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<p>I'm curious about what happens if an explosive substance detonates in space. On Earth, I guess a good chunk of the energy released is carried away by shock waves in the atmosphere. But in space, the medium that supported the propagation of the shock wave on Earth is much more rarefied, so how does that work? </p> | 5,646 | [
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<p>Since phase transition is closely connected with symmetry, I am wondering whether it is possible to determine the universality class of phase transitions just by symmetry? </p>
<p>Actually, I found it is quite boring to calculate critical exponent numerically
and I want to find a new method. </p> | 5,647 | [
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<p>When I asked here why neutrons in nucleus (with protons) don't decay I was told that it would require energy for the neutron to decay, it wouldn't give energy. And since that wasn't really what I wanted to hear, since I already knew that, I'm now asking a similar question.</p>
<p>I know that strong force holds protons and neutrons together in nucleus, but how does that effect a neutron in a way that it doesn't decay? </p> | 5,648 | [
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<p>A problem in Bohr's day was understanding why an orbiting electron does not continuously radiate an EM field. An orbiting electron is a moving charge and according to Maxwell, this should generate an electric and magnetic field. Bohr said that photons (packets of EM energy) are only emitted when an electron jumps from a high quantum state to a lower one. Does that mean that an electron in a stable atom really isnt moving? Conversely, does Bohr's model mean that all motion requires energy transitions from one qunatum sate to another?</p> | 132 | [
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<p>there is some confusion to me in the case of "<strong>motion of block on a frictionless wedge</strong>"</p>
<p>Below is a simple diagram!</p>
<p><img src="http://i.stack.imgur.com/ET6uv.jpg" alt="enter image description here"></p>
<p>Let us consider a situation as above in which there is a block of mass $m$ moving with velocity $v$ in positive $x$ direction on a frictionless wedge as given above.</p>
<p>Now since block doesn't have any velocity in $y$ direction initially so it moves horizontally as long as it is on table But <strong>when at the instant it encounters the wedge it it gains some some velocity in upward direction and starts to move upward (i.e in positive $y$ direction) with velocity decreasing with time</strong>. it approaches some height and after that comes back down the wedge.</p>
<ol>
<li><p>What gives the block that upward velocity (i.e which force(s) changes the velocity in $y$ direction.) because of which it goes to some height and after that comes back?</p></li>
<li><p>What is the velocity of Block (net velocity which is along wedge). At the Instant it just starts to slide on the wedge?</p></li>
</ol> | 5,649 | [
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<p>I've been reading about nanotubes lately, and I keep seeing the $ (n,m) $ notation. How does this describe a nanotube's structure? How do I determine which is $n$ and which is $m$ ?</p>
<p>I'm familiar with matrix notation referring to rows and columns, but I couldn't connect it with the nanotube structure which is, albeit predictable, not quite like a row-column grid.</p> | 5,650 | [
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<p>Because in theory, visible light contains more energy than the microwaves actually used to cook your food. </p>
<p>1) Does it have to do with resonance frequency of the $H_2O$ molecule? <br /></p>
<p>2) How can microwaves be dangerous when they are about the same wavelength as Radio waves? (unless it has to do more with intensity)</p> | 5,651 | [
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<blockquote>
<p>A sample of $^{24}_{12}\mathrm{Mg}$ is bombarded by a monoenergetic proton. If the resulting nucleus in a $^{24}_{12}\mathrm{Mg}(p,\gamma)$ reaction; $^{24}_{12}\mathrm{Mg}(p,\gamma)$ has its first energy level at 2.5 MeV. What is the minimum energy of the of the proton to excite this level?</p>
</blockquote>
<hr>
<p>As I understand this problem, I assume that this is non-relativistic. I transformed the reaction from:</p>
<p>$^{24}_{12}\mathrm{Mg}(p,\gamma)$</p>
<p>into:</p>
<p>$^{24}_{12}\mathrm{Mg} + ^{1}_{1}\mathrm{H} \to \gamma + ^{25}_{13}\mathrm{Al}$</p>
<p>I found out that the product is not existing (because there is no 25,13 aluminum in our given table)</p>
<p>The formula for non-relativistic would be:</p>
<p>$$Q= KE_b + KE_y - KE_a + E$$</p>
<p>as part of my understanding to this problem, the 2.5 MeV will be the $KE_y$ and the variable to be find will be the internal energy ($E$).</p>
<p>Solving for $Q$</p>
<pre><code>Q= (summation of excess mass of resultant) - (summation of excess mass of product)
Q= (-13.933 + 7.289) - (0)
</code></pre>
<p>i made the summation of excess mass of product into 0 because gamma has no excess mass(in our table) and the product is not existing (in our table)</p>
<pre><code>Q= -6.644 MeV
KEb = Q / (1+ [mass of b / mass of Y])
= -6.644 / (1 + 0)
= -6.644 MeV
</code></pre>
<p>from the formula of non-relativistic:</p>
<pre><code>E = -6.644 + 6.644 - 2.5
E = -2.5 MeV
</code></pre>
<p>for the velocity, as my professor did on her example, </p>
<pre><code>momentum (p)= mV
myVy = mbVb
Vy = (mbVb) / my
</code></pre>
<p>but in this problem, the unknown is Va (velocity of a or velocity of proton)</p>
<p>I am confused in this problem because of the terms "monoenergetic proton" and "first energy level" which is not that familiar to us. I am doubtful to how I understand the problem. Does my understanding correct? does my internal energy (E) correct? i don't know how to find the velocity. please help me to solve this problem.. this will be passed the same day we are going to take exam on this topic. our homework serve as our reviewer so please, I don't want to be confused in our examination day. </p> | 5,652 | [
