, can anyone tell me what a bilateral constraint is? Can't seem to find any information on the exact definition, just uses of it such as "considering only bilateral holonomic constraints..." Is this just saying there will be equal and opposite reaction forces? What are some examples of bilateral and non-bilateral constraints?


title: Pre-gauge-fixed superspace action of the RNS superstring

text: When writing down the the action of the RNS superstring in superspace, all of the sources I have checked (BBS, GSW, Polchinski) seem to just write down the action in conformal gauge, that is $$ S_{\text{RNS}}:=\mathrm{i}\, \frac{T}{4}\int _W\mathrm{d}^2\sigma \mathrm{d}^2\theta \, \bar{D}Y\cdot DY, $$ where $W$ is the superworldsheet, $Y$ is a superfield on $W$: $$ Y:=X+\bar{\theta}\psi +\frac{1}{2}\bar{\theta}\theta B, $$ $D$ is the 'supercovariant derivaitve': $$ D_A:=\frac{\partial}{\partial \bar{\theta}^A}+(\rho ^\alpha \theta )_A\partial _\alpha, $$ $\rho ^\alpha$ are generators of the $(-,+)$ Clifford algebra: $$ \\{ \rho ^\alpha ,\rho ^\beta \\}=2\eta ^{\alpha \beta}, $$ and the bar denotes the Dirac conjugate. On the other hand, for the bosonic string, we have the Polyakov action: $$ S_{\text{P}}:=-\frac{T}{2}\int _W\mathrm{d}^2\sigma \, \sqrt{-h}\nabla _\alpha X\cdot \nabla ^\alpha X, $$ where $h_{\alpha \beta}$ is the metric on the worldsheet $W$ and $\nabla _\alpha$ the corresponding Levi-Civita covariant derivative (which for scalar fields happens to agree with just the usual partial derivative). If we take $h_{\alpha \beta}=\eta _{\alpha \beta}$ (conformal gauge), then this reduces to the Bosonic part (ignoring the auxilary field $B$) of $S_{\text{RNS}}$. I was wondering: what is the appropriate generalization of $S_{\text{RNS}}$ to a theory defined on a supermanifold with a 'supermetric'? For that matter, what is the right notion of a supermetric on a supermanifold and do we have an analogous Fundamental Theorem of Super-Riemannian Geometry that, given a supemetric, gives us a canonical supercovariant$^1$ derivative? This generalization should be analogous to the pre-gauge-fixed form of $S_{\text{P}}$ given above in which the metric and all covariant derivatives appear explicitly. * * * $^1$ While $D$ is called the "supercovariant derivative", it clearly cannot be the right notion, at least not in general, because it makes no reference to a supermetric.


title: Celsius to Fahrenheit confusion: why there is no 1 to x ratio

text: Why is it not possible to know ratio of 1 degree to how many Fahrenheit and use that to convert from or to Celsius/Fahrenheit. I mean what is really happening. Fahrenheit increases linearly and so does Celsius. So why there is no ratio.


title: How can I model computationally or experimentally the flow in my aquarium tank?

text: I have a 60-gallon aquarium tank and I have always wondered about the flow of water in the tank. Let's represent the flow of water in terms of a direction field representing velocity, so if we see the tank from one side (mainly the wide side) we will see a direction field like in the image below: ![flow vector field](http://i.stack.imgur.com/FOXaY.gif) But I know there are areas in the tank that have loops where the water is circular, or where the water is simply stagnant and there is no flow (bacteria can form there). How would I be able to see the flow of water in my tank? What equipment do I need? Better yet, is there some model I can use to determine the flow in my tank?


title: Elastic potential

text: I have a doubt: elastic potential energy is given by: $U=\frac{k}{2}x^2+K$ but does elastic potential exist? (for example: potential gravitational energy is given by $U=mgz+K$ and gravitational potential is given by $V=-U$) ($K$ is the constant of integration)


title: Existence of time in some other universe

text: Is it necessary that time should exist in another universe if it (universe) is there? How do we perceive timelessness?


title: What is the mass distribution within the sun?

text: Jupiter is roughly 1/1000 the total mass of the sun. To get some idea of what effect Jupiter's gravity may have on the sun I'd like to know the approximate mass distribution of the sun. (i.e) the approximate mass of the core, the radiative zone, the convective zone and of the photosphere?


title: Goldfish perspective

text: What does the world look like from the Goldfish point of view, from inside a spherical aquarium? If our eyes were inside, would we be able to see the straight lines, focus on different objects and what would a light point source looks like? (elaborate on the curvatures, with or without the flat water above)


title: Photons and Relativity

text: Consider a Photon from Sun and travels with a velocity $c$. Now think we are that photon. For us, it looks like Sun is moving away from us with a velocity $c$. So, why don't we get attracted back towards Sun, because the mass of Sun would be infinite for us since it moves away from us with a velocity $c$.


title: Need to write a function that computes the UV index

text: I have found a lot of people talking about how to compute the UV index, but have not found an example that shows the actual equations, so I could adapt them into an algorithm. Probably the closest is this example here. The most confusing part of the process is why there are so many cases that present hypothetical data rather than a real example. For instance, here is the EPA's explanation. They seem to compute a real number. But they say the weighing factors are fake.


title: What are the implications for quantum gravity if the LHC sees no higgs?

text: Following the news in serious "non-hype" physics blogs I`ve learnd that as things are now one needs a lot of patience and more data to learn what happens in the higgs sector. There are already a number of good questions and discussions about what the absence of a higgs would mean for the standard model or more general for theoretical physics on this site. So I want to "extrapolate" this a bit and ask about what would be the implications for the different quantum gravity approaches if no higgs boson(s) (are) seen at the LHC ?


title: Quarks as preons for the whole standard model

text: This is a sequel to an earlier question about Alejandro Rivero's correspondence, the "super-bootstrap". The correspondence itself was introduced in his "Supersymmetry with composite bosons"; see the tables on page 3. Briefly, we consider all possible pairings of quarks and antiquarks, but only from the first five flavors, "because" the top quark decays too quickly to hadronize. If we add up the electric charges, we get exactly the right number of combinations to get all six flavors of quark and all three charged leptons. There are also enough combinations to match all three neutral leptons (with an extra state left over); and there are also some exotics with charge 4/3. One may repeat the exercise with the full electroweak quantum numbers and it still works, more or less. We seem to be getting back all the standard model fermions by pairing up five flavors of quark - except that two quarks should create a boson. So in addition we postulate that the standard model fermions are superpartners to these pairings: a lepton is a "mesino", a quark is (or mixes with) a "diquarkino". For months I've been thinking how to obtain this set of relationships within a supersymmetric theory. There are two problems: first, how do you get elementary and composite particles into the same superfield; second, what about gauginos, superpartners of the gauge bosons? But it has finally occurred to me that maybe you don't need supersymmetry at all - you just need some extra fermionic preons. Suppose there are five fundamental quarks, _udscb_ , and fermionic preons _n_ , _n'_ ,... which don't feel the strong force. And suppose that there is a new confining force, the ultrastrong force, and that _udscb_ and _n_ , _n'_ ,... all feel it - they all have ultracolor charge. Then there will be "ultrahadrons", some of the form " _qqnn..._ " or " _qbar qnn.._ ", and these can be the diquarkinos and mesinos of the correspondence. Leptons would be mesino-like ultrahadrons, the top quark would be a diquarkino-like ultrahadron, and the _udscb_ quarks might also mix with other diquarkino ultrahadrons. This seems like a very simple idea, and yet I see no sign of it in the literature on preon models. I would therefore be interested in any anticipation of the idea that _does_ exist, and in any arguments which have a bearing on its viability.


title: How to sum several thrust vector on a single object (like a plane)

text: I'm a developer. I tried (mostly for fun) to create a realistic physic engine. I try to sum all forces (here thrust) applying to an object in two composantes: an acceleration vector and the angular acceleration. I read a lot about torque and thrust but there is still some stuff that I can't figure out. Let's take a exemple: Imagine a plane(or space craft) with two motors each placed at the opposite side with respect to the center of gravity. If they have equals thrust I figure that I can modelise them by a single thrust vector placed on the center of gravity equals to the sum of the thrust provide by the two engines. Is that right? What's happen if one motor delivered 10% less thrust that the other one, how my thrust decrease, how to compute resulting torque ?   Did I loose or gain efficiency if my thrusters are placed at the back of the plane and why? I try to find a mathematic model that could work with any number of thruster placed at different position of the air-craft.


title: How do I read the simple, but contradictory, Lagrangian ($\mathcal{L} = x + v$)?

text: I understand the lagrangian formulation of classical mechanics, to a degree. I can derive the Euler-Lagrange equations from the "least" action principle, and equivalently can determine the equations of motion from a given lagrangian. I can handle lagrangian exercises in textbooks with ease. I don't quite grok it, though. If you erased it from my mind, I wouldn't be able to re-invent it. So I'm going back through it. The way I understand the Euler-Lagrange equation is thus: In classical physics, by observation, there is a stationary quantity. This stationary quantity is called the "action", and it is the sum of energies over time (alternatively, a product of energy and time). Again by observation, energies can be calculated given the positions and velocities of all elements of the system. Call such a function $\mathcal{L}(x, \dot{x})$, treat it as a black box. Then we have $Action = \int dt \mathcal{L}(x, \dot{x})$. Making this stationary, we derive $$\frac{\partial\mathcal{L}}{\partial x} = \frac{d}{dt}\frac{\partial\mathcal{L}}{\partial \dot{x}}$$ Informally, this says to me that the paths taken in actual physics are ones where small perturbations in a particle's position are canceled out precisely by the changes in velocity before and after that position (where the velocity changes are caused by the position perturbation). This has an almost economic elegance to it. But it still doesn't completely make sense to me what a Lagrangian is. Its unit is energy, sure, but I also don't quite grok energy beyond the abstract. So I figured I'd play with a few simple Lagrangians, hoping to break the formulation and learn something from how the pieces fell. Take, for example, this trivial function: $$\mathcal{L}(x, v) = x + v$$ It describes an unphysical world, surely. Energy is far from conserved. But I figured constructing a weird but simple lagrangian would give me insight as to the nature of the formulation. Let's derive the equations of motion: $$\frac{\partial\mathcal{L}}{\partial x} = 1$$ $$\frac{\partial\mathcal{L}}{\partial v} = 1$$ $$\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial v} = 0$$ So the path taken is the path satisfying $$1 = 0$$ ...Huh. The laws of motion are unsatisfiable. I'm not sure how to take that. What, precisely, went wrong here? I'm looking for a geometric or intuitive explanation -- the algebraic troubles I understand. What sort of world was I trying to construct? Whence the contradiction? More generally, when you are handed a Lagrangian, what does it really signify? I can integrate it to get the action (an abstract number assigned to a path) or I can plug it into the Euler-Lagrange equations to figure out motion, but what does it mean in its native form? How do I read a Lagrangian without twisting its arm?


