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<p>Simple question. A system with a uniform electric field everywhere in space has translational invariance in the directions perpendicular to the electric field but no translational invariance parallel to it. This system also has rotational invariance in the plane perpendicular to the electric field.</p>
<p>What about a uniform magnetic field? Judging from the Hamiltonian $(\vec{p}-q\vec{A})^2/2m$, it would seem that the symmetries depend on our choice of gauge. Choosing a different $\vec{A}$ breaks different symmetries. Is there a most symmetric choice for $\vec{A}$?</p>
| 3,091 |
<p>One of my favourite ever pictures taken from space is a picture of the 2009 eruption of Sarychev Peak (Ostrov Matua island, Japan) taken by an ISS astronaut during a lucky fly-over.</p>
<p><a href="http://earthobservatory.nasa.gov/IOTD/view.php?id=38985"><img src="http://i.stack.imgur.com/O1FfD.jpg" alt="enter image description here"></a></p>
<p><sup>Image Source: <a href="http://earthobservatory.nasa.gov/IOTD/view.php?id=38985">Earth Observatory Image of the Day</a>. I heartily recommend clicking through and seeing the animation.</sup></p>
<p>I always assumed that the hole in the cloud cover around the island was caused by some sort of shockwave originating in the eruption plume, most probably as a result of increased temperatures evaporating the cloud, or some such. However, upon revisiting the IOTD page, I was surprised to learn that this is not necessarily the case and that the origin of this hole is a matter of some controversy:</p>
<blockquote>
<p>**Editor’s note: Following the publication of this photograph, the atmospheric and volcanic features it captured generated debate among meteorologists, geoscientists, and volcanologists who viewed it. Post-publication, scientists have proposed—and disagreed about—three possible explanations for the hole in the cloud deck above the volcano.</p>
<p>One explanation is that the hole in the clouds has nothing to do with the eruption at all. In places where islands are surrounded by oceans with cool surface temperatures, it is common for a sheet of clouds to form and drift with the low-level winds. When the cloud layer encounters an island, the moist air closer to the surface is forced upward. Because the air above the marine layer is dry, the clouds evaporate, leaving a hole in the cloud deck. These openings, or wakes, in the clouds can extend far downwind of the island, sometimes wrapping into swirling eddies called von Karman vortices.</p>
<p>The other two possibilities that scientists have offered appeared in the original caption. One is that the shockwave from the eruption shoved up the overlying atmosphere and disturbed the cloud deck, either making a hole or widening an existing opening. The final possibility is that as the plume rises, air flows down around the sides like water flowing off the back of a surfacing dolphin. As air sinks, it tends to warm; clouds in the air evaporate.</p>
</blockquote>
<p><strong>Has this controversy been settled?</strong> Is there a convincing, accepted explanation for the origin of the hole in the cloud cover?</p>
| 3,092 |
<p>What are some good examples? For example, chaotic systems where you can show that quantum nondeterminism sets the initial conditions. </p>
| 3,093 |
<p>Let us assume one dimensional heat transfer, for example a finite length wire starting at point $0$ and ending at point $\ell$. If the current passes the wire, the Joule heat $I^{2}R$ will be generated and dissipated into the wire and its thermal surroundings. Had the wire had a constant temperature $T$, the half of the power $I^{2}R / 2$ will be passing the left end, the other half will be passing the right end.</p>
<p>Will the situation change if the non-zero temperature gradient $\nabla T $ is present before the Joule heating starts? I cannot grasp, which principle has "higher priority" in this case - be it either principle of dissipation of heat which should be considered "a random walk" or the second thermodynamic principle which states that on average more heat will flow from colder to hotter parts.</p>
<p>Motivation for this question are heat transfer equations in thermoelectricity. Thank you in advance for any answer of insightful comments!</p>
| 3,094 |
<p>Suppose there is a space with constant magnetic field, and a charged particle is moving in that space with a constant velocity, ofcourse it experiences a magnetic force and gets deflected. <br>
But the particle it not necessarily moving wrt all frames. There may be some frame for which the particle doesn't move at all but still gets deflected. How is this possible, there must be some velocity for the particle to experience any deflection under influence of a magnetic field?<br>
Also a still particle in the same space doesn't experience any deflection. True, but the same particle may be moving wrt to some other frame without experiencing a magnetic force, so here we have a velocity but no magnetic force case.</p>
<p>I remember I have used the relation $|F| = qv|B|$ a lot before, but I am afraid to use it any more, these things are really confusing.</p>
| 70 |
<p>In particular, I am asking if two distinct many-body systems (e.g. system A and system B) separated at some arbitrary distance will necessarily be found to contain entangled particles (such that particle A1 is entangled with particle B1...etc) <strong><em>simply as a consequence of them both being forced into a ground state</em></strong>? Specifically, can two experimenters go off to separate labs with no preparations made and then force their systems to the ground state and expect to <strong><em>later</em></strong> find through conventional testing of entanglement and sharing of statistical results that in fact they did share entangled particles? Or will any entanglement only be coincidental? The reason I ask is because I'm trying to find a way in which entanglement can be acheived by two separated labs remotely (without having to meet up..though classical comms would be allowed) since entangled states are so fragile. It is impractical, it seems, to meet up to prepare entanglement in one lab and then try to keep it from falling apart as you migrate to separate labs to take advantage of its utility or to test its sensitivity to distance (I'm assuming there is none, of course). Has this been achieved and if so, how is it achieved? Also, I have read a paper that documents that two distinct <strong><em>subsystems</em></strong> of a many-body system are entangled when the many-body system as a whole is in the ground state. What I am asking here is that if that can be possible because the particles making up the many-body system do not have identity, why can't it be true of the particles comprising two separate many-body systems? Why are they not equally indistinguishable, if in fact they are not? Thanks</p>
| 3,095 |
<p>The mass of a neutron is greater than mass of a proton so how is it possible in positron emission for a proton to form a neutron and a positron?</p>
| 737 |
<p>We know that current is passed through a circuit if there is a potential difference across the two terminals of the conductor. But, in the case of a short circuit, we say that there is no potential difference between the two terminals and a large amount of current is passed through it. This is a violation of Ohm's law. Isn't it wrong to say that there is no potential difference between the terminals?</p>
| 71 |
<p>For small angles $\theta$ the rotation along a particular axis $n$ is given by</p>
<p>$R(n,\theta)(r)=Id+ \theta (n \times r)+ o(\epsilon)$.
Now, the rotation operator in Quantum Mechanics is given by
$R(n,\theta)(r)=r-\frac{i}{\hbar} \theta \langle n , L \rangle r + o(\epsilon)$</p>
<p>But if I check this for $n=e_z$ we have:
$R(n,\theta)(r)=r+ \theta (e_z \times r)+ o(\epsilon)=r+ \theta (x e_y - y e_x)+ o(\epsilon)$
and
$R(n,\theta)(r)=r-\frac{i}{\hbar} \theta l_z r + o(\epsilon)=r-\theta (x \partial_y - y \partial_x)r + o(\epsilon)=r-\theta (x e_y - y e_x) + o(\epsilon)$</p>
<p>so obviously the last two expressions of the last two rows differ in a sign and I do not see why.</p>
| 3,096 |
<p>Having just heard about the asteroid <a href="http://en.wikipedia.org/wiki/2010_TK7" rel="nofollow">2010 TK7</a> in <em><a href="http://www.bbc.co.uk/news/science-environment-14307987" rel="nofollow">Trojan asteroid seen in Earth's orbit by Wise telescope</a></em>, I want to know more about its orbit. The BBC article says it moves above and below the ecliptic. I was under the impression that everything orbits in ellipses around the Sun, so what causes this motion?</p>
<p>Also the Wikipedia article about it states that it shuttles between <a href="http://en.wikipedia.org/wiki/Lagrangian_point#The_Lagrangian_points" rel="nofollow">Lagrangian points</a> 3 and 4, but just above that that it orbits 60 degrees in front of Earth which seems contradictory? And that its path oscillates? What does that mean and what exactly is a Lagrangian point?</p>
<p>You can see animations in <em><a href="http://www.ibtimes.com/articles/189342/20110729/earth-finds-dance-partner-in-a-trojan-asteroid.htm" rel="nofollow">Earth Finds Dance Partner in a Trojan Asteroid</a></em> of <a href="http://en.wikipedia.org/wiki/2010_TK7" rel="nofollow">2010 TK7</a>. </p>
<p>It is circling the Sun because of the Sun's gravity, but why is it spiraling in its orbit? Does that mean there is an object in the center, around which it spirals? Does it disobey <a href="http://en.wikipedia.org/wiki/Newton%27s_laws_of_motion#Newton.27s_first_law" rel="nofollow">Newton's first law</a>?</p>
<blockquote>
<p>A body in motion will continue its motion unless it is opposed by another force.</p>
</blockquote>
<p>Also, is there some 3D model or program that can demonstrate its orbit?</p>
| 3,097 |
<p>Forgive the elementary nature of this question:</p>
<p>Because a new moon occurs when the moon is positioned between the earth and sun, doesn't this also mean that somewhere on the Earth, a solar eclipse (or partial eclipse) is happening?</p>
<p>What, then, is the difference between a solar eclipse and a new moon?</p>
| 3,098 |
<p>I am simulating a waveguide in COMSOL, a FEM solver. My model looks like this (it is similar to a standard Quantum Cascade Laser geometry):</p>
<p><img src="http://i.stack.imgur.com/Ca9Rj.png" alt="geometry"></p>
<p>Therefore there is a very thin (30nm) layer of gold sandwiched by two regions consisting of Gallium Arsenide ($ \epsilon_r \simeq 13$). When I perform a mode analysis around 3THz (the frequency of my laser, $\simeq 100 \mu m$), the mode most likely to propagate (lowest losses, matching effective mode index) is the one below (E field norm is plotted here):</p>
<p><img src="http://i.stack.imgur.com/fprKN.png" alt="mode"></p>
<p>Now, I would expect this behaviour <strong>if</strong> the gold layer in between was thick enough, because the plasma frequency of the gold, according to the Drude model, is well above 3THz. The absorption coefficient is equal to</p>
<p>$$\alpha = \frac{2 \kappa \omega}{c}$$</p>
<p>and the skin depth, defined as $\delta = 2/ \alpha$, is around 40-50nm at this frequency. Therefore, due to the skin effect, the reflectivity of the thin gold layer would be very low and the wave would leak out into the bottom layer.</p>
<p>The E field does penetrate the gold, but decays very rapidly. This is the norm of E in the gold layer (0 to 0.03 on the y-axis), zoomed in a lot and the color range adjusted (notice $|E_{max}|>300$ on the original plot):</p>
<p><img src="http://i.stack.imgur.com/hR5lV.png" alt="gold zoom"></p>
<p>This is what I got from the support:</p>
<blockquote>
<p>The reflection appears to be the result of the normal flux conservation boundary condition $Dy_1=Dy_2$. Since in the gold Er is much larger than in the material, $Ey_1<<Ey_2$. However, the tangential components of the electric field (which should indeed decay after one skin depth) are almost continuous thorough the gold thickness.</p>
</blockquote>
<p>But when I look at the displacement current (y component), I get a clearly discontinuous plot:</p>
<p><img src="http://i.stack.imgur.com/CX4tS.png" alt="Dy"></p>
<p>I got another reply, commenting on this issue:</p>
<blockquote>
<p>This discontinuity is attributed to transition between a conductor and a dielectric. The D1-D2=pho_s condition is derived from the charge conservation law, J1-J2=dt phos_s, where J1 and J2 are the sum of displacement and conduction currents. In case that the conductivity in both domains is zero, you get the electric flux conservation. When passing through a conductor-dielectric boundary, the induction current becomes discontinuous and therefor, the displacement current will too be discontinuous. The displacement current is directly related to the flux density. Thus you see the jump in the normal flux component. If you plot emw.Ex or emw.Jy you will see that they are continuous. The different current components (emw.Jiy, emw.Jdy) and emw.Dy will be discountinuous.</p>
</blockquote>
<p>If I understand correctly, this just means that there will be charge accumulation on the Au/GaAs interface due to the existence of a Schottky junction.</p>
<p><strong>Am I wrong in my assumptions, am I misunderstanding the skin effect, am I plotting the wrong thing? If not, why am I getting the wrong result?</strong> From the technical point of view, the mesh in the software is small enough to resolve the thin layer, so it can't be the problem.</p>
| 3,099 |
<p>It seems that we are <a href="http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation#CMBR_dipole_anisotropy">moving relative to the universe at the speed of ~ 600 km/s</a>.