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<p>Why do same charges repel each other and opposite charges attract each other (please explain the phenomenon using real laws of nature (QED) not with the approximation model)? </p> | 5,653 | [
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<p>What is a formula for velocity of a spacecraft in a circular orbit? Actually, is there a formula or I can find it from equation of motion?</p> | 5,654 | [
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<p>I've been told that, from Maxwell's equations, one can find that the propagation of change in the Electromagnetic Field travels at a speed $\frac{1}{\sqrt{\mu_0 \epsilon_0}}$ (the values of which can be empirically found, and, when plugged into the expression, yield the empirically found speed of light)</p>
<p>I'm really not sure how I would go about finding $v = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$ simply from Maxwell's equations in the following form, in SI units --</p>
<p>$\nabla \cdot \mathbf{E} = \frac {\rho} {\epsilon_0}$
</p>
<p>$\nabla \cdot \mathbf{B} = 0$
</p>
<p>$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$
</p>
<p>$\nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right)$
</p>
<p>Is what I believe true? (that the speed of propagation is derivable from Maxwell's Equations)</p>
<p>If not, what else is needed?</p>
<p>If so, can you provide a clear and cogent derivation?</p> | 5,655 | [
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<p>Are all elementary particles of the same type EXACTLY the same? Is there some variation in what an electron is, for example, or are they all the same?</p> | 5,656 | [
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<p>I am trying to find the geometric structure factor and my work here is clearly wrong. I will put my wrong answer and then I will throw up the link to wikipedia for the correct answer, because I cannot tell the difference.</p>
<p>My attempt: BCC has a four atom basis. If x,y,z vectors are taken to be along the edges of the conventional cube like in the picture below (all credit due to Aschroft and Mermin) :</p>
<p><img src="http://i.stack.imgur.com/Qycg9.png" alt="enter image description here"></p>
<p>But the basis are located at:</p>
<p><img src="http://i.stack.imgur.com/3vP1X.png" alt="enter image description here"></p>
<p>Therefore the geometric structure factor is:</p>
<p>$$F_{hkl} = \sum_{j=i}^{N} f_j e^{i \Delta \vec{k} \bullet \vec{r_j} }$$</p>
<p>But because the structure is monatomic, $f_j = f$ for all j. Also, $r_j$ denotes the location of the jth atom in the cell (I.E. $r_1 = \vec{0}, r_2 = \vec{ a_1} ,r_3 = \vec{ a_2}, r_4 = \vec{ a_3}$ in the picture )</p>
<p>$\Delta \vec{k}$ is just some vector that is an element of Reciprocal vector space. That is,
$$\Delta \vec{k} = h \vec{b_1} + k \vec{b_2} + l \vec{b_3}$$</p>
<p>but because $\vec{a_i} \bullet \vec{b_j} = 2 \pi \delta_{ij}$ , $F_{hkl}$ reduces to:</p>
<p>$$F_{hkl} = f [ e^0 + e^{i(h \vec{b_1} + k \vec{b_2} + l \vec{b_3}) \bullet \vec{a_1}} + e^{i(h \vec{b_1} + k \vec{b_2} + l \vec{b_3}) \bullet \vec{a_2}} + e^{i(h \vec{b_1} + k \vec{b_2} + l \vec{b_3}) \bullet \vec{a_3}} ] = f [ 1 + e^{i2 \pi h} + e^{i2 \pi k} + e^{i2 \pi l} ] = 4f$$</p>
<p>This answer is incorrect. The correct answer can be seen at <a href="https://en.wikipedia.org/wiki/Structure_factor" rel="nofollow">this link to Wikipedia</a>, scrolling down to fcc. From what I can tell, they must have defined their reciprocal lattice vector differently from me but I cannot see why. </p> | 5,657 | [
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<p>Cosmic rays can have energies going into the $10^{20}$ eV domain. <a href="http://en.wikipedia.org/wiki/Asteroid">Asteroids</a> and <a href="http://en.wikipedia.org/wiki/Meteoroid">meteoroids</a> originating in the solar system are probably limited in their speed because they all started out from the same lump of matter having more or less the same speed, but what about rocks from other galaxies ? Could they reach relativistic speeds in relation to the solar system? Are there astronomical events that could be interpreted as a collision between such an asteroid and a planet or star?</p>
<p>Just a mole of hydrogen atoms with that kind of energy would have three times the energy of the Chicxulub impact event, so I hope there's a reason for them not existing!</p> | 5,658 | [
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<p>Can somebody help me in deriving the Hamiltonian of system starting from Euclidean Lagrangian? </p>
<p>Say we are given the Minkowski Lagrangian</p>
<p>$$L_m = \frac{\dot{\phi}^2}{2} - V(\phi).$$</p>
<p>The Hamiltonian can then be found by Legendre transformation</p>
<p>$$H = \dot{\phi}\frac{\partial L_m}{\partial\dot{\phi}} - L_m,$$</p>
<p>which equals $$H = \frac{1}{2}\dot{\phi}^2 + V(\phi)$$ which was not hard.</p>
<p>Now consider the corresponding Euclidean Lagrangian</p>
<p>$$L_e = -\frac{\dot{\phi}^2}{2} - V(\phi).$$</p>
<p>How do I calculate the Hamiltonian in this formalism? The above way applied naively will not give the correct result.</p> | 5,659 | [
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<p>In gamma-gamma physics, what could be some way of either minimising the mass of the intermediary fermions or minimising the time for which those intermediary fermions could exist?</p> | 5,660 | [
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<p>Let's have Pauli-Lubanski operator:</p>
<p>$$
\hat {W}^{\alpha} = \frac{1}{2}\varepsilon^{\alpha \beta \gamma \delta}\hat {J}_{\beta \gamma}\hat {P}_{\delta} = \frac{1}{2}\varepsilon^{\alpha \beta \gamma \delta}\hat {S}_{\beta \gamma}\hat {P}_{\delta},
$$</p>
<p>where $\hat {J}_{\beta \gamma} = \hat {L}_{\beta \gamma} + \hat {S}_{\beta \gamma}$ and $\hat {S}_{\beta \gamma}$ is the spin-tensor.</p>
<p>In some books there is a following expression for square of this operator:</p>
<p>$$
\hat {W}_{\lambda}\hat {W}^{\lambda } = \hat {P}_{\alpha}\hat {P}_{\beta }\hat {S}^{\alpha \mu}\hat {S}_{\beta \mu} -\frac{1}{2}\hat {P}_{\alpha}\hat {P}^{\alpha}\hat {S}_{\alpha \beta}\hat {S}^{\alpha \beta }.
