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<p>I'm trying to understand the classical derivation of Friedmann equation but I'm missing one step.</p>
<p>So, I start with accelerations, where $a$ is a scale factor</p>
<p>$\ddot{a}=-\frac{GM}{a^{2}}$</p>
<p>$\ddot{a}=-\frac{4\pi G}{3}\rho a$</p>
<p>Now multiply both sides by $\dot{a}$, and here is the question: Why LHS is</p>
<p>$\dot{a}\ddot{a}=\frac{1}{2}\frac{d}{dt}\left(\dot{a}^{2}\right)$</p>
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<p><img src="http://i.stack.imgur.com/O9S2S.png" alt="enter image description here"></p>
<p>Say I have two boxes which both contain, say, 25 red particles (as shown in picture). These particles are randomly placed in a 2d grid, and in one the total area $A_{1}=20000$ and the other has area $A_{2}=10000$. </p>
<p>If the average distance between any two particles before was $d_{1}$, what is the average (expected) distance between any two particles after, $d_{2}$? </p>
<p>I have worked it out as:</p>
<p>Since we have vertically compressed the box by factor 0.5:
<img src="http://i.stack.imgur.com/kdKaW.png" alt="enter image description here"></p>
<p>Then we get a factor of $\frac{\sqrt{1.25}}{\sqrt{2}}$ = 0.7905.... of the distance $d_{1}$. However, I am not sure if this is correct because surely the particles would be
closer together in the x-direction as well.</p>
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<p>I have a puzzle I can not even understand.
A graviton is generally understood in $D$ dimensions as a field with some independent components or degrees of freedom (DOF), from a traceless symmetric tensor minus constraints, we get:</p>
<ol>
<li><p>A massless graviton has $D(D-3)/2$ d.o.f. in $D$-dimensional spacetime.</p></li>
<li><p>A massive graviton has $D(D-1)/2-1$ d.o.f. in $D$-dimensional spacetime. </p></li>
</ol>
<p>Issue: In classical gravity, given by General Relativity, we have a metric (a symmetric tensor) and the Einstein Field Equations(EFE) provide its dynamics. The metric has 10 independent components, and EFE provide 10 equations. Bianchi identities reduce the number of independent components by 4. Hence, we have 6 independent components.
However, for $D=4$, we get</p>
<ol>
<li><p>2 independent components.</p></li>
<li><p>5 independent components. </p></li>
</ol>
<p>Is the mismatch between "independent" components of gravitational degrees of freedom (graviton components) one of the reasons why General Relativity can not be understood as a quantum theory for the graviton? </p>
<p>Of course, a massive graviton is a different thing that GR but even a naive counting of graviton d.o.f. is not compatible with GR and it should, should't it? At least from the perturbative approach. Where did I make the mistake? </p>
| 8,521 |
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] |
<p>I've seen the preferred basis problem referred to in many places, but have not seen a clear explanation of what the problem is. For example, <a href="http://physics.stackexchange.com/questions/65177/is-the-preferred-basis-problem-solved">this question</a> asks whether the problem has been solved, but I wasn't able to glean anything from the discussion on what the problem was. Zurek 2001 gives the following:</p>
<blockquote>
<p>the original MWI does not address the "preferred basis question" posed by Einstein</p>
</blockquote>
<p>with a footnote quoting Einstein as saying this:</p>
<blockquote>
<p>When the system is a macrosystem and when ψ1 and ψ2 are 'narrow' with respect to the macrocoordinates, then in by far the greater number of cases this is no longer true for ψ = ψ1 + ψ2. Narrowness with respect to macrocoordinates is not only independent of the principles of quantum mechanics, but, moreover, incompatible with them.</p>
</blockquote>
<p>The first sentence seems obvious to me. I'm baffled by the second sentence. Some states are narrow and some are not. What's the problem, and what does this have to do with a basis or a preferred basis?</p>
<p>Zurek, "Decoherence, einselection, and the quantum origins of the classical," <a href="http://arxiv.org/abs/quant-ph/0105127" rel="nofollow">http://arxiv.org/abs/quant-ph/0105127</a></p>
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<p><img src="http://i.stack.imgur.com/PxLaK.png" alt="enter image description here">Two particles, $P_1$ and $P_2$, of equal masses $m$ are linked by a spring of stiffness $k$ and natural length $a$. They are sliding freely without friction along a horizontal fixed hoop of radius $R$. </p>
<p>I need some help in writing Lagrangian for generalized coordinates $\theta_1$ and $\theta_2$ given as $\theta_1=\frac{\phi_1+\phi_2}{2}$, $\theta_2=\frac{\phi_1-\phi_2}{2}$. </p>
<p>I wrote that position vectors for $P_1$ and $P_2$ are $\vec r_1$ and $vec r_2$, respectively.<br>
$\vec r_1=-R\cos\phi_1 \vec i+R\sin\phi_1\vec j$<br>
$\vec r_2=R\cos\phi_2\vec i+R\sin\phi_2\vec j$<br>
From this<br>
$\dot{\vec r_1}=R\dot\phi_1\sin\phi_1\vec i+R\dot\phi_1\cos\phi_1\vec j$<br>
$\dot{\vec r_2}=-R\dot\phi_2\sin\phi_2\vec i+R\dot\phi_2\cos\phi_2\vec j$ </p>
<p>Kinetic energy is </p>
<p>$T=\frac{1}{2}m(\dot{\vec r_1})^2+\frac{1}{2}m(\dot{\vec r_2})^2=\frac{1}{2}mR^2(\dot\phi_1^2+\dot\phi_2^2)=mR^2(\dot\theta_1^2+\dot\theta_2^2)$ </p>
<p>I need help about forces that are doing work.<br>
I know that there is a force in the string, $F=ke$ where $e$ is extension, but I don't know how to determine its direction. </p>
<p>I wrote that<br>
$\vec {P_1P_2}=\vec r_2-\vec r_1$<br>
$|\vec{P_1P_2}|=R\sqrt{2(1+cos(\phi_1+\phi_2))}$ and $|\vec {P_1P_2}|=a+e$ </p>
<p>This is all I did. I'm not sure if normal reaction does any work. </p>
<p>Any help would be appreciated. </p>
<p>Left particle is $P_1$, right one is $P_2$, I forgot to write it on the picture.</p>
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<p>I'm reading this <a href="http://arxiv.org/abs/hep-th/9410093" rel="nofollow">paper</a> where to determine the coefficient $C^{\phi\phi O}(x_{12},\partial_2)$ of the OPE (p.10)
$$\phi^\alpha (x_1)\phi^\beta (x_2)=C_\phi \frac{\delta^{\alpha\beta}}{x_{12}^{2\eta}}+C^{\phi\phi O}(x_{12},\partial_2)O(x_2)\delta^{\alpha\beta}$$
they require consistency of the OPE with the 3-point correlation function, i.e. comparing the fixed form of the 3-point correlation function to what happens when we apply the OPE to two channels in the 3-point function and then use the form of the 2-point function.</p>
<p>Anyway, to solve the form of the coefficient that satisfies eq. 2.29 they use the integral representation
$$\frac{1}{(ab)^\rho}=\frac{1}{B(\rho,\rho)}\int_0^1 dt \frac{[t(1-t)]^{\rho-1}}{[at+(1-t)b]^{2\rho}}$$
which comes from the book Table of Integrals, Series and Products by I.S. Gradshteyn and I.M. Ryzhik. However, looking at that book (I only have access to the 7th ed.) I only found the relation
$$B(\mu,\nu)\frac{1}{(a+c)^\mu(b+c)^\nu}=\int_0^1 dx x^{\mu-1}(1-x)^{\nu-1}\frac{1}{[ax+(1-x)b+c]^{\mu+\nu}}$$
which reduces to the desired relation when $c=0$ however, in the book it says that the relation is valid for $c>0$, so how can they use that relation? </p>
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<p>Due to electron "spin", a small magnetic field is produced. Maxwell's equations imply that magnetic fields are due to changes in electric fields. Is the magnetic field produced then because the electric field is "spinning" with the "spinning" electron, in the quantum sense of "spinning" and this change in electric field is generating the magnetic field?</p>
<p>Can one generalize to say that the magnetic field thus would spin when a magnet is spun?</p>
| 8,525 |
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<p>I am looking for a good explanation and/or reference quotation explaining why infrared absorption technique is essentially nonlinear (eg. for carbon monoxide quantification).</p>
<p>When using UV/visible/near-IR absorption technique, Beer-Lambert Law often is valid and then you have quite straight way to convert transmittance into absorbance which linearly respond to concentration (provided BLL limitations are not met).</p>
<p>Above technique relies on electronic transitions, when IR absorption technique relies on vibrational and rotational transitions for molecules which has permanent or transient dipole moments. I would like to figure out why such technique is essentially nonlinear.</p>
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<p>I guess it depends on the heat or the type of the material but can you give some examples or formulas to calculate it ?</p>
<p>The best example would be the average speed of the air molecules (all types in the air) at room temperature or water molecules at human body temperature.</p>
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<p>An example of a test: Ask each variant whether its estimate of the electron mass lies within $\pm\,x\%$ of the known value. This surely can't take long per theory. Although $10^{500}$ is huge, whittling them down could be essentially a background task for a few thousand computers. But is it being done, and if not, why not?</p>
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<p>Consider general pure state
$|\psi\rangle$
in some hilbert space $\mathcal{H}$ (which could be a tensor product of other Hilbert spaces)</p>
<p><strong>I would like to know whether there is a way to systematically find the group of all operators on $\mathcal{H}$ under which $|\psi\rangle$ is invariant.</strong></p>
<p>A little more precisely: the group I'm looking for is
$$
G=\{A \in H(\mathcal{H}): A|\psi\rangle = |\psi\rangle\}
$$
Where $ H(\mathcal{H})$ is the set of Hermitian operators acting on the Hilbert space.</p>
<p>This group I defined here is a stabilizer group in the <a href="http://mathworld.wolfram.com/Stabilizer.html" rel="nofollow">usual sense in mathematics</a>, and a slight generalization of the <a href="http://arxiv.org/abs/quant-ph/9705052" rel="nofollow">ones popularized by Gottesman in quantum information</a> as we're considering a general state. </p>
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<p>Flow rate is calculated using only the parallel component of the velocity vector to the area vector. Why is this? How can I mathematically prove this? Namely, how do I prove any perpendicular component of the velocity vector is not contributing to any volume output? </p>
<p>I know this is the result of the dot product; but I want to know why the dot product is valid reasoning for this scenario.</p>
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<p>According to NASA </p>
<blockquote>
<p>a black hole is anything but empty space. Rather, it is a great amount of matter packed into a very small area</p>
</blockquote>
<p>According to the documentary <a href="http://www.imdb.com/title/tt3443616/">Space Unraveling The Cosmos</a> about Black Holes</p>
<blockquote>
<p>Gravity here is so strong even light cannot escape</p>
</blockquote>
<p>Which leads me to believe that Black Holes basically are masses so compact but great which gives them very strong gravitational influence (Newton's Gravitational Law) that light is not allowed to go out of, hence a "Black" hole.</p>
<p>If my above understanding of Black Holes is correct, then why is those Spiral Galaxies pictured using Hubble Telescope is showing a big shiny ball instead of a "Black" hole ? What is the obvious clue that I am missing here ?</p>
<p>Here is the image of Andromeda Galaxy image taken using Hubble.</p>
<p><img src="http://i.stack.imgur.com/YkdZ2.jpg" alt="Andromeda Image taken using HST"></p>
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<p>Does the strong (nuclear) force ever contribute to decay ?