title: How far is the horizon, if a 2m tall man watches to the sea?

text: How far is the horizon, if a $r=2 m$ tall man watches to the sea? I have calculated that it would be even just about 6 km. if R = radius of earth( $6370 \cdot10^3$ m ). By pythagorean theorem we get. $(r+R)^2 = x^2 + R^2$ => $\sqrt((r+R)^2 - R^2) = \sqrt((2+6370 \cdot 10^3)^2 - (6370x10^3)^2) = 5048$ m ~ $5$ km = x Is this right?


title: Linear rising potential from a Gribov propagator

text: It is common wisdom that a gluon propagator (Gribov-)like $$ G(p^2)=\frac{a+bp^2}{cp^4+dp^2+e} $$ should give rise to a linear rising potential. So far, I have not seen a proof of this and I would like to get a mathematical derivation, or a reference, displaying how a potential emerges from it. Thanks.


title: Mass of empty AdS$_5$

text: Five dimensional empty AdS$_5$ space has mass $$ E = \frac{3 \pi \ell^2}{32 G}. $$ > Is the above equation correct? Let's do some dimensional analysis to confirm. In natural units, in 5 dimensions $[G] = -3$ where $[...]$ is the mass dimension. Also $[\ell]=-1$. Therefore $\left[ \frac{\ell^2}{G} \right] = 1$. So the dimensions seem to work out fine. Here's my second question: > The limit of $\ell \to \infty$ of $AdS_5$ space is flat space. Isn't it > weird that the mass diverges in this limit? I would have presumed that mass > should vanish in this limit, since flat space has vanishing mass? Are we > using two different definitions of mass? EDIT: Due to some requests in the comments, I will include the derivation of the formula above. I use the boundary stress tensor given by Brown-York (derived from the Einstein action alongwith the Gibbons-Hawking boundary term. $$ t_{ij} = \frac{1}{8\pi G} \left[ K_{ij} - \gamma_{ij} K + \frac{2}{\sqrt{-\gamma}} \frac{\delta S_{ct}}{\delta \gamma^{ij}} \right] $$ Here $K_{ij} = \nabla_{(i} n_{j)}$ is the extrinsic curvature. $n^\mu$ is the unit normal vector to the boundary. $\gamma$ is the boundary metric and $K = \gamma^{ij} K_{ij} $. $S_{ct}$ is the counterterm action included to make all the B-Y charges finite, which are defined as $$ Q_\xi = \int_{\cal B} d^d x \sqrt{\sigma} u^i \xi^j t_{ij} $$ Here $\sigma_{ab}$ is the metric on a spatial hypersurface and $u^i$ is a time-like unit vector normal to the hypersurface. $\xi^j$ is a Killing vector of the boundary metric. Now, for $AdS_5$, the counterterm action is given by $$ S_{ct} = -\frac{3}{\ell} \int d^4 x \sqrt{-\gamma} \left( 1 + \frac{\ell^2}{12} R(\gamma) \right) $$ The B-Y tensor is then $$ t_{ij} = \frac{1}{8\pi G} \left[ K_{ij} - \gamma_{ij} K - \frac{3}{\ell} \gamma_{ij} + \frac{\ell}{2} \left( R_{ij} - \frac{1}{2} \gamma_{ij} R \right) \right] $$ We can now work in the Fefferman-Graham coordinates for $AdS_5$ space where the metric is $$ ds^2 = \frac{\ell^2 d\rho^2}{4\rho^2} - \frac{(1+\rho)^2}{4\rho} dt^2 + \frac{\ell^2 ( 1 - \rho)^2}{4\rho} d\Omega_3^2 $$ Thus $$ t_{ij} = - \frac{ \rho }{4\ell \pi G} \left( \gamma_{ij}^{(0)} + \gamma_{ij}^{(2)} \right) + {\cal O}(\rho^2) $$ where $$ \gamma_{ij}^{(0)} dx^i dx^j = -\frac{1}{4} dt^2 + \frac{\ell^2}{4} d \Omega_3^2 $$ $$ \gamma_{ij}^{(2)} dx^i dx^j = - \frac{1}{2} dt^2 - \frac{\ell^2}{2} d \Omega_3^2 \\\ $$ We also have $$ u = \frac{2 \sqrt{\rho}}{1+\rho} \partial_t,~~ \xi = \partial_t,~~\sqrt{\sigma} = \frac{\ell^3 (1 - \rho)^3 }{8 \rho^{3/2} } \sin^2\theta \sin \phi $$ Plugging all this in, we find that the B-Y charge corresponding to the Killing vector $\partial_t$ is $$ Q_t = \frac{3 \pi \ell^2 }{32 G} $$ This is where I got the formula from. I interpret this as the mass of $AdS_5$ space. Disclaimer - I have intentionally left out several computations to reduce the length of the problem. I have not referred any paper and all computations have been done by me.


title: What is the longest distance that can be jumped after swinging from a rope?

text: In the movie _Mission Impossible 3_ , the main character Ethan Hunt tries to enter a building in Shanghai by swing through the sky, as shown below: ![enter image description here](http://i.stack.imgur.com/1CxqB.png) The jump consists of 2 sections, the red part, which is an arc, and the orange part, which is a parabola. A question naturally arises: **_For a target building with a given height, what is the furthest horizontal distance it can be placed such that we can jump to its roof using this method?_** Before answering this question, we must consider the following fact: ![enter image description here](http://i.stack.imgur.com/Vppya.png) Observing the path when we release at different heights, we see that if we release very late, we can reach great heights, but with reduced horizontal distance. If we release early, we increase horizontal distance, but with reduced height. Because of this trade off, the area that can be reached is restricted within the yellow line。 The question is then: (1) **What is the equation x(z) that describes the orange line above?** (2) **For a given point (x,z) that is inside the region bounded by the orange line, what is the proper release height h(x,z) such that the parabola will go through (x,z). The answer should be a set of h. This is the same as asking what is the proper release height if we want to reach a building with height z and is located at x distance away from the center building.** For the ease of discussion, I propose that we use the follow coordinate system and notation : ![enter image description here](http://i.stack.imgur.com/82StP.png) * * * Here are some calculations that are already done: At the release point, $$\text{Upward shooting angle}=\psi$$ $$v=\sqrt{2gh}$$ $$h(\psi)=r\cos\psi-H$$ $$\psi(h)=\arccos\frac{h+H}{r}$$ Upward and horizontal speeds are: $$v_{u}=v\sin\psi$$ $$v_{h}=v\cos\psi$$ Time for Ethan to fly through the orange part is: $$t=\frac{2v_{u}}{g}=\frac{2v\sin\psi}{g}$$ Thus $$d_{h}=v_{h}\cdot t$$ $$=v\cos\psi\frac{2v\sin\psi}{g}$$ $$=\frac{v^{2}}{g}\sin2\psi$$ If we were to use $h$ as the variable, then $$d(h)=\frac{v^{2}}{g}\sin\left(2\arccos\frac{h+H}{r}\right)$$ $$d(h)=\frac{\left(\sqrt{2gh}\right)^{2}}{2g}\left(\frac{h+H}{r}\right)\sqrt{1-\left(\frac{h+H}{r}\right)^{2}}$$ $$d(h)=h\left(\frac{h+H}{r}\right)\sqrt{1-\left(\frac{h+H}{r}\right)^{2}}$$


title: What is the exact gravitational force between two masses including relativistic effects?

text: I was wondering if there is a closed-form formula for the force between two masses $m_1$ and $m_2$ if relativistic effects are included. My understanding is that the classic formula $G \frac{m_1 m_2}{r^2}$ is just an approximation (which is good enough for probably even going to the Moon), but what would the "correct" formula be according to the general theory of relativity? Does a closed formula even exist? For example for an idealized situation of just two spherical masses with homogenous mass distribution?


title: Why do we want to entangle qubits?

text: The title is pretty much all I want to ask. Why are qubits entangled? To my knowledge (which isn't that deep) a quantum register can be realized without entangling the qubits.