$$
But I don't understand how to get it, because I can't claim, that $\hat {S}_{\alpha \beta}$ is antisymmetrical. So, I only get, using expression
$$
\varepsilon_{\alpha \beta \gamma \lambda}\varepsilon^{\lambda \mu \nu \sigma} = \delta^{\mu}_{\alpha}\varepsilon^{\nu \sigma}_{\beta \gamma} + \delta^{\mu}_{\gamma}\varepsilon^{\nu \sigma}_{\beta \alpha} + \delta^{\mu}_{\beta}\varepsilon^{\nu \sigma}_{\gamma \alpha}=\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}\delta^{\sigma}_{\gamma} - \delta^{\mu}_{\alpha}\delta^{\nu}_{\gamma}\delta^{\sigma}_{\beta} + \delta^{\nu}_{\alpha}\delta^{\sigma}_{\beta}\delta^{\mu}_{\gamma} - \delta^{\nu}_{\alpha}\delta^{\mu}_{\beta}\delta^{\sigma}_{\gamma} + \delta^{\sigma}_{\alpha}\delta^{\mu}_{\beta}\delta^{\nu}_{\gamma} - \delta^{\sigma}_{\alpha}\delta^{\nu}_{\beta}\delta^{\mu}_{\gamma},
$$
following:
$$
\hat {W}_{\lambda}\hat {W}^{\lambda} = \frac{1}{4}\varepsilon_{\lambda \alpha \beta \gamma }\hat {J}_{\alpha \beta}\hat {P}_{\gamma }\varepsilon^{\lambda \mu \nu \sigma}\hat {J}_{\mu \nu}\hat {P}_{\sigma} = -\varepsilon_{\alpha \beta \gamma \lambda}\varepsilon^{\lambda \mu \nu \sigma}\hat {J}_{\alpha \beta}\hat {P}_{\gamma }\hat {J}_{\mu \nu}\hat {P}_{\sigma} =
$$</p>
<p>$$
= \frac{1}{4}\left(\delta^{\mu}_{\alpha}\varepsilon^{\nu \sigma}_{\beta \gamma} + \delta^{\mu}_{\gamma}\varepsilon^{\nu \sigma}_{\beta \alpha} + \delta^{\mu}_{\beta}\varepsilon^{\nu \sigma}_{\gamma \alpha}\right)\hat {J}_{\alpha \beta}\hat {P}_{\gamma }\hat {J}_{\mu \nu}\hat {P}_{\sigma} =
$$</p>
<p>$$
=-\frac{1}{4}\left( \hat {J}_{\alpha \beta}\hat {P}_{\gamma}\hat {J}^{\alpha \beta}\hat {P}^{\gamma} - \hat {J}_{\alpha \beta}\hat {P}_{\nu}\hat {J}^{\alpha \nu}\hat {P}^{\beta} + \hat {J}_{\alpha \beta}\hat {P}_{\gamma}\hat {J}^{\gamma \alpha}\hat {P}^{\beta} - \hat {J}_{\alpha \beta}\hat {P}_{\gamma}\hat {J}^{\beta \alpha}\hat {P}^{\gamma} + \hat {J}_{\alpha \beta}\hat {P}_{\gamma}\hat {J}^{\beta \gamma }\hat {P}^{\alpha} - \hat {J}_{\alpha \beta}\hat {P}_{\gamma}\hat {J}^{\gamma \beta}\hat {P}^{\alpha}\right).
$$
So, if spin-tensor operator is antisymmetrical, I'll get the answer easily using Poincare algebra. But if not, I can't continue. </p>
<p>Can you help me to get the answer? Maybe, I can get the answer, because there is the identity of spin-tensor operator and orbital momentum tensor operator?</p> | 5,661 | [
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<p>In momentum and energy is low enough, we end up with the same number of neutrons, protons and electrons after a collision as before it. This can be considered an approximate conservation law. Shouldn't there be a corresponding (approximate) symmetry corresponding to each of these particle groups?</p> | 5,662 | [
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<ol>
<li>Most images you see of the solar system are 2D and all planets orbit in the same plane. In a 3D view, are really all planets orbiting in similar planes? Is there a reason for this? I'd expect that the orbits distributed all around the sun, in 3D.</li>
<li>Has an object made by man (a probe) ever left the Solar System?</li>
</ol> | 5,663 | [
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<p><a href="http://en.wikipedia.org/wiki/Plasma_%28physics%29" rel="nofollow">Plasma</a> is ionized gas which as far as I know only occurs at high temperatures. When plasma cools down it tends to recombine with the electrons present and turn back into gas. But what if the disassociated electrons in the plasma were removed and the plasma were allowed to cool down in a vacuum, while being held in place by a strong magnetic field. Would this substance still be plasma? Is this possible? </p> | 5,664 | [
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<p>I'm studying the Method of Images and I seemed to have come to a conundrum. Method of Images takes advantage of grounded objects, (I am currently studying spheres), to set boundary conditions. However, how would one use the idea of MoI to set the potential of a conducting sphere as constant?</p>
<p>Since $E = \nabla V$, a constant potential would be the electric field inside would be constant? Therefore, a density of charge inside the sphere?</p> | 5,665 | [
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<p>In equation 6.2.7, Polchinski defines his reduced Green's functions $G'$ on the 2-manifold to satisfy the equation,</p>
<p>$$ \frac{-1}{2\pi \alpha '}\nabla ^2 G'(\sigma_1, \sigma_2) = \frac{1}{\sqrt{g}}\delta ^2 (\sigma_1 - \sigma_2) - X_0^2 $$</p>
<p>(..where $\sigma_1$ and $\sigma_2$ are two points on the manifold and $X_0$ is the zero-eigenvalue of the Laplacian..Why is he assuming that there is only one zero mode?..)</p>
<p>Now at various place he has written down the solutions to the equation like,</p>
<ul>
<li>on a $S^2$ it is given by 6.2.9,</li>
</ul>
<p>$$G' = - \frac{\alpha'}{2}ln \vert z_1 - z_2\vert ^2 + f(z_1,\bar{z_1}) + f(z_2,\bar{z_2})$$</p>
<p>where $f(z,\bar{z}) = \frac{\alpha'X_0^2}{4} \int d^2 z' exp(2\omega(z,\bar{z}))ln \vert z - z'\vert ^2 + k$ </p>
<ul>
<li>For the disk it is given by 6.2.32,</li>
</ul>
<p>$$G' = -\frac{\alpha'}{2}ln \vert z_1 - z_2\vert ^2 + \frac{\alpha'}{2}ln \vert z_1 - \bar{z_2}\vert ^2 $$</p>
<ul>
<li>For $\mathbb{RP}^2$ it is given by,</li>
</ul>
<p>$$G ' = -\frac{\alpha'}{2}ln \vert z_1 - z_2\vert ^2 + \frac{\alpha'}{2}ln \vert 1 +z_1\bar{z_2}\vert ^2 $$</p>
<p>I would like to know how these functions are derived. </p>
<ul>
