title: Are there any good references on the "gravitational" curvature of spacetime of a moving mass being distorted due to special relativity?

text: In this Wikipedia paragraph suggesting an explanation for the phenomenon of inertia, it claims: > Another physicist, Vern Smalley, has derived the Lorentz transformation for > mass by assuming that the gravitational field identified by Einstein is > distorted during acceleration. Einstein's assumption was that the field > would not be deformed during acceleration, but Smalley explored what > happened if it was. He found that during acceleration, the gravitational > field of a mass was compressed in the direction of the acceleration and > rarefied in the opposite direction. The gravitational field potential of the > mass was already propagating at the speed of light, and by forcing the field > to move faster, it was resisting the acceleration by resisting compression > and rarefaction. This resistance to acceleration is what we call inertia. > Throughout the process of acceleration, even up to the speed of light, the > total quantity of gravitational field material remains constant, but the > shape of the field identifies the inertial mass. ref: Smalley, "Deriving > Mass Inertia and Time Interval Dilation." Who is Vern Smalley? But, more importantly, are there any good references on the "gravitational" curvature of spacetime of a moving mass being distorted due to special relativity?


title: Parabolic motion and air drag

text: Are this equations correct, in order to calculate the parabolic motion of an arrow with the computation of the drag with the air? $$ \begin{cases} x(t)=\left(v_0-\frac{1/2C_DA\rho v_0^2}{m}t\right)\cos(\theta)t\\\ y(t)=\left(v_0-\frac{1/2C_DA\rho v_0^2}{m}t\right)\sin(\theta)t-\frac{1}{2}gt^2+h \end{cases} $$ Update: correction. $$ \vec{r}= \begin{vmatrix} \left(v_0-\frac{C_DA\rho v_0^2}{4m}t\right)\cos(\theta)t \\\ \left(v_0-\frac{C_DA\rho v_0^2}{4m}t\right)\sin(\theta)t-\frac{1}{2} g t^2+h \\\ 0 \end{vmatrix} $$ $$ \vec{a} = \ddot{\vec{r}} = \begin{vmatrix} -\frac{C_DA\rho v_0^2}{2m}\cos(\theta) \\\ -\frac{C_DA\rho v_0^2}{2m}\sin(\theta) -g \end{vmatrix} $$ Update for AlanSE: \begin{equation} \begin{split} m\frac{d^2 x(t)}{d t^2}&=-\frac{C_DA\rho}{2}\sqrt{\left(\frac{dx(t)}{dt}\right)^2+\left(\frac{dy(t)}{dt}\right)^2}\frac{dx(t)}{dt},\\\ m\frac{d^2 y(t)}{dt^2}&=-mg-\frac{C_DA\rho}{2}\sqrt{\left(\frac{dx(t)}{dt}\right)^2+\left(\frac{dy(t)}{dt}\right)^2}\frac{d y(t)}{dt}. \end{split} \end{equation} Related Cauchy problem: \begin{equation} \begin{cases} \displaystyle x(0)=0,\\\ \displaystyle y(0)=h. \end{cases} \end{equation}


title: How do you find the magnetic field corresponding to an electric field?

text: If we are given the electric field $\vec E$ how can I find the corresponding magnetic field? I think I can use Maxwell's equations? In particular, $\nabla\times \vec E= -{\partial \vec B\over \partial t}$? But is it completely determined? Since we only have a partial derivative?


title: Torque in an accelerated frame of reference

text: I'm taking a mechanics course this year and are currently studying rotational motions for my finals. The book we have explains it mostly fine, but there is one question I can't seam to find an answer to in it, nor on the internet. It's probably just my fault for being to bad at figuring out the best search words though. ^^, So here goes: Does torque and spin work the same way in an accelerated frame of reference as in an inertial frame of reference. The reason I'm wondering is because I know how to handle torque in an inertial frame of reference (I.e. from a point away from the system I'm analysing), but the lecturer in the course I'm taking seams to prefer to use the center of mass as the reference point, which i can see why will work if the system is not accelerating. My question is then, if my center of mass is accelerating, will using the center of mass as the reference point when calculating torque and spin still work? Thanks in advance


title: If you removed every particle from space...?

text: I'm trying to find something Einstein (I think) said about time...It was something like.. "If you removed every particle from space and were left with only one pocket watch (clock, timepiece?), time would still exist and the watch would be ticking?" That is not exactly it so I can't find it in google. Do you know of anything he said similar to that? It is not this one... “It was formerly believed that if all material things disappeared, time and space would be left. According to the relativity theory, however, time and space disappear together with the things.”


title: Statistical physics of molecular dissociation of a diatomic gas

text: Say there are $N$ atoms of type $A$ in a box of volume $V$ and they are undergoing a reversible association-dissociation reaction $A + A = A_2$. Let an $A$ atom have mass $m$, and hence the molecule $A_2$ has mass $2m$. Assume that the temperature $T$ is high-enough that Boltzmann statistics applies. Say $N_1$ be the number of $A$ atoms and say $N_2$ be the number of $A_2$ molecules at equilibrium. Then $N_1 + 2N_2 = N$. If there had to be $N$ $A$-type atoms in the the box then I guess the partition function $Z$ would be given by, $$Z = \frac{1}{N!} [ \int e^{-\frac{\beta p^2}{2m}} \frac{d^3p d^3q}{h^3} ] ^N = \frac{1}{N!} [ \frac{V}{h^3} (2\pi m kT)^{\frac{3}{2}} ] ^N .$$ Say that the dissociation energy of the reaction is $E_0$. Say $N_1$ be the number of $A$ atoms and say $N_2$ be the number of $A_2$ molecules at equilibrium. Then $N_1 + 2N_2 = N$. I want to write the partition function for the above mixed system in equilibrium. Using the expression for $Z$ as stated above, I am guessing that this partition function (say $Z_N$) would look like, $$Z_N = \sum _ {N_1 + 2N_2 = N} e^{-\beta N_2 E_0} \frac{1}{N_1 !} [ \frac{V}{h^3} (2\pi m kT)^{\frac{3}{2}} ] ^{N_1} \frac{1}{N_2 !} [ \frac{V}{h^3} (4\pi m kT)^{\frac{3}{2}} ] ^{N_2}. $$ (In the above I am using the idea that for $N_2$ molecules to have formed $N_2$ $E_0$ (binding) energy would have been released and hence that much of extra energy has to be accounted for which is not captured in the partition functions over the $N_1$ atoms and $N_2$ molecules...)   * Is the above expression for the partition function correct ?  One further introduces the grand canonical partition function (say $Z_{\mu}$ ) for the above system with a chemical potential $\mu$ to "take care of the constraint $N_1 + 2N_2 = N$", $$Z_{\mu} = \sum _ {N=0} ^\infty e^{\beta \mu N} Z_N.$$   * Firstly I don't understand the meaning of the $\mu$ in the above. (I can't relate it to what I would usually think as chemical potential) I mean..isn't the above expression for $Z_{\mu}$ independent of $N_1$ and $N_2$?    * How does the $Z_{\mu}$ help find the expectation value for $N_1$ and $N_2$?   * If my expressions for $Z_N$ is correct then does $Z_{\mu}$ have a closed form expression?  All this is to understand the claim that as expectation values the following relationship apparently holds, $$\frac{N_1 ^2} {N_2} \sim e ^{-\frac{|E_0|}{kT}} .$$ I would be glad to know of other insights along the above.


title: QM interpretation of Larmor precession

text: Typically when I think of a spin moment in an external field, I visualize it classically in terms of a magnetic moment precessing around the external field vector. Now how is this described quantum mechanically? My guess would be that the external field causes transitions between the up-state and down-state at the Larmor frequency. However if the Zeeman energy was much greater than the thermal energy, wouldn't the spin need to stay in the low energy state? Does that mean the spin ceases precession in the classical picture?


title: Do we live in an integer dimension?

text: I have read that there exist non-integer fractal dimensions and the images generated from these dimensions look organic and they seem to provide a new way of describing the world around us, which raises the question. How many dimensions (spatially or otherwise) do we live in? 3, 3.1, 3.5 as well as 1 time dimension or could time itself exist in a fractal dimension of its own.


title: Lie derivative of a scalar and PDE

text: I am reading about differential geometry, and in particular the Lie derivative and its relation to (relativistic) hydrodynamics. In particular, I was wondering if, given two scalar functions $f(\mathbf{x}),~g(\mathbf{x})$, it is possible to solve the following equation: $$\mathcal{L}_Z[f]=Z^\mu\partial_\mu f=g,$$ where $Z=Z^\mu\partial_\mu$ is an arbitrary vector field. Since I am thinking of hydrodynamics here, I am considering the manifold associated to $Z$ to be Euclidean, but I would be interested to know how it works for arbitrary manifolds. Is their a theorem from our mathematician friends that guaranties the existence of a solution (I am okay with existence, I don't necessarily need a way to find the solution)? If there is no such proof, what are the restriction one needs to impose on $Z$? If there is a proof, is it also valid for tensor quantities, ie is there a solution to $$\mathcal{L}_Z[f^{\mu_1...\mu_n}_{\nu_1...\nu_m}]=g^{\mu_1...\mu_n}_{\nu_1...\nu_m}.$$


title: How is the size of the particles is determined?

text: What is the size of atomic and subatomic particles, like proton, neutron, photon etc? Is it defined based on some quantum characterics as _de Broglie_ wavelength or _Compton_ wavelength?


title: Is rate of temperature change constant?

text: Is the rate of change in temperature for an object constant? For example, from 0ºC to 25ºC, or from 25ºC to -10ºC? Does it take the same amount of time to increase temperature from 1º to 2º as 24º to 25º, or 0 to -1 as -9 to -10.


title: Can silicon droplets bouncing on a vibrating surface be a model for Quantum Mechanics?

text: In   * this video on youtube it is claimed that silicon droplets bouncing on a vibrating surface show behaviour in analogy to particle/wave duality in Quantum Mechanics. Is this true? Did they really reproduce the double slit experiment (the video doesn't show that)? And what happens to the silicon droplets on a single slit? What about Quantum tunneling? Which effects of Quantum Mechanic can be "emulated" with bouncing silicon droplets and which can't?


title: How to interpret the magnetic vector potential?

text: In electromagnetism, we can re-write the electric field in terms of the electric scalar potential, and the magnetic vector potential. That is: $E = -\nabla\phi - \frac{\partial A}{\partial t}$, where $A$ is such that $B = \nabla \times A$. I have an intuitive understanding of $\phi$ as the electric potential, as I am familiar with the formula $F = -\nabla V$, where $V$ is the potential energy. Therefore since $E = F/q$, it is easy to see how $\phi$ can be interpreted as the electric potential, in the electrostatic case. I also know that $F = \frac{dp}{dt}$, where $p$ is momentum, and thus this leads me to believe that $A$ should be somehow connected to momentum, maybe like a "potential momentum". Is there such an intuitive way to understand what $A$ is physically?


title: Is it theoretically possible for the orientation angle of a projectile to remain exactly equal to the orientation of velocity?