This is the speed of our galaxy relative to the cosmic microwave background.</p>
<p>Where does this rest frame come from? Is this special in any way (i.e., an absolute frame)? </p>
<p>EDIT: I think the more important question is "where does the CMB rest frame come from?".</p>
| 3,100 |
<p>This question has come about because of my discussion with Steve B in the link below. </p>
<p>Related: <a href="http://physics.stackexchange.com/questions/72227/why-is-glass-much-more-transparent-than-water">Why is glass much more transparent than water?</a></p>
<p>For conductors, I can clearly see how resistivity $\rho\,\,(=1/\sigma)$ can depend on frequency from Ohm’s law, $\mathbf{J}=\sigma\mathbf{E}$. So if the E-field is an electromagnetic wave impinging on a conductor, clearly the resistivity is frequency dependent. In a similar fashion, the frequency dependence of the electric permittivity $\epsilon=\epsilon_0n^2(\omega)$ can be derived through the frequency dependence of the electric polarization and impinging electromagnetic wave (see <a href="http://physics.stackexchange.com/questions/68885/how-does-epsilon-relate-to-the-dampened-harmonic-motion-of-electrons/68925#68925">How Does $\epsilon$ Relate to the Dampened Harmonic Motion of Electrons?</a>). </p>
<ol>
<li><p>What does it mean physically for a dielectric to have a frequency dependent resistivity from (i) classical and (ii) quantum viewpoints? I am especially interested in the optical frequency range.</p></li>
<li><p>Can a simple mathematical relationship be derived similar to the frequency dependent resistivity (for conductors) and electric permittivity (for dielectrics)?</p></li>
</ol>
<p>Thank you in advance for any help on this question</p>
| 3,101 |
<p><strong>The problem statement:</strong> </p>
<blockquote>
<p>Measurement detects a position of a proton with accuracy of $\pm10pm$.
How much is the position uncertainty $1s$ later? Assume the speed of a
proton $v\ll c$.</p>
</blockquote>
<p><strong>What i understand:</strong></p>
<p>I know that in general it holds that: </p>
<p>\begin{align}
\Delta x \Delta p \geq \frac{\hbar}{2}
\end{align}</p>
<p>This means i can calculate momentum uncertainty for the first measurement: </p>
<p>\begin{align}
\Delta x_1\Delta p_1 &\geq \frac{\hbar}{2}\\
\Delta p_1 &\geq \frac{\hbar}{2\Delta x_1}\\
\Delta p_1 &\geq \frac{1.055\times10^{-34}Js}{2\cdot 10\times10^{-12}m \rlap{~\longleftarrow \substack{\text{should I insert 20pm}\\\text{instead of 10pm?}}}}\\
\Delta p_1 &\geq 5.275\times10^{-24} \frac{kg m}{s}\\
\Delta p_1 &\geq 9.845 keV/c
\end{align}</p>
<p><strong>Question:</strong></p>
<p>Using the position uncertainty $\Delta x_1$ I calculated the momentum uncertainty in first measurement $\Delta p_1$.</p>
<p>How do I calculate the position uncertainty $\Delta x_2$ after $1s$? I am not sure what happens $1s$ later but is a momentum uncertainty conserved so that it holds $\Delta p_1 = \Delta p_2$? I know that the problem expects me to use the clasic relation $p=m_ev$ somehow but how? </p>
| 3,102 |
<p>If we placed <code>p-type</code> and <code>n-type</code> semiconductors close enough to be touching (see fig. 1), would this arrangement work as a diode? Please explain.</p>
<p><img src="http://i.stack.imgur.com/n5UzB.jpg" alt="connecting p-type and n-type semiconductors"></p>
<p><sup>Fig. 1 - Connecting p-type and n-type semiconductors</sup></p>
| 3,103 |
<p>I live in a tall building (20 floors) on a mountain. Because the water pressure from the water company is not enough, there is a water pump at the last floor which is activated each time someone is using fresh water in his apartment.</p>
<p>To me, that's a big waste of energy, and I think that in theory it would be possible to reuse the used water which is going down to pump some fresh water up.</p>
<p>I wonder if such device already exist on the market, and I would like to know how such device would be called. Would there be a problem to operate with such a height difference?</p>
| 3,104 |
<p>I remember from introductory Quantum Mechanics, that hydrogen atom is one of those systems that we can solve without too much ( embarrassing ) approximations.</p>
<p>After a number of postulates, QM succeeds at giving right numbers about energy levels, which is very good news.</p>
<p>We got rid of the orbit that electron was supposed to follow in a classical way ( Rutherford-Bohr ), and we got orbitals, that are the probability distribution of finding electron in space.</p>
<p>So this tiny charged particle doesn't emit radiation, notwithstanding its "accelerated motion" ( Larmor ), which is precisely what happens in real world.</p>
<p>I know that certain "classic questions" are pointless in the realm of QM but giving no answers it makes people asking the same questions over and over. </p>
<ul>
<li>If the electron doesn't follow a classic orbit, what kind of alternative "motion" we can imagine?</li>
<li>Is it logical that while the electron is around nucleus it has to move in some way or another?</li>
<li>Is it correct to describe electron motion as being in different places around nucleus at different instants, in a random way?</li>
</ul>
| 3,105 |
<p>Using zeta regularization one can get a formula for regularizing the integral $ \int_{a}^{\infty}x^{m-s}\text dx $ for any $m$. </p>
<p>However, I have not seen anywhere. For example, I do not know why in physics you can use zeta regularization to define infinite products $ \prod _{n=0}^{\infty}a_{n} $, but they do not use zeta regularization for the regularization of divergent integrals like $ \int_{a}^{\infty}x^{m-s}dx $ for any $m$ even though there is a known formula</p>
<p>$$ \begin{array}{l}
\int\nolimits_{a}^{\infty }x^{m-s} \text dx
=&\frac{m-s}{2} \int\nolimits_{a}^{\infty }x^{m-1-s} \text dx +\zeta (s-m)-\sum\limits_{i=1}^{a}i^{m-s} +a^{m-s} \\
&-\sum\limits_{r=1}^{\infty }\frac{B_{2r} \Gamma (m-s+1)}{(2r)!\Gamma (m-2r+2-s)} (m-2r+1-s)\int\nolimits_{a}^{\infty }x^{m-2r-s} \text dx, \end{array} $$</p>
<p>which I think is better than Pauli-Villars or dimensional regularization. Why do people ignore it? Is there a plot against zeta regularization?</p>
| 3,106 |
<p>The <a href="http://en.wikipedia.org/wiki/Husimi_Q_representation" rel="nofollow">Husimi $Q$ function</a> of a quantum state $\rho $ is defined as
$ Q (\alpha)=\langle \alpha \vert \rho \vert \alpha \rangle $, where $\alpha = (x, p) $ is a phase space coordinate and $\vert \alpha \rangle$ is a coherent state.</p>
<p>Is the off-diagonal generalization
$ Q (\alpha, \beta)=\langle \alpha \vert \rho \vert \beta \rangle $
used for anything? Does it have a name?</p>
<p>This is an interesting object because it essentially measures coherence (or decoherence) in the overcomplete basis of wavepackets .</p>
| 3,107 |
<p>Heat is to be transfer from $CO_2$ gas to fluid.
$CO_2$ is being heated by solar energy.</p>
<p>How can we calculate heat transfer from the hot $CO_2$ gas to the moving fluid?</p>
| 3,108 |
<p>Suppose I have a charge moving back and forth above an infinite, grounded, conducting plane. Can I calculate the total radiated power by using image charges? That is, are the scalar and vector potentials the same in the upper-half space for all time for both the image charge "picture" and the standard picture?</p>
| 3,109 |
<p>I'd want to know the basic rules to apply the conservation laws in nuclear reactions (nuclear fusion, nuclear fission, radioactive decays, etc.) to determine parity and angular momentum of the products. I know that these principles lay deep in quantum mechanics, but I don't need a very thorough explanation, just the rules. I have searched the Internet a lot but I didn't find simple and clear information. I know that this question would probably require a quite long answer, so if you'd post just a exhaustive link about this it would be enough.
Thank you</p>
| 3,110 |
<p>In this discussion: <a href="http://chat.stackexchange.com/rooms/4243/discussion-between-arnold-neumaier-and-ron-maimon">http://chat.stackexchange.com/rooms/4243/discussion-between-arnold-neumaier-and-ron-maimon</a> Arnold Neumaier suggested that there might be a close link between classical and quantum integrability, while I think there are many more classically integrable systems than quantum integrable ones.</p>
<p>The reason is that classically integrable systems are easy to make up--- you make up an infinite number of action and angle variables, and change canonical coordinates in some complicated way to x,p pairs, and say this x-p version is your system of interest. But quantum systems don't admit the same canonical transformation structure as classical systems, so there might be systems which have an integrable classical limit, but no real sign of integrability outside of the classical limit.</p>
<p>But I don't know any examples! Most of the 1+1d integrable stuff is for cases where the classical and quantum integrability are linked up, for the obvious reason that people are interested in finding integrable systems, not examples where they are not. The reason I think finding an example is not trivial is because the classical integrability guarantees that the motion is not classically chaotic, and that the asymptotic quantum energy states are pretty regular. So I don't think one can look for a counterexample in finite dimensions, where all high enough energy states are permanently semi-classical.</p>
<p>But consider a field theory on a lattice in 2+1 dimensions (continuous time). The lattice is so that the dynamics can be arbitrary, no continuum limit, no renormalization. Even if you have an integrable classical dynamics for the field theory, the energy can still dissipate over larger volumes (this isn't 1+1 d), and eventually the classical field will be weak enough that the classical limit is no longer valid, and you see the quanta. This allows the possibility that every finite energy state to eventually leave the semi-classical domain, and turns quantum, and then the integrability is lost.</p>
<p>So is there a 2+1 (or 3+1) dimensional lattice scalar field theory where the classical dynamics is integrable, but the quantum mechanical system is not?</p>
<p>By saying that the quantum system is not integrable, I mean:</p>
<ul>
<li>the many particle S-matrix doesn't factorize or simplify in any significant way (aside from the weak asymptotic relations implied by having a classical integrable limit)</li>
<li>there are only a finite number of quantum conserved currents (but an infinite number of conserved currents in the classical limit).</li>
</ul>
| 3,111 |
<p>A girl is riding a bicycle along a straight road at constant speed, and passes a friend standing at a bus stop (event #$1$). At a time of $60$ s later the friend catches a bus (event #$2$)
If the distance separating the events is 126 m in the frame of the girl on the bicycle, what is the bicycle's speed?</p>
<p>$$u = u' + v$$ </p>
<p>can be written as: </p>
<p>$$Δx/t = Δx'/t + v$$ </p>
<p>$$v = Δx/t - Δx'/t$$ </p>
<p>$$v = 0m/60s - 126m /60s$$ </p>
<p>$$v = -126 m/ 60s$$ </p>
<p>$$v = -2.1 m/s$$ </p>
<p>Just wondering if the negative holds any significance? I know we're talking about speed which is scalar but how come the calculation gives a negative? Sorry I am just beginning to learn about relativity.</p>
| 3,112 |
<p>Quantum interpretations like <a href="http://physics.stackexchange.com/q/34074/36915%5C">superdeterminism</a>, nonlocal hidden variables, etc. are regularly dismissed by the vast majority of physicists because they require "cosmic conspiracies" which can potentially involve anything and everything within the universe, including distant quasars billions of light years away (e.g. from Bell inequality violations), events arbitrarily long ago in the past or far away in the future (e.g. from delayed choice experiments), the "will" of experimenters and their brain states (from complementarity), etc. . This conspiracy has also got to be good enough to escape detection from within. Then, proponents are accused of "paranoia" and as such, not worth debating with any further. Other than premature accusations of "paranoia", are there any good objections against "cosmic conspiracies"?</p>
<p>Let me put it this way. Other than "gut feeling" metaphysical <em>pre</em>-judices, what other good objections against cosmic conspiracies are there?</p>
| 3,113 |
<p>As the cone cells are different in numbers in people, how can we say that everyone is seeing the color as same? for example the color you are saying as red may be not the one i see as red..</p>
| 72 |
<p>The range of electromagnetic radiation is indefinite.</p>
<p>When was that established? Doesn't Hubble's limit have an effect?</p>
| 3,114 |
<p>I am working on a physics problem, but my issue is math-related. My professor skips some steps based on 'intuition' that I lack:</p>
<p>In a conservative system, to find out the nature of equilibrium points, we are looking at the potential energy functional, in general form given by $P[f(s)] = \int F(f(s)) ds$. (The problem is about deflection of beams, $s$ being the coordinate of the deflected elastic line.)</p>
<p>To me, it makes sense to perturb the function $f(s)$ to $f(s)+\epsilon g(s)$, $\epsilon$ being a small number. Then, $P[f + \epsilon g]$ is an ordinary function of $\epsilon$, so Taylor expansion works:</p>
<p>$$
\Delta P = P[f+\epsilon g] - P[f] = \epsilon \frac{dP[f+\epsilon g]}{d\epsilon}\big|_{\epsilon=0} +
\frac{1}{2}\epsilon^2 \frac{d^2P[f+\epsilon g]}{d\epsilon^2}\big|_{\epsilon=0}
+O(\epsilon^3).
$$</p>
<p>Equilibrium requires that the first term on the RHS is zero, and for a stable equilibrium, the potential energy should be minimal, so it is necessary that
$$
\frac{1}{2}\epsilon^2 \frac{d^2P[f+\epsilon g]}{d\epsilon^2}\big|_{\epsilon=0} \geq 0 \qquad (1)
$$
for all $g$.</p>
<p>My problem arises when my professor says the above is more formal and involved, and that it would suffice to just collect powers of $f(s)$ and write the functional as
$$
P[f(s)] = P_0 + P_1 + P_2 + \cdots,
$$
where $P_i$ has $i$th-order terms in $f$. Then, apparently, equilibrium dictates that $P_1=0$ and a minimum requires that
$$
P_2 \geq 0. \qquad (2)
$$</p>
<p>Could anyone please show me how to go from (1) to (2)? I have tried a second Taylor expansion, now in $F$, but it confuses me, especially since it is not a variable but a function.</p>
| 3,115 |
<p>This might be a naive question, but how can an object such as a black hole singularity have infinite density but finite mass? (For example, we can approximate the mass of a black hole based on Kepler's Laws and use info from surrounding movements of stars to determine the central mass, but the black hole, excluding the event horizon, has infinite density.)</p>
| 73 |
<p>This <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydr.html" rel="nofollow">link</a> describes a method for determining the most probable radius of an electron for a Hydrogen atom in the ground state.</p>
<p>It states that :</p>
<blockquote>
<p><em>The radial probability density for the hydrogen ground state is obtained by multiplying the square of the wavefunction by a spherical shell volume element.</em></p>
</blockquote>
<p>When I went to solve this problem myself I multiplied the square of the given wavefunction by the volume of a sphere, which gave me the wrong answer as I know it should be the Bohr radius. </p>
<p>When I thought about this problem it seemed reasonable to multiply it by the volume of a sphere rather than the surface area of a sphere (my gut feeling). The link doesn't really explain to me <strong><em>why</em></strong> it uses a surface area and not a volume of a sphere and I would like some help in getting an intuition for why the surface area is used and gives the right answer and the volume of a sphere does not.</p>
| 3,116 |
<p>In a <a href="http://en.wikipedia.org/wiki/Metric_expansion_of_space" rel="nofollow">Wikipedia article</a> I read that the "Metric Expansion of Space" exceeds the speed of light.
If this is true then we must be being disconnected from very remote parts of the universe since gravity can only travel at the speed of light.</p>
<p>In his book "A Brief History of Time", Hawking seems to still believe in the big crunch. Well, it does appear that way, as he devotes a lot of space about the vector of time reversing due to the big crunch.</p>
<p>But how can the big crunch still be a valid theory if the above statement is true, or is it not true?</p>
<p>This question is not about dark energy,although i do appreciate the replies.</p>
| 3,117 |
<p>As I understand lasers, you start off with a few photons that are in an identical state, and other photons that are created later tend to have the same quantum numbers due to Einstein-Bose statistics. Isn't each photon that "joins" the group of preexisting ones a clone of the previous ones? Why doesn't this violate the no-cloning theorem?</p>
| 3,118 |
<p>In terms of the different light phenomenon (reflection, diffraction, rectilinear propagation and dispersion), what are some problems with Huygen's wave model of light?</p>
| 3,119 |
<p>How can I derive the Einstein's relation $D=k_{b}TB$, where $D$ is the diffusion coefficient and B is the mobility coefficient, from the concept of osmotic pressure?</p>
| 3,120 |
<p>I have a question. Is it possible to perform computer simulations based on the M-Theory?</p>
<p>I was looking for such simulators or source codes but I have not found anything.</p>
<p>However, M-theory must have a mathematical model that can be saved as a code to be executed by the computer.</p>
<p>Something like this might work? Leaving aside the problem of computational complexity.</p>
| 3,121 |
<p>How can you calculate the work done by a force (of unknown quantity) exerted on a 10kg block on an inclined plane. The force is pointing upwards and parallel to the incline (which is inclined 30 degrees with respect to the horizontal).</p>
<p>a. frctionless plane
b. coefficient of friction = 0.12</p>
<p>So the forces acting the block are the normal force, its weight, the friction force (for letter (b)), and the force exerted upwards the incline. All are given or can be solved almost instantly except for the force upwards denoted by F.</p>
<p>How do I solve this problem? I am not sure what value of acceleration to use in the axis of the incline for F=ma. Sorry I could not provide a diagram for this. </p>
| 3,122 |
<p>The <a href="http://www.celestrak.com/NORAD/documentation/tle-fmt.asp" rel="nofollow">Celestrak website</a> provides information on reading the TLE ( Two-Line Element Set ) format. In <code>Line 1, Column 34-43 & 45-52</code> give information about <code>First Time Derivative of the Mean Motion</code> and <code>Second Time Derivative of Mean Motion</code>.</p>
<p>Is there any way to calculate or estimate these two parameters for a spacecraft, if you don't have its TLE at disposal to simply read them from there ? And what do they actually mean ?</p>
| 3,123 |
<p>How to calculate the quasiparticle current density starting from bogoliubov - de gennes equation:
$$\left( \begin{array}{cc}H_{0} - E_{F} & -i\sigma_{y}\Delta \\ i\sigma_{y}\Delta^{*} & E_{F} - H_{0}^{*}\end{array}\right)\Psi = \mathcal{E}\Psi$$
I am not assuming that the single particle hamiltonian $H_{0}$ is $x,y,z,$ dependent and has electromagnetic field dependence.
How to start? What is the definition? </p>
| 3,124 |
<p>I am attempting to design a obstacle avoidance system with the <a href="http://en.wikipedia.org/wiki/Arduino" rel="nofollow">Arduino</a>. My position (for now) is going to be stationary. I will be detecting an incoming object and I want to use the below known variables to predict if it will hit me or not. Unfortunately this requires physics and I am weak. </p>
<p>I plan on doing the following calculations:</p>
<p>Object detected at point 1 = Known Distance and azimuth. </p>
<p>Object detected at point 2 = Known Distance and azimuth and speed.</p>
<p>Object detected at point 3 = Known Distance and azimuth and speed and acceleration.</p>
<p>Now that I have the above values I think I can plot a trajectory but I will need to account for gravity. I can't account for gravity until I know my objects mass. </p>
<p>So, given that I have Velocity and acceleration, is there anyway to calculate, taking into account for gravity, a graph of its trajectory in a 2D and 3D space. (Please use 2D for the example, I'll figure 3D out later.)</p>
| 3,125 |
<p>My question has 2 parts:</p>
<ol>
<li><p>I just followed the derivation of Navier Stokes (for Control Volume CFD analysis) and was able to understand most parts. However, the book I use (by <a href="http://rads.stackoverflow.com/amzn/click/0131274988" rel="nofollow">Versteeg</a>) does not derive it in its entirety. He pulls a lot of results directly from Schlichtling and continues his analysis. I want to understand the derivation in its full form. <strong>Is there any resource other than Schlichtling (My library doesn't have it) for deriving the NV equations in their full form? I would prefer an online (free) PDF or similar.</strong> (I'm not sure my library would have many books on this including the ones discussed <a href="http://math.stackexchange.com/questions/35562/reference-for-navier-stokes-equation">here</a>)</p></li>
<li><p>After the derivation, most books follow it up with :</p></li>
</ol>
<p>$$\underbrace{\frac{\partial (\rho\phi) }{\partial t}}_{\text{Rate of increase of }\phi} +\underbrace{\text{div}(\rho\phi \vec u)}_{\text{Convective Term}} = \underbrace{\text{div}(\Gamma \text{ grad}(\phi))}_{\text{Diffusive Term}} + \underbrace{S_\phi}_{\text{Source Term}}$$
where $\phi$ is property per unit mass, $\vec u$ is velocity vector and $\Gamma$ is diffusive term (Like viscosity or thermal conductivity).</p>
<p><strong>What I can't understand is the use of the terms "convective" and "diffusive"? What do they mean? What is the physical interpretation of these terms?</strong></p>
<p>Their dictionary meanings seem to exacerbate the situation:</p>
<p><code>(convection) the transfer of heat through a fluid (liquid or gas) caused by molecular motion.</code></p>
<p><code>(diffusion) The spreading of something more widely or the intermingling of substances by the natural movement of their particles</code></p>
| 3,126 |
<blockquote>
<p>Find the electric potential in the middle of a square with side length $a$ and charge $Q$.</p>
</blockquote>
<p>If I put the origin in the middle of the square, for the potential I get: $$4\frac{1}{4\pi\epsilon_0}\int_0^\frac{a}{2}\int_0^\frac{a}{2}\frac{\sigma}{\sqrt{x^2 + y^2}}dxdy$$</p>
<p>,where $\sigma$ is surface charge density. Then I'd proceed to change to polar coordinates, so I get something like: $$\frac{\sigma}{\pi\epsilon_0}\int_0^{\frac{\pi}{2}} \int_0^{\frac{a}{2\cos\varphi}} dr d\varphi=\frac{\sigma}{\pi\epsilon_0}\frac{a}{2}\int_0^{\frac{\pi}{2}}\frac{d\varphi}{\cos \varphi}$$
which is a divergent integral. </p>
<p>What am I doing wrong?</p>
<p>I think I can imagine changing the approach in such a way that the limits of integration are different, but what's wrong with this one?</p>
| 3,127 |
<p>How are contact forces electromagnetic? What are they: Coulombic forces between polarized molecules in contact? Or they are the magnetic forces due to the motion of subatomic particles? I also would like to know what the feeling of 'Mechanical Contact' means? First I thought that it only means normal reaction force, but I think that if a sufficiently strong laser beam falls on our hand we will also feel the 'Mechanical Contact' with that beam. Will this be due to the normal reaction force exerted on our hand? If it is so, are electromagnetic waves capable of exerting normal reactions on another body? If yes then owing to which mechanism?</p>
<p>In short, what are contact forces? I also would like to know to what extent these forces act (can they act between two protons in contact)?</p>
| 3,128 |
<p>The are several ways, in which one can write the Klein-Gordon equation, the most straightforward being probably the following:</p>
<p>$$
\hbar^2 \partial_t^2 \psi(x) = (\hbar^2 c^2 \Delta + m^2c^4) \psi(x)
$$</p>
<p>However it is possible to use the d'Alembert operator $\square$ and write KG equation like this:</p>
<p>$$
(\square+\mu^2)\psi = 0
$$</p>
<p>where $\mu := \frac{mc}{\hbar}$. the $\mu$ in natural units ($c = \hbar = 1$) is obviously just the mass of the particle, however I wonder if is it possible to interpret it somehow in non-natural units (e.g.: $[\mu] = m^{-1}$). I mean something along the lines of $mc^2$, which we can interpret as energy and therefore easily analyze particle collisions in terms of possible masses that appear in such events. Just a soft question, because I don't have a lot of experience with relativistic quantum mechanics.</p>
| 3,129 |
<p>A virtual creation with total mass-energy = $E$ is allowed as long as that virtual creation doesn’t last longer than $E/h$. Can the uncertainty principle also be used to estimate the mass-energy in the spontaneous creation of a universe - a spontaneous creation that has now lasted $13.6$ billion years? If so, the principle would require that universe to have a mass-energy less than $1.54\times 10^{-51}$ Joules. Is there a flaw in this?</p>
| 27 |
<p>Kind of an odd, random question that popped into my head. Tidal energy - earth's ocean movement, volcanism on some of Jupiter's moons, etc. - obviously comes from the gravitational interaction between large bodies. On earth the interactions with the moon are pulling water around the surface, creating some amount of heat due to friction, etc. </p>
<p>My question is, where does that energy come from exactly? More specifically, what potential energy source is getting depleted to do that work? Is the earth minutely slowing down in its spin - or are the orbits of earth and the moon subtly altered over time by the counteractive movement and friction of liquids and gasses? </p>
| 3,130 |
<p><strong>Problem/Solution</strong></p>
<p><img src="http://img215.imageshack.us/img215/6696/problem2f.jpg" alt="">!
<img src="http://img59.imageshack.us/img59/4281/sol2pp.jpg" alt="">!</p>
<p>In the third FBD, they made the whole system as one object. What happens if I <em>don't</em> want to do that? What if I jsut apply Newton's second Law to the mass M? I tried it out but it didn't work. Could someone point out my mistake? Note that my other force diagram are like the ones in the solutions. Note that $n_1'$ and $n_2'$ are the reaction forces for mass $m_1$ and $m_2$ respectively.</p>
<p><img src="http://img528.imageshack.us/img528/799/progk.jpg" alt="">!</p>
<p>Writing out Newton's Second law for all the other massses</p>
<p>For M</p>
<p>$\begin{cases}
\sum F_x = F - n'_2 = Ma \\
\sum F_y = n = n_1' + Mg + m_2g
\end{cases}$</p>
<p>For $m_1$</p>
<p>$\begin{cases}
\sum F_x = T = m_1 a \\
\sum F_y = n = m_1g
\end{cases}$</p>
<p>For $m_2$</p>
<p>$\begin{cases}
\sum F_x = n_2 = m_2 a \\
\sum F_y = T = m_2g
\end{cases}$</p>
<p>If I eliminate the system of equations, I would get something absurd like $F = (M + m_2)a$</p>
<p>Could someoene please help me nail down this concept? I really appreciate it. </p>
| 3,131 |
<h2>Why are quasicrystals projections from higher dimensional regular crystal lattices?</h2>
<p>See for example <a href="http://en.wikipedia.org/wiki/Quasi_crystal#History" rel="nofollow">wikipedia</a>: </p>
<blockquote>
<p>»Mathematically, quasicrystals have been shown to be derivable from a
general method, which treats them as projections of a
higher-dimensional lattice.«</p>
</blockquote>
<p>Mathematics aside, is there a physical reason, why this has to be the case?</p>
| 3,132 |
<p>I <a href="http://physics.stackexchange.com/q/21851/2451">read</a> some <a href="http://physics.stackexchange.com/q/48292/2451">answers</a> regarding <a href="http://en.wikipedia.org/wiki/Negative_temperature" rel="nofollow">negative temperatures</a> but I think my question is new. I want to know that what is the physical significance of negative temperature. </p>
<p>Suppose I say a body has temperature -2 K. Can I interpret it physically? </p>
| 3,133 |
<p>I have always been told that the geomagnetic field acts "as if" there were a bar magnet inside the Earth. I now this is not true, but I feel that knowing what is actually going on would help me understand the answer to the question that I'm about to ask:</p>
<p>Why does geomagnetic reversal happen? Is it something gradual or relatively fast?</p>
<p>(What is the origin of the geomagnetic field? Can we answer my questions starting from this mechanism?</p>
| 3,134 |
<p>In stat mech we calculated the <a href="https://en.wikipedia.org/wiki/Radial_distribution_function">radial distribution function</a> (a.k.a. pair correlation function) for a classical gas by using perturbation theory for the <a href="https://en.wikipedia.org/wiki/BBGKY_hierarchy">BBGKY hierarchy</a>. (I could post more details of the calculation if you want, but it is a rather long winded but standard perturbation theory type calculation.) The result we got was</p>
<p>$$ g_2 (r) = \mathrm{e}^{-u(r)/T} \left[ 1 + n_0 \int\mathrm{d}^3r'\ f(r')f(|\vec{r}-\vec{r}'|)\right],$$</p>
<p>where $u(r)$ is the interaction potential, $T$ is the temprature, $n_0$ is the density and $f(r)=\mathrm{e}^{-u(r)/T}-1$ is the Mayer function. $g_2$ roughly measures the probability of finding two particles seperated by a distance $r$. $n_0$ is the small parameter of the perturbation theory.</p>
<p>If you then apply this result to hard sphere (infinite repulsive potential of diameter $a$), you get this:
<img src="http://i.stack.imgur.com/95oKU.png" alt="enter image description here"></p>
<p>Now it makes perfect sense that $g_2$ is zero for $r<a$. Also the asymptote to one at large $r$ is part of the definition of $g_2$, meaning that particles are uncorrelated at large distances. The problem is the peak at $r\sim a$ which implies that you are more likely to find particles clustered together, despite the complete absence of any attractive forces! <em>Why is that</em>?</p>
<p>Our lecturer seems to think it is because when two particles collide they stop, then bounce, hence spending more time in the vicinity of each other than for an ideal gas. But this seems dubious because perfect hard sphere collisions are instantaneous. I can imagine three possibilities:</p>
<ul>
<li>This argument could be formalized as a limit of soft sphere scattering and is the correct explanation of the correlation,</li>
<li>there is some other (presumably entropic) explanation,</li>
<li>the correlation doesn't exist - the perturbation theory gives a qualitatively wrong picture (seems unlikely in this case).</li>
</ul>
<p>So what is it?</p>
| 3,135 |
<p>I did some maths and physics up to the age of 18, and hold an amateur radio licence. This thing has puzzled me for a while - does reception of an electromagnetic wave imply an interaction with the transmitter? Does it drain some of the transmitter's energy?</p>
| 3,136 |
<p>What is degenerate in degenerate electron gas state? Why is it called degenerate?</p>
| 3,137 |
<p>I'm looking for an equation to find the tension on the ends of a cable suspended between two poles (one higher than the other) with no load but the cable itself.</p>
<p><img src="http://i.stack.imgur.com/1rkOa.png" alt="catenary curve section"></p>
<p>I determined that the tension would be different on each end, that the shape of the suspended cable would be a <a href="http://en.wikipedia.org/wiki/Catenary" rel="nofollow">catenary curve</a> truncated at one end, and that the following would be the variables necessary in the equation:</p>
<ul>
<li>cable weight</li>
<li>cable length</li>
<li>vertical distance between ends</li>
<li>horizontal distance between ends</li>
</ul>
<p>Some links that could be helpful:</p>
<ul>
<li><a href="http://www.scribd.com/doc/12736614/Sag-Tension-Calculation" rel="nofollow">http://www.scribd.com/doc/12736614/Sag-Tension-Calculation</a></li>
<li><a href="http://mysite.du.edu/~jcalvert/math/catenary.htm" rel="nofollow">http://mysite.du.edu/~jcalvert/math/catenary.htm</a></li>
</ul>
| 3,138 |
<p>Almost all uncertainties (for example the position-momentum uncertainty or time-energy uncertainty) are greater than ${\hbar}/{2} $. But what is the derivation of this <a href="http://en.wikipedia.org/wiki/Uncertainty_principle" rel="nofollow">uncertainty</a> by Heisenberg? Is there any sort of intuitive explanation behind the magnitude of uncertainty? I know why uncertainty happens but I do not why the value. It'd be great if somebody could provide a simple explanation.</p>
<p>$$ \mathrm{uncertainty} \geq \frac{\hbar}{2} $$</p>
| 74 |
<p>With a hypothetical system, where the moon would be always on the opposite side of the planet than the sun, in a way that the moon would only be visible at night on the planet.</p>
<p>I don't know if this is possible, but if it was possible how they would behave. For example, I think that if this being possible, maybe at dusk/dawn both moon and sun would appear on the horizon at opposite sides.</p>
<p>And the last thing, could this system have a periodic event where both star and satellite would appear entirety above the horizon line. </p>
<p>Another thing I am not sure, is if the place at the planet would affect this, but if it doesn't matter I can work with both scenarios.</p>
<p>Old question:</p>
<p>Is there any way to calculate how a planetary system would behave, where the satellite and the star appear simultaneously in the sky only once a year?</p>
<p>I'm writing a novel where this day would be of great importance, but would like to have some facts as realistic as possible, for example the duration of one day and one year for this scenario to be possible.</p>
| 3,139 |
<p>I have a homework assignment about rigid body dynamics. Take a disc of radius $r=2m$ with uniform mass density $\rho=1$ $kg/m^2$ in the x-y plane, resting in an inertial frame. At some instant, a force of $F = (0,0,1)kN$ is applied at the point $A=(0,r,0)$. What's the acceleration of the point $A$ at that instant?</p>
<p>This should be straightforward computation, but this is new to me and I think I'm making a fundamental mistake somewhere. Could you help me find it? In particular I'm not sure I'm treating the angular velocity $\omega$ correctly. I can assume the equations of motion for a rigid body, as well as the equation for the acceleration of an arbitrary point of the body given the acceleration of the center of mass and the angular acceleration. </p>
<p>The disc is just a circle of
homogenous mass $\rho = 1$, so </p>
<p>$$M = \rho \int_{R=0}^r\int_{\theta=0}^{2\pi} R d\theta dR =
\left.\frac{\rho 2\pi R^2}{r}\right|_{R=0}^2 = \left.\pi
R^2\right|_{R=0}^r = 4\pi~\mbox{kg}$$</p>
<p>where though the the problem gives $r=2$ I'm just going to leave it as symbols. Without
integration, I suppose we might have noticed that $M = \rho \pi
r^2$. Now we compute the tensor of inertia, </p>
<p>$$\begin{pmatrix}
\int_V \rho x_2^2 +x_3^2 dv & -\int_V \rho x_1 x_2 dv& -\int_V \rho x_1x_3dv
\\
-\int_V \rho x_1 x_2dv & \int_V \rho x_1^2 +x_3^2dv & -\int_V \rho x_2 x_3dv
\\
-\int_V \rho x_1 x_3 dv& -\int_V \rho x_2 x_3dv & \int_V \rho x_1^2 +x_2^2dv
\end{pmatrix}$$</p>
<p>Since our object is
symmetric about the x-y axes, we can eliminate the
cross-moments. Since our object has no mass in the z direction, we can
eliminate $x_3$ in the tensor, so</p>
<p>\begin{align*}
I &= \begin{pmatrix}
\int_V \rho x_2^2 dv & 0& 0
\\
0 & \int_V \rho x_1^2 dv & 0
\\
0 & 0 & \int_V \rho x_1^2 +x_2^2dv
\end{pmatrix}
\\
&= \begin{pmatrix}
\rho \int r^3 sin^2\theta drd\theta & 0& 0
\\
0 & \rho \int r^3 cos^2\theta drd\theta & 0
\\
0 & 0 & \rho \int r^3 dr d\theta
\end{pmatrix}
\\
&= \begin{pmatrix}
\frac{\rho}{2} \int r^3 (1-\cos(2\theta)) drd\theta & 0& 0
\\
0 & \rho \int r^3 (1+\cos(2\theta)) drd\theta & 0
\\
0 & 0 & \frac{\pi \rho r^4}{2}
\end{pmatrix}
\\
&= \begin{pmatrix}
\frac{\rho}{2}\left[ \frac{2\pi r^4}{4} - \frac{1}{2} \sin(4\pi)\right] & 0& 0
\\
0 & \frac{\rho}{2}\left[ \frac{2\pi r^4}{4} + \frac{1}{2} \sin(4\pi)\right] & 0
\\
0 & 0 & \frac{\pi \rho r^4}{2}
\end{pmatrix}
\\
&= \begin{pmatrix}
\frac{\rho \pi r^4}{4} & 0& 0
\\
0 & \frac{\rho \pi r^4}{4} & 0
\\
0 & 0 & \frac{\pi \rho r^4}{2}
\end{pmatrix}
\end{align*}</p>
<p>The
angular momentum vector should be $\begin{pmatrix} \omega_x,& 0,&
0 \end{pmatrix}^T$, taking the cross product of the force vector and the vector from the center of mass to $A$. Now we plug this in
to the equations of motion to solve for the linear and angular acceleration, $a_c$ and $\alpha$. They
are, </p>
<p>\begin{align*}
(Mv_c)' &= S \implies & a_c = \frac{1}{M}\begin{pmatrix} 0,& 0,&
F \end{pmatrix}^T
\\
(I\omega)' &= M_c = \tilde r \times F \implies & \dot \omega = \frac{r}{I_{11}}
e_y\times |F|e_z = \frac{r|F|}{I_{11}} e_x
\end{align*}</p>
<p>We should now be able to plug this into the equation for angular
acceleration (where the angular velocity at the instant of impulse is $0$ so we ignore it),</p>
<p>\begin{align*}
a_A &= a_c + \alpha \times e_y
\\
&= \frac{|F|}{M}e_z + \dot \omega \times(r e_y)
\\
&= |F|(\frac{1}{M} + \frac{r^2}{I_{11}} )e_z
\\
&= |F|(\frac{1}{\rho \pi r^2} + \frac{4}{\rho \pi r^2} )e_z
\end{align*}</p>
<p>Which I guess seems natural enough. But by newton's formula we would have </p>
<p>\begin{equation*}
M a_A = 5 |F|
\end{equation*}</p>
<p>But does it make sense that the force should be $5F$? Not really...</p>
| 3,140 |
<p>It's not difficult to see that the graviton in $D$ spacetime dimensions has $(D-3)D/2$ polarizations. In $D=4$ there are two $\epsilon^{\pm}_{\mu\nu}$. What I find curious is that in $D=4$ I can actually pick $\epsilon^{\pm}_{\mu\nu}=\epsilon^{\pm}_{\mu}\epsilon^{\pm}_{\mu}$ where $\epsilon^{\pm}_{\mu}$ are the two polarizations (of definite elicity $\pm1$) for a massless spin-1 particle like the photon. In higher dimension this doesn't seem possible since the photon has $D-2$ polarizations so that the number $(D-2)(D-1)/2$ of $\epsilon^{\lambda}_{\mu}\epsilon^{\lambda^\prime}_{\mu}$ pairs doesn't match the number $(D-3)D/2$ of graviton polarization. Well, unless somehow I consider only a smaller subset of them, say adding a constraint or removing one of them
$$(D-2)(D-1)/2-1=(D-3)D/2$$
as in $D=4$ where $\epsilon^{+}_{\mu}\epsilon^{-}_{\mu}$ is discarded having zero elicity.</p>
<p>Is there an analogous constructions for $\epsilon^{\lambda}_{\mu\nu}$ in terms of the spin-1 polarizations $\epsilon^\sigma_\mu$ in higher dimensions?
I suspect that something similar may happen only if I involve polarizations of higher elicity as well.</p>
| 3,141 |
<p>It's often stated that the central charge c of a CFT counts the degrees of freedom: it adds up when stacking different fields, decreases as you integrate out UV dof from one fixed point to another, etc...
But now I am puzzled by the fact that certain fields have negative central charge, for example a b/c system has $c=-26$. How can they be seen has negatives degrees of freedom? Is it because they are fictional dof, remnant of a gauge symmetry? On their own, they would describe a non unitary theory, incoherent at the quantum level.</p>
| 3,142 |
<p>On a similar note: when using Gauss' Law, do you even begin with Coulomb's law, or does one take it as given that flux is the surface integral of the Electric field in the direction of the normal to the surface at a point?</p>
| 327 |
<p>Einstein was able to make some predictions before GR was fully formulated. e.g. gravitational time dilation.</p>
<p>Such predictions before the full theory are referred to as "semi-heuristic derivations" here: <a href="http://motls.blogspot.cz/2012/09/albert-einstein-1911-12-1922-23.html" rel="nofollow">http://motls.blogspot.cz/2012/09/albert-einstein-1911-12-1922-23.html</a></p>
<p>and have also been called "generic features" by Smolin I think about loop quantum gravity although I can't find the right reference right now.</p>
<p>What such phenomena are predicted today about theories that we don't yet have in full form ?</p>
| 3,143 |
<p>Is it true to say Space time <a href="http://en.wikipedia.org/wiki/Curvature" rel="nofollow">curvature</a> and <a href="http://en.wikipedia.org/wiki/Matter" rel="nofollow">Matter</a> are just the same thing, part of the same coin and that therefore <em>Space time curvature</em> $\Leftrightarrow$ <em>Matter</em>? In other words is <em>Space time curvature</em> just <em>Matter</em> and <em>Matter</em> just <em>Space time curvature</em>?</p>
| 3,144 |
<p>this is simple.</p>
<p>what i actually want to ask is, when they do the subatomic particle collision experiments, how do they produce one single subatomic particle, e.g proton, neutron? how do they rip one single electron or photon off something? where do they store the single particle?</p>
<p>still, how do they keep the subatomic particle on track, on the designed trajectory? </p>
<p>even further, how do they make sure 2 such small particles to have a head-on collision? it's really a long shot, for they are incredibly small. how do they aim? </p>
| 3,145 |
<p>I am looking at <a href="http://www.mcgrating.com/references/ref02.pdf" rel="nofollow">this paper</a> (Multicoated gratings, J. Opt. Soc. Am., 1981) and I am getting confused around equation 22. I do not completely understand where he comes up with the equation </p>
<p>$$\xi^j_q=b_q^j(R^{-1}V_q^j)\qquad \text{(22)}$$</p>
<p>And then what is the meaning of the $b^{j+1}$. I initially thought they were the eigenvalues of the T matrix he defines, but all he says is that it is a vector of components $b^q_j$. </p>
<p>This includes the $\bf b^0$ and $\bf b^{Q+1}$ which I do not see how they are vectors?</p>
| 3,146 |
<p>We know that if Single Line to Fault occurs, then fault current flows to the earth. I want to know whether the current will return to the source or not. For the current to flow we need a closed path.</p>
<p>How it is possible for current to flow in a open circuit if you say that current does not return to the source. </p>
| 3,147 |
<p>The planetary orbits have been studied as ellipses but the solar system is in motion in relation to the distant stars. Their path is along the tip of an <strong>helix</strong> and the ecliptic plane is a convenient plane of projection. I think that the studies were never conducted under this viewpoint. </p>
<p>The sunlight we see now was emitted more than 8 minutes ago when the Sun was ‘below’ the ecliptic but we see it centered in the plane. I wonder why we do not see any consequence of this.</p>
<p>I’m following a line of reasoning that the motion can have consequences and I revisited the anomalous precession of the perihelion of Mercury, settled long time ago by Einstein, and I found this perturbing equation:</p>
<p>$$\frac{43}{5557} = 2\pi\frac{369.2\ \text{km}\ \text{s}^{−1}}{299792.458\ \text{km}\ \text{s}^{−1}}$$</p>
<p>Where 5557 is the predicted theoretical value (in <a href="http://mathpages.com/rr/s6-02/6-02.htm" rel="nofollow">mathpages</a>) for the advance and the 43 is the anomaly.</p>
<p>The simplicity of the formula $$\text{error}/\text{theoric}=2\pi\ V/c$$ and because $V$ is <a href="http://arxiv.org/abs/1303.5087" rel="nofollow"> <strong>0.054%</strong> off the central measured value of the speed of solar system</a> - $369(\pm0.9$) - makes me wonder if this can be more than a coincidence.</p>
<p>Any kind of reasoning on the why's will be helpful.</p>
| 3,148 |
<p>Sorry I have a stupid question in Polchinski's string theory book vol 1, p46.
For a holomorphic function $T(z)$ with a general operator $\mathcal{A}$, there is a Laurent expansion
$$T(z) A(0,0) \sim \sum_{n=0}^{\infty} \frac{1}{z^{n+1}} \mathcal{A}^{(n)}(0,0). \tag{2.4.11}$$
Under transformation $\delta \mathcal{A}=-\epsilon v^a \partial_a \mathcal{A}$, why the OPE is determined as
$$T(z) A(0,0) \sim \cdots + \frac{h}{z^2}\mathcal{A}(0,0) + \frac{1}{z} \partial \mathcal{A}(0,0)+\cdots? \tag{2.4.14}$$
How to derive this equation? </p>
| 3,149 |
<p>My question relates to something that I´ve seen in many books and appears in all its glory here: <a href="http://books.google.com.br/books?id=nnuW_kVJ500C&pg=PA198&lpg=PA198&dq=the%20left-hand%20side%20of%20%286.72%29&source=bl&ots=vpxrfzRO2V&sig=ZdPtvv_zqoLv8MTvE5Brc87zhOs&hl=pt-BR&sa=X&ei=HPgSUPuuC4Km8gTkkYHADg&ved=0CD0Q6AEwAQ#v=onepage&q=the%20left-hand%20side%20of%20%286.72%29&f=false" rel="nofollow">Ryder, pg 198</a></p>
<p>My question is about eq. 6.74. Which I repeat below:</p>
<p>$$i \int {\cal D}\phi \frac{\delta \hat{Z} [\phi] }{\delta \phi} exp \left(i\int J(x) \phi(x) dx\right) = i\; exp \left(i\int J(x) \phi(x) dx\right) \hat{Z}[\phi] \Bigg|_{\phi\rightarrow\infty}+ \int {\cal D}\phi J(x) \hat{Z}[\phi] exp \left(i\int J(x) \phi(x) dx\right)$$</p>
<p>$\phi$ is a scalar field, J is a source, $x = x_{\mu}$ in 4D Minkowsky space and $\hat{Z}[\phi] = \frac{e^{iS}}{\int {\cal D}\phi\; e^{iS}}$</p>
<p>The author is clearly doing a integral by parts and the first term on the right hand side is a kind of surface term for the path integral. He then considers this term to be zero and the second one gives us:</p>
<p>$$i \int {\cal D}\phi \frac{\delta \hat{Z} [\phi] }{\delta \phi} exp \left(i\int J(x) \phi(x) dx\right) = J(x) Z[J]$$</p>
<p>The trick thing here is that integral limits for $\int{\cal D}\phi$ are not very obvious (at least not to me). You are in fact summing up for all field configurations. So, there are actually two problems in my mind:</p>
<ol>
<li><p>For what configuration of $\phi$ is the surface term calculated? (the author says it is $\phi \rightarrow \infty$)</p></li>
<li><p>Assuming the author is right about taking huge $\phi$: why is this term zero?</p></li>
</ol>
<p>This applies to path integrals in general: can we do the usual trick of throwing out surface terms safely?</p>
| 3,150 |
<p>I am familiar with the tools that appear in (linear) perturbation theory for general relativity, that is namely that one writes:</p>
<p>$$g_{\mu \nu} = g^{(0)}_{\mu \nu} + \epsilon g^{(1)}_{\mu \nu} + \mathcal{O}(\epsilon^2) \tag{*}$$</p>
<p>Where $g^{(0)}_{\mu \nu}$ is typically assumed, and then one assumes a perturbation of some kind and solves some equations. Let us now perturb the background space-time manifold, $(M,\mathbf{g}^{(0)})$, by a (massless) scalar field, we have that the equations of motion for the scalar field $\Phi$ are given by the Klein-Gordon equation:</p>
<p>$$(\square + \xi R) \Phi := (\nabla_{\mu} \nabla^{\mu} + \xi R) \Phi = 0 \tag{**}$$</p>
<p>Where the covariant derivative is taken with respect to the background metric tensor $g^{(0)}_{\mu \nu}$. Suppose we can solve these.</p>
<p>How does one translate the solution for $\Phi$ into terms of the metric components $g^{(1)}_{\mu \nu}$? In the sense that we have considered a physical perturbation on the space-time and now the metric tensor field must be modified via $(*)$. What if we generalise a bit and consider a spin $\sigma$ perturbation; electromagnetic, Dirac or gravitational (via the Teukolsky equation)? Following the excellent review article of Kokkotas and Schmidt (<a href="http://arxiv.org/abs/gr-qc/9909058" rel="nofollow">http://arxiv.org/abs/gr-qc/9909058</a>) he says that (on page 10): "The variation of the Einstein equations:</p>
<p>$$\delta G_{\mu \nu} = 4 \pi \delta T_{\mu \nu}$$</p>
<p>by assuming a decomposition into tensor spherical harmonics for each (in my notation) $g^{(1)}_{\mu \nu}$ of the form $\chi(t,r,\theta,\phi) = \sum_{\ell m} \chi_{\ell m}(r,t) Y_{\ell}^{m}(\theta,\phi)$ the perturbation is reduced to a single equation." I believe this immediatly in light of separation of variables and equation $(**)$ above. But they do not describe how the metric components in $g^{(1)}_{\mu \nu}$ are recovered. Is there a general procedure by which I can match a spin $\sigma$ perturbation solution (given that I can solve the EoM for the spin $\sigma$ field) with the metric?</p>
<p><strong>Question:</strong> <strong>How does one match $\Phi$ in $(**)$ with $g^{(1)}_{\mu \nu}$? How do the quasi-normal modes in terms of $\omega$ explicitly couple to the metric?</strong></p>
<p>Edit: I have re-written the question slightly in the hope that it is now more cogent.</p>
<p>Thanks!</p>
| 3,151 |
<p>I have very basic question regarding sound I tried to search it over google but couldn’t find the right answer, my question:</p>
<pre><code>What are those characteristics by which every sound can identified uniquely?
</code></pre>
<p>For e.g. pitch is one of the characteristics of sound but let’s say a note C# can also be played on a guitar and piano with same pitch but the resulting <em>sound</em> that we hear is different so what are those characteristics which defines every sound.</p>
| 819 |
<p>I wonder if it is possible to generate electric energy from the radiation of radioactive materials like nuclear waste? If it is then wouldn't that also mean that it could be used as an energy source for a very long time since the radiation takes a long time to decay?</p>
| 3,152 |
<p>I've been teaching myself DC electronics as a hobby and, although I have a feeling i'm "missing something obvious", I was wondering if someone could help me out. If two components of differing resistance are wired in series, they likewise drop different voltages; yet when they are wired in parallel they apparently share a single magnitude of voltage drop. I would like to better understand why a common drop is exhibited across components of differing internal resistance wired in parallel? </p>
| 3,153 |
<p>This is a difficult question to phrase, so please bear with me. I found some cheap sunglasses and pulled out the plastic lenses which are polarized. For clarity, I have labeled them as lens-1 and -2 with faces A and B. </p>
<p><img src="http://i.stack.imgur.com/iKSi9.jpg" alt="enter image description here">
There are three cases I need help with in explaining what is going on. I wrote a letter “A” on a sheet of white paper that I am looking at through these lenses. </p>
<p><strong>Case (1): Sides 1A and 2A (or sides 1B and 2B)</strong>
<img src="http://i.stack.imgur.com/7C5SM.jpg" alt="enter image description here"></p>
<p>If put side 1A in front and 2A in back (or vice versa), there is a color change depending on the rotation. With $0^0$ or $180^0$ rotation, there is a brownish color while a $90^0$ or $270^0$ rotations produces a greyish color. If on the other hand, one switches the A-sides to B’s, you get the exact same colors as above (as expected).</p>
<p>Here is where I was really surprised. If you have combinations of 1A and 2B, I found to my surprise that it matters which lens goes in front. </p>
<p><strong>Case (2): Side 1A in front with 2B behind (or sides 1B and 2A)</strong>
<img src="http://i.stack.imgur.com/O9g1e.jpg" alt="enter image description here"><br>
With lens-1A in front, you can barely see the letter A with rotations of $0^0$ or $180^0$ whereas with rotations of $90^0$ or $270^0$ it appears to have perpendicular polarizations. </p>
<p><strong>Case (3): Side 1A behind while 2B is in front (or sides 1B and 2A)</strong>
<img src="http://i.stack.imgur.com/4sSqu.jpg" alt="enter image description here">
Reversing the order of 1A and 2B allows me to clearly see the letter A with rotations of $0^0$ or $180^0$. On the other hand, rotations of $90^0$ or $270^0$ appear to have perpendicular polarizations. So apparently, the order in which lenses 1A and 2B appear matter. </p>
<p>Here are my questions:</p>
<ol>
<li><p>There are 4 different color changes when these lenses are overlapped – why?</p></li>
<li><p>Why does the order of the lenses (either in front or behind) in case (2) and (3) matter? I would have expected the same outcome, however, that is not correct.</p></li>
<li><p>What role does polarization play in these lenses? I think that case (3) is a situation of linear polarizers but the order dependence has thrown me through a loop.</p></li>
</ol>
<p>Thank you in advance for any help.</p>
| 3,154 |
<p>Assume an alternate universe with same physical laws as here. In this universe nothing exists except 3 observers, each in a transparent box with a clock. Observer A is travelling at .5c. Observer C is travelling in the opposite direction at .5c. Observer B is at rest. </p>
<p>In the absence of outside landmarks each would feel they were at rest and their clocks were functioning normally. Suppose they pass each other. To A it would appear B had zoomed by at .5c and C at c. To B it would seem A and C had zoomed by at .5c in opposite directions. </p>
<p>Suppose they noted the functioning of each others' clocks as compared to their own in passing, and they met later to compare notes on their observations. What observations would they have concerning clock function? If C shone a light beam ahead of himself, how would it appear to A?</p>
| 3,155 |
<p>everyone. I got a bit stuck on 2(iii), this is supposed to be a easy question, but i don't know how you get the square term? I thought you just do the Fourier transform, but then I got some exponential out of it and I don't know what to do? Can anyone suggest or shed some light on the problem? THANKS
If you just plug in infinity into the integral, how can you avoid the problem of getting zero?</p>
<p><img src="http://i.stack.imgur.com/8A8e4.png" alt="enter image description here"></p>
<p><img src="http://i.stack.imgur.com/6jd2U.png" alt="enter image description here"></p>
<p>You may use the integral provided.</p>
| 3,156 |
<p>I've heard this in many quantum mechanics talks and lectures, nevertheless I don't seem to grasp the idea behind it.</p>
<p>What I mean is, at which point is that our modern understanding of quantum mechanics led to a technological development so fundamental for today's computers that we could not have got it working other way?</p>
<p>Why is it not enough with Maxwell, Bohr, Lorentz, (Liénard)?</p>
| 3,157 |
<p>Given a nail, and a magnet it is possible to magnetize the nail by patiently rubbing the nail until it is magnetized; albeit the field strength may be less than that of the source. </p>
<p>With several score kilograms of space debris in orbit, all ferromagnetic material up there may each well have been subjected to several thousand revolutions around their primary; some sooner than the other. Would such ferromagnetic debris be magnetized by their orbit within the magnetosphere?</p>
<p>p.s. If yes, I would assume the material that magnetize quicker might attract those that are yet to be magnetized to form a larger mass, and so on ad infinitum. What prevents such an occurence? Is it precluded merely because of the volume of open space and the possible repulsion if the similar poles face each other? </p>
| 3,158 |
<p>I place $N$ Brownian particles in $V$ liters of solution, shake until I assume that the particles are "well-mixed", and sample and randomly sample an $S$ liter volume. What is the probability distribution for the number of particles in my sample of $S$ liters? </p>
| 3,159 |
<p>What happens at the zone boundaries of the brillouin zones in the tight binding model?
How does the band gap originate in the TB model?</p>
| 3,160 |
<p>I have read some articles such as <a href="http://sinnott.mse.ufl.edu/Backgrounds/theo01_CNT.html" rel="nofollow">http://sinnott.mse.ufl.edu/Backgrounds/theo01_CNT.html</a> and <a href="https://www.rose-hulman.edu/math/seminar/seminarfiles/2006-07/abstract2006-11-01.pdf" rel="nofollow">https://www.rose-hulman.edu/math/seminar/seminarfiles/2006-07/abstract2006-11-01.pdf</a> which talk about how to roll up a graphene sheet into a nanotube. Now I am confused about the chiral angle and $n,m$ lattice vectors they talk about and the complex math equations make it a little blurry to me. Can someone provide a simple an layman explanation of how this works and how it relates to things like this <a href="http://demonstrations.wolfram.com/RollingUpASheetOfGraphene/" rel="nofollow">http://demonstrations.wolfram.com/RollingUpASheetOfGraphene/</a></p>
| 3,161 |
<p>In Chemistry, I was taught that there are three main states of matter: solid, liquid, and gas, and that heat and pressure determine that state. For some substances, the line is blurry between them.</p>
<p>Some materials don't seem intuitively to do this--nor have I been able to find data on them. For example, what is a reasonable estimate of a melting point for brick? What is the boiling point of paper? When will a carpet sublimate?</p>
<p>The common theme seems to be that these are all composite materials. Certainly all the elements have melting points (as applicable) and boiling points. Many compounds do too. However, something like cardboard is a mixture of fiber, glue, pigment, and possibly other things. Each of these might be made up of several compounds, with each compound having its own boiling point.</p>
<p>My suspicion is that for composite materials, individual compounds would exhibit properties roughly individually--so to melt wood, the water would boil off first, and then maybe it would start melting into a glucose-protein slag. Is this truly the right idea for what happens?</p>
| 3,162 |
<p>Why does the vacuum polarization in 2D massless Fermion QED, </p>
<p>$$ i\Pi^{\mu\nu}(q) = i(\eta^{\mu\nu}-\frac{q^\mu q^\nu}{q^2})\frac{e^2}{\pi}, $$</p>
<p>have the structure of a photon mass term, as is claimed on Peskin chapter 19 page 653?</p>
| 3,163 |
<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/9857/nature-of-an-observer">nature of an observer</a> </p>
</blockquote>
<p>For instance, in the double slit experiment, what is exactly defined as an observer?
I remember from somewhere, light is also an observer? </p>
| 75 |
<p>What exactly is a <a href="http://en.wikipedia.org/wiki/Boson" rel="nofollow">boson</a>?</p>
<p>Is the <a href="http://en.wikipedia.org/wiki/Higgs_boson" rel="nofollow">Higgs boson</a> the cause of gravity or a result of it? Does the collision of particles at the LHC create a gravity field or waves or somehow interact with the gravity field of the earth?</p>
<p>The Higgs Boson is supposed to be quite massive and equivalent to a large number of protons. Were many particles needed to create it or only a few travelling at high speeds? Was the high energy converted into the large mass?</p>
<p>Why is the particle so short lived and what does it decay into?</p>
| 3,164 |
<p>To me this is very confusing, but I hope we can discuss it and find a solid answer to the question.</p>
<p>If you were somewhere where there was absolutely nothing, what colour would your eyes see?</p>
| 3,165 |
<p>I'm studying quantum field theory and I encountered some problems of diffusion of particles by an external potential. Until now I have to do with diffusion of the type particle-particle obtaining the Feynman rules from the Lagrangian of the theory and I don't know how to implement the presence of an external potential. How can I implement it?</p>
| 3,166 |
<p>Let me base the discussion on the <a href="http://en.wikipedia.org/wiki/File%3aKim_EtAl_Quantum_Eraser.svg">pictorial description of the delayed choice quantum eraser experiment on wikipedia</a>.</p>
<p><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/2/21/Kim_EtAl_Quantum_Eraser.svg/1000px-Kim_EtAl_Quantum_Eraser.svg.png" alt="quantum eraser diagram"></p>
<p>First suppose that we do precisely the same thing with the lower parts of the blue and red beams as we do with the upper parts, i.e. we put them through a lens which ends at a new detector $E_0$.</p>
<blockquote>
<p><strong>Question</strong>. Would there be the interference pattern at $D_0$ (or - I suppose equivalently - at $E_0$)? If there woudn't be - why?</p>
</blockquote>
<p>I thought there would be, but then it occured to me we could use it to communicate faster than light - by shooting the lower parts of the red and blue beams to a distant partner. Then the partner would send the mesage either by preparing the setup above (and so causing the patterns at $D_0$ to interfere), or by preparing the setup of the original delayed quantum choice eraser (and so causing the interference pattern at $D_0$ to vanish.) Actually as I've described it, it would be even a message from the future.</p>
<blockquote>
<p><strong>Question</strong> If there would be interference pattern, what's wrong in the above faster-than-light communication protocol?</p>
</blockquote>
| 3,167 |
<p>I am trying to understand the structure of the fermions in non-abelian gauge theories. Disclaimer: my question might be very trivial (I suspect the answer could simply be "a change of basis"), but I would be grateful is someone could shed some light if there's something deeper lurking around the corner.</p>
<p>Let's consider the Yang-Mills Lagrangian </p>
<p>$$ \mathcal{L} = -\frac{1}{4} (F_{\mu \nu}^a)^2 + \bar{\psi}(i \gamma^\mu D_\mu -m)\psi $$</p>
<p>where $ D_\mu = \mathbf{1} \partial_\mu - i g A_\mu^a t^a$, then $\psi$ needs to have both Dirac ($\mu$) and colour (a) degrees of freedom. I get confused when I change the regular approach to the problem and I am not sure if it is a real issue or if I'm just overcomplicating/overthinking this.</p>
<p>For instance, let's consider $SU(N)$ YM where the generators $t^a$ are $N \times N$ matrices, and there are $N^2-1$ of them. Therefore, the covariant derivative $D_\mu$ is an $N \times N$ matrix, and the index $a$ above runs over $N^2-1$ value. </p>
<p>When we start by contracting $D_\mu$ with the $\gamma$-matrices, </p>
<p>$$ (\gamma^\mu D_\mu)_{ij} = \delta_{ij} \gamma^\mu \partial_\mu - i g (\gamma^\mu A_\mu^a) (t^a)_{ij} $$</p>
<p>one gets an $N \times N$ matrix of $4 \times 4$ matrices. The corresponding $4N$-component object this matrix acts upon is the $N$ Dirac spinors arranged in a column. </p>
<p>However, notice that if we start in the following way instead:</p>
<p>$$ (D_\mu \psi)_i = \partial_\mu \psi_i - i g A_\mu^a (t^a)_{ij} \psi_j $$</p>
<p>we get that the covariant derivative acts on $\Psi \equiv (\psi_1, \cdots, \psi_N)$. For all we know, $\Psi$ has no spinor structure since we haven't contracted with the gamma matrices yet. </p>
<p>Contracting with the $\gamma^\mu$, one gets a $4 \times 4$ matrix of $N \times N$ matrices that encodes the same information as before. This time, it would seem that we only have one Dirac spinor, where each component is a $N$-valued singlet. </p>
<p>However, it seems that the matrix $\gamma^\mu D_\mu$ and the $4N$-component spinor look different, even though all we did was to change the order in which we constructed things. </p>
<p>In the first case, we get $N$ spinors corresponding to the $N$ colors of the adjoint representation. In $SU(3)$, this would be like saying we effectively have 3 spinors which correspond to the red/blue/green colors as $(\psi_R, \psi_B, \psi_G)$. In the second case, this identification fails since we only have one big complicated object. </p>
<p>What went wrong? Is this difference simply a change of basis for $\psi$? Is there something relevant we can learn from looking at the YM Lagrangian in these two different ways? </p>
<p>Also, in the first case, when we get $N$ spinors, I am confused about their signification. I always assumed that in QCD, $\psi$ would correspond to a quark, which is a fermion by itself. Does it mean that quarks are fermions that can be described by fermionic fields/degrees of freedom that we call color? </p>
| 3,168 |
<p>Given a 4-dimensional compact manifold (torsion free), the Euler characteristic is defined as:</p>
<p>$$E_4 ~=~ \int \epsilon_{abcd}R^{ab} \wedge R^{cd}$$</p>
<p>with $R^{ab}$ is the curvature 2-form. Perturb the connection 1-form (represent by $\delta \omega^{ab}$), $E_4$ should be unchanged. How can I proof that, and what will be change if the manifold is not torsion free anymore? </p>
<p>Given that:</p>
<p>Connection 1-form from torsion-free condition: </p>
<p>$$T^a ~=~ De^a ~=~ de^a + \omega^a_b \wedge e^b ~=~ 0$$ </p>
<p>and </p>
<p>$$\omega^{ab} ~=~ \delta^{ac} \omega^b_c.$$</p>
<p>Curvature 2-form:</p>
<p>$$R^{ab} ~=~ D\omega^{ab} ~=~ d\omega^{ab} + \omega^a_c \wedge \omega^{cb}.$$</p>
<p>Gauss-Bonnet term appears as a topological term in some theoretical gravitational actions. I don't see why it's unchanged, so I post this question of mine here.</p>
| 3,169 |
<p>I have been reading up on the production methods of graphene, and one that I found interesting in particular was the thermal exfoliation of graphite oxide. From what I gather the basic idea is that the oxides are located in between the graphene layers of graphite, and that you then heat the material up, causing gasses such as CO and CO<sub>2</sub> to form in between the layers. This sort of lifts the layers from each other, and then they become separate. However, what I don't get is why the layers don't simply re-aggregate on top of each other. </p>
<p>The reason I wonder about this is because thermal exfoliation is done (at least it is possible to do so) without putting the graphite oxide in a liquid or anything; it can just be done 'dry'. Now liquid exfoliation, on the other hand, is based on using ultrasound to separate graphite layers, which are submerged in a fluid with surface energies such that it is unfavorable for the graphite to re-aggregate.
So is it simply that this re-aggregation is not troublesome when your graphite is not in a liquid? I suppose if this is the case, then the only reason for liquid exfoliation is if it is more efficient/effective/cheap than doing the thermal version.</p>
| 3,170 |
<p>From Peskin & Schroeder <em>QFT</em> page 35:</p>
<blockquote>
<p><em>The Lagrangian formulation of field theory makes it especially easy to discuss Lorentz invariance. And equation of motion is automatically Lorentz invariant by the above definition if it follows from a Lagrangian that is a Lorentz scalar. This is an immediate consequence of the principle of least action: If boosts leave the Lagrangian unchanged, the boost of an extremum in the action will be another extremum.</em></p>
</blockquote>
<ol>
<li><p>Could anyone please help me translate the statement of this paragraph into a rigorous mathematical proof with symbols (and, in addition, to generalize it to proper orthochronous Lorentz transformations and not just boosts)? </p></li>
<li><p>Maybe as warm up: for boosts, how does one show that the boost of an extremum in the action will be another extremum?</p></li>
</ol>
| 3,171 |
<p><em>I realise that this might be conventially very difficult to answer because there's no KG or Newtons in space, only particles.</em></p>
<p>As far as I understand, every object creates a 'pull' due to the forces of gravity, that'll be felt noticeably by smaller objects (as well as the larger object being attracted 'less' to it). If we fire a bullet in a straight line on earth it'll curve down towards the earth and hit it because of the force of gravity.</p>
<p>If we say had a bullet fired in a straight line in space, and an object 1km away at say 30 degrees from the main path, is there any way we can work out how massive that object would have to be for the bullet to hit it.</p>
<p>Say the bullet is fired at 10m/s and is on Earth 20g - it's not really the figures I'd just wonder if it's possible.</p>
| 3,172 |
<p>Assuming you have a flat poster with no curvature, why is it that when you pin it to the wall (with thumbtacks) it gains curvature as seen in the picture below. When I put the poster up it was entirely flat to the wall with no curvature, but over time it somehow curls and develops this sort of curvature (it looks like positive curvature at the corner).</p>
<p><img src="http://i.stack.imgur.com/oPDQ9.jpg" alt="enter image description here"></p>
<p>Is there a physics-based explanation for why this happens?</p>
| 3,173 |
<p>The charge and current density fields in classical electromagnetism are scalar real number fields on space time manifold. But these fields diverge/become infinite in case of point charges, how is this justified and mathematically consistent ?</p>
| 3,174 |
<p>In Neil DeGrass Tyson's epic video, at 2:26:50</p>
<p><a href="https://www.youtube.com/watch?v=AdHlVY8pEk0&list=PL1L9zQimONkUOxXJyM0xcJja9ItQu1P8l&src_vid=AdHlVY8pEk0&feature=iv&annotation_id=annotation_4179310099#t=12s" rel="nofollow">https://www.youtube.com/watch?v=AdHlVY8pEk0&list=PL1L9zQimONkUOxXJyM0xcJja9ItQu1P8l&src_vid=AdHlVY8pEk0&feature=iv&annotation_id=annotation_4179310099#t=12s</a></p>
<p>He mentions how if net energy is negative, the spacetime curvature is spherical, and if it's net positive, saddle-shaped.</p>
<p>He uses the terms "flat" and "saddle-shaped" which are 2d, but he's actually just using that as an analogy for 3d curvature, which we can't really visualize, correct? If so, what does it mean for 3d space to be "saddle-shaped" or "spherical"? </p>
<p>1) II'm guessing spherical implies that if we keep traveling in one direction in space, it would be possible to end up back where you started? (kind of like how you can end up in the same position traversing the surface of a planet if the 2d plane is curved?</p>
<p>2) If net energy is 0. But they also say there is far more dark energy and dark matter than energy and matter. So isn't this a contradiction? How does the accounting work out?</p>
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<p>How is <a href="http://en.wikipedia.org/wiki/Graphene" rel="nofollow">graphene</a> a 2D substance? It has length, width and some thickness to it, else it would be invisible. Why is it considered a 2D substance?</p>
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<p>Are they equivalent?</p>
<p>I came up this question because they both explain states beyond Loughlin state.</p>
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<p>Please bear in mind that I'm neither a physics guy nor particularly a math guy. So I will probably need a bit more hand-holding than just a complex equation.</p>
<p>For a game, I'm trying to calculate a <em>reasonable</em> simulation of a human body on a trampoline. It seems to me that there are two phases to consider once the body has contacted the trampoline: the deceleration of the body once it has contacted the surface, which gets stored as potential energy in the trampoline, and then the combination of the release of that potential energy once the body has reached the nadir plus any extra force that body might apply (assuming he has bent his knees prior to that point).</p>
<p>I'm interested in understanding the interaction between the trampoline's storage and release of energy and the body's own contribution (addition) to that energy with its own kinetic energy.</p>
<p>CLARIFICATION:
In short, I want to know when the best time to crouch is (pull your knees up) and when the best time to release (jump) is. Anyone who's been on a trampoline knows you can get better height by crouching and jumping at the right time - you can also bring yourself to a complete stop by crouching and jumping at different times. I can't seem to find anything documented though about what these times actually are. Since I couldn't find the information from the sports / athletics side, I thought I'd come over here to the physics side and see if anyone knew.</p>
<p>I'm surely not being very clear, which only further reveals my own shortcomings in this matter. Any guidance would be appreciated. I understand that this forum typically operates at a much higher level than what I'm looking for, but I thought I'd ask here, since this is where the great minds seem to gather.</p>
<p>EDIT
I can very easily do a "good enough" simulation of a mass "bouncing"off the trampoline by taking its incoming velocity and simply inverting it and multiplying by an arbitrary restitution amount to simulate the springiness of the material. Nothing more complicated really need from the perspective of a game.</p>
<p>The part that's killing me is getting the body to continue moving past the plane of the trampoline while decelerating to 0 and then accelerate again up to final "release velocity". The part that baffles me is the part that happens between the time the body touches the material to the time it leaves contact, taking into account any additional force added by the person through his legs (I've decided not to model any additional lift sure to arm motion etc, to keep both the simulation and the user controls thereof straightforward).</p>
<p>SECOND EDIT
I've read through the answers here and they're very good! And surprisingly, I seem to be following about 75% of it.</p>
<p>The challenge I'm having is that I'm principally interested in Velocity, rather than Energy, even though it's obviously energy that is behind all of this. Every 30th of a second (every "step") I need to know the velocity of the body so that I can plot its new position. It's really easy to plot the position of the body once it has left "contact": starting at some arbitrary position and velocity, v = v + g, y = y + v. See, it's a very different way of representing the same facts when you're dealing with visual graphics than physics equations.</p>
<p>So now y = the plane of the trampoline. v = v * c, where 'c' is a factor of how far the springs will compress, less than 1. y = y + v until v approaches 0 at which point we start the "bounce back". Here I suppose v = v * (1/c) * s, where 's' represents whatever factor of 'loss' we want to simulate so that we don't get 100% bounce back. Once y > the plane of the trampoline, we can then forget the more complicated stuff and go back to letting gravity take over.</p>
<p>Obviously this whole bit ignores mass, which is unfortunate, but I have to keep it simple for now. The "mass" factor gets rolled into the numbers for 's' and 'c' that I fudge until it 'feels' right.</p>
<p>Okay, now that you guys have stopped shuddering in utter horror at this wishy-washy approach, the bit I'm trying to figure out is the part about the human body bending his legs and then performing the jumping motion to add his force to the rebound of the trampoline.</p>
<p>It seems to me, having spent a little time on a trampoline, that you can jump at the right time and at the wrong time. </p>
<p>So, from a physics perspective, when is the optimal time to crouch? When is the optimal time to jump / release?</p>
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<p><strong>Imagine if there was a hole between two earth poles, let's say 100 meters wide, as deep as the Earth's height between north and south poles - what would happen if we threw an object from the North Pole, weighing 100 kg?</strong></p>
<p>Would it lose speed as it approached Earth's center/core, would it speed up while it approached core and would it slow down or speed up as it leaves earth's center/core ...</p>
<p>Would it gain much speed to go out of atmosphere and get into Earth's orbit, or even further?</p>
| 76 |
<p>I have a homework question based on the following diagram:</p>
<p><img src="http://gyazo.com/f0de3827222c4c5651df70ae25b657e9.png" alt="velocity diagram"></p>
<p>I need to find the <strong>angular velocity of the object as seen by an observer at the origin</strong> of the frame. The question says that the observed angular velocity is given by $\omega_0 = \Delta \varphi_0 / \Delta t$, where $\Delta t = t_2 - t_1$ and $\Delta \varphi_0 = \varphi_0(t_1^*) - \varphi_0(t_2^*)$ is the change of the angle $\varphi$, the polar angle which the object had at times $t_1^*$ and $t_2^*$. These are the times when the light detected by the observer (at $t_1$ and $t_2$) was emitted. The object is located at $(x_1, y_1)$ at time $t_1^*$ and it is located at $(x_2, y_2)$ at time $t_2^*$.</p>
<p>I started to answer this question by just finding an expression for $\Delta \varphi_0$ in terms of the coordinates ($\Delta \varphi_0 = \arctan{(y_1/x_1)} - \arctan{(y_2/x_2)}$) and dividing this by an expression for $\Delta t$. However, I was told that this was not correct. Apparently, I have to get an expression for the angular momentum in terms of the angle $\theta$ in the diagram. My lecturer told me that this was could be done with some basic geometry. I can't quite see what to do though. Also, I was told that the angle $\Delta \varphi_0$ was assumed to be very small. I have a feeling that what the lecturer wants us to derive is the expression $\omega = \frac{|\vec{v}|\sin{(\theta)}}{|\vec{r}|}$, but I can't see how to get this. Can anyone help?</p>
<p>Edit: This question is from a special relativity course, but I believe this can be answered without using any knowledge of special relativity. As I said, the lecturer told me that it was basically just geometry.</p>
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