title: knowledge of an internal observer

text: I would like to discuss the consequences of the concept of an internal observer in quantum theory. If we assume that we have a universe that evolves unitarily at a global scale and an observer is defined as a subsystem of that universe, what can this observer learn about the universe? Some assumptions are that the evolution of the universe is generated by interaction. Interactions are local and the observer gathers information by interacting with his environment. But most importantly, there is no measurement postulate that the observer could use to perform a quantum measurement. I would argue that everything the observer learns about the universe is contained in the state history of the subsystem he has interacted with. If one assumes that he can derive a law for the evolution of this state history, then he can only reconstruct the state history up to isomorphisms. Meaning, if he finds plausible state history psi(t) which evolves unitarily like $\psi(t) = U(t,t_0)[\psi(t_0)]$, then any $\phi(t)$ for which a bijection $f$ exists so that $\phi(t) = f(\psi(t))$ and $\psi(t) = f^{-1}(\phi(t))$ and which evolves with the same law $\phi(t) = U(t,t_0)[\phi(t_0)]$ is an equivalent description of the state history of the system he is part of. (Note that psi is not necessarily a ket. I would like to stay agnostic of the actual representation for now) A very simple example for such a bijection would be the multiplication with a nonzero complex number. Another would be a spacetime symmetry transformation. Do you think this argumentation is correct so far? I will continue the argument to the point where it actually gets more interesting, but I would like to see some confirmation so that we can later discuss the consequences.


title: What arrangement of sound waves would be needed to heat air in a typical sized room?

text: From what I understand, sound is simply the jostling of the molecules that make up the air in a specific pattern, widely known as waves. I also know that these are longitudinal waves. If we were to look at "heat", it'd simply be the jiggling of molecules that make up the air that are moving in all directions. Mathematically put, the sum of all the vectors in 3D space that would describe each molecule would equal zero. Now if we are to look at sound waves, we could in principle, describe each molecule with a vector as well. For example: The air that is in the compression portion at a given dt would have a vector that is larger in magnitude than the uncompressed parts. We can also have standing sound waves by reflecting them off the walls of our hypothetical room, let's say is 20 ft^3. Say we have special speakers on each of the three walls that are orthogonal to each other. What overlap of sound waves from all the speakers would be the best to heat the room? Would I be correct in saying that the arrangement of sound waves would have to make all the molecules in the room move in all directions because that is the concept of what heat is? Going further with the example, couldn't we overlap sound waves in such a way that a certain part or parts of the room are colder or hotter? Can't we cancel out sound waves in certain places and therefore efficiently control heating? Couldn't this save energy because you wouldn't have to heat the entire room? Preferably, I'm looking for a mathematical answer.


title: Find the action from given equations of motion

text: Is there a systematic procedure to generally obtain an appropriate action that corresponds to any given equations of motion (if I know that it exists)?


title: Is entropy a dynamical quantity

text: I am confused whether Entropy is a dynamical quantity or not. Gibbs entropy, and quantum mechanical Von Neumann entropy.


title: Is it possible to find the number of gas atoms/molecules in a box when the number is small?

text: Given very low number of particles in a system (e.g. in the 100s), is there a way to accurately measure the number of particles in the system? Assume temperature, pressure and volume is constant and we are using state of the art pressure and temperature sensors. Is it possible to do it from observing the speed distribution of particles?


title: What is the volume of gases liberated when a battery is charged?

text: Apparently Hydrogen/Oxygen are liberated when a Lead-acid battery is charged. If true, how does one calculate the **expected** volume & rate at which each gas is liberated when a battery is charged?


title: Has anyone developed tunable notch filters in the visible RGB wavelengths which have very fast response times?

text: I am interested in whether the combination of a spectrometer and a tunable notch filter with notch frequencies at the visible red, green and blue wavelengths which have very fast response times could be used to deflect laser beams aimed at airplane windows. The purpose of the spectrometer is to detect the wavelength(lambda) of the incoming laser beam and then tune the optical notch filter to the notch wavelength lambda. I am currently reading a paper, "Independently tunable optical notch filter based on double ring resonator structure" by J. Zhang, S. Guo, X. Li published in Optik Volume 124, 2013, pages 1307-1310. This paper describes a tunable optical notch filter based on the Mach-Zender interferometer(MZI) with cascaded double ring resonator structures.Does anyone know whether the optical notch filter described in this paper could be used to deflect red, green or blue laser beams aimed at airplane windows? I am also reading a paper, "Demonstration of a Tunable Microwave-Photonic Notch Filter Using Low-Loss Silicon Ring Resonators" by M. Rasras, K. Tu, D. Gill, Y. Chen, A. White, S. Patel, A. Pomerene, D. Carothers, J. Beattie, M. Beals, J. Michel and L.C. Kimerling published in IEEE Journal of Lightwave Technology Volume 27, No.12, June 15 2009. This paper discusses how to use thermo-optic phase shifters to set the coupling ratios, all pass filter phases and the Mach-Zender input-output couplers. While thermo-optic phase shifters can be tuned in milliseconds, subnanosecond tuning speeds should be achievable using technology currently being explored for the high-speed modulators such as carrie injection or carrier depletion in a reversed biased junction. The tuning speed must be fast as possible for the laser beam deflector to react as fast as possible to laser beams aimed at airplane windows


title: Would a person not be able to distinguish left from right in outer space?

text: Is it true that human in outer space can't differ right side and left side, with no other objects for reference?


title: Why am I not burned by a strong wind?

text: So I was thinking... If heat I feel is just lots of particles going wild and transferring their energy to other bodies, why am I not burned by the wind? When I thought about it more I figured out that wind usually carries some humidity, and since particles of liquid are moving same speed as the wind, they are basically static relative to each other, so no energy is transferred between them (wind and water particles). And if that water sticks to my skin and wind blows, it'll evaporate thus taking energy from my skin and make me feel cold. Thing is, I don't think that's really the case but even if it is, if I somehow dry out the wind, will it burn me if it's strong enough? And winds can reach some pretty high velocities (though I must admit I'm not sure if they are comparable to movement of atoms in warm bodies etc...). So. Bottom line. Can I be burned by wind in some perfect scenario?


title: Vertical column of water for maintaining gas pressure in horizontal water tube?

text: I have an inverted T-Tube arrangement with the vertical tube of a height to add pressure to the continuously moving fluid of the horizontal tube. The purpose is to maintain dissolved gases in the fluid of the horizontal tube. How can this be accomplished with the least amount of energy applied?


title: Length of a curve in D dimensional euclidean space

text: In a book I am reading on special relativity, the infinitesimal line element is defined as $dl^2=\delta_{ij}dx^idx^j$ (Einstein summation convention) where $\delta_{ij}$ is the euclidean metric. Next, if we have some curve C between two points $P_1$ and $P_2$ in this space then the length of the curve is given as $\Delta L = \int_{P_1}^{P_2}dl$ I am having trouble deriving the next statement, which I quote: > A curve in D-dimensional Euclidean space can be described as a subspace of > the D-dimensional spce where the D co-ordinates $x^i$ are given by single > valued functions of some parameter $t$, in which case the length of the > curve from $P_1=x(t_1)$ to $P_2=x(t_2)$ can be written $$\Delta L = > \int_{t_1}^{t_2}\sqrt{\delta_{ij} \dot{x}^i \dot{x}^j} dt \qquad > \mbox{where}\; \dot{x}^i\equiv \frac{dx^i}{dt}$$


title: Issue with running a Van De Graaff generator

text: I am following this online video: http://www.youtube.com/watch?v=fsT69XnjxgU but my Van De Graaff generator is not working. I am using pvc pipes for the body, nylon and electrical tape as two rollers, and rubber band as the belt. On top, I have tried two different cans, one metal sphere but none work. What could be the issue? Thanks.


title: Non-Dimentionalization of equations

text: I am trying to understand a paper in which they mention; " We non-dimentionalize the problem hereafter by selecting mass, length and time units such that the fluid density $\rho\equiv1$ the gravitational acceleration g $\equiv1$ and initial tank depth $h_0\equiv1$ ". The reson for doing this is to calculate a pressure from a plot of y vs $P/\rho g h_0$ . So if the value in the plot is 20, what is the value in Pascal for a $\rho = 1025 kg/m^3$ , $g = 9.81 m/s^2$ and $h = 50 m$ . Thanks in advance.


title: What happens when half of an orbiting object's matter is annihilated?

text: Suppose object A orbits object B, and that there are no outside gravitational influences non-Sun gravity influences, and that object A is completely spherical and its mass is evenly distributed. Now, suppose half of object A's matter is annihilated evenly from the surface of object A. Given the above scenario, would the orbit of object A remain exactly the same? If not, would the variation be due only to the decrease in gravitational influence of object A on object B (e.g., less wobble)?


title: ER = EPR and Time Travel

text: In Maldacena-Susskind paper arXiv:1306.0533, they propose an idea of $$\text{ER = EPR}$$ the relation between the wormhole and the quantum entanglement. which ER means **Einstein Rosen (ER) bridges** , the general-relativity- allowed solutions to the equations of motion that connect far away regions through relatively short wormholes. EPR means the quantum entanglement, where quantum mechanics gives rise to **Einstein Podolsky Rosen (EPR)** correlations. > If **ER = EPR** , then what is Time Travel in this relation, **Time Travel = > ?** I am expecting the answer to be some no-go theorem in the quantum entanglement. I have some basic thoughts and ideas, but willing to hear constructive comments.


title: What are the proper domains of the position and squared angular momentum operator?

text: I am looking at the position operator on a compact set $K \subset \mathbb{R}^n$ and the squared angular momentum operator (so essentially the Laplace-Beltrami operator where I just look at the angular parts so this operator acts on $\mathbb{S}^2$). My question is: What are the canonical domains that we adjust to these two operators?


title: Difference in perception of unpolarized and polarized light

text: What is the difference in perception of polarized light and unpolarized light? What difference does polarized light cause to our eyes?


