title: Defining Orderly/Chaotic states in terms of entropy?

text: I'm trying to properly understand the meaning of entropy, and how the universe is moving from an orderly state to a chaotic one. If a glass of wine (for example) only has meaning to a human, what makes a shattered spilled glass any less orderly than a full whole glass? Or if a particle is observed, and it's probability wave collapses, isn't this moving from a 'chaotic' (uncertain and in many places), to an 'orderly' (certain, definite position) state? Also, isn't the formation of stars, planets, galaxies etc... a move from a more chaotic state to a more orderly one? Just trying to get some perspective into what the true meaning of entropy is and why it's important.


title: How laminar or turbulent is air?

text: Consider an outdoors scenario, with good weather and no sensible air currents at the floor level. How turbulent or laminar is the air surrounding this environment?


title: Scalar product of torsional forms - how are the standard identities modified?

text: It is known that for any smooth, orientable, compact manifold $X$ without boundary and $\alpha \in \Omega^{r}(X), \beta \in \Omega^{r-1}(X)$ it holds \begin{equation} (d\beta,\alpha)= (\beta, d^{\dagger}\alpha), \end{equation} where \begin{equation} (d\beta,\alpha) = \int_X d\beta \wedge \star \alpha \end{equation} and $d^{\dagger}$ (acting on a $r-$form) such that \begin{equation} d^{\dagger} = (-1)^{mr+m+1}\star d \star \end{equation} with $m=\dim X$ for $X$ Riemannian and with an additional $(-1)$ for Lorentzian manifolds. This is actually not so hard to prove and I don't see any further assumptions required. Now, suppose I have a manifold $X$ with the assumptions above (if possible!) but which admits torsional cycles. For instance, to the corresponding torsional cycle one can then consider a globally well-defined $r-$form $\alpha_r$ s.t. $d\alpha_r = k \beta_{r+1}$ with some integer $k$. Is it then still possible to write for example \begin{equation} (d\alpha_r, d\alpha_r) = (\alpha_r,d^{\dagger}d\alpha_r)? \end{equation} Can the existence of torsional cycles spoil this relation? Is the integration well-defined at all? I mean, I could imagine that one constructs a torus by twisting one end of a cylinder when identifying both ends. Then, intuitively, I would doubt that this surface is orientable and allows integration of forms over this manifold. Unfortunately, I almost don't know anything about torsional cycles and forms. Hence, I'm sorry if my question is sloppy and not precise. This is also, why I'm asking it here in the physics section: 1. At this level, I probably can't follow a mathematician. 2. This question is related to field theory. I'd be grateful for any help and hints. Thanks!


title: Is there a known optical design for a beam compressor?

text: With my little knowledge of optics I have come across some 'known' designs such as the Double Gauss for example, is there a 'beam compressor'. My requirements are to reduce an incoming parallel wavefront (source at infinity) to either an outgoing parallel beam (ie afocal), by a factor of around 100 within 10cm or coming to focus at an f-number around 4 or 5. I have access to Zemax but am currently just searching in the dark worried I might be re-inventing the wheel...


title: About mass defect

text: Here's how my book explains mass defect: > _Particles inside the nucleus interact with each other - they feel > attraction. The potential energy $U$ of such attraction is negative, because > in absence of these forces we consider the potential energy to be zero. So > we can write the total energy as:_ $$E=E_{rest}+U$$ _Dividing $E$ by $c^2$ > we obtain the mass, and because $U <0$ the mass of the nucleus is less than > the sum of individual nucleons._ Now, I have problem with the $U$ term. We know that we can choose the zero level for PE arbitrarily. Thus, $E$ can't be defined well (up to constant). However, real measurements "obey" the standard convention of zero PE at infinity. So how can I solve the contradiction? (Obviously, I'm wrong, but I fail to understand why). This question leads me to a more general question regarding the $E=mc^2$ relation. It follows that $m$ has no certain value when we're dealing with potential energies. Only the **change** in mass matters, because only the change in potential energy has physical meaning (and can be defined precisely). But mass is a quantity which we measure everyday very precisely, and there's no ambiguity in its value, despite the fact that the systems we measure include quite often some potential energy.


title: Time period for spring connected body

text: ![two different spring systems](http://i.stack.imgur.com/xRtob.png) Two identical springs with spring constant $k$ are connected to identical masses of mass $M$, as shown in the figures above. The ratio of the period for the springs connected in parallel (Figure 1) to the period for the springs connected in the series (Figure 2) is $ 1/2 $ What would be the better way to solve this? I have used this law $$\begin{equation} T = 2 \pi \sqrt{\frac{l}{g}} \end{equation}$$ and assumed, $2l$ for the $2^{nd}$ picture but got wrong answer.


title: Canonical transformations and conservation of energy

text: I have an important doubt about the nature of canonical transformations in hamiltonian mechanics. Suppose I have a one-degree-of-freedom lagrangian system, whose hamiltonian depends explicitly on time: $$\frac{\partial{\mathcal{H}(p, q, t)}}{\partial{t}} \neq 0$$ so _in principle energy is not a conserved quantity_. Then I find a canonical transformation, $Q(q, t), P(p, t)$ such that the new hamiltonian, $\mathcal{H}'$ has no explicit time dependency: $$\mathcal{H}' = \mathcal{H}'(Q, P)$$ Can I say then that **indeed energy is a conserved quantity**? If the answer is yes, then it's a bit counter-intuitive for me, specially if it is more or less easy to find such transformations. And if the answer is no, then that makes me think that canonical transformations don't conserve the nature of the system.


title: What's a pseudo-rotation?

text: I'm sorry for this lexical, probably extremely elementary, question. But what is a pseudo-rotation? I just read this term for the first time, in the beginning of the 4th chapter book of CFT by Di Francesco & al. I would say it may be an hyperbolic rotation or a rotation followed by a parity operation (with determinant equals to -1). Couldn't find it on google, so it doesn't seem to be a standard terminology, otherwise please forgive my ignorance.


title: Water ripples and nonzero divergence

text: In 2-d, one ripple would mean the velocity of water particles move out radially forming a circular wavefront. The Navier Stokes equations say the divergence of velocity has to be zero, but this circular radiation pattern has nonzero divergence. What am I missing here, since water ripples clearly exist?


title: What is the Lorentz tensor with a superscript and subscript index?

text: I have been reading about symmetries of systems' actions, e.g. the Polyakov action, and I have encountered Lorentz transformations of the form: $\Lambda^{\mu}_{\nu} X^{\nu}$. I am moderately familiar with $\Lambda$, the Lorentz matrix. If the indices are in superscript then it is the inverse of $\Lambda_{\mu \nu}$. However, what is $\Lambda^{\mu}_{\nu}$ in terms of the Lorentz matrix? I have chosen mathematical physics as the tag, as I do not think any discipline of pure mathematics is appropriate based on the context. Please correct me if I am wrong.


title: What planets are visible to the naked eye from Mars?

text: Here on Earth we are blessed with being able to see some other planets, Mars & Venus etc, with the naked eye on a fairly regular basis thanks to the distance between the planets. What about from Mars? What planets would be visible to the naked eye on a regular basis from Mars? Earth would obviously be one of them, as we can see it, but are any other planets close enough to mars at any point to be visible?


title: Adding many more neutrons to a nucleus decreases stability?

text: If you take any large nucleus and add protons to it, the electrostatic repulsion between them will make the nucleus more unstable, because the electrostatic force between them is more repulsive at a greater distance than the strong force is attractive So how come if you add more neutrons, which dont have a charge and so there is no electrostatic force, the nucleus still becomes more unstable? Also why arent there groups of neutrons bound together, so purely neutron nuclei (because they are neutral, im guessing they couldnt form atoms, because of the electrons needed for an atom)


title: Cosmological triangle with PLANCK results

text: Is there an updated version of the cosmological triangle with recent PLANCK results included? ![](http://i.stack.imgur.com/Q31Pq.png)


title: What is the influence of the nuclear leakage in Japan

text: The recent news says that > Japanese authorities confirmed Saturday that radiation had leaked from a > quake-hit nuclear plant after an explosion destroyed a building at the site. What will be the influence of the nuclear leakage?


title: Relativistic charged particles in a constant uniform magnetic field

text: How can i derive the dynamic of a relativistic charged particle in a uniform magnetic field $B=(0,0,B)$?


title: Maximum potential of a van de graaff generator

text: I know that we can calculate the maximum potential of a van de graaff generator by ( radius* electric field in which corona discharge begin to form in the surrounding gas( according to wikipedia 30 kv/cm)). What if i covered the sphere with Mica with a dielectric strength ( according to wikipedia) 1.18 Mv/cm , will that increase the maximum potential of the sphere? I know it will be hard to discharge the sphere in this case , but is it possible?


title: Euler equation of fluid dynamics

text: I'm trying to obtain Euler equation for a perfect fluid in laminar or stationary flow. A particle fluid is submitted at volume forces and surface force. The fist, in my case, is giving only by gravity and the second by pressure. By Newton's second law I obtain: $$\vec{F}_V + \vec{F}_s = m\frac{d\vec{v}}{dt}.$$ An element of volume force is given by $$d\vec{F}_V = dm\vec{g}=\rho d\omega\vec{g}$$ and an element of surface force is given by $$d\vec{F}_S = -pd\vec{S}.$$ Integrating I obtain $$ \int_V \rho \,d\omega\vec{g} - \int_S p\,d\vec{S} = \frac{d\vec{v}}{dt}\int_V \rho\, d\omega$$. Now Euler equation is written in local form as $$\rho\vec{g} - \nabla p = \rho \frac{d\vec{v}}{dt}.$$ My question is this: where the gradient of $p$ comes from? I must have the following identity $$-\int_S pd\vec{S} = -\int_V \nabla p\,d\omega.$$ Why the transformation from a surface integral to a volume integral is given by the gradient and not by the divergence? I'm doing something wrong in the previous calculations?


title: Can bosons that are composed of several fermions occupy the same state?

text: It is generally assumed that there is no limit on how many bosons are allowed to occupy the same quantum mechanical state. However, almost every boson encountered in every-day physics is not a fundamental particle (with the photon being the most prominent exception). They are instead composed of a number of fermions, which can _not_ occupy the same state. Is it possible for more than one of these composite bosons to be in the same state even though their constituents are not allowed to be in the same state? If the answer is "yes", how does this not contradict the more fundamental viewpoint considering fermions?


title: Why is the velocity on the top of a wheel twice the velocity of its axle?

text: When a wheel is rolling, not skidding, and its axle moves at velocity $\vec{v}$, then a point on the top of its circumference will move at velocity $2\vec{v}$, shown below. ![Velocity wheel top and wheel axis](http://i.imgur.com/qLXEZ.gif) I really don't understand this. I'm quite familiar with the geometry of a circle, but I don't understand how it's being applied on this case. Also:   1. Why doesn't the relation between the velocities depend on the radius of the wheel?    2. When is this relation valid? Does it take friction into account or it doesn't matter? If it doesn't matter, why?   3. A consequence of all this is that the point in the bottom has no velocity. Why? That makes no sense to me. This is a very confusing topic to me. Here is a very nice page full of pictures and animations about the physics of a wheel but I still don't get it.


title: Rayleigh-Benard Convection

text: I found this nice paper about RB convection. However I am confused by what is going on page 6. In particular why we are suddenly using Helmholtz equation to find spatially periodic solutions. Aren't we working with convection, so why are we looking at it from a wave like point of view? Or maybe I'm just missing the point all together. Furthermore I would like to run a quick experiment to collect data. I was planning on using a heating place, a small glass tube filled with olive oil and a thermometer. Any tips? Suggestions?


title: Calculating partition function of ultra-relativistic 1D gas

text: This is a problem (Problem 3.16) from the book Statistical Mechanics 2nd Ed. by Pathria. In the problem I have to calculate the partition function of an ultra-relativistic 1D gas ($E_i=cp_i$) consisting in $3N$ particles moving in one dimension. I know that the partition function is given by $$Q_{3N}=\frac{1}{(3N)!h^{3N}} \int e^{-\beta H(q,p)}d\omega,$$ where $d\omega$ denotes a volume element of the phase space. In this case $$d\omega=dq_1dp_1\cdots dq_{3N}dp_{3N},$$ and $$H(q,p)=\sum_{i=1}^{3N}cp_i.$$ Then, making the substitution I find that $$Q_{3N}=\frac{L^{3N}}{(3N)!h^{3N}} \left[\int_{-\infty}^{+\infty} e^{-\beta c p_j}dp_j\right]^{3N}.$$ $L$ being the "length" of the space available. But I'm pretty sure that this integral does not converge. Where am I wrong?


title: Can quantum communication really replace electromagnetic waves for telecommunication medium in future?

text: Currently I am planning to get masters degree. So I am thinking about a subject in which I have to get masters degree. Following are my questions to leading physicists..   1. Which technology is the future of telecommunication? Currently electromagnetic waves only rules the world for telecommunication. But now a days there is very vast amount of research going on in quantum mechanics and particle physics. According my knowledge the quantum entanglement and delocalization is the base for teleportation and future of communication. I want from the experts to suggest for choosing the topic for masters degree. Is the quantum communication is possible in real world one day? Can it replace our way of telecommunication through electromagnetic waves? If i will do masters in electromagnetism, will i stay behind in time?


title: Physical intuition for deformation quantization of Poisson manifolds

text: First of all, I know almost nothing about physics. I was reading Kontsevich´s paper on Deformation quantization of Poisson manifolds, however I could not figure out what´s the intuition for such operation. Why there is the word "quantization" and the Planck constant in the star product? Where's the physics in such formal deformation of the algebra $\mathscr{A} = \mathscr{C}^{\infty}(M)$ ? In fact, the unique thing that I can understand is the "deformation" part, since (if I'm not wrong) it looks like a deformation of $\mathscr{A}$ along the 2-cocycle $\\{ \cdot, \cdot\\}$ of the Hochschild cochain complex. Another thing, that I never got: why is the deformation done only in the global sections and not in the whole sheaf $\mathscr{C}^{\infty}$?


title: Deccelerate and Accelerate to achieve timed journey

text: I have a vehicle traveling at initial speed $100\:\mathrm{m/s}$. Expected destination speed is also $100\:\mathrm{m/s}$. Maximum deceleration is $3$. Maximum acceleration is $10$. Distance to the target is $2000\:\mathrm{m}$. The vehicle should reach the target in 40s. The vehicle should follow a linear path, decelerate($a_1$) to a certain speed($v_1$) and accelerate back($a_2$) to reach the target precisely on $40\:\mathrm{s}$. Note: $v_1 > 0$, $a_1 < 3$, $a_2 < 10$. How do we find $a_1$, $a_2$ and $v_1$ ??


Once a quantum partition function is in path integral form, does it contain any operators? I.e. The quantum partition function is $Z=tr(e^{-\beta H})$ where H is an operator, the Hamiltonian of the system. But if I put this into the path integral formalism so that we have something like $Z= \int D(\bar{\gamma},\gamma) e^{-\int_0^\beta d\tau H(\bar{\gamma},\gamma)}$, is the $H(\bar{\gamma},\gamma)$ an operator? Thanks!


title: How much power would a space craft's magnetic shield require?

text: I've read over the decades that a magnetic shield might protect a spacecraft from cosmic radiation. Its a fascinating idea that might only be theory or science fiction at the moment. In regards to that here is an article just in case someone isn't sure what I mean: http://physicsworld.com/cws/article/news/2008/nov/06/magnetic-shield-could- protect-spacecraft The article even mentions simulations about midway into the article. Does anyone know if research has continued? My big question though, is how much electrical power would be required to protect a space vessel in this way? How large would the field be in order to be useful? Better yet, please use some practical objects that anyone can wrap their head around: What if you were using similar technology to protect the Apollo Command Module? Or the Discovery One from 2001 (140 meters long). _In laymans terms, how much electrical power would it take to create a magnetic shield?_


title: how do compression and expansion of air transfer energy(sound) and why it is adiabatic not isothermal?

text: Suppose a vibrating fork exerts force to the air particles to compress which leads to the increase of internal energy (heat).This heat or energy given by the fork is given to the next layer of air and in this way, this energy makes sound. But if it was so, it will be isothermal. And Laplace told it was adiabatic. How can it be? I am confused. The heat energy infact is the energy given by the fork. If it is adiabatic, how can the air pass this energy ?


title: propagation of sound energy

text: In my previous questions like 'how do compressions and relaxation carry energy...' and 'how are they not isothermal...' and 'where does the heat come from in the compressed gas...',i've made enquiries reagarding the phenomenon of sound energy propagation and got answers from experts.So i am now writting my concept:when a vibrating fork compresses the air,it does work on it increasing the K.E. of the layer.When the fork goes opp. side,it works on the compressed layer and relaxes it increasing theP.E.Meanwhile,during expansion,the layer works on the next layer and compresses it using the excess kinetic energy.And the next layer does the same to its next layer by compressing using that K.E. given to it by the fork via the first layer.Ultimately,the last layer works on our ear drum to produce sound using the kinetic energy.The question is am i right in my assumption?Plz help...


title: Work for constant volume compression by adding moles of gas

text: I often see the expression $W = V \Delta P$ for the work of a constant volume compression where there are a fixed number of moles and the compression is caused by heating. Is this the work equation for a constant volume, isothermal process where the pressure is increased by adding moles of a gas?


title: What was the motivation behind suggesting the trichromatic theory of vision?

text: ## Background In this thread, I asked whether it is true that the colors red, green and blue, through additive mixture, can make up _any_ color. Turns out they can't. However, when reading about the trichromatic theory of color vision (see a quote from a popular textbook in psychology in the linked thread), it is stated that this is the case, and also that it was the motivation for proposing the theory in the first place. For example, this is taken from the Wikipedia entry on trichromacy: > Trichromatic color vision is the ability of humans and some other animals to > see different colors, mediated by interactions among three types of color- > sensing cone cells. This concept is described in the following depiction (taken from the very same Wikipedia article) where the x-axis denotes wavelength of light, the y-axis denotes neural response, and the three curves represent three different types of cones in the eye, each with its own neural response profile to different light. ![enter image description here](http://i.stack.imgur.com/GhubI.png) Now, the article states: > The trichromatic color theory began in the 18th century, when Thomas Young > proposed that color vision was a result of three different photoreceptor > cells. Hermann von Helmholtz later expanded on Young's ideas using color- > matching experiments which showed that people with normal vision needed > three wavelengths to create the _normal range of colors_ [My emphasis]. > Physiological evidence for trichromatic theory was later given by Gunnar > Svaetichin (1956). I don't know if there is some hidden meaning embedded in the phrase "normal range of colors" but supposing that isn't the case (please inform me if so), there seems to be somewhat of a contradiction going on here. Different sources states that the fact that the trichromatic theory of color vision was proposed in the first place was because three colors were found to be able to create any color through additive mixture (if there were other reasons, I haven't heard about them); however, this statement isn't true. WTF? ## TL;DR On what grounds was the trichromatic color vision theory proposed in the first place? Was it based on a misconception (red, green and blue can make up any perceivable color through additive mixture) that just happened to lead to the right answer (there being three different type of cones)?


title: Equation of motion for the reduced density matrix

text: The equation of motion for the density matrix of a many body isolated quantum system is the von Neumann's equation: $\dot{\rho }(t)=i[\rho (t),H]$. How about the equation of motion for the reduced density matrix of one particle, is there a general procedure to obtain it?


title: Is this possible? LCD right in front of the eye

text: I came across these swimming goggles with a LCD display integrated right on the goggle itself. http://www.psfk.com/2011/09/smart-goggles-enable-swimmers-to-track-and-record- swim-pace.html/my_pace_goggles2 Does this look possible as shown in the design? Would I need a lens over the LCD to make it easier to focus on? How thick and/or curved would that lens need to be? Any advice appreciated!


title: How do I find equations for the distance of the planets relative to one another?

text: I know I should be able to piece together some basic Newtonian equations for this, but I'm not sure where to start. I want to be able to choose one planet as the center and calculate its distance from another planet over time as they both rotate the sun. First I need to even find equations for the motion of the planets, then... magic? Thanks for any guidance!


title: Availability of the Larkin's paper: Application of superconductivity theory methods to the problem of the masses of elementary particles (1961)

text: Discussing with colleagues (mostly condensed matter ones), we were astonished there is no mention about the work by Larkin in the announcement of the Nobel Prize committee this year. The title of the Larkin's paper -- APPLICATION OF SUPERCONDUCTIVITY THEORY METHODS TO THE PROBLEM OF THE MASSES OF ELEMENTARY PARTICLES -- sounds promising, but I have no idea about its content. It seems to be extremely difficult to find. I'm not even sure it has been translated. So the question is: can anyone provide me with the original publication, some translation in some west-european language, or some extracts of the original paper by Larkin ? More details about this paper can be found following this link. _This question does not intend to open any polemics regarding the attribution of this year Nobel prize. Larkin died in 2005 by the way._


title: Are neutrino stars theoretically possible?

text: Since neutrinos   1. have a small mass and   2. are affected by gravity, wouldn't it be theoretically possible to have such a large quantity of them so close to each other, that they would form a _kind_ of a stellar object, i.e. one that would keep itself from dissolving due to the large gravity. If such objects were possible, how would they interact with rest of the world? Would they be invisible (dark matter?) because of neutrinos' lack of electromagnetic charge? Would this lack also make ordinary matter pass through them, or would the Pauli exclusion principle prevent this passing through due to the high density of neutrinos?


title: Why does mode hopping occur through a change of temperature?

text: Why does mode hopping occur when the temperature of a diode is changed? Why is a similar effect seen when we change the electric current?


title: Physics Equivalent of Physical Chemistry (P-Chem)

text: Before I take P-Chem I would like to understand how physicists view the same material. What Physics course(s) should I take and/or books should I read to learn the same material from a physicist's perspective? Would thermodynamics be equivalent? Background: I have done the math: Calculus 1-3 and differential equations and have already taken general physics 1+2 and 2 semesters of modern physics (relativity, QM...).


title: Why isn't pressure used for flight?

text: Why isn't pressure used as flight? I've heard that 2L bottles can hold a pressure of up to 90 PSI safely. Since $F = PA$, if the nozzle of a pressure rocket is about 4 inches squared in area, that would be a thrust of 360 pounds!? Is there something wrong with my math, or why don't we just pressurize air and use it for flight?


title: Action of conformal generators on fields

text: I am calculating the action of the conformal generators on fields, to be more precise on wavefunctions. For now, I'm classical. I will just paste the part of my report on this to show what I am talking about: Now let us use the fact that $\Phi_a(x) = U_P(-x)\Phi_a(0) = e^{iP_\mu x^\mu}\Phi_a(0)$. We can write the action of the generators as follows: \begin{align*} G\Phi_a(x) &= Ge^{iP_\mu x^\mu}\Phi_a(0) \\\ &= e^{iP_\mu x^\mu}e^{-iP_\mu x^\mu}Ge^{iP_\mu x^\mu}\Phi_a(0) \\\ &= e^{iP_\mu x^\mu}\tilde{G}\Phi_a(0) \end{align*} where have defined $\tilde{G} = e^{-iP_\mu x^\mu}Ge^{iP_\mu x^\mu}$. We can now calculate the $\tilde{G}$ using the Hausdorff formula: $e^{-A}Be^{A} = B+[B,A] + \frac{1}{2!}[[B,A],A] + \frac{1}{3!}[[[B,A],A],A] + \dots $ We obtain: \begin{align*} \tilde{\J}_{\mu\nu} &= e^{-ix^\rho P_\rho}\J_{\mu\nu}e^{ix^\rho P_\rho} = \J_{\mu\nu} + ix^\rho[\J_{\mu\nu}, P_\rho] \\\ &= \J_{\mu\nu} + x^\rho(\eta_{\rho\mu}P_\nu - \eta_{\rho\nu}P_\mu) \\\ &= \J_{\mu\nu} + x_\mu P_\nu - x_\nu P_\mu \\\ &= \J_{\mu\nu} - i(x_\mu \partial_\nu - x_\nu \partial_\mu) \\\ \\\ \tilde{D} &= e^{-ix^\rho P_\rho}De^{ix^\rho P_\rho} = D + ix^\rho[D,P_\rho] = D - x^\rho P_\rho = D + ix^\rho\partial_\rho \\\ \\\ \tilde{K}_\mu &= e^{-ix^\rho P_\rho}K_\mu e^{ix^\rho P_\rho} = K_\mu + ix^\rho[K_\mu,P_\rho] - \frac{1}{2}x^\rho x^\nu[[K_\mu,P_\rho], P_\nu] \\\ &= K_\mu - 2x^\rho(\eta_{\mu\rho}D - \mathcal{J}_{\mu\rho}) + x^\rho x^\nu(-\eta_{\mu\rho}P_\nu + (\eta_{\nu\mu}P_\rho - \eta_{\nu\rho}P_\mu)) \\\ &= K_\mu - 2x_\mu D + 2x^\rho\mathcal{J}_{\mu\rho} + (2 x_\mu x^\nu P_\nu - x_\rho x^\rho P_\mu) \\\ &= K_\mu - 2x_\mu D + 2x^\rho\mathcal{J}_{\mu\rho} - i(2 x_\mu x^\nu \partial_\nu - x_\rho x^\rho \partial_\mu) \end{align*} The action of the generators are therefore given by: \begin{align*} {\J}_{\mu\nu}\Phi_a(x) &= e^{iP_\mu x^\mu}(\J_{\mu\nu} - i(x_\mu \partial_\nu - x_\nu \partial_\mu))\Phi_a(0) \\\ &= (\Sigma_{\mu\nu})_a^{\;\;b}\Phi_b(x) + i(x_\mu \partial_\nu - x_\nu \partial_\mu))\Phi_a(x) \\\ D\Phi_a(x) &= e^{iP_\mu x^\mu}(D + ix^\rho\partial_\rho)\Phi_a(0) \\\ &= D_a^{\;\;b}\Phi_b(x) - ix^\rho\partial_\rho\Phi_a(x) \\\ K_\mu &= e^{iP_\mu x^\mu}(K_\mu - 2x_\mu D + 2x^\rho\mathcal{J}_{\mu\rho} - i(2 x_\mu x^\nu \partial_\nu - x_\rho x^\rho \partial_\mu))\Phi_a(0) \\\ &= (K_\mu)_a^{\;\;b}\Phi_b(x) + 2x_\mu D_a^{\;\;b}\Phi_b(x) - 2x^\rho(\mathcal{\Sigma}_{\mu\rho})_a^{\;\;b}\Phi_b(x) - i(2 x_\mu x^\nu \partial_\nu - x_\rho x^\rho \partial_\mu)\Phi_a(x) \end{align*} As you can see, every time from the first to the second line there is a minus sign appearing precisely where there is an x. This is what happens in Salam's paper Finite component field representations of the conformal group and it has to happen in my case for it to be consistent with what I wrote before. BUT I don't understand why this change of sign appears. $e^{iP}$ and $P$ commute, so why is this happening. I have been annoyed by this for a moment now. Please help me!


title: If friction disregard area, why the direction you drag a long object matters?

text: I am talking here about dry friction between solid objects, for example a ruler and a table, not anything lubricated or fluid. I noticed that with a ruler and a table for example, if you drag the ruler like it was a knife, it is much easier to do it than if still holding the ruler in the same position you drag it sideways (like if you are scrapping something). Also I noticed that when I am washing dishes, if I leave two metal objects (ie: to flat metal areas) in contact, they don't move much, but the same objects, if I try to find deformations that make the area of contact between them smaller, then they can be easily pushed around from rest, or spun. This also applies to tyre sizes (ie: for dramatic effect, dragster cars with HUGE rear tyres and tiny front tyres). Also I did some experiments with a paper, ie: holding it down with a finger make it much easer to slide than if I make sure more of its area is in contact, but doing that also is a downforce on the paper, so I guess I can sum that on the normal force. My best guess is that it all has to do with the normal force, but I am not sure at all... Can someone quench my curiosity here?


title: Potential Energy of a Particle in One Dimension?

text: Since a particle in one dimension can only move in a straight line. Is it possible to have potential energy? And how would the kinetic energy and potential energy differ in higher dimensions?


title: How is a Higgs boson created?

text: I have read a lot on Higgs bosons, yet I do not fully comprehend how they are created and how they are "flicked off" the Higgs field. I have also had trouble comprehending why a Higgs boson quickly becomes unstable and decays into more common particles. How is a Higgs boson created and how and why does it quickly decay?


title: What is the symmetry associated with the local particle number conservation law for fluid?

text: According to Noether's theorem, every continuous symmetry (of the action) yields a conservation law. In fluid, there is a local particle number conservation law, which is $$\partial{\rho}/\partial{t}+\nabla \cdot \vec{j} ~=~0,$$ where $\rho$ and $\vec{j}$ is the density and current respectively. I just wonder is there any symmetry associated with this conservation law?


title: How can one reconcile the temperature of a black hole with asymptotic flatness?

text: A stationary observer very close to the horizon of a black hole is immersed in a thermal bath of temperature that diverges as the horizon is approached. $$T^{-1} = 4\pi \sqrt{2M(r-2M)}$$ The temperature observed by a stationary observer at infinity can then be obtained through the gravitational redshift formula (see http://en.wikipedia.org/wiki/Hawking_radiation#Emission_process) to be $$T^{-1} = 8 \pi M$$ which is what is often quoted as the temperature of a black hole. As QGR points out here in an answer to my related question here, the resulting non-zero stress-energy tensor at infinity is incompatible with the asymptotic flatness of the Schwarzschild spacetime. What exactly is going wrong here?


title: Homework: A parachutist jumps from a plane

text: I need help with the following problem. I think I almost have it . . . A parachutist bails out and freely falls a distance of $y_1$. Then the parachute opens and thereafter, the parachutist decelerates at a rate of $a_2$. She reaches the ground with a speed of $v_2$. Find her average speed for the fall. (The answer should be $18.5\ \text{m/s}$.) Note: In the diagram, replace "x" with "y". :) ![http://i.imgur.com/OPTZfWI.png](http://i.stack.imgur.com/5lnSe.png) $ \text{Let upwards be the positive y direction.}\\\ \text{Givens:}\\\ y_1 = -59.7\ \text{m}\\\ v_0 = 0\ \text{m/s}\\\ v_2 = -3.41\ \text{m/s}\\\ a_1 = -9.8\ \text{m/s$^2$}\\\ a_2 = 1.60\ \text{m/s$^2$}\\\ \text{Solve for $t_1$:}\\\ y_1 - y_0 = \frac{1}{2}a_1t^2+v_0t\qquad\text{for}\ \ y_1 = 0; v_0 = 0; t=t_1.\\\ t_1 = \pm\sqrt{\frac{2y_1}{a_1}}\\\ t_1 \approx \pm 3.4905\ \text{s}.\\\ \text{Plug $t_1$ into the velocity equation for $v_1$ to find $v_1$:}\\\ v_1 = a_1t_1 + v_0\\\ v_1 \approx -34.207.\\\ \text{Plug $v_1$ into the velocity equation for $v_2$ to find $t_2$:}\\\ v_2 = a_2t_2 + v_1\\\ t_2 = \frac{v_2 - v_1}{a_2}\\\ t_2 \approx 19.248\ \text{s}.\\\ \text{Plug $t_2$ into the position equation for $y_2$ to find $y_2$.}\\\ y_2 = \frac{1}{2}a_2t_2^2+v_1t_2 + x_1\\\ y_2 \approx -422\ \text{m}.\\\ \text{Now solve for $\lvert\overline{v}\rvert$:}\\\ \lvert\overline{v}\rvert = \lvert\frac{x_2 - x_0}{t_2 - t_0}\rvert\\\ \lvert\overline{v}\rvert \approx 21.9\ \text{m/s} \neq 18.5\ \text{m/s}. $ My solution is incorrect. What am I doing wrong?


title: The color purple in a rainbow

text: In a rainbow the colors order is red then orange (made from red and yellow, thus making sense that it appears in between them) the yellow followed by green after which comes blue (again green formed from yellow and blue). The final color is purple, which is next to the blue' but not in any contact with the red. The question is - is there any actual connection between the colors and there source as known to as (such as orange from yellow and red, the last two being the source) to its appearance in a rainbow? If so, why doesn't the purple follow the same logic? How is the purple seen were it is seen in a rainbow (where there is no red nearby)? thx


title: Explicit evaluation of a radially ordered product

text: I am trying to understand the application of the operator product expansion to calculate the radially ordered product in the complex plain of $T_{zz}(z)\partial_w X^{\rho}(w)$ which should result in $$ \langle R(T_{zz}(z)\partial_w X^{\rho}(w))\rangle = -l_s^2\frac{1}{(z-w)^2}\partial_w X^{\rho}(w) - l_s^2\frac{1}{(z-w)}\partial_z^2 X^{\rho}(z) + \cdots $$ but embarassingly I encounter a stumbling block right at the beginning. After inserting $T_{zz}(z) \doteq \, :\eta_{\mu\nu}\partial_z X^{\mu}\partial_zX^{\nu}:$ one has $$ \langle R(T_{zz}(z)\partial_w X^{\rho}(w))\rangle = R(:\eta_{\mu\nu}\partial_z X^{\mu}(z)\partial_zX^{\nu}(z):\partial_w X^{\rho}(w)) $$ which can obviously be further expanded to $$ ... = \eta_{\mu\nu}\langle \partial_z X^{\mu}(z)\partial_w X^{\rho}(w)\rangle \partial_z X^{\nu}(z) \+ \eta_{\mu\nu}\langle \partial_z X^{\nu}(z)\partial_w X^{\rho}(w)\rangle \partial_z X^{\mu}(z) $$ It is exactly this last step I dont understand. If this initial stumbling block is removed, I can understand the rest of the derivation, so can somebody help me remove it? To generalize a bit, it seems I do not yet properly understand how such expressions involving normal and radial (time) ordered products are generally evaluated. So if somebody could give me a more general hint about this, I would probably be able to see how the last expression in my particular example is obtained.


title: Baryonic Missing Mass

text: A recent article from a popular astronomy website tells of discovery of missing mass (not dark matter) that has puzzled astronomers for some time. Apparently, the discovery involves enhanced electron density in filaments associated with superclusters of galaxies. How were astronomers able to determine that this baryonic mass was missing in the first place, and what percentage of total baryonic mass did it entail?


title: Grassmann Variables Representation?

text: It might be a silly question, but I was never mathematically introduced to the topic. Is there a representation for Grassmann Variables using real field. For example, gamma matrices have a representation, is it not possible for Grassmann Variables? The reason for a representation is, then probably it will be easier to derive some of the properties.


title: Simple projectile motion problem

text: A person wants to throw an object from the top of a tower $9,0m$ high towards a target which is $3,5m$ far from the place where the person is launching the object. Suppose that this object is thrown horizontally. What I want to find out is:   1. What initial speed does the object need to have in order to hit the target?   2. What is the acceleration of the object one moment before it touches the ground? I suppose that the answer to the second question is simply $9,81m/s^2$, but I am not sure about it because it looks so simple :D As far as the first question is concerned, I was thinking about using the formula $y_{MAX} = \frac{v_0^2 + sin^2\theta}{2g}$, with $y_{MAX}$ being $3,5m$ and $\theta$ being $45°$, but the result in this case would be extremely unrealistic ^^ Am I going in the right direction (in both questions)?


What observables are indicative of BCS Cooper pair condensation? "Thought" experiments and "numerical" experiments are allowed. This question is motivated by the question Has BCS Cooper pair condensate been observed in experiment? , and by our recent research on anyon superfluidity where anyons are emergent from a fermion system.


title: Eigenvalues of Infinite Dimensional Matrix

text: If I take a infinite-dimensional square matrix, what can I say about its eigenvalue spectrum? Will they have a discrete infinity of eigenvalues or continuous infinity of them?


title: Why Quantum Mechanics as a non-fundamental effective theory?

text: **My question:** What (physical or mathematical) reasons (not philosophical) do some physicists ('t Hooft, Penrose, Smolin,...) argue/have in order to think that Quantum Mechanics could be substituted by another theory in the future? Namely...Why should it be an effective and non-fundamental (``non- ultimate'') theory? 't Hooft has spent some years trying to "prove" that Quantum Mechanics can arise from a classical theory with information loss through some particular examples (cellular automata, beable theory,...). Penrose advocated long ago that gravity should have a role in the quantum wave function collapse (the so called objective reduction), Smolin wrote a paper titled , I believe, Can Quantum Mechanics be the approximation of another theory?. What I am asking for is WHY, if every observation and experiment is consistent (till now) with Quantum Mechanics/Quantum Field Theory, people are trying to go beyond traditional QM/QFT for mainly "philosophical?" reasons. Is it due to the collapse of wavefunctions? Is it due to the probabilistic interpretation? Maybe is it due to entanglement? The absence of quantum gravity? Giving up philosophical prejudices, I want to UNDERSTAND the reason/s behind those works... **Remark:** Currently, there is no experiment against it! Quite to the contrary, entanglement, uncertainty relationships, QFT calculations and precision measurements and lots of "quantum effects" rest into the correction of Quantum Mechanics. From a pure positivist framework is quite crazy such an opposition!


title: Can you use a wormhole to travel through space not time?

text: I want to know if you could theoretically travel from your house to work via a wormhole but stay in the present day...without changing time. Kind of like teleportation but harnessing the energy of a black hole or wormhole. My background is chemistry not physics so I'd prefer a mathematical explanation. Thanks.


title: Eliminating IR light reflection perceived by a steep viewing angle

text: I am having a problem with reflection on an acrylic surface, in the IR part of the spectrum. This reflection is interfering with an algorithm that looks at objects, as it makes two show up when only one exists. **Some background:** An IR sensitive (and IR filtered) camera must view an IR backlit stage below at a steep angle, the walls of which are perpendicular to the IR light source, the cameras are at about 30 degrees tilted back with respect to the walls of the stage. Because the walls of the stage must be translucent to visible light, they are made of acrylic (just about the same optical properties as glass). A problem arises when an object comes close to the surface of the glass, as a reflection of that object appears from the acrylic. This confuses the software that uses the camera into thinking there are two objects. To get rid of this problem, a matte finish is applied to the inner surface of the wall to blur the reflection enough, and this works. However it doesn't work perfectly, and the inner surface of the wall would ideally be perfectly smooth. **The first question is: How can I eliminate the reflection due to the internal reflection, in the IR part of the spectrum, of my object?** **Specifications:**   * A wall of acrylic is reflecting an object at a steep angle in the IR spectrum   * The inner wall must be kept smooth (although could have some filter added, and then a smoothing film added over top, perhaps, if it would still elimiate the reflection)   * The viewing angle is steep and cannot be changed   * Visible light must be able to pass from the outside of the wall to the inside. **Proposed solution is now:** Using an IR filter applied to the outside wall to prevent the reflection from coming in and up to the camera as strongly, this leaves the inside smooth. Or applying a matte to the outside of the wall to blur the reflection, which also leaves the inside able to be perfectly smooth. **A second question is: will the proposed solution work?**


title: Is it possible to use the observation of wave function collapse as a basis for superluminal communication?

text: I'm guessing the answer is no ;) but figured I'd ask anyway. Suppose I have one partner in an entangled pair and my friend on Alpha C has the other. I measure an observable (spin?) and the wave function collapses to a single eigenvalue. Does this cause the wave function for that observable to collapse for the remote partner, and if so, can the wave function collapse event itself be used as a superluminal signal? I assume the answer is no simply because he can't tell if the wave function has collapsed without making a measurement . . . which would cause the wave function to collapse. The whole idea falls apart if wave function collapse per se is not something that can be observed directly but is only inferred from measurement, I suppose.


title: The most general form of the metric for a homogeneous, isotropic and static space-time

text: What is the most general form of the metric for a homogeneous, isotropic and static space-time? For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) signature) $$ds^2=dt^2+a^2(t)g_{ij}(\vec x)dx^idx^j$$ Now the static condition. If I'm not mistaken, it means that the metric must be time-independent and invariant under time reversal $t\to -t$. So does that mean that the most general metric that satisfies all these 3 criteria is $$ds^2=dt^2+g_{ij}(\vec x)dx^idx^j$$ for some spatial metric $g_{ij}(\vec x)$? Thank you. (My apologies for the mistaken definition. I may have misunderstood what's been said in class...)


title: Is radiation pressure constant in this experiment, and does force of a spring change?

text: Let's say a light clock consists of two parallel mirrors, some photons bouncing between the mirrors, and a spring that pulls the mirrors together with the same force that the photons push them apart. Now we start accelerating the mirrors and the spring, in the direction parallel to the mirrors. Whenever a photon falls off from the end of the light clock, we put a similar photon at the front part of the light clock. Now my question is: does the distance between the mirrors change? (Actually I'm interested whether there are any changes in the forces in the spring)


title: How to calculate the charge and the potential across a charged capacitor charging an uncharged capacitor?

text: Initially, there is a single capacitor $A$ attached to a power source that charges it to a certain voltage $V$. Once it is charged, it is instantaneously removed and placed in a circuit with an uncharged capacitor. How do the charges and voltages redistribute over these two capacitors? It seems that these capacitors would end up being in series and thus would have the same charge equal to the capacitance of $A$ multiplied by the voltage of the battery. Beyond that, I would assume that voltage can be calculated by $V=q/C$ for each capacitor? Is this the correct approach?


title: Does heat always rise?

text: Does heat always rise in a gravitational field? I recently read that heat could be traveling down to the deep part of the ocean. Is there some new or old physics that makes heat sink?


title: Casimir force using Pauli-Villars regularization

text: In Zee's Quantum field theory in a nutshell, 2nd edition, p. 74 he claims that: $$ \sum_a c_a \Lambda_a \sum_n \frac{\omega_n}{\omega_n + \Lambda_a} = - \sum_a c_a \Lambda_a \sum_n \frac{\Lambda_a}{\omega_n + \Lambda_a} $$ can be proved using $$ \sum_a c_a \Lambda_a = 0 $$ This doesn't seem to make any sense. Note that $\omega_n$ can be both larger and smaller than $\Lambda_a$. Perhaps I should note that the $\Lambda_a$ are taken to be "large". Also, $\omega_n \propto n$ and the sum over n goes from 1 to $\infty$. Has anyone gone through this derivation successfully?


title: Independent systems and Lagrangians

text: **Definition 1:** The notion of independent systems has a precise meaning in probabilities. It states that the (joint) probability or finding the system ($S_1S_2$) in the configuration ($C_1C_2$) is equal to the probability of finding the system ($S_1$) in the configuration ($C_1$) times the probability of finding the system ($S_2$) in the configuration ($C_2$). **Definition 2:** However, in we consider fields systems, the practical tool is Lagrangians. So I should say that 2 systems are independent if : $$ Lagrangian (S_1S_2) = Lagrangian (S_1) + Lagrangian (S_2)$$ **The question:** Now, what is the relationship between these 2 definitions? They could be only compatible, or they could be equivalent in the field domain. Is there a way to "demonstrate" the latter from the former ?


title: What is the work done by an ideal gas?

text: What is the work done by an ideal gas when final pressure and volume are both different from its initial pressure and volume or when both pressure and volume changes ?


title: Where do high-energy neutrinos come from?

text: Last week the IceCube South Pole Neutrino Observatory published a press release reporting the possible discovery of two neutrinos with energies of over 1 PeV. Would anyone here be willing to help me understand the significance of this announcement? Although it was reported in the press that these neutrinos originated from outside our galaxy, the original press release seemed to say that this might possibly be the first sighting of 'astrophysical' neutrinos. Are there other observations of extragalactic neutrinos? Are there any conjectures as to processes which might generate? How would a neutrino even be accelerated if it is unaffected by magnetic fields?


title: Mathematical explanation of quantum teleportation

text: I am now studying quantum teleportation. I get what the process is like but I'm wondering why it happens this way. You've got two entangled particles A and B whose wavefunctions are entangled. You also have a third particle C, which is the one you want to teleport. You get C entangled with either of the two and final result is that the wavefuction of one of the two A and B (depending on which you entangled C with) becomes the same as the C's wavefunction is.   1. Why does it occur this way (mathematically explained, if possible)? Why does the wavefunction of A or B becomes that of C?   2. What is the equation explaining this process (addition of wavefunctions, maybe)?


title: Electric Field in Dieletric

text: I have a simple doubt. The textbook I'm working with says that if $E_0$ is the electric field in some region without diletric, then if we introduce a dieletric of constant $k$ the new field will simply be $E=k^{-1}E_0$. Is it always true? I mean, suppose we have concentric spheres of radii $a$ and $b$ being $a$ the radius of the inner sphere and $b$ the radius of the outer sphere and charge $Q_a$ in the inner and $Q_b$ in the outer. If there's no dieletric between them the field in the space between at a distance $r$ from the center will be given by Gauss' Law as: $$E_0=\frac{1}{4\pi\epsilon_0} \frac{Q_a}{r^2}$$ Now if we insert a dieletric on the space between is it true that the field on the dieletric will _simply_ be: $$E=\frac{1}{4\pi k \epsilon_0}\frac{Q_a}{r^2}$$ Thanks very much in advance.


title: How can I model buckyballs sticking together? (Tiny spherical magnets)

text: Buckyballs are a desktoy that consist of 216 tiny spherical magnets. You can makes all kinds of interesting shapes and structures out of them (do a Google search). I want to model them in a physics simulation, but I'm not sure how to do it. Right now, I'm thinking calculate the sum of the forces between each of the buckyball's poles and each of the other buckyballs' poles, and update its linear momentum. Then calculate the torque that each of those forces causes about the magnet's center, and use that to update angular momentum. The problem I'm having is how to model the magnets touching. It's easy in real life to make a long chain or a hexagon. How should I model the magnets actually running into each other to mimic this behavior?


title: Alcubierre Drive - Clarification on relativistic effects

text: On the Wikipedia article on the Alcubierre drive, it says: > Since the ship is not moving within this bubble, but carried along as the > region itself moves, conventional relativistic effects such as time dilation > do not apply in the way they would in the case of a ship moving at high > velocity through flat spacetime relative to other objects. And... > Also, this method of travel does not actually involve moving faster than > light in a local sense, since a light beam within the bubble would still > always move faster than the ship; it is only "faster than light" in the > sense that, thanks to the contraction of the space in front of it, the ship > could reach its destination faster than a light beam restricted to > travelling outside the warp bubble. I'm confused about the statement _"conventional relativistic effects such as time dilation do not apply"_. Say Bob lives on Earth, and Jill lives on a planet in Andromeda, and we'll say for the sake of argument that they're stationary. If I were to travel from Bob to Jill using an Alcubierre drive such that the journey would take me, say, 1 week from my reference frame... how long would Jill have to wait from her reference frame? Do the time dilation effects cancel out altogether? Would she only wait 1 week?


title: Need some help interpreting a formula inspired from Coulomb's law

text: It has been more than a decade since I did all vector related math and physics so pardon me if my question does not make sense. I am reading some article that says it was inspired from Coulomb's law and gives the following expression to calculate the partial force that one particle is exerting on another at time $n$: $f^{i,j}_n = (c - |p^{i}_n - p^{j}_n|)\frac{p^{j}_n - p^{i}_n}{|p^{i}_n - p^{j}_n|}$ where $c$ is the range of the particle over which it's force can spread and $p^{i}_n$ is the location of particle $i$ in the form of $(x,y)$. I checked up the Wikipedia article on Coulomb's law but this formula does not make sense to me. What exactly is the formula trying to achieve? Can someone please help me understand this?


title: Is this the reason solids suspended in turbulent fluids don't settle?

text: While I know (and can observe) that solids don't settle easily in a turbulent flow, I struggle with understanding why exactly this is the case. Here' my problem:   Conceptually, turbulence means high Re-number (or vice ersa), and that means the inertail forces are far stronger than friction/viscous forces. But friction keeps the particle suspended, and I would see bouyancy and gravity as inertial forces.   In a non-moving fluid, the only forces acting on a particle are friction (pointing upwards with a sinking particle) and gravity/buoancy, with a net force pointing up or down and/or movement of the particle.   In a perfect turbulent flow, I would expect no dominant direction of flow. So, the fluid 'tugs' at my particle in one direction one moment and somwhere else the next moment - the net force from the friction over a long enough time will be zero, but bouyancy has a preferred direction. Here's how I would explain why solids don't settle in turbulent flows and I ask you to poke holes into this explanation:   In turbulent conditions, friction is generally proportional to $v^2$, with $v$ beeing the velocity of the particle relative to the fluid. So compared to the same particle in the same fluid but in laminar conditions, the friction resulting from a specific downward (or upward) velocity is higher. This is compunded by the fact that for a particle sinking, any surrounding upward flow will have a stronger effect than a surrounding downward flow with the same speed relative to the particle. Hence, solids settle far slower in turbulent flows.


title: If subatomic particles pop into existence all the time, why don't I gain weight?

text: Watching Discovery's first episode of the first season of Curiosity (entitled "Did God Create the Universe?" by Stephen Hawking), I heard this information: > [...] _you enter a world where conjuring something out of nothing **is** > possible (at least, for a short while). That's because at this scale > particles, such as protons, behave according to the laws of nature we call > "quantum mechanics", and they really can appear at random, stick around for > a while, and then vanish again to reappear somewhere else._ ... and this isn't the only time I've heard this. I imagine countless billions (trillions!) of particles popping into existence all the time in the smallest of spaces for the shortest periods of time. **If subatomic particles pop into existence all the time and in all locations, why doesn't the weight of my body (or anything) change?**


title: Are quantum fluctuations completely uncaused events?

text: I read here that particles/antiparticles appear and annihilate each other spontaneously in empty space. Since particles appear and disappear in empty space, it would seem that empty space has some form of energy. Otherwise these particles' fluctuations appear to be completely uncaused events, which defies reason. Does space have some form of energy, or is this phenomenon completely uncaused?


title: Regular initial data

text: I have a very basic question. What exactly is meant by "regular" initial data in general relativity? Does it mean smooth? at least $C^{2}$? All literature on the subject just uses this term without exact explanation, so it seems that it's something very trivial, but I just wanted to know for sure.


title: A simple thought experiment about traversable wormholes

text: Let's say I have a tube, of large radius (about 5 - 7 meters in diameter), with traversable wormholes at the ends. The wormholes are arranged as such that if something falls inside one hole from inside of the tube, it will come out at the other end still inside the tube. Now, let's say I empty all air from the tube (to make a "vacuum tube," if you will), set it upright and somehow manage to get a rock (or whatever other object) in there. My question now is, does the situation described above rule out the existence of traversable wormholes? Or, if not, since the rock is falling through the wormhole over and over again, will it always be accelerating at the same rate? Or will its velocity only always be approaching light speed? In either case, would the rock's mass increase to the point that it overpowers Earth's gravity, or even collapses into a singularity? Or is there something that would prevent that from happening?


title: Concentration of Proteins

text: > You release a billion protein molecules at position $x = 0$ in the middle of > a narrow capillary test tube. The molecules’ diffusion constant is $10^{−6} > \ cm^2 s^{−1}$. An electric field pulls the molecules to the right (larger > x) with a drift velocity of $1\ μms^{−1}$. Nevertheless, after $80 \ s$ you > see that a few protein molecules are actually to the left of where you > released them. How could this happen? What is the ending concentration > exactly at $x = 0$? > > [Note: This is a one-dimensional problem, so you should express your answer > in terms of the concentration integrated over the cross-sectional area of > the tube, a quantity with dimensions $L ^{−1}.$] In this problem I tried to use diffusion equation $$ \frac{\partial C}{\partial t} = D \frac{\partial ^2 C}{\partial x^2} + v\frac{\partial C}{\partial x} $$ where $v$ is drift velocity. Firstly, does this equation help me, or am I in the wrong woods? Secondly, I have trouble finding the initial conditions, since we have two parameters.


title: What are some critiques of Jaynes' approach to statistical mechanics?

text: Suggested here: What are the justifying foundations of statistical mechanics without appealing to the ergodic hypothesis? I was wondering about _good_ critiques of Jaynes' approach to statistical mechanics. @Yvan did a good job in pointing out a couple of them, but I would like to have them fleshed out a bit, especially by someone who is not biased _towards_ Jaynes. As I think Jaynes' thoughts on this matter shifted a bit through the years, let me summarise what I think his position is: > Logically, one must start with two things: > >   1. There is a microscopic theory of the phenomenon under consideration --- > for the moment that can be the existence of a (quantum or classical) > hamiltonian formulation, which then ensures the existence of a preferred > Louiville form. Thus it makes sense to discuss probabilities of trajectories > (ensemble), independently of where those probabilities come from. > >   2. There exists a viable macroscopic, coarse-grained description, which is > only the case if an experiment says so --- the key is what Jaynes sometimes > calls "reproducibility". If a phenomenon is not readily reproduced then > clearly one has not gained sufficient control over enough variables --- e.g. > it was an experimental fact that controlling temperature and volume of a gas > was sufficient to determine its pressure. > > > > Then it is logically true that one could develop a quantitative > theory/relationship of the macroscopic degrees of freedom or observables, > and the claim is that one should set up an ensemble over the microscopic > trajectories subject to the constraints of the macroscopic observations and > a unique one is chosen by maximising entropy. > > With the ensemble in hand, one could then proceed to make predictions about > other observables, failure to then observe them means either your > microscopic theory is wrong, or the set of variables chosen is not correct, > and the circle of Science is complete. Importantly this makes no reference to ergodicity, and in fact this works out of equilibrium --- equilibrated systems just tends to help with experimental reproducibility. Personally I see it as morally dimensional analysis, writ large. Yvan pointed out that there is a problem with classical hamiltonian systems because they have an (uncountable) underlying configuration space, and there are technical problems with defining entropy on them. My opinion is that this does not matter, because all physical hamiltonians are really quantum, and those come with a canonical choice. (Field theories need regularisation, both UV and IR as always, to render the number of degrees of freedom finite to be "physical".) However, I'm probably just being naive and I would certainly welcome education on this. My preferred reference (never published, I think): Where do we stand on maximum entropy?


title: Quantum statistics of branes

text: Quantum statistics of particles (bosons, fermions, anyons) arises due to the possible topologies of curves in D-dimensional spacetime winding around each other What happens if we replace particles by branes? It seems like their quantum statistics should be described by something like a generalization of TQFT in which the "spacetime" (worldbrane) is equipped with an embedding into an "ambient" manifold (actual spacetime). The inclusion of non-trivial topology for the "ambient" manifold introduces additional effects, to 1st approximation describable by inclusion of k-form fluxes coupling to the brane. To 2nd approximation, however, there is probably non-trivial coupling between these fluxes and the "generalized quantum statistics" A simple example of non-trivial "brane quantum statistics" is the multiplication of quantum amplitudes of strings by the exponential of the euler charactestic times a constant. In string theory this corresponds to changing the string coupling constant / dilaton background. > Were such generalized TQFTs studied? Which non-trivial examples are there > for branes in string theory?


title: Uncertainty in Acceleration given the uncertainty in position

text: I had a motion detector record the position of a dynamics cart and automatically plot Position vs Time and Velocity vs Time plots in Logger Pro on the computer. If the instrument uncertainty in position $x$ was given by $±0.05$ (or more generally any value $±e$) what will be the propagated instrument uncertainty on the second derivative of position, $\ddot{x}$ -- i.e. acceleration? Thanks in advance


title: Nuclear Magnetic Resonance (NMR) Conceptual Questions

text: Let $M$ be the magnetic moment of a system. Below are the Bloch equations, including the relaxation terms. $$\frac{\partial M_x}{\partial t}=({\bf M} \times \gamma {\bf H_0})_x-\frac{M_x}{T_2} $$ $$ \frac{\partial M_y}{\partial t}=({\bf M} \times \gamma {\bf H_0})_y-\frac{M_y}{T_2} $$ $$\frac{\partial M_z}{\partial t}=({\bf M} \times \gamma {\bf H_0})_z+\frac{(M_{\infty}-M_z)}{T_1} $$ At $t=0$, $ {\bf M}=(0,0,M_{\infty})$. Also, ${\bf H_0}=H_0 {\bf k'}$ where primed coordinates are in the lab frame. Now suppose an on resonance pulse is applied along the i direction of the rotating frame for $ T_{\frac{\pi}{2}} =0.005$ milliseconds, then it is turned off to watch the free induction decay. $T_2=5$ milliseconds, $T_1=5000$ milliseconds. So, naturally we will have nutation due to the pulse, $T_2$ decay of the transverse magnetization, and $T_1$ recovery of the longitudinal magnetization. Due to the timescales, they will proceed sequentially. I'm trying to sketch the time evolution of the above three components of the magnetic moment in both the rotating frame and lab frame, and understand exactly how these processes are related.


title: Can you lift a basket up while standing inside it?

text: Suppose you stand in a basket and you try to pull the handles of the basket. Will you be able to lift the basket from ground? Similarly, Baron Münchhausen allegedly pulled himself and the horse on which he was sitting out of a swamp by his own hair. Is this possible?


title: How do I learn higher level physics?

text: I'm a chemical engineering student (just completed BS and am started the PhD program), but I'm very interested in particle physics as a hobby. I'm dismayed though with the sheer amount of information and math that is required to understand this stuff. I feel like a physicist could learn all of my engineering knowledge in a couple of weeks, but it would take years to learn concepts such as:   * Gauge symmetry   * Lagrangians, Hamiltonians   * SUSY   * Noether's theorem   * Tensors   * Feynman diagrams   * General relativity   * Standard model   * Holographic principle I don't even know what most of these things are -- I just see them in answers to questions, look them up on Wikipedia, glaze over at all the math I've never heard of, and continue on just as confused as before. And it's not that I'm bad at math either -- I scored quite well on the AIME in high school. It's just that I've never learned these things (quaternions, spinors? No idea). So when do most physics students learn concepts like this? Undergrad, grad school?


title: Object submerged in a fluid

text: 1)A sphere of a given mass is put in a non viscous liquid. The sphere is released, and it moves down in the liquid. My doubt is, is the mechanical energy of the system decreasing? The answer given in the book is no, but I dont have access to the reason. I could not think why, because, according to me, when an object is freely falling, only then is the mechanical energy conserved, but here it is experiencing an upward buoyant force, so I think this should mean that mechanical energy is not same. Is it increasing? If so, why? 2)a) The situation is the same as above, now the question is, is the gravitational potential energy of the liquid continuously increasing? b) the liquid is now viscous, and the sphere is falling at terminal velocity. Now, is the gravitational potential energy of the liquid continuously increasing?? To for both 2a and 2b, the answer is yes. But again, I dont see how this is happening, as the level of liquid has already risen. now if the sphere moves down further, why should it have an impact on the gravitational potential energy of the liquid? Please help!


title: Dimensional regularization: removing more than just logarithmic divergencies?

text: I have followed two courses on QFT, which both involved renormalization by dimensional regularization. My confusion is that one of the professors claimed that dimensional regularization can only be used to regularize logarithmically divergent integrals, whereas the other professor claimed that the scheme can renormalize higher order divergencies. Let me make my question more precise. Consider the standard integral which is computed in a dimensional regularization scheme: $$ I(n,\alpha) = \int \frac{d^np}{(p^2 + m^2)^{\alpha}} = i\pi^{n/2}\frac{\Gamma(\alpha-n/2)}{\Gamma(\alpha)}(m^2)^{n/2 - \alpha}. $$ For $\alpha=2$ and $n=4\pm\epsilon$ (depends on convention) the integral is logarithmically divergent and we can continue standardly by expanding the gamma, obtaining a $1/\epsilon$ pole and adding a counterterm to the Lagrangian. Generally, the integral is only convergent for $\alpha > n/2$. In the case of $\alpha = 1$ for example my first professor claimed that one has to introduce a Pauli-Villards cut-off in order to renormalize correctly. My second professor claims that one may use: $$ \int d^n q \frac{\partial}{\partial q^{\mu}}(q^{\mu}f(q)) = \int d^n q q^{\mu}\left(\frac{\partial}{\partial q^{\mu}} f(q)\right) + n\int d^n q f(q), $$ such that one throws away the boundary term as it is a surface integral by Gauss' theorem. He states explicitly that this may be done if $f(q)$ vanishes quickly enough at infinity, but also that "analytic continuation will be implemented by ignoring the surface term irrespective of the asymptotic behaviour of the integral." If this is true/legit, one may write: $$ \int d^n q q^{\mu}\left(\frac{\partial}{\partial q^{\mu}} f(q)\right) = - n\int d^n q f(q), $$ such that one can express divergent integrals in terms of finite ones. Is the last approach correct? Is the first approach wrong? Could you comment on these two approaches?


I was reading a thread on Reddit about "what is a skill someone can learn in 6 months that will impress employers on a resume". I pose you the same question, but physicist wannabe oriented. Which are those real world useful skill for physicists that just aren´t taught on college?


title: Why (in relatively non-technical terms) are Calabi-Yau manifolds favored for compactified dimensions in string theory?

text: I was hoping for an answer in general terms avoiding things like holonomy, Chern classes, Kahler manifolds, fibre bundles and terms of similar ilk. Simply, what are the compelling reasons for restricting the landscape to admittedly bizarre Calabi-Yau manifolds? I have Yau's semi-popular book but haven't read it yet, nor, obviously, String Theory Demystified :)


title: Concept of a point particle in quantum mechanics

text: A point particle is usually thought of as structureless and without dimension. However, given that Heisenberg's uncertainty principle prohibits us from knowing the position of a particle exactly, what is the significance of the concept of a _point particle_? Are point particles equivalent to elementary particles?


title: Fractional statistics

text: A common way to show that anyons exhibit fractional statistics in 2D is by arguing that the paths of two anyons winding round each other cannot be continuously deformed to zero. This seems to assume that the particles cannot pass through each other. Why is this assumption valid?


title: is it possible to gain from charging cold battery

text: Is it possible to get higher load of energy into a battery if you charge it when its cold eg. +9C? the battery is emptier, but does it let the power in? Is it safe?


title: Fermi surface questions

text: In order to define the fermi surface we must need to know about the momentum space. But I found a little bit about momentum space. Can you elaborate it pleaese? what is the meaning of the line > the k-vector has dimensions of reciprocal length, so k is the frequency > analogue of r, just as angular frequency ω is the inverse quantity and > frequency analogue of time t.


What type of PDE are Navier-Stokes equations, and Schrödinger equation? I mean, are they parabolic, hyperbolic, elliptic PDEs?


title: Magnetism-Related Terminology

text: A few questions about a magnet and a paperclip:   1. What do you call a material that attracts another material via magnetism? (i.e. the magnet)   2. What do you call the material that is attracted in #1? (i.e. the paperclip)   3. What properties make #1 (the magnet) and #2 (the paperclip) get along so well? Electron configurations? These are pretty elementary...please be gentle. I think I'm looking for basic answers that might be found in a level 200 course.


title: Combining metric tensors/curvature tensors

text: I was thinking about the following scenario: Consider a particle which causes a metric $g_{\mu\nu}$ on an otherwise Minkowski spacetime (or any manifold). Now, consider another particle, somewhere in the vicinity of the first particle, which causes a metric $h_{\mu\nu}$ on a spacetime which would have been Minkowski if not for these two particles. Then, what what would the metric in the vicinity of these two points be? I am guessing that it is: $$(g_{\mu\nu}-\eta_{\mu\nu})+(h_{\mu\nu}-\eta_{\mu\nu}) + \eta_{\mu\nu} = g_{\mu\nu}+h_{\mu\nu} - \eta_{\mu\nu}$$ Also, does the Riemann Curvature Tensor $R_{\mu\nu\rho}^\sigma$add up directly? I don't think it should because the Einstein tensor $G_{\mu\nu}$ does (I think) and it is dependent on the Ricci Curvature AND the spacetime metric tensor. Thanks in advance.


title: Question about charge conservation at vertices of web diagram

text: My question is about web diagram in paper of Aganagic et al. How to get the $\alpha-\beta$ line rather than $-\alpha+\beta$ line of rightmost diagram of fig.3? More precisely, how to show the charge conservation at each vertex? ![enter image description here](http://i.stack.imgur.com/BjrVw.png) Something similar happens in the earlier paper Fig. 13.


title: Change of variables in an interval expression

text: This question is a continuation of How to calculate a scalar curvature fast? . Let's have Lorentz-Fock spacetime with an interval $$ d \hat {s}^{2} = \frac{t_{0}^{2}R^{2}}{\hat {t}^{4}}\left( d \hat {t}^{2} - \frac{\hat {t}^{2}}{R^{2}}d\hat {l}^{2}\right), \qquad (1) $$ and $$ d\hat {l}^{2} = \frac{d \hat {r}^{2}}{1 + \frac{\hat {r}^{2}}{R^{2}}} + \hat {r}^{2}\left( d\theta^{2} + \sin^{2}(\theta )d\varphi^{2}\right) , \quad t_{0} = \frac{R}{c} . $$ Recently, I realized that expression $(1)$ is looks like interval for Frieedman-Robertson-Walker model. Let's have $$ r = R \sinh(\psi ). $$ So $(1)$ can be transformed as $$ d \hat {s}^{2} = \frac{t_{0}^{2}R^{2}}{\hat {t}^{4}}\left( d \hat {t}^{2} - \hat {t}^{2}\left[d \psi^{2} + \sinh^{2}(\psi)\left( d\theta^{2} + \sin^{2}(\theta )d\varphi^{2}\right) \right]\right) \qquad (2). $$ This expression is almost corresponding Frieedman-Robertson-Walker metric. To fully comply I need to **change variables** : $$ \tau = \frac{t_{0}^{2}}{\hat {t}}, d \tau = -\frac{t_{0}^{2}d \hat {t}}{\hat t^{2}}, $$ and $(2)$ can be rewrite as $$ d\hat {s}^{2} = c^{2}d \tau^{2} - c^{2}\tau^{2}\left[d \psi^{2} + \sinh^{2}(\psi)\left( d\theta^{2} + \sin^{2}(\theta )d\varphi^{2}\right) \right], $$ which is equal to FRW metric. So, my questions are following:   1. Is it possible to make the change of variables specified?   2. What can I do with singularity of $\tau $ for $\hat {t} -> 0$?


title: Is dark matter really present around the sun?

text: Recently I read an article that there is dark matter around the sun but if it is so, than why can we see it clearly. If it is called matter than it shall show some hindrance in radiation we receive and the solar flares must bounce back against them though they are coming straight to us. To stay close to sun it has to be rotating at a very very high speed, so does it? Does it pose some kind of gravitational effect on the rest of the system? Is it constant or does it increase or decrease in size?


title: Finite, square, potential well

text: Lets say we have a finite square well symetric around $y$ axis (picture below). ![enter image description here](http://i.stack.imgur.com/3PGLK.png) I know how and why general solutions to the second order ODE (stationary Schrödinger equation) are as follows for the regions I, II and III. \begin{align} \text{I:}& & \psi_{\text{I}}&= Ae^{\kappa x} \\\ \text{III:}& & \psi_{\text{III}}&= Be^{-\kappa x} \\\ \text{II:}& & \psi_{\text{II}}&= C \cos(k x) + D\sin(kx) \end{align} But now i got to a point where i have to start applying a boundary conditions to get a speciffic solution. So i start with the 1st boundary condition which is $\psi_{\text{I}}\left(-\frac{d}{2}\right)=\psi_{\text{II}}\left(-\frac{d}{2}\right)$ for the left potential shift and $\psi_{\text{II}}\left(\frac{d}{2}\right)=\psi_{\text{III}}\left(\frac{d}{2}\right)$ for the right potential shift. These leave me with a system of 2 equations (one for left and one for right potential shift): \begin{align} {\scriptsize\text{left potential shift:}}& & Ae^{-\kappa \frac{d}{2}} &= C \cos\left(k\tfrac{d}{2}\right) - D\sin\left(k \tfrac{d}{2}\right)\\\ {\scriptsize \text{right potential shift:}}& & Be^{-\kappa \frac{d}{2}} &= C \cos\left(k\tfrac{d}{2}\right) + D\sin\left(k \tfrac{d}{2}\right)\\\ \end{align} * * * **Question 1:** From here on authors of most books don't seem to explain much. Most of them only say that we must use $\boxed{C\\!=\\!0}$ to solve for _odd solutions_ and $\boxed{D\\!=\\!0}$ to solve for _even solutions_. What is this argument based on?


title: Reducing deposition on power supply wires

text: From my house there are two wires, i.e. a live and a ground wire which are placed on the power supply lines by a hook. They are not tightly jointed because of some reasons. These two wires are also in contact with air. So, after a few days some material gets deposited on the wire. As a result of which, when wind blows, wire starts shaking and power supply starts fluctuating too much throughout the day. So my questions are:   * What is the material that gets deposited on the wire?   * How to reduce this deposition so as to reduce the power fluctuation?


title: Distinctive properties of liquids

text: The molecules are closely arranged in solids, loosely arranged in liquids and are free to move in gases... But, Why are liquids **(especially water)** exhibiting these distinctive properties such as _incompressibility_ , surface tension and wetting phenomena?


title: Force analysis of silver atom in Stern–Gerlach experiment

text: In this experiment we only consider the force at z direction, but $\vec B$ field gradient doesn't exclusively exist at z direction according to Maxwell's equations. So why don't we see the splitting in other directions?


title: How to understand worldsheet fermion as a section?

text: I am reading Witten's paper on topological string, and I found some mathematical notation is hard to understand for me. Consider the nonlinear sigma model in 2 dimensions governed by maps $\Phi : \Sigma \rightarrow X$ with $\Sigma$ being a Riemann surface and $X$ a Riemann manifold of metric $g$. $z, \bar{z}$ are local coordinate on $\Sigma$ and $\phi^I$ is coordinate on $X$. $K$ and $\bar{K}$ are canonical and anticanonical line bundles of $\Sigma$ (the bundle of one forms of types (1,0) and (0,1) repectively), and let $K^{1/2}$ and $\bar{K}^{1/2}$.The fermi fields of the model are $\psi_{+}^I$, a section of $K^{1/2}\otimes\Phi^*(TX)$. I can not understand the sections of $K^{1/2}$, $\Phi^*(TX)$ and $K^{1/2}\otimes\Phi^*(TX)$. From my point of view, the element of $K$ should be of the following form $\alpha_z dz\in K$, and what is the element of $K^{1/2}$? the pull back of tangent space should be of form $\Phi^*(\beta^i \frac{\partial}{\partial \phi^i})=\beta^i \frac{1}{\frac{\partial \phi^i}{\partial z} }\frac{\partial}{\partial z}$. But in some notes the author seems give that the (0,1) form $\psi_-^i$ with values in $\Phi^*(T^{1,0} X)$ can be written as $\psi_{\bar{z}}^i$ satisfying $\psi \supset \psi_{\bar{z}}^id\bar{z}\otimes \frac{\partial}{\partial \phi^i}$. This contradicts with my naive point of view. where did I make mistakes? How to understand the sections of $K^{1/2}$, $\Phi^*(TX)$ and $K^{1/2}\otimes\Phi^*(TX)$? Thanks in advance.


title: Proof that Our Planet is 1D

text: I have took a discrete mathematics course this summer and there we talked about power of groups and functions,and yesterday I though and realize that if we can map all the 3D coordinates with a one on one and surjective function from $(x,y,z)$ to $(a)$, we can prove that our planet is 1D, that's mind blowing. Is that proof is right?


title: Forces on two boxes

text: This is a problem from my problem sheet but we haven't yet covered the material or received our books. There is a small box of mass $m_{1}$ resting on a plank, of mass $m_{2}$ and length $L$, which itself rests on a frictionless, horizontal surface. Both box and board are stationary when a constant force $F$ is applied to the board. Take $g$ to be the acceleration due to gravity and $F_{\text{f}}$ to be the magnitude of the frictional force between the board and the box. Assuming the coefficient of static friction between the box and the board is unknown, what is the magnitude, and the direction, of the acceleration of the box in terms of $F_{\text{f}}$ (relative to the surface)? Instinct and experiment tells me that $F_{\text{f}}$ must act in the direction of $F$ or the box would accelerate backwards off the box, and that the magnitude of the acceleration must be $$F_{\text{f}} + F = m_{1}a_{1}$$ taking $a_{1}$ to be the acceleration of the box. So $$a_{1} = \frac{F+F_{\text{f}}}{m_{1}}$$ However, later in a later part of the question, it is implied that $F_{\text{f}}$ can be expressed independently of $F$. Is my reasoning correct, and if so how can I express it a little more convincingly. If not then why not? Sorry for asking such a basic question but no one on my course has yet answered it. * * * Edit: Here are all the problems and my new answers to them so you can see what work I've done. A picture of the problem can be found on page 26, in question 13 here, though the problem is different. **(a) Assume that the coefficient of static friction between the board and the box is not known at this point. What is the magnitude, and the direction, of the acceleration of the box in terms of $F_{\text{f}}$?** If $a_{\text{box}}$ is the acceleration on the box, then $F_{\text{f}}=m_{1}a_{\text{box}}$ so $a_{\text{box}}=\frac{F_{\text{f}}}{m_{1}}$ This must act in the direction of $F$ or the box would slide or accelerate off the negative side of the board (taking the direction of $F$ to be positive) **(b) Now take the coefficient of static friction to be $\mu_{S}$. What is the largest possible magnitude of acceleration of the box? Express your answer in terms of some or all of the variables $F, \mu_{S}, m_{1}, m_{2}$ and $ g$.** So $ m_{1} a_{\text{box}} = F_{\text{f}} \leq \mu_{S} F_{n}$ where $ F_{n} $ is the normal force on the box. $ F_{n} = m_{1}g $ so $ m_{1} a_{\text{box}} \leq \mu_{S} m_{1}g$ $a_{\text{box}} \leq \mu_{S}g$ and therefore $\max{a_{\text{box}}} = \mu_{S}g$ **(c) Write down the sum of all horizontal forces on the board, taking the positive direction to be towards the right. Give your answer in terms of $F$ and $F_{f}$.** The total force on the board, $F_{\text{board}}$, is equal to the accelerating force, $F$, plus the friction from the box acting on the board, $-F_{f}$ i.e. $F_{\text{board}}=F-F_{f}$ **(d) Find an expression for the acceleration of the board when the force of static friction reaches its maximum possible value in terms of $F, \mu_{S}, m_{1}, m_{2}$ and $ g$.** $\max{F_{\text{f}}}=\mu_{S}m_{1}g$ so the total force on the board when static friction is at its maximum is $F_{\text{board}}=F-\mu_{S}m_{1}g$. Because the mass of the box $m_{1}$ acts through the board, $F_{\text{board}}=(m_{1}+m_{2})a_{\text{board}}$, where $a_{\text{board}}$ is the acceleration of the board. This means $(m_{1}+m_{2})a_{\text{board}} = F-\mu_{S}m_{1}g$ so $a_{\text{board}} = \frac{F-\mu_{S}m_{1}g}{m_{1}+m_{2}}$ **(e) What is the minimum value of the constant force, $F$, applied to the board, needed to ensure that the accelerations satisfy $|a_{\text{board}}| >|a_{\text{box}}|$? Express your answer in terms of some or all of the variables $\mu_{S}, m_{1}, m_{2}, g$ and $L$. Do _not_ include $F_{\text{f}}$ in your answer.** Assuming that the force of static friction is at its maximum when slippage occurs, $\left|\frac{F-\mu_{S}m_{1}g}{m_{1}+m_{2}}\right|> |\mu_{S}g|$ If $F<\mu_{S}m_{1}g$ then $\mu_{S}m_{1}g - F > \mu_{S}m_{1}g + \mu_{S}m_{2}g $ then $-F > \mu_{S}m_{2}g$ and $ F < -\mu_{S}m_{2}g$. This implies F is negative when it is actually positive so $F \geq \mu_{S}m_{1}g$ and then $F-\mu_{S}m_{1}g > \mu_{S}m_{1}g + \mu_{S}m_{2}g $ and $F>\mu_{S}g(2m_{1}+m_{2})$


title: Direction of the Area Vector (with regards to magnetic dipole)

text: I'm learning about torque on a conductive coil in a magnetic field. I have been taught that $\vec\tau = \vec\mu \times \vec{B}$, where $\vec\mu$ is the magnetic dipole moment. Also, $\mu = I\vec{A}$, where $\vec A$ is the area vector of the loop. To find the direction of the area vector, I am told to use the right hand rule with regards to the current in the loop (curl your fingers in the direction of current, and your thumb points in the direction of the area vector). **My question is:** Why does this give the correct direction for the area vector? Is the area vector just defined to be this way to avoid nasty usage of minus signs, or is there some other reason for this? My guess is that whoever formalized this law/equation (not sure what correct term is for this instance) started with the direction of torque, and worked backwards defining the direction of $\vec\mu$ and $\vec{A}$ to reduce or eliminate stray minus signs in the equations. However, this is, of course, just a guess; I want to know what the true reason is.


title: Third-order phase transition in Landau theory

text: $F=\frac{a}{2}m^2+\frac{u}{4}m^4+\frac{v}{6}m^6-hm$, where F is the free energy, m is the order parameter, h is the external field, $a=a_0(T-T_c)$, and $a_0>0,u>0$ and $v>0$.We know this free energy expansion describes a second order phase transition. How to write down a free energy such that the transition is a third-order phase transition?


title: How is Momenergy frame independent?

text: momenergy = mass * spacetime displacement/proper time for that displacement What I dont understand is how can the momenergy be frame independent? The unit-4 vector always points in the direction of the worldline? Agreed, the magnitude of momenergy will be invariant wrt to any frame. But wont the worldline of a particle be different in different frames? So wont the direction of the spacetime interval be frame dependent?


title: Does electromagnetic radiation make sense in one dimension?

text: I'm trying to do a simple simulation of a 1D charged quantum particle, which gets irradiated by an electromagnetic wave — in context of non-relativistic QM. The Schrödinger equation for such a particle would be $$i\hbar\frac{\partial \Psi(x,t)}{\partial t}=\frac1{2m}\left(-i\hbar\frac{\partial}{\partial x}-qA(x,t)\right)^2\Psi(x,t)+q\left(\phi_0(x)+\phi(x,t)\right)\Psi(x,t),$$ where $A(x,t)$ is (what I believe to be) an $x$ component of vector potential $\vec A$ of radiation, $\phi_0(x)$ is static scalar potential of non- irradiated system, and $\phi(x,t)$ is scalar potential of radiation. What I can't seem to understand is how electromagnetic radiation should be incorporated in this equation. I suppose it shouldn't depend on coordinate, i.e. in non-relativistic context speed of light should be considered infinite, thus wavelength should be infinite. From basic knowledge of electromagnetism I suppose that such a wave would be represented by: $$E=E_0 \cos(\omega t)$$ $$B=B_0 \cos(\omega t)$$ But simulation is to be 1D, while electromagnetic wave has three dimensions: one for direction of propagation (which I guess is irrelevant because of infinite wavelength), second for $\vec E$ and third one for $\vec B$. To add the complexity, I need to determine how vector potential $\vec A$ and scalar potential $\phi$ would look in such case, which I might understand once I realize how to orient the radiation in terms of $\vec B$ and $\vec E$. So, my question is: does it even make sense to talk about external electromagnetic radiation in 1D case? If yes, how should the field be oriented in this case to make a sensible quantum simulation? What would its vector potential $\vec A$ and scalar potential $\phi$ look like?


title: Considerations and questions for a DIY heat pipe; how to choose a working fluid?

text: I plan to build a DIY heat pipe! For more info on what a heat pipe is, see Wikipedia. I would like to use it to cool a desktop graphics card I have. I am having trouble deciding how to choose the working fluid for the pipe. I plan to build it based on this design. Using gravity to return the fluid to the hot end. Working fluid considerations My graphics card consumes 250 watts at 100% load and 15-20 watts at idle, the maximum temperature is 90º Celsius and the temperature at idle is 30º Celsius.   * How does this translate to a choice of working fluid? I would assume the fluid has to have a phase transition temperature just lower than that of the lowest operating temperature of the card, which can be changed by based on the PV=nRT law? So just whatever substance will boil between 30–90ºC at the given pressure in the pipe?   * How can I decide how much fluid to use? As long as the pipe is filled with only the vapor of the working fluid at the proper pressure, then would the only amount of fluid you need be just enough to cover the surface area of the boiling point on the hot end? Would it be a good idea to add a little more?


title: Raising a toolbox with a rope

text: Suppose you use a rope to hoist a box of tools vertically at constant speed v. The rope exerts a constant upward force of magnitude F_up on the box, and gravity exerts a constant downward force (the weight of the box). No other forces act on the box. To raise the box twice as fast, the force of the rope on the box would have to have:- A) the same magnitude $F_{up}$ B) a magnitude of $2F_{up}$ The answer seems to be (A), and I don't understand why! It says, if it's constant velocity, both upward and downward force = 0. Since the downward force is always the same, the upward force must be the same, too, regardless of the box's speed. My take on that: But how are you gonna change the speed of the box without exerting force? If the force is the same, while there is one in the other direction, there would be no change of speed. So, can you explain to me the correct answer? EDIT: "Consider ... (1) starting the motion, (2) maintaining the motion, and (3) slowing the toolbox to a stop ..." (1) Starting. The object needs force greater than weight to go upward. (2) Maintaining. The object needs force equal to the weight to maintain motion upward. (3) Slowing. The object needs force less than weight for it to stop. Ummm.. I don't know what to conclude with that? I understand one important thing: It can't be $F_{up} = W$, or in other words $F_{up} = F_{down}$, because there would be no acceleration in any direction. If there is no acceleration at any direction, I can't get the velocity to be twice as fast, or any amount faster. EDIT: Thank you all. @Manishearth I could imagine this, but the book doesn't mention the "jerk", nothing about it in the question or in the explanation for the answer. (This is an example in the book). That's why I got confused. Anyway, again, thank you all for the clarification.


title: When do the von Neumann projections occur and what causes them?

text: (Transferred as a separate question from comments in Scott Aaronson’s gravitational decoherence question) Reversing gravitational decoherence The modern answer seems to be that they never occur, and that therefore nothing causes them. This leads on (or back) to Everett and MWI. A closely related idea is that there is only one collapse, just as there is only one wave function, but it covers the whole universe, all time and all space. An older answer, at least sometimes called Copenhagen, is that collapses are merely Bayesian updating, and hence do not occur as physical processes. A third answer, associated with Wigner and von Neumann himself, is that the observer causes collapses by the act of observation. GRW suggest that collapses occur gradually, due to a nonlinear modification of the Schrodinger equation. I do not find these answers satisfactory. See related question, “Can you count collapses?” I am interested in answers that do not modify QM, (unlike GRW), but (like GRW) nevertheless regard collapse as a real physical process that can be observed, and counted.   Can someone provide references to papers that discuss collapses that meet this criterion?   Are there other answers to the collapse question not mentioned above?


title: Can you count “collapses”? How many collapses in the observable universe?

text: If that’s too hard, how many collapses in 100 cc’s of boiling water in one second? In biology, the very first robin that is scientifically described is preserved and called the “type robin”. The “type robin” for collapse was described by Einstein in 1905, and won him the Nobel prize. It is called the photoelectric effect. (Later collapse was formalized by von Neumann as a mathematical projection.)   A very similar effect is the building up of an Airy circle in a telescope. Another is the point by point emergence of an interference pattern in a two- slit experiment, perhaps very slowly. In these examples, “collapses” can be counted. When one photon is absorbed and detected, one collapse happens. When a second photon is absorbed, a second collapse happens.   The collapse is caused by the photon hitting the detector. (Or the collapse is caused by the silicon atom absorbing the photon.) Collapses like these are countable and it makes sense to ask and answer how many there are in a given 4-volume of spacetime.   So, how many?


title: supressing certain decay paths and enhancing others with interference

text: In a scattering reaction, there are many possible final states for the products, each with different production rates. > **Question:** Is there a way in which we could in general supress certain > rates of products of the reaction and enhance others? can we use coherence > and interference of the initial states to modulate such rates? As an application of such possibility, i'm wondering if we could enhance the production rate of antimatter (for the purpose of the question, let's consider antihydrogen) by a few orders of magnitude by such manipulations?


title: Could one transmit a signal with equally-tuned casimir plates across the quantum field?

text: It seems, one could exploit the Casimir effect to send messages across arbitrarily-large distances with carefully-tuned Casimir plates. Obviously, relativity would preclude FTL information transfer, _but_ as long as one did not try to _measure_ the signal timing and impose an external reference frame of time, then there would be no violation of the speed of light limit, and would, in _theory_ , be communicated instantaneously (presumably along an extra-dimensional Calabi-Yau surface/"brane").


title: Why is current not 0 in a regular resistor - battery circuit immediately after you closed a circuit?

text: In regular open circuits with either a capacitor or inductor element, (when capacitor is uncharged) with a battery, when a switch is closed to complete the circuit the current is said to be 0 because current doesn't jump immediately. But in a circuit with just resistors, as soon as a switch is closed the current isn't 0? Example is this question from 2008 AP Physics C Exam http://apcentral.collegeboard.com/apc/public/repository/ap08_physics_c_em_frq.pdf http://www.collegeboard.com/prod_downloads/ap/students/physics/ap08_physics_c_e&m_sgs_rev.pdf Go to Question 2 for details.


title: How do Anamorphic images work?

text: I found the following anamorphic image on this site ![enter image description here](http://i.stack.imgur.com/c2f3G.jpg) Can anyone explain the physics behind the working of such images? Can we create our own anamorphic images? If so how? Thanks in advance.


title: What is the origin and nature of the "hydrogen envelope" located on the sun side of a comet coma?

text: It is tempting to assume the origin of the hydrogen is water vapor from the comet, in which case, what process "breaks up" the water molecule (or perhaps ammonia molecule?)? Is the hydrogen actually hydrogen ions? If the hydrogen is hydrogen ions, then could the source actually be solar plasma brought by the solar wind and trapped by the comets magnetic field?


title: Transfered thermal energy to a gas with varying mass

text: In my physics book (and wikipedia) it states that the thermal energy transferred to an object is: $Q = c \ m \ dT$, where $Q$ is the transferred thermal energy, $c$ is the specific heat capcity of the material, $m$ is the object's mass and $dT$ is the change in temperature. So far so good, but what I don't understand is that, as an objects temperature is changing, then surely its mass would change as well, right? So how should one use the formula in that case? Is it as simple as the average mass between the two temperatures? Or should one just use the mass from the first state?


title: From 3D velocity to coordinates

text: I want to calculate the 3D position $T_x, T_y, T_z$ of an object with respect to a coordinate system if I have the mean velocity (norm) $v$ and its 3D rotation $(\omega, \phi, \kappa)$ with respect to the same coordinate system. I want to use a naive motion model, as simple as possible. So, I assume linear constant velocity and given the time I calculate $T=vt$ where $T$ is the 3D position, $v$ is the mean velocity and $t$ is the time. Then I use sphrerical coordinates to calculate the translation components $T_x=Tsin(\phi)cos(\theta)$   $T_y=Tsin(\phi)sin(\theta)$   $T_z=Tcos(\phi)$ Are the values $\phi$ and $\theta$ the same as the ones from $\omega,\phi,\kappa$? Is it a correct approach, although simple? I don't want to use angular velocity etc. Any help appreciated!


title: Is voltage electric potential or electric potential difference?

text: On Wikipedia, voltage is defined to be the electric potential difference. However, I am still not certain as to whether voltage is the electric potential ($PE/q$) or electric potential change ($\Delta PE/q$). My confusion partially comes from talking about circuits and defining the amount of volts a point in a circuit has. So, is voltage electric potential or its change? If voltage is the change of electric potential, then how can one point in a circuit be measured in volts?


title: At what launch angle will a (model) rocket keep flying straight?

text: When a model rocket is launched straight up with an angle of 90degrees to the ground it will do a 180 flip when it reaches the apex of its flight and then dive straight down to Earth. If I were to launch a rocket at 0degrees to the ground it would fly straight and eventually angle down until the nose hits the ground. At what angle does the rocket stop doing the 180degree flip and, instead, continue flying in a parabola until it reaches the ground?


title: Length of day of a gas giant

text: How can the rotational speed, or the length of a day be determined or estimated in a planet which is composed entirely of non homogeneous fluids? There must be internal forces (pressure gradients, Coriolis, and others) that move the gases around, so what do they use as a point of reference in a planet which has no solid or fixed surface?


title: Optical absorption in multilayer structure

text: Assume a hypothetical 3 media/ 1 layer structure with the following indices of refraction: $$n_1 = 2, n_2 = 2+i0.5, n_3 = 1$$ where the thickness of the layer is 100 nm and wavelength = 1000 nm. Using Fresnel equations to derive the total transmittance and reflectance I get that about ~38% of incident power is absorbed in the layer. If we now add another boundary, so that the structure now looks like: $$n_1 = 2, n_2 = 2+i0.5, n_3 = 1, n_4 = 3$$ The absorption in the first layer as a function of the second layer thickness now looks like: ![enter image description here](http://i.stack.imgur.com/MwzQb.png) My question is how is that the absorption now goes below the ~38%? I understand that when deriving the transmission and reflectance I am summing up the electric field amplitude, some of which might cancel due to the reflections in the second layer but intuitively this still doesn't make sense.