text: This question is sparked by my answer to this question: Is this simulation following real physics? After examining the math, I don't see how it is theoretically possible for the situation simulated to happen. The situation in question is whether or not a spear in projectile motion can stay oriented over the entire path such that the angle is equal to the angle of the velocity vector. I start with proposing that the rate of change of the angle of the velocity vector must be a constant - there can be no torque applied during the projectile motion, only the initial angular velocity imparted at the release. So taking a projectile, with say an initial velocity of <20, 20> - this would be at a 45 degree angle, the case where the condition intuitively seems most likely: $\frac{dy}{dt} = -9.8t + 20$, $\frac{dx}{dt} = 20$ We find $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{-9.8t + 20}{20}$ And an expression for the angle, then: $\theta = \arctan{\frac{dy}{dx}}$ Graphing the derivative of this function over the time span of the projectile shows that it is NOT a constant. This implies that there would have to be some sort of torque being applied mid-air if the angular displacement is directly proportional to $\theta$. However, it IS a very shallow bowl. First of all, is this analysis correct? Second, if this implies that the condition is impossible, what happens when one throws a javelin or spear perfectly? Why is it so natural to do so?


title: Problem with forces acting upon a sloped plane

text: We have a structure that is sloped at 20 Degrees to the horizontal and weights 160N. We then have a force (wind) which is acting along the horizontal plane which has been calculated to be of about 8000N. The problem we are having is that we are unable to find what amount of ballast that needs to be added at the bottom of the structure to stop the structure from being blown away. The structure will not be bolted to the ground. At the moment we do not have an exact amount for the coefficient of friction, but the structure will be placed on a flat concrete surface. We have tried to find a solution by breaking the forces along the vertical and horizontal components, but we where unsure if we where heading in the right direction. Below is an image which describes the setting: ![Problem Description](http://i.stack.imgur.com/KQA35.png) Could someone please advise us as to which mathematical model we should use? Thanks in advance


title: How to calculate the intensity of the interference of two waves in a given point?

text: There are two different point sources which produce spherical waves with the same power, amplitude, ω, wavenumber and phase. I can calculate the intensity of each wave in a point: $$ I_1 = P / (4 \pi r_1^2) $$ $$ I_2 = P / (4 \pi r_2^2) $$ But how can I calculate the intensity of the resulting wave in that point?


title: How would I explain a farad being a second per ohm?

text: The farad has many alternative representations in SI. I can comprehend some of them. For example, a capacitance of 1 farad means that it will take a single coulomb to create a single volt between the capacitor plates. It could also mean that it will take one second of continuous 1A current to create that volt between the plates. In this manner, how would I imagine the second/ohm representation?


title: Permanently charging a sphere by induction using a high voltage

text: I want to charge a sphere using a van de graff generator , it would sound easy , but using a 1 MV or a 900 KV is really hard, it is enough to break like 1 meter of air so whenever i try to ground the sphere there is spark . So i wanted to know if it will help if i covered the sphere with a layer of plastic or an insulator , or if there was a vacuum between the the generator and the sphere. And another thing, can a graphite rod be charged like a metal one ?


title: $\alpha$ decay to more than one nuclear state

text: I do not understand how $\alpha$ decay can be a probabilistic process such that there are multiple products from the decay. For example: $^{241}\mathrm{Cm}$ decays to the excited states of $^{237}\mathrm{Pu}$, * 0.202MeV, spin: 5/2 with probability 13%, 0.125MeV, spin 3.2 with probability 17% and 0.145MeV, spin 1/2 with probability 70% Also there is no decay to the $^{237}\mathrm{Pu}$ ground state. I dont not understand how the decay has this choice? * class notes


title: If you view the Earth from far enough away can you observe its past?

text: From my understanding of light, you are always looking into the past based on how much time it takes the light to reach you from what you are observing. For example when you see a star burn out, if the star was 5 light years away then the star actually burnt out 5 years ago. So I am 27 years old, if I was 27 light years away from Earth and had a telescope strong enough to view Earth, could I theoretically view myself being born?


title: How the inverse square law in electrodynamics is related to photon mass?

text: I have read somewhere that one of the tests of the inverse square law is to assume nonzero mass for photon and then, by finding a maximum limit for it , determine a maximum possible error in $\frac{1}{r^{2+\epsilon} }$ for $\epsilon$ . My question is: (in the context of classical physics)   * How are these two related and what is the formula relating them?   * Why measuring photon mass is easier than testing the inverse square law directly? (Indeed it seems more challenging!)


title: Light Ray Reflection from concave mirror

text: Suppose a ray of light hits a concave mirror and is parallel to principle axis but far away from it such that it doesn't follow _paraxial ray approximation_. Will it pass through focus or between focus and radius of curvature or between pole and focus? Here pole, focus and radius of curvature mean the same thing as in paraxial ray approximation .


title: Can statistical mechanics be formulated generally in terms of phase space?

text: In many statistical mechanics books, notably Landau and Lifschitz' volume in the course on theoretical physics, the quantities central to statistical mechanics such as entropy are defined in terms of phase space densities. This is because (classically) particles in the real world can be described in terms of $n$ coordinates and their corresponding momenta. Also, in order to derive the Gibbs distribution, L&L argue that a mechanical system has only seven independent additive conserved quantities. However, many statistical mechanics models (such as spin models) are not formulated in terms of phase space and Hamiltonian mechanics at all. One still wants to be able to talk about entropy, probability distributions, etc. when discussing these models. Can the phase space formulation be generalized to systems that are not 'particles in a box' but can have arbitrary configurations and energies?


title: Why do focal lengths affect magnification?

text: For compound lenses, the image formed by first lens acts as the imaginaryobject for the second lens. In telescopes, the objective lens projects an image on its focal point which works as the object for the eyepiece. Per the property of convex lenses, the eyepiece magnifies the image. If the focal length of the eyepiece is smaller we'll get a higher magnification. Now if the focal length of the objective lens is increased, it'll again project a small image on its focal point. So for two objective lenses with different focal length, it seems the image size should about the same. So why does the magnification change?


title: Three equal mass points on the string

text: Help me please to solve my problem. I have three equal point masses on the line string. $L$ is a distance between masses. $P$ is a pulling of the string. I know that the potential energy of that system is equal to $$E=\dfrac{P}{L}(x_1^2+x_2^2+x_3^2-x_1x_2-x_2x_3),$$ where $x_1(t), x_2(t)$ and $x_3(t)$ are mass deviation from the equilibrium position. But I cant understand, how this formula is came out? Help me please, tell me, how can I get this formula?


title: Distance and velocity question

text: I know that speed is the derivative of distance. So integrating speed should give you distance. Let's suppose we have a speed which obeys this function: $$ v(x) = 2^{2^x} $$ So at time 0 the speed is 2 m/s, at time 1 4 m/s, at time 2 16 m/s, and so on. I would like to calculate the amount of time required to travel a distance $Y$ (let's say 1 km for the sake of the example). I would be correct to do this? $ \int 2^{2^x}x dx $ from 0 to t = 1000 m. EDIT: Yes, I just realized the error!


title: Can a compass be used in a submarine?

text: This question is motivated by several "myths" I have heard recently - I want to know if there is a problem with using a compass inside a submarine. I can see two different issues, so I'm wondering if anyone can clarify (if you do some googling on this issue, you will see all answers from "yes" to "no", most of which include very suspicious physics). 1) Does the submarine somehow "shield" the magnetic field of the Earth? Well, it seems obvious that a submarine is not a Faraday cage, but it is a giant hunk of metal so there could in principle be an issue. I guess they are generally made of something like steel, so it would magnetize (weakly), but the magnetization should match the field of the Earth so the compass should still read true, right? 2) Electrical activity in the submarine can produce magnetic fields larger than the Earth. This is likely true - on this site you can see that non- nuclear submarines used batteries capabible of producing currents around 15,000 A (of course this site called that the "power" of the battery, so maybe it's not a good reference). Anyway, a current of 15,000 A would match the field of the Earth (assuming 0.5 G) at a distance $$B=\frac{\mu_0 I}{2 \pi r}\rightarrow r\sim 60 m$$ In any case, this seems like a pretty reasonable argument - maybe it does not apply to the nuclear case however. Anyone have any ideas? COMMENT: Is there a tag for this kind of question? Something about using basic physics to "mythbust"?


title: Will a black hole eventually turn into a neutron star?

text: As far as i understand, black holes radiate away energy in form of Hawking Radiation. Thus, they lose mass, i suppose. Is there a point where the mass becomes too small for the object to still be a black hole? What happens then, will it turn into a neutron star supported by degeneracy pressure? Or is it all about density, and once a black hole is formed, it will stay a black hole (i guess until it completely radiates away)?


title: Double decay $\beta\beta$ observation and neutrino mass

text: If the double decay $\beta\beta$ will be detected this means the neutrino is a Majorana particle coincident with its antiparticle. At the moment the half life of this decay is put to $\tau\ge10^{25}$years. The limit of the Majorana neutrino mass is about 0.2–0.4 eV. What is the relation between the probability of $\beta\beta$ decay and the neutrino mass? Thanks.


title: Diffraction of electron through graphite film

text: "Electrons of wavelength 434 nm are directed into a thin film of graphite. Will they diffract?" Ignoring relativistic effects, I used de Brogile's relation, $p = \frac{h}{\lambda}$, to determine that the velocity of the electrons is around $1680 \,\mathrm{m\,s^{-1}}$, which I believe is rather small. I'm aware that electrons do diffract when projected through a thin metal foil (George Thomson's experiment) and through crystal (Davisson and Germer's experiment). I would like to know if the velocity of the electron affects the diffraction of the electrons (i.e. is there a "threshold" velocity?) and if changing the metal foil from George Thomson's experiment into a graphite film would affect anything. Thanks in advance!


title: Maxwell equations

text: $$\oint B.dl = \mu_0\left(I+\epsilon_0\frac{\partial\Phi_E}{\partial t}\right)$$ Please explain the applications , and implications of the modified Ampere's circuital law with Maxwell's addition. Especially, significance of Maxwell's work


title: Magnetic field and AC current

text: If I run an ac current under a plate of iron filings, what pattern should the filings form? If I use the plug for an appliance, will the magnetic fields in the two wires cancel each other out (more or less)?


title: Steel filings just like iron filings?

text: I am having problems with an experiment involving field lines. I filed a piece of steel to get "iron filings." Ought these steel filings be vastly different from the iron filings typically used in this sort of experiment?