title: Emf of a rod sliding on two conducting rails

text: The following is the question. ![enter image description here](http://i.stack.imgur.com/2v9Ro.png) Actually, I do not know how to find the answer. I have tried to find out the solution or hints from the lecture note given to me but it is quite difficult to understand. The following is what I have tried. ![enter image description here](http://i.stack.imgur.com/YBctA.jpg) Please give me some hints. Thank a lot. (Ans : 0.14V)


title: Physical applications of the mathematical curvature

text: I was studying multivariable calculus last semester and had one of the topics talking about a curvature, but we had no applications on it. So how does it help in physics? E.g. curvature of curve: $k=|T(t)|/|r'(t)|.$


title: Centre of gravity for a very tall building

text: Imagine a very tall building. Centre of gravity is defined as the point of application of force of gravity, right? Now, is the centre of gravity the same as centre of mass of this building? I am including the change in g with height, so force of gravity varys with height


title: Double double-slit experiment

text: Suppose I have a _double_ double slit experiment. That is, I have an electron gun in the center, that shoots entangled pairs of electrons in opposite directions, one to each double slit. I tried to portray the situation in the following image: ![enter image description here](http://i.stack.imgur.com/ryd8f.png) Since the electrons are entangled, determining the slit through which the electron on the right goes, fixes the slit through which the electron on the left went. From what I know of quantum mechanics, I think that placing a measuring device on the slits on the right, destroys the interference pattern on _both_ double-slits. I find this very unintuitive, even more so than the classical EPR thought experiment, because there is a directly observable effect. You can _see_ the interference pattern disappear on the left after placing the detectors on the right. Am I correct? Does determining the slit through which the electron on the right went destroy the interference pattern on the left?


title: How to block coupling between adjacent waveguides?

text: I have a number of nonlinear waveguides, say six. I don't want coupling between specific two waveguides. Other than keeping them large distance away, what are the other possibilities to ensure no coupling between the waveguides? Will using a specific material composition of the waveguides help?


title: Quantum symmetries that are not classical symmetries

text: An anomaly is a symmetry of the classical action that fails to be a symmetry of the path integral, due to non-invariance of the path integral measure. Does it ever occur that the opposite thing happens, i.e. that the classical action does not possess a symmetry, but the combined transformation of the action and the measure leaves the path integral invariant? Is there a name for such a symmetry of the quantized theory that does not exist in the classical theory?


title: Fermi level with Landau levels

text: So my question is regarding where the Fermi energy is when you have 2D electron gas in an applied magnetic field. My book explains that, using the Landau gauge, you find that the 2D density of states (normally the constant $g_{2D} = m/\pi \hbar^2$) collapses into a series of Dirac delta functions $g_{2D}\delta(E - E_n)$ for the harmonic oscillator levels $E_n = \hbar \omega_c (n + 1/2)$ ($\omega_c$ being the cyclotron frequency) because in this gauge, all the states with the same $n$ are degenerate. This is illustrated in the top half of this picture. They also say that, due to scattering, the peaks can't be defined with such high precision, so you actually get shapes more like the bottom half of that picture. This all makes sense to me. But I get confused when I try to find the Fermi energy for a given density and nonzero magnetic field (at $T = 0$, always...) _without the scattering that widens the peaks_. With no magnetic field, I would simply integrate the 2D DoS from $0$ to $E_F$ and set that equal to my number density, and solving for $E_F$, I'd find it somewhere. But with a magnetic field I run into a problem: My understanding of the Fermi energy is that it's the energy such that all states at or below it are occupied and all those above are unoccupied. But this seems to only offer certain values of the number density for this case! When you integrate over a range of these Dirac deltas, you essentially pick up $g_{2D}\hbar \omega_c$ states for each delta you integrate over, so it seems like in the case of an applied B field, $T = 0$, and no scattering that widens the deltas to something more realistic, you can only have number densities of the form $n_{2D} = m g_{2D}\hbar \omega_c$, where $m$ is an integer. I'm sure I'm not explaining this well and that I'm probably wrong, but what am I missing? Thanks!


title: Inside vs outside humidity

text: I have a practical problem that I want to reduce the humidity inside our house. I know the inside and outside temperature (degrees C) and the relative humidity (both inside and outside). How do I know if opening the window will make it less humid inside? Background. During winter it is cold and humid where we live, both inside and outside. The temperature never gets below freezing outside, although close on some nights. We want to keep the inside as dry as possible and want to know if it is worth airing the room by opening the window.


title: Why did Otto Hahn use neutrons to generate transuranium?

text: In physics textbooks with chapters about nuclear fission there is often a historical introduction about Otto Hahn. That he tried to generate transuranium but discovered the nuclear fission. If you start with uran (atomic number 92) you need to increase the atomic number i.e. the number of protons in the nucleus. However if you just add (naively) a neutron you just and up with another isotope (but not with another element i.e. not transuranium). So which facts were known in 1938 to Otto Hahn which lead him to try neutrons to generate transuranium? What was his reason to try it with neutrons?


title: Slater determinant space vs real space

text: Could someone explain to me what this snippet of text means? > Although it is possible for DMC to be used as a benchmark for quantum- > chemistry methods and vice versa, DMC does not operate in a Slater > determinant space, but rather a real space representation of the > wavefunction. As such, it would require quantum-chemical calculations to be > converged to high accuracy with respect to the basis set size before any > meaningful comparisons could be drawn. DMC stands for Diffusion Monte Carlo. I know what a Slater determinant is, but what is a Slater determinant space? And real space is just $\Psi(\mathbf{r}, \mathbf{t})$, right?


title: Properties of the Faraday tensor for constant fields

text: I'm doing a special relativity past exam paper and have got caught up with something that I hope someone can help me with! I have to show that for **constant fields** , the magnitude of A, the acceleration 4-vector, is constant. Given that the 4-force in the presence of electric and magnetic fields is $f^{\mu}=eF^{\mu\nu}U_{\nu}$ we can use $A^{\mu}=\frac{f^{\mu}}{m}$ to get $\frac{d|A^2|}{d\tau}=2A_\mu\frac{dA^\mu}{d\tau}=2\frac{e}{m}F^{\mu\nu}A_{\mu}A_{\nu}$. Now apparently this last expression equals zero but I cant work out or find any justification for this, can anyone help?


title: Calculating Fraunhofer diffraction patterns

text: How does one calculate the Fraunhofer diffraction pattern for the following arrangement of slits: |...|...........|...| ..a.....3a......a (Four slits arranged linearly, spaced a distance a, 3a and a apart.) The width of the single slits can be neglected, so that the transmission function can be expressed as sum of delta-functions.


title: Why doesn’t gravity break down in a large black hole?

text: By popular theory gravity didn’t exist at the start of the Big Bang, but came into existence some moments later. I think the other forces came into existence a little latter. When a black hole crushes matter to a singularity (infinite density), at some point shouldn’t the forces cease to exist including gravity? Kent


title: Energy momentum tensor from Noether's theorem

text: in the book Quantum Field Theory by Itzykson and Zuber the following derivation for the stress-energy tensor is proposed (p.22): Assume a Lagrangian density depending on the spacetime coordinates $x$ only through fields and their gradients. Under a translation we have $$\mathcal L (x+a)=\mathcal L[\phi_i(x+a),\partial_\mu\phi_i(x+a)].$$ Consider an infinitesimal $x$-dependent transformation $$\delta\phi_i=\delta a^\mu(x)\partial_\mu\phi_i(x),$$ $$\delta\partial_u\phi_i(x)=\delta a^\nu \partial_\nu\partial_\mu\phi_i(x)+\partial_\mu[\delta a^\nu (x)]\partial_\nu \phi_i(x).$$ The proof then proceeds with variation of the action and integration by parts. But why do we consider an $x$-dependent local transformation instead of a global transformation?


title: how calculate resistance between two points for arbitrary resistor grid?

text: I need to calculate resistance in an electrical grid.   Grid is a graph. Edges of a graph are resistors. How calculate resistance between any two vertexes for **arbitrary** graph?   I know Kirchoff law and Ohm law. I can calculate resistance between any points for given grid, but not for arbitrary. Is there programs to solve this problem?   Actually, my goal is to write algorithm that calculate resistance for arbitrary graph.   Any help is appreciated.


title: In the dual-slit experiment, can the time it takes for a photon to arrive reveal the path it took?

text: It seems like this experiment is possible:   1. Fire a photon, and start a timer   2. The photon travels through the slit(s)   3. The photon strikes a plate -- the timer is stopped. Based on where on the plate the photon struck, the time it took, and the speed of light, it seems possible to deduce through which slit the photon passed. Or, would the times correspond with a direct line from the source to destination?