<li>Also how is it that the dependence on the $f$ for the first case drops out in equation 6.2.17? If I plug in the functions I see a remnant factor in the exponent of the form, $\sum_{i,j,i<j,=1}^n k_i k_j (f(\sigma_i) + f(\sigma_j))+ \sum_{i=1}^n k_i^2 f(\sigma_i)$</li>
</ul> | 5,666 | [
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<p>I know that when there are excess positive charges in a conductor, for example, a metal sphere, the positive charges will spread out over its surface. However, I am confused about how this excess charge spreads out over the surface, if protons cannot move and only electrons can move.<br>
Can someone please inform me on how the excess positive charge spreads out over the surfaces of conductors?</p> | 5,667 | [
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<p>I just finished watching Into the Universe with Stephen Hawking (2010). Specifically the third episode titled 'The Story of Everything.'</p>
<p>In the episode Hawking is explaining the mainstream theories of the events just after the Big Bang. What confuses me is that he just said at the beginning that the laws of physics were different and time (among other things) did not exist as we know it during this early stage in the life of the universe, but then goes on to say certain events took certain amounts of time.</p>
<p>I don't know if the numbers match, but the same issue exists on <a href="http://en.wikipedia.org/wiki/Chronology_of_the_universe" rel="nofollow">this Wikipedia article</a>.</p>
<p>How can we even guess the time frames of these events if time itself is largely dependent on the current state of the universe?</p> | 5,668 | [
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<p>I know that water expands in the freezer, but I'm curious about which materials <em>contract</em> in response to cold temperatures --- and most importantly, which ones undergo the most drastic changes?</p> | 5,669 | [
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<p>Was one of Hilbert questions regarding physics to make an axiomatic foundation for physics? Regardless of Godels work could some Physics principles that are 'basic' and 'presently verifiable' be treated as axioms in a logic system along with Math axioms and Set theory or maybe with Category theory as a basis. If 'axiomatized' one could prove a physics theory with a type of Physics 'proving' algorithm. Like Leibnitz's dream of a 'calculating machine' that could calculate any question's answer.</p> | 203 | [
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<p>I am a non-expert in this field, just have a layman's interest in the subject. Has anyone ever considered the possibility of magnetic monopoles (one positive and one negative charge) being confined together like quarks, and hence that could be the reason monopoles have never been observed? At the macro level the total magnetic charge would always be zero due to the charges being tightly bound. </p> | 5,670 | [
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<p><img src="http://imgur.com/KwER0d8" alt="Diagram"></p>
<p>I have a 5 mm (diameter) solid ball lens that I intend to place in front of a 36mm X 24mm camera sensor (I have already made preparations for the image to be focused and for the sensor to not be flooded with light). I want to calculate the resulting horizontal and vertical field of view from the ball lens when I take a picture.</p>
<p>Please see the link for a diagram of the setup if the image does not work.</p>
<p><img src="http://i.imgur.com/KwER0d8.jpg" alt="http://imgur.com/KwER0d8"></p>
<p>I have already calculated the back focal length of the ball lens to be 1.479 mm.</p> | 5,671 | [
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<p>When some photon detector detects a photon, is it an instantaneous process (because a photon can be thought of as a point particle), or does the detection require a finite amount of time depending on the wavelength of the photon?</p>
<p>EDIT: I guess what I am wondering is if a photon has a wavelength and travels at a finite speed, then if a photon had a wavelength of 300,000,000m, would its interaction with the detector last 1s? Or does the uncertainty principle say that a photon with wavelength 300,000,000m (and therefore energy E), it cannot be known exactly when it hit the detector with an accuracy better than 1s. Or is it more like this: suppose there is a stream of photons moving towards the detector with wavelengths of 300,000,000m and they reach the detector at a rate of 10 photons/second and the detector has a shutter speed such that the shutter is open for 1s at a time, then it would record 10 photon hits (records all the photons). But if the shutter speed is only 0.5s, then it would record 2.5 hits on average?</p>
<p>EDIT2: I'm not interested in the practical functioning of the detector and amplification delays. I'm looking at and ideal case (suppose the photon is 'detected' the instant an electron is released from the first photomultiplier plate). It is a question regarding the theory of the measurement, not the practical implementation.</p> | 5,672 | [
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<p>I am trying to confirm a conservation law I cam across in a paper (Janssen 1983 "On a fourth-order envelope equation for deep-water waves" Journal Fluid Mechanics), and am having difficulty. </p>
<p>In particular, I'm trying to confirm the conservation of linear momentum $P$, where </p>
<p>$P = \frac{i}{2} \int AA^*_x - A^*A_x \ dx$,</p>