Or is the weak nuclear force the only decaying force ?</p>
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<p>Little background first:<br>
I take part in preparing environmental reports for wind turbines/farms and now I'm exploring shadow flickering effect of moving blades. </p>
<p>From some sources (in Polish) I got to know that the shadow casted by turbine blades is solid, black only to some distance and above this distance the shadow is blurred. This distance was described as approx. $108\ * \ width$ of blade. As an explanation of this arguments the <a href="http://upload.wikimedia.org/wikipedia/commons/2/2e/Diagram_of_umbra,_penumbra_&_antumbra.png" rel="nofollow">umbra-penumbra-antumbra diagram</a> was presented.</p>
<p>I wanted to find where from this $108$ value comes from.<br>
I found that for spherical bodies, when the occulting object is smaller than the star, the length (L) of the umbra's cone-shaped shadow is <a href="http://en.wikipedia.org/wiki/Eclipse#Umbra.2C_penumbra_and_antumbra" rel="nofollow">given by</a>:<br>
$L=\frac{rR_o}{R_s-R_o}$<br>
where $R_s$ is the radius of the star, $R_o$ is the occulting object's radius, and $r$ is the distance from the star to the occulting object.<br>
I've also found that this equation comes from simple triangles similarity in Sun-Earth arrangement.</p>
<p>When I use the radius of the Sun and the distance from the sun to the Earth, the equation actually simplifies ($R_s >> R_o=3.5m$) to: $L=215R_o \approxeq108D_o$ ($D_o$ - object diameter)</p>
<p>My question is:<br>
Can this equation be used to approximate the length of solid part (umbra part) of a shadow of objects on the Earth?</p>
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<p>I was given a graph of square of the wave function of a hydrogen atom, against the distance of the electron from the nucleus (denoted by r).</p>
<p>What I know is that the square of the wave function gives the probability of finding an electron at a particular position, but could anyone explain to me that why does the area under this graph has to do with the probablity of locating an electron? Because I thought that on the graph where the highest value of square of wave function is, the corresponding value of r will be the answer. Isn't that so??</p>
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<p>I am using cross correlation for sound localization. I have a four microphone array (arranged linearly). My problem is, sometimes the time diff found by cross correlation between a given pair of mics jumps to some eratic value. For example, if for the first three times it gives a value of 14 then the next time it suddenly jumps to some totally different value like 153 for the sound generated from the same point. Is this normal or am I doing anything wrong? My data collection procedure is as follows:</p>
<ol>
<li>Start listening</li>
<li>Keep listening till trigger event occurs (which is a peak in the sound amplitude)</li>
<li>As soon as the trigger event is noticed, collect sound sample for the next 300ms from all mics</li>
<li>Calculate cross-correlation to find time diff</li>
</ol>
<p>In order to detect the trigger event, I am listening for 10ms and checking the mic data if there is a peak in it. If yes, I record for the next 300ms and process it.</p>
<p>The ambient noise in my setup is very minimal so I dont think that is effecting the calculations. If anyone has come across a similar situation, could you please shed some light?</p>
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<p>I have a question regarding hadron fluctuations. </p>
<p>For instance on page 85 in Feynman's "Photon-Hadron Interactions" equation 15.2 reads:</p>
<p>$$\tag1\vert \omega\rangle = \vert \omega_0\rangle+\frac{\delta}{m_\omega-m_\rho-\frac{\mathrm{i}\Gamma_\omega-\mathrm{i}\Gamma_\rho}{2}}\vert \rho_0\rangle$$
where $\vert\delta\vert\approx 3.7$ MeV (see p. 86 same book).</p>
<p>My question is where I can find an analogous expansion for the proton (say), i.e. something like </p>
<p>$$\tag2\vert p\rangle = c_0\vert p_0\rangle+c_1\vert h\rangle+\cdots$$
where $h=$ some hadron.</p>
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<p>Concerning fiber optics, I'd like to know how a 2D <a href="http://en.wikipedia.org/wiki/Microelectromechanical_systems" rel="nofollow">MEMS</a> based optical cross-connect is able to switch optical signals. I've read that these mirrors are controlled with electrical signals to be either "on" or "off" to redirect light to output ports. </p>
<p>As far as I know, all-optical based switches don't use buffers on the incoming data. However, an electrical signal must be generated to control the MEMS mirrors in order to redirect the signal to the output port. How can this electrical signal be fast enough to switch the optical signal? I assume this electrical signal comes from the header of the packet, but I'm not sure how an optical switch obtains that information and generates an electrical signal.</p>
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<p>This is about Polchinski's eq(2.3.11).</p>
<p>It says that
$$Res_{z\rightarrow z_0}j(z)\mathcal A(z_0,\bar z_0)+\bar{Res}_{z\rightarrow z_0}\tilde j(\bar z)\mathcal A(z_0,\bar z_0)=\frac1{i\epsilon}\delta \mathcal A(z_0,\bar z_0)$$
from (2.3.10),
$$\oint_{\partial R}(jdz-\tilde jd\bar z)\mathcal A(\sigma_0)=\frac{2\pi}{i\epsilon}\delta\mathcal A(\sigma_0).$$
``Res'' is the coefficient of $\frac1{z-z_0}$. But how did we know that $j\sim\frac{1}{z-z_0}$, i.e., it has a pole at $z=z_0$? I know nothing about $j$ except that it is a function of $z$.</p>
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<p>I have several confusions regarding <a href="http://en.wikipedia.org/wiki/Escape_velocity" rel="nofollow">escape velocity</a>. I am sure I am missing something(s) obvious or maybe I am taught wrong. </p>
<ol>
<li><p>Lets say we throw an object of any mass at exactly escape velocity of earth calculated from $v^2=\frac{2GM}{r}$ which is almost $11 \text{km s}^{-1} $ but I am talking about exact escape speed. That ball initially has $KE=\frac{1}{2}mv^2$ and $PE=mgh$.