title: Point of Lagrange multipliers

text: I am trying to understand how for a constrained system the introduction of Lagrange multipliers facilitates the incorporation of the holonomic constraints. I am using _Classical Mechanics_ by John Taylor (2005). On page 276 it says that $$\delta S = \int ( \frac{\partial L}{\partial x} - \frac{d}{dt} \frac{\partial L}{\partial \dot x})\delta x dt + \int ( \frac{\partial L}{\partial y} - \frac{d}{dt} \frac{\partial L}{\partial \dot y})\delta y dt = 0 \quad (1)$$ for any displacements $\delta x$ and $\delta y$ _consistent with the constraints_. This I understand. By writing the constraint equation $f(x,y)=C$ as $$\delta f = \frac{\partial f}{\partial x}\delta x + \frac{\partial f}{\partial y}\delta y = 0, \quad (2)$$ this constraint is incorporated in the Hamilton principle, giving $$\delta S = \int ( \frac{\partial L}{\partial x} +\lambda (t)\frac{\partial f}{\partial x} - \frac{d}{dt} \frac{\partial L}{\partial \dot x})\delta x dt + \int ( \frac{\partial L}{\partial y} +\lambda (t)\frac{\partial f}{\partial y} - \frac{d}{dt} \frac{\partial L}{\partial \dot y})\delta y dt = 0. \quad (3)$$ At this point I would expect this equation hold for all $\delta x$ and $\delta y$, though the author notes that still this integral is zero for _any displacement consistent with the constraints_. What is the point of putting it in the integrand? Apparently, by some choice of $\lambda (t)$, this equation can be reasoned to lead to two (new) Lagrange equations in this 2-coordinate case, i.e. $$\frac{\partial L}{\partial x} + \lambda \frac{\partial f}{\partial x} = \frac{d}{dt}\frac{\partial L}{\partial \dot x} \quad (4)$$ $$\frac{\partial L}{\partial y} + \lambda \frac{\partial f}{\partial y} = \frac{d}{dt}\frac{\partial L}{\partial \dot y} \quad (5)$$ which is said to still have to be supplemented with the original constraint equation $f(x,y)=C$ in order to have enough equations to be able to solve the system. I understand the basic ideas that one needs enough equations for the number of unknowns that a problem has, and that - conveniently - Lagrange multipliers used in this way can be related to the magnitude of constraint forces. But why make the original formulation more complicated when you still need to supply the original constraint equation? Also, to include the constraints (by means of $\delta f$) in $\delta S$, would you really need to multiply with an unknown $\lambda$? Why not set $\lambda=1$? I suspect that my difficulty in understanding this is related to me not understanding either why, in the more purely mathematical problem of maximizing $h(\vec x)$ given $g(\vec x)=0$ by means of $\Lambda(\vec x,\lambda)=f(\vec x)-\lambda g(\vec x)$, one needs more than just the equations $\frac{\partial \Lambda}{\partial \vec x}=\vec 0$ (*) (viz. $\frac{\partial \Lambda}{\partial \lambda}=0$, too) Isn't the problem already contained in Eqs. (*)? Thanks very much for helping me out.


title: Boundary Condition for Perfect Conductor in Uniform Magnetic Field

text: When I was studying the perfect conductor scattering ( **Section 10.1** ) in Jackson's book, I was confused by the calculation for magnetic dipole induced by the incident wave. He simply said like "set the $\mu$=0 in **equation (5.115)**......" Actually, **equation (5.115)** $$ M=\frac{3}{\mu_0}\frac{\mu-\mu_0}{\mu+2\mu_0}B_0 $$ is derived under the boundary condition that the surface current=0, i.e. the tangential component of **H** is continuous. But for perfect conductor, this condition is not valid. So any body can help me to figure out how can I get the magnetic dipole in this situation? Thanks.


title: At what frequency would a quartz crystal shatter?

text: Would it be possible to emit a sound at such a frequency that a quartz crystal inside of a watch would shatter? (Rendering the watch useless)


title: Hall conductivity from Kubo: Bulk or edge?

text: Using the Kubo formula, Thouless, Kohmoto, Nightingale, and den Nijs (TKNN, PRL 49 405-408 (1982)), proved that upon summing all the contributions of the filled states of an insulator, the Hall conductivity must be an integer (the Chern number) times e^2/h. Their theorem uses periodic boundary conditions along x and y and therefore avoids the discussion on edges. An intuitive picture of the quantum hall effect (QHE) is, however, based on currents on the edge (people always talk about skipping orbits going around the edge). My question now is: Why do some people say that the edges are the whole story of the QHE whereas the formulation of TKNN make it look like this is not important? Alternatively, how can I convince myself that the conclusions from the TKNN derivation hold also for open boundary conditions? Also, from the TKNN derivation, the obtained Chern number seems to me a property of the bulk. Is there a simple proof that translates this Chern number to the number of edge channels in an open boundary condition setting? Many thanks.


title: Finding the force of an object?

text: Sorry if my initial question was seen as a "homework question" but I don't really know how to go about asking this type of question. I have a feeling I have to use Newtons law but there is no acceleration, all I have is mass and distances. I wasn't really asking you to give me an answer but more a less a starting point or pointer to how I would go about resolving it. ![diagram](http://i.stack.imgur.com/fcA6d.png)


title: Using the Normalization Condition with Wavefunction

text: I'm very confused with this problem and I was looking for some guidance. $$\psi(x) = Ae^{ikx}e^{-x^2/2a^2}$$ Use the normalization condition to find A. So I understand that you use the normalization condition where $$\int_\infty^\infty|\psi(x)|^2 dx = 1$$ But the integral is really complicated and doesn't really lead me anywhere. Can anyone suggest a hint? Am I missing something? $$\int_\infty^\infty (Ae^{ikx}e^{-x^2/2a^2})^2dx$$ $$\int_\infty^\infty (A^2 e^{2ikx}e^{-x^2/a^2})dx$$


title: Terminal Velocity Question

text: **Question** : A ball of mass m is thrown vertically upward with initial velocity $v_0$. Air resistance is proportional to the square of the velocity. If the terminal velocity of the ball is $v_t$, show that when the ball returns to its original position its velocity satisfies $\frac{1}{v_1^2}=\frac{1}{v_0^2}+\frac{1}{v_t^2}$ I began by splitting the whole motion in two: i) Upwards and ii) Downwards... i) Upwards: Let y be the upward displacement from the starting point y=0, h be the maximum height of the ball, k is a constant. Using Newton's second law of motion, **F=ma** , I got: ${mv}\frac{dv}{dy}=mg-mkv^2 \Rightarrow {v}\frac{dv}{dy}=g-kv^2 \Rightarrow \int dy = -\int \frac{vdv}{g+kv^2}$ Once I insert the bounds for the integrals I get: $h=\frac{-1}{2k}log(\frac{kv_0^2+g}{g})$ ii) Downwards: Now this is where I've gotten stuck and the concept of terminal velocity has gotten me very confused. Can anyone tell me where to proceed from here (if what I've gotten so far is correct) in order to get $\frac{1}{v_1^2}=\frac{1}{v_0^2}+\frac{1}{v_t^2}$ HELP!!!


title: Evaluating the Ricci tensor effectively

text: If given a metric of the form $$ds^2=\alpha^2(dr^2+r^2d\theta^2)$$ where $\alpha=\alpha(r)$, then can one immediately conclude that $$R_{\theta\theta}=r^2R_{rr}$$ where $R_{ab}$ is the Ricci tensor, without doing any explicit calculations? I can show that this is true by the long- winded way of computing both explicitly, but it would seem that there may be a more elegant way?


title: What experiments have been proposed to discriminate between interpretations of quantum mechanics?

text: There are a lot of potentially correct interpretations of quantum mechanics. While I've heard descriptions of a lot of them, I've never heard of an experiment being done to test any of them aside from the old Bell test experiments. What ongoing experiments are looking to test interpretations of quantum mechanics? What about experiments that have been proposed, but not actually performed?


title: Wavefunction operators and the observable

text: So I got this from the exam I had yesterday. I couldn't really answer it other and it played on my mind through the night Show that if a wave function $\psi$ , is an eigenfunction of an operator [Q], then the observable Q is sharp for that wave function. I said that if it is an eigenfunction then performing the operator must be a whole multiple of the wavefunction and then tried to bull my way through it. What would you guys have accepted as an answer. I mean I know that $\Delta Q=\sqrt{\overline{Q^2}- \overline{Q}^2}$ But how do I translate my knowledge across to give an answer that if the observable is sharp this reduces to 0?


title: How multiple objects in contact are resolved in an inelastic collision, when edge normals don't "line up"

text: In a case I understand, let's say I have an object A moving at velocity V toward 3 objects in contact B, C, and D: ![enter image description here](http://i.stack.imgur.com/g83ZY.png) The momentum of A is the mass of A times its velocity. To figure out how the inelastic collision ends up when A hits B, I sum the masses of A, B, C, and D, and I divide the old momentum of A by the sum of those masses. That is the velocity all 4 objects end up with. Easy and extendable! But when objects are in contact along edges with normals not parallel to the original momentum: ![enter image description here](http://i.stack.imgur.com/Nrjgw.png) Such as here, with C mucking up the nice neat solution I had for the system above. I know that when objects collide, they are pushed away along the normal of the edge of contact. So D is going to move up and right at an angle, and presumably C and E need to get pushed downward to conserve momentum. Further, I still have to deal with my momentum from A pushing things to the right. I'm pretty sure D will be moving to the right slower than A, B, and C. And E won't be moving to the right at all, assuming there's no friction. Right? What is the summed mass to use in the equation I used to solve my first problem? Can I simplify this into equations dealing with the "branching off" of the "main path" (D branching off, and C + E branching off?)? I need to understand this as a general case, not simply the solution to this one problem... Any possible set of convex polygons in contact being hit. How does this work? I'm also wondering how to deal with this with partially elastic collisions... Seems like it'd get pretty crazy, especially with systems more complicated than my 2nd example.


