title: How Does Dark Matter Form Lumps?

text: As far as we know, the particles of dark matter can interact with each other only by gravitation. No electromagnetics, no weak force, no strong force. So, let's suppose a local slight concentration of dark matter comes about by chance motions and begins to gravitate. The particles would fall "inward" towards the center of the concentration. However, with no interaction to dissipate angular momentum, they would just orbit the center of the concentration and fly right back out to the vicinity of where they started resulting in no increase in density. Random motions would eventually wipe out the slight local concentration and we are left with a uniform distribution again. > How does dark matter form lumps?


title: Gradient of the potential originated from two similar magnetic vector potentials is not the same

text: The magnetic vector potential $\textbf{A}$ can be defined up to a gradient of a field. Adding or subtracting such gradient should not change the physics of the problem. The same reasoning is applied for the time derivative of $\textbf{A}$. Imagine a simple vector potential such that $$\frac{\partial\textbf{A}}{\partial t} = (0, -z, y)$$ Inside a finite conductor (e.g., a cube), such field originates a charge accumulation at the boundaries, that creates a total electric field $$\textbf{E} = -\nabla \phi -\frac{\partial\textbf{A}}{\partial t} $$ where the gradient of the potential is due to the charge accumulation. So.. When $\textbf{A}$ is shifted, there shouldn't be any change, but what I see doing FEM simulations is that $\phi$ and $\nabla \phi$ change radically, producing the same total electric field in the two cases. A second vector potential is such that: $$\frac{\partial\textbf{A}}{\partial t} = (0, -z-1, y-1)$$ Looking at the electric field equation, that makes sense. But if the gradient of the potential is due to the charge accumulation, then there are two different charge densities for the two presented situations. Since I believe the physics is right, there should be an error in my reasoning. Can someone please point where?


title: Caustic and Singularities in General Relativity

text: What is the relation between the formation of caustics of a family of null geodesics and the existence of an incomplete null geodesic?


title: A Question on Singularity

text: I am not aware of GR, but due to curiosity i have a question in my mind. Please let me know if it is inappropriate to ask here. My question is about singularity. I am under the assumption that singularity means in mathematical terms equivalent to a discontinuity in a function. My question is what type of discontinuity are the ones corresponding to Penrose- Hawking singularity theorems and also to the the naked singularity ? by type i mean   1. removable (left and right limits exist and are equal but not equal to the function value at that point).   2. a jump type discontinuity (left and right limits exists and are not equal to each other).   3. Oscillating discontinuity (as described as described in the above given link)


title: BCS theory, Richardson model and Superconductivity

text: I'm studying Richardson Model in second quantization. There are many initial points that I don't understand:   * We supposed that an attractive force between 2 electrons exists, due to electron-phonon interaction: an electron generates a phonon that is absorbed by the other electron, and we prove that the resulting potential is attractive. But is this really stronger than the Coulomb repulsive potential? How can one estimate this?   * Why does it neglect the attractive force between electrons with opposite spin? ($\pm$) They have an opposite magnetic momentum too...   * Why do we only consider the attractions between electrons with opposite spin?   * Why do we suppose $k_f ~R ~\ll ~1$ , where $k_f$ is (Fermi momentum)/$\hbar$, and $R$ the interaction distance?


title: Why won't my door close in the winter?

text: Please, take this question seriously, because this is a real problem to me. I have a door in my flat. A closet door, to be specific. And there is a problem with it. In the summer everything is ok, the standard wood door opens and closes as predicted, but in the winter time it drives me crazy! It just won't close! Seems that it doesn't fit where it belongs. Damn door just doesn't allow me to welcome guests and take a cup of mulled wine with them. So the question is, what physical processes are standing behind that unfitting? I believe that sun and gravity may be involved, but I don't really understand it. So, please, help. If I'll know the problem, I'll figure out a solution. P.S Maybe it is important that I live in Kazan', Russia


title: Scale dependence of energy dissipation in viscous flow via AdS/CFT

text: The famous AdS/CFT calculation of the shear viscosity/entropy ratio for strongly coupled $N=4$ SYM relates the shear viscosity to the absorption cross section for fluctuations of the metric onto a black hole in AdS which in turn can be related to the absorption cross section for a massless scalar field. On the AdS side it seems clear that dissipation and relaxation back to an equilibrium configuration after a disturbance means that the disturbance has been absorbed by the black hole, and this would seem to imply that the disturbance propagates from small $z$ to large $z$ in Poincare coordinates where $z \rightarrow 0$ is the UV boundary. That is, the energy of the disturbance propagates from the UV to the IR where it is dissipated. First question: is this the right physical picture? Second question: this seems to imply on the CFT side that the dissipation of energy due to shear viscosity also involves a transfer of energy from the UV to the IR. Is this consistent with what is known in fluid mechanics for viscous flow? If so, can you provide a reference?


title: Calculating uncertainties for a final result

text: Say you are dividing 2 times with uncertainties: $$\frac{t_1}{t_2} ~=~ \frac{0.551s \pm 0.002s}{ 0.712s \pm 0.002s}.$$ After doing the calculations you get: $$\frac{t_1}{t_2} ~=~ 0.774 \pm \ldots?$$ How do you calculate your uncertainty with the two uncertainties $\pm0.002s$?


title: History of Man Acquired knowledge of solar system

text: I know the basics of solar system like how earth moves around sun, and that we have so many planets, elliptical orbit of earth, and how far is sun from earth etc etc. I want to take a step back and understand how mankind has been able to construct this knowledge of solar system piece by piece.Questions: \-- Sun is something we see on daily basis, so it would not be strange if humankind got curious to know about it, and made theories,observations,maths to prove it. How did Copernicus and others came up with the theory that earth moves around sun in elliptical orbit. How did they prove it? Has there been any actual way by which we have actually observed sun being in center and earth moving around in elliptical orbit. or we made such theory and result of this theory have been proved by other observations like day,night,change in weather at a point and at different points on earth. \-- How did human have an inkling that there are other stuff on space apart from sun and moon and some stars, like mars,jupiter and other planets, because these planets are not visible from earth. Did human accidently bump into these planets while moving around in spacecraft like they bumped into America. \-- When was the first man made satellite made? When did the first spacecraft go into space. Weren't people afraid to step their foot outside their spacecraft in an utter uncertainty now knowing what waits them on an alien planet, living condition on it etc. \-- How has the distance between sun and earth been calculated? and speed of light? and size of sun? \-- What are the most used tools used to acquiring such knowledge apart from telescope? What is the range of most powerful telescopes that man has made? \-- And why are humans so curious to know more and more about Mars in particular? Is it because it is relatively easier to study Mars for some scientific reason? how we do scientist intend to put their Mars knowledge to practical use? \-- What is the difference between spacecraft and rocket? What is the application of each? Thanks,


title: Is the curvature of space-time a smooth function everywhere ? (except at black holes)

text: Is the curvature of space-time a smooth function everywhere (except at black holes) in view of General relativity. By 'smooth' it is meant that it possesses derivatives of all order at a given point.


title: Derivation of the Lagrangian method using discretized time axis

text: I'm watching this video lecture by Leonard Susskind of Stanford: http://www.youtube.com/watch?v=3apIZCpmdls After some preliminaries, at 34 minutes he jumps into a discretization of the time axis and proceeds to mess around with it and the Lagrangian, completely losing me in the process. I haven't come across this discrete method before. What is this called and are there any resources on the net (or textbooks) that use this approach rather than a continuous approach? Is it necessary to do this discretely rather than continuously?


title: Resources on Solar Neutrino Physics

text: For a project I'm doing for an introductory Astrophysics course, I'm trying to understand the nuances of solar neutrino physics. However, most of what is said while explaining neutrino oscillations whizzes past me, as I have no background in particle/nuclear physics. Can anyone please tell me what are the concepts I need to understand in order to quantitatively understand the physics of Solar Neutrinos? Book and paper recommendations would be much appreciated.


title: Boil Water,The Fastest Way

text: Few days ago I had to boil water to cook for myself a meal, and then I wondered what is the fastest way to boil that pot of water , any suggestions?(of course I had like to get some formulas and relation between the amount of water and the shape of the pot).


title: If one is talking about antineutron production, what is the charge exchange?

text: In this question was given the antineutron production in the way $$ p + \bar{p} \rightarrow n + \bar{n}. $$ Internet search gives one source only where the charge exchange is mentioned. They talk about the "value for the $ \sigma (p + \bar{p} \rightarrow n + \bar{n}) $". So I couldn't understand how does charge exchange works. Where comes the missing energy from? What is the full equation? Can one give later primary sources?


title: Unsure how resistors are connected? (Series/Parallel)

text: ![enter image description here](http://i.stack.imgur.com/vDRnq.jpg) I am currently attempting this practice question in my tutorial and while I am able to reduce the circuit below to ![enter image description here](http://i.stack.imgur.com/ZCFK4.png) I do not know whether the 6/6/6/6 pane is connected in series or parallel, it should not be parallal else R(Total) = 3+3+3 and Vo will be 2/3 * 12 = 4 The answer im expecting is 4/3V so my current attempt is wrong. I also doubt it is a Wye-Delta transformation. Can anyone explain how the resistors are connected and why? Much thanks in advance.


title: Why do we need the material derivative?

text: I'm studying fluid mechanics, and I got the impression that the material derivative is nothing more than "differentiating along a path" and so I got confused on why do we need it. Basically, let $D\subset \mathbb{R}^3$ be the region containing the fluid and let $f : D\times \mathbb{R}\to \mathbb{R}$ be a time dependent function on $D$. Suppose then $\gamma : I\subset \mathbb{R}\to D$ is a trajectory on the fluid. The material derivative of $f$ along $\gamma$ is defined by $$\dfrac{D}{Dt}f(\gamma(t),t) = \dfrac{\partial f}{\partial t}(\gamma(t),t) + (\mathbf{u}\cdot \nabla)f(\gamma(t),t)$$ Where $\mathbf{u}$ is the spatial velocity field of the fluid. But that expression is nothing more nothing less than simply differentiating $f(\gamma(t),t)$ with respect to $t$, or better, differentiating the function $f\circ (\gamma, I)$ where $I$ is the identity in $\mathbb{R}$. Indeed we have $$\dfrac{d}{dt}f(\gamma(t),t) = \nabla f(\gamma(t),t)\cdot \gamma'(t) + \dfrac{\partial f}{\partial t}(\gamma(t),t) = \dfrac{D}{Dt} f(\gamma(t),t)$$ So since $\dfrac{d}{dt}= \dfrac{D}{Dt}$ why do we need the material derivative? Why do we define it, since in truth it is just the well know derivative of a composition just? What are the advantages of defining it? **EDIT:** I think I got it now. When computing $\dfrac{d}{dt}$ as I said a composition is need, in other words, we _need_ a path. But since all information is contained in $\mathbf{u}$ we can dispose the path by defining: $$\dfrac{D}{Dt} f(a,t) = \dfrac{\partial f}{\partial t}(a,t) + (\mathbf{u}\cdot \nabla) f(a,t)$$ And that coincides with $\dfrac{d}{dt}$ if $a = \gamma(t_0)$ for some $\gamma$ and some $t_0$. Is that really the point we are making when defining the material derivative? Is the same we would get if we had a path going through there with velocity $\mathbf{u}$ but we dispose the path since $\mathbf{u}$ already contains all needed info.