Wikipedia says and I quote </p>
<blockquote>
<p>In physics, escape velocity is the speed at which the kinetic energy plus the gravitational potential energy of an object is zero. </p>
</blockquote>
<p>How is that possible?</p></li>
<li><p>As $F=\frac{GmM}{r^2}$ no matter how much the particle travels away from earth's surface it will always be accelerated towards earth. With increasing $r$ the $F$ will decrease but it will never reach $0$. That means that there will be no point where the particle will stop and will continue to move with slower and slower <em>speed will never reaches zero</em>. Am I right?</p></li>
<li>A request: Please explain exactly what happens to particle's $KE$ and $PE$ at different points such as at $r=0$ and at $r=\infty$. </li>
</ol>
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<p>I am studying Quantum electrodynamics (QED) and the core part of this theory is the light-matter interaction (quantized light and matter). I referred many papers and books (Molecular quantum electrodynamics, Interaction Hamiltonian of quantum optics etc), and in all the places the interaction Hamiltonian is derived based on the dipole approximation or multipole. However if we are to find the interaction Hamiltonian for more realistic situation like quantumdots/nano wire to light interaction, what should be considered in the first place?</p>
<p>Thank you very much.</p>
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<p>I was having one of those obnoxious conversations with a friend where he was arguing that a fat skydiver would reach the ground faster than a skinny skydiver. To me it seemed as obvious that the world is round (that is: not very obvious unless you have some education).</p>
<p>We weren't getting anywhere so I punched in the data in Wolfram Alpha and I was surprised that indeed a fat person falls faster. I don't quite grasp why a heavier person with the same surface area has less <code>slowdown factor</code>. Can anybody explain this to me?</p>
<p><img src="http://i.stack.imgur.com/LjDM9.png" alt="enter image description here"></p>
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<p>I'm familiar with the ideal gas law $$PV=nRT$$ but I don't think it applies to liquids like water. If I'm wrong, please correct me! If I'm right, then what equation of state applies to liquids such as water?</p>
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<p>I learned that the wavefunction for the hydrogen atom can be solved analytically (we did the derivation in class), but that for more complicated atoms it is "impossible" to solve and that only approximations can be obtained.</p>
<p>Well, I'm not sure I like the vague wording with "impossible".</p>
<p>Basically, a hydrogen atom can be solved in terms of elementary functions. Elementary functions, as far as I know, are just arbitrarily defined (Is $\sqrt{}$ one? What about $\text{erfc}$?). For instance, I could say that $\sin{x}$ cannot be solved exactly except in a few special cases (x = some rational multiples of $\pi$).</p>
<p>To analogize, a general quintic equation cannot be "solved" exactly using elementary functions -- that is, unless you define a Bring radical as a new elementary function. To me, it's really arbitrary and all of these solutions come down to numerical, iterative refinement. Solved exactly vs solved approximately seems to be an ill-defined concept.</p>
<p>So what I am asking -- given enough computer time -- can I solve the Schrodinger equation of, say, Lithium, to whatever precision I like? Much as how I can solve $\sin{x}$ to however many decimal places I want? Or is there some <em>other</em> limit to such a calculation?</p>
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<p>Along the idea of frame-dragging; </p>
<p>Will the rotation of a black hole, which has some velocity <em>v</em> and angular momentum, influence its path in 3D space?</p>
<p>I've seen the fact that depending on the direction of the rotational velocity, the Kerr metric can give different revolution periods for orbiting masses. (e.g. as in our solar system)</p>
<p>So why (or why not) is this so? </p>
<p>How does the metric show that the direction of the velocity can influence orbital periods?</p>
| 8,544 |
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<p>i'm working on a paper about symmetron cosmology. symmetron is a scalar field that by its symmetry breaking can explain the dark energy.</p>
<p>the action is:
<img src="http://i.stack.imgur.com/5WGnl.jpg" alt="enter image description here"></p>
<p>ans A , V are assumed to be:</p>
<p><img src="http://i.stack.imgur.com/THXhv.jpg" alt="enter image description here">
<img src="http://i.stack.imgur.com/BJqVP.jpg" alt="enter image description here"></p>
<p>where M and <img src="http://i.stack.imgur.com/Saayr.jpg" alt="enter image description here"> are mass scales and lambda in a coupling. </p>
<p>somewhere in this paper (<a href="http://arxiv.org/abs/1107.2112" rel="nofollow">http://arxiv.org/abs/1107.2112</a>) "we get interested in the case where the field becomes tachyonic around current cosmic density,</p>
<p><img src="http://i.stack.imgur.com/qFKaH.jpg" alt="enter image description here"></p>
<p>my questions are:</p>
<blockquote>
<ol>
<li>what is the meaning of a field becomes tachyonic?</li>
<li>where the above approximation come from?</li>
</ol>
</blockquote>
<p>thank you so mach</p>
| 8,545 |
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<p>I am sure there is a simple explanation for my confusion, but I am a little puzzled:</p>
<p>We are dealing with two parallel electron cannons that each produces a straight beam of electrons. They are placed at a distance d next to each other, which means there will be a repulsive electrostatic force between the electrons of the two beams. Also, according to the Maxwell equations, every moving electron will produce a magnetic field which then exerts another force on the other beam's moving electrons (similar to the effect that two currents in wires will cause an attractive force between the wires). </p>
<p>However, if we change the reference frame such that we are moving at the same speed alongside the electrons, there will be no magnetic field and no Lorentz Force. We are only left with the electrostatic force.</p>
<p>Do you guys know what might be the fallacy?</p>
| 8,546 |
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<p>The Hamiltonian for one-dimensional Ising model is given by,
\begin{equation}
\mathcal{H} = -J\sum_{<ij>} S_iS_j; \quad i,j=1,2,...,N+1
\end{equation}
where $<ij>$ denotes that there is nearest neighbor approximation. The partition function is given by,
\begin{equation}
\mathcal{Z}=\sum_{\{S_i\}} e^{-\beta \mathcal{H}(S_i)}
\end{equation}
with $\beta = \frac{1}{k_BT}$, where $k_B$ is the Boltzmann constant and $T$ is the temperature.
Now my questions are:</p>
<p><strong>1. How to compute the partition function when $S_{N+1}=+1$ while other spins ($S_i$ for $i=1,2,...,N$) may take value $+1$ or $-1$?</strong></p>
<p><strong>2. How to compute the partition function when $S_{N+1}=-1$ while other spins ($S_i$ for $i=1,2,...,N$) may take value $+1$ or $-1$?</strong></p>
| 8,547 |
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<p>Whereas I have seen frogs fly in high magnetic fields and Sumo ringers stand on high-temperature superconductors it seems that these are not the only forms of levitation.</p>
<p>Recently a group of physicicsts managed to levitate coffee without a magnetic field (<a href="http://www.pnas.org/content/early/2013/07/10/1301860110.full.pdf+html" rel="nofollow">PNAS link</a>, only <a href="http://www.pnas.org/content/early/2013/07/10/1301860110.abstract" rel="nofollow">abstract</a> is free though). So while I am still trying to grasp how this is even possible my question is a bit more far fetched:</p>
<p>Can I levitate using this approach? </p>
| 8,548 |
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] |
<p>Does speed of writing data on an optical disk ( like DVD ) affect the speed of reading data from that disk ? even a little?</p>
| 8,549 |
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<p>Say we have a bar immersed in liquid, and that the bar in being rotated at a given constant angular speed. What force is exerted on the liquid by the faces of the bar moving through the liquid (so the force is tangent to the circular motion of the bar). My guess is that the expression involves the distance from the centre of the bar, $r$, and the angular velocity $\omega$. But I have no clue what exactly the relationship should be; can anybody help?</p>
| 8,550 |
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<p>The wave equation is often written in the form</p>
<p>$$(\partial^2_t-\Delta)u=0,$$ </p>
<p>involving the <a href="http://en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami_operator" rel="nofollow">Laplace-Beltrami operator</a> $\Delta$. However, the Laplace-Beltrami operator $\Delta$ is defined only in the presence of Riemannian metric $g$. Indeed, one can think of $\Delta u$ as of difference between $u$ and its mean value in the balls, and the notion of ball only makes sense in the presence of a metric.</p>
<p>What is the physical meaning of metric $g$ here? My guess is it's connected with the property of isotropy of space: in the vacuum, the waves should propagate uniformly in all directions, and this situation corresponds to Euclidean metric. The case of general Riemannian (= ‘perturbed Euclidean’) metric should then correspond to the propagation of waves in a medium of some sort, so that the property of ambient space isotropy is not satisfied, hence balls in it do not look like usual ‘Euclidean’ balls, and wave equation takes its general form $(\partial^2_t-\Delta)u=0$.</p>
<p>Is my guess true? If not, what is the correct interpretation of metric in the general wave equation?</p>
| 8,551 |
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<p>I'm trying to understand simple things about electricity reading allaboutcircuits.com web. <a href="http://www.allaboutcircuits.com/vol_1/chpt_9/2.html" rel="nofollow">This chapter</a> includes image which´s principle I don't understand. Here it is:
<img src="http://i.stack.imgur.com/gizaD.png" alt="enter image description here"></p>
<p>When supplying constant voltage to circuit where resistance and current change, how is it possible that change of resistance affects voltage on resistor?</p>
<p>Please help.</p>
| 8,552 |
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<p>I <a href="http://www.independent.co.uk/environment/sunflowers-to-clean-radioactive-soil-in-japan-2303086.html">read yesterday</a> that sunflowers were used and to cleanup radioactivity at Chernobyl and the Atomic Bomb sites in Japan and may be used as part of a campaign to clean up the Fukushima area. </p>
<p>But my questions are:</p>
<ol>
<li>How does a sunflower absorb radioactivity?</li>
<li>Is there a way to measure how well the sunflowers are working? </li>
<li>Can sunflowers planted in not particularly radioactive places concentrate otherwise harmless radiation?</li>
</ol>
<p><em>I was tempted to ask this on Skeptics.SE, but there's not much to be skeptical about, except whether or not <a href="http://www.rense.com/general93/sunf.htm">it's a good idea</a></em> </p>
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<p>According to impulse principle the impulse is the same as the change in the object's momentum:
$\bar I = \delta p$</p>
<p>Because the momentum can be calculated like this:
$\bar p = m\bar v$. If we solve speed from that: $\bar v = \frac{\bar p}m$</p>
<p>We can conclude that a certain impulse causes certain change in speed as long as the mass is a constant:
$\delta \bar v = \bar v_2 - \bar v_1 = \frac{\bar p_2}m - \frac{\bar p_1}m = \frac{\bar p_2 - \bar p_1}m = \frac{\delta \bar p}m = \frac{\bar I}m$</p>
<p>However, if we solve how much energy it takes to cause a certain change in speed it depends on the original speed:
$W = \delta E = E_2 - E_1 = \frac 1 2mv_2^2 - \frac 1 2mv_1^2 = \frac 1 2m(v_1 + \delta v)^2 - \frac 1 2mv_1^2 = mv_1\delta v + \frac 1 2m(\delta v)^2$</p>
<p>So my question is: Does the energy needed to generate an impulse of certain magnitude depend on the original speed of the object?</p>
| 8,554 |
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<p>I don't know how to say it, but in the <a href="http://www.youtube.com/watch?v=xE7xRgfPjAI">TV dominatrices</a> and the <a href="http://rads.stackoverflow.com/amzn/click/039333810X">popular science books</a> we see the <a href="http://en.wikipedia.org/wiki/String_theory">string theory</a> as "the best theory to explain everything", and as "the only game in town"... etc. And yet, in <a href="http://home.web.cern.ch/">CERN</a> -The largest physics laboratory-, they are just using the <a href="http://en.wikipedia.org/wiki/Standard_Model">Standard Model</a> (SM) as the only way to find the <a href="http://en.wikipedia.org/wiki/Theory_of_everything">theory of everything</a> (ToE), and sometime they call it: "<em>theory of almost everything</em>".</p>
<p>Also, they didn't even mention anything about string theory. (as if it doesn't exist!)</p>
<p>So my question is:</p>
<p><strong>Why?</strong> Isn't it easier for them to take the string theory as a Theory of Everything? </p>
<p>Also, the Standard Model is not able to explain the gravitational force from the SM equation! So, the SM is essentially <em>incomplete</em>! Why do they keep working on the SM? Am I missing something here?</p>
<p>-P.S.: I'm an engineer, not a physicist! So, I don't know much about the details.</p>
| 8,555 |
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] |
<p>Can you give me examples where the standard physics curriculum in high school or college is inaccurate? When did you realize you had been fed lies?</p>
| 8,556 |
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<p>Super string theory establishes that at each point of our 4 dimensional spacetime, there is attached a 6-dimensional Calabi-Yau (CY) manifold which encodes the geometry of the additional compactified spatial dimensions needed to formulate the theory. It is postulated that the size of these compactified dimensions is of the order of the Planck scale. </p>
<p>On the other hand, I've always understood that in quantum gravity, we are confronted with the big problem of wild quantum fluctuations -due to the Heisenberg principle- at the Planck scale.These fluctuations should lead to wild shifts and fluctuations in the metric coefficients and hence, they would make hard to properly talk in the terms of differential geometry (manifold, distance, metric, curvature) that are the terms in which we talk about CY manifolds.</p>
<p>I'd like to know how string theory solves this contradiction that has arisen due to my lack of knowledge of the full technical details of the theory. Is it possible that, the conformal invariance required for the string worldsheet during its propagation, prevents the pathological effects of these quantum fluctuations allowing us to talk in terms of geometry even at the Planck scale?</p>
| 8,557 |
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<p><a href="http://books.google.com/books?id=hFqxLgPol4EC&printsec=frontcover&dq=equations%20of%20states%20of%20fluids&hl=en&sa=X&ei=lvy5U6jOCe-X0AXd-4GQAg&ved=0CBwQ6AEwAA#v=onepage&q=equations%20of%20states%20of%20fluids&f=false" rel="nofollow">Sengers</a> describes the deviation function as the difference of the actual mixture property and as determined from a model. If this model happens to be the ideal gas model, then it is also called residual function.