title: Reflection of a Plane Wave at Oblique Incidence

text: I am still trying to prepare for my electromagnetics preliminary exam, but I find myself stuck on understanding the origins of Eqs. 10.105a, 10.105b, 10.106a, and 10.106b on page 453 of Elements of Electromagnetics 3rd edition (these can be seen on page 44 of the PDF located here). The equations, reproduced below, are $$ \mathbf{E}_{is}=E_{i0}\left(\cos\theta_{i}\mathbf{a}_{x}-\sin\theta_{i}\mathbf{a}_{z}\right)e^{-j\beta_{1}\left(x\sin\theta_{i}+z\cos\theta_{i}\right)}\\\\\mathbf{H}_{is}=\frac{E_{i0}}{\eta_{1}}e^{-j\beta_{1}\left(x\sin\theta_{i}+z\cos\theta_{i}\right)}\mathbf{a}_{y}\\\\\mathbf{E}_{rs}=E_{r0}\left(\cos\theta_{r}\mathbf{a}_{x}+\sin\theta_{r}\mathbf{a}_{z}\right)e^{-j\beta_{1}\left(x\sin\theta_{r}-z\cos\theta_{r}\right)}\\\\\mathbf{H}_{rs}=-\frac{E_{r0}}{\eta_{1}}e^{-j\beta_{1}\left(x\sin\theta_{r}-z\cos\theta_{r}\right)}\mathbf{a}_{y} $$ These represent the phasor forms of the incident electric field, incident magnetic field, reflected electric field, and reflected magnetic field, respectively. Here, the direction of propagation of the wave is at an angle $\theta_i$ with respect to the normal to the surface of the boundary between two lossless dielectric materials. So I tried to understand the origins of these equations, as I've been doing through the course of going through this book, and the results of my work appear below ![My derivation](http://i.stack.imgur.com/YcDc1.png) I think I have the right idea, but my signs are off. But it's confusing because I have the sign to the $x$ component of $\mathbf{E}_{i}$ correct, but not the sign to the $z$ component of $\mathbf{E}_{i}$ But both of those terms just came from the drawing and I don't understand where a negative sign could come in for the $z$ component. Perhaps my $r$ vector, given by $$r = x\mathbf{a}_{x} + y\mathbf{a}_y + z\mathbf{a}_z$$ is somehow wrong? Once again, any help is appreciated. EDIT: Here's what the image should look like. ![Updated graphic](http://i.stack.imgur.com/hWDy0.png) Therefore the derivation I had previously provided becomes ![My updated derivation](http://i.stack.imgur.com/E9Dtj.png) But note that I'm off by a minus sign in the complex exponential part. Looking at the updated diagram, I think it's easy to see that $\mathbf{k}_i$ has a $+\mathbf{a}_x$ component and a $+\mathbf{a}_z$ component. So that leads me to believe that my $\mathbf{r}$ is wrong. Should it perhaps be $\mathbf{r} = -x\mathbf{a}_x + y\mathbf{y} - z\mathbf{a}_z$? If so, why?


title: Which is this formula Feynman talks about in the QED book?

text: I am reading the fantastic QED Feynman book. He talks in chapter 3 about a formula he considers too complicated to be written in the book. I would like to know which formula he talks about, although I have a vague idea. Please don't shy away from writing it properly, or giving a non-popular description or reference, I am conversant with basic QM, although not with the variational formulation nor QFT in general. He talks about a formula for E(A to B), the amplitude for an electron to go from B to A in a space-time diagram), and states that it > (...) _can be represented by a giant sum of a lot of different ways an > electron could go from point A to point B in space-time. The electron could > take a 'one-hop flight' going directly from A to B; it could take a 'two-hop > flight', stopping at an intermediate point C; it could take a 'three-hop > flight'_ (...) _The amplitude for each 'stop' is represented by_ $n^2$ (...) > _The formula for E(A to B) is thus a series of therms_ (...) _for all > possible intermediate points_ Additionally, I would like to know whether this book is another literal transcription of some lectures video footage, as it happens with, for instance, the 1964 Messenger Lectures at Cornell (published as _The Pleasure of Finding Things Out_ ). This is secondary and can be answered in a comment for instance, if you like it so.


title: How fast must a torus spin to have a felt gravity?

text: Given a rotating _torus_ (like a gyroscope or a rotary) with a radius of _1km_ , how fast must it rotate to have someone inside the torus feel normal gravity of _1G_?


title: Airplane with banner in a windy day

text: Will the banner of this airplane be always in the proper direction if the airplane flies in any direction on a windy day? ![enter image description here](http://i.stack.imgur.com/VSRhi.jpg)


title: Four Fermion Interactions

text: Given an action with a term like \begin{equation}S_{I}\sim \int\int (\psi^{\dagger}\psi)V(\psi^{\dagger}\psi)\end{equation} How do you evaluate this with a Fermionic path integral? I know the fields are Grassmann valued so does that mean that \begin{equation}e^{-S_{I}}=1-S_{I}\end{equation} because higher powers of $\psi$ and $\psi^{\dagger}$ are zero? The reason I'm confused is that there is an integral involved, so when you consider $n$ powers of the action, you have to do $n$ integrals over $n$ different regions of spacetime. (i.e. is $(S_{I})^{n}=0$ for $n>1$?)


title: Photons hitting the Black Hole Singularity

text: I was watching one of Dr. Susskind's Lectures on String theory and he was talking about black hole singularities (see the 40 minute mark of this lecture http://www.youtube.com/watch?v=SkEvsxg5Tu4&list=PL3E633552E58EB230). I'm curious, given that the singularity becomes a 'time' and not a 'place', if at the moment you cross the horizon, you emit a photon, according to the explanation given, does this mean that it would take an infinite amount of time for the photon to hit the singularity? And this would also mean that the faster you move after crossing the horizon, the longer it would take to hit the singularity (which seems kind of cool)? I'm just curious to know if I'm interpreting the explanation correctly.


title: Reciprocating vs. rotating machines

text: Why is the RPM achieved on a reciprocating machine in general lesser than that obtained in a rotating machine? For example, an IC engine typically provides an RPM of about 2500 while a turbine can go as high as 80,000 RPM. The inertia and the repetitive change in direction of motion in case of reciprocating machine is one aspect, but is there an analytical justification/limitation on the RPM?


title: Can you speed up radioactive decay of plutonium?

text: We all know the issue of deep geological repositories for fuel rods. Is there a currently feasible way to speed up the rod's decay to render them harmless in less than 10 years?


title: Logic behind topological orders

text: Long-range entanglement (LRE) is the main feature of topological orders. The string-net condensation model was constructed to exhibit LRE. But the many-body systems of such models do not look like any earthly materials at all but are closer to quantum gravity models. In quantum gravity, nobody sees the detailed structure beyond the Planck scale. But in condensed matter, the microscopic structures are "crystal" clear. Then how can such strange models be used to explain LRE in laboratory materials?


title: Lagrangian for Goldstone mode + topological excitation

text: The XY-model Hamiltonian is the following, $${\cal H}~=~-J\sum_{\langle i,j\rangle} \cos (\theta_i -\theta_j).$$ The Goldstone mode corresponds to term $(\nabla \theta)^2$ in the effective Lagrangian. > Then what's the form of effective Lagrangian that produces both Goldstone > mode and topological excitation (vortices)? > > And how to derive the effective Lagrangian from the XY-Hamiltonian?


title: Apparent weight on a hill and in a ditch

text: first time posting. I'm trying to solve these 2 problems. I have a solutions I just don't know if I'm right...seems odd that they are the same answer... > 1) If you are driving over a hill of radius of curvature **r**. How fast > must you drive for you to lift up off your chair. **My solution** : Since at that point **Fn** is 0, then **Fnet = Fc = mg** so we have **mv^2/r = mg** which rearranges to give you a required speed of **v=sqrt(gr)** ...this is fine, it makes sense to me. It's the fact that I got the same answer for the next one that is throwing me off > 2) If you are driving your car into a dip in the road of curvature **r**. > How fast must you drive for you to feel twice as heavy. **My solution** : ok so here I figured **|Fn| = 2mg** since it's normally **mg** so then I have this... **Fnet = Fc = Fn - Fg** .... **mv^2/r = 2mg - mg = mg** so again I get **v=sqrt(gr)** Like...it seems right to me...?


title: super-jump air balloon

text: We have the following objects:   * a 80 kg person   * a rope of negligible weight   * a balloon filled with helium, which can lift for around the same weight, 80kg. My question is, which of the following outcomes will occur (without considering weather conditions)?   1. The balloon can actually lift a bit more than 80 kg.   2. The balloon cannot lift more than 79.9 kg Most importantly, suppose this person manages to do a 10 meters jump, on the way down, would he accelerate too fast for a "safe" landing? * * * ![enter image description here](http://i.stack.imgur.com/6rcI5.jpg)


title: Are these guard rails truly strong enough?

text: I'm lay about physics, but I'd like to learn about it behind this question. A friend criticises and claims that the single strip of traffic barriers (as shown below) would fail and is useless. For example, can it veritably protect and stop vehicles from falling off the bridge, especially provided that many double-decker buses and large trucks traverse them at speeds of around 100 km/hr.   Is she right? Yet what does physics reveal? The left picture: Ting Kau Bridge. I purposely include a double-decker Citybus for reference.   The bottom picture: Tsing Ma Bridge, depicted with a typical Hong Kong taxi. ![enter image description here](http://i.stack.imgur.com/AaVrU.png)


title: Isentropic Processes

text: I'm having trouble understanding why reversible adiabatic processes are isentropic. I understand that in a reversible adiabatic process there is no heat exchange and so $dQ = TdS = 0$. However, if I have a thermally isolated ideal gas in a piston, and I reversibly compress the gas, even though there is no heat exchange (the process is reversible and adiabatic), I have increased the temperature of the gas! Doesn't that mean I've increased it's entropy? It occurs to me that the decrease in volume probably exactly matches the increase in temperature in terms of entropy changes so that the total entropy change is 0. But still, if I try and do some calculations, I'm stuck. Say I change the volume from V1 to V2 by reversibly compressing the piston. Aren't I changing the pressure and temperature simultaneously as I do that? How do I integrate the ideal gas law here?


title: How to calculate the force acting on a magnet due to a solenoid

text: Can I assume both magnet and solenoid as magnetic dipole and use Coulombs law to find the force acting the magnet?


title: Where to learn Temperature Dependent Conductivity induced by Electron-Phonon Interaction?

text: I want to learn how to calculate the temperature dependent conductivity induced by electron-phonon interaction. I know in low temperature, the resistance in metal $\rho$ is proportional to $T^5$, $T$ is the temperature. In high temperature region, $\rho$ is linear to $T$. Could someone tell me which text book and which chapter can I find related calculations? Thanks in advance.


title: Coulomb gauge fixing and "normalizability"

text: **The Setup** Let Greek indices be summed over $0,1,\dots, d$ and Latin indices over $1,2,\dots, d$. Consider a vector potential $A_\mu$ on $\mathbb R^{d,1}$ defined to gauge transform as $$ A_\mu\to A_\mu'=A_\mu+\partial_\mu\theta $$ for some real-valued function $\theta$ on $\mathbb R^{d,1}$. The usual claim about Coulomb gauge fixing is that the condition $$ \partial^i A_i = 0 $$ serves to fix the gauge in the sense that $\partial^iA_i' = 0$ only if $\theta = 0$. The usual argument for this (as far as I am aware) is that $\partial^i A'_i =\partial^iA_i + \partial^i\partial_i\theta$, so the Coulomb gauge conditions on $A_\mu$ and $A_\mu'$ give $\partial^i\partial_i\theta=0$, but the only sufficiently smooth, _normalizable_ (Lesbegue-integrable?) solution to this (Laplace's) equation on $\mathbb R^d$ is $\theta(t,\vec x)=0$ for all $\vec x\in\mathbb R^d$. **My Question** What, if any, is the physical justification for the smoothness and normalizability constraints on the gauge function $\theta$? **EDIT 01/26/2013** Motivated by some of the comments, I'd like to add the following question: are there physically interesting examples in which the gauge function $\theta$ fails to be smooth and/or normalizable? References with more details would be appreciated. Lubos mentioned that perhaps monopoles or solitons could be involved in such cases; I'd like to know more! Cheers!


title: How is energy transferred between atoms in a collision?

text: Consider two bare protons. One (A) is stationary (relative to some arbitrary massless observer); the other (B) is approaching A at 1 m/s. When they collide, I assume that they bounce. What is the precise mechanism by which energy is transferred from A to B? For bonus points: Is the mechanism the same if each proton has one electron (that is, a normal hydrogen atom)?


title: Interpretation of the 1D transverve field Ising model vacuum state in a spin-language

text: The 1D transverse field Ising model, \begin{equation} H=-J\sum_{i}\sigma_i^z\sigma_{i+1}^z-h\sum_{i}\sigma^x_i, \end{equation} can be solved via the Jordan-Wigner (JW) transformation (for further reference about the explicit form of the JW transformation, one may look at the following link). Here, $\sigma^z,\sigma^x$ are the Pauli matrices and the sum is carried over an infinite chain (consider $J>0$). After the JW non-local mapping to spinless fermionic operators (and a Bogoliubov transformation), one obtains a quadratic Hamiltonian that is diagonal in momentum space : \begin{equation} H=\sum_k \epsilon(k,h)(a^\dagger_ka_k+1/2) \end{equation} Where we express $h$ in units of $J$ and $\epsilon(k,h)$ represents the dispersion relation of the excitations of the system and depends on the transverse field $h$ and wave-vector $k$. $a^{(\dagger)}_k$ is the usual annihilation (creation) operator, which annihilates (creates) a spinless fermion with momentum $k$. For $h<1$ this dispersion relation is gapped, meaning that there is a finite energy cost to create $a^\dagger_k$ excitation with respect to the vacuum ground-state $\left|~0\right>$. Here is my **question** : How do you explicitly write down this $\left|~0\right>$ ground-state in terms of the original $\sigma^z$ basis ? Explicitly : For $h=0$, we know that the ground-state corresponds to all spins up or down, i.e. $\left|~0\right>$=$\left|~\cdots\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\cdots\right>$ or $\left|~0\right>$=$\left|~\cdots\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\cdots\right>$. Now, how do you find the explicity form of $\left|~0\right>$ for $h\neq0$ ?


title: Would a magnetic rod through Mars shield it from solar wind?

text: Would a magnetic rod going all the way through Mars shield it from solar wind?


title: A question on charge renormalization in QED

text: Let us work with charge renormalization in QED. Consider 2-point photon correlation function $\Pi_2(q^2)$ at one loop level. We normalize the coupling constant at $q^2=0$ (point of normalization). What is the effective coupling constant $e(q^2)$ at $−q^2 ≫ m^2$ at one loop level. Does coupling constant decrease or increase with $q^2$?


title: Compressible fluid flow through a branched pipe junctions

text: Backdrop: Designing a dust extraction system (LEV) with a branched junction. What principles allow for the calculation of volume flow rate and pressure at the inlets if the volume flow rate and pressure is known at the outlet using a _compressible fluid_? I am familiar with Continuity and Bernoulli equations using incompressible fluids but clearly in this state they are inadequate here. Alternatively is anyone aware of simple approximations for such calculations? ![enter image description here](http://i.stack.imgur.com/LfcLe.jpg)


title: A simple logical problem regarding pressure and force

text: Suppose a 2 meter long pipe, with area of cross section is $1 m^2$. The pipe is vertically placed, with the bottom end closed. Inside the pipe, at the closed bottom end, there is $1 m^3$ of air, closed with a disc. This disc is movable, like a piston in a cylinder (the cylinder is the pipe). Now, I'm pressing on the disc downwards and compressing the air to 5 bar of pressure (the volume of the air decreased). Now I'm pouring water on to the disc, $1000$ $kg$ of water. Now in the pipe,there is the compressed air at the very bottom, then the piston on it, and on the piston, filled with water of $1 m$ height ($1 m^3$ _i.e._ , $1$ ton of weight). If I release the pressure of the compressed air at the bottom very suddenly, UP TO WHAT HEIGHT will the water column will throw off(like a bullet)? Consider the atmospheric pressure also, which is acting downwards,when it is shooting up..


title: How does the correlation length of weather emerge?

text: The question is pretty simple: If I know the weather where I stand, I can estimate the weather 5 meters or 1 km away away pretty well, but I'll have a hard time guessing what the weather is, say, 50 km away. Therefore, it seems that the climatic system has a length-scale. Where does it come from? Navier-Stokes equations do not feature an internal length scale, and it doesn't seem that the scale comes from earth's radius either.


title: Hubble's law for the Kasner solution

text: I'm puzzled with the following question: find an analog of the Hubble's law for the Kasner solution. Kasner metric is a solution to the vacuum Einstein equations $$ds^2=dt^2-\sum_{i=1}^3a^2_i(t)(dx^i)^2$$ with coefficients $a_i(t)=t^{p_i}$, where $p_i$ are constant parameters satisfying $\sum_ip_i=1$ and $\sum_i (p_i)^2=1$. This solution describes anisotropic space. It is known that in the Robertson-Walker metric, when coefficients along all axes are the same $a_i=a$ wavelengths of a photon at times $t$ and $t_0$ are related by the scale parameter $a(t)$ as $$\frac{\lambda(t)}{\lambda(t_0)}=\frac{a(t)}{a(t_0)}$$   this is true if the rate if the space expanding $\dot{a}/a$ is significantly smaller then the frequency of the photon. The question is how to generalize this statement in the case of the Kasner metric. A natural idea is to suppose that the statement is generalized directly to the propagation along "main" axes $$\frac{\lambda_i(t)}{\lambda_i(t_0)}=\frac{a_i(t)}{a_i(t_0)}$$ and "linearly" for a generic direction, i.e. for $\vec{\lambda}=\lambda_i \vec{e_i}$ $$\frac{|\vec{\lambda}(t)|}{|\vec{\lambda}(t_0)|}=\frac{\dot{a}_i(t)\lambda_i}{a_i(t)\lambda_i}$$ However, I'm not able to confirm this conjecture by any quantitative argument. I've tried to show that the EOM in a curved space-time $\nabla^\mu F_{\mu\nu}=0$ admit a solution of a kind $A_\mu(x)\sim e_{\mu}\exp{\left(i g_{\mu\nu}q^\mu x^\nu\right)}$ with $q^\mu$ being a rescaled "minkowskian" momentum $q^i(t)=\frac{k_i}{a_i(t)}, \eta^{\mu\nu}k_\mu k_\nu=0$ but failed to succeed. After some time I've realized that one should probably look not for an exact solution of this type (even in the Robertson-Walker case the relation $\lambda(t)\sim a(t)$ is approximate) but for a "high-frequency" approximate solution. But still I'm not currently able to find it. So my questions are: 1) Is the suggested generalization of the Hubble's law correct? 2) If so, how to provide quantitative evidence for it? I would be satisfied even with calculations for a massless scalar field if they are simpler to perform in order to achieve the goal.


title: Is voltage important when creating a magnetic field?

text: Would a DC circuit that has a high current and low voltage have a powerful magnetic field? I'm trying to create a powerful solenoid. In order to create a powerful magnetic field, I'm focusing more on the current(I). Increasing the number of turns would cause a stronger field, but at what cost?


title: Why is the following gauge transformation singular?

text: Suppose I have a single particle Hamiltonian: $H=\frac{p^2}{2m}+\frac{\hbar k_0}{m}\vec{\sigma}\cdot\vec{p}$, either for boson or fermion. I do a gauge transformation $e^{-ik_0\vec{\sigma}\cdot\vec{r}}$ to the field operator, then the Hamiltonian will be $H'=\frac{p^2}{2m}$. My question is, is there anything wrong with this gauge transformation?


title: Entropy of a chain

text: A chain has N segments which can be oriented in either the x or y directions. For each segment oriented along y, there is an energy penalty of $\epsilon$. We also know the end segment is at $(L_x, L_y)$. How can we define the entropy S as a function of E,N,Lx, and Ly? I know $S=k*ln(\Omega)$, but is can you actually solve for the possible arrangements of the chain? I'm assume overlapping and crossing cannot be done.


title: Is it really impossible for Bumblebees to fly?

text: According to some source or other (I forget which now) it is theoretically impossible for bumblebees to fly by virtue of their size/bulk/aerodynamic properties. Is this old adage apocraphyal or true? And if its true, how come they do fly?


title: Does the Holographic Principle disagree with the eternal inflation model?

text: Does the Holographic Principle disagree with the eternal inflation model or it's just a new way to study quantum gravity?