<p>where $^*$ denotes complex conjugate and spatial integrals are taken to be over all space in this question. Now, the governing equation takes the form </p>
<p>$A_t = \mathcal{N}(A,A^*) - P.V. \ i\alpha\ A\int_{-\infty}^{\infty} \frac{\partial|A|^2}{\partial x'}\frac{1}{x-x'} \ dx'$,</p>
<p>where $\mathcal{N}$ is a (nonlinear) operator describing the rest of the dynamics, which is not important for this question, P.V. denotes the principal value and $\alpha$ is some real constant. Note, this integral is proportional to the Hilbert transform. </p>
<p>Now, I want to look at the time evolution of the momentum $P$. To that end, we have </p>
<p>$\frac{d P}{dt} = \frac{i}{2} \int \dot{A}A^*_x +A\dot{A}_x^* -\dot{A}^*A_x -A^*\dot{A}_x \ dx$ </p>
<p>Substituting in the relation for our governing equation, and using integration by parts (we assume the field A is compact) we find</p>
<p>$\frac{dP}{dt}=\frac{\alpha}{2} \int (|A|^2)_x \mathcal{H} (|A|^2_x) \ dx$</p>
<p>where </p>
<p>$\mathcal{H}(|A|^2_x) \equiv P.V. \int_{-\infty}^{\infty} \frac{\partial|A|^2}{\partial x'}\frac{1}{x-x'} \ dx'$.</p>
<p>Now, this term is non-zero, where as the author claims this integral is conserved. I don't see why this integral should vanish, but perhaps I'm not exploiting a property of the Hilbert transform. </p>
<p>Am I missing something obvious? </p>
<p>Thanks,</p>
<p>Nick </p> | 5,673 | [
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<p>I am learning supersymmetry right now. I am mostly following Bailin and Love. I try to connect all the steps from the book and complete the derivations in order to get comfortable with the calculations but I feel the lack of a problem book with a lot of solved and unsolved problems, something like the Schaum's series which I can use together with Bailin and Love. Do you happen to know any such book that you think would be helpful for me?</p> | 5,674 | [
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<p>This question has been asked multiple times here and all over the internet yet I can't find a conclusive answer:</p>
<ul>
<li><p>Some claim that heavier objects do fall faster: <a href="http://physics.stackexchange.com/q/3534/">Don't heavier objects actually fall faster because they exert their own gravity?</a> </p></li>
<li><p>Others claim that all objects fall at the same speed regardless of their mass: <a href="http://physics.stackexchange.com/q/5973/">Free falling of object with no air resistance</a></p></li>
</ul>
<p>Which one is the right one?</p> | 134 | [
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<p>I am reading through Cavagna's <a href="http://arxiv.org/abs/cond-mat/0505032" rel="nofollow">Spin glass theory for pedestrians</a>, but I am stuck at equation (35).</p>
<p>I'll try to provide a little context. Given two spin configurations $\sigma$ and $\tau$, we define their overlap as:</p>
<p>$$q_{\sigma\tau} = \frac{1}{N}\sum_{i=1}^N \sigma_i \tau_i$$</p>
<p>The configuration space can be split into sets of configurations called <em>pure states</em>, which are characterized by the <em>clustering property</em>, which states that for the set of configurations in a pure state, correlations drop to zero as the distance increases:</p>
<p>$$\langle\sigma_i \sigma_j\rangle \rightarrow \langle\sigma_i\rangle \langle\sigma_j\rangle \qquad \text{for}\quad |i-j|\rightarrow \infty$$</p>
<p>The overlap between pure states $\alpha$, $\beta$ is defined as:</p>
<p>$$q_{\sigma\tau} = \frac{1}{N}\sum_{i=1}^N \langle\sigma_i\rangle_\alpha \langle\sigma_i\rangle_\beta$$</p>
<p>An overlap distribution is defined as:
$$P(q) = \frac{1}{Z^2} \int D\sigma D\tau e^{-\beta H(\sigma)} e^{-\beta H(\tau)} \delta(q-q_{\sigma \tau})$$</p>
<p>In the text it is claimed that because of the clustering property,</p>
<p>$$P(q) = \sum _{\alpha \beta} w_\alpha w_\beta \delta(q-q_{\alpha\beta})$$</p>
<p>where $w_\alpha = Z_\alpha/Z$ (where $Z_\alpha$ is the partition function of all spin configurations in the pure state $\alpha$), and similarly for $w_\beta$. This is the formula I don't understand. I don't know how to derive this formula from the previous definitions and equations.</p> | 5,675 | [
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<p>What exactly is meant by <em>Lorentz invariance</em>?</p>
<p>Is it just an experimental observation, or is there a theory that postulates it?</p>
<p>What quantities do we expect to be Lorentz invariant?
Charge? Charge densities? Forces? Lagrangians?</p> | 5,676 | [
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<p>For example, the Lagrangian formulation.
I may be missing something, i.e. not having done it in enough detail, but here is my issue:</p>
<p>from the definition of the lagrangian ($\mathcal{L}$) and from the principle of least action ($S(t)$), we get that $$ S(t) = \int \mathcal{L} \ dt$$ has to be minimised.</p>
<p>From this, one can derive the Euler-Lagrange equation, $$\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{x}}\right) = \frac{\partial L}{\partial x}$$</p>
<p>and using $\mathcal{L} = T - V = m\dot{x}^2 - V(x)$ we get
$$ m\ddot{x} = -V'(x),$$ i.e. Newton's second law of motion.</p>
<p><strong>Now:</strong>
Does the fact that we get Newton's second law mean that the Lagrangian formulation (and therefore the principle of least action) is correct? </p>
<p>I mean, could it not be a coincidence that we get this equation? But then again, is there a way that this equation could <em>not</em> hold?