title: Ion Optics: Electric and Magnetic field. A comparison with Light Optics

text: When we compare ion optics with light optics, normally we consider electric field. For example Snell's law. $n_1\sin\theta_1$=$n_2\sin\theta_2$. When an electron move from one electric potential to another, its tangential velocity remains the same and so we can write $v_1\sin\theta_1$=$v_2\sin\theta_2$. By the law of conservation of energy, velocity is proportional so square root of electric potential so we can write $\sqrt{V_1}\sin\theta_1$=$\sqrt{V_2}\sin\theta_2$. My First Question is: Can we do the same in magnetic field as well? If I say, the velocity of electron in a magnetic field is proportional to the magnetic field strength, can I write the same equation for an electron entering from one magnetic field to another as $B_1\sin\theta_1$=$B_2\sin\theta_2$ ? Is this right? But I never seen in any ion optics textbooks saying this or at even referring to this! Second question: To guide the ions/electrons, we use electric fields most of the times. Magnetic field also can do the same. But mostly people use magnetic field only in case if mass separation is needed (mass spectrometry). Other cases to guide the ions/electors we use electrostatic lenses. I understand that it is the convenient way. But I didn't understand the real basic reason why electric field act more like an optical lens. What is the basic property of the electric field which make it more close to light optics when compared to magnetic field?


title: Why do canvas water bags and doors expanded when wet?

text: I recently asked a question on a canvas water bag and I realized that there was something really interesting that I didn’t understand. When a dry water bag gets filled, it leaks a lot of water; however, after sometime the water stops leaking and some of the water seeps and is cooled by evaporative cooling. My question here is that in order for the water to stop leaking, the canvas fibers must “expand” to stop the water. It can’t be a linear or thermal expansion since this is occurring at the same temperature. Is it the unfolding of canvas fibers or something else? That is, **what type of expansion occurs in a water bag to stop it from leaking?** Furthermore, is this the same process that occurs when doors swell during the rainy part of the year? That is, **why do doors swell when they get wet?**


title: Why are anti-de Sitter spaces so interesting when we believe the universe is expansionary?

text: Perhaps this is a naive question, but in my recent (admittedly limited) readings about AdS spaces, I keep wondering why they seem to be such a hotbed for theoretical research (AdS/CFT correspondence, etc.). To my understanding, an AdS space has constant negative curvature in a vacuum, which should yield an attractive universe, not one with accelerating expansion. An AdS space can be thought of as having a negative cosmological constant, while a universe with accelerating expansion would imply that such a constant be positive. Since we observe that our universe's expansion is accelerating, it seems that if anything, we should be seeking to model it as a de Sitter space. Am I mistaken? What aspects of our universe do AdS spaces attempt to model?


title: Why do waves occur?

text: I've seen a couple derivations of wave equations, but I was never convinced that waves would actually exist. Consider a spring with a compression pulse in a part of it (that is, assume it has an elongated region, followed by a compressed region, followed by an elongated region). It is rather surprising that the configuration of forces and speeds(*) in this situation is such that, after a small while, the pulse will be shaped the same but be a bit further on the spring. I am looking for an intuitive explanation on why that would be so, or barring that, a derivation that is as clear as possible (preferably on a spring, or on the simplest medium of propagation possible, to avoid adding extra hypotheses). (*) Speed seems to be important, since it probably decides which way a wave is traveling. I had never thought of that.


title: Does firing electrons at an object change its colour?

text: If one had an electron gun capable of firing electrons with a specific energy at an object, can it change its colour? I think it is possible because the emission spectrum of the material would allow it to emit photons of specific energy, which our photo receptors in our eyes perceive as colour.


title: When drift velocity equals thermal velocity?

text: In some papers, I can see the drift velocity of electrons equaling thermal velocity. Can anyone tell me when both almost equal each other?


title: Chiral coupling in string-nets

text: In Xiao-Gang Wen's review of topological order http://arxiv.org/abs/1210.1281 , he states in footnote 52 that string-nets are so far unable to produce the chiral coupling between the SU(2) gauge boson and the fermions. Is it known how this chiral coupling can be expressed in lattice gauge theory, and the remaining question is one of emergence; or is it the case that its expression in lattice gauge theory is unknown, analogous to the chiral fermion question which took many years to solve?


If airplanes' highest speed depends directly on the air it is moving through and not the ground, does it mean we can build a time machine if only the air moves near the speed of light?


title: Does this equation mean anything?

text: I saw this equation on a coffee mug, and I am wondering what it means. I am not far enough along in physics to understand it. Does it actually mean anything or is it gibberish? ![enter image description here](http://i.stack.imgur.com/ifd4q.jpg) I can't embed the image because I am a new user. (But I can, L.M.)


title: Solar System Capture of Orphan Planets

text: NASA recently announced (http://www.nasa.gov/topics/universe/features/planet20110518.html) that orphan planets, planets no longer orbiting a star, may be more numerous than the stars in our galaxy. The Sun's Hill sphere, its gravitational influence within the Milky Way, is ~2.2 ly. So it's possible, however unlikely, that a Jupiter size orphan planet could attain solar system orbit, say with a semi-major axis of one light year and a ~1.6E7 year period. The barycenter of this body and the Sun would therefore be ~60.3 AU from the Sun. Could we detect the Sun's motion around the hypothetical Jupiter barycenter by its gradual anomalous dispacement versus distant stars and therefore detect the existence of such a far-off massive planet?


title: Why is the Haar measure times the volume of the eigenvalue simplex considered a good measure of Hilbert space volume?

text: In particular, why do we need both of these to find the volume? And should I be thinking of it as an actual volume or not? This Hilbert space volume is talked about in this paper. It says > There exist the unique, uniform measure $\nu$ on $P$ induced by the Haar > measure on the group of unitary matrices $U(N)$. Integration over the set > $P$ thus amounts to an integration of the corresponding angles and phases in > N-dimensional complex space that determine the families of orthonormal > projectors. And > Since the simplex $\Delta$ is a subset of the ($N-1$)-dimensional > hyperplane, there exist a natural measure on $\Delta$ which is deﬁned as a > usual normalized Lebesgue measure $\mathcal{L}_{N-1}$ on $R^{N-1}$. The volume of the Hilbert space is then defined to be $\mu=\nu \times \mathcal{L}_{N-1}$, but why?


title: 1-form formulation of quantized electromagnetism

text: In a perpetual round of reformulations, I've put quantized electromagnetism into a 1-form notation. I'm looking for references that do anything similar, both to avoid reinventing the wheel and perhaps to cite. Taking the electromagnetic field $\hat A$ to be an (operator-valued distributional) 1-form, we can write a smeared operator $\hat A_H$ in terms of the inner product on the exterior algebra as $\bigl<\hat A,H\bigr>$ (taking the test function $H$ also to be a 1-form, but satisfying Schwartz space-like smoothness conditions in real space and in fourier space instead of being a distribution). $\hat A$ projects into annihilation and creation parts, $\hat A^+$ and $\hat A^-$ respectively, $\hat A=\hat A^+ +\hat A^-$, for which the action on the vacuum is defined by $\hat A^+\left|0\right>=0$, and we have the commutation relations $\Bigl[\bigl<\hat A^+,H\bigr>,\bigl<\hat A^-,J\bigr>\Bigr]=\left<H,E(J)\right>$, where $\widetilde{E(J)}(k)=2\pi\delta(k^2)\theta(k_0)\tilde J(k)$ projects to the positive frequency forward light-cone. The commutation relations are not positive semi-definite for arbitrary test functions $H$, $\Bigl[\bigl<\hat A^+,H^*\bigr>,\bigl<\hat A^-,H\bigr>\Bigr]\not\ge 0$, which is fixed by the Gupta-bleuler condition, which can be stated in this formalism as $\delta\hat A^+\left|\psi\right>=0$, for _all_ states $\left|\psi\right>$, not just for the vacuum state. By the Hodge decomposition theorem, we can write the test function $H$ uniquely as $H=d\phi+\delta F+\omega$, where $\phi$ is a 0-form, $F$ is a 2-form, and $\omega$ is a harmonic 1-form, so we can write $$\bigl<\hat A^+\\!,H\bigr>\left|\psi\right>\\! =\bigl<\hat A^+\\!,d\phi+\delta F+\omega\bigr>\left|\psi\right>\\! =\Bigl(\\!\bigl<\delta\hat A^+,\phi\bigr>+\bigl<\hat A^+\\!,\delta F+\omega\bigr>\\!\Bigr) \left|\psi\right>\\! =\bigl<\hat A^+\\!,\delta F+\omega\bigr>\left|\psi\right>.$$ The harmonic 1-form $\omega$ has to be zero to satisfy the Schwartz space condition, and by the left action of $\hat A^-$ on arbitrary states we have the same projection of the arbitrary test function $H$ to $\delta F$, so all we are left with is the electromagnetic field observable $\hat\Phi_F=\hat A_{\delta F}=\bigl<\hat A,\delta F\bigr>$. Unsurprisingly, in the free field case, in the absence of charges, the electromagnetic potential observable is exactly equivalent to just the electromagnetic field observable, for which we can verify that $\Bigl[\bigl<\hat A^+,\delta F^*\bigr>,\bigl<\hat A^-,\delta F\bigr>\Bigr]\ge 0$, using which we can use the GNS construction of a free field Hilbert space. So this is a reference request. Is there any literature that uses this kind of mathematical formalism for the quantized electromagnetic field? Even _vaguely_ the same! My sense is that AQFT has moved to more abstract methods, while more practical interacting QFT has become historically committed to index notations that are little changed from 50 years ago, even though the methods of interacting quantum fields have changed in many other ways, and that mathematicians who take on the structures of QFT, as Folland does in http://www.amazon.com/Quantum-Theory-Mathematical-Surveys- Monographs/dp/0821847058, make strenuous efforts not to let their notation and methods stray too far from the mainline.


title: Will a rotating body gain linear acceleration in water?

text: If a ball is floating in water and it has some angular velocity, will it gain some linear acceleration from the drag on it as it rotates? **Edit** : ![Ball in water](http://i.stack.imgur.com/r2roF.png) This is how I pictured it. I guess my reasoning is somewhat like a tyre trying to drive in mud? I should probably ask as well, if this DOES happen, what is this force called? Equations would be cool too. **Edit 2** : OK so this seems to be a pretty definitive yes on the linear acceleration front. But how can I _calculate_ this acceleration? Just so no one is put off answering, this isn't homework. It's just something I'm trying to simulate for a game I'm writing.


title: Is Einstein's Special Relativity completely inclusive of Newton's 3 laws of motion?

text: Relativity has always been explained to me (in books I've read, etc) as a superset of newton's laws - that is; it encapsulates all of Newton's mechanics in addition to other effects (observer effect, time dilation, space-time geometry, etc). I can kind of imagine 2 of Newton's 3 laws of motion to be incapsulated within Einstein's Special Relativity, but the one about "for every action there is an equal and opposite reaction", I'm struggling to find a place for that within Special Relativity. Does this sit outside Special Relativity's explanatory power or is it inferred within somehow?


title: Combining Proportions to get Newton's Law of Universal Gravitation

text: I've read a little on the history of Newton's Law of Gravitation and noticed that the formula can be separated into 3 distinct parts that lead to the end result of $F_g = G \frac{m_1 m_2}{r^2}$; the masses ($m_1 m_2$), the inverse square relation ($\frac{1}{r^2}$), and the gravitational constant ($G$). While looking through readings, however, I noticed that a considerable amount of focus was placed on the discovery and development of the inverse square law, with little or no focus on how the other parts of the formula were "glued" together to form the final result. Thus I felt I should open up a discussion to see if there is anyone out there that can shed some light on how these 3 distinct sections/parts of this formula came together to form the whole. From what little I've read and through naive reasoning I just logically deducted the following to be what had happened for each part: **1.)** $m_1$ and $m_2 -$ Related through proportioning techniques, equating ratios of say 2 different sets of masses between celestial bodies with similar orbits. (ie. $\frac{m_1}{M_1}=\frac{M_2}{m_2}$, generalizing the quantity $m_1m_2$) **2.)** $\frac{1}{r^2} -$ Found empirically using Kepler's Laws and geometric techniques **3.)** $G -$ Found much later by Henry Cavendish via his famous Cavendish experiment (though not numerically calculated, so I've read) The mystery now is how all of these parts come together at the end.


title: Loop of wire in a varying magnetic field

text: Let us suppose we have a loop of wire with some definite resistance in a magnetic field. Let the magnetic field be varying. This varying magnetic field will st up an EMF in the wire. My questions are as follows: 1). Energy is dissipated across the wire resistance. Where does this energy come from? 2). The current generated in the loop will also generate a varying magnetic field. Will this magnetic field induce another EMF in the loop setting up another varying magnetic field? Will this go on indefinitely?


title: Which are other anomalies like Divergence of 1/r^2?

text: As one might have learned in the basic science (ex. Electrodynamic theory), when we apply the divergence theorem to the vector function like 1/r^2 with it pointing in the radial direction (like electric filed of a point charge) the direct divergence gives value zero and the integration over a closed surface gives a non zero value. Which violates the divergence theorem. The explanation given is that the function blows at the origin(Delta function). Can anyone please point out other anomalies like this (like in this case violation of the divergence theorem) in the physics or maths? It is very important for me and will be really helpful.


title: Does friction decrease as objects move faster against each other?

text: I was told that the faster two objects move against each other, the less the friction between them would be… compared to if they were moving slower. In physics class, we always use the coefficient of static and kinetic friction. But I’m thinking that the friction doesn’t suddenly jump from say a factor of 1.5 to 1.2 then stays at 1.2 no matter how fast the objects move against each other. Is the decrease in friction continuous and does it approach zero as the speed between the objects approaches infinity?


title: 2013 Russian meteor event, Sonic Boom or Explosion Boom?

text: In the lastest meteor event in Russia there was a big BOOM heard long after the meteor had passed. In this youtube video we can see at 4:30 the Meteor passing.. then hearing the BOOM at 7:00 , 150 seconds later, so what cause the boom? Sonic Boom from the meteor entering the atmosphere? or the big explosion that happens a bit later on its decent?


title: Dispersion relation in continuum mechanics

text: I'm looking at the vibration of a solid having a lattice structure, they obey the following equation: $$\rho\partial_t^2u_i = C_{ijkl}\nabla_j\nabla_ku_{l}$$ with $u(\vec{x},t)$ the displacement to the equilibrium position, $C_{ijkl}$ the stiffness matrix, $\rho$ the mass density. Knowing the following shape for the stiffness (in the Voigt notation):$$\begin{pmatrix}a&b&b&0&0&0\\\b&a&b&0&0&0\\\b&b&a&0&0&0\\\0&0&0&c&0&0\\\0&0&0&0&c&0\\\0&0&0&0&0&c\end{pmatrix}$$ I do end with the following differential equation: $$\begin{pmatrix}\rho\partial_t^2-a\partial_x^2& (b+c)\partial_x\partial_y&(b+c)\partial_x\partial_z\\\ (b+c)\partial_x\partial_y&\rho\partial_t^2-a\partial_y^2&(b+c)\partial_y\partial_z\\\ (b+c)\partial_x\partial_z&(b+c)\partial_y\partial_z&\rho\partial_t^2-a\partial_z^2&\end{pmatrix}\begin{pmatrix}u_x\\\u_y\\\u_z\end{pmatrix}=0$$ And I would like to derive a relation dispersion from there but I am not sure of the following steps, should go to Fourrier space and look for calculate the values of $\omega$ that would make the determinant vanish, or diagonalize it first?


title: Higher To Lower Electric Potential

text: The question I am working on is: > "An electron moving parallel to the x axis has an initial speed of $4.65 > \cdot 10^6~m/s$ at the origin. Its speed is reduced to $1.27 \cdot 10^5 > ~m/s$ at the point x = 2.00 cm. > > (a) Calculate the electric potential difference between the origin and that > point. > > (b) Which point is at the higher potential?" I am having difficulty visualizing this situation. Obviously, because of the reduction in speed, the kinetic energy is reducing; but, because of conservation of energy, the reduction of kinetic energy corresponds to an increase in potential energy. Is something giving the electron an initial "push," and a positive source charge distribution is pulling the electron as it passes by, thereby reducing the speed of the electron? Also, what confuses me is this paragraph: > We assume that the electron is moving in a vacuum. Its speed is reduced > because it is >moving to a higher electric potential energy. Since its > charge is negative, this means >that it is moving to a more negative > potential; that is, its starting position is at a higher potential (a higher > voltage) than its final position. What do they mean by "more negative potential," what does a negative potential physically mean? I realize that at our initial position, we arbitrarily choose the initial electric potential the value zero, and so if we chose another point in the electric field, relative to our initial point our electric potential will either be positive or negative--assuming that our second point is not in the equipotential region that the first point is. I suppose I am also having trouble understanding what situation a negative electric potential corresponds to, and what situation a positive electric potential corresponds to. I'd appreciate the help! Thanks in advance! EDIT: Also, is it true that if a charge has negative electric potential energy, then it is not necessary that it has lost that energy. In other words, negative electric potential energy can mean a gain in that sort of energy? I've heard electric potential being described as being "the measure of potential energy per unit charge." From this, I have developed the understanding that electric potential is a scalar that is assigned a different value at every point in an electric field.It almost seems like a conversion factor. It's how much potential energy a charged particle could possession if it were there. It's a potential of a potential. Again, sorry for all of the comments. I hope you can help me * * * Another question, if I push a negative test charge near a negative source charge, the test charge will be gaining potential energy, and that potential energy will be positive, right? Alternatively, if I move a negative test charge near a positive source charge, making sure it doesn't accelerate towards it, it will loss potential energy, showing up as a negative potential energy, right? And the mathematics will show that these are true? I guess the root of my problem is seeing a generalized situation, and the mathematics that accompany it. * * * Actually, I have one more question. I was able to answer the question I originally posed, except for part (b). I thought, because the electric potential energy was increasing at that point, that the electric potential at that point would be higher. After all, electric potential is defined one way as, "the amount of electric potential energy that a unitary point charge at that location would have." So, if the electron has more PE at the final position, wouldn't that mean the electric potential was greater at that final position too?


title: Induce a Fayet-Iliopoulos term

text: In a supersymmetric U(1) gauge theory, if I leave off the Fayet-Iliopoulos term $\kappa [V]_D$, what keeps it from being induced in loop corrections?


title: What happens to speed and frequency of a light beam moving in transparent medium when observed from different inertial frame of reference?

text: Suppose a transparent medium where speed of light is $c/n$, an inertial frame of reference $K$ which is stationary relatively to the medium and an inertial frame of reference $K'$ which is moving with speed $V$ according to $K$ along the $x$ axis. Also suppose that a light beam is travelling through the medium and its frequency measured from $K$ is $f$ and from $K'$ is $f'$ and the angle between the $x$ axis and the bean is $\theta$ and $\theta'$ respectively. If we know $n,f,\theta'$(or $\theta$ or both) and $V$ how can we calculate $f'$? * * * From a book I have read the transformation was the following: $$f'~=~\frac{f}{\gamma (1+\beta n\cos\theta')},$$ where $$\gamma~=~\frac{1}{\sqrt{1-\beta^2}},\qquad \text{and}\qquad \beta~=~v/c.$$ I can prove it with momentum-energy Lorentz transformations only if I suppose that light is moving with same speed ($c/n$) both in $K$ and $K'$ which (I think) is wrong according to speed transformation. Does the speed of light when moving through a transparent medium change when observed form a different inertial frame of reference or not?


title: Any difference between "Mueller matrix" and "Scattering matrix"?

text: I find in some references 4x4 Mueller matrix and in other references 4x4 Scattering matrix. Are they different or identical? If they are different, I would like to know the book or any research paper describing the relation between them.


title: Does KI have to be a salt of stable I-127 isotope?

text: I was recently watching a TV show where some people exposed to radiation were asked to take KI. I googled for the information and discovered this and also the Wikipedia article. From what I read, I understand that KI is a salt of stable Iodine. Why does this have to be so? Can there be KI with I-131 isotope? Or is the therapeutic KI salt used to treat radiation poisoning specifically enriched to have I-127 isotope?


title: Would a craft travelling increasingly close to the speed of light appear to be decelerating?

text: I've been pondering the implications of time dilation. Is it right to assume that if a craft was travelling at a speed very close to the speed of light (>= 0.9999c for example), that to an external observer it would appear to be moving very much slower? Example: at 0.9999c, time dilation is 70.71, so while the craft is travelling at 186,263.37 miles/s, doesn't that mean to a (stationery) observer it's only moving at 2,634.10 miles/s? And at 0.99999c, time dilation is 223.61, so while the craft is now travelling at 186,280.14 miles/s, doesn't that mean to the observer it's now only moving at 833.07 miles/s? The obvious implication is that as the craft approaches the asimptote of the speed of light, external observations (beyond around 0.9c) see it decrease in velocity towards the asimptote of being stationary. Is this reasoning correct?


title: Question about bosonization method

text: I have a question about bosonization method used in 1D system. Generally the bosonized field is assume to the following form \begin{equation} \psi = e^{i\phi}, \quad \phi = \phi^\dagger \end{equation} On the Haldane's method, however, the bosonized field is assumed to take the following slightly different form \begin{equation} \psi = \sqrt{\rho} e^{i\phi}, \quad \rho = \rho_0 + \Pi \end{equation} where $\Pi$ and $\phi$ are conjugate pair. My question is that why these two methods are totally different but both of them are widely used in literatures. How can I understand that these two methods are equivalent? Maybe these two methods are identical, but this is a very strong claim and need justification.


title: How about retrodiction, if climate prediction is difficult?

text: What I mean by the question is if _retrodiction_ has been attempted back to, say, 50 or 100 years starting with the presently known state of climate as initial conditions using the same equations and methods that are popularly used to _predict_ climate 50 or 100 years ahead? If yes, how well retrodiction turned out? If not, why not?


title: Homemade particle accelerator

text: Watching a documentary, I heard Michio Kaku telling that when he was young, he built a home-made particle accelerator. What I would like to know is if I, myself can do so? And how?


title: Is it possible to do a PhD in theoretical physics after a BSc in Mathematics

text: I would like to ask if it is useful to have a solid maths background (but only 2 courses of "general physics" and 2 of "mathematical physics") as BSc in order to be a successful researcher in theoretical physics. Also, is it advisable to start with a BSc in physics instead of one in maths (it should be noted that in a physics BSc it will be possible to do few maths courses: only 3 analysis courses; 1 geometry; 1 mathematical methods of physics)?


title: Baker and Copson "The Mathematical Theory of Huygens Principle" 1987 page 17

text: In Baker and Copson "The Mathematical Theory of Huygens Principle" page 17, how is this equation derived (after the first paragraph): $ R_0 r sin(\theta) d\theta = R dR $ I don't even know where $d\theta$ or $dR $ came from, although the figure on p 16 helps a little. Baker reproduced these results from "Wave-motions with Discontinuities at Wave-fronts" A. E. H. Love, proc. london math. soc. (1904) s2-1:291-344. But this paper does not clear things up for me either. Thanks if you can help. page 16 and top of page 17: ![page 16](http://i.stack.imgur.com/rPVmI.jpg) ![enter image description here](http://i.stack.imgur.com/SIzds.jpg)


title: Double slit experiment observation

text: In the double-slit experiment, if you shoot particles through the slits one by one and observe which slit they travel through, is there still an interference pattern on the screen behind the slits? If not, then how is our observation of the particle different than any other object in the universe being affected by it? Doesn't our observation of an event have essentially the same effect as any other object in the universe being affected by the event?


title: Operator Product Expansion (OPE) in Conformal Field Theory

text: We denote local operators of a conformal field theory (CFT) as $\mathcal{O}_i$ where $i$ runs over the set of all operators. Formally, the operator product expansion (OPE) is given by, $$\mathcal{O}_i(z,\bar{z})\mathcal{O}_j(\omega,\bar{\omega}) = \sum_{k} C^k_{ij}(z-\omega,\bar{z}-\bar{\omega})\mathcal{O}_k(\omega,\bar{\omega})$$ where we have left the $\langle...\rangle$ implicit. My question is primarily, in practice, how does one determine the operator product expansion for an explicit case? Is the OPE similar to a Laurent expansion? In which case, can one determine the residues from the OPE? I would appreciate an explicit non- trivial example. * * * I'm following _String Theory and M-Theory_ by Becker, Becker and Schwarz, and the OPE is dealt with somewhat rapidly.


title: Show that two families of curves are orthogonal (without using orthogonal trajectories)

text: I'm reading through Hartle's General Relativity and came across this question: > Consider the following coordinate transformation from rectangular > coordinates $(x,y)$, labeling points in the plane to a new set of > coordinates $(m,n)$: $$x = mn,$$ $$y = (1/2)(m^2 - n^2).$$ (c) Do the curves > of constant m and constant n intersect at right angles? I determined that the curves of constant $m$ are orthogonal trajectories to the curves of constant $n$, but the answer in the solutions manual simply states "The curves intersect at right angles because there are no cross terms $dmdn$ in the metric." I don't understand where this comes from. What does he mean? Where can I learn more about this? I imagine that my method of orthognal trajectories will get unwieldy with more variables.


title: What if all the electrons leave a metal?

text: I was studying photoelectric effect. Then I thought that what will happen if all the electrons from a metal piece come out as photoelectrons by using a light source of particular frequency? Will the metal still exist?


title: Acceleration of a steady line vortex

text: In a question, I have to find the acceleration of a fluid parcel in a steady line vortex. I am given that $u_\theta=\frac{A_0}{r}$. So for a steady line vortex, the parcels are following circular paths therefore in cylindrical coordinates, $u_r=u_z=0$ and the acceleration comes from the Lagrangian derivative $\frac{D\vec{u}}{Dt}=\frac{\partial\vec{u}}{\partial{t}}+(\vec{u}\dot{}\vec{\nabla})\vec{u}$ evaluating this for the velocity given above gives a zero acceleration as the only term from $\vec{u}\dot{}\vec{\nabla}$ that gives a non zero derivative is $\partial/\partial{r}$ but then $u_r=0$ which kills it. However this doesnt ring true as from just considering circular motion, $\vec{a}=-\frac{u_\theta^2}{r}\hat{r}=-\frac{A_0^2}{r^3}\hat{r}$. If I use the identity $(\vec{u}\dot{}\vec{\nabla})\vec{u}=-\vec{\omega}\times\vec{u}+\vec{\nabla}(\frac{\vec{u}\dot{}\vec{u}}{2})$ then it can be shown that the vorticity is zero and the expected acceleration is obtained. How can this not work when evaluating the acceleration before the identity is used!? Three of us have puzzled over this and got nowhere!


title: Why does water not evaporate in below 0 degrees?

text: I have learnt that, most objects still have energy in their atoms until the temperature reaches absolute zero. In other words, everything has energy around us. Now, that in mind if I go to antartica, the land is solid ice however below due to pressure it remains water which is cold yet it still moves. That in mind, that should tell me particles must be hitting each other and such that in mind, water should evaporate yet it does not. I asked a similar question: Why does evaporation take place? Yet this is an phenomena which I never truly understand as it too should follow the basic rules of physics as well so it too should evaporate as a result of Maxwell-Boltzmann distribution.


title: A block, a string and Newton's third law

text: ![enter image description here](http://i.stack.imgur.com/DSkxy.jpg) So this is a general force diagram of the system shown. My question is, according to the third law, if the block is exerting a force of magnitude mg on the thread in the down direction, then the thread should exert an equal and opposite reaction on the block. So, it should exert a force of mg in the upward direction, and nothing should move, as the block experiences equal and opposite forces by the thread and the earth. But this doesn't happen always in real life. Can you please help?


title: Cycling down a mountain

text: I am Dutch, so I cycle a lot, hence the following question. Say I am standing at the top of a mountain on my bicycle. I have an amount of energy $E$ available. Say that after the slope of the mountain ends, there is an infinite long, straight, smooth road ahead. How do I get the furthest on this flat road? Options: a) using all my energy $E$ to peddle while I am coming down the mountain b) roll gently down the mountain and start peddling precisely when I am at the end of the mountain c) roll gently down the mountain and down the straight road and only when I am starting to get to a hold, I start cycling very slowly, keeping my acceleration and speed to a minimum d) some kind of hybrid of the above. In this story I would like to keep it as realistic as possible: there's friction, both from my tires and from air resistance. I once heard that air resistance goes $\propto v^2$, such that I am tempted to opt for option c). I am interested in your opinions/thoughts. Help me spend as little energy as possible while cycling!


title: Heat transfer in a pipe

text: I have a gas (assuming air) at T = 500 K that enters a cylindrical pipe. The outlet target temperature is 330K. There will be heat transfer via: Forced convection from the gas to the inside of the pipe, conduction through the pipe thickness, convection to the ambient air environment ($T_{\infty} = 300 K) $ and radiation from the pipe surface. What would be an appropriate assumption for the surface conditions of this heat sink pipe? Would the temperature be constant and the heat transfer a linear function of the pipe length? It seems intuitive that the pipe surface temperature changes along the pipe, being hotter closer to the hotter inlet.


title: Is Sapphire crystal flexible?

text: It is widely rumoured that the next iPhone's screen cover will be a sheet of synthetic Sapphire crystal. A video purporting to show a leaked part has emerged. The chap in the video (Marques Brownlee) talks about the optical quality of the screen, its scratch resistance and lastly, its flexibility. The screen seems to be about 1mm thick. To my eyes, he's not really "going for it" when he tries to scratch the screen, but he does definitely bend it to an extreme degree. The bending can be seen here – I find it excruciating to watch because I'm imagining it as glass and thinking about what it will do to his hands it if cracks. I'm curious: is Sapphire crystal actually flexible to this degree? If it's not, then the whole video, or the featured part must be a hoax or a fake.


title: Magnet insulation and Permanent magnets

text: Is magnetic insulation possible? I have seen its not.If so Why? Permanent magnets are not permanent? I read about that permanent magnets lose their magnetism after some time. What does this mean.How much time or usage does it take to make it a non magnet.


title: State-dependent diffusions: Fick's law vs. Fokker-Planck's, which and why?

text: Consider a "state-dependent diffusion": a diffusion process for which the diffusion coefficient $D(x)$ depends on the (stochastic) state $x$ of the system. (An example is provided by the diffusion of tracers in a spatially inhomogeneous bath, e.g. with varying viscosity or temperature.) What is the associated flux? If $c(x)$ is the probability to find the system in the state $x$ at a given time, should the flux read $j(x)=-D(x)\nabla c(x)$, as in Fick's law? Or rather $j(x)=-\nabla(Dc)(x)$, as in the Fokker-Planck equation? (I'm using the terms "Fick" and "Fokker-Planck" here only as tags for each alternative.) It is well known that this question does not have a definite answer, and should be answered case by case. My actual question is threefold:   1. Do you know examples of physically relevant state-dependent diffusions? What category do they fall in? Fick, or Fokker-Planck?   2. Do you have an intuition for the physics underlying this alternative?    3. The Fokker-Planck case has the peculiarity to lead to equilibrium distributions which are not of the Boltzmann-Gibbs form (the steady-state $c(x)$ depends on $D(x)$ and not just on the state's energy). What does this tell us about the foundations of statistical mechanics?


title: Motorbike with parachute vs normal skydive

text: Had a strange discussion in the coffeeroom today. The background was a story about a guy that took his motorcycle to transport himself to the airstrip to do a skydive. Since there is no place to put a parachute on the motorbike, he put the parachute on his back (since it is like a backpack). And apparently the parachute almost opened during the drive, but he was lucky and nothing happened. Now we come to the actual question, what would happen if you ride a motorbike and the parachute opens? Is it really that different from a skydive? Let's compare the two cases. The first case, the skydive, I guess free-fall speed is around 150-200km/h, and then you open the parachute and the speed drops and eventually speed is so low that you can survive the touchdown. And all the motion is vertical. In the second case, the motorbike, we can have the driver going at the same speed 150-200km/h. And we assume that he travels on a big open space, with flat tarmac, so there is nothing that can come in his way (like trees, or other cars). Also he is wearing full protective clothing, like a proper race driver. In this case all the motion is horizontal. What happens when the parachute opens? Will the biker die? or will he glide on the ground until friction and the parachute stops him? Is there really any difference if you are going vertically or horizontally? Except for the proximity to the ground? ![Figure with biker and skydiver](http://i.stack.imgur.com/jX5u7.png)


title: Electrical energy storage in superconductors

text: I am a first year A-level student and I am doing a project about the possibility of storing electrical energy in a superconductor. I have researched and I am aware of the critical current density and the critical magnetic field of different superconductors, where the magnetic field created by the wire (Ampere’s law) interacts with the magnetic field of the superconductor (Meissner effect). But, if I had a loop of superconductor and I cool it down enough (with that I mean enough so its resistance becomes completely 0) then could I use a huge voltage and a relatively small current (not big enough to reach its critical current) in order to store electrical energy in the superconductor? If it is not possible I would like to know why not and if it is possible I would like to know why we are not using it nowadays as a form of storing electrical energy (is it only the practical issues of cooling it down)? Thank you very much


title: Is relational QM experimentally distinguishable from standard QM?

text: Some people advocate for relational quantum mechanics, which is an interpretation (or at least Wikipedia calls it an interpretation) of QM where quantum states are thought of as representing a relationship between "subject" and "object" systems, rather than as objectively-real states of the object system. In particular, RQM claims that there is no quantum state for the whole universe; quantum states only arise when we divide the universe into two pieces and describe the relationship between the two. It's been described by its proponents as "one-world" (or even "zero-world") as opposed to the many- worlds that seem to inevitably result if you apply standard QM to the whole universe. That sounds like it would have to produce a difference in physical predictions between RQM and standard QM. They might be indistinguishable by any experiment we could realistically do locally, but on a cosmological scale the fact that RQM (somehow) forbids a whole-universe quantum state from existing seems like it should result in some different behavior from standard QM. Is that the case? Is there a known thought experiment that shows predictions of RQM diverging from those of standard QM—in other words, is it really a different theory, not an interpretation? Or if they always give the same predictions, how can they be conceptually reconciled at the cosmological scale?


title: How to model/simulate pressures and flows in a network of pipes

text: I'm having a hard time finding information on how to model/simulate this. I attached a couple files, both of which show an example tank & pump network. It's just nonsense that I made up for this post, but it shows what kind of system I'm trying to simulate. (I just put it on the web instead) http://gneissshirt.com/Example%20for%20pipe%20model.pdf http://gneissshirt.com/Example%20for%20pipe%20model.vsd I've been looking on the web to find information on how to do this, but I keep running into formulas that are way more complex than what I need to do. I just want to know what the pressure and flow is at any node in the system, with 80% or so accuracy. There's no need to consider fluid compression, and we can just use a constant for the resistance experienced through valves or narrowing pipes. We don't need to consider friction of fluid against piping. We can ignore heat. It's important to note, again, that I don't need complete accuracy, it's much more the behavior of the simulated system that is important. I've tried this going in both directions, that is following a flow path down from the tanks/sources and propogating pressure changes through the system, and I've also tried it directly from the node in question, and considering that each node has a number of source nodes in terms of pressures and flows. Both of these simulated SW systems become highly erratic the further downstream from the pumps that you get, and the more on/off valves that I put into the system. Even if you just told me where to look or what book to read, that would be a huge help. If there's example source code out there somewhere that does this, I will buy you a 60 oz beer of your choice. Thanks in advance. ~~~~ (post edit 4/12) Thanks for the answers so far, I've ended up spending quite a bit of time looking around for a solution that already exists, because I was just _convinced_ that many have run into this problem before, and knew that it was solvable in a general sense. Part of the reason I was so interested in a general solution is that I can rip through 12 more of these simulations that I have to do quickly and accurately if that's the case. I've found another piece of software called EPAnet, which seems like a simpler solution than what was listed below, and I'm looking through their SDK at the moment and playing around with it. I'll keep you posted, the few that are interested ... this might not be a big deal to most, but to the few that it matters to it will be huge, so I'll write up what I find.   ~~~~ (edit 4/13) I've spent the last two days building a solid wrapper around the EPAnet programmer's toolkit, and making a scratch program that does some of the things I want to do. (Incidentally, I've broken a new record for the # of times in a day I can make LabVIEW crash quickly and silently.) It's really quite complex underneath, and I'm delighted by how much mathematical power I can stand on. I don't think they've fleshed out all the issues, so I've been working around some of them. But it's looking like it's 80% chance so far that it's going to work for what I want.


title: Invariance, covariance and symmetry

text: Though often heard, often read, often felt being overused, I wonder what are the precise definitions of invariance and covariance. Could you please give me an example from quantum field theory? Thanks!!!


title: Avoiding Pseudo-tensors when addressing global conservation of energy in GR

text: Discussions about global conservation of energy in GR often invoke the use of the stress-energy-momentum pseudo-tensor to offer up a sort of generalization of the concept of energy defined in a way that it _will_ be conserved, but this is somewhat controversial to say the least. Rather than dwell on the pseudo-tensor formalism, I was wondering if anyone had any thoughts on the following approach. Prior to Noether's theorem, conservation of energy wasn't just a truism despite the fact that new forms of energy were often invented to keep it in place. This is because there was a still a condition the laws of nature had to satisfy, and in many ways this condition is a precursor to Noether's theorem. The condition is simply that it had to be possible to define this new energy such that no time evolution of the system would bring the system to a state where all other energy levels were the same, but this new energy had changed, or conversely, a state where this new energy remained the same, but the sum of the remaining energy had changed. For example, if I wanted to define gravitational potential energy as $GMm/r$, the laws of gravity could not allow a situation where, after some time evolution, a particle in orbit around a planet had returned to its exact position (thus maintaining the same potential) but its kinetic energy had somehow increased. This is of course just saying that it physics must be such that there can exist a conserved scalar quantity, which later in time we found out to mean time invariant laws of physics. The reason I formulated this in a convoluted manner instead is the following. In GR we invoke pseudo-tensors because Noether's theorem tells us energy should not be conserved given that the metric is not static, but we try to recover some semblance of energy conservation by asking if the dynamics of that metric do not somehow encode a quantity somehow resembling energy. Couldn't we ask instead if some sort of equivalent to energy was conserved in GR by asking : > In GR, do there exist initial conditions $\rho(t_0,x,y,z)$ and $g_{\mu > \nu}(t_0,x,y,z)$) such that after some time we recover the same > $\rho(t,x,y,z) = \rho(t_0,x,y,z)$ but with a different $g_{\mu \nu}$, or > conversely, we recover the same $g_{\mu \nu}$ but with different $\rho$ ? Obviously here we only ask about $\rho$ and not the entire $T_{\mu \nu}$ tensor. For starters, setting $T_{\mu \nu}$ immediately determines $g_{\mu \nu}$ so what I'm asking would be impossible. This isn't a problem because we're looking to generalize the concept of a conserved scalar, so it makes more sense to ask about the scalar field $\rho(t,x,y,z)$ than it does to ask about the tensor field $T_{\mu \nu}$. What this means is that you'd have to play around with how the momenta change in order to find a situation in which the answer is yes. For the Friedmann equations, the answer is obviously no so some semblance of energy conservation still exists ($\rho \propto a^{-3(1+w)}$, therefore the $ \rho$s will return to the same values if the universe ever collapses in on itself and returns to previous values of $a$, though I'm not sure if this is necessarily the case for time varying values of $w$). Can this be shown to generally be the case for General Relativity ?


title: Vasiliev gravity and "holographic" entanglement

text: It has been proposed that AdS/CFT arises because of the entanglement structure of quantum field theories, e.g. see the discussion which occurred right here. Until now I have been skeptical of the idea, on the grounds that so much else is going on in the duality. A CFT only has a gravity dual if it has the right properties, or so I thought. But I've just started looking into Vasiliev's "higher spin gravity" or "higher spin gauge theory", and eventually I ran across "Holographic dual of free field theory", which claims to construct a dual Vasiliev-type theory for _every_ free QFT! Looking at the paper's only figure, I began to wonder if the tensor-network MERA diagrams could somehow be interpreted as Witten diagrams (which are the boundary-to-bulk Feynman diagrams used in AdS/CFT). Or perhaps the "discrete version of anti de Sitter space" which the MERA diagrams can provide, could be the background space on which a "discrete Witten diagram" is drawn. Suddenly I'm not so sure about my skepticism. Does anyone have an argument either way?


title: Why does in string theory the amount of supersymmetry have to be $\cal{N} \leq 2 $?

text: Why is this, that in string theory the maximum amount of supersymmetry is $\cal{N} = 2$, whereas in supergravity one can have up to $\cal{N} = 8$ ?


title: Why does cold metal seem colder than cold air?

text: (I apologize for this elementary question. I don't know much about physics.) Let's say that I put a metal pot in the refrigerator for several hours. At this point, I guess, the pot and the air (in the refrigerator) have the same temperature. Now, I touch this pot. It feels very cold. But when I "touch" the air (that is inside the fridge) it doesn't "feel" as cold. I don't feel the same "ouch!" that I feel when I touch the pot. Why is that? Why does the metal seem colder than air although they both have the same temperature? (I know that gas has less particles in it in one unit of volume compared to solids and liquids, but since "temperature" means "the average kinetic energy", these fewer air particles are supposed to hit my hand in a velocity that's going to compensate for their lower number, aren't they?) A related question, for clarification: If I use a thermometer to measure the temperature of the pot & air (let's assume it's a thermometer that has a probe that can touch objects), will it show the same reading for both? If so, what makes the thermometer different than my hand? I mean, my hand is sort of a thermometer, so why would it fail whereas a non-human thermometer would work?


title: Which is the lightest thing in this universe? Is that a photon or neutrino?

text: I hear a lot of people saying that neutrino is the lightest subatomic particle but according to me a photon must be the lightest as nothing can travel faster than light because it gets heavier and heavier and bends time.


title: Dimensionial Regularization (integrating over extra dimensions)

text: I tried to replicate a result about dimensional regularizations in the review article of Howard Georgi, _Effective Field Theory_ at: people.fas.harvard.edu/~hgeorgi/review.pdf. On page 17, he writes about Integrals of the form: $$I = \int \frac{d^4l}{(4\pi)^4}\frac{1}{(l^2 + A^2)^\alpha}$$ He then regularizes the integral in the form $$I = \int \frac{d^{4 + \delta}l}{\mu^\delta(4\pi)^{4+\delta}}\frac{1}{(l_\delta^2 +l^2 + A^2)^\alpha}$$ and integrates seperately over the extra dimensions $\delta$ getting: $$I = \int \frac{d^4l}{(4\pi)^4} \left(\int \frac{d^{\delta}l_\delta}{(4\pi\mu)^{\delta}}\frac{1}{(l_\delta^2 +l^2 + A^2)^\alpha}\right) = \\\ \int\frac{d^4l}{(2\pi)^4}\left(\frac{1}{(l^2 + A^2)^\alpha}\frac{\Gamma(\alpha - {\delta\over 2})}{\Gamma(\alpha)}\left(\frac{l^2 + A^2}{4\pi\mu^2}\right)^{\delta\over 2}\right)$$ This seems to make perfect sense to me, but when I try to calculate the integral over the extra dimensions $\delta$, I run into problems. First of all, the angular part of the Integration over $\delta$ is $$\frac{\pi ^{\delta /2} \delta }{\Gamma \left(\frac{\delta }{2}\right)}$$ The radial part of the integral is: $$\int \frac{d^{\delta}l_\delta}{(4\pi\mu)^{\delta}}\frac{l_\delta^{\delta-1}}{(l_\delta^2 +l^2 + A^2)^\alpha} = \frac{\Gamma \left(\frac{\delta }{2}\right) \Gamma \left(\alpha -\frac{\delta }{2}\right) \left(A^2+k^2\right)^{\frac{1}{2} (\delta -2 \alpha )}}{2 \Gamma (\alpha )}$$ Together I get: $$\frac{\pi ^{\delta /2} \delta \Gamma \left(\alpha -\frac{\delta }{2}\right) \left(A^2+k^2\right)^{\frac{1}{2} (\delta -2 \alpha )}}{2 \Gamma (\alpha )}$$ All in all I get a extra factor of $\delta$ which makes the integral zero in the limit $\delta\rightarrow 0$. But it should not give zero? I can calculate dimensionally regulated integrals the usual way, integrating over all dimensions d at once, reproducing all the known results. But I fail at the Integration over the extra dimensions $\delta$. Where is my error? (How) Is it possible to integrate over extra dimension $\delta$ seperately?


title: Is the total cross section a Lorentz Invariant?

text: In Peskin and Schroeder's book (P&S), on the botton of page 106, the authors say that the total cross section transforms as its only non-invariant factor, namely: $$ {1 \over E_{A} E_{B} |v_A - v_B|} $$ Where $E_i$ and $v_i$ are energies and velocities of the incoming particles ($i=A,B$). The authors then conclude that the cross section itself is not invariant. They actually go as far as rationalizing that it transforms as an area should (invariant to boosts in one direction but not on the other two). This is at odds with many other sources (here is an example, look for equation 3.18 and 3.19), where an invariant cross section is obtained and that very factor is found out to be: $$ {1 \over F} = {1 \over \sqrt{(p_A.p_B)^2-m_A^2 m_B^2}} = {1 \over \sqrt{|E_A\vec{p}_B-E_B\vec{p}_A|^2-|\vec{p}_A\times\vec{p}_B|^2}} $$ Where $F$ is the so called Møller's invariant ﬂux factor ($p_i$ are the four momenta, and $m_i$ the masses). And the conclusion here is that the cross section is Lorentz invariant. Of course, the second expression reduces to the first in any frame that $\vec{p}_A\times\vec{p}_B = 0$ (in particular the center of mass frame or the usual colinear bean "laboratory frame"). I am under the impression that P&S assume such a frame in more than one step of the calculation and that's why their result is frame dependent, but that means that their conclusion is wrong. Am I missing something?


title: Physical origins of the Heisenberg model of ferromagnetism

text: I am trying to understand physical intuition behind the Ising and Heisenberg models (thus I am not sure if my question is appropriate for this mostly mathematical site). I will concentrate of the Heisenberg model. In the Heisenberg model of ferromagnetism the atoms are assumed to be arranged in a lattice. To the $i$-th atom corresponds a spin operator $\vec S_i$ (here $i$ belongs to the lattice). The Hamiltonian is given by $H=-J\sum_{<ij>}\vec S_i\cdot\vec S_j$ where the sum runs over all pairs of atoms connected by an edge, $J>0$ is a constant. As far as I understand from the literature that this form of the Hamiltonian comes from the exchange interaction which, in turn, comes from the fact that each (neutral) atom is a boson (see e.g. vol. IX of Landau-Lifshitz, paragraph 72). **QUESTION. In addition to exchange interaction there exists a different spin- spin interaction which has a different form. Why it was neglected?** The spin-spin interaction between two atoms with magnetic moments $\vec\mu_1,\vec\mu_2$ and large separation $\vec R$ is equal to $$\frac{\vec\mu_1\cdot\vec\mu_2}{R^3}-3\frac{(\vec\mu_1\cdot\vec R)(\vec\mu_2\cdot\vec R)}{R^5}.$$ After quantization, $\vec \mu_i$ is replaced by $\vec S_i$ times a constant. As it is claimed in "Statistical mechanics" by R. Feynman, Section 7.1, the spin-spin interaction is very small in comparison to the exchange interaction. However this seems to be true only on short distances since the exchange interaction decays exponentially with the distance $R$ between the atoms, while the spin-spin decays as $1/R^3$. Hence when we are on 3d-lattice the sum over the lattice of terms of order $1/R^3$ diverges logarithmically. **This cannot be neglected from the experimental point of view too. When the ferromagnetic material is magnetized, it creates a magnetic field, which will interact with another magnet or moving charge. This interaction has already nothing to do with the exchange interaction, as far as I understand.**


title: Rotation in an 'empty' universe

text: > **Possible Duplicate:**   >  Is rotational motion relative to space? Assume a universe with the same physics as ours, but containing only one rotating (charge-free) body - let's say the size of the Earth. Would the rotation cause a bulge at the equator of the rotating body?


title: Four-dimensionalism vs energy economy

text: Four-dimensionalism claims that the universe is basically one huge space-time worm and that everything exists at once (however you want to say that since "internal time" is then just another coordinate in this worm and I'm talking about the look from the outside). OTOH, all processes in the universe seem to follow an energy economic principle (second law of thermodynamics), that is, everything tries to reach a state of lowest energy. Now I'm wondering if that isn't enough to refute the idea of 4D-ism: For it to be the correct theory about space-time, you'd need a lot more matter (at least) because every point in time must be materialized as a "solid" state. That means for everything can happen within what we perceive as "one second", one state must be encoded. Quantum processes happen at attoseconds (10^-18s), so you need _at least_ 10^18 states just to encode one second of the 4D worm. So my argument is that, just to prove the theory, you're "wasting" insane amounts of whatever the universe is made of. Is that correct or am I missing something? NOTE: English is not my mother language, so I may have use the wrong terms but I hope you get the idea.


title: Apollo and orbital mechanics: orbital decay if the Trans Earth Injection (TEI) burn had failed

text: I'm reading Jim Lovell (Apollo 8 and 13) and Jeffrey Kluger's book _Apollo 13_ , which is a fantastic read about a long past era I only have kindergarten memories of. On page 54 there is a paragraph that reads (emphasis mine): > Also, unlike the LOI burn, during TEI there would be no free-return > slingshot to send the ship home in the event that the engine failed to > light. If the hydrazine, the dimethylhydrazine, and nitrogen tetroxide did > not mix and burn and discharge just so, Frank Borman, Jim Lovell, and Bill > Anders would become permanent satellites of Earth's lunar satellite, > expiring from suffocation in about a week and then **continue to circle the > moon, once every two hours, for hundreds -- no thousands; no, millions -- of > years.** Apollo 8 was in an elliptical orbit of 169.1 × 60.6 miles. I am skeptic about how long such an orbit would be stable. My skills in orbital mechanics are rudimentary at best and I know there is no atmosphere to cause orbital decay due to atmospheric drag. But I have heard about other effects causing orbits to decay over time, such as variations in the moon's gravitational field (density anomalies, mascons). This may cause lunar satellites to eventually crash into the moon. Can someone estimate how long this would be for the case of the Apollo 8 lunar orbit?


title: How was Dorsey able to measure speed of light with 3 significant digits?

text: How was Dorsey able to measure speed of light with 3 significant digits in the year of 1907?


title: How is the dynamic equilibrium nature of fermi-dirac distribution of particles facilitated?

text: I read this in _Kittel: Introduction to Solid State Physics_ about deriving that product of electron and hole concentration as independent at a given temperature by the law of mass action. For this model, it is assumed wherein the electron-hole pairs are generated using photons of black body radiation at rate $A(T)$, with $T$ being temperature, within the semiconductor and the recombination rate of pair at $B(T)np$. At equilibrium $\frac{dn}{dt} = A(T) - B(T)np=0 \implies np = \frac{A(T)}{B(T)}$ Is this model a specific case of facilitating dynamic equilibrium nature of Fermi-Dirac distribution or the principle way of how fermions interact through photons?


title: Integrating over a gauge field in the field integral formalism

text: I'm currently trying to study a chapter in Altland & Simons, "Condensed Matter Field Theory" (2nd edition) and I'm stuck at the end of section 9.5.2, page 579. Given the euclidean Chern-Simons action for a gauge field $a_µ$ that is coupled to a current $j_µ$ $$ S[a_µ,j_µ] = ∫d^3x (j_µ a_µ \+ \frac{iθ}4ε_{µνλ}a_µ ∂_ν a_λ) $$ the task is to integrate out the gauge field and obtain the effective action for the current. Since this is a gauge field, we have to take care about the superfluous gauge degree of freedom. Altland & Simons note that one way to do it would be to introduce a gauge fixing term $α (∂_µ a_µ)^2$ and let $α\to ∞$ at the end. However, this does not seem to work. In momentum space, the Chern-Simons action plus gauge fixing terms is proportional to $$∫ d^3q\ a_µ(-q) \left( \begin{array}{ccc} \alpha q_0^2 & -i q_2 & i q_1 \\\ i q_2 & \alpha q_1^2 & -i q_0 \\\ -i q_1 & i q_0 & \alpha q_2^2 \end{array} \right)_{µν} a_ν(q) .$$ To get the effective action for the current, I just have to invert this matrix, which we call $A_{µν}$, and send $α\to ∞$. But this can't be. For instance, one entry of the inverse matrix reads $$ A^{-1}_{01} = \frac{-q_0 q_1-i q_2^3 \alpha }{q_1^2 q_2^2 q_0^2 \alpha ^3- α(q_0^4+q_1^4+q_2^4) } $$ and this vanishes in the limit $α\to∞$. Same for the other entries. This is bad. My question, hence > How to properly perform the functional integral over a gauge field $a_µ$ > with a gauge fixing contribution $α(∂_µ a_µ)^2$ where $α\to ∞$? I am aware that there are other methods, for instance to integrate only over the transverse degrees of freedom, as Altland & Simons note. I don't mind learning about them as well, but I would like to understand the one presented here in particular. Not to mention that I may have made a simple mistake in the calculation above.


title: Gabriele Veneziano, strong nuclear force and beta-function

text: Background to the question: From _The History of String Theory_ : > Gabriele Veneziano, a research fellow at CERN (a European particle > accelerator lab) in 1968, observed a strange coincidence - many properties > of the strong nuclear force are perfectly described by the Euler beta- > function, an obscure formula devised for purely mathematical reasons two > hundred years earlier by Leonhard Euler. It also mentioned in the YouTube video _Part 01: String theory_. It made me wonder about the way physiscits work. For example, A beta function is an instance of the functions that satisfy the symmetric functional equation that beta function satisfies. Listening to Gabriele Veneziano in the video it is as if there is something not right with using already existing mathematical functions. What could be the possible reasons for such a view? What is the the math book in the story? And why is it such a revelation to have found a beta function? First year calculus or engineering mathematics books have it. One can assume that acolyte theoretical physicists would have special mathemtical functions in their bag of tricks (Feynman anyone?) Or is there something more to their use of the beta function that is not mentioned?


title: Electric Field of a sphere

text: Is there a point in space where the electric field is half of the electric field on the surface of the sphere? I just got curious if it would be possible.


title: wind vs air resistance

text: I'm wondering which offers more resistance: pulling an object at some speed through air, or holding the object steady against wind at the same speed. I think initially people would think same resistance. Then I thought that the air that is flowing has probably been compressed under it's own speed (or rather, in order to get to it's speed), and therefore offers more resistance. Also, the wind may be colder (thus denser) if it's the wind I'm used to, but disregarding temperature influence, do you think that my theory is correct? Can we quantify the resistance gained by windspeed? Thanks in advance.


title: Interacting classical strings?

text: May _classical_ strings be interacting? I would guess no, I can not see any way to break a classical closed string in two of them (the "pants" diagram); but maybe I'm missing something.


title: Difference between baryonic, inertial and gravitational mass? (and relation to dark matter)

text: Inertial mass: determined from Newton's `f = ma` Gravitational mass: determined via Newton's law of gravity Baryonic mass: sum of mass of all baryons.   1. Which ones will include mass of dark matter?   2. Since dark matter does not interact with matter, therefore it should not have "normal" inertial mass.    3. Does dark matter interact with itself and have its own "dark matter" inertial mass?


title: Why are spacecraft made to "spin" after launch?

text: At some point after launch, usually just before or after separation from the last booster stage, spacecraft are often made to "spin" (about the axis of their trajectory)? See e.g this You Tube video. What is the reason for this spin?


title: Is there any way to anticipate exceptional Aurora Borealis viewing conditions?

text: I've always wanted to see the Aurora Borealis, but never had the opportunity, since it is so rarely visible in my area (Pennsylvania). However, last week was apparently a rare opportunity of exceptional visibility. Unfortunately, I missed my opportunity, as I did not find out about it until the next day. Are there any good resources or strategies for anticipating opportunities like this? Are coronal mass ejections like the one that precipitated this event predictable? If so, are such predictions available to the amateur public?


title: Show that the electric field, $\mathbf{\vec{E}}$ cannot be generated by any static distribution of charges

text: > Show that the electric field $$\mathbf{\vec{E}}=\begin{pmatrix}0 \\\ E_0x > \\\ 0\end{pmatrix}$$ where $E_0$ is a constant, cannot be generated by any > static distribution of charges. I understand that an electrostatic field is irrotational and the divergence of the electric field is the charge density over epsilon 0 but I have no idea why that makes the answer.


title: Could submarine SONAR kill a diver?

text: Could a diver swimming next to a submarine be killed or seriously injured by its SONAR? What physical aspect of SONAR affects the human body in a potential harmful way?


title: How 4-vector nature of the value is connected with it's conservation law?

text: In electrodynamics Poynting vector and energy flux of field don't create 4-vector. Also they aren't conserved independently from substance (conservation law includes summand connected with current density). In linearized gravity mass density and mass current density as components of stress-energy tensor also aren't conserved. And they also aren't components of 4-vector. Is facts of non-conservation of values above connected with the absense of 4-vector nature?


title: Angular momentum of hydrogen from $n,l,m$ values

text: Given a wavefunction for hydrogen $\psi(n,l,m)$ it is possible to calculate its associated energy from $E=-13.6/n^2$. Does a similar equation exist for $L^2$ and $L_z$? That is, if we are given the $n$, $l$ and $m$ values, is it possible to calculate the energy and angular momentum, without needing to know any further numbers?


title: Why Least Distance of Distinct Vision is treated as image distance in the lens equation.

text: Least distance of distinct vision is the minimum object's distance that is able to produce a distinct image on the retina. Yet, it is treated as image's distance while applying lens equation. Say, for an elderly person, least distance of distinct vision is $50$ cm. SO, if I am to calculate the focal length of the corrective lens, would I consider the object's distance as $50$ cm, or would it be $25$ cm - the normal 'near point' for a healthy eye?


title: Subtleties in the exact solution to the 1D quantum XY model, in particular the Bogoliubov transformation

text: I am writing programs to construct the spectra of models with known exact solutions, and soon noticed some subtleties that are not often mentioned in most references. These subtleties are not important in the thermodynamic limit, but since I am writing numerical recipes to finite size systems, they do matter. Now there is still one puzzle concerning **Bogoliubov transformation** (or so I believe) that I cannot resolve on my own. Take for example, the 1D anisotropic XY model in a transverse field (including the case of Ising model), given by the following Hamiltonian $$ H = - J \sum_{j=1}^N \left( g \sigma_j^z + \frac{1+\eta}{2} \sigma_j^x \sigma_{j+1}^x + \frac{1-\eta}{2} \sigma_j^y \sigma_{j+1}^y\right), $$ assuming periodic boundary condition $\vec{\sigma}_{N+1} = \vec{\sigma}_1$. This model can be solved by applying Jordan-Wigner transformation, Fourier transform, and Bogoliubov transformation successively. The final diagonal form of the Hamiltonian, as given in many references, is $$ H = \sum_k \varepsilon(k) (b^\dagger_k b_k - \frac{1}{2} ) $$ where $ \varepsilon(k) = 2 J \sqrt{(g - \cos(ka))^2 + \eta^2 \sin^2(ka)}$. Two known subtleties:   1. The original Hilbert space of periodic spin chain is mapped to **TWO** sectors of Fock space: odd/even parity sector ($n = \sum c^\dagger_j c_j$ odd/even) with periodic/anti-periodic boundary condition $c_1 = \pm c_{N+1}$ respectively.   2. The momentum $k$ takes values of integer/half-integer multiples of $2\pi/Na$ in the odd/even parity sector respectively. Another subtitles that (almost) immediately follows (but unresolved) is that the dispersion relation given above actually only gives the **absolute value** of a single fermion's energy. The true single fermion energy should be given by $$ \varepsilon(k) = \pm 2 J \sqrt{(g - \cos(ka))^2 + \eta^2 \sin^2(ka)}. $$ This undetermined **sign** matters in a finite size system because there are two parity sectors with different choices of $k$, hence the "presence" and "absence" of fermions with "opposite energies" are not equivalent (in fact, they can't have exactly opposite energies in general). Even if this sign ambiguity does not affect total energy $E_n$ itself, it most certainly still disrupts the dispersion relation of the total spectrum $E_{n,k}$ (this is the case when $g=\eta=0$). I _believe_ this sign is determined in the Bogoliubov transformation, but I do not know how. The Hamiltonian right before the Bogoliubov transformation is $$ H = J \sum_k \left\\{ 2(g- \cos(ka)) c^\dagger_k c_k + i \eta \sin(ka) (c^\dagger_{-k} c^\dagger_k + c_{-k} c_k ) \right\\} + \text{const.} $$ For example, in the isotropic case $\eta = 0$, by comparing test cases with results from exact diagonalization, I know the correct single fermion energy is given by $ \varepsilon(k) = 2J (g - \cos(ka)) $, not any other variants. However, when making the Bogoliubov transformation: $$ c_k = \cos \frac{\theta_k}{2} b_k + i \sin \frac{\theta_k}{2} b^\dagger_{-k}, $$ and assuming $\cos \frac{\theta_k}{2} = \cos \frac{\theta_{-k}}{2}, \; \sin \frac{\theta_k}{2} = -\sin \frac{\theta_{-k}}{2} $, we get $$ H_{\eta=0} = J \sum_k \left\\{ (g-\cos(ka) ) \left[ 2 \cos \theta_k b^\dagger_k b_k + i \sin \theta_k ( b^\dagger_k b^\dagger_{-k} - b_{-k} b_k ) \right] \right\\} + \text{const.} $$ The only restriction that I can see is $\sin\theta_k = 0$. If that's the case, then $\cos\theta_k$ can take $\pm 1$ however it likes. This is NOT what happens! **Only** $\cos\theta_k = 1$ gives the correct solution! If someone could point out what I am or might be missing here, or point me to some relevant reference (most references don't discuss finite size case in such details), that would be great. Thanks!


title: About the recent discovery of tetraquark boundstates

text: I am referring to this, http://home.web.cern.ch/about/updates/2014/04/lhcb- confirms-existence-exotic-hadron So how does this work if we stick to keeping quarks in the 3 dimensional fundamental representation of $SU(3)$? This bound-state seems to have 2 anti-quarks and 2 quarks. So with just 3 colours how do we make the whole thing anti-symmetric with respect to the colour quantum number? Is there anything called "anti-colour" quantum number that an anti-quark can posses so that there are a total of $(3\times 2)^2$ colour options to choose from for the 2 quarks and 2 anti-quarks? I have never heard of such a thing! The point is that unlike the $U(1)$ charge, the non-Abelian charge doesn't occur in the Lagrangian for the quarks. The Lagrangian only sees the different flavours, the gauge groups and the gauge coupling constant.


title: Free Electron Lasers (FEL) and limits on laser wavelength

text: Free Electron Lasers are based on Synchroton coherent radiation when a collimated electron beam suffers trajectory curvature due to a strong bouncing magnetic field inside an undulator. Free Electron Lasers are currently being built that will produce coherent beams of X-ray laser light. Since these lasers do not rely on any specific nature-given discrete transition, the question that i have is: **Can a Free Electron Laser be built to create gamma-ray laser light?** if not, why not. If yes, how bigger would have to be the Linear accelerator and the undulator? what are the figures of relevance?


title: A simple thermodynamic question?

text: I tried this question a lot of time and few day's earlier I also solved it correctly but I don't know why I am not able to solve it now. I can't get what link I am missing. ques: 2 moles of helium gas undergo a cyclic process as shown. Assuming the gas to be ideal calculate the net change in heat energy, the net work done, the net change in internal energy ![enter image description here](http://i.stack.imgur.com/vP9J5.png) If any one would just be able to give me the basics to approach the ques this will also help me?


title: Physical and Geometrical interpretation of Differential Forms

text: I have a doubt about the physical and geometrical interpretation of differential forms. I've been studying differential forms on Spivak's Calculus on Manifolds, but my real intent is to use those concepts in physics. I was very amazed when I found out that a force can be described as a $1$-form that given a vector gives me the work to move a particle along the vector, cf. this Phys.SE post. I really believe that there's much more use for those concepts in Physics. For instance, writing Maxwell's equations in a more general form. The problem with those books like Calculus on Manifolds is that all the ones I've found don't care too much about the physical and geometrical interpretation of those concepts. For instance, although they comment about how this fits and geometry and so on, it's not the focus to justify why forms relates to density and how can this be used to model things in physics. What I want then is to ask if any of you can recommend be books that explains how those concepts fits in physics, how can they be used to give precise descriptions of physical phenomena. Thanks in advance, and sorry if the question is to silly.


title: Can a super conducting magnet & gravity create a cheap source of energy?

text: I have a question about whether this would produce a cheap source of energy. If you created two superconductive magnets configured in a figure of eight layout, in a cold enough environment so you needed no extra power for cooling and the magnets would be in persistent state (so after initial charge would need no extra power). If you then made these magnets levitate a very heavy piece of iron could you then use the gravity of the earth (or moon if the experiment was set up on the cold moon) to generate power when then power was switched from one magnet to the other magnet. E.g. The iron block 1 would rise, power would move to magnet 2, iron block 1 would fall (maybe turning a flywheel on the way down), iron block 2 would rise etc. In my mind this configuration would not cost any additional energy ( as long as the environment was cold enough) and you could get virtually free energy by utilising the gravitational field of the earth or moon... Has this idea got legs?