title: Is it true that the Schrödinger equation only applies to spin-1/2 particles?

text: I recently came across a claim that the Schrödinger equation only describes spin-1/2 particles. Is this true? I realize that the question may be ill-posed as some would consider the general Schrödinger equation to be $$ i\hbar \frac{\partial}{\partial t} |\psi\rangle = H|\psi\rangle $$ which is tautological if an arbitrary Hamiltonian is allowed. I assume what was meant by "Schrödinger equation" is essentially a quantized version of the Newtonian relation $T = \frac{p^2}{2m}$. I know the Dirac equation can be reduced to such a form if we take the nonrelativistic limit and assume there is no magnetic field, and besides the Schrödinger equation does predict the gross line spectrum of hydrogen. So the claim that the Schrödinger equation (approximately) describes spin-1/2 particles would seem to be justified. Now, I often see it claimed that the Schrödinger equation is essentially the non-relativistic limit of the Klein--Gordon equation, which Schrödinger published instead of the Klein--Gordon equation because the latter predicted the hydrogen atom fine structure incorrectly. So is it not the case that the Schrödinger equation is also valid as a non-relativistic approximation for the behaviour of spin-0 particles? For spin 1, obviously the Schrödinger equation cannot describe the photon, which is always relativistic. However it seems incorrect to me that the Schrödinger equation should not be able to describe, say, a deuteron.


title: Blackbody radiation Color

text: An ideal blackbody absorbs all incident radiation. Josef Stefan found that the intensity $R$ (power per unit area) radiated by an ideal blackbody is given by $$ R = \sigma T^4 $$ Q1) Since an ideal blackbody absorbs all incident radiation, and if it is in thermal equilibrium, doesn't that mean that $R$ is just whatever the incident intensity of radiation coming in is? So like whatever radiation hits it is what is given off? In other words if I shine a flashlight on an ideal blackbody, and I know that the flashlight gives off 100 W per m$^2$, and my ideal blackbody is a sheet of 3 m$^2$, then the emitted power from the black body is 100*3 = 300 W, or an intensity of 300 W / 3 m$^2$ = 100 W/m$^2$ which is the same as the flash light? Q2) Does a blackbody mean that the object is black? I understand the keyhole thing where you shine light in a keyhole and its trapped there, making it essentially a blackbody. But what if an actual black object absorbed all incident radiation. That means that 1) it will keep absorbing radiation and getting hotter and hotter forever. or 2) the black object will absorb all incident radiation (we said it was an ideal black body) but then it emits radiation given by the equation above. Does that mean that the color black actually has a frequency of light?


title: A step in zeta function regularization

text: I'm just wondering about the mathematical step $$\sum_{n=1}^\infty n\exp[-\epsilon n\sqrt x]=\frac1{\epsilon^2 x}-\frac1{12}+\mathcal O(\epsilon).$$ Why is this equality so? I see that $$\sum_{n=1}^\infty n\exp[-\epsilon n\sqrt x]=-\frac1{\sqrt x}\frac{\partial}{\partial\epsilon}\frac{1}{1-\exp[-\epsilon\sqrt x]}\simeq-\frac1{\sqrt x}\frac{\partial}{\partial\epsilon}\frac{1}{\epsilon\sqrt x}=\frac1{\epsilon^2x}.$$ But how about the $-\frac1{12}$?


title: How can the Huygens-Fresnel principle be derived from the Maxwell equations?

text: The Huygens-Fresnel principle states that every point to which a luminous disturbance reaches becomes a source of a spherical wave. I have been trying to understand this considering a infinite screen with a microscopic hole $dS$ on which a plane wave is incident, but I see no obvious way to describe the resulting propagating wave. How can it be derived from the Maxwell equations?


title: Infinite square well

text: 1\. Given that for an infinite square well problem, $\psi(x,0)=\frac{6}{a^3}x(a-x)$, I can show by Fourier transform that the probability of measuring $E_n$ for $n$ even is 0. But is there a physical reason for this? 2\. The ground state, $E_1$ is most probable, but why is its probability $0.99855...\approx 1$? i.e. why sooo close? Cheers.


title: Is Einstein's 1916 General Relativity paper a recommended way to start learning about the subject?

text: If a person has a good grounding in classical mechanics, electrodynamics and special relativity, is Einstein's 1916 paper a recommended way of learning about the subject? After looking through it briefly, I like what I see because he explains all about tensors from first principles. On the other hand, I'm not too sure if his commentary on the following is outdated:   * Ehrenfest paradox    * Mach's Principle   * other sections


title: Accelerometer on a wheel's rim

text: If I put a 3-axis accelerometer on a vertical wheel's rim which acceleration is going to give to me? Will the gravity interfere my results?


title: Do particle accelerators produce (dangerous) radiation?

text: I was under the impression that particle accelerators were pretty harmless, but some article said that they produce harmful radiation when you're in the tunnel. Given that the Internet... isn't always correct, is this true?


title: Direction of force of friction?

text: I know that friction opposes the direction of the velocity, but the problem is when I don't know what the direction of the velocity is. When I'm drawing a FBD for complex systems with ramps and pullies and stuff (with angular forces considered), I don't really know if the mass is accelerating/moving down or up and hence I don't know if the friction is to the left or to the right. How do I figure it out? Is there any tips you have for me for figuring that out? Thanks in advance.


title: Deriving field equation in Yang Mills theory

text: Trying to show that $$D_\mu\vec{F^{\mu \nu}} = \partial_{\mu}\vec{F^{\mu \nu}} + g \vec{A_\mu} \times \vec{F^{\mu \nu}} = 4 \pi \vec{J^\nu},$$ or (correct me if I'm wrong) $$ \partial_{\mu} F^{\mu \nu}_a + g f_{abc}A_\mu^bF ^{c\mu \nu} = 4 \pi J^{\nu}_a.$$ by varying the action $$S= \int \left( \frac{-1}{4}F^a_{\mu \nu} F_a^{\mu \nu} +J_a^\mu A_\mu^a \right) d^3x dt$$ I know how to do this for regular E&M (starting directly with the Euler- Lagrange equations actually), but not sure how to deal with that extra term arising from the operator $D_\mu$.


title: Body's Electrical Resistance

text: I am not someone specialized on physics, I am just curious on why our Body Electrical Resistance measure as shown by a multimeter varies so much. When allocating the 2 probes each on one hand, the resistance varies from low 100k's of Ohms up to 700k Ohms and even more. Can someone illustrate in a easy way for a physics dummy this phenomena? Thanks


In general what will holding an anti-hydrogen atom for more than a 1/10th of second allow scientists to discover? Specifically, given that they can hold one for <1/10th of a second, what would they discover that have not previously been able to determine. Or if not known, what have they been able to discover to date?


title: Big bang and time

text: I heard Carl Sagan talking about the Universe 15 Billion years ago, and the Big Bang. He made the statement that it was the biggest explosion of all time (at first I thought this a subtle pun). This leads me to my question. What would time have been like at +1 "moment" after the big bang? What I'm trying to ask is, and I hate to say it because I'm afraid I'll sound foolish, did time flow at the same rate? Wouldn't all that mass in one place have distorted space/time (and why didn't it "rip" it)? If I were inside of that mass with a stop watch, I'm guessing I wouldn't have been able to measure a difference because time would have effected me the same way as it effected all the other space/mass in the area. I'm guessing I would have to have something inside (the initial Big Bang mass) and something outside measuring time and see if there was a difference (intuitively this feels weird to consider, could I actually place something "outside" the "Big Bang mass"). Maybe I've said too much, or made the question too complicated. I apologize if this is the case. ## Update A black hole is a lot of Mass collapsed into a small space. I believe that as mass increases time dilation increases. I remember hearing that if you fell into a black hole, you'd never experience the last second of your life... If this is true of black holes, how did time pass in the mass/energy that is responsible for the big bang? As the big bang occurred, did time speed up with the expansion of the universe? I'm trying to explain what I'm getting at by asking more questions related to what I was asking. I'm trying to understand what time itself looked like. As I understand it time prevents everything from happening at once. If time was a line, were the ends smashed together into a point before the Big Bang ? Maybe as mass/energy expanded the "time line" expanded too?


title: Kinematics - angle of projection given relation between speed at highest point and half of that

text: For a projectile launched from a point on horizontal ground, the speed when it is at the greatest height is $\sqrt{\frac{2}{5}}$ times the speed when it is at half of its greatest height. Determine the angle of projection. I attempted to define the velocity vectors, after which i multiply it by $\sqrt{\frac{2}{5}}$, which simply gives me $45^\circ$ as the angle of projection when i split the components, however it actually is $60^\circ$. Kindly tell me how to find the answer to this, because i am getting $45^\circ$ after several attempts...


title: Does the conservation of energy not contradict Newton's $3^{\text{rd}}$ law?

text: Suppose "ball A" is flying with force F. The ball makes contact with "ball B" that has force 0. If the force applied to ball B is equal to the force that is applied to ball A then isn't there a total force of 2F? Doesn't that contradict the conservation of energy?


title: gravitational waves and orders of magnitude

text: I have a question about numerical magnitude in eq. 6.110 of this reference. So, using this reference sheet for this PSR1913+16 binary, I get the following numbers: $$M = 2.8 \times 10^{30} \mbox{kg}$$ $$R = 2 \times 10^{9} \mbox{m}$$ so the power according to 6.110 is $$ \frac{2}{5} \frac{G^4 M^5}{R^5} \approx \frac{(6.6 \times 10^{-11})^4 (2.8 \times 10^{30})^5}{(2 \times 10^9)^5} \approx 10^{65} \mbox{watts} $$ Which doesn't match the $10^{24}$ watts that mentions that page are radiated by gravitational radiation... by 41 orders of magnitude! What mistake I'm making in the numerical evaluation of the radiated power of this binary?


title: maintaining atmosperic pressure on mars

text: how would mankind be able to maintain an earth like atmospheric pressure on mars since mars only has 1/3rd the mass? I am not, nor have i ever attended college, but am very curious.