The only difference between systems is the potential $V(x)$ (right?), so as long as it is included in a general form the equation should hold. (?).</p>
<p><em>In light of this</em>, is there any physical meaning as to why the Lagrangian is defined as the difference between kinetic and potential energy, $\mathcal{L} = T - V $?</p> | 5,677 | [
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<p>In other words, given a magical room with walls that produce no vibration and transmit zero vibration from the outside, and nothing on the inside except room temperature air, what would be the noise level in dB SPL (<a href="http://en.wikipedia.org/wiki/Sound_pressure" rel="nofollow">sound pressure level</a>) from the <a href="https://en.wikipedia.org/wiki/Thermal_fluctuations" rel="nofollow">thermal motion</a> of the air itself? (Similar to the <a href="https://en.wikipedia.org/wiki/Noise_floor" rel="nofollow">noise floor</a> of electronics being determined by <a href="https://en.wikipedia.org/wiki/Thermal_noise" rel="nofollow">thermal noise</a> in the conductors.) What is the quietest possible anechoic chamber?</p>
<p>For reference: Sound pressure is defined as the root-mean-square value of the instantaneous pressure, measured in pascal = N/m². SPL is the same number, but expressed in decibels relative to 20 µPa.</p>
<p>(I assume that it has a white spectrum, but I could be wrong. Thermal noise in electronics is white, but other types of electronic noise are pink, and blackbody thermal radiation has a bandpass spectrum.)</p>
<p><a href="http://www.int-res.com/articles/theme/m395p005.pdf" rel="nofollow">Here's an explanation</a> in the context of underwater acoustics. Not sure how this applies to air:</p>
<blockquote>
<p>Mellen (1952) developed a theoretical model for thermal noise based on classical statistical mechanics, reasoning that the average energy per degree of freedom is kT (where k is Boltzmann’s constant and T is absolute temperature). The number of degrees of freedom is equal to the number of compressional modes, yielding an expression for the plane-wave pressure owing to thermal noise in water. For non-directional hydrophones and typical ocean temperatures, the background level due to thermal noise is given by:</p>
<p>NL = −15 + 20 log <em>f</em> (in dB re 1 µPa)</p>
<p>where <em>f</em> is given in kHz with <em>f</em> >> 1, and NL is the noise level in a 1 Hz band. Note that thermal noise <a href="https://en.wikipedia.org/wiki/Colors_of_noise#Violet_noise" rel="nofollow">increases at the rate of 20 dB decade<sup>−1</sup></a>. There are few measurements in the high-frequency band to suggest deviations from the predicted levels. </p>
</blockquote>
<p>Citation is R. H. Mellen, The Thermal-Noise Limit in the Detection of Underwater Acoustic Signals, J. Acoust. Soc. Am. 24, 478-480 (1952).</p> | 5,678 | [
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] |
<p>Where could I find open-source up-to-date cosmological datasets? Are there any available as a data stream with an API?</p>
<p>I already know about these: </p>
<p><a href="http://cas.sdss.org" rel="nofollow">http://cas.sdss.org</a></p>
<p><a href="http://www.ukidss.org/archive/archive.html" rel="nofollow">http://www.ukidss.org/archive/archive.html</a></p>
<p><a href="https://archive.stsci.edu" rel="nofollow">https://archive.stsci.edu</a></p> | 204 | [
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<p>I've looked through several papers that talk about the anomalous integer quantum Hall effect of graphene (such as <a href="http://journals.aps.org/prl/pdf/10.1103/PhysRevLett.95.146801" rel="nofollow">http://journals.aps.org/prl/pdf/10.1103/PhysRevLett.95.146801</a>), and they all state that the Hall effect is anomalous for graphene because the n=0 Landau level has half the degeneracy of the n>1 levels, with little to no explanation and without citing any (helpful) sources.</p>
<p>One of the papers (<a href="http://journals.aps.org/prb/pdf/10.1103/PhysRevB.75.165411" rel="nofollow">http://journals.aps.org/prb/pdf/10.1103/PhysRevB.75.165411</a>) states that "the Dirac fermions have acquired an effective gap in a form of a “relativistic mass” due to the Coulomb interaction. Such a gap reduces the degeneracy of <i>only</i> the zeroth LL." Could someone explain this in more depth, specifically <i>why</i> the Coulomb interaction only breaks the degeneracy of the zeroth LL?</p> | 5,679 | [
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<p>I am a beginner in electrical engineering. Often times (most cases actually), the underlying physics aren't really explained to us and we are just left to assume that it works "because it works." This is never enough for me in classes etc and I always end up doing a follow up of physics side of the spectrum.</p>
<p>My question is, you have a battery with excess electrons built up on the negative lead, and excess holes on the positive lead, why is it that in our universe then connecting a conductive compound (like.. copper) to just the negative lead does not produce current, or vice versa connecting the wire to just the positive lead. </p>
<p>After all, if we move a magnet passed a conductor, there is a tiny induced current. How can the magnet do this, but an excess of electrons or electron holes repelling each other cannot?</p> | 5,680 | [
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<p><strong>This is the question-</strong></p>
<p>Consider a cylinder of mass $M$ resting on a rough horizontal rug that is pulled out from under it with acceleration $a$ perpendicular to the axis of the cylinder. What is $F$ (friction) at a point <em>P</em>? It is assumed that the cylinder does not slip.</p>
<p><img src="http://i.stack.imgur.com/b05sS.png" alt="Diagram"></p>
<p>Options-</p>
<p>A) $F = M g$</p>
<p>B) $F = M a$</p>
<p>C) $F = \frac{M}{2} a $</p>
<p>D) $F = \frac{M}{3} a $</p>
<p><strong>My attempt-</strong></p>
<p>Since rolling is nothing but rotation about the point of contact (P in this case), according to me P should be at relative rest, therefore $F=Ma$. This should have been a short and crisp question, help me out please answer given is $F=\frac{M}{3} a$, and if you can perhaps please point me out to articles for such rolling friction question, I get stumbled whenever they ask friction in rolling. Thanks in advance..</p>
<p><strong>Edit-</strong>
This idea also came to my mind-
Moment of inertia for a solid cylinder= I = $\frac{MR^2}{2}$</p>
<p>Let the frictional force be F, then,</p>
<p>$F*R=(torque)=I*\alpha$</p>
<p>$F*R=\frac{(M*R^2)*(a)}{2R}$
(as the cylinder is in pure rolling)</p>
<p>therefore- $F=\frac{Ma}{2}$</p>
<p>which is again incorrect any pointers to where I went wrong here again would be appreciated.. thanks</p> | 5,681 | [
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<p>I am trying to understand 2D electrostatics of $n$ point charges. Roughly, </p>
<p>$$ H = \sum_{i=1}^N n_i \ln |z- z_i|$$</p>
<p>However, I keep bumping across the Gaudin model instead with this Hamiltonian</p>
<p>$$ H_j = \sum_{i \neq j} \frac{n_i}{z_j- z_i}$$</p>
<p>and these are often put into context with other integrable systems, such as Calogero-Moser, or Ruisenaars-Schneider. They talk about how $[H_i, H_j] = 0$ as operators and a torus fibration of the phase space.</p>
<p>Is there a larger family of integrable systems that contains both cases I mentioned above?</p>