<a href="http://search.proquest.com.ezproxy.library.wisc.edu/pqdtft/docview/304245257/655B848409BD438DPQ/1?accountid=465" rel="nofollow">Lemmon</a> describes excess property as the amount that a mixture property changes solely due to the mixing of the components. Does this mean that a deviation function is dependent on the type of model that you specify whereas excess property is independent of that? What are the differences between these two terms?</p>
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<p>So, I've been reading about the photoelectric effect for my modern physics class, and I was confused about how Ohm's law works in relation to it. Say we have a photoelectric apparatus that simply generates some current and then applies a stopping potential V to bring the current to zero. If we increase the intensity, this increases the current. However, then V = IR, and resistance remains the same, so shouldn't voltage increase? This is against the fact that intensity and stopping potential are independent, so does this mean that Ohm's law just fails in this case? Thanks so much.</p>
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<p>Autorefractors are being used by eye opticians for eye diagnosis. I searched internet for articles and wiki page as well but I wasn't satisfied. I am interested to know how do they actually work. How the machine is able to focus sharp on retina automatically. How do they figure out spherical/cylindrical aberration for human eye. I even asked optician as well but he tricked out of this position saying it does all happen automatically. LOL!</p>
<p>A detailed answer would be much appreciated. Please don't hesitate to write any technical details or mathematical derivations. I would love to know how these machines actually work.</p>
<p>Thanks in advance!</p>
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<p>Can someone outline the difference between <a href="http://en.wikipedia.org/wiki/Supersymmetry" rel="nofollow">supersymmetry</a> and <a href="http://en.wikipedia.org/wiki/Supersymmetric_quantum_mechanics" rel="nofollow">supersymmetric quantum mechanics</a>? I often hear the two used interchangeably but I'm almost certain they are not the same. </p>
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] |
<p>I recently saw this picture posted on <a href="https://twitter.com/makeusabrew/status/355995584190091265" rel="nofollow">Twitter</a> which shows a so-called <a href="http://en.wikipedia.org/wiki/Ice_spike" rel="nofollow">ice spike</a> rising from an ice cube tray.</p>
<p><img src="http://i.stack.imgur.com/DwBoZ.jpg" alt="Ice Spike from Ice Cube Tray"></p>
<p>I have read the Wikipedia page, but it doesn't mean much to me. My instinct was that it is caused by vibrations from the freezer setting up a resonant frequency on the surface of the water, and the spike has gradually "built up" as the water freezes. But I'm not a physicist, so I don't know if this makes sense. I'm ideally looking for a layman's explanation of why this happens.</p>
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<p>This is a technical (may be trivial?) question about this <a href="http://arxiv.org/abs/1208.1030" rel="nofollow">sigma scalar field advertised by Chamseddine and Connes</a> to improve the electroweak vacuum stability involved by the weak mass of the already found Higgs-like particle. </p>
<p><strong>Is it a scalar or a pseudo scalar field?</strong> </p>
<p>(motivation: I was looking for a possible connection with Majoron models)</p>
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<p>In Polchinski's string theory vol 1, p. 23, it is said
"It is a complicated question whether the bosonic string has any stable vacuum, and the answer is not known."</p>
<p>The book was published on 1998. What is the current answer of this question? Is bosonic string a realistic theory nowadays? </p>
<p>In addition, the ground state here is $$|0;k \rangle $$, not $$| \mathrm{vacuum} \rangle$$. Is that because the latter one is absolute nothing so we don't study?</p>
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<p>I am a first year and general student. I Am just curious about the vacuum or space.
I just don't get what actually space is.
Well, I know our scientists are still working on it but still I just want to know what is the accepted and best definition of space or a point in space that is the basic building block of this medium that allows matter to exist and its properties.</p>
<p>Is space accepted as absolute zero or we believe that space itself lies within another medium having higher dimension or something like this.</p>
<p>One more question when we say universe is expanding, in reference to what we are saying it is expanding?
I mean is space itself stretching (which means space in between basic building blocks of matter is increasing, and if it is like this then how can we measure it as even our standard of measurement will be effected and hence net change will be zero)
<strong>OR</strong> just heavenly bodies are going away from each other but then what is the reference ?
Let us suppose it is Milkyway, but is Milkyway itself not moving and gaining velocity in the direction of the objective (the heavenly body which we say is going far from us)?
Hence net change in velocity here should be zero?</p>
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<p>I have two questions, which are connected with each other.</p>
<p><strong>The first question</strong>.</p>
<p>In a classical relativistic (SRT) case for one particle can be defined (in a reason of "antisymmetric" nature of angular momentum) 4-tensor
$$
L_{\alpha \beta} = x_{\alpha }p_{\beta} - x_{\beta }p_{\alpha } = (\mathbf L, \mathbf G),
$$
where $\mathbf L = [\mathbf r \times \mathbf p ], \quad \mathbf G = \frac{E}{c}\mathbf r - ct\mathbf p $ are angular momentum and center of energy vectors. It is called angular momentum tensor.</p>
<p>In a case of many particles it's convenient to introduce Pauli-Lubanski 4-vector of spin:
$$
S_{\nu} = \frac{1}{2}\varepsilon_{\nu \alpha \beta \gamma }L^{\alpha \beta}U^{\gamma},
$$
where $U^{\gamma}$ is summary 4-speed. The physical meaning of this vector is representation of own angular momentum of particles (without summand connected with motion of system as one body):</p>
<p>$$
S_{0} = (\mathbf L \cdot \mathbf U), \quad \mathbf S = \mathbf L U_{0} - [\mathbf G \times \mathbf U] = \frac{E}{M}\left( \mathbf L - [\mathbf R_{E} \times \mathbf P]\right).
$$ </p>
<p>By the way, for one particle in classical case
$$
\mathbf S = \mathbf L - [\mathbf R_{E}\times \mathbf P ] = [\mathbf R_{E}\times \mathbf P ] - [\mathbf R_{E}\times \mathbf P ] = 0 .
$$
For many particles in general case $\mathbf L $ include orbital angular momentum, so $\mathbf L - [\mathbf R_{E}\times \mathbf P ] \neq 0$.</p>
<p>There, if we introduce an operator representation, it will be easy to show, that
$$
[\hat {L}_{x}, \hat {L}_{y}] = [\hat {S}_{x}, \hat {S}_{y}] = i\hat {L}_{z},
$$
where equality of first and second commutators is possible in the rest frame (add: i.e., for a case $\hat {\mathbf P}| \Psi_{\mathbf P = 0} \rangle = 0|\Psi_{\mathbf P = 0} \rangle $). That, after simple transformations, leads us to the discrete spectra of eigenvalues of $\hat {L}_{z}$ (or $\hat {s}_{z}$ in a case of the rest frame):
$$
m = s, s - 1, ..., -s,
$$ </p>
<p>where s is the "full momentum" (or spin in the rest frame).</p>
<p>So, the question: can we talk about classical analogue of spin in a case of quantum mechanics, when we analyze the quantum system (not one particle)? In a classical case the analogue of spin is an own angular momentum determined above.</p>
<p><strong>The second question</strong>.</p>
<p>In a field theory it is well known that angular-spin tensor, which is a current of a Lorentz-symmetry of Lagrangian (Noether formalism). Does it connect with a spin of the quantum field in terms of first question? If the answer is positive, there is another question: has Belifante procedure an analogue of a quantum case?</p>
| 8,566 |
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<p>I can see that $a_{0}$ is not an independent field and Gauss law is a constraint on the theory arising from field equations. But, I don't get the geometrical picture.</p>
<p>Let $A$ be the space of all field configurations and let $G$ be the group of gauge transformations. Then, the actual configuration space $C=A/G$ where we consider $A$ modulo the action of gauge transformations. So, first of all, why don't we do our variational calculus to obtain field equations on configuration space? It has always been taught to me that I should consider variation of "paths" in configuration space only while applying Hamilton's principle. Instead, we first obtain field equations regarding $A$ as our configuration space and then "project" them perpendicular to gauge orbits using Gauss law. Are these two methods equivalent?</p>
<p>Moreover, I don't understand how Gauss law really projects my fields orthogonal to gauge orbits. According to Manton's book (see the relevant pages scanned below) on topological solitons, he starts with a variation of the fields $\phi$ and $a$ in a small interval of time. Then, he claims that a "naive contribution to the kinetic energy" would be some expression involving these variations (see below) which is non-zero for infinitesimal gauge transformations also. But, when I look at the expression of kinetic energy (which is $T=\int(\frac{1}{2}e_{i}e_{i}+\frac{1}{2}\bar{D_{0}\phi}D_{0}\phi)d^{d}x$) in terms of fields, then I find that the expression is gauge invariant. I don't see why the particular expression</p>
<p>$\frac{1}{2}\int\frac{1}{(\delta t)^{2}}(\delta\vec{a}.\delta\vec{a}+\bar{\delta\phi}\delta\phi)d^{d}x$ </p>
<p>be the change in kinetic energy.</p>
<p>In short, I want to understand this geometric insight into Gauss law whose details are attached below.</p>
<p>Also, I wonder if this Gauss law has a relation to the Gauss law I learned while doing high school electrodynamics. Any information or resource on this would be very helpful.</p>
<p><img src="http://i.stack.imgur.com/3rP4Y.jpg" alt="enter image description here"></p>
<p><img src="http://i.stack.imgur.com/nRlWL.jpg" alt="enter image description here"></p>
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<p>So if light travels from one media to another with a different refraction index, what may happen happen? Refraction, reflection or total reflection? I am quite confused as to the differences between these three concepts. Does 1 of these always happen in this situation? What exactly is total reflection and how/why does it occur?</p>
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<p>When a permanent magnet attracts some object, lets say a steel ball, energy is converted into for instance kinetic energy and heat when attraction happens, and they eventually collide. Does this imply that energy is drawn from the magnetic field and the magnet is depleted, making it weaker and weaker for each magnetic attraction performed?</p>
<p>(if the answer requires Quantum Mechanical explanations, please elaborate :))</p>
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<h1>Setup</h1>
<p>In the typical treatment of time-dependent perturbation theory in quantum mechanics, one arrives at the set of equations</p>
<p>$$
i \dot{a}^{(r + 1)}_m(t) = \sum_n \langle m |H_1(t)|n \rangle e^{i \omega_{mn} t} a^{(r)}_m(t)
$$
for integers $r > 1$ and $\dot{a}^{(0)}_m(t) = 0$. Here, the Hamilton is $H(t) = H_0 + \lambda H_1(t)$, the states $\{|n\rangle\}$ satisfy $H_0 |n\rangle = \varepsilon_n |n\rangle$ and $\omega_{mn} = \varepsilon_m - \varepsilon_n$. The expansion coefficients are defined by
$$
a_m(t) = \sum_{r = 0}^{\infty} \lambda^r a^{(r)}_m(t)
$$
where $a_m(t)$ are the coefficients in the expansion
$$
|\psi(t)\rangle = \sum_n a_n(t) e^{-i \varepsilon_n t} |n\rangle
$$
where
$$
i \frac{d |\psi(t)\rangle}{dt} = H(t) |\psi(t)\rangle
$$
Note then that $a^{(0)}_m(t) = a^{(0)}_m(0) = a^{(0)}_m$, say.</p>
<h1>Question</h1>
<p>The literate then claims that if $|\psi(0)\rangle$ is known, then $a^{(0)}_m = \langle m |\psi(0)\rangle$.</p>
<p>Does this really follow from the above? I feel an additional assumption (perhaps a boundary condition on $t$) is required to conclude the association of the initial state with the zeroth order coefficient. (Though obviously we have $a_m(0) = \langle m |\psi(0)\rangle).$</p>
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<p>The orbit of satellites around Earth eventually decays, or so I read. This is typically caused either by atmospheric drag, or by tides. I would assume most satellites have a limited service life in orbit. Hence the question - What is the typical orbital life of a satellite? How long before it's successor must be launched into orbit ?</p>
<p>EDIT:
For instance, a weather satellite</p>
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<p>I think my pure-math head is messing with me on the question below: my physics and CS friends both seemed to think it was a simple computational thing, and my program says the method works, but now I've confused myself about rotations vs. angular velocities.</p>
<p>The problem: you have data from gyroscopes on a rigid body that give you the angular speeds of the object around three orthogonal axes through the body. These measured angular speeds are constant, call them $\omega_x$, $\omega_y$, and $\omega_z$.</p>
<p>Sine the angular speeds are constant, the axis of rotation of the object is also constant, and it's not too much trouble to find it by considering the combined angular velocity vector $\Omega$ = ($\omega_x$,$\omega_y$,$\omega_z$). </p>
<p>I am then confused when I think about this in terms of rotations. There's a unique axis of rotation because every composition of rotations is equivalent to a single rotation about some axis. It seems, then that you should be able to determine this axis by determining the eigenvectors of the combined matrix ABC, where A, B, C each represent the rotation about one of the axes.</p>
<p>But of course, rotations in general don't commute. I'm forced to conclude that what makes the unique determination of an axis work is that infinitesimal rotations commute.</p>
<p>So, if I'm right about this, my question is -- how can I see that infinitesimal rotations commute? My intuition fails me here.</p>
<p>If they don't - what's the link between the rotation and angular velocity vector way of looking at this that removes the usual problems dealing with rotations that don't commute? </p>
| 8,571 |
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<p>Let's imagine a diesel engine's cylinder head would be heated by external means, to some hundred degree celsius, and the fuel would be replaced by water.</p>
<p>We've made modifications so that water would'nt destroy the fuel system and the cylinders would sustain the high temperature.</p>
<p>Now the injector would squirt some water into compressed hot air. It would instantly vaporize and the steam expands with the moving piston. </p>
<p>Would this give a working engine? What degree of efficiency can be expected compared to internal combustion?</p>
| 8,572 |
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] |
<p>Learning about the fourier transform and its connection to filtering/convolution got me curious about naturally occurring filters.</p>
<p>Why/how is it that brick walls naturally act as a low-pass filter (which requires something as seemingly complicated as convolution with the <code>sinc</code> function) to sound waves?</p>
| 8,573 |
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<p>Imagine that two electrons interact by exchanging a virtual photon.</p>
<p>I know that the total energy and linear momentum of the two electrons is conserved by the interaction.</p>
<p>Is the total (orbital) angular momentum of the electrons conserved as well?</p>
<p>If the photon transfers energy and momentum between the electrons does it also transfer angular momentum between them?</p>
| 8,574 |
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<p>This question is the continuation of <a href="http://physics.stackexchange.com/questions/122980/lsz-reduction-theorem-derivation-in-weinberg-qft">this one</a>.</p>
<p>For simplicity, let's use $(1)$ from the linked question (it is called n-point Green function and in particle case coincides with internal diagram),
$$
G(p_{1},...,p_{n}) = \int d^{4}x_{1}...d^{4}x_{n}e^{-i\prod_{i =1}^{n}p_{j}x_{j}} \langle |\hat {T}\left( \hat {O}_{l}(x_{1})\hat {A}_{2}(x_{2})...\hat A_{n}(x_{n})\right) |\rangle , \quad (1)
$$
(in accordance with the linked question $\hat {O}_{l}(x)$ transforms as some free field $\hat {\Psi}_{l}(x)$ under the irreducible rep of the Lorentz group, and the other operators are local functions of the free fields) </p>
<p>for the case when $\hat {A}_{2}(x) = \hat {O}_{l'}^{\dagger}(x)$ and the other operators $\hat {A}_{i}(x_{i})$ is equal to the unity. Then it's not hard to show that
$$
G (p_{1}, p_{2}) \to \frac{-2i|N|^{2}\sqrt{\mathbf p_{1}^{2} + m^{2}} }{p_{1}^{2} - m^{2} - i\varepsilon}\sum_{\sigma}u^{\sigma}_{l}(\mathbf p_{1})(u_{l'}^{\sigma}(\mathbf p_{1}))^{*}\delta (p_{1} + p_{2})(2 \pi )^{4}, \qquad (2)
$$
which coincide with propagator of the free field $\langle |\hat {T}\left(\hat {\Psi}_{l}(x) \hat {\Psi}_{l'}(y)\right)|\rangle$ in the momentum representation excepting $|N|^{2}$ factor. </p>
<p>For the simplicity lets assume that $\hat {O}_{l}(x) = \hat {O}(x)$, i.e. we working with the scalar operator. Then it's not hard to show that $|N|^{2} = (2 \pi )^{3}\left|\langle \mathbf p| \hat {O}(0) |0\rangle \right|^{2} = \frac{Z}{2\sqrt{\mathbf p^{2} + m^{2}}}$ (this result may be easily compared with the paragraph about renormalization of the field strength of Peskin and Schroeder QFT; the result above coincides with the result from Peskin QFT up to the normalization condition). </p>
<p>So, the question: why in the LSZ formalism $|N|^{2} \neq \frac{1}{2\sqrt{\mathbf p^{2} + m^{2}}}$ as for case of free theory? I.e., why $Z \neq 1$?</p>
| 8,575 |
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<p>In quantum computation there are several principal quantum gates that have corresponding matrix representations. One of these is the Z gate, whose matrix is $\left[\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}\right]$.</p>
<p>... anyway, I've found the eigenvalues (equal to +1, -1) using the characteristic equation, and used them to derive the corresponding eigenvectors, which come together quite nicely in a 2×2 matrix $\left[\begin{smallmatrix} 1 & 0 \\ 0 & 1\end{smallmatrix}\right]$, equal to the identity. So, in diagonalizing this matrix, I find that the diagonal matrix $D$ is the same matrix as the one for gate $Z$.</p>
<p>... the next step and where I'm stuck is to find the corresponding point on the Bloch sphere for this gate. In order to do that, I need to compute how to take the diagonalized matrix call it $D_z$ and derive two things: <strong>(a)</strong> its diagonal representation $| 0 \rangle \langle 0 | - | 1 \rangle \langle 1 |$, and <strong>(b)</strong> the normalized eigenvalues $a, b$ for $Z$, where $Z = a|0\rangle + b|1\rangle$ and must be orthonormal i.e. $a^2 + b^2 = 1$. The $a$ and $b$ terms correspond to the probabilities of measuring 0 or 1 for the state, respectively (I think).</p>
<p>After I have the values for $a$ and $b$, I'll be able to locate the gate on the Bloch sphere because the calculation of its coordinates on the sphere is straightforward: $a = \cos(\theta / 2)$, and $b = e^{i\phi}\sin(\theta/2)$. </p>
| 8,576 |
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<p>The tachyonic string mode in perturbative bosonic string theory indicates that the "vacuum", flat Minkowski $\mathbb{R}^{25,1}$, is not really a vacuum. What is conjectured about tachyon condensation in this theory? Do we expect the theory to have a vacuum? Is there any way the condensate might generate fermions dynamically?</p>
<p>[Edit] Since Chris Gerig asked for more background: Tachyons, sadly for Star Trek writers, are typically an indication that the state you think is the vacuum of your system is, in fact, not the vacuum. This is because you can act on the ground state with a tachyon creation operator and get a new state with lower energy. For example, the Higgs potential is $$V(\phi) = \frac{1}{2}\lambda (|\phi|^2 - c^2)^2 = -\lambda c^2 \phi^2 + \frac{\lambda}{2} \phi^4 + const$$ If you do perturbation theory around $\phi_0 = 0$ instead of around one of the true minimum $\phi_0 = C$, with $|C| = \mu^2$ , you'll find the creation operators in $\phi_{pert} = \phi - \phi_0$ create tachyons, with negative mass-squared $m^2 = -\lambda c^2$. Create enough of these tachyons, and you'll turn the state $|\phi_0 = 0\rangle$ which you thought was the vacuum into one of the true vacua $|\phi= C\rangle$.</p>
<p>Bosonic string theory in 26d has tachons: the most basic closed and open string excitations, created by vertex operators with no derivatives. So it's a natural question to wonder about: what state do we get if we add tachyons to the false perturbative vacuum? Does this process converge? This is a pretty hard question to answer, since in bosonic string theory we don't have SUSY-protected quantities that we can compute to check our predictions. Which is why I asked what had been conjectured.</p>
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<p>As many cosmologists have pointed out, the universe is extremely fine-tuned for life in so many respects, and life, especially intelligent life, is extremely unlikely to emerge. The strong anthropic principle tries to explain it statistically by an extremely huge multiverse ensemble. However, are there alternative explanations? One is an intelligent designer, but given how hard it is to create intelligent life, It would probably have to engage in significant trial-and-error by simulating many prototype universes before stumbling upon one that works. This is probably true no matter how intelligent It is. So, the multiverse reemerges through the back door. This still applies even if the Designer is a blind naturalistic algorithm. Can you think up of some mechanism for intelligent life to emerge which doesn't require a huge multiverse ensemble?</p>
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<p>Sometimes when I'm filling up a container with water from the tap, bubbles will form on the top of the water. They look a little like soap sud bubbles, but there's no soap suds present. I notice, sometimes this will happen more than others.</p>
<p>Why does water sometimes form bubbles when I use the tap to fill containers?</p>
<p>Are there certain materials that will make them more likely to form, eg plastic, or glass?</p>
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<p>What is the difference between <a href="http://en.wikipedia.org/wiki/Proper_time" rel="nofollow">proper time</a> and the observer time?</p>
<p>Whilst thinking about Black holes, when we see the Schwarzschild metric </p>
<p>$$c^2\tau ^2 = \left ( 1 - \frac{r_{s}}{r} \right )c^2t^2 - \frac{r^2}{1-\frac{r_{s}}{r}} - r^2d\Omega ^2,$$</p>
<p>and compare it with Einstein's special relativity equation,
$$c^2\tau ^2 = c^2t^2 - x^2,$$ </p>
<p>we find that at the horizon of a black hole or at the schwarzschild radius for any <strong>infinitesimally small time spend by any object at the horizon the observers time tends to infinite</strong></p>
<p>Why and how is this so? it doesnt make sense whilst trying to imagine it?</p>
| 8,580 |
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<p>When we connect 2 light bulbs to a voltage source in parallel, the equivalent Resistance of the bulbs is R/2 (1/Req = 1/R + 1/R). If we connect these lights in parallel we have Req = 2R; </p>
<p>I= V/Req is valid for both circuits, and P=I*V (with V being the same for circuit 1 and 2)
This means the parallel circuit produces more light.</p>
<p>How does this change for a circuit with a current source instead of a voltage source?
Which of these 2 is brightest with a current source?</p>
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<p>I'm using modal decomposition to predict the steady state response of a beam structure to harmonic loading. The structure itself is very lightly damped, but we know from experiments that the aerodynamic drag force as it moves through the air is very significant.</p>
<p>I have the mass and stiffness matrices for the beam ($\boldsymbol{[M]}$ and $\boldsymbol{[K]}$) and so I am able to obtain the eigenvectors and eigenvalues ($\boldsymbol{[V]}$ and $\boldsymbol{[D]}$). The eigenvalues yields a vector containing the natural frequencies of the system (the square root of the diagonal entries of $\boldsymbol{[D]}$, I'll call it $\boldsymbol{\omega}$.</p>
<p>The system is excited by a force vector in the global reference frame $\boldsymbol{F_0}$ at an excitation frequency $\omega_F$.</p>
<p>There is a rotation matrix which rotates the beams from the local to the global reference frame $\boldsymbol{[R]}$. </p>
<p>I have successfully used the procedure to calculate the steady state response without aerodynamic damping:</p>
<ul>
<li>Get eigenvalues and eigenvectors $\boldsymbol{[V]}$ and $\boldsymbol{[D]}$</li>
<li>Get generalised force vector $\boldsymbol{Q}=\boldsymbol{[V]}^T \boldsymbol{F_0}$</li>
<li>Calculate modal participation factor vector $\boldsymbol{q}$. The individual elements are calculated as follows:</li>
</ul>
<p>$$q_i=\frac{Q_i}{{\omega_i}^2}\frac{1}{\sqrt{[(1-(\omega_F/\omega_i)^2)^2+(2\zeta_i\omega_F/\omega_i)^2]}}$$</p>
<ul>
<li>Calculate response from $\boldsymbol{x}=\boldsymbol{[V]q}$</li>
</ul>
<p>Of course, this drastically overpredicts the amplitude of the oscillations because there is no aerodynamic damping. I've found a few examples for single d.o.f. systems in which the energy lost from the system is calculated (by integrating the drag force x the velocity). The integration results in the following equation for energy loss:
$$\Delta E=\frac{8}{3}\frac{1}{2}\rho A C_D \omega^2 X^3$$
This can then be converted to an equivalent damping coefficient as follows:
$$C_{eq}=\frac{\Delta E}{\pi \omega X^2}$$
Which can be converted to a damping coefficient by doing:
$$\zeta=\frac{C_{eq}}{2 \omega_n}$$
I've modelled a simple spring-mass system with air resistance and it works fine (I have to iterate to find the final position because the amplitude calculation requires knowledge of the damping which requires knowledge of the amplitude so there is a circular reference, but that's no biggy!)</p>
<p>Could someone please explain the equivalent for a multi-d.o.f. system? My initial forays have been unsuccessful, with much higher drag coefficients than would really occur.</p>
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<p>The Sun's magnetic field goes all the way out to the heliopause which is a very long distance to travel and magnetic fields consist of virtual photons so how long do these virtual photons exist for to reach the heliosphere?</p>
<p>Edit
My problem is that if virtual photons travel at light speed then they must live for hours in some magnetic fields, which would not make them very virtual.
If virtual photons travel faster than light then by turning a magnetic field on or off could be used for FTL comunication. Where have I gone wrong?</p>
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<p>If there are two 10 x 2 x 1 foot rectangles in space and they are lined up so if they hit each other there will be no spots that are not hit in the front of the rectangle. Then they are pushed forward with the same exact same force, (this is space and there is no resistance) and they hit, will the tiny shockwaves of sound moving through them hit each other (they are exactly the same kind of sound wave), will the sound hit it's opposite would they cancel themselves into heat on contact?</p>
<p>I know that the sound does not cancel itself out. You cannot say the wave of atoms hitting each other collides with a wave that is exactly the same and stops both of them in a word. Can you?</p>
| 8,584 |
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<p>A DC solenoid creates a magnetic field B.
A highly permeability magnetic core is added to solenoid to increase B.</p>
<p>Now, there is change in flux, due to the increase in magnetic field. What would be the result for the circuit? Would more power drawn support the new stronger field? </p>
<p>Or P would be the same before/after?</p>
| 8,585 |
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<p>I don't know how to ask this more clearly than in the title.</p>
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] |
<p>To maintain the surface tension which formed our original bubble (in order to keep the bubble from breaking), we may change the temperature/pressure of air on both sides of the bubble varyingly, with different rates.</p>
<p>Is such a configuration possible? Examples of any such system?</p>
| 8,587 |
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<p>Whenever I read about the curvature of spacetime as an explanation for gravity, I see pictures of a sheet (spacetime) with various masses indenting the sheet to form "gravity wells." Objects which are gravitationally attracted are said to roll down the curved sheet of spacetime into the gravity well. This is troubling to me, because, in-order for objects on the locally slanted spacetime sheet to accelerate, gravity must be assumed. Therefore I ask; does the explanation of gravity as the curvature of spacetime assume gravity? If yes, what is the point of the theory? If No, what am I missing?</p>
| 13 |
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<p>In three dimensions, the density of states of a free electron is the square root of the energy of the electron. Can somebody explain the relationship between this dependence and the shape/formation of sub-bands in k-space? In other words, how does the shape and quantity of bands always lead to this same dependence across various materials?</p>
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<p>I am trying to figure out where is the focal point and where is the image. I read some information online about the point where you see the image is the focal point, but however, my supervisor mentioned the point where you see the image is on "point 2" on the drawing. But as I remembered from school, the "point one" is where the retina is in order to have a good vision in our eyes.</p>
<p>The main purpose I am asking this because I want to get as much light as possible to a small area, and me and him were trying to figure out <strong>which point has the <em>maximum light</em></strong>. I managed to obtain a focused image on a sheet of paper when doing the experiment (I believe that's point 1). Please help, Thanks a lot!!</p>
<p>Basically, could you tell me which point has the maximum intensity of light and if possible, point one or two is the image is. Thanks
<img src="http://i.stack.imgur.com/cxcp0.jpg" alt="Convex lens, determine where is maximum light"></p>
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] |
<p>I was recently reading a popular science book called <a href="http://www.amazon.co.uk/The-Canon-Beautiful-Basics-Science/dp/0571239722" rel="nofollow">The Canon - The Beautiful Basics of Science</a> by Natalie Angier, and it talks about subatomic particles like protons, neutrons and electrons in chapter 3. I came across this section on subatomic charges that made me wonder about the nature of the positive and negative charges that we associate with protons and electrons respectively.</p>
<blockquote>
<p>When you talk about a fully charged battery, you probably have in mind
a battery loaded with a stored source of energy that you can slip into
the compartment of your digital camera to take many exciting closeups
of flowers. In saying that the proton and electron are charged
particles while the neutron is not, however, doesn't mean that the
proton and electron are little batteries of energy compared to the
neutron. A particles's charge is not a measure of the particles's
energy content. Instead, the definition is almost circular. <strong>A particle
is deemed charged by its capacity to attract or repel other charged
particles.</strong></p>
</blockquote>
<p>I found this definition/description a bit lacking, and I still don't grasp the nature of a "subatomic charge", or what do physicists mean when they say that a proton is positively charged and electron is negatively charged?</p>
| 8,590 |
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<p>I am working on a point charge simulator, and I was wondering what forces can affect point charges (assuming that they operate in a closed system, with no externally generated light, and initial velocities). I have thought of the following "classes" of forces so far:</p>
<ul>
<li>Electric Field</li>
<li>Magnetic Field</li>
<li>Gravitational Field</li>
<li>Linear Momentum</li>
<li>Kinetic Energy</li>
</ul>
<p>Thanks!</p>
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<p>On bicycles.se a question came up about whether one cyclist drafting another causes the lead cyclist to be slowed down. A contributor suggested that the opposite might be the case, that the leading cyclist would be 'helped' too. Clearly, in the real world of cycling there are winds, potholes, traffic and other variables, plus the lead cyclist would not necessarily notice a small bit of extra help. Therefore 'probably not' is not the answer I am looking for, a bit of theory would help.</p>
<p>Another 'drafting situation' happened at Monza today in the qualifying for the F1. The Ferrari team had one of their cars give the other an aerodynamic tow along one of the straights, this helped their number 1 driver get to 4th on the grid, a position he would not necessarily have achieved otherwise, in a race where fractions of a second do matter. If you can better visualize F1 cars than bicycles, then today at Monza is another situation where drafting went on. We know it helps the guy behind, but does it also help the guy up front?</p>
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<p>I have two vectors $\vec{A}$ and $\vec{B}$ and I need to find the x- and y-components of $\vec{C} = \vec{A} + \vec{B}$. Here's what I have so far:</p>
<p>$$|\vec{A}| = 50.0 \mathrm{m}, \theta = -20.0^\circ$$
$$|\vec{B}| = 70.0 \mathrm{m}, \theta = 50.0^\circ$$</p>
<p>$$C_x = |\vec{A}| \cos (\theta) + |\vec{B}| \sin (\theta)$$
$$C_y = |\vec{A}| \sin (\theta) + |\vec{B}| \sin (\theta)$$</p>
<p>Now, according to my professor, this is the solution for C_x:</p>
<p>$$C_x = 50.0 \cos(-20.0^\circ) + 70.0 \cos(50.0%^\circ)$$
$$C_x = 46.98^\circ + 45.0^\circ$$
$$C_x = 92.0^\circ$$</p>
<p>What I'm wondering is how the rounding works here. I got $46.98$ for $50.0 \cos(-20.0^\circ)$ and $44.99$ for $70.0 \cos(50.0^\circ)$. Why is $44.99$ rounded to $45.0$? If anything, shouldn't it be rounded to $45.00$? What am I missing here? </p>
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<p>Suppose there is a material that interacts with 50% of high-energy neutrinos. To what temperature such material will heat up naturally here on Earth? What power can be extracted from it?</p>
| 8,594 |
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<p>If Lagrangian of the motion is </p>
<p>$$\mathcal{L}=\frac{1}{2}m\left(a^2\dot\phi^2+a^2\dot\theta^2\sin^2\phi\right)+mga\cos\phi,$$ </p>
<p>how can I show that total mechanical energy is conserved? I've read this: </p>
<blockquote>
<p><em>If the time $t$, does not appear [explicitly] in Lagrangian $\mathcal{L}$, then the Hamiltonian $\mathcal{H}$ is conserved. This is the energy conservation unless the potential energy depends on velocity.</em></p>
</blockquote>
<p>Potential energy of this motion doesn't depend on velocity. Also, $t$ does not appear explicitly in Lagrangian. Is this enough to say that total mechanical energy is conserved?</p>
| 8,595 |
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<p>A speedway turn, with radius of curvature $R$, is banked at an angle $\theta$ above the horizontal. If there is a coefficient of static friction, $\mu_{s}$ between the tires and the track, what are the maximum and minimum speeds at which turn can be taken? </p>
<p>My solution:
Since coefficient of kinetic friction, $\mu_{k}$ $\leq$ $\mu_{s}$ , I have shown that maximum speed $$ v = \sqrt {\frac{R\, g \sin\theta+ \mu_{s}\cos\theta} {\cos\theta-\mu_{s} \sin\theta}}$$, where $g$ = gravitational constant</p>
<p>Any advice on finding the minimum speed? </p>
<p>Thank you!</p>
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<p>I recently read (sorry but I don’t have a reference) that interference is not only about destructive and constructive interference but moving energy from destructive to constructive regions according to conservation of energy. The place where this really bothers me is when light is being transmitted through glass. As light moves through the glass there is forward scattering of light where there is constructive interference between the primary and secondary waves; so light propagates forward. On the other hand, the light that back-scatters (or side scatters as well) destructively interferes with the primary waves (I am assuming that the light has already passed the surface atoms so reflection has already been accounted for). Said another way, I can set up a thin film such that the reflectance is zero (R = 0) and transmittance is one (T = 1), don’t I need energy in the waves to cause destructive interference to begin with? So how does interference account for this energy exchange?</p>
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<p>This is a question of topological insulator.</p>
<p>Liang Fu and C. L. Kane proposed a method to judge whether an inversion symmetric insulator is a topological insulator or not in their article(<strong>L. Fu and C.L. Kane, Phys. Rev. B 76, 045302 (2007)</strong>).
The method is just to determine the parity of the occupied band eigenstates at the eight or four(in two dimensions) time-reversal invariant momenta $\Gamma_i$in the Brillouin zone. The Z2 invariant is determined by quantity $${{\delta }_{i}}=\prod\limits_{m=1}^{N}{{{\xi }_{2m}}\left( {{\Gamma }_{i}} \right)}$$Where ${{\xi _{2m}}\left( {{\Gamma _i}} \right)}
$is the parity eigenvalue of 2m band at $\Gamma_i$ point.</p>
<p>My question is how to determine the parity of band state at these points from first principle band calculation(like Wien2K band calculation)?</p>
| 8,598 |
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<p>Given the following 16 matrix multiplications of the Dirac gamma matrices
\begin{align}
G_{\mu\nu}
=
\dfrac{1}{2}
\begin{pmatrix}
I && \gamma_{0} && i\gamma_{123} && i\gamma_{0123}
\\
i\gamma_{23} && i\gamma_{023} && i\gamma_{1} && \gamma_{01}
\\
i\gamma_{31} && i\gamma_{031} && i\gamma_{2} && \gamma_{02}
\\
i\gamma_{12} && i\gamma_{012} && i\gamma_{3} && \gamma_{03}
\end{pmatrix},
\end{align}
where $\gamma_{01} = \gamma_{0}\gamma_{1}$:</p>
<p>What is
\begin{align}
\sum_{\mu=0}^{3}\sum_{\nu=0}^{3}
\langle G_{\mu\nu} \rangle^2
=
?
\end{align}
The Bra-Ket notation is used in this question:
\begin{align}
\langle G_{\mu\nu} \rangle
=
\langle \Psi | G_{\mu\nu} | \Psi \rangle
,
\end{align}
where $\Psi$ is the Dirac bi-spinor of four complex components.</p>
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<p>I've heard that special relativity makes the concept of magnetic fields irrelevant, replacing them with relativistic effects between charges moving in different velocity frames. Is this true? If so, how does this work?</p>
<p>[Temporary note, to be deleted later: This is my first attempt to compose a self-answered tutorial question, which to my surprise Stack Exchange seems to encourage rather strongly. So, here goes...]</p>
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<p>In the Standard Model, the Higgs mass doesn't really have any theoretical constraints. It could have basically any value and nothing 'breaks'.</p>
<p>However, in MSSM models, we often see the tree level constraint
$$m_h < m_z \cos(2\beta)$$
which is modified by 1-loop stop corrections.</p>
<p>What is happening differently in the SUSY EWSB to give this constraint? Is it because of SUSY or would you also get this by having two Higgs doublets with different VEVs?</p>
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<p>Inspired by <a href="http://physics.stackexchange.com/q/6419/3816">this question</a> and specifically <a href="http://physics.stackexchange.com/questions/6419/electric-engine-transmission/6425#6425">this answer to it</a>.</p>
<p>From my experience there's always some very specific limit to how much torque an electric motor can output. For example, an electric drill will often have a manually switched mechanical transmission - if one needs to drill some relatively weak materials (like wood) he will use the setting that outputs lets torque at higher RPM and if one want to drill steel or mix cement mortar he will use the settings for more torque at lower RPM.</p>
<p>The relation between RPM and torque is more or less clear if one imagines a set of two gears of different diameters and thinks that their radiuses are lever arms - three times more RPM automatically induces three times lower torque and vice versa.</p>
<p>But where does the limit to any given electric motor torque come from? Say I have some specific motor right now in front of me and it can output 40 Newton-meters at 500 RPM. Why exactly 40 and not more?</p>
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<p>I am trying to learn about radioactive energies and wonder if, because these also seem to come under the topic of radiation, can these energies become electromagnetic. I'm pretty much a beginner, so if you could keep the answer pretty simple, it would help.</p>
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<p>In Arnold's <em>Mathematical Methods of Classical Mechanics,</em> he derives the Hamilton-Jacobi equation (HJE) using a generating function $S_1(Q, q)$ to get</p>
<p>$$
H\left(\frac{\partial S_1(Q, q)}{\partial q}, q, t \right) ~=~ K(Q, t).
$$</p>
<p>However, this is different from what I've seen in other physics texts. For example, Goldstein uses the generating function $S_2(q, P, t)$ to get the equation</p>
<p>$$
H\left(\frac{\partial S_2(q, P, t)}{\partial q}, q, t\right) ~=~ - \frac{\partial S_2(q, P, t)}{\partial t}.
$$</p>
<p>Why is there this difference? Are the two equations saying the same thing? </p>
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<p>I was reading the lecture notes from</p>
<p><a href="http://nic.desy.de/sites2009/site_nic/content/e44192/e62778/e91179/e91180/hmc_tutorial_eng.pdf" rel="nofollow">http://nic.desy.de/sites2009/site_nic/content/e44192/e62778/e91179/e91180/hmc_tutorial_eng.pdf</a></p>
<p>I am a little bit confused how points on the lattice gets assigned by a unique index j given in the lecture as</p>
<p>$$j = \sum_{i = 0}^{D-1} n_{i} L^{i}$$</p>
<p>Can any one explain what does it means. It is even not equal to total lattice sites V.</p>
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<p>Quick question regarding superficial degrees of freedom and Ward identities.</p>
<p>For instance in Peskin and Schroeder it is stated that the photon-self energy is superficially quadratically UV divergent but due to the Ward identity it is only logarithmically divergent. I don't see this argument.</p>
<p>The self-energy is given by</p>
<p>$\Pi^{1-loop}=(g^{\mu\nu}p^2-p^\mu p^\nu)\Pi(p^2)$</p>
<p>How does the Ward identity, or in other words, gauge invariance kill of the divergences?</p>
<p>Best,
A friendly helper</p>
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<p>A black hole collapses and emits Hawking radiation with average wavelength 1/M. The observer at I^+ sees it, so do the local observers (infalling or hovering). None of these cases should violate the equivalence principle. What is the size of the 'local' patch associated with the observer, the size beyond which he is allowed to see deviations from flat space because curvature effects become measureable? </p>
<p>I'm confused because on the one hand I'd have thought this size goes to infinity if the observer moves towards I^+, but on the other it seems to me it should stay at 1/M so that the (average) Hawking radiation has wavelengths the size of the locally flat patch. What am I getting wrong?</p>
<p>(Forget about firewalls, just thermal radiation.)</p>
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] |
<p>Just what the title states please... and perhaps naive too... </p>
<p>Wikipedia pegs the mass of our Earth at 5.9722 × 10^24 kg.</p>
<p>Does this figure include the mass of the Atmosphere?</p>
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] |
<p>Newton's theory explained the changes in sea level by the
effect of gravity (moon/sun).
Now we are trying to detect gravitational waves from distant
cosmic surces.
But, apart from the differences in "signal frequency"
(extremely small in case of sea effects (10-4 / 10-5 Hz or so),
maybe higher in case of "signals" from cosmic sources (in the order of 1 Hz?)
what is the difference in principle?
At least in principle, can we say that the change in sea level
is an effect of gravity waves? </p>
| 8,609 |
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<p>During my visit to the doctors', I recollect having seen the sign which says </p>
<blockquote>
<p>Please remove all metallic objects from the body before the commencement of the X-Ray</p>
</blockquote>
<p>How do metallic objects interfere with X-Rays? Don't they have metallic implants for certain kind of surgeries?</p>
| 8,610 |
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] |
<p>Are there any applications of quantum information theory to physics?</p>
| 8,611 |
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<p>It is known that when a conductor is placed in an electric field, the charges redistribute themselves such that $E=0$ inside the conductor. I was also told that the same is NOT true for the 2D and 1D analogues. Why is this true? And what happens in these cases?</p>
<p>Thank you.</p>
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<p>I was wondering if there's an underlying physical reason why detection in microphones and antennas is a cardioid, or if it's just that a cardioid happens to be the mathematical object that best approximates the shape of the detection pattern.</p>
<p>In other words, is there an "ab initio" reason, or it's an empirical fact?</p>
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<p>I'm reading about the auto-refrigeration effect and can't find a really good explanation. Is the idea that when you have your condensed liquid, and then release it into a low pressure environment, some of it evaporates instantly, leaving behind a cold liquid?</p>
<p>Let me explain what I'm thinking again: pressurized coolant is in a tube. It was pressurized by the compressor and cooled down moderately by the condenser until it barely became a liquid. Now, it reaches the throttling valve, which lets a little bit of this liquid out at a time into a lower pressure environment. Some of the liquid evaporates instantly (due to the lower pressure and thus lower boiling point), but some of it is left behind as a liquid. The part that evaporates instantly is hotter than the pressurized liquid, while the liquid left behind is colder than the pressurized liquid. The part that evaporated instantly moves through the evaporator quickly while the super cold liquid is left to slowly evaporate and move through the evaporator. This means the super cold liquid lingers in the evaporator and we can then run a fan over it and get nice AC or refrigeration.</p>
<p>The key point I'm getting at is that when the pressurized liquid moves through the throttling valve, the auto-refrigeration effect is really a way of splitting the hot vapor "part" away from the cold liquid "part". The vapor and liquid combined equal the same temperature as the source pressurized liquid, but the hotter part is instantly separated from the colder part (via vapor), and we take advantage of the colder part (that remains a cold liquid for enough time to be useful to us).</p>
<p>Am I right? Thanks!</p>
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] |
<p>I have been reading the article review for Optical Quantum Memory (<a href="http://arxiv.org/abs/1002.4659" rel="nofollow">http://arxiv.org/abs/1002.4659</a>) when I came across this section about <a href="http://www.google.com/search?as_q=dlcz+protocol" rel="nofollow">DLCZ protocol</a>. I don't understand about Beam splitter. For what I can interpret, the beam splitter overlaps both photons which emitted simultaneously by two atomic ensembles. My question is how does the beam splitter mixes two photons together so they become entangled?</p>
<p>P.S. My English is not good. I am sorry. </p>
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<p>Imagine $\xi_{\nu}$ is a <a href="http://en.wikipedia.org/wiki/Killing_vector_field" rel="nofollow">Killing vector field</a> on a manifold. Does $\xi_{\nu}\xi^{\nu}$ remain constant along any isometric curve defined by the Killing vector field? </p>
<p>My guess is that yes since as you move along an isometric curve every point "around" you looks pretty much as it did a differential step ahead, but I am don't have a full grasp on this and would thank some clarification.</p>
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