title: Speed Distribution of The Particles

text: I want to know the distribution of the particles's speed. The particles what I mean are nucleons and electrons of element. Consume there is 1kg of iron on room temperature and it's shape is sphere. And also it's in the vacuum environment. **Question.** How can I get the distribution of particles's speed?


title: Electromagnetic Radiation

text: If there was a way to see the electromagnetic waves that is all around us, i.e. see the electric and magnetic fields spreading away from an antenna say, how would it look? I mean if we were to freeze the picture to a single instant of time, and somehow construct a 3D diagram of the spreading electromagnetic disturbance how would it look. Considering that in nature, a change in electric flux creates a change in magnetic flux and vice versa and they sustain each other as they spread out, how would the picture look? starting from a single circle of electric field, would it create infinite circles of magnetic field perpendicular to itself around its circular periphery and those magnetic fields in turn give rise to infinite circular electric fields around themselves? This is a poor explanation possibly but the point is just how the EM wave propagates and how it looks like in reality?


title: How can there be a path to ground with thick shoes and a carpet?

text: I'm connecting a test light to one pin of an halogen lamp. When I touch the metallic part on the back of the test light, the light glows, as it is supposed to. However, I have thick shoes and I am standing on a carpet. How could I possibly provide a path to ground? I have tried:   * to stand on a plastic sheet -- the light still glows with the same intensity   * to touch a radiator with the other hand -- the light is much more intense   * to interpose my shoe between the metallic part of the test light and me -- the light does not glow   * to interpose a paper sheet -- the light still glows Thank you for any response


title: Uncertainty-principle and the Maxwell formalism of electromagnetic waves

text: An electromagnetic wave (like a propagating photon) is known to carry it's electric and magnetic field-vectors perpendicular and each depending on the differential change of the other thus "creating" each other and therefore appearing in-phase and reaching their minima/maxima together. I'm interested to know whether there was any uncertainty as a principle discussed in the underlying principles of the Maxwell equations, like $\Delta E \Delta B \geq \hbar$. I appreciate links, hints and answers.


title: given the infinitesimal generator can we deduce the symmetry?

text: for example given the differential operator $ \partial _{x} $ we know that it belong to the translation $ y=x+a $ for some 'a' given the differential operator $ x\partial _{x} $ we know that we are dealing with dilations $ y=ax$ however for any particular first order differential operator $ a(x) \partial _{x} +b(x) $ for some polynomials a(x) and b(x) can we obtain the symmetry ??


title: Equivalences and derivations in Newtonian/Classical Mechanics

text: In Newtonian mechanics there are several "laws" and axioms:   * Newton's Laws   * Conservation of: Mass, Energy, Momentum, Angular Momentum  I know some are equivalent (e.g., conservation of momentum and Newton's second law) and some are derived from others (e.g. first Newton's law is a special case of the second). What is the entire directed graph of derivations here, and as a result, what are the minimal sets of assumptions for Newtonian Mechanics?


title: Discrete Fourier Transform: Why do we only consider a full cycle?

text: I am studying Fourier analysis and I am still new to this topic. If I understand that the maximum frequency that can be used in a DFT is given by $N/2$, where $N$ is the number of samples in our discrete signal. This always assumes that the sinusoid is an harmonic of $f$, where $f = 1$ with $\sin(2 \pi f t)$. In other words, we assume the first harmonic is a full cycle. How can this be justified? I am trying to make sense of this. Why don't we use a half cycle for example as in the case of DCTs? Why does the 1st harmonic have to be a full cycle? EDIT: asking the question made me think;-). I would actually say that this is because the discrete signal is considered to be periodic and thus the harmonics have to be periodic as well thus $f = 1$. But could someone confirm please that this is the right reasoning?


title: How do electrical devices suck electricity?

text: If the electric potential is 220V and some device needs 1500watts then how does it suck exactly that amount of electrical energy from outlet?


title: Is there a way to create a flickering frequency to be dependent on speed of the person looking at it?

text: Is there a way to make a screen or a road sign flash at different rates, depending on the velocity of the observer looking at it?   * I would like to achieve a state where two observers going at different speeds would see the screen flash at different rates at the same time.   * another thing i would like to check if possible is - is there a way to make the observer see a different image depending on his/her velocity? (without a radar).


title: Is light so important in special relativity?

text: I'm an amateur physics enthusiast with no formal university education in Physics. So my question might sound very naive, so forgive me. I had this question in the back of my mind since the wrong discovery of neutrinos travelling faster than light. My understanding of special relativity is as following. You can measure the physical parameters like length, time, mass of objects in a frame of reference (A) from another frame of reference (B) having a constant relative velocity with A by only looking into the information signals coming from A to B. Since the light(electromagnetic radiation) is the fastest known carrier of information between two frames and since light has a constant relative velocity with any moving inertial frame, the length/time measurements between moving inertial frames are formulated based on its velocity. Back in 2011 when the 'faster than light' neutrinos were reported one point I heard in popular science reports was that it will invalidate the special theory of relativity. But why is it so? Any particle faster than light would simply interchange the constant $c$ in the special relativity with some new constant(provided this particle also has a constant relative velocity with inertial frame). Doesn't the special relativity theory still holds with the information being exchanged with this new fast particle instead of light? As long as these particles don't travel with infinite velocity all measurements from moving frames will have the time dilation and length contraction. Is my understanding right? Edit 1: I see another question here and the answer to that seems like my understanding is right. Special relativity theory can be derived from any particle travelling with a constant speed.


title: How does stuff glow in the dark?

text: Many things have glow in the dark properties (glow sticks, paint, toys ..), and I am wondering what is the physics behind them. How do these materials store light energy and emit it later when dark? What dictates the wavelength(s) of the glow? Is there a limit on how much energy can be stored and recovered in these materials? Do they give out heat as well as light?


title: Why does the amount of energy transferred depend on distance rather than time?

text: The change in energy of an object can be determined by the work equation, where work is the change in energy: $$ W = F \cdot d $$ I conceptualize the transfer of energy as simply a series of small "packets" of energy being transferred at every Planck length. These small "packets" of energy add up to the total energy transferred (i.e. work). I'm not sure if this conceptualization is correct, so correct me if I am wrong. However, it makes me wonder why the amount of energy transferred is dependent on _distance_ and not _time_. $$ m_1 = 10~kg \\\ m_2 = 20~kg \\\ W_1 = (10~N)\cdot(5~m) = 50~J \\\ W_2 = (10~N)\cdot(5~m) = 50~J \\\ W_1 = W_2 \\\ t_1 \neq t_2 $$ If I apply a _constant force_ on an object, why isn't the energy transferred at a _constant rate_ with respect to time? The energy transfer rate varies dependent on how long it takes to cover the set distance. In other words: why is the energy transferred consistent per unit of _distance_ , and not per unit of _time_?


title: has AdS/CFT any predictive power in the natural context?

text: The question is rather simple as it is in the title. Has AdS/CFT any predictive power in the case of the space-time as we know it and in the case of reality as we know it experimentally? I may add: AdS/CFT in the interpretation given from the perspective of heavy ion collisions has been falsified (I doubt you can say anything different without getting into fitting parameters trouble). So, how does AdS/CFT relate to reality?


title: Would being underwater help survive a nuclear bomb?

text: If I jump in my pool, on the river near my house knowing that a nuclear bomb, or atomic or H-Bomb exploded around 10 km from my house, would I survive? The way I see it is that water will protect me from the heat, so then I will be able to surface up after the explosion and escape.


title: Can a sound mathematical formula become a science theory?

text: Can a sound mathematical formula become a science theory if it is constructed using a pattern creation process from sense-data, applied to observations by an inductive mapping, in contrast to deductively 'prove' ? Or in other words: Can a sound mathematical formula become a science theory if it is not deductively derivable from any axioms (there are no axioms) ? I have read on an internet forum: "Mathematics does work from axioms, but basic perceptions do not. Mathematics has axioms and the calculations made from those axioms (using the symbols and operations that are defined by the system) can be considered to be theorems of the system. They are deductively 'provable' within the system. But the mathematics system is, in its pure form, entirely abstract. It has no explicit correlation with anything in the real world. In contrast, perception is an inductive process whereby the conclusions are not deductively derivable from any axioms. (There are no axioms.) Instead they are constructed using a pattern creation process from sense-data. In science, mathematics is applied to observations by an inductive mapping. If it is seen that the data can be described by using sound mathematical formula then that becomes a 'theory'. This theory can subsequently be used to make predictions and be tested against further observations." So, my physics question now is: Can the sound mathematical formula 'the ear differentiates and quadrates', inductively be applied to provide in dedail descriptions of perceptions of the hearing of beats by the auditory sense, so that the mathematical formula 'the ear differentiates and quadrates' is an candidate to become a 'theory', so that it can subsequently be used to make predictions and be tested against further observations ?


title: Can ice have a higher entropy than water?

text: I've leant that entropy is a state of randomness, and that solids have a more structured form, therefore having less entropy. However, I saw a YouTube comment stating the following: > a liquid NOT ALWAYS means higher entropy than a solid it depends...of the > context for example, in the south pole, ice means higher entropy, because > Mother Nature sets the equilibrium for liquid water to become ice. Is there justification for this statement? Is it true that even in a more ordered substance like the ice, there is more entropy?


title: Do people confuse all-order perturbation theory with non-perturbative physics in QFT?

text: I've heard about people talking about the 1/N expansion as a mothod of nonperturbative QCD. This seems strange. In the 1/N expansion all you have is Feynman diagrams, which are by definition perturbative. Just resumming them to infinite orders doesn't really count as nonperturbative physics. Is it really jusified to think resummation has anything to do with noperturbative physics? Is there some subtle connection betweeen the two that I missed?