<p>This is not electromagnetism, since that Hamiltonian looks rather different (I will try to look up the source for that). What I have written down is just the electrostatic potential.</p>
<hr>
<p>The other strange part is the second Hamiltonian is a real part of the derivative $\frac{d}{dz}$ of my first example.</p> | 5,682 | [
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<blockquote>
<p>Are there physical models of spacetimes, which have bounded (four dimensional) holes in them? </p>
</blockquote>
<p>And do the Einstein equations give restrictions to such phenomena? </p>
<p>Here by holes I mean constructions which are bounded in size in spacetime and which for example might be characterized by nontrivial higher fundamental groups. </p>
<p>On a related note:</p>
<blockquote>
<p>Could relatively localized and special configurations of (classical) spacetime be interpreted as matter? </p>
</blockquote>
<p>I.e. can field-like behavior emerge in characteristic structures of spacetime itself, like for example holes in the above sense, or localized areas of wild curvature? Gravitational waves certainly go in this direction, although they act on such a huge scale, that in their case we probably wouldn't recognize an organized, maybe even life-like behavior as such.</p> | 5,683 | [
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<p>Elastomeres are "defined" as:</p>
<p>"linear-chain polymers with widely spaced cross-links attaching each molecule to its neighbours" </p>
<p>Now I found sentences (talking about glass transition):</p>
<p>"This means that at room temperature the secondary bonds are melted and the molecules can slide relative to each other with ease. Were it not for the cross-links, the material would be a viscous liquid, but the cross-links give it a degree of mechanical stability."</p>
<p>So, what exactly are these bonds that melt? and which ones do not? An example would be nice. Chemistry is not my strongest field so be rigorous.</p>
<p>The quotations are from cellular solids by gibson and ashby.</p> | 5,684 | [
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<p>I am in AMO Physics and work a lot with optics. I just wanted to get an idea of what coupling efficiencies one "should" get in a "reasonable time"* by coupling light into a fiber using different couplers, like collimators, mounting plates, mounted lens systems, etc. I understand that this dependents on a lot of factors, so I will narrow it down for my specific case but would appreciate if ppl report some numbers with a short info about their setups.</p>
<p>We use a cage system system with 5 mm lens and <a href="http://www.apc.com/site/apc/index.cfm" rel="nofollow">APC</a> fiber plate to couple 633 nm laser light into an APC single mode fiber. Pretty much everything of the optical equipment is from Thorlabs.com.</p>
<p>The reason for my question: My <a href="http://en.wikipedia.org/wiki/Principal_investigator" rel="nofollow">PI</a> told me I should get efficiencies up 80% percent with our coupling scheme but I can only get 30%. I hope to get a better feeling for coupling efficiencies with this answer. </p>
<p>Thanks for your input, </p>
<p>n3rd</p>
<p>*With "should" and "reasonable time" I refer to coupling efficiencies one can achieve at time scales on the order of tens of minutes rather than hours, days, weeks. </p> | 5,685 | [
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<p>The maximum obtainable angular resolution of an optical system with some given aperture is well known, but it seems to me that this isn't a real theoretical limit. The assumption is that you are going to take a picture using the system and no further processing will take place. However, given the known point spread function of the optical system, you could perform a deconvolution calculation. The final resolution of the processed image should be limited by the noise. So, what then seems to matter is the observation time (the longer you integrate the signal the better the signal/noise ratio will be).</p>
<p>So, what is the correct theoretical limit of the maximum resolution in terms of brightness of the two sources to be resolved, the aperture and the observation time (assuming that the only noise comes from the fluctuations in the finite numbers of photons from the sources)?</p> | 5,686 | [
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<p>Can we see any galaxies/stars that are newer than our own galaxy?</p>
<p>As light takes (c) amount of times to reach us - so relatively speaking, light which would have left the newer galaxy (and far enough from us) still would not have reached us. Therefore, can we ever see a galaxy that is newer than our own one. If we can see a new galaxy that is newer than our own one, how can we determine its age as light would not have reached us yet.</p>
<p>EG : Light travels from galaxy (x) to us (y) and it has taken 100 million light years. However if (x) was less than 100 million years younger than us, how could it have reached us and can we determine it and its age ?</p> | 5,687 | [
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<p>The <a href="http://www.nasa.gov/mission_pages/newhorizons/main/index.html">New Horizons Spacecraft</a> is scheduled to whiz by Pluto around 2015, and my understanding is that it is going to do exactly that — whiz by it.</p>
<p>Where will it go after that? Or what else can it do once that mission is completed?</p> | 5,688 | [
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<p>In reference to the <a href="http://www.nasa.gov/mission_pages/kepler/news/kepscicon-briefing.html">Kepler 22b news</a>:</p>
<blockquote>
<p>The Kepler team had to wait for three passes of the planet before
upping its status from "candidate" to "confirmed".</p>
</blockquote>
<p>This is possible because the planet has a orbital period ('year') of 289 days, and so it has transitioned more than once (as above) since the Kepler mission started.</p>
<p>However I've seem some confirmed exoplanets that have orbital periods in 10s and 100s of years, and therefore could only have been detected once by Kepler.</p>
<p>As far as I'm aware all methods of discovering exoplanets require the planet to pass in from of their parent star (i.e. we can't use a telescope to 'zoom in' and see them directly)?</p>
<p>So how are exoplanets confirmed, when we haven't been searching long enough to cross in front of their parent star? How long does the transition even last?</p> | 5,689 | [
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<p>Shortly, I will be beginning my third year at University in Computer Science, I am a software developer and I will be required to work on a final year project. </p>
<p>My idea for my final year project is to write a desktop application which would allow the user to specify a date/time and it would return all of the positions of the planets at that point in time.</p>
<p>I was hoping someone could point me in the right direction of how I would begin to do the math or calculate this, as I'm aware that position is relative and a bit open ended - any scope refinements to my project would be gladly accepted too as I'd like to incorporate other things.</p>
<p>Just to reiterate: This is not a programming question, and I'm not asking for someone to 'do my project for me' but any helpful pointers would be appreciated.</p> | 5,690 | [
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] |
<p>So, here's a diagram with some galaxies.</p>
<p>I realize there are 500 billion galaxies out there (likely many more), but is there are fairly up-to-date diagram of all the galaxies, or a representative sample, at all observable distances?</p>
<p><img src="http://i.stack.imgur.com/uCejG.gif" alt="Hubble Diagram"></p> | 5,691 | [
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<p>It seems there is a term called "center of gravity of a band" in solid state physics or chemistry which I'm confused about. Could anyone give a formal definition of the term or point to some reference that have a clear explanation of it's meaning?</p> | 5,692 | [
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<p>I was just reading about quantum entanglement and the example was the Double-Slit Quantum Eraser Experiment. Then this was used as a basis for saying that particles might be half a universe apart and still be just as connected.</p>
<p>So I was just wondering if entanglement experiments have been done to show that this is really the case or if it is considered "academic" to conclude that there's no difference when the distances are huge? </p>
<p>As a side query, quantum mechanical interference seems to be based on the existence of probability - does this mean probability should be thought of as something as tangible as matter/energy?</p> | 5,693 | [
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<p>Considering that all frames are equivalent, isn't it up to the observer to say "The earth turns around the sun" or "The sun turns around the earth"? Isn't this more or less like arguing about which meridian should be considered the null-meridian, or whether time should be counted according to years before/after Jesus Christ or the conquest of Mecca? </p> | 205 | [
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<p>I've seen the rings from powder diffraction images, and I read that each line is made up of a lot of dots, I was wondering if these dots are reciprocal lattice points of the structure. And if we weren't looking at a powder but just a 2D hexagonal lattice, then the xrd pattern would simply be its reciprocal lattice? </p> | 5,694 | [
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<p>In P. Di Francesco, P. Mathieu, D. Snchal they fix the generators of the conformal group acting on a scalar field by somewhat arbitrarily defining
$$\Phi'(x)=\Phi(x)-i\omega_a G_a\Phi(x)$$
and by arbitrary I mean the sign. The "full" transformation would then be given by the exponentiation
$$\Phi'(x)=e^{-i\omega_a G_a}\Phi(x)$$
the generators are given in eq. 4.18 and the commutation relations are in eq. 4.19. But by looking at the action of the group in the coordinates, I think we could get the same representation 4.18 by requiring
$$x'^\mu=e^{ i\omega_aT_a}x^\mu=\alpha^\mu+\omega^\mu_{~~\nu} x^\nu + \lambda x^\mu +2(\beta \cdot x)x^\mu-\beta^\mu x^2$$
but the problem is that, in order to reproduce 4.18 the sign of the exponential is not the same as the one for $G_a$. I would have expected that the field transform in a similar way:
$$\Phi'(x)=e^{i\omega_a G_a} \Phi(x)$$
In fact it seems that there is not a "standard" definition. For example in this <a href="http://www.sciencedirect.com/science/article/pii/0003491669902784" rel="nofollow">article</a> by Mack and Salam, as well as in this <a href="http://rads.stackoverflow.com/amzn/click/0792341589" rel="nofollow">book</a> by Fradkin and Palchik, the generators as well as the commutation relations disagree among each other.</p>
<p>Anyway I would like to know how the generators should be defined so that everything is consistent from the Lie theory/representation theory point of view.</p> | 5,695 | [
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<p>Currently I'm doing Advanced Classical Mechanics courses. I'm finding it hard to understand due to the lack of knowledge in linear algebra, multi variable calculus and other chapters.</p>
<p>Can anyone suggest a mathematical book which is dedicated to teaching all the math that is used in physics?</p> | 185 | [
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<p>Are earthquakes getting less frequent across the centuries? I know that more seismic stations have register more earthquakes in the last century, but that doesn't imply there were more. I am interested on the geophysical side of it.</p>
<p>The logic is that things settle down, so there is less stuff to shuttle. Add to it that the earth (at its core) get colder with the passage of time. </p> | 5,696 | [
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<p>It is well known that scattering cross-sections computed at tree level correspond to cross-sections in the classical theory. For example the tree level cross-section for electron-electron scaterring in QED corresponds to scattering of classical point charges. The naive explanation for this is that the power of hbar in a term of the perturbative expansion is the number of loops in the diagram.</p>
<p>However, it is not clear to me how to state this correspondence in general. In the above example the classical theory regards electrons as particles and photons as a field. This seems arbitrary. Moreover, if we consider for example phi^4 theory than the interaction of the phi-quanta is mediated by nothing except the phi-field itself. What is the corresponding classical theory? Does it contain both phi-particles and a phi-field?</p>
<p>Also, does this correspondence extend to anything besides scattering theory?</p>
<p>Summing up, my question is:</p>
<blockquote>
<p>What is the precise statement of the correspondence between tree-level QFT and behavior of classical fields and particles?</p>
</blockquote> | 5,697 | [
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<p>An elementary example to explain what I mean. Consider introducing a classical point particle with a Lagrangian $L(\mathbf{q} ,\dot{\mathbf{q}}, t)$. The most general gauge transformation is $L \mapsto L + \frac{d}{dt} \Lambda(\mathbf{q},t)$ which implies the usual transformations of the canonical momentum $p \to p+ \nabla_q \Lambda$. Generalizing this derivative as an extended one gives the connection of electromagnetism. Once the particle motion is quantized, we recognize this as a local $U(1)$ "internal" symmetry of the quantum-mechanical phase.</p>
<p>Is this a fundamental property of gauges symmetries - being implied by non-dynamical symmetries of the action? By "non-dynamical symmetries" I mean those coming from the structure of and the freedom of labeling for the degrees of freedom under consideration. </p>
<p><strong>EDIT</strong>: After reflecting on the comments below, I'd re-formulate the question as: </p>
<blockquote>
<p>Do non-dynamical symmetries of a local Hamiltonian exhaust all
possible types of gauges fields that it can couple to in a
gauge-invariant manner?</p>
</blockquote>
<p><strong>EDIT-2</strong>: The reason I'm asking is that it appears that the very possibility for a particle to couple to electromagnetism and gravity come from the applicability of the action formalism and is already built-in as the symmetry under addition of a total time derivative (which as I understand to be one of the possible general definitions of gauge symmetry).</p>
<p>Some comments suggest the answer is a trivial yes, presumably because non-dynamical symmetries are gauge symmