title: Is there a theortical limit to the amount of sound-energy air can contain?

text: is there a theoretical limit to the amount of sound energy air can contain? In case, there is a limit, what is that limit?


title: What is "quantum locking"?

text: I've always assumed that "quantum locking" was a term invented by the writers of Dr Who, but this video suggests otherwise. What is _quantum locking_? Is it real?


title: Massless $\phi^3$ theory in $d=6$ dimensions

text: I am asked to calculate renormalization for a massless $\phi^3$ theory in $d=6$ dimensional space using dimensional regularization. I'm having trouble finding the three-point vertex correction as one has to choose correct renormalization conditions. Does anyone know where I can read about this problem?


title: Analogues of mass, space, or time?

text: I began wondering about this when working through this question: is there anything analogous to spacetime or matter? Both spacetime (note: most of what I know of about the concept of spacetime comes from special and general Relativity) and mass are fundamental things in this universe. That is, our most fundamental physical laws contains measurements of position, time, and mass; they cannot be derived from anything. We know that space and time are inherently linked thanks to special and general relativity, but I can't think of anything else like spacetime. Is there even any other quantity that can be separated into dimensions? Mass is similar in that we can't really get to a lower level of how much of something there is (volume is mostly empty space, not stuff, so it can vary when the amount of stuff remains constant). The thing that sets it apart from spacetime, though, is that space and time make up a coordinate system, in which everything is measured with respect to a defined origin (zero). Mass, however, is an absolute scale, so it seems to me that it is something fundamentally different. Is there anything else like spacetime? Or mass? When thinking of fundamental quantities, I went to the base SI units. Besides position, mass, and time, they measure current (simplified to electric charge/time, and we've already discussed time, so let's just say electric charge here), thermodynamic temperature, luminous intensity, and number of moles (which is related to mass by a factor that's unique to each element, so we'll leave that as well and just use mass as the only measure for the amount of a substance). So, electric charge, thermodynamic temperature, and luminous intensity seem to be the only logical choices for analogues. These are all manifested forms of energy (the quantifiable ability to cause change), and they are all absolute in scale. How does energy fit into all of this? (My instinct is to say it resembles mass, but then what is the "matter" of energy?)


title: Analytical problems with Green's function

text: I have a question about the right definition of the Green's function in physics. Why do we introduce (or not) an infinitesimal, positive number $\eta$ to the following definition: $$\left[ i\hbar\frac{\partial}{\partial t} - \hat{H}(\mathbf{r}) \pm i\eta\right]G(\mathbf{r},t;\mathbf{r'},t') = \delta(\mathbf{r} - \mathbf{r'})\delta(t-t')$$


title: How does the Alcubierre drive warp space in the vicinity of it's destination?

text: ![enter image description here](http://i.stack.imgur.com/kWreF.gif) I don't understand how this could work. So lets say the drive is set to go 30,000,000 meters in a straight line to get to a distant planet. Now lets say the ship size is 20 meters. Now let's make the assumption that the warp field in the photo is 30 meters (10 meters bigger than the ship). So the warp bubble is applied. Now the 30 meters of space the ship occupied is cut. But it only could apply the warp bubble to this confined region which means 30 meters are cut instead of 30 million and now 29,999,970 meters are left and the ship is not even close to the planet it wants to go to (barely moves). Are there any solutions to this problem?


title: What provides the centrifugal force for planets orbiting a star?

text: this is a question I had when I was a kid. I'm a bit ashamed because I think I am missing out on something very obvious since I have the same question despite almost being an engineer now! From Newtonian physics, I understand how although the gravitational force pulls the planet towards a star, and the planet 'falls' towards the star, due the angular momentum, it also moves laterally. In case of a planetary orbit, this is just enough to keep it moving around it in an elliptical orbit. (With speed and radius being such that angular momentum and energy is conserved) But let's consider a single planet and star system where the planet moves around the star in a circular path for simplicity. If gravitational force provides centripetal force, what can account for the centrifugal force? I have a feeling it is related to some inaccurate view I have, of centrifugal force. The xkcd comic and wikipedia article talk about two concepts: fictitious and reactive centrifugal force. But it would be safe to assume the star(EDIT: sorry, I meant to choose a frame of reference such that the centripetal force on the planet has to be cancelled by a centrifugal force on the planet in the direction: star to planet) as a stationary frame of reference for the purpose of this question, right? Which means that it is not a fictitious force, right? Or is it not considered a centrifugal force if you take the star as the reference frame?


title: Why do books write $X(f)$ when they mean actually mean $\lvert X(f)\rvert$?

text: All books write $X(f)$ in plots - the Fourier transform of $x(t)$ - when they actually mean $\lvert X(f)\rvert$, without even mentioning in passing that they are dropping the mod sign. And also they consider only the mod of FT in the analysis, completely ignoring the phase term - especially communications analysis like AM and FM. Why? It is well known that in image processing phase also place an important role? Is there any proof or valid reason that phase is unimportant in communication engineering?


title: Velocity is "Distance/Time" in a particular direction OR simply "Displacement/Time"?

text: Is velocity:   1. Distance/Time in a particular direction, or    2. simply Displacement/Time? Or both 1 and 2 is the same thing?


title: How do you determine the degree of localization of a wavefunction?

text: Suppose that there is a wavefunction $\Psi (x,0)$ where 0 is referring to $t$. Let us also say that $a(k) = \frac{C\alpha}{\sqrt{\pi}}\exp(-\alpha^2k^2)$ is the spectral contents (spectral amplitudes) where $k$ is defined as wavenumber $k$. $\alpha$ and $C$ are constants. My question is, why do we calculate $\Delta x$ by looking at where the value of $\Psi (x)$ diminish by $1/e$ from the maximum possible value of $\Psi (x)$? Also, although the width of the $\Psi (x)$ packet is $4\alpha$, we define $\Delta x$ as $\alpha$. Why is it like this? By the way, $\Delta x$ is used as in uncertainty principle.


title: Demonstrate magnets adhere to conservation of energy pursuant to the laws of thermodynamics

text: I am looking for a way to demonstrate that magnets adhere to the laws of thermodynamics, in particular the requirement that energy in a closed system be conserved. To adhere to the requirement that energy not be lost, I would expect that the energy required to create a magnet would be offset by the energy exerted when that magnet exerts force. My (elementary, if you will pardon the pun) understanding of magnets is that they exhibit a magnetic field because of a kind of polarization of the electrons, that looks something like this for a "perfectly" magnetized metal:               ---------------------     ---------------------     ---------------------      whereas an unmagnetized object would not have this polarization (if that is the right word), and might "look" something like this:               |\/-/-\|-/|\/-/-\|-/     /|--/-\|-/|\/-|\\|\-     \-/-/-\-/|\/-|/-\|-\      The former object would exhibit the maximum force possible for the given material. The latter object would exhibit no magnetic force. One would expect that the energy required to align the electrons would never exceed (but where inefficient it may be less than) the force that is exerted by the magnetic field. As an example, suppose the polarization of a magnet from a completely unpolarized state uses 1 kJ of energy. To adhere to the laws of thermodynamics, the maximum amount of force the magnet can exert is 1 kJ, after which point it would be depolarized. One would expect the strength of the magnetic field to dissipate as it exerts force. Is there a demonstration that one can perform to help visualize the conservation of energy by way of "charging" and "discharging" a magnet?


title: Simple applications of group theory which can be understood by a senior undergrad

text: I am looking for references (books or web links) which have "simple" examples on the use of group theory in physics or science in general. I have looked at many books on the subject unfortunately they usually require extensive technical coverage of the basics, i.e. the 1st 100 pages or something, to be able to start discussing applications. I believe that there is an easy way to explain anything (it's just hard to find it).


title: polyphase circuits

text: I was going through a physics book which contains a short description of Nikola Tesla . The description mentioned about poly phase circuits (power supply). Out of curiosity I googled it up and I found that in a poly phase supply we use alternating currents of more than one frequency, say, w1, w2,w3,w4... Can anyone tell me what was the logic behind the invention of polyphase power supply,or what forced Tesla to invent this polyphase power supply system?


title: Very basic question on spin

text: Can anyone give a simple explanation for what the fractions and integers mean in particle physics when describing spin? I've seen on another forum (the naked scientist) that it should not be thought of as angular momentum as described for spinning objects like the earth, so I don't understand what it actually is. I also don't understand how it was measured from collisions i.e. if it was calculated from the momentum and energy of the collisions of particles?


title: If light diverges, how can there be a virtual image?

text: I have seen this question: Why can you see virtual images? but answers evade the question. Light needs to hit the retina in order, what is the meaning of "dashed" lines in most visualizations that track the light back near the original object? Can someone clarify? Is it reflected?


title: Reducing General Relativity to Special Relativity in limiting case

text: I understand that general relativity is applicable to gravitational fields and special relativity is applicable to case when there is no gravity. But is there a derivation on how to reduce General Relativity to Special Relativity in limiting case, much like how General Relativity is reduced to Newtonian gravity in weak-gravity case? **Edit: By _reducing_ I mean, how can we derive the Lorentz transformation from General Relativity under appropriate limits?**


title: What will happen if we use a speed greater than light speed and find a body'motion and energy relative to it?

text: In Einstein's papers, he used light speed as a reference speed. What if we use a greater finite speed and do the same calculations. Won't this greater speed then be the limit.


title: Can x and y axis have different scales?

text: In a graph on reflection, can i give the x axis a scale of 4 cms = 10 degrees , and the y-axis a scale of 2 cms = 10 degrees? Will it affect the readings?


title: what is the 2D gravity potential?

text: In 3D, I can calculate the total force due to gravity acting on a point on the surface of the unit sphere of constant density, where I choose units so that all physical constants (as well as the density of the sphere) is 1: $$F = 4\int_{-1}^1 \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \int_{-\sqrt{1-x^2-y^2}}^{\sqrt{1-x^2-y^2}} \tfrac{x+1}{\left[(x+1)^2+y^2+z^2\right]^{3/2}} dz\, dy\, dx = \frac{16\pi}{3}.$$ This force agrees with what we get if we treat the sphere as a point particle at the sphere's center of gravity with lumped mass $\frac{4\pi}{3}$. So far so good. But now if I try to calculate the force of gravity of a point on the boundary of the unit disk in 2D, I get infinity: $$F = 2\int_{-1}^1 \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \tfrac{x+1}{\left[(x+1)^2+y^2\right]^{3/2}} dy\, dx = \infty?$$ Intuitively, if I think of the total force as the sum of contributions from nested circular rings around the point of interest, the circumference of the rings scales like $r$, the force density like $\frac{1}{r^2}$, so the contribution of each ring scales like $\frac{1}{r}$ which diverges as $r\to 0$. What is going on here? How is the gravity potential derived? Is it an accident that in 3D, $\frac{G m_1 m_2}{r}$ is harmonic on the punctured space $\mathbb{R}^3 \backslash\\{0\\}$? Is the "right" gravity potential in 2D something like $G m_1 m_2 \log r$? If so, why, and isn't it a paradox if point masses in 2D orbit according to a different law than coplanar point masses in 3D?


title: What is the spectrum of the Hamiltonian of the universe?

text: Let us assume that there exists a Hamiltonian that (together with the initial state) describes the whole universe. Then my question is : What is the spectrum of this Hamiltonian and what are the multiplicities of the spectrum ? More precisely, what is the representation of the Hamiltonian in Spectral multiplicity theory (see http://en.wikipedia.org/wiki/Self-adjoint_operator, section "Spectral multiplicity theory").


title: Which mathematical operation does the right hand rule for current come from?

text: I am currently wondering about this famous rule: ![picture](http://ffden-2.phys.uaf.edu/212_fall2003.web.dir/Daniel_Lenord/righthand.jpg) Where does it come from mathematically that when you point with your thumb in the direction of the current, your curved fingers will point in the direction of the $B$-field? In other words, which mathematical operation does it come from? Probably it has to do with some vector product or Stokes theorem application, but I am not quite sure about it.


title: Expand metric $g_{ij}$ about flat space

text: I expand metric $g_{ij}$ about flat space $\delta_{ij}$ $$g_{ij}=\delta_{ij}+h_{ij}$$ where $|h_{ij}|\ll 1$. I would like to find $R_{ij}$, to linear order, in terms of $h_{ij}$, but I dont know what terms are negligible. Which $\partial h \partial h$, $\partial h$, $hh, h \partial h $, $\partial \partial h$ are negligible?


title: Charge and conducting thick slab

text: I have thought over this problem but I haven't found the solution: There is an electric charge $q$ at a distance $d$ from a conducting slab with thickness $t$, the problem is to find the potential everywhere, assumming that the slab is grounded. So, on the side of the slab where the charge is I think one may use the image method solution to find the potential in this region of space. Inside the slab $\phi =0$, but the question remains how to find the potential on the other side? The method of images does not seem to work on that side(?), and other solutions seem difficult to implement. Charge conservation seems to imply that the total charge induced on each surface of the slab is $\pm q$. Is there any easy or semi-easy way to do it? I guess my question would be: what is the most straightforward way to do it? Any help is greatly appreciated.


title: Is wearing 3D glasses from the cinema as sunglasses hurtful to the eyes?

text: I heard a few times that using them as sunglasses is hurting the eye since UV light is not filtered, but the pupil is wider than it would be w/o wearing them because the visible light is dimmed. I have always assumed since atmospheric (i.e. not reflected from flat surfaces) sunlight is unpolarised and since the glasses used in 3D cinema only allow circular polarised light to pass they do indeed filter out a large portion of the intensity of the incoming (unpolarised) light. Does this not work for the UV part of the spectrum or is my initial assumption wrong?


title: Expectation value of the stress-energy tensor in 2-D CFT

text: Due to a previous question, I am confused with the expectation value of the stress-energy tensor in a 2-D conformal field theory. Let's take the example of string theory, to sketch the problem. Defining the action by $S = \frac{1}{2 \pi \alpha'}\int d^2z \partial X^\mu \bar \partial X_\mu$, the (euclidean) expectation value of an operator is : $$\langle\mathcal F[X]\rangle=\int [dX]~ exp(-S)~F[X]\tag{0}$$ My questions are : 1) In string theory, what is : $$\langle \partial X^\mu \partial X_\mu\rangle \tag{1}$$ 2) In string theory, what is : $$\langle :\partial X^\mu \partial X_\mu:\rangle \tag{2}$$ 3) In a general 2-D conformal field theory, what is : $$\langle T^{\alpha \beta}\rangle \tag{3}$$


title: Mass fixed to a wheel - mechanics problem

text: I am struggling with a problem from classical mechanics. Imagine a massless wheel (to make it simpler) with a mass $m$ fixed to it rolling without slipping on a horizontal ground. If we now try to find the equations of motion of the wheel (for instance the angle $\alpha$ it turns) we will find that all the forces are independent of velocity, so $\alpha ''=f(\alpha)$. After doing that I decided to solve this problem using Euler-Langrange equations (since friction does no work). I came up with $L=\frac{1}{2} m R^{2} (\frac{d\alpha}{dt})^{2} (1+\cos \alpha)-mgR(1+\cos \alpha)$ which, upon solving, gives $\alpha ''$ as a function of both $\alpha$ and $\alpha '$. What is my problem?


title: What future technologies does particle physics and string theory promise?

text: What practical application can we expect from particle physics a century or two from now? What use can we make of quark-gluon plasmas or strange quarks? How can we harness W- and Z-bosons or the Higgs boson? Nuclear physics has given us nuclear plants and the promise of fusion power in the near future. What about particle physics? If we extend our timeframe, what promise does string theory give us? Can we make use of black holes?


title: Brachistochrone problem in general relativity

text: This question Brachistochrone Problem for Inhomogeneous Potential has the obvious extension. Namely the same question, when gravity is treated according to general relativity. To make it specific let's consider the case of Schwarzschild metric. Given to points, outside the event horizon, what is the trajectory (or the worldline) of a test body, which goes from the first point to the second for least time. All considered from the point of view of a distant observer at rest (with respect to the star/back hole). Of course other metrics or the general case are also interesting. I couldn't find much by searching on the net. There is one article with the tittle suggesting relevance (actually more but this one seems very relevant), but it is over 60 pages of technicalities. So it would be nice to see a shorter answer.


title: Glauber's Formula

text: In the Cohen-Tannoudji Quantum Physics book, Complement BII, says: [...] two operators $A$ and $B$ with both commute with their commutator. An argument modeled on the preceding one shows that, if we have: \begin{align} [A,C]=[B,C]=0 \end{align} with $C=[A,B]$, then: \begin{align} [A,F(B)]=[A,B]F'(B) \end{align} Then this last property, is used to proof Glauber's Formula. \begin{align*} e^Ae^B=e^{A+B}e^{\frac{1}{2}[A,B]} \end{align*} I understand this proof, but I couldn't find a way to demonstrate, \begin{align} [A,C]=[B,C]=0 \end{align} with $C=[A,B]$, then: \begin{align} [A,F(B)]=[A,B]F'(B) \end{align} I would like to know to do this, so I can understand better the Glauber's Formula proof.


title: Static Friction - Only thing that can accelerate a train?

text: I'm a computer programmer that never studied physics in school and now it's coming back to bite me a bit in some of the stuff I'm being asked to program. I'm trying to self study some physics and I've got a few open source intro physics books and understanding it for the most part but I'm a bit confused on this statement I've stumbled upon in a section about static friction. It's already gone over the formula for static friction and so forth. It gets into a section explaining that the weight of a train increases static friction between the wheels and the tracks. Alright, makes sense. But then it says this: > The reason locomotives are built to be so heavy is for traction. The upward > normal force of the rails on the wheels, FN, cancels the downward force of > gravity, FW, so ignoring plus and minus signs, these two forces are equal in > absolute value, FN = FW. Given this amount of normal force, the maximum > force of static friction is Fs = sFN = sFW. This static frictional force, of > the rails pushing forward on the wheels, is the only force that can > accelerate the train, pull it uphill, or cancel out the force of air > resistance while cruising at constant speed. The coefficient of static > friction for steel on steel is about 1/4, so no locomotive can pull with a > force greater than about 1/4 of its own weight. If the engine is capable of > supplying more than that amount of force, the result will be simply to break > static friction and spin the wheels.   >   >  \- "Newtonian Physics", Light and Matter - Book 1, p158 B. Crowell > http://www.lightandmatter.com/bk1.pdf I'm confused as to how static friction is the only thing that can move the train forward. I thought static friction was what kept it in place in the first place. There's another force - that I can't think of the name of, but I've heard of somewhere - that I thought was more what they're describing here, where the weight of the wheels pushing down and forward slightly on the tracks causes the tracks to push up and forward (from the opposite side). Can someone explain to me what this is saying?


title: Rocket/Thrust/Gas/Free Expansion of Gas

text: We know, the rockets in space use Newton's 3rd law to increase their velocity and hence move. What I don't understand is how it is possible in space aka vacuum-state without air? From what I know, Joule's "Free Expansion of Gas" says that free-expansion compresses the gas and is therefore "affected" by vacuum so it can't make the rocket move as the gas will have zero press/force. Could someone please explain me how rockets do really work and the above- mentioned statement? Actually, please have a look at this site: http://cluesforum.info/viewtopic.php?f=23&t=1632 Not: The site appears to include some conspiracy theory thingummies, but made me wonder anyway.


title: From where we can view earth moving?

text: Earth moves with a speed around $100000 \,\,km/h$, which will be amazing, but from where we can see that moving speed? If we far away from earth (outside of earth gravity) then can view its moving? is any video already taken? (I failed to search that kind of video).


title: How to demonstrate that there are just 14 types of Bravais lattice?

text: Is it possible to demonstrate that there are just 14 types of Bravais lattice without the knowledge of group theory?


title: is redshift still true if object emitting is not moving away but the observer is

text: If something is moving away fast it's color would appear redder than its actual color due to phenomenon known as Doppler effect. But what if the object isn't moving but i am moving away faster would that still affect the way waves travel. Optional:- And why. Object emitting isn't moving. And won't that means that doppler effect is subjective. What does that say about light and its wavelength?


title: What is the percentage of stars with planetary systems

text: We have discovered quite a number of exoplanets to date. The Kepler spacecraft has examined 150,000 stars and found 1,059 exoplanets. We know that Kepler, as well as all other exoplanet searches to date, can only find planets that cross in front of their star. That means that a large percentage of existing planets will not be detected. In addition, a lot of planets may be too small for the sensitivity of our detectors. From these facts, has anyone calculated the probability of a star having a planetary system? I am guessing it is likely quite close to 1, i.e. all stars have planets, but I have not seen anyone figures for it.


title: Expansion of $L(v^2 + 2\vec{v}\cdot\vec{\epsilon}+\epsilon^2)$

text: How can I find the expansion of the Lagragian (it it only dependent on $v^2$) $L(v^2 + 2\vec{v}\cdot\vec{\epsilon}+\epsilon^2)$ in powers of $\vec{\epsilon}$ ? (From L.Landau, E. Lifshitz, Mechanics , page 7)


title: Dimensional regularization - integral

text: How can I derive the following formula? $$\int d^{d+1} k \frac{e^{i K X}}{K^2} = \frac{\Gamma (d-1)}{(4\pi)^{d/2} \Gamma (d/2) |X|^{d-1}}, \quad K^2 = k_0^2 + \vec k^2, KX = k_0 \tau + \vec k \vec x$$ What I tried so far:   1. Integrate over $k_0$ while promoting $k_0$ to complex variable, so I can use residue theorem. Then I move to n-spherical coordinates. I obtain: $$ \propto {\rm sign} (\tau) \int d \Omega \int dr \, r^{d-3} e^{-r |\tau| + ir |\vec x| \cos \phi_1}$$ And I'm stuck.   2. I change variables to n-spherical ones and I end up with this: $$\int d \Omega \int dr \; r^{d-2} e^{ir |\vec x| \cos \phi_1}$$ And I'm stuck as well.


title: Natural units of information

text: In physics entropy is usually measured in nats. I wonder is there a possible model of a physical system which has entropy of discrete number of nats? How particles and degrees of freedom should be arranged so this to happen? I would be interested in both   * example of an analog (non-discrete) system   * example of a quantum system


title: DIfference in Pitch Caused by Water Temperature?

text: I have recently been intrigued by the following question: What is the difference between the pitch of the noise of dripping water between hot and cold water? For example, would cold water create a higher pitched noise while dripping into a pot of water? Or vice versa?


title: Moment of inertia around two axes

text: In the this chapter of an online pdf we are given an equation for the deflection of a beam: $$\frac{d^2y}{dx^2}=\frac{\overline{M}}{E I}$$ where $E$ is the modulus of elasticity, $\overline{M}$ is the moment as function of $x$ and $I$ is the moment of inertia. It works nicely for a beam with one free end as the beam would be rotating around a single axis. However when they get to the worked example with both ends fixed they leave the moment of inertia uncalulated (left just as $I$). I am wondering how one would calculate the moment of inertia for such a beam. Is it even defined?


title: How can kinetic energy be proportional to the square of velocity, when velocity is relative?

text: Let's start with kinetic energy (from los Wikipedias) > _The kinetic energy of an object is the energy which it possesses due to its > motion. It is defined as the work needed to accelerate a body of a given > mass from rest to its stated velocity. Having gained this energy during its > acceleration, the body maintains this kinetic energy unless its speed > changes. The same amount of work is done by the body in decelerating from > its current speed to a state of rest. In classical mechanics, the kinetic > energy of a non-rotating object of mass m traveling at a speed $v$ is > $\frac{1}{2}mv^2$._ Let's say you & your bike have mass of 100kgs, then your kinetic energy at 10m/s would be $$ E_a = 1/2 \times 100 \times 10^2 = 5000J = 5kJ$$ If you apply another 5kJ of energy, you don't get to 20m/s though, you only get to: $$ E_b = 10000J =1/2 \times 100 \times V_b^2$$ $$\implies V_b = \sqrt{10000 / (1/2 \times 100)} = √200 = 14.14m/s$$ Let's say you and a buddy are both coasting along at 10m/s though, from their perspective you've just burned 5kJ but only accelerated 4.1m/s, _even though you seemed stationary_. Imagine you and your mate are in space drifting along together, at an unknown speed. Your mate fires his burners and accelerates away from you. There's a big screen on his ship showing how many joules of energy he just burned, and you can measure his resulting relative velocity just fine. The question is, Will 5kJ of energy always produce 10m/s of relative velocity,,assuming 100kg spaceships? If 5kJ always produces 10m/s, Why does the second 5kJ only produce 4.1m/s? What is going on here?


title: Will a space traveller slow down due to space expansion?

text: Photons of relic radiation loose their energy as they propagate through space. Will a space traveler loose their peculiar velocity as he travels through vast distances? Will he stop somewhere or still he would be able to reach the stars at unlimited distance without burning the engines?


title: How does light travel create time travel violating causality?

text: Saw a question about faster than light travel... I still have the same question though none of the answers offered any resolution for me. It is so summarily assumed by all physicists and commentaries that exceeding the speed of light would turn back the clock. I can't see the relation. Doubling the amount of any speed halves the time taken to travel a given distance. Keep doubling the speed and that time is halved (or otherwise divided). Divide any quantity (time in this case) and you always end up with a fraction of it but never zero and certainly never a negative amount as would be the case for the causality conflict. So it seems to me that whatever speed one attains, there is always a positive time element in the travel no matter how tiny!! The speed of light is only unique to me in that it is the fastest observed speed but is otherwise just another speed quantity set by nature (just like the speed of sound etc) could it be that other elements in nature are travelling faster than light but we lack the means to detect or measure them (like the rebellious neutrino)? I also don't understand time as an independent element that can be slowed sped up etc. It seems to me that time is simply a relative measure of the ever- changing state of matter relative to other states of matter. If every thing in the universe stopped- that is all state of matter everywhere frozen, all electrons frozen in place etc wouldn't we observe that time had stopped? Isn't it therefore our observation of the changing state of matter around us that gives the perception (perhaps illusion) of time? I can therefore only understand time as a subjective sense of changing states relative to an observer! It should be the rate of change of these states that slow down or speed up (in relation to the observer or instrument) and not the universal rate of change or universal time that changes! It would also debunk any notion of time travel, as it would involve the manipulation of every particle in the universe to a previous of future state... Disclaimer.. I hate calculations, stink at them and have no idea what mathematical formulas are used to arrive at the accepted conclusions so I'm not trying to dispute any findings etc by the experts, just trying to align my lay understanding to their conclusions.


title: Starting postdoc in VHE Astrophysics with VERITAS and HAWK, what books to read?

text: I will be starting a postdoc in Astrophysics, particularly VHE gamma-rays with the VERITAS and HAWK observatories. I am coming from particle physics and am looking for some good books to get me up to speed. I have taken 1 astrophysics course, but it didn't even have a book, so I would like some options on both general overviews, and some more detailed to VHE gamma-ray in specific. Thanks so much!


title: false false beliefs in physics

text: Reading Common false beliefs in Physics I feel that there are also "fake false beliefs", ie, that the "false belief" does not actually exists, it is a rhetorical argument or just a basis for argumentation, but not actually a belief... while the meta-belief on its existence actually exists. Prototypical example is mid-age "flat-earth". All the discussion with Colon was about the diameter of the earth, not about the flatness or sphericity. And still, you can hear here and there that the mid-age philosophers had the "false belief" of a flat earth. A more dubious example: does people actually think that summer is related to the distance to the sun? Or is it a rhetorical trick of the teacher, to induce this thinking during the classroom only to correct it later? Most countrymen clearly perceive that summer is about height of the sun, not distance.


title: K3 gravitational instanton

text: Could you please recommend a sufficiently elementary introduction to K3 gravitational instanton in general relativity and the problem of finding its explicit form? Under 'sufficiently elementary' I mean the texts geared at physicists with basic knowledge of general realtivity but not much more than that (i.e. with almost no knowledge of modern differential geometry and algebraic geometry). To give a hint, I find the description of the problem in the book _Solitons, Instantons and Twistors_ by Maciej Dunajski way too terse. Many thanks in advance!


title: Lagrangian depends on second derivative of field

text: In case of the gauge-fixed Faddeev-Popov Lagrangian: $$ \mathcal{L}=-\frac{1}{4}F_{\mu\nu}\,^{a}F^{\mu\nu a}+\bar{\psi}\left(i\gamma^{\mu}D_{\mu}-m\right)\psi-\frac{\xi}{2}B^{a}B^{a}+B^{a}\partial^{\mu}A_{\mu}\,^{a}+\bar{c}^{a}\left(-\partial^{\mu}D_{\mu}\,^{ac}\right)c^{c} $$ (for example in Peskin and Schröder equation 16.44) If you expand the last term (for the ghost fields) you get: $$ \bar{c}^{a}\left(-\partial^{\mu}D_{\mu}\,^{ac}\right)c^{c} = -\bar{c}^{a}\partial^{2}c^{a}-gf^{abc}\bar{c}^{a}\left(\partial^{\mu}A_{\mu}\,^{b}\right)c^{c}-gf^{abc}\bar{c}^{a}A_{\mu}\,^{b}\partial^{\mu}c^{c} $$ And so, the Lagrangian has a term proportional to the second derivative of $c^a$. In this case, how does one find the classical equations of motion for the various ghost fields and their adjoints? I found the following equations of motion so far: $$ D_{\beta}\,^{dc}F^{\beta\sigma}\,^{c}=-g\bar{\psi}\gamma^{\sigma}t^{d}\psi+\partial^{\sigma}B^{d}+gf^{dac}\left(\partial^{\sigma}\bar{c}^{a}\right)c^{c} = 0 $$ $$ \sum_{j}\partial_{\sigma}\bar{\psi}_{\alpha,\, j}i\gamma^{\sigma}\,_{ji}-\sum_{\beta}\sum_{j}\bar{\psi}_{\beta,\, j}\left(gA_{\mu}\,^{a}\gamma^{\mu}\,_{ji}t^{a}\,_{\beta\alpha}-m\delta_{ji}\delta_{\beta\alpha}\right)=0 $$ $$ \left(i\gamma^{\mu}D_{\mu}-m\right)\psi=0 $$ $$ B^{b}=\frac{1}{\xi}\partial^{\mu}A_{\mu}\,^{b} $$ $$ \partial^{\mu}\left(D_{\mu}\,^{dc}c^{c}\right)=0 $$ $$ f^{abd}\left(\partial_{\sigma}\bar{c}^{a}\right)A^{\sigma}\,^{b}=0 $$ But it is the last equation that I suspect is false (I saw the equation $ D_\mu\,^{ad} \partial^\mu \bar{c}^d = 0 $ in some exercise sheet (http://www.itp.phys.ethz.ch/education/fs14/qftII/Series7-3.pdf Exercise 3) and I also saw the equation $D^\mu\,^{ad}\partial_\mu B^d = igf^{dbc}(\partial^\mu\bar{c}^b)D_\mu\,^{dc} c^c$ which I don't understand how they were derived.) EDIT: Thanks to Qmechanic's answer I was able to derive the correct equations of motions (as noted in the comment to that answer) but I still don't know where to "obtain" the last equation I mentioned which connects the auxiliary field with the ghost fields.


title: Inelastic nucleon-nucleon cross section at LHC energies

text: I am trying to reproduce the number of participants $N_{part}$ in Pb-Pb Collisions at LHC using a Glauber Monte Carlo simulation, specifically aiming to reproduce the values given in Phys. Rev. Lett. 106, 032301 (2011) by K. Aamodt et al. (the ALICE Collaboration). To do this, I am looking for the inelastic nucleon-nucleon cross section. A paper on the software written for the PHOBOS experiment (available on arXiv here) suggest that it is (at LHC energies) " _expected to be around $\sigma_{NN} = 72$ mb (with large uncertainty from the unknown elastic cross section)_ ". I have been using this web-interface provided by the GSI to get an estimate of $N_{part}$. The website suggests to use $60$mb at $\sqrt{s_{NN}} = 5500 \text{GeV}$. However, even using the 72mb suggested above, for 0%-5% centrality I only get about $376$ participant nucleons, compared with $N_{part} = 382.8$ for $2.76$ TeV published in the ALICE paper. I looked at the PDG website as well, but since (as hinted above) the plots do not show data for the elastic part of the cross-section, no value is provided for the energy I'm looking at. The fit funciton provided does not seem to give a reasonable value for LHC energies. > Am I right in assuming that we simply don't know the value of $\sigma_{NN}$ > yet and that it must thus be estimated for such calculations? If that is the > case, is there a common way to do so?


title: Eigen modes in a room

text: In a rectangular room 10*8*5 how can the eigen modes belonging to octave and third octave bands centered at 50Hz be found ? I have the formula to calculate the number of modes but don't have the idea how to find the modes. Do I need to use Matlab for it ?


title: What if the surface charge on a conductor is positive?

text: This site says that if the field at the surface at the conductor has a parallel component, then the surface charge will move, which is impossible if the conductor is at equilibrium. But I learnt that positive ions can't move in a conductor, only electrons can. I asked my prof and he said that if the field has a parallel component, the parallel components will cause the electrons to move, which will result in movement of positive charge. ![enter image description here](http://i.stack.imgur.com/DocHi.png) 1) But an electric field is a vector field, so each point has a different vector associated with it. So what would happen if the vector field had parallel components where positive ions are located but is completely perpendicular where electrons are located (look at the above image)? Isn't this a scenario where the entire field isn't perpendicular but there is no movement? 2) If a perpendicular field is acting on the electron, what's stopping it from leaving the conductor? Normally, if an electron tries to leave because of a field, the positive charges redistribute themselves to cancel out the field, preventing it from leaving. But if the field isn't cancelled out, theres nothing holding the electron back is there?


title: Capacitance and Light Bulb And Dielectric Slab

text: An electric lamp having coil of negligible inductance connected in series with a capacitor and an AC source is glowing with certain brightness. How does the brightness of the lamp change on reducing the (i) capacitance, and (ii) the frequency ? What if the diaelectric slab is introduced between the capacitor.


title: Virgo SuperCluster and space time

text: Is the Virgo Supercluster affected at all by the expansion of space? What about the larger filament that Virgo resides in?I don't quite understand how gravity affects space time locally in groups of galaxies, and am wondering if the expansion of space is lumpy. It would be nice to know we are not all being stretched... :)


title: How to find the electric field of a charged tube?

text: Let's say there is a charged tube(cylinder with no top or bottom) with radius $a$, length $l$ and charge $q$ and a point which is collinear with the centre of the charged tube. Anyway, since we can think the tube as infinite charged rings we first should calculate the electric field of a single ring.(the distance between ring and point charge is $r$ and the distance between the centre of the ring and point charge is $d$) $$ dE= (k.dq)/r^2 $$ where $dq$ is infinitesimal charge. $$ dE_x=(k.dq)/r^2cos(\theta) $$ $$ dE_x=(k.d.dq)/(a^2+d^2)^{(3⁄2)} $$ $$ E_x=∫(k.d.dq)/(a^2+d^2)^{(3⁄2)} =(k.d)/(a^2+d^2)^{(3⁄2)}∫dq=kdq/(a^2+d^2)^{(3⁄2)} $$ Finally, we found the electric field of a single ring. However, I am stuck at how to integrate the equation to find the total electric field of the charged tube. How should I integrate it? P.S: It may seem like the other question I asked but they are different in concept of what the question is.


title: Speed of light and distance

text: Our measure of distance (the meter) is defined in terms of how far light in a vacuum travels in a specific time. When light travels through another medium, we say it travels at a different speed. Why not say it travels at the same speed, and the distance in the medium is different that the distance outside? Is it only because the 2nd explanation feels less intuitive than the first, or is there more to it?


title: Gauge formalism in rigid body mechanics

text: When doing calculations in rigid body mechanics, it is necessary to choose an origin to calculate torques and angular momenta. However, the underlying dynamics does not depend upon the choice of that origin. This is very reminiscent of gauge theories. Could we interpret the choice of origins as a gauge fixing? Has gauge formalism been applied to rigid body mechanics?


title: Spherical charge distribution along a wire of length $L$ in which its center coincides with the origin of the system

text: I've already tried to find the expression but I don't find the expected result for the quantity of charge when I use the expression I found. For example, I checked it by integrating the expression that I found from $r=0$ to $r=\frac{L}{2}$, it should lead to $Q=\frac{\lambda L}{2}$ but it is not what I find.


title: Capacitor with big space of insulation and polymer inside

text: The capacitor has insulator and it is covered by two conductors from top and bottom. I started to think if this insulation part can be wide like 1 cm big - yes, it can but then permittivity has to be great for great capacitance. If the insulator is air, then I think the capacitor is very weak. Capacitance is \begin{equation} c = \epsilon/d \end{equation} where $c$ is capacitance, $\epsilon$ permittivity and $d$ distance between plates. Here some list of possible materials and their permittivity ![enter image description here](http://i.stack.imgur.com/oFyeM.png) The gap is 0.01m (d) so the permittivity should be more than 100 for capacitance to be more than one. I think the capacitance must be more than one to be useful. I think Schrödinger equation can be used here to simulate some things. ## Case Assume you have insulator as polymer which has permittivity $\epsilon = 100 000$. Distance between plates is $d = 0.01 m$ so capacitance is $c = 1000$. **How can you simulate the movement of electron from the one plate to another in different situations by Schrödinger equation?**


title: Can anyone clarify Brian Greene's relation of holographic parallel universes with Plato's cave/

text: The string theorist Brian Greene recently came up with the book The Hidden Realities. In Chapter 9, and also in the article http://discovermagazine.com/2011/jun/03-our-universe-may-be-a-giant-hologram , he argued that our universe might be a hologram with a holographic parallel universe and compares our world to shadows in Plato's cave. Can anyone point me to where I can read up more on this/ Are we living in a black hole. Are we puppets pulled by strings from some distant surface


title: Book request for an abstract treatment of QM without using any particle formalism

text: I am an electronics and communication engineer, specializing in signal processing. I have some touch with the mathematics concerning communication systems and also with signal processing. I want to utilize this knowledge to study and understand Quantum Mechanics from the perspective of an engineer. I am not interested in reading about the historical development of QM and i am also not interested in the particle formalism. I know things have started from the wave-particle duality but my current interests are not study QM from that angle. What I am interested is to start studying a treatment of QM from very abstract notions such as, 'what is an observable ? (without referring to any particular physical system)' and 'what is meant by incompatibility observables ?' and then go on with what is a state vector and its mathematical properties. I am okay to deal with the mathematics and abstract notions but I some how do not like the notion of a particle, velocity and momentum and such physical things as they directly contradict my intuition which is based on classical mechanics ( basic stuff and not the mathematical treatment involving phase space as i am not much aware of it). I request you to give some suggestions on advantages and pitfalls in venturing into such a thing. I also request you to provide me good reference books or text books which give such a treatment of QM without assuming any previous knowledge of QM.


title: Why are downed power lines dangerous?

text: A friend of mine was telling me about a storm that knocked down a power line over at their place, and it got me to thinking. Why are downed power lines dangerous? I don't see any good reason for it. It seems to me that if they can put a GFI on a hair dryer that will kill power within a fraction of a second if it falls into the sink or tub and shorts out, they ought to be able to do the same thing to a power line. Is there any reason why the basic concept can't work at that scale? Or if not, why aren't they standard equipment on power lines?


title: Phase Shift of Tunneling Wave

text: What is the phase shift of a wave that tunnels through a barrier, meaning the difference in phase between the incoming (in front of the barrier) and the outgoing (behind the barrier) waves? For example, in the situations of a wave function tunneling through a classically forbidden region, where the wave is no longer oscillating, but decaying exponentially within the barrier. Or of a beam of light incident at the critical angle of an interface in a frustrated total internal reflection setup.


title: Determining the group associated with a given potential?

text: I'm trying to understand how symmetry groups are related to potentials of the Schrodinger equation. In particular, I wish to know if it is possible to find the symmetry group of this potential $$V(x) = A_0 +A_1x +A_2x^2 -\frac{9}{4}x^4$$ where $A_0$,$A_1$,$A_2$ $\in \mathbb{R}$ I've tried to see if it is related to the SO(3) group and the unitary group U(1), but neither seem possible. I asked this question because coming from a pure math background, I am having a really difficult time trying to understand this.


title: Beginner Physics - Explaining longitudinal waves

text: I am having difficulty grasping the concept of a longitudinal wave. My textbook definition "In longitudinal waves, the vibration is backwards and forwards in the direction of motion of the wavefront" If it vibrates backwards and then forwards, would it not be in the same position it originally was? Do we assume it vibrates forward at a rate faster than that at which it vibrates backwards? Further, what am I supposed to 'visualize' when I think of waves, in a physics sense? What is the purpose of a longitudinal wave? Sorry if I am asking a lot of questions,


title: How would the composition of the Venusian atmosphere visually affect the sunrise and sunset there?

text: The Venusian atmosphere is according to CalTech's webpage What is the atmosphere of Venus like? is 90 times heavier, than Earth's, containing predominantly carbon dioxide, some nitrogen, sulfuric acid clouds in its atmospheric chemical soup. What affect would Venus' heavier atmospheric mass and composition have on how the sunset and sunrise would be viewed visually?


title: Why does a cooling motorcycle make popping noises?

text: I was sitting in my parked car tonight when a Harley-Davidson motorcycle parked next to me. With my window down, I could hear that as the motorcycle was cooling down, it was giving off some type of “popping noises.” My first guess was that as the metal of the exhaust pipes cooled, thermal contraction occurs such that it produces some sound (even though I really have no idea why this would be true). I am hoping someone could correctly explain the cause of this popping noise.


title: Acceleration of a car in a collision?

text: For example because of newton law $F=ma$ force is acceleration times mass but if I go at a high constant velocity ($900\: \mathrm{miles/hour}$) in my car for examples I have $0$ acceleration but if I hit an obstacle by newton's law it will have no effect on me because $F=ma$ and mass of my car x 0 = 0 But if I do the experiment I will find that I actually hit the obstacle and it does an force on me why?


title: Modeling sound resonance in an arbitrary cavity

text: I am trying to solve a challenging problem, and I'm hoping for some advice on how to proceed. I want to model sound waves in a cavity for the purpose of determining resonance. The plan is to answer the question numerically, but I need some better bearings regarding the physics involved. Here is a summary of my main questions:   1. How should I model this situation for a _closed_ cavity?   2. What does resonance in a closed cavity even look like? (as opposed to a "whistling" kind of cavity)   3. What wave equation should I use? I'm assuming spherical for an arbitrary cavity.   4. I don't even know if using diff eq's (wave equation) is the best approach here. Would it be possible, or even advisable, to try modeling noise in the cavity and see what kind of response I get?   5. I don't want my solution to be dependent on the location of the source inside the cavity. Rather, I want to understand what the natural frequency of the space is. Is that even possible, or am I living in a fantasy here? I want to start by solving this problem for the cube, so that I understand all the parameters involved, and solve for general cavities from there. My naive guess at the solution for cubes (or rectangular prisms) is to look at standing waves in the 3 dimensions (though I don't know which harmonic would be dominant). The problem with this approach is that it does not translate to arbitrary cavities very well.


title: Changing the resistance of a circuit

text: I am wondering, when we change the resistance in a circuit, does the voltage tend to increase or the ampere decrease? More specifically, under what conditions does what happen? E.g. If we have a high-resistance voltmeter, does increasing the resistance of the circuit itself necessary increase the reading, or might the current just decrease further?


title: What is the reference spectrum?

text: What is the reference spectrum? I need to know how to calculate the reference spectrum of a wavelength 500nm.


title: How is explained the lower ionization potential for atoms compared to closed shell nobel atoms?

text: Why adding just one electron changes tremendously the ionization potential from any of the nobel atoms ? If it is the screening why adding a second electron increases the ionization potential ? http://en.wikipedia.org/wiki/File:IonizationEnergyAtomicWeight.PNG


title: energy of electrons outside an atom

text: The higher the quantum number(energy levels)m the higher the energy. What does the energy refers to? Kinetic energy, potential energy, or the total mechanic energy?


title: Why the electric potential of earth is zero?

text: For a localized charge distribution the potential is set to zero far away from the charge distribution (at infinity) Now, when grounding a conductor, i.e. connecting it to Earth, it is said that we are setting its potential to zero. Why Earth's potential is zero? And if it is zero, is it zero even if the potential is still zero at infinity?


title: Decomposition of elastic constants of a crystalline material

text: I have performed a calculation the tensor of elastic constants (or stiffness tensor) for a given crystalline material. From there, I calculated various elastic properties, such as Young’s modulus, shear modulus, etc. I have, however, received advice to analyze it by > further decomposing the elastic constant into contributions from the bare > terms and phonon relaxation terms I don't have any more context than that, and I have not been able to identify what these two terms (“bare terms” and “phonon relaxation”) refer to. Does anyone here know? * * * PS: I don't know if it helps, but the elastic tensor was calculated using quantum chemistry calculations, by measuring the energy of the solid as a function of strain.


title: Why 2s state is lower in energy  that 2p state in atoms?

text: The s orbital have higher probability to be closer to the core and feels larger attraction than the p orbital and on average is further away and in addition p has repulsive potentilal l(l+1)h^2/2mr^2. But is there name for this effect and is it this the only way to explain ? What is the usual energy difference between these levels and how is measured, the transitions are highly forbidden ? Does quantum electrodynamics separate s from p ? In Lamb shift I think I does not but how about the multielectron atoms ? In real life if hydrogen was not a molecule but atom is s lower that p, and is it opposite in all multielectron atoms ? There is no hund rule for this ?


title: Stresses on an inclined plane

text: # Background This issue has been bothering me for a few days now. It's actually part of my homework, but I've already finished them and in a sense they're not part of the question here. What I would like to know is more of a theoretical question about this formulation. Now my professor's book and many online resources (sample) maintain the same mathematical relation, hence my dilemma. # Problem Consider the following: ![Problem](http://i.imgur.com/bTaKei0.png) Under axial loading, the stresses can now be computed for any section. The question posed is what if we make an imaginary section at an _inclined_ , its normal defined by an angle $\theta$. I will not write every detail here, but its clear that we can use $A$ as a placeholder for the area and we can define $\sigma_x$ to be $\sigma_x = \frac{F}{A}$ and therefore in the diagram below we will use only stress. ![Problem](http://i.imgur.com/u4PU86j.png?1) Now, I will refrain working with vector quantities, I will only use scalar relations. If you take a look at this triangle, its easy to see that the area is projected to the section and we can get the relationship: $$ A' = \frac{A}{\cos(\theta)} $$ Now the rest of the equations go by the books: $$ T = \frac{F}{A'} = \frac{F}{A}\,\cos(\theta) = \sigma_x\,\cos(\theta) $$ $$ T_n = T\,\cos(\theta) $$ $$ T_s = T\,\sin(\theta) $$ My dilemma starts here. First of all _notice_ how in each of the above expressions the value for the length of the object is not taken into consideration. Also, we're working with areas and forces hence the length itself is not part of the equation at all even as a _contained_ information. Now that would be all fine, if it were not for the following section(to simplify, I've taken the origin to be at the center of the object). ![Section C](http://i.imgur.com/CZYfjZA.png) Now as you can see this section of the object gets very interesting. Clearly the formula for $A'$ does not hold anymore because the area itself is shrinking on the left side. To make things even worse, notice how the area on the right appears. The two areas are clearly linked, in this case, the sum of the two should be equal to the original area - from the fact that the height appearing on the right is equal to the height subtracting from the left. Calculating the angle at which the breaking point is made, that is when the section plane cuts the corner is easily calculated using: $$ \tan(\theta_k) = \frac{l}{h} $$ Where $ \theta_k $ stands for the critical angle, just so we can refer to it. Now the question that is on my mind is what is the distribution of the stresses on this section. Clearly there is stress on the left, there's stress on the right. Now I am confused as to the distribution of stress at this angle. # My Work Now, I've attempted a solution here. First, I am in a tight spot between assumptions. Clearly the original force $F$ must still be present here. But the force distribution is confusing me at this point. The best idea I could come up with, is the following. I would be grateful for your review. ## FBD ![Problem 3](http://i.imgur.com/0DIhQMU.png) I've only written here $F_1$ and $F_2$ for visualization purposes, because personally I don't think that we will need the $F_2$. Knowing that $F_1 = F$ it's obvious that they will cancel each other out, and they will also cancel the moments they create against the newly formed centroid on the trapezoid. Hence I think the only remaining task is to compute the areas of each individual piece and compute stresses. ## Areas Computing the individual areas for the cuts, since we won't need the area at where supposedly $F_2$ was located, we will ignore that for the time being. Though it will come naturally as we try to find the left area. The inclined plane area is the easiest to find, clearly, if we project the length $l$ to the top of the smaller rectangle we can see that using trigonometric relations we can find: $$l\,'=\frac{l}{\sin(\theta)}$$ Multiply that by the base and you've got the area $A'=b l\,'$. To find the areas of the left and right sections one must notice that the amount we deviate from the original height we add to the right side. So if we're to project the length like we did last time we get this: ![Area2](http://i.imgur.com/9m1MbQ0.png) From the figure above it's clear that the following relation holds true: $$ h = h' + 2\,h_2 $$ From trig we know $h'=l\cot(\theta)$. From this, $h_2$ is defined by: $$ h_2 = \frac{1}{2}(h - l\cot(\theta)) $$ Now $ h_1 = h - h_2 $ and after some mathmagic we should get the following: $$ h_{1/2} = \frac{1}{2}(h\pm l\,\cot(\theta)) $$ Now let's multiply this by the base and we've got ourselves $A_1 = h_1\,b$. ## Stress Now we have two stresses, namely $\sigma_{x_1}$ and $T$. Let's analyze the relations. The forces clearly have an eccentric deviation from the centerline. Now, is this going to be a problem? I seem to think that it won't be, and I base my logic on the fact that they will cancel the bending moments as we said above. The formulas are: $$\sigma_{x_1} = \frac{F}{A_1}$$ $$T = \frac{F}{A'} $$ Now obviously these are general equations, one could always dig deeper and I believe we could even express these stresses as a function of elementary stresses, namely $\sigma_x$ and some form of this other _tangential_ stress $F/(l\,b)$. ## Appendix To be completely accurate, the formula for $\theta_k$ seems to be a bit more complex, since there are four corners on the cut (in our case). Knowing that paired corners are symmetric to the x-axis we can use this equation: $$\theta_k(n) = |\arctan(\frac{l}{h}) + \pi\,n|$$ where $n\in(-2,-1,0,1)$. Is this ugly or is it just me? # Closing I hope to get some review on the ideas here, I'd really love to know how would this get solved correctly. And please someone, help me fix the appendix formula.


title: A physical quantity that is a real combination and commutability

text: Suppose that a matrix $$A ~=~ x_1 B + x_2 C$$ is a linear combination of two self-adjoint matrices $B$ and $C$. I'm interested in when $A$ represents a physical quantity. When the linear combination is a complex combination, then $B$ and $C$ have to be commutable for $A$ to represent any physical quantity, cf. this Phys.SE post. Now suppose that $x_1$ and $x_2$ are real. What happens in this case? If $B$ and $C$ are noncommutable, does $A$ still represent physical quantity?


title: How to find proper speed (relate to homogeneous cosmic background) of the galaxy by given redshift z and observing angular velocity?

text: The galaxy moves of the celestial sphere. It is given that proper speed is transverse to the observer and it must to find this speed in the moment of light emission. The motion is in the FLRW universe. Does it relate to the comoving distance?


title: Showing the Poincare invariance of a term

text: I know that this is a simple question! But I would like to know the details. How we can show that the term $$A_\mu(x)\dot{x}^\mu$$ is global and local Poincare invariant? Where $A_\mu(x)$ is supposed to be a four-covector and This term is a part of reparametrization-invariant square- root Lagrangian $\mathcal{S}=\int d\tau[-mc\sqrt{-(\dot x^\mu)^2}+\frac{e}{c}A_\mu\dot x^\mu]$ where $x^\mu$ are parametric equations of the physical trajectory $x_i(t)$. We assume $\frac{dt}{d\tau}>0$ and $A^\mu$ is electromagnetic four-vector potential.


title: Where does this equation originate from? (found in the Big Bang Theory)

text: Recently, I've been watching "The Big Bang Theory" again and as some of you might know, it's a series with a lot of scientific jokes in it - mostly about Physics or Mathematics. I understand most of the things mentioned in the series and whenever I don't understand a joke, I just look up the knowledge I'm missing on wikipedia - e.g. I learnt about Schrödinger's cat in this way. However, at one point, I did not know how to proceed, which is why I'm asking this question. At a Physics quiz, the participants are asked to "solve" the following equation: !["Solve the equation!"](http://i.stack.imgur.com/GXfQe.jpg) The solution turns out to be $-8 \pi \alpha$. My questions are: What is the meaning of this equation? How does one obtain it? And of course, how does one solve it and is the solution given by the university janitor (in the series) correct? Also, I am sorry if this is an equation found in Physics, I didn't know.


title: Why can we assume independent variables when using Lagrange multipliers in nonholonomic systems?

text: I'm studying from Goldstein's _Classical Mechanics_. In section 2.4, he discusses nonholonomic systems. We assume that the constraints can be put in the form $f_\alpha(q, \dot{q}, t) =0$, $\alpha = 1 \dots m$. Then it also holds that $\sum \lambda_\alpha f_\alpha = 0$. Using Hamilton's principle (i.e. that the action must be stationary), we get that $$\delta \int_1^2 L\ dt = \int_1^2 dt\ \sum_{k=1}^n \left(\frac{\partial L}{\partial q_k} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q_k}}\right)\delta q_k = 0. $$ But we can't get Lagrange's equations from this because the $\delta q_k$ aren't independent. However, if we add this with $\sum \lambda_\alpha f_\alpha = 0$, it follows that $$\delta \int_{t_1}^{t_2} \left(L +\sum_{\alpha=1}^m \lambda_\alpha f_\alpha\right)\ dt = 0.$$ And then Goldstein says that > The variation can now be performed with the $n\, \delta q_i$ and $m\, > \lambda_\alpha$ for $m+n$ independent variables. Why have the variables suddenly become independent? First we had $n$ dependent variables, why do we now have $m+n$ independent ones?


title: A Puzzle about $SO(3)$

text: Lie algebra of nonabelian group is $[T^a,T^b]=if^{abc}T^c$. For $SO(3)$ case, is the representation $T^a_{ij}=-i\epsilon^{aij}$ fundamental or adjoint? The fundamental representation is defined as identifying $T^a$ the original generator. The adjoint representation is defined as identifying $T^a$ as structure constant. For $SO(3)$ case, generators seem to be same as structure contant. Reference:   1. Srednicki QFT


title: Is it possible for the universe to be infinite considering Zeno's and Hilbert's paradox?

text: I'm only in high school (junior) so goes easy on me. But, how do physicist refute these paradoxes? Considering there are a number of theories regarding the origin of the universe and some postulate that the universe is infinite, I assume they've come across this. Just looking for a direct answer to what should be a relatively simple question. Here are the two paradoxes (for those unaware): Zeno's paradox "A similar paradox arises if the past is infinite. If there exists an infinite past, we would never have the present day. If there was an infinite set of past events and each event requires the previous event to occur, would we ever have the present? Of course not. This is because if today is dependent upon the fact that yesterday happened, and there is an infinite set of these dependencies (i.e. forever) - today will have not occurred. This is similar to the library book example mentioned earlier." Hilbert's hotel "If the universe did not have a beginning, then the past would be infinite, i.e. there would be an infinite number of past times. There cannot, however, be an infinite number of anything, and so the past cannot be infinite, and so the universe must have had a beginning. Why think that there cannot be an infinite number of anything? There are two types of infinites, potential infinites and actual infinites. Potential infinites are purely conceptual, and clearly both can and do exist. Mathematicians employ the concept of infinity to solve equations. We can imagine things being infinite. Actual infinites, though, arguably, cannot exist. For an actual infinite to exist it is not sufficient that we can imagine an infinite number of things; for an actual infinite to exist there must be an infinite number of things. This, however, leads to certain logical problems. The most famous problem that arises from the existence of an actual infinite is the Hilbert’s Hotel paradox. Hilbert’s Hotel is a (hypothetical) hotel with an infinite number of rooms, each of which is occupied by a guest. As there are an infinite number of rooms and an infinite number of guests, every room is occupied; the hotel cannot accommodate another guest. However, if a new guest arrives, then it is possible to free up a room for them by moving the guest in room number 1 to room number 2, and the guest in room number 2 to room number 3, and so on. As for every room n there is a room n + 1, every guest can be moved into a different room, thus leaving room number 1 vacant. The new guest, then, can be accommodated after all. This is clearly paradoxical; it is not possible that a hotel both can and cannot accommodate a new guest. Hilbert’s Hotel, therefore, is not possible."


title: Metric tensor under coordinate transformation x to y(x)

text: Say I have a metric representation $g_{\mu\nu}$ in a coordinate system $x$ and I want to find the representation of the metric in a new set of coordinates $y = y(x)$. I know how to do this if you are given $x(y)$, as in this post. $g_{\mu' \nu'} = \frac{\partial x^{\mu}}{\partial y^{\mu'}} \frac{\partial x^{\nu}}{\partial y^{\nu'}} g_{\mu \nu}$ -------- $(1)$ But what if I'm only given $y=y(x)$, and it's tricky to figure out $x=x(y)$? Is there a method that uses partial derivatives $\frac{\partial y^{\mu'}}{\partial x^{\mu}}$ instead? Or is $(1)$ the only way?


title: Cause of buoyant force?

text: Can you explain to me what causes the buoyant force? Is this a result of a density gradient, or is it like a normal force with solid objects?


title: what is the basic form of the 'fire'?

text: > **Possible Duplicate:**   >  Is fire matter or energy? What is the basic form of fire? physics defines every entity by a basic form either solid or liquid or as a gas, example: water is liquid, ice as solid, water vapor: gas so what is fire? when I queried is 'Plasma a fire?' I landed in below link, where it is said that fire isn't plasma, but not defined which basic form it belongs to! the link: Is fire plasma? What I understand is, it's a form of energy caused by reaction of gases(?) so it's basically a gas! is it true?


title: Why is planar geometry preferred to observe ordinary Hall effect?

text: In the Physics Today article by Avron et.al. " _A Topological Look at the Quantum Hall Effect_ " Physics Today (2003) it is suggested that to observe ordinary Hall effect, planar geometry is preferred to bar geometry as below. I am trying to understanding the reason behind this. Can someone help me? Any reference would be helpful too. > hasty experiment . . . though without success.”Hall made a fresh start and > designed a different experiment, aimed at measuring, instead, the magneto- > resistance—that is, the change of the electrical resistance due to the > magnetic field. As we now know, that is a much harder experiment, and it too > failed. Maxwell appeared to be safe. Hall then decided to repeat Rowland’s > experiment.Following his mentor’s suggestion, Hall replaced the original > metal conducting bar with a thin gold leaf, to compensate for the weakness > of the available magnetic field.


title: Has anyone tried to do Galileo's tower of Pisa experiment at a molecular or atomic scale

text: Galileo is reputed to have demonstrated that unequal masses fall at an equal rate by dropping them off the tower of Pisa and observing that they hit the ground simultaneously. Has anyone tried to push this experiment to the limit at small scale or is the result observable from some other experiment?


title: What determines a state of matter?

text: Earlier I studied about the three states of matter-gaseous,liquid ans solid. Then, I came to know about Plasma and Bose-Einstein Condensate. Now, scientists are trying to explain superconductivity as a state of matter. But, I would like to know on what factors does scientists rely to decide whether a particular state is a state of matter?


title: How does weight add up to press on things?

text: I think I understand how pressure works with gases. More molecules bouncing around -> more random impacts -> stronger force. But I realized to my embarrassment that I don't understand how solid things press on each other, on a molecular level. Say I put a block of iron on my head. If I put another one on top of it, I feel twice the weight. The two blocks together can tear through thin paper where one block can't. But the contact between my head (or paper) and the blocks is just a very thin layer of atoms of the lower block's structure. If the lower block doesn't move when I put the upper one on it, what causes this thin layer to "press" on my head (or paper) more? When the two blocks together tear through thin paper, where does the force come from that acts on the paper molecules - it can't be gravity from the upper block, right? And how come that whatever this source of pressure is only depends on the weight of the upper block, and not on what it's made out of, iron or wood?


title: What is the result of applying a fourier transform n times to a distribution?

text: For a function applying the fourier transform twice is equivalent to the parity transformation, applying it three times is the same as applying the inverse of the fourier transform, and applying four times, is the identity transformation. Wikipedia has a good description of this. What do we get when we apply the transform multiple times to a distribution? For example applying it twice would look like this: $ \mathscr F \mathscr F T(f) = T(\mathscr F \mathscr F f) = T( f_-) = ? $ Where $f_- = f(-x)$. Here $\mathscr F$ is the fourier transform $T$ is a distribution, and $f$ is a function.


title: Solution of multiple materials

text: ## Introduction I'm trying to make some strawberry liqueur. The recipe I got says I should put alcohol, strawberries and sugar in a bowl and leave it soaking for 4-5 weeks. I read that making liqueur is not a chemical reaction, but only a solvent (mostly alcohol) acting on some fruit, herb or nut. So, intuitively, I would add the sugar only after the soaking time. I wonder if that makes a difference. ## Question If I dissolve material A in alcohol until it reaches saturation, how much of material B can I still dissolve in the solution?   * Nothing   * As much as with pure alcohol   * Something in between None of the materials are gaseous and both are soluble in alcohol.


title: Special relativity: how to prove that $g = L^t g L$?

text: We have $$X^\textrm{t}gX = 0 \iff X^\textrm{t}L^\textrm{t}gLX = 0,$$ where $X$ is a column vector of length four, $L$ is a non-singular $4 \times 4$ matrix, 't' denotes matrix transpose, and $$g = \left(\begin{matrix} 1 & 0 & 0 & 0 \\\ 0 & -1 & 0 & 0 \\\ 0 & 0 & -1 & 0 \\\ 0 & 0 & 0 & -1 \end{matrix}\right)\,.$$ doesn't it immediately follow that $g = L^\textrm{t}gL,$ since $$X^\textrm{t}gX = 0 \iff X^\textrm{t}\left(L^\textrm{t}gL\right)X = 0?$$ Why or why not? I ask because the proof in my book takes up an entire page, so I have a feeling that this argument is not sound. It shouldn't matter for the question, but $X$ and $L$ come from the following equations: $$X = \left(\begin{matrix} ct_2 \\\ x_2 \\\ y_2 \\\ z_2 \end{matrix}\right) - \left(\begin{matrix} ct_1 \\\ x_1 \\\ y_z \\\ z_1 \end{matrix}\right)\,,$$ where $t_1$, $x_1$, $y_1$, $z_1$ and $t_2$, $x_2$, $y_2$, $z_2$ are the inertial coordinates of two events, and $$\left(\begin{matrix} ct \\\ x \\\ y \\\ z \end{matrix}\right) = L \,\left(\begin{matrix} ct^\prime \\\ x^\prime \\\ y^\prime \\\ z^\prime \end{matrix}\right) + C,$$ which gives the Lorentz transformation from the primed to the unprimed inertial coordinate system.


title: What is "Energy" of a vacuum in the context of quantum theory?

text: I'm not a physicist or cosmologist, so I hope I am asking the right question with the right words here. My question regards the word "energy" as it pertains to quantum vacuum states, and to concepts such as Lawrence Krause's "something from nothing" theory. When I hear the word "nothing" I envision a "void" that is an empty set. But Krause, Eva Silverstein and J. Richard Gott (2 hr YouTube video of debate on "nothing" hosted by Neil deGrasse Tyson - last url segment = watch?v=sNh- pY3hJnY) seem to be saying, if I understand correctly, that this void has a non-zero energy state, and that quantum fluctuations create "something" out of nothing because of this. I have to say that this sounds a bit to me like hand waving and weasel words, but perhaps my ignorance is committing an injustice. But if this is what is happening, is this "nothing" actually a void? Energy is something, and that would make "nothing" contain something. And I fail to see how this could be so. The question: What, exactly, IS "energy." In usage in this context it sounds a lot like phlebotinum.


title: Why are some jenga pieces easier to remove than others?

text: Jenga is a game place with wooden blocks stacked on top of one another in an alternating pattern. Players take turns removing blocks from any layer and placing them on top. ![](http://i.stack.imgur.com/NxBEl.jpg) As the game progresses the tower gets higher and higher until it collapses. ![](http://upload.wikimedia.org/wikipedia/commons/3/34/Jenga.gif) For a given configuration of blocks is there a way to calculate wither the tower is going to collapse? I also want to know why some pieces are easier to remove than others...


title: stopping, moving of mobile phone when vibrating

text: A mobile phone move aside when it vibrates. How is that happening ? and most importantly is it possible to make any changes to the vibration motor to stop moving when vibrating or any other methods to stop that ? I have already read following links Would a phone move upon vibration in a completely uniform situation? How does a mobile phone vibrate without any external force?


title: How does physicists calculate the gravitational self collapsing force of a star?

text: The nuclear fusion taking place inside the stars opposes its gravitational self collapsing force. But, how does physicists calculate it? I just know the classical gravitational theory and not a bit of general relativity. So, I would like to take the situation in a classical way. For example, in terms of Lyttleton-Bondi Model for the Expansion of the Universe the expansion of the universe is explained if matter has a net charge. Imagine a spherical volume containing un-ionized atomic hydrogen gas of uniform density. Assume the proton charge is $ke$ (where $e$ is the charge of an electron (this situation is taken for net charge of matter)). How to calculate the gravitational self collapsing force of this so that I could calculate the value of $k$ for which the electrostatic repulsion opposes it and the volume doesn't collapse on itself? I know how to calculate the gravitational force between masses or simple shapes(found using integration). But,I have no idea on how to calculate this. (Even though Lyttleton-Bondi Model for the Expansion of the Universe model is discarded I would like to proceed with this as I don't know any high level math or other theories except classical mechanics taught in high school).


title: Mass-energy equivalence and Newton's Second Law of motion

text: According to Einstein's Mass-energy equivalence, $ E = mc^2$ OR $ m = \frac E{c^2}$..... (1) and According to Newton's Second Law of motion, $ F = ma$ OR $m = \frac Fa$ ..... (2) If we compare eq. (1) and eq. (2), we obtain; $\frac E{c^2} = \frac Fa$..... (3) If we multiply both the sides of eq. (3) with $c^2$, we get; $E = \frac Fac^2$ ..... (4) Is the above relation valid?


title: Why is solar eclipse so important for sunlight spectrum analysis?

text: People wait for decades to catch the chance of a solar eclipse to observe the sun. Why cannot they do it every noon?


title: Optimal speed for the water wheel

text: The hydroelectricity plants extract the potential energy of highly deployed massive object (water) as it falls down. Without turbine, all that energy would be converted into speed (kinetic energy) at the bottom of the waterfall and further into heat. The turbine produces energy by slowing water down. The efficiency of turbine, how much energy is extracted by turbine, can be charactarized by the exhaust speed: the faster is the output stream, the less efficient our turbine is since not all speed/energy is extracted. So, slower the turbine spins, the higher is its the efficiency. The extraction is 100% when turbine does not spin and no electricity is produced at all. So, there must be a trade-off between the efficiency and amount of the output, the trade-off determined by the turbine spinning speed (exhaust speed). How is it decided? I read that large modern water turbines operate at mechanical efficiencies greater than 90%. Since couple of percent losses are inevitable whatever you do, it seems that they say that theoretical efficiency is 100%. Identical efficiency is provided by switching power supply converters, which are 100% efficient in theory. I understand the secret exploited by SMPS. My question is how similar, 100% energy extraction, is achieved through the turbines, which seem to operate linearly (spinning at the same pace) rather than switching mode pumping. What is the water release speed when 100% energy extraction is achieved? This question is actually is not limited to water wheels. Today wind turbines are becoming more popular and I am curious how do you extract all power from the wind flow. If turbine spins quckly, the air is realased at high speed, which means that you do not slow down the flow, which means that it makes no work. On the other hand, if if you stop the flow completely, your turbine stops and no power is extracted either. What is the optimal turbine speed? tags: engeneering, efficiency


title: Potential in Relativistic Scalar Field Theory

text: My intention is to establish a Soliton equation. I have cropped a page from Mark Srednicki page no 576. I have understand the equation (92.1) but don't understand that how they guessed the potential in equation (92.2).![enter image description here](http://i.stack.imgur.com/YIlt3.jpg) EDIT: Contrast the above potential the author used this potential in equation (2) $$U(\phi)= \frac{1}{8} \phi^2 (\phi -2)^2.$$ My query is , are we using different potential just for our convenient using? or I need to redefine the $\phi^4$ theory to get the potential ?


title: Time reversed laser

text: Recently, I read an article on time reversed laser. I don't know why they call it a time reversed. I have a doubt that why they use two laser in the device. And what is an anti-laser? The device absorbs the particular frequency of light which means some energy is lost. What happens to the lost energy?


title: Does inertia increase with speed?

text: I have heard that when the speed of the object increase, the mass of the object also increase. (Why does an object with higher speed gain more (relativistic) mass?) So inertia which is related to mass, increase with speed? So, if I accelerate on a bus, my mass will increase and my inertia will increase for a while on the bus, until the bus stops?


title: Speed of air through valve

text: Right now I'm working on a hot air balloon in Physics formulas. At the moment it's flying up like it should, but once it is at a certain height we want it to go back down. After some research about hot air balloons we found out that they drop down by opening a valve at the top. So what we want to do is once the balloon gets at a certain height it will open the valve and drop down again. We can get the size of the valve and that it opens at a certain height. How can we calculate the speed of cold air going into the valve? We know the speed of the balloon, the size, temperature of the air in and outside of the balloon, air pressure etc. Is there some kind of formula we can use to calculate how fast the cold air will flow in and then cool of the air in the balloon making it drop down.


title: $\phi ^4$ theory explaining

text: In $φ^4$ theory we often write the Lagrangian as $$\mathcal{L}=\frac{1}{2}\partial^\mu \phi \partial_\mu \phi -\frac{m^2}{2}\phi^2 -\frac{\lambda}{4!}\phi^4 \tag {1}$$ If I want to write from the Relativistic Lagrangian then it takes $$\mathcal{L}=\frac{1}{2}\partial^\mu \phi \partial_\mu \phi -V \tag{2}$$ but how will I convert this equation to equation like (1) ? EDIT: I just want to get Equation (1) from equation (2) EDIT by joshphysics: What motivates choosing to study the $\phi^4$ potential as opposed to other potentials?


title: Why is the void devoid of an electric field?

text: For a region devoid of charge, maxwell's equation yields $\nabla \cdot \mathbf{E} = 0$ which still allows a constant field. So why is in electrostatics for the vacuum always $\mathbf{E} = 0$ assumed?


title: Theta parameter of QCD: Very small, periodic in $2\pi$ or both?

text: I've been learning about the theta parameter of QCD and I'm confused about the fact that it's supposed to be very small but at the same time some sources say that the Yang-Mills theory should be invariant to 2$\pi$ shifts in $\theta$ Sources that says that $\theta$ is small: (1)The CP Puzzle in the Strong Interactions (See page 6) (2) TASI Lectures on The Strong CP Problem (See page 19) Sources that say that Yang-Mills theory is invariant under $2\pi$: (3) Notes on Supersymmetry (See page 6) (4) Non-Perturbative Dynamics Of Four-Dimensional Supersymmetric Field Theories (See page 4) Now source (1) (amongst others) tells us that there exists a term of the form: $\theta n = \theta (1/32 \pi^2) \int d^4 x \ \epsilon^{\mu\nu\rho\sigma} \ F^a_{\mu\nu} F^a_{\rho\sigma}$ in the Lagrangian of our QCD, where the $\int d^4 x \ \epsilon^{\mu\nu\rho\sigma} \ F^a_{\mu\nu} F^a_{\rho\sigma}$ part gives CP violation. Thus we require $\theta = 0$ if want no CP violation. However to me this implies that the theory is not invariant under a shift in $2\pi$ of $\theta$ as $\theta = 2\pi$ would not make the above term disappear (you'd end up with $2\pi n$ rather than $0 \times n = 0$). Similarly how can we say that $\theta$ has to be small if $\theta$ can be shifted by any multiple of $2\pi$ to give the same theory? Is it the case that when we say '$\theta$ is small' we actually mean '$\theta$ modulo $2\pi$ is small'. Also is it the case that the above equation some how disappears for all values of $\theta$ that are multiples of $2\pi$, and not only $\theta = 0$? Theta vacuum effects on QCD phase diagram \- page 2 seems to imply that the CP violating term disappears for all value of $\theta$ that are multiples of $\pi$.


title: Simplified partial trace of two operators

text: If I have two operators A and B living in the Composite Hilbert Space $H_I \bigotimes H_{II} $ and I want to take the partial trace of $C=AB$ over the subspace $H_I$, i.e., $Tr_I[AB]$, is there any identity that can help me do this in terms of $Tr_I[A]$ and $Tr_I[B]$. Actually what I am interested in is the partial trace of the commutator $[A,B]$.


title: Explain the electric dipole moment vector

text: For a pure dipole situated at the origin, I don't understand why the dipole moment vector is $$ \vec{p} = p\cos \theta \hat{r} - p\sin \theta \hat{\phi} $$ Since the vector does not have any $r$ in it, the value of the vector is same any distance away from the origin if the angle is constant, which doesn't make sense to me. It would be very helpful if someone could derive it because Griffiths uses this value without explanation.


title: Earth's stationary iron core

text: Why is the earth's iron core stationary, while the liquid matal circles around it creating the magnetic shield. Don't understand how can the entire planet rotate where as the planet's center is stationary. Anyone could explain this?


title: Non-computable numbers in Physics

text: It is known that there are real numbers that can't be calculated (non- computable numbers). Quite probably that some physical phenomena (it is possible still undetected) depend on this numbers. Whether means it, what we can't construct the theory of this phenomenon and-or (if probably to construct) to do predictions (i.e. to check up this theory in practice)?


title: What tests can I do at home to identify any metal?

text: Regarding a question on another Stack Exchange site which identifies a weighted Lego brick to be some kind of metal, I wanted to know how to identify a metal, perhaps limited to household objects like scales or a bowl of water to measure volume.


title: Why are bubbles formed in the empty space of soda when turned upside down?

text: I know this may actually be a chemistry question, but I don't know where else to ask. When I have a soda plastic bottle, I turn it upside down so the soda fills the empty space, and bubbles start growing on the walls of the plastic. So, why does this happen? I can think of something like a film forming around the empty space, and it reacts with the liquid, if so, what is it? why does it react that way?


title: Sources to learn about Greens functions

text: For a physics major, what are the best books/references on Greens functions for self-studying? My mathematical background is on the level of Mathematical Methods in the physical sciences by Mary Boas.


title: What physical forces pull/press water upwards in vegetation?

text: Each spring enormous amounts of water rise up in trees and other vegetation. What causes this stream upwards? Edit: I was under the impression that capillary action is a key factor: the original question therefore was: what are the fundamental forces involved in capillary action? Can perhaps anybody make an estimation of the amount of gravity energy that is involved in this process earth-wide each season?


title: Expressing a particle's matter wave in terms of its momentum

text: I'm following Zettili's QM book and on p. 39 the following manipulation is done, Given a localized wave function (called a wave packet), it can be expressed as $$\psi(x,t) = \frac{1}{\sqrt{ 2 \pi}} \int_{-\infty}^{\infty} \phi(k) e^{i(kx-\omega t)} dk$$ Now use the de broglie relations: $p = \hbar k$ and $E = \hbar \omega$ and define $\tilde{\phi}(p) = \phi(\frac{k}{\hbar})$. This should yield $$\psi(x,t) = \frac{1}{\sqrt{ 2 \pi \hbar}}\int_{-\infty}^{\infty} \tilde{\phi}(p) e^{i(px-E t)/ \hbar} dp$$ but I get $$\psi(x,t) = \frac{1}{\sqrt{ 2 \pi} \hbar}\int_{-\infty}^{\infty} \phi\biggl(\frac{p}{h}\biggr) e^{i(px-E t)/ \hbar} dp$$ when I make the change-of-variable $k=\frac{p}{\hbar}$. What am I missing?


title: How to derive inverse Fourier transform for periodic functions (in crystal lattice)?

text: I would like to derive the following two well-known formulas that work for crystal lattice [1]: $$ F[f(\mathbf{x})] \equiv \tilde f(\mathbf{G}) = {1\over\Omega_\mathrm{cell}} \int_{\Omega_\mathrm{cell}} f(\mathbf{x}) e^{-i\mathbf{G} \cdot \mathbf{x}}\,d^3 x $$ $$ F^{-1}[\tilde f(\mathbf{G})] = f(\mathbf{x}) = \sum_{\mathbf{G}} \tilde f(\mathbf{G}) e^{+i\mathbf{G} \cdot \mathbf{x}} $$ Specifically, I want to derive them from the general 3D Fourier transform (the other way is to simply plug the second formula into the first, one obtains a delta function and obtains an identity --- see this question where I have worked this out in details, but here I don't want to use this approach). How do I derive the second formula? Following [1], here is how to derive the first formula from the basic definition of a 3D Fourier transform, divided by the volume of the crystal $\Omega_\mathrm{crystal}$ (to make it finite): $$ F[f(\mathbf{x})] \equiv \tilde f(\boldsymbol\omega) = {1\over\Omega_\mathrm{crystal}}\int_{\Omega_\mathrm{crystal}} f(\mathbf{x}) e^{-i\boldsymbol\omega \cdot \mathbf{x}}\,d^3 x = $$ $$ = {1\over\Omega_\mathrm{crystal}} \sum_\mathbf{n} \int_{\Omega_\mathrm{cell}} f(\mathbf{x}+\mathbf{T}(\mathbf{n})) e^{-i\boldsymbol\omega \cdot (\mathbf{x}+\mathbf{T}(\mathbf{n}))}\,d^3 x = $$ $$ = {1\over\Omega_\mathrm{crystal}} \sum_\mathbf{n} \int_{\Omega_\mathrm{cell}} f(\mathbf{x}) e^{-i\boldsymbol\omega \cdot (\mathbf{x}+\mathbf{T}(\mathbf{n}))}\,d^3 x = $$ $$ = {1\over\Omega_\mathrm{crystal}} \sum_\mathbf{n} e^{-i\boldsymbol\omega \cdot \mathbf{T}(\mathbf{n})} \int_{\Omega_\mathrm{cell}} f(\mathbf{x}) e^{-i\boldsymbol\omega \cdot \mathbf{x}}\,d^3 x = $$ $$ = {1\over\Omega_\mathrm{crystal}} N_\mathrm{cell} \int_{\Omega_\mathrm{cell}} f(\mathbf{x}) e^{-i\boldsymbol\omega \cdot \mathbf{x}}\,d^3 x = $$ $$ = {1\over\Omega_\mathrm{cell}} \int_{\Omega_\mathrm{cell}} f(\mathbf{x}) e^{-i\boldsymbol\omega \cdot \mathbf{x}}\,d^3 x $$ In here, the function $f(\mathbf{x})$ is periodic: $f(\mathbf{x}+\mathbf{T}(n_1, n_2, n_3)) = f(\mathbf{x})$ and the sum $\sum_\mathbf{n} e^{-i\boldsymbol\omega \cdot \mathbf{T}(\mathbf{n})} = \sum_\mathbf{n} 1 = N_\mathrm{cell}$ for $\boldsymbol\omega=\mathbf{G}$, where $\mathbf{G}$ are reciprocal space vectors (defined by $e^{i\mathbf{G} \cdot \mathbf{T}(\mathbf{n})} = 1$). For $\boldsymbol\omega\neq\mathbf{G}$, the sum is bounded, and so in the limit $\Omega_\mathrm{crystal}\to\infty$ the factor before the integral sign above goes to zero. For the second formula, there is no hint in [1] how to proceed. Here is my best effort so far: $$ F^{-1}[\tilde f(\boldsymbol\omega)] = f(\mathbf{x}) = {\Omega_\mathrm{crystal}\over(2\pi)^3}\int_{-\infty}^{\infty} \tilde f(\boldsymbol\omega) e^{+i\boldsymbol\omega \cdot \mathbf{x}}\,d^3 \omega = $$ $$ = {\Omega_\mathrm{cell}N_\mathrm{cell}\over(2\pi)^3}\int_{-\infty}^{\infty} \tilde f(\boldsymbol\omega) e^{+i\boldsymbol\omega \cdot \mathbf{x}}\,d^3 \omega = $$ $$ = {N_\mathrm{cell}\over\Omega_\mathrm{BZ}} \sum_{\mathbf{G}} \int_{\Omega_\mathrm{BZ}} \tilde f(\mathbf{G}+\boldsymbol\omega) e^{+i(\mathbf{G}+\boldsymbol\omega) \cdot \mathbf{x}}\,d^3 \omega = $$ $$ = {N_\mathrm{cell}\over\Omega_\mathrm{BZ}} \sum_{\mathbf{G}} e^{+i\mathbf{G} \cdot \mathbf{x}} \int_{\Omega_\mathrm{BZ}} \tilde f(\mathbf{G}+\boldsymbol\omega) e^{+i\boldsymbol\omega \cdot \mathbf{x}}\,d^3 \omega = \cdots $$ Here $\Omega_\mathrm{BZ} = {(2\pi)^3 \over \Omega_\mathrm{cell}}$ is the volume of the Brillouin zone. I have moved the integration over $\boldsymbol\omega$ to the Brillouin zone. As you can see, it's quite close, but I can't figure out how to finish it. Any ideas? [1] Martin, R. M. (2004). Electronic Structure -- Basic Theory and Practical Methods (p. 642). Cambridge University Press.


title: Definition of mean free time in the Drude model

text: In the Drude model they derive a formule for the conductivity of a conductor. I wonder though how the main free time $\tau$ is defined in this formula. Wikipedia says that it is "the average time between subsequent collisions". But I have two possible interpretations of this:   1. the average time an electron travels before colliding (which it seems to imply).   2. The average amount of time electrons have been travelling at a given time $t$ ($\tau$ will be substantially smaller in this definition). The first definition seems to me how they describe it, while the second definition seems to be implied by the formulas. I wonder the same for "mean free path", which seems to be analogous.


title: How to represent a multiple in the frequency domain

text: So, in my previous question Where does this formula for prediction of a multiple wave come from?, I get that using this picture: ![enter image description here](http://i.stack.imgur.com/NPiCC.png) we have so far written the time it takes for a multiple to travel in the form: $$T_2 = T_1 + \left( \frac{1}{v} \frac{2h}{\cos (\theta)} \right)$$ where $$\cos(\theta) = \frac{h}{\sqrt{\frac{x^2}{4} + h^2}}.$$ AFter substituting in for $\cos(\theta)$, this can then be re-written as $$T_2 = T_1 + \sqrt{\tau^2 + \frac{x^2}{v^2}}$$ where $\tau = \frac{4h^2}{v^2}$. From here though, it then says that we can represent this in the frequency domain as: $$R h(\omega) e^{2 \pi i f (T_1 + \sqrt{\tau^2 + \frac{x^2}{v^2}}}$$ where $R$ is some constant including decay and $h(\omega)$ is the source wavelet. I don't understand how they have managed to transform the previous equation into this one. Where has the exponential come from?


title: What is the condition for accelerating charge to radiate?

text: I was always taught that any accelerating charge produces radiation, but I don't think this condition is sufficient condition. For instance, any free charge on Earth is accelerated due to Earth orbiting the Sun but it doesn't produce radiation.


title: A question about verifying the transverse of electric field

text: I came accross a question about verifying the transverse of electric field in Peskin and Schroeder's QFT p179. Given > $$ \mathcal{A}^{\mu}(\mathbf{k}) = \frac{ -e}{| \mathbf{k} | } \left( > \frac{p'^{\mu}}{k \cdot p'} - \frac{p^{\mu}}{k \cdot p} \right) \tag{6.7} $$ and > $$ \mathcal{E} = -i \mathbf{k} \mathcal{A}^0 (\mathbf{k}) + i k^0 > \mathcal{A}(\mathbf{k}) \tag{6.9} $$ it is said > you can easily check that the electric field is transverse: $\mathbf{k} > \cdot \mathcal{E}(\mathbf{k}) =0 $ I tried to vertify this almost trivial relation and I didn't get it... Here is my attempt $$ \mathbf{k} \cdot \mathcal{E}(\mathbf{k}) = \left( \frac{ -e}{| \mathbf{k} | } \right) \left[ - i \mathbf{k} \frac{ \mathbf{k} p'^0 - k^0 \mathbf{p}'}{ k \cdot p'} + i \mathbf{k} \frac{ \mathbf{k} p^0 - k^0 \mathbf{p}}{ k \cdot p} \right] \tag{1}$$ I may use $p^{\mu}=(p^0, \mathbf{0})$ (since the particle is initially at rest) and $k^0=|\mathbf{k}|$. It seems that Eq. (1) is still not necessarily zero. Would you provide, at least, any hint how to show Eq. (1) is zero?


title: Does length contraction affect travel time?

text: If an astronaut leaves planet $A$ for planet $B$ at speed $v$, will the time (measured by the astronaut's clock) that it takes for the astronaut to reach planet $B$ be less than the distance between the planets divided by the speed because of length contraction?


title: Resonant inductive coupling and Schumann resonances

text: I was reading about WiTricity (http://en.wikipedia.org/wiki/WiTricity) a technology developed by MIT to wirelessly transmit electricity through resonance, and I have this question: Given the phenomenon of resonant inductive coupling which wikipedia defines as: > the near field wireless transmission of electrical energy between two coils > that are tuned to resonate at the same frequency. > http://en.wikipedia.org/wiki/Resonant_inductive_coupling And the Schumann resonances of the earth ( ~7.83Hz, see wikipedia), would it be theoretically possible to create a coil that resonates at the same frequency or one of it's harmonics (7.83, 14.3, 20.8, 27.3 and 33.8 Hz) to generate electricity? I have a feeling that these wavelengths may be too big to capture via resonance (they are as large as the circumference of the earth if I understand it correctly), so alternatively would it be possible to create a coil that resonates with one of the EM waves that the sun sends our way?


title: Is PHD in Theoretical Physics(Quantum Mechanics) on my own possible?

text: Do you think it is possible to earn a PHD degree in Theoretical Physics on your own? I have a M. Sc. in Applied Computer Science but I really want to go in the direction of Physics. Is there a way one could perform a significant work in this field without a team of other PHD-students and/or lab experiments? Is it even officially possible to acquire the PHD degree in Physics if one has B. Sc. in Applied Mathematics and Ms. Sc. in Applied CS only?


title: Kinematics simple question

text: Question: In order to protect himself from the rain, a person is standing holding an umbrella at right angle to the horizontal surface. The rain is falling at 10m/s when the velocity of the wind is zero. Suddenly, wind begins to blow at a speed of 20m/s, towards 30 degrees south of west. Now by what angle does the person have to turn his umbrella in order to protect himself from the rain? What I did: Split the speed of wind into components: $$v(x) = 20\cos30$$ $$v(y) = 20\sin30 + 10$$ Thus, I found the angle that the resulting rain makes with the vertical: $$\tan\theta = \frac{10\sqrt{3}}{20} = \frac{\sqrt{3}}{2}$$ $$\theta = \arctan\frac{\sqrt{3}}{2}$$ However, the answer is given to be $\arctan2$, where have I made a mistake?


title: What sets the resolution on analog film?

text: When taking a picture with old fashioned film what sets the resolution of the picture? Is it the wavelength, or the chemical makeup of the film?


title: Problems that Lagranges equations of the 1st kind can solve whereas the 2nd kind can't?

text: Can anyone give examples of mechanics problems which can be solved by Lagrange equations of the first kind, but not the second kind?


title: Difference between the 'phonons and virtual photons'

text: I understand what are virtual photons and the difference between the real and virtual photons. However, I am not able to clearly distinguish the difference between the 'phonons and virtual photons'. Both are involved in non radiative energy transfer.


title: Finding work done by rotational force?

text: > A disk with a rotational inertia of 5.0 kg·m2 and a radius of 0.25 m rotates > on a fixed axis perpendicular to the disk and through its center. A force of > 2.0 N is applied tangentially to the rim. As the disk turns through half a > revolution, the work done by the force is: This was another question on my test that I missed. I didn't know where to get started. I know the work done is the integral of the torque. However, I don't know how to get a function that I can integrate from this example. Any ideas?


title: UV reflective surfaces

text: Do surfaces that reflect visible light efficiently also reflect UV light? If not, are there surfaces that do? (I have a large array of UV LEDs that I need to make larger and more diffuse, so I'm considering reflecting it off nearby surfaces and into some diffuse material.)


title: Why do we take the first derivative of EPR/ESR spectra?

text: Apologies if this question is a bit too chemistry-flavoured. In electron paramagnetic resonance spectroscopy, there's a practically ubiquitous convention of plotting the first derivative of the absorption with respect to field, rather than the simple absorption. Why is this?