title: Missing centrifugal acceleration

text: I am trying to get correct equations for acceleration of a point in reference frame A, given position, velocity and acceleration in rotating reference frame B. Let $\mathbf{x}_A(t)$, $\mathbf{v}_A(t)$, $\mathbf{a}_A(t)$ denote the position, velocity and acceleration of the point in reference frame A respectively. Let $\mathbf{x}_B(t)$, $\mathbf{v}_B(t)$, $\mathbf{a}_B(t)$ denote the position, velocity and acceleration of the point in rotating reference frame B respectively. If rotation of reference frame A into reference frame B is presented with rotation matrix $\mathbf{R}(t)$: \begin{equation} \mathbf{x}_A(t) = \mathbf{R}(t)\mathbf{x}_B(t) \end{equation} If derivative is taken from previous equation, velocity of the point in reference frame A is : \begin{align} \mathbf{v}_A(t)=\mathbf{\dot{x}}_A(t) &= \mathbf{\dot{R}}(t)\mathbf{x}_B(t) + \mathbf{R}(t)\mathbf{\dot{x}}_B(t)\\\ &=\mathbf{\dot{R}}(t)\mathbf{x}_B(t) + \mathbf{R}(t)\mathbf{v}_B(t) \end{align} If derivative is taken from previous equation, acceleration of the point in reference frame A is : \begin{align} \mathbf{a}_B(t)=\mathbf{\ddot{x}}_B(t) &= \mathbf{\ddot{R}}(t)\mathbf{x}_B(t) \+ \mathbf{\dot{R}}(t)\mathbf{\dot{x}}_B(t) \+ \mathbf{\dot{R}}(t)\mathbf{v}_B(t) \+ \mathbf{R}(t)\mathbf{\dot{v}}_B(t)\\\ &= \mathbf{\ddot{R}}(t)\mathbf{x}_B(t) \+ 2\mathbf{\dot{R}}(t)\mathbf{v}_B(t) \+ \mathbf{R}(t)\mathbf{a}_B(t)\\\ \end{align} In most textbooks, there is cross product presentation of equations for position (not sure about correctness of below equations): \begin{equation} \mathbf{x}_A(t) = \mathbf{\phi}(t)\times\mathbf{x}_B(t) \end{equation} velocity: \begin{equation} \mathbf{v}_A(t) = \mathbf{\phi}(t)\times\mathbf{v}_B(t) + \mathbf{\Omega}(t)\times\mathbf{x}_B(t) \end{equation} acceleration: \begin{equation} \mathbf{a}_A(t) = \mathbf{\phi}(t)\times\mathbf{a}_B(t) \+ 2\mathbf{\Omega}(t)\times\mathbf{v}_B(t) \+ \mathbf{\dot{\Omega}}(t)\times\mathbf{x}_B(t) \end{equation} Previous equations I derived using derivative property of cross product $\frac{d}{dt}(\mathbf{a}\times\mathbf{b}) = \left(\frac{d}{dt}\mathbf{a}\right)\times\mathbf{b} + \mathbf{a}\times\left(\frac{d}{dt}\mathbf{b}\right)$ From both of these sets of equations, I'm missing centrifugal acceleration $\mathbf{\Omega}(t)\times\mathbf{\Omega}(t)\times\mathbf{x}_B(t)$. I know I may have lack of some fundamental knowledge and I hope you could help me with that.


title: Peskin and Schroeder Equation 3.23

text: I've been trying (for a while) to prove that $S^{\mu\nu}:=\frac{i}{4}\left[\gamma^\mu,\,\gamma^\nu\right]$ is a representation of the Lorentz Lie algebra, that is, to prove that it satisfies the commutation relations of equation (3.17): $$ \left[J^{\mu\nu},\,J^{\rho\sigma}\right]=i\left(\eta^{\nu\rho}J^{\mu\sigma}-\eta^{\mu\rho}J^{\nu\sigma}-\eta^{\nu\sigma}J^{\mu\rho}+\eta^{\mu\sigma}J^{\nu\rho}\right)$$ where $\gamma$ are the gamma matrices defined by the relation $\left\\{\gamma^\mu,\,\gamma^\nu\right\\}=2\eta^{\mu\nu}\mathbb{1}_{n\times n}$. Peskin says "by repeated use of the defining relation of the gamma matrices it is easy to verify that these matrices satisfy the commutation relations." But I failed at that:![my attempt](http://i.stack.imgur.com/yKndu.jpg)


title: 3D Surface Related Multiple Elmination (SRME)

text: Let's assume we are looking at first order multiples of the same surface (i.e no peg leg, etc). I am looking at marine 3D SRME and I am a little confused as to how this works. I understand that in the 2D sense, you can interpolate between two source points to calculate the source point of your reflection point and then you can go about doing your convolutions to create your multiple model, however I don't understand how this works in the 3D sense as you don't have a source point every where along your 3D survey. How do you therefore choose your free surface reflection point to generate your "secondary primary" wavelet to do your SRME? If you want me to clarify ANYTHING, please ask as I really need an answer ASAP. Also, if this is the wrong place for this question, could you please direct me to the correct SE website, thank you.


title: "Book domino" propagation speed?

text: I was watching this video on the Guardian website. As it can be seen, the "wavefront" of the fallen books travel with a fairly constant speed, which I guess depends on the mass of the book $m$, on its height $h$, and of course on the spacing between two neighbour books $l$. How would you estimate the "collapsing front speed" of the books?


title: What determines the shape of the WIMP cross-section vs mass limit curves?

text: In figure 5 of http://arxiv.org/abs/arXiv:1310.8214 (PDF: http://arxiv.org/pdf/1310.8214v1.pdf), the experiments all seem to reach the lowest cross-sections when the WIMP is in the $40-100\, \mathrm{GeV}/c^2$ range. What is the physical reason for the scale of this minimum?


title: How are the finite speed of light and the atomic nature of mater related to the end of Moore's law?

text: In this article from 2007, Moore talks about the end of his Law. Can someone throw more light as to how the finite speed of light and the atomic nature of mater are related to the end of Moore's law?


title: Questions about Michelson interferometer

text: I have been doing experiment on Michelson experiment, but I don't quite understand why white light results in an interferogram with very few fringes, and why are they necessarily Gaussian? I know that white light has a very short coherence length, but is there any reason behind this? ![enter image description here](http://i.stack.imgur.com/UPkHH.png)


title: Force from a HEMIspherical shell

text: I am working on this problem and just trying to figure out what my prof/ TA did in the solution sheet -- but also to make sure I understand what I am doing and some of the mathematics involved. A spherical shell will exert no net force on a particle inside it. But what of a HEMIspherical shell? That is, (working from the question asked on here before about gravity on a hemispherical planet) how to show rigorously what kind of forces will act on a particle in that situation. If you do this problem and ad on the force for the other "half" of course, it should all come out to zero. This is also a bit different from the straight up hemisphere since we aren't assuming that its exactly 1/2. So, given a spherical shell of radius R and surface density σ, and a particle at position a from the center where a is between 0 and R. Assume the position a is along the z (vertical) axis. What is the force on the particle from the smaller hemisphere, described by the part that extends from the top (at z=R) down to where the particle is (at z=a) and where is the center of mass of that hemisphere? I am using θ as the angle between the origin and the point on the shell that intersects a line drawn through a that is perpendicular to the z axis. (I wish I knew how to post a diagram here, that would make it easier I think, but I hope people get the idea). Repeat for the larger part of the shell. Now, I know intuitively (and from the symmetry principle) that the rings, each with a mass of dM = 2πRsinθRdθ will exert a net zero force in the x-y plane as we are assuming symmetry here. So, I only need the force in the z-direction. Anyhow, first I want to know the total mass of the smaller part of the shell. That's going to be an integral from 0 to θ. Since the radius of the strip is R cosθ, and the force vector is along the z-axis, I should get this: $$ \int dF_z = \int_0^{\theta} \frac{Gm2\pi R \sin \theta R d \theta}{r^2}\cos\theta $$ where r is the distance from my particle at a to the piece of the shell I want. So far so good. Now, the distance r is going to be determined by the law of cosines, if I want to get it in terms of θ. So I can say $r^2 = a^2 + R^2 - 2aR \cos \theta$. I'm also multiplying the Force by cosθ because it is in the z direction, which in terms of R and a is $\frac{R^2+a^2-r^2}{2aR}$ If I differentiate $r^2$ -- because I am setting up an integral here -- I get $2rdr = 2aR \sin \theta d \theta$. A little algebra gets me $ \sin \theta d \theta = \frac{2rdr}{2aR} = \frac{rdr}{aR}$ and I should be able to plug that back into the integral and get: $$ \int dF_z = \sigma Gm2\pi R^2 \int_0^{\theta} \frac{rdr}{aR} \frac{1}{r^2} \frac{R^2+a^2-r^2}{2aR} $$ and we have to change the limits, which here would be arccos(a/R) so I have: $$ \int dF_z = \sigma Gm2\pi R^2 \int_0^{\cos^{-1} \frac{a}{R}} \frac{rdr}{aR} \frac{1}{r^2} \frac{R^2+a^2-r^2}{2aR} = \sigma Gm\pi \int_0^{\cos^{-1} \frac{a}{R}} \frac{rdr}{a^2} \frac{1}{r^2} (R^2+a^2-r^2)$$ $$= \sigma Gm\pi \int_0^{\cos^{-1} \frac{a}{R}} \frac{R^2+a^2-r^2}{a^2} \frac{1}{r} dr = \frac{\sigma Gm\pi}{a^2} \int_0^{\cos^{-1} \frac{a}{R}} \frac{R^2+a^2-r^2}{r} dr$$ and because $ \sigma = \frac{M}{4 \pi R^2}$ the integral looks like: $$= \frac{M Gm}{4R^2a^2} \int_0^{\cos^{-1} \frac{a}{R}} \frac{R^2+a^2-r^2}{r} dr$$ So my question is, how am I doing so far? I feel like there is something I am missing -- because the answer sheet says that I should get $$\frac{M Gm}{4Ra^2}2(\sqrt{R^2-a^2}-2R$$ and I am not entirely sure if I got the final integral right. thanks in advance.


title: Understanding direction of magnetic field

text: I'm going through my notes from class on the Biot-Savart Law and looking at these examples I am wondering why the $\vec{B}$ field in the last one is in the $\hat{\phi}$ direction while the first two are in the $\hat{k}$ direction. It seems to me that they should all be in the $\hat{\phi}$ direction.![notes](http://i.stack.imgur.com/Zmu9D.jpg) Can some one help shed light on this subject for me? Thanks


title: Legendre transformation of Entropy as a natural function of T, V, n

text: I have come across a question asking to get a Legendre transformation of entropy which is a natural function of 1/T, V, n starting from entropy being a natural function of T, V and n. However, I have a concern here since T and S are already conjugate variables. In my opinion writing as in the described way is not possible. I would be happy if someone can elaborate on my argument and correct me if I am wrong. Thanks in advance


title: Does the success of quantum teleportation means quantum cloning is possible too?

text: Well, I thought this might just be possible. Instead of having Bob and Alice having an entangled pair, they have multiple (> 2) amount of entangled quantum object (I'm not sure if this is possible too, but I assume it is because in quantum teleportation, when the measurement is taken B is simultaneously entangled with A and C). Alice has another qubit, which she proceeds to scan A and B together, and she gives the information to Bob, where he applies a suitable transformation to the rest of the quantum objects to the same qubit (C, D, E, F, G...) Is this possible?


title: Torque of a rotating fluid on a weighing scale

text: I was trying to do an experiment to measure torque (Moment force) of a rotating fluid. I placed a cylindrical vessel of dia-12cm with 1liter of water on a weighing scale and using a spoon I rotated to my anticlockwise at quiet good speed to find any change in the weight on the scale. I did find change in weight when I dipped my spoon in the liquid, but no change in the weight of liquid during water moving around(Without spoon). Why is that? There should be change in weight (increase) due to moment force (Torque) which is directing towards weighing scale as per right hand rule for torque right...?you can see a whirl in the middle of the vessel which says the torque is at the center but force?. ![enter image description here](http://i.stack.imgur.com/gacKb.png)


title: Where does the photon's energy come from in X-ray bremsstrahlung?

text: In an X-ray tube, bremsstrahlung is generated by accelerating electrons toward a target metal. As a photon is emitted, something loses energy. I suppose that the target metal remains fixed and is re-used -- does this mean that the electron loses kinetic energy? Could it lose all of its energy and start to orbit the target atom?


title: Experiment with spring and two light sources emitting light in opposite directions

text: Two (almost massless) light sources connected with a (almost massless) spring emit light beams in opposite directions. What is the speed at which the spring extends?


title: How strong can tidal forces get?

text: I am imagining a planet the size of Earth being in close proximity to something of high mass such as a gas giant or a star, but ultimately I'd be more interested in _how_ to figure this out myself. So (as it states in the title), I am trying to find out how strong, realistically tidal forces could be, before a planet is destroyed. One way of stating this question would be: ignoring other factors, how close could the earth get to the sun before it would be destroyed by tidal forces? Would the tidal forces be strong enough to be noticed directly by anything on the surface? I am honestly not sure which quantities are relevant. Is the important quantity the yield strength of rock? Or does the molten core make this slightly irrelevant? My other guess would be that it has something to do with the gravitational binding energy. It seems like if the tidal potential energy was on the order of the gravitational binding energy, then bad things would happen, but that seems rather subjective. I guess it depends on what I mean by destroyed, but at least that is an upper bound though, right?


title: What does the term 'hyperbolic model' mean?

text: I am reading this non-linear discrete dynamical system paper. The authors mention the term hyperbolic model. What does that mean?


What conditions do a bunch of atoms need to satisfy to have a temperature? Suppose that we have a beam of helium atoms travelling in a common straight line, equally spaced with the same velocity. If we place a thermometer in the path of this beam, the atoms will impinge on it and heat it up, to register a temperature. However, if we move the thermometer along with the beam, the indicated temperature can be reduced, ultimately to zero. Now add a second collimated beam, exactly opposite to the first one, on a very close parallel path. In this case, whichever way we move the thermometer, a positive temperature will register, and it seems reasonable to assign the minimal such reading to be the temperature of this "beam dipole". Is this the simplest possible, or least-entropic, system that can be said to have a temperature? **Added in response to replies:** I am troubled by the need for entropy to be maximized as a condition for temperature to be meaningful. The above scenario has the unrealistic quality of an idealized micro-experiment (as Maxwell's demon might carry out), which puts it out of range of classical statistical mechanics. So let's consider a macro-scale (even industrial-scale) alternative, at the risk of introducing features that may distract from the essential idea of defining temperature. Suppose that we have a gas-phase chemical reactor. The aim is to combine the primary reactants and cool the product as quickly as possible so as to minimize secondary reactions among the products. This is done by injecting two symmetrically opposed, narrow, very-high-speed jets of cooled gaseous reactants under a bath of inert liquid coolant. The reaction rates at any given pressure, temperature, and concentration of reactants and products are well known. (For convenience, we may assume that the reaction is neither exothermic nor endothermic, but the change in entropy favours the reaction at the designed parameter settings.) When the reaction rate is measured, there is an implied temperature profile in the reaction chamber, assuming a classical chemical-thermodynamic--fluid-dynamic model of the process (which would surely be very complicated, but whose details need not concern us!). _New question_ : Is this implied temperature profile fictitious?


title: How can I relate the energy of a gamma particle to electromagnetic radiation power?

text: I have a distinct feeling that dimensional analysis says these two (let's pick 1 MeV and +30dBm) can not be directly compared. What is the missing link between the two? I know one is energy and one is power. I suppose I'll have to get the energy of +30dBm radiated by an ideal +3dBm antenna at a distance of 2cm impacting an area of $10\times10\,mm$. $1.6\times10^{-13}$ joules for the 1MeV gamma [unknown] for the radio wave (for reference it will be in the low GHz frequency)


title: Is it true that the angular momentum of electromagnetic waves in an anisotropic medium is an integral of motion?

text: Extending my previous question Angular moment and EM wave, does it make sense to talk about the angular momentum of electromagnetic waves in an anisotropic medium? It is not obvious that the angular momentum is conserved in this case. However, if the anisotropy is introduced by the external magnetic field (eg, magnetized plasmas), the projection of the angular momentum of a wave packet on the direction of the magnetic field might be conserved.


title: In optics, how does the vacuum state compare to thermal radiation?

text: In quantum optics, a perfect absorber of light is said to emit the "vacuum field". In practice, any beam dump will be at finite temperature, so it will emit blackbody radiation. How do these fields compare? Is there some critical frequency (for a given temperature) above which the vacuum field dominates?


title: The quantum state just after a position measurement

text: The wave function of a free particle is given as, $$\psi(x) ~=~ e^{-{ x }^{ 2 }/{ a }^{ 2 }}.$$ Then a position measurement is made and the position of the particle is found to be at $x=a$. My question is: What is the state of the particle just after this measurement? Is it equal to $\psi (x=a)$ ?


title: Evaluating the Einstein-Hilbert action

text: The Einstein-Hilbert action is given by, $$I = \frac{1}{16\pi G} \int_{M} \mathrm{d}^d x \, \sqrt{-g} \, R \, \, + \, \, \frac{1}{8\pi G}\int_{\partial M} \mathrm{d}^{d-1}x \, \sqrt{-h} \, K$$ including the Gibbons-Hawking-York boundary term. A well-known derivation of the entropy of the Schwarzschild metric requires the evaluation of the boundary term. However, one must introduce a radial regulator $R$, and subtract off a counter-term which is the Gibbons-Hawking action of emtpy space with the same boundary. The final result is finite as $R\to\infty$. I am attempting to calculate the full action for a solution for which $R\neq0$, hence I need to also compute the pure Einstein-Hilbert piece. However, I must introduce a regulator, and the final action is not finite as I take it to infinity. My question: **is there an analogous procedure to "tame" the infinity of the pure Einstein-Hilbert piece, perhaps similar to the treatment of the Gibbons-Hawking term?** * * * I actually have to introduce two regulators. For my solution, the Ricci scalar is independent of a particular coordinate, $x_1$, so I get a factor of $x_1$ after integration evaluated at $\pm\infty$, so I introduced regulators, such that $L^{-} < x_1 < L^{+}$. As I take $L^{\pm}\to \pm \infty$, of course it is divergent.


title: Gravity and the Standard Model

text: Gravity is ignored in the SM. The proton rest mass is ~0.938 GeV/$c^2$. LHC protons will move with 7 TeV energy, presumably with a relativistic mass about 7,450 times rest mass. A cosmic ray with the highest energy was detected at about $6\times 10^{21}$ eV. If it was a proton, its relativistic mass would have been about $6.4\times 10^{12}$ times rest mass. My question is, at what energy levels would it be necessary to include gravity in the SM? I recognize this isn't an issue for forseeable particle accelerators, but astronomical events like supernova are capable of generating extremely high energy particles.


title: Fate of light ray

text: ![enter image description here](http://i.stack.imgur.com/VQWpx.png) This question has been on my mind since I first learned optics. Where will the light ray go? First, I thought of it taking a U turn. But, I don't think that will happen. Now I have studied a bit about wave optics and I think that the light will go... everywhere? Can someone please explain it in realm of ray optics? I wouldn't mind a bit about wave optics. Just don't go deep down there. Thanks.


title: How about an exact solution for the position of a planet as a function of time?

text: Recently I was surprised to discover that no exact solution for the position of a planet as a function of time exists. I am referring to the two-body problem in a gravitational field where Newtons law of gravity holds. Well known are proofs that the planet will move in an ellipse, Keplers laws can be derived fairly easy etc., but for the exact position of the planet on the ellipse as a function of time, no formula exists, only numerical approximations. Is this correct?   Can anybody elaborate on the deeper reason(s) that this relatively simple case cannot be solved?


title: What does it mean a temperature of billions of degrees?

text: I read a few days ago that in the LHC temperatures of billions of degrees were achieved. I'm curious to know what does it really mean such a temperature? The concept of temperature is easy to grasp when the numbers are familiar, such as 100 C, but when it's in the range of millions it's difficult to understand. Also, if you can provide some explanation of how the temperature is estimated it would be very helpful.


title: Symmetry of wave pulse

text: How can one decide whether a wave pulse is symmetrical by looking at its equation? $$y(x,t)=\frac{0.8}{[4x+5t]^2} $$ represents a moving pulse will it be symmetric?


title: Electron-electron interaction in solids

text: We always consider a Coulomb interactions between electrons in the Hamiltonian to modelize a solid. Why not to take into account retardation effects of the electromagnetic interaction which go beyond the static Coulomb interaction? Because the electronic velocity is usually quite small compare with c?


title: How to find positions of $n$ masses in Newton mechanics?

text: I ran into a problem while doing research. The problem can be described as: consider the original $n$-body problem, and if we fix the position of them(unknowns), no interaction among them, they don't ever move at all. I want to find out their positions. Now I put a test point with a small mass somewhere (known to me) in the system. I can measure the force it receives, I can write down one equation : $$ \sum_{j=1,...,n}\frac{G m_{0}m_j\mathbf{r}_{0j}}{||r_{0j}||^3} = \mathbf{F}_{0}, $$ where $G$ is a constant, $m_0, m_j$ are known to me, $\mathbf{r}_{0j} = \mathbf{x}_j - \mathbf{x}_0$ is the distance vector. $\mathbf{F}_{0}$ can be measured. $\mathbf{x}_0$ is the position of the test point, and $\mathbf{x}_j$ is the location of the $j^{\rm th}$ object. if I move the test point $0$ to many other places, I can get many equations (as many as I want). My question is: how can I solve the positions of the bodies $x_j$ from these equations? Numerical solution is enough for me, I mainly use python programming language, any suggestion on how to solve this non-linear equation system?


title: Do all closed systems, only considering kinematic/mechanical principles, exhibit time reversal symmetry?

text: It makes a lot of sense to me to imagine a cannonball flying through space as not so much experiencing a macroscopic non-conservative drag force, but as pushing a bunch of air molecules and giving away its momentum and energy to them, thereby losing speed. I'd also imagine that if one took a snapshot of the cannonball in mid-flight, replicated the system exactly down to each molecule (assuming this is possible), and then reversed all of the velocities and angular velocities, and pressed 'play', the cannonball would end up flying back to where it was launched. Is this right? Drag, therefore, is simply an emergent phenomenon that is a result of our simplification of the entire system, isn't it? And is really the result of kinematic principles and classical mechanics. In thinking about this I noticed that if we think of drag on mechanical terms, conservation of mechanical energy is held -- and in the result, time-reversal symmetry (right?). So conservation of mechanical energy is therefore "linked" to time-reversal symmetry -- would this be related to how mechanical energy is the conserved quantity associated with time translation symmetry, as well? What about something like a sugar cube dissolving in tea? If I dropped a sugar cube into tea, and then watched it dissolve, froze that system and recreated it while swapping **all velocities** of all of the molecules, would the sugar cube re-gather and then pop out of the cup?


title: buiding circuits from color superconductors

text: caveat: the sort of exotic matter engineering in here is currently beyond the reach of our technology, but, that having been said: Has their been any research on building models of these sorts of circuits? Are there computations that would be more effective in circuits built from color superconductors than would be in ones built from electric superconductors?


title: Can Black Holes be the Dark Matter?

text: Seems to fit the definition: interacts with gravity, doesn’t radiate energy (except Hawking Radiation) and could create gravity lensing without absorbing very much of the light. Could 80% of the original matter created in the Big Bang have ended up in lots of black holes thereby giving us the amount of mass to balance the universes gravity? Kent


title: What is p_T? (transverse momentum?)

text: I've been looking at a few papers in experimental physics (from the ATLAS collaboration, for example) and I've often run across phrases such as "high- p_T electron." What exactly is p_T? Is it simply momentum, but with the component parallel to the main beam projected off? Also, why is p_T, as opposed to p, an important characteristic of a particle in a collision process?


title: Complex Representation of a gauge group and a Chiral Gauge Theory

text: In this John Preskill et al paper, a statement is made in page 1: > We will refer to a gauge theory with fermions transforming as a **complex > representation of the gauge group** as a **chiral gauge theory** , because > the gauged symmetry is a chiral symmetry, rather than a vector-like symmetry > (such as QCD). But my question is: why does a **Complex Representation of gauge group** imply a **Chiral Gauge Theory**? If fundamental representation of SU(3) is a complex representation (with complex conjugate anti-fundamental Rep), then isn't QCD with fundamental representation of SU(3) a perfect counter example where the gauge symmetry is vector-like, instead of chiral??? ps. See this page, or learn that from Wiki: > In physics, a complex representation is a group representation of a group > (or Lie algebra) on a complex vector space that is neither real nor > pseudoreal. In other words, the group elements are expressed as complex > matrices, and the complex conjugate of a complex representation is a > different, non-equivalent representation. For compact groups, the Frobenius- > Schur indicator can be used to tell whether a representation is real, > complex, or pseudo-real. > > For example, the N-dimensional fundamental representation of SU(N) for N > greater than two is a complex representation whose complex conjugate is > often called the antifundamental representation.


title: Can a third type of electrical charge exist?

text: Upon reading my book on physics, it mentions that there are only two discovered types of electric charges. I wonder if there could be a third type of elusive charge, and what type of effects could it have upon matter or similarly?


title: Is the new Hawking black hole all about photon launch angles?

text: The new Jan 2014 Hawking paper (arXiv:1401.5761v1) asserts on page 3: > The absence of event horizons means that there are no black holes - in the > sense of regimes from which light can't escape to infinity. There are > however apparent horizons which persist for a period of time. If light can _always_ escape, then by simple geometry the only path by which light can be emitted from very deep within a black hole is straight up. That is, I am reading Hawking's above statement as an assertion that when an object falls into a black hole, it will see watch its view of the outside world shrink to a smaller and smaller point-like angle. However, critically and in contrast to prior views, that final point will _never_ completely disappear. An object swallowed billions of years ago thus will retain some incredibly tiny solid angle over which a sufficiently vertical photon will eventually reemerge into the outside world, at some vastly lower frequency due to gravity shift. The size of this solid angle should be directly linked to the apparent temperature of the black hole to an outside observer. Here's why: A very large black hole, say a galactic center, will be like a super-hot but perfectly insulated furnace in which only an incredibly tiny porthole allows an exterior observer to see the interior. Since the apparent temperature will be the average of that tiny porthole over the entire furnace surface, the result is an apparent temperature that is appallingly close to absolute zero. Conversely, a very small black hole in roughly the mountain-of-mass range will only barely have enough gravity to keep even photons emitted horizontally trapped in a circular orbit. These would be Hawking's exploding black holes. The solid angle of emission would be nearly hemispherical, meaning that the hole has lost essentially all of its gravitational insulation. I'm not sure quantum mechanics is even needed to describe what happens next, since it would be like suddenly opening the side of a perfectly insulating chamber containing incredible densities of energy. So, boom, big time. Incidentally, if matter falling into a black hole always retains even a trace of connection to the outside world, I'm pretty sure you can't just dismiss deep conservation laws based on symmetries. Hawking seems to imply something like that when he asserts on page 3 that: > The chaotic collapsed object will radiate deterministically but chaotically. A "chaotic collapsed object" is _not_ a pure energy object. So when a small antimatter-fed black hole loses its last gravitational insulation, I'm betting the resulting radiation burst signature would differ somehow that of a matter- fed black hole of the same mass. One think I really like about Hawking's new black hole is that is seems to show a path for getting rid of the extremely paradox-prone distinction between singularity mass and event horizon mass. Here's just one small example: For a galactic center black hole, how exactly does "pump" mass-energy from the singularity back out to the event horizon, assuming that a reasonable and self-consistent location for that horizon can even be defined? If black holes evaporate, you have to do that _somehow_ , and the mechanism is not very clear. Hawking instead seems to be heading toward some form of reunification of interior of the black hole with the event horizon. In his new scheme, even as the interior becomes astronomically introspective and isolated, it never _quite_ takes the final leap of cutting itself off from the apparent horizon at which the vast majority of photons fall back. My own guess is that a careful re-examination of Kruskal-Szekeres coordinates in terms of quantities that are forced to remain "in touch" with and meaningful to the outside world will provide some insights on that. I would even wager based on these more symmetric ideas that small black holes need to be a lot "bouncier" than we give them credit for. For example, smaller ones may be nearly time symmetric when they when they form at the center of collapsing stars, and so can be dismantled at that stage almost as easily as they can be created. Hawking again seems to be deeply interested in exploring higher symmetries between collapse and re-radiation: > This suggests that black holes should be redefined as metastable bound > states of the gravitational field. It will also mean that the CFT on the > boundary of anti-deSitter space will be dual to the whole anti deSitter > space, and not merely the region outside the horizon. My own guess, a prediction I guess, is that the strange bounce seen at the last moment in the cores of collapsing stars is nothing more than a small black hole forming briefly _and then just as quickly self-destructing_. Why not? Small Hawking black holes already blow up, so this would just be a generalization of that concept to a larger range of masses, albeit with higher levels of time symmetries that make even larger holes unstable. Persistent black holes would form only if a threshold of mass is reached that allows gravitational insulation to dominate and keep the hole stable. The increased insulation would show up inside the holes as reduced solid angles of emission, and outside the holes as abrupt and precipitous drops in the apparent temperature of the surface of the hole. Such "too big to bounce" black holes would exhibit the equivalent of sudden and massive phase changes, transforming within a few milliseconds from very hot objects into very cold ones. So, my question: Are my above conceptual interpretations of Hawking's paper reasonably accurate? * * * Related prior questions:   1. Why does Stephen Hawking say black holes don't exist?   2. Photon Escape Angle From Black Hole


title: Centripetal issue when considering gravity

text: Forgive me if my question seems silly, but I am quite baffled. Suppose you have a satellite orbiting a horizontal swing planted into the ground and we want to find the velocity with which the satellite must be moving in order to succeed in taking a fully circular path around the swing (no failure followed by parabolic fall). Indeed if we solve for F = m(v^2)/R where F can be equated to g, we can also solve for the velocity; all nice and good, and the nature of this method makes some intuitive sense but there is a problem (at least for me): the force that the satellite experiences is an inward force towards the center, and therefore the force described in the aforementioned equation is the centripetal force. However, this conflicts with the method of solving the question, since at the top of the path, the centripetal Force and the gravitational force will be both pointing downwards. Therefore shouldn't the satellite actually be "doubly" compelled to accelerate to the ground? But clearly when we for for v with g, we assume that they cancel each other out - hence they point in opposite directions. The idea that the cancel also "explains" the intuition that at the top of the circular path, the satellite will be feeling the most "weightless" it will throughout its revolution. Additionally from a different reference frame / non centrifugal one, it makes sense that at the top, the most force will be on the satellite to turn its path downward as it is after that instant that the satellite sees its downward acceleration. I am, essentially confused with how to handle centripetal/centrifugal "fictitious" forces in this worked example.


title: How does a dehumidifier work?

text: I have read sites like this, but I am just not getting it yet. I'm looking to understand the variables involved, which I think are:   * air moisture content   * air flow (cfm?)   * condenser surface area?   * temperature of the heat exchange medium (fluid)? Is there more? Are there terms used besides these? * * * ### Background: The problem I'm trying to solve I have a water foundation and I want to regulate the humidity of the air that flow past / out of it. I think I would have an area about 14" diameter and 6-10" tall to put the dehumidifier system. I'm not presuming that I'll find one off-the-shelf, but better if I can. * * * I'm looking forward to understanding about the theory of how they work.


title: Two-Person Problem

text: The following is a thought experiment I've been stuck on. (No, this isn't homework. I made it up.) Here it is:   **Two-Person Problem**   Let's say we have a superposed particle going through a tube. There are two people on each side of the tube, as well as two detectors, one 1 yard away from the start of the tube, and the other 2 yards away from the start of the tube, each on different sides. Also they are two people on each side. Now, when the particle reaches the first detector, it is collapsed to person A, but still superposed to person B. Let's say that the result was $|0 \rangle$. My question is, is it possible to be collapsed and superposed at the same time? Even worse, when the particle reaches the second detector, it is possible it could have a result of $|1 \rangle$? And no, in the experiment they are not allowed to communicate with each other.      What's happening here!


title: Relation between classical and quantum information

text: It is known that for a classical system the amount of information needed to store its state is the same as the amount of information that can be stored in that system. This amount is equal to Shannon's entropy or its classical generalizations. On the other hand the amount of information needed to describe the state of a quantum system is greater than the amount of information that can be stored in such quantum system. My impression is that to describe the state of a quantum system of $N$ qubits one needs $2^N$ bits of information, on the other hand one can only store $N$ bits of information inside such system, which is in my impression equal to von Neumann's entropy of the system. Thus I have two possible explanations of such discrepancy. a) The true entropy of a quantum system is exactly von Neumann's entropy. The difference is only due the fact that a classical storage can only independently operate with digits not less than one bit, thus to store, say 4 variables of 1/4 bit each a classical system needs 4 bits (at least one bit for each variable) while a quantum system can use parts of a digit independently and thus store all 4 variables in one binary digit. b) The true entropy of a quantum system of $N$ qubits is exactly $2^N$ bits but the amount of **extractable** information is only $N$ bits. Which explanation do you consider more correct or are there possible other explanations?


title: Work done by gravitational field

text: To calculate the work done by a gravitational field, the equation is written as $$W=GMm(r_\text{initial}^{-2} - r_\text{final}^{-2})$$ Suppose a small mass has distance $R$ from a big mass. So here a gravitational force exists between the 2 masses. If we displace the small mass a very short distance $\mathrm{d}r$, it will need an opposite equal force against the gravitational force. Considering this fact, my question is, if these two forces are equal how we will displace the thing?


title: Superposition and particle mediated interactions

text: Suppose particle A interacts with particle B by means of an exchange particle E. Presumably, since this is quantum physics, the exchange particle emission and receipt are just one of many possible paths by which the two particles can interact. Does that mean that A is in a continual state of superposition of having emitted an exchange particle in every direction? And that B then is in a continual state of superposition of having absorbed it (and thus moved if there was a force involved) and not? Somehow that seems like the most sensible option, because the alternative is that A knows where B is and sends it E at some frequency determined by the strength of the fields. The process above would also entangle A and B as they cannot disagree over whether they exchanged E.


title: Creating Energy!

text: I have this doubt in my mind for a long time, see if we rotate a fan in space: it would keep on rotating forever and would never stop. Due to no air friction present in space. I could never really accept this fact. So i thought of the following situation: I take a fan and rotate it, and this fan is on the top of a turbine. So wouldn't the turbine generate electricity forever? Or if you cant except a turbine, lets say i attach and axle to the fan which in turn has magnets attached to it, There are other magnets surrounding it and remember the thing is rotating, so it would make energy...? or if i make a ball go around an axle. I hope i have made myself clear, that according to me the ball/fan would not stop due to absence of friction. Would it? Why? Sorry if this sounds stupid. I am just a 15 year old young student, enthusiastic about physics so i have a habit of challenging all the theories before accepting them... **Update** From the question i mean to say that will this cause the law of conservation of energy to fail?(ofcourse it won't and i know it) but I want someone who could prove it to me without saying that ....as energy is conserved.... Got it, what i am trying to ask?


title: Have the values of constants ever changed before?

text: Example sub-questions:   * Was the speed of light always that value?   * Was the acceleration due to gravity always been 9.8m/s/s?   * Etc By "change" I mean both actual change in the physical world because of multiple factors and "change" in the numerical value (because the initial value discovered was not precise or accurate).


title: Power transfer in a transformer

text: Can it be proved using the concept of induced emf that power supplied at the primary coil equals power consumed at the secondary. I tried following. Let primary coil be called 1 and secondary be called 2. Assuming there is no resistance in the primary and a purely resistive load on the secondary. Kirchoff's eqns for the two coils are: $V_1-L_1dI_1/dt+MdI_2/dt=0$ $MdI_1/dt-L_2dI_2/dt-I_2R=0$ Multiplying 1st eqn by $I_1$ and 2nd by $I_2$ and adding the two eqns one gets$V_1I_1-L_1I_1dI_1/dt+MI_1dI_2/dt+MI_2dI_1/dt-L_2I_2dI_2/dt-I_2^2R=0$ $L_1I_1dI_1/dt$ represents rate of change of energy of magnetic field due to coil 1 $L_2I_2dI_2/dt$ represents rate of change of energy of magnetic field due to coil 2 $I_2^2R$ represents power dissipated in the resistor. The sum of these three terms should equal $V_1I_1$. but I am getting the extra term of $MI_1dI_2/dt+MI_2dI_1/dt$. Can anybody help me out.


title: Question regarding operators and cylindrical coordinates

text: I have the following problem in my hand: I need to arrive from the Cartesian expression $$x_{j}{\partial_{k}}x_{j}{\partial_{k}}-x_{j}{\partial_{k}}x_{k}{\partial_{j}}$$ to this expression: $$r^{2}\bigtriangledown ^{2}-r^{2}\frac{\partial^2 }{\partial r^2}-2r\frac{\partial }{\partial r},$$ given that I can use the identity $${\partial_{k}}x_{j}=\delta _{jk} +x_{j}{\partial_{k}}.$$ This is part of a derivation to bridge kinetic energy to angular momentum. I appreciate all help. Thank you


title: The potentiality of the electric field

text: Could you please explain using just words why electric the field is potentially? I know the proof using integral: $$A = \int_{12}q\vec{E}\cdot{d}\vec{r} = qQ\int_{12}\frac{\vec{r}\cdot{d}\vec{r}}{r^3} = qQ\int_{r_1}^{r_2}\frac{dr}{r^2} = qQ\left(\frac{1}{r_1} - \frac{1}{r_2}\right)$$ (so the work depends only on the coordinates of the beginning and the end). I can hardly understand it and I want to know how to prove it using only words (maybe with simple school math).


title: What causes the permittivity and permeability of vacuum?

text: When light travels through a material, it gets "slowed down" (at least its net speed decreases). The atoms in the material "disturb" the light in some way which causes it to make stops on its path. This is expressed in the material's permittivity and permeability: Its ability to transmit electromagnetic waves. But then what causes the limited permittivity and permeability of vacuum? There are no atoms there to disturb the light, so what is keeping it from travelling at infinite speed? Is there a special property of space itself which determines the speed of light? Or is it the virtual particles in vacuum which interact with the light?


title: Optical depth in astrophysics

text: How can be the exponential decay of the intensity $I$ via the optical depth $\tau$ be derived? $$I(\tau)=I(0)e^{-\tau} $$


title: Representations of the Poincare group

text: Which type of states carry the irreducible unitary representations of the Poincare group? Multi-particle states or Single-particle states?


title: Definition of "direction"

text: Is there an actual definition of "direction" (that is, spatial direction) in physics, or is it just one of those terms that's left undefined? In physics textbooks it's always just taken for granted that the reader knows what it means (and it is true that just about everyone does indeed have an "intuitive" idea of what it is). But it would be more satisfying to have an concrete definition, if possible.


title: Ionization of Electrons Intensity Relationship

text: Why can't light eject electrons out of atoms (ie. do ionization radiation)? Although the energy of light photons are low (more or less 2 eV), can't 5 photons consecutively hit the electron and make it go out of orbit?


title: What if $F\neq \frac{dp}{dt}$?

text: I was thinking of this idea that maybe there are esoteric cases where the force is not given in classical mechanics as $F=dp/dt$ but as some function of $F=F(p,q,\dot{p},\dot{q})$ E.g, something like: $k\cdot \frac{dp}{dq}$ with a suitable constant $k$, or any other sort of function of p,q and its time derivatives. Are there any toy models that theorists suggest on this idea?


title: Can one define a "particle" as space-localized object in quantum field theory?

text: In Peskin and Schroeder, while discussing creation and annihilation operators for a Klein-Gordon field (p.22), the authors say, as we all know the creation operator $a_p^{\dagger}$ acts on vacuum to produce a state with momentum $p$ having energy equal to $\sqrt{p^2+m^2}$. Then they state: "It is quite natural to call these excitations particles, since they are discrete entities that have the proper relativistic energy-momentum relation. (By a particle we do not mean something that must be localized in space; the creation operator creates particles in momentum eigenstates)" This clashes with my previous conception of a particle to be something localized in space; a point particle located at point $(x,y,z)$ for example, or an extended massive object occupying certain region in space. Form above should I conclude that no such concept of space-localized point-particle exists in quantum field theory? I am not familiar with the process of visualizing something in coordinate space which is known to be localized in momentum space. Please help.


title: Why should a battery not give current in order to measure EMF?

text: Emf is the "potential difference (PD) across the terminals of a battery when it is giving no current to the circuit." What does "when it is giving no current mean"? Will the PD across the terminals of battery when it is not giving current to the circuit equal to the case when the battery is giving current to the circuit? If the internal resistance is coming into play to answer this question then I want to ask that "Internal resistance is present is present in both the cases, either the battery is giving no current or giving it to the circuit, then why do we not call the PD across battery terminals "emf" when it is giving current."


title: Demonstrating Quantum effects with home experiments using green diode laser

text: I would like to preform home experiments to demonstrate quantum effects to my kids. I need a list of experiments on Laser photons that could be easily conducted at home to demonstrate quantum mechanical effects. I'm not interested in experiments that can be explained by considering light as electromagnetic waves. I would appreciate helpful suggestions or a good reference.


title: Molecular dynamics simulation: fluctuating dipole model implementation

text: I'm conducting a molecular dynamics simulation for silica. Some time ago I turned to the fluctuating dipole model, and after much effort I'm still having problems implementing it. In short, all oxygen atoms in the system are polarizable, and their dipole moments depend on their position with respect to all the other atoms in the system. More particularly, I use TS potential (http://digitallibrary.sissa.it/bitstream/handle/1963/2874/tangney.pdf?sequence=2), where dipoles are found iteratively at each time step. This means that when evaluating forces acting on atoms, I have to take into account this potential energy dependency on coordinates. Before, I was using simple pairwise potential models, and so I would set my program to compute forces using analytic formulas obtained by differentiating potential energy expression. Now I'm at a loss: how to implement this new potential? In all the articles that I've found they only give you formulas, but not the algorithm. As I see it, when I compute forces, acting on a certain atom, I have to take into account the change of the dipole of this atom, the change of dipoles of all the neighboring atoms, then tge change of dipoles of still more atoms, and so on, as they depend on each other. After all, it is because of this interdependency that the dipoles are found iteratively at each time step. Clearly, I can't compute forces iteratively for each atom, because the computational complexity of the algorithm would be way too high. Should I use some simple functions to account for the change of dipoles? This doesn't look like a good idea either, cause dipoles are calculated iteratively, with high precision, and then, where it actually matters (computing forces), we would use crude functions? So how do I implement this model? Also, is it possible to compute forces analytically, as I did before, or is it necessary to compute them using finite difference formula for derivative? I haven't found the answer for my question in literature, but if you know some article, or site, or book, where this material is highlighted, please, direct me to that source. Thank you for your time! ================================================================================== UPDATE: Thank you for your answer. Unfortunately, this was not my question. I didn't ask how to compute dipoles, but how to compute forces given that those dipoles vary with movement considerably. I tried to compute forces in straightforward manner (not taking into account dipoles interdependance via their distances, just compute the dipoles on each steps, then compute the forces as if those dipoles are static), but the results I got were not physically correct. To analyze the situation, I set up a simulation of a system consisting of just two atoms: Si and O. They have opposite charges, and so they oscillate. And the energy time dependence graphic looks like this: http://yadi.sk/d/9IfGEouw58I4c (sorry for the link, it appears that I don't have enough points to post pictures yet) The curve on the top represents kinetic energy, the one in the middle represents potential energy without taking dipole interaction into account, and the one at the bottom represents potential energy of the system, where dipole interaction was taken into account. You can clearly see from this graphic, that the system is doing what it shouldn't do: climbing up the potential slope. So I decided this is due to the fact that I didn't take dipole moment coordinate dependence into account. For instance, at a given time point, we compute the forces, and they are directed so as to move both atoms toward each other. But when we do move them towards each other (even slightly), the dipole moment changes, and we find out that we actually ended up with higher potential energy than before! During the next time step the situation is the same. So the question is, how to take this effect into account, cause what few ways I can think of are either way too computationally intense, or way too crude.


title: Calculating absolute speed of a relative object

text: Say you pick a point on the Earth. The earth spins at some constant $x (I don't know how fast, sorry). Now, that point has a different relative speed to a fixed point on Earth's orbit as it revolves around the Sun. This point _also_ has a relative speed to the galaxy from a fixed observer, and etc down to an arbitrary point in the Universe. How would you calculate the absolute/actual speed of this point from a theoretical relative point in the Universe traveling at constant speed 0. Would this be considered the absolute speed on the universe that all objects should be compared against? (I'm not a physics person really, just got into an interesting conversation with a buddy at work)


title: Do siphons really require air pressure to function?

text: Can the siphon mechanism work in vacuums? There seems to have been experiments where exotic liquids (low vapor pressure) were used in siphons mounted in vacuum chambers. The idea is that the falling section of the liquid literally **pulls** the rest of the fluid channel using its weight via intermolecular forces--equivalent to the fluid having a tensile strength. Whereas when we observe normal siphons we are quick to attribute the continuous flow to air pressure which pushses up the rising section.


title: What is the equation for condition of obtaining a bright spot in a wedge shaped film?

text: I want to know if the condition for obtaining a bright spot in a wedge shaped film is $2 \mu t\cos(r+\theta)=(n+\frac{1}{2})λ$ or $2 \mu t \cos(r+\theta)=(2n-1)λ/2$ . $t$ = thickness   $r$ = angle of refraction   $\theta$ = angle of inclination   $n$ = 0,1,2....


title: Flavour diagonal SUSY breaking

text: Because there is a single Yukawa matrix for the SM leptons, the lepton mass and flavour states can be aligned, by diagonalization, even if the Yukawa matrix had off-diagonal elements. SUSY breaking, however, introduces soft-breaking masses for the sleptons, which could be off-diagonal and introduce flavour-changing currents, e.g. $e\to \mu \gamma$. What, in general, is the connection between the soft- breaking mass and the Yukawa coupling? I know that in mSUGRA they can be connected by the Kahler potential, but is there any symmetry principle that can connect the two? I understand that the connection in mSUGRA arises only from minimality and nothing deeper? Surely this alignment cannot be an accident?


title: What are the fields produced around a current carrying conductor?

text: If you consider a current carrying conductor, every instant an electron enters the conductor, another electron will be leaving the conductor. Thus, the current carrying conductor will not be charged (i.e, it would not have any net positive or negative charge). And magnetic field will be produced around it due to motion of electrons. It is important to notice that, if assume only electrons to be moving, and kernels (positive nuclei) to be static, magnetic field will be produced only due to electrons. Thus, only electric field produced due to either of electrons and positive nuclei gets cancelled. As there is no magnetic field due to positive nuclei, we can't expect magnetic field to get cancelled. [it is difficult to think, so as to how electric vector field gets cancelled, with electrons moving and positive nuclei at rest, any explanation or corrections with respect to this, will be appreciable] The speed at which energy or signals travel down a conductor is actually the speed of the **electromagnetic wave** , not the movement of electrons. [this is an modified statement extracted from wiki encyclopedia- **speed of electricity**] By this, _if it mean there is both electric and magnetic field around the conductor, due to electromagnetic wave, it would be contrary to the above statements regarding electric field to be zero around the conductor, due to zero net charge._ Does it mean that electric field and magnetic field exists around the current carrying conductor?   Or   Does it mean that only magnetic field exists around the current carrying conductor? _NOTE_   By the discussion until now (2/11/2013), I have found difference in answers with respect to AC and DC. So, from here on wards, I want the question to be discussed on both AC and DC. Every one are suggested to update their answers with respect to both the cases (AC and DC). [All the statements made by me, are up to my view, taking into account DC. Any correction advisory is welcome] _LINKS_   * This link http://en.wikipedia.org/wiki/Relativistic_electromagnetism provides you an idea about fields around a current carrying conductor, with respect to different frame of reference.    * I went through quite similar question Does a current carrying wire produce electric field outside? . I found that first question was assuming electric field to be existing around the current carrying conductor, and the second answer was assuming electric field to be not existing around current carrying conductor.    * I went through the link:http://www.ultracad.com/articles/propagationtime.pdf, it expresses the presence of both electric and magnetic field around a current carrying conductor.    * I took a look on this question, Confusion between Electric field and Magnetic field of a charged particle., it makes a clear attempt to say that, there is no electric field around a current carrying conductor.


title: Does the Earth's core have tides?

text: Does the moon produce a measurable tidal-effect on the Earth's (liquid) core? If so, how strong is it? Would it play a factor in other geological effects like earthquakes, volcanoes, etc?


title: How does "warp drive" not violate Special Relativity causality constraints?

text: I'm talking about this nonsense: http://www.washingtonpost.com/news/post- nation/wp/2014/06/11/this-is-the-amazing-design-for-nasas-star-trek-style- space-ship-the-ixs-enterprise/ Now, I'm aware that there are problems with the practicalities (or possibilities) of methods involving exotic matter with negative mass, and that kind of thing has been addressed here: Doesn't Warp theory violate causality? But in this case my question is different and more specific. Sweeping all the other problems of constructing such a beast under the rug, wouldn't this violate causality? SR says that the simultaneity and relative ordering of events is dependent on the relative motion of the observer, but that no matter how it works out, if event A causes event B, no observer will see B first. One of the many problems with FTL travel (or signals) is that if any information is sent faster than light, there will exist an inertial frame in which the signal arrives before it is sent, a blatant violation of causality. (I think we can all agree, can we not, that the threshold should be extremely high before breaking assumptions that we're in a causal universe?) So how is any scheme to travel faster than light, even if it may locally satisfy the equations of GR, not a blatant violation causality according to SR? Is my understanding of GR incomplete and there's something in it that obviates the causality implications of SR? Or am I correct that FTL claims should be dismissed with extreme prejudice on this basis alone? (Pending a serious replicable experiment showing causality to be breakable.)


title: Does geodesics from solving full field equations are same as path from energy-momentum tensor?

text: As we know, if we had an energy-momentum tensor in all space-time we could obtain the metric tensor by solving field equations. Also i think if we had an energy-momentum tensor then we have distribution of matter in space-time which means we know its motion and its path. Now my question is that if we calculate the geodesics from the metric, does this geodesic is same as the path obtained from energy-momentum tensor?


title: What is the maximum efficiency of a trebuchet?

text: Using purely gravitational potential energy, what is the highest efficiency one can achieve with a trebuchet counter-weight type of machine? Efficiency defined here as transformation of potential energy in the counterweight to kinetic energy of the trajectory. * * * Edit: To be more specific, we can use the following idealization:   * no friction   * no air resistance   * no elastic/material losses But any of the "standard" trebuchet designs are allowed: simple counterweight, hinged counterweight, vertical counterweight, etc.


title: Cross-section of a wave packet

text: In text books, wave packets are one-dimensional drawings. But we live in a three-dimensional world. Suppose a wave packet from a HI-cloud (frequency 1420 MHz) is approaching the earth, distance about 10 million miles and it will arrive here in a few seconds. How long is this packet (FWHM)? How large is the cross-section? Is the diameter zero? Or 1/10 of a wavelength or three time the wavelength? Or even bigger? Any good ideas? (This question is partly inspired by this Phys.SE answer.)


title: Can we model Chemical Reactions using Quantum Mechanics? If so, what is the most complex reaction we can model?

text: Not a physicist or Chemist, just interested in QM and it's applications. I've been reading lately about Quantum Chemistry and it occurred to me that since we can model electron orbitals in QM and solve for the shape of the orbitals that result (the S, P, D, F etc orbitals), which explains why neutral atoms can form bonds together (valence electron wave function spreading out into a lower energy state between atoms, the difference going into kinetic energy). So, it occurs to me that we should be able to represent chemical reactions as some kind of interaction between groups of atoms. So, is this realistically (computationally) possible? And if so, how complex of reactions can we represent in QM? Are we struggling to model H2 or the 4H + O2 => 2H2O reaction, or are computers at the point where we can model the methylation of Benzene or ATP synthesis? Bonus points if anyone can link to visualisation videos of reactions that demonstrate their Quantum nature! thanks!


title: How to find the principal point in an image?

text: I need to find the principal point in an image. Its a point where the principal axis intersects the image sensor. Due to misalignment this point is not at the center of image always(or image sensor). I need to precisely determine its location using any of the optical methods available(if any). Any suggestions are greatly appreciated. More information: The camera is giving me a live feed over a screen and I am able to store, analyze any part of it in real time or later. ![enter image description here](http://i.stack.imgur.com/JeXNa.png)


title: Partially polarized light in laboratory

text: Does anybody know a way to create (in laboratory, but with no super- sophisticated equipement) a beam of light with partial linear polarization whose degree of polarization can be adjusted (at least over the range 0-0.5)?


title: Wigner $3j$ symbols

text: I am trying to determine the expansion that requires using $3j$ symbols; however, I am running into some conceptual snags. First, the expansion produces symbols that have m's that do not agree with the quantum number requirement that $m=-l, -l+1,..., l$. I assume I can simply throw these out. Second, I am determining $m=m_1$ by evaluating $m_1+m_2=m_3$. I will never have a nonzero term because $m_1+m_2+m_3=0$. Can someone explain?


title: Conservation of energy in objects at terminal velocities

text: In vacuum, object free falling under gravity, the sum of Gravitational Potential Energy(GPE) and Kinetic Energy (KE) is a constant. The GPE is a decreasing side of a quadratic and KE is a increasing one , thus making it's sum a constant straight line. Now, in real world, accounting for air drag, the object attains terminal velocity, and falls down at a uniform rate. So GPE goes down linearly (instead of a quadratic) and KE stays constant since velocity is constant. So the total energy of the system appears as though it is reducing continuously. Since that is not true, as energy is lost in viscous drag. My question is, can someone explain me in detail how the energy is being redistributed. Also, the energy being lost here is not a constant term but actually a linearly increasing function with respect to time. p.s. ![](http://i.imgur.com/f1Vdj.png) In the image above. The left side shows the plot of an object in freefall accelerating under gravity. The 3 plots being GPE, KE, TE versus Time. The right side shows the plot of an object in terminal velocity state. **EDIT :** Trying to be more specific, it is very clear, energy is lost due to viscous drag which is a function of velocity. Now since the object is in terminal velocity state, the drag has a constant value. But for the total energy to remain constant, the energy dissipated must increase with time. So how is a constant drag value causing energy to be dissipated in a linearly increasing fashion?


title: Are circular polarizations a basis for any light polarization?

text: I was trying to solve an exercise in classical optics, consisting in finding the type polarization of certain field profiles. And, by analyzing this one: $$E_x(z,t) = |E|sin(kz-\omega t)\\\ E_y(z,t)=|E|sin(kz-\omega t +\frac{\pi}{4})$$ I first found that it was the superposition of a elliptical polarization and a linear one by splitting the $sin$ in two, then I plotted it and discovered that in fact it's an inclined ellipse. It brought me to the conclusion that superposing a defined longitudinal and an elliptical gives another elliptical one. Now, I recall hearing that the elementary polarization are the two circular ones, which should imply that any elliptical or longitudinal can be decomposed into a sum of circular polarization, am I right? Because formally it's not evident to me. ps: Also, what do the terminology TM and TE labeling the polarization of single photons refer to? I encountered them a few times, but they were never defined.


title: General question on aligning a quantization axis

text: I have a general question on how to work with quantization axis. Here is the setup: > I am looking at a single two-level atom placed at the origin $(0, 0, 0)$, > which is unperturbed in the sense that no magnetic field is applied to it. I now send in a coherent **EM** -field, resonant with the transition of the atom. With a right-handed coordinate system in mind, the field is linearly polarized along $y$ and propagagtes along $+x$. Since there is no magnetic field applied, I am allowed to choose the quantization as I wish.   1. The easiest choice is to choose the $y$-axis directly such that the **EM** -field drives the $\pi$-transition of the atom.   2. If I had chosen the $z$-axis instead, then the **EM** -field would drive $\sigma$-transitions instead. Since no magnetic field is applied, they are equal to the $\pi$-transition, so we get exactly the same signal as we should.   3. Now say I had instead chosen to put my quantization axis along in the $(y, z)$-plane with an angle $A$ relative the $y$-axis. Is it valid to to decompose the quantization axis into a $\cos(A)$ part (driving the $\pi$-transition) and a $\sin(A)$ part (driving the $\sigma$-transitions)?


title: Does glass get stronger the longer its under water?

text: I was reading the internets and I saw an image of of random facts. Yes, I know, its a humor site, but based on the image item #23 I have some questions. Does glass really get stronger the longer its submerged? If so, why and whats the upper limit of its strength? Does depth matter?


title: Could light travel more slowly than the "universal speed limit"? Could this imply quantization of spacetime?

text: One description of relativistic effects that I've heard/read goes something like this: Everything moves through spacetime at a constant speed. An object's direction of travel through spacetime can change, though. For instance, if you are at rest, all of your motion through spacetime is through time. If you are walking, though, some of your motion through spacetime will be allocated to traveling through space. Since less of your motion through spacetime will be allocated to traveling through time, your local time will be slower than the local time of your surroundings. I don't know if this description is a good model or not. It could just be an imperfect analogy as far as I know. If it is a good model, though, it seems to me like light might only be devoting some of its total motion through spacetime to moving through space: it seems that light also travels through time. (This might be incorrect because of the relativity of simultaneity. I don't really know.) If nothing can go through space faster than light but light isn't devoting all of its motion through spacetime to traveling through space, could that mean that spacetime is quantized?


title: What does (simple) $j/cm^2$ represent AND how does this result $6.959j/cm^2$?

text: According to the image shown below, this specific Laser Hair Treatment device claims that it has a concentration of $6.959 j/cm^2$. So far by research I have found that it needs around **$6\mbox{ to }7 J/cm^2$ for better hair growth**. You can find a slight description of it here. However the specific product which has shown here also claims that it **uses $443$ $J$ of energy** per treatment which is **20 minutes** long, and **80 Laser emitters** with **$678\pm 8$ nm of wavelength** which covers an area of **582cm2 of the scalp** out of $720cm^2$ of total scalp area. Here is the image of the Laser Hair Treatment Device which shows its specifications. ![Here is the image of the Laser Hair Treatment Device which shows its specifications](http://i.stack.imgur.com/Hd3pM.jpg). And here is the paragraph which they have mentioned its specifications and energy distribution. > The key to treating hair loss is lasers, but all lasers are not created > equal. We use specially developed, high efficiency lasers that allow us to > deliver the maximum amount of light and cover $582 cm^2$ of the $720 cm^2$ > total scalp area in the average human head. We don't want to bore you with > numbers, but with over $440$ Joules per treatment and an optimised > wavelength of $678\pm 8$ nm, each of the $80$ lasers is precisely tailored > for maximum hair growth. You can find the entire description about this device, here. According to my memory and knowledge I tried my best to get the result of $6.969 J/cm^2$ or something closer to it using these data and I failed. These are my questions.   1. What is (simple) $j/cm^2$ which is indicated here as $6.959 j/cm^2$? And why is it shown as $j/cm^2$ instead of $J/cm^2$?   2. Even so, how do I get the result of $6.959$ ($J/cm^2$ or $j/cm^2$) or something closer to it using these given data, as this device manufactures claim? Your help is highly appreciated. Thank you!


title: Ground states of the Hamiltonian of a two spin system

text: For the spin system shown in this graph (http://i.stack.imgur.com/3lg1R.png), the Hamiltonian is $$S^{(1)}_z\cdot S^{(1)}_z=\frac{1}{4}\begin{pmatrix} 1 & 0 &0 &0 \\\ 0&-1 &0 &0 \\\ 0 &0 &-1 &0 \\\ 0 &0 &0 &1 \end{pmatrix}.$$ The eigenvalues of this Hamiltonian are $-\frac{1}{4}$ when the eigenstate is $$\begin{pmatrix} 0 \\\ 1 \\\ 0 \\\ 0 \end{pmatrix}$$ or $$\begin{pmatrix} 0 \\\ 0 \\\ 1 \\\ 0 \end{pmatrix}$$ and $\frac{1}{4}$ when the eigenstate is $$\begin{pmatrix} 1 \\\ 0 \\\ 0 \\\ 0 \end{pmatrix}$$ or $$\begin{pmatrix} 0 \\\ 0 \\\ 0 \\\ 1 \end{pmatrix}.$$ Moreover, the expectation value of the Hamiltonian for $$\begin{pmatrix} 0 \\\ 1 \\\ 0 \\\ 0 \end{pmatrix}$$ and $$\begin{pmatrix} 0 \\\ 0 \\\ 1 \\\ 0 \end{pmatrix}$$ is $-\frac{1}{4}$ and for $$\begin{pmatrix} 1 \\\ 0 \\\ 0 \\\ 0 \end{pmatrix}$$ and $$\begin{pmatrix} 0 \\\ 0 \\\ 0 \\\ 1 \end{pmatrix}$$ is $\frac{1}{4}$. Based on these information can we say that $$\begin{pmatrix} 0 \\\ 1 \\\ 0 \\\ 0 \end{pmatrix}$$ and $$\begin{pmatrix} 0 \\\ 0 \\\ 1 \\\ 0 \end{pmatrix}$$ are the ground states of the Hamiltonian while $$\begin{pmatrix} 1 \\\ 0 \\\ 0 \\\ 0 \end{pmatrix}$$ and $$\begin{pmatrix} 0 \\\ 0 \\\ 0 \\\ 1 \end{pmatrix}$$ are the first excited states?


title: Without apparatus can we say that the system is measured(decohered) by the environment?

text: "Einselection" and "tridecompositional uniqueness theorem" seem to resolve the preferred basis problem. But the premise is that there are three parts in discussion.(system, apparatus, environment) However, it seems that in many situations we don't have the role of apparatus, and thus there are just system and environment. For instance, we usually say that the system is monitored by its environment and thus in a state with determinate classical physical value. In these situations, in which there are just two parts(system and environment) and the Schmidt form of the total system is not unique, can we say that the system is measured(decohered) by the environment?


title: split gravitational force into x, y, and z componenets

text: I am writing a program for a computer science class in which I am doing an n-body simulation in 3-dimensional space. Currently, I have figured out the gravitational force along the hypotenuse between two bodies. Now I have to split these up into the x, y, and z components, and this is where I am having trouble. Force between two objects (force_t)=G*m1*m2/r^2 Typically, force_x=force_t*cos(angle between x-axis and hypotenuse) force_y=force_t*sin(angle between x-axis and hypotenuse) So, I can figure out my triangle. I know that my two bodies are located at (x1, y1, z1) and (x2, y2, z2), so I have my triangle. And I am fairly certain that the above equations will hold for x and y, even though I'm in three dimensions, but I'm just not sure what the corresponding equation for the z component will be. My thought was that the combination of the three forces has to equal the total force, so perhaps I solve for the z component algebraically using the pythagorean theorem, but I have no way to check whether or not this is true. Anyway, any help would be appreciated. Thanks!


title: The Uncertainty Principle and Black Holes

text: What are the consequences of applying the uncertainty principle to black holes? Does the uncertainty principle need to be modified in the context of a black hole and if so what are the implications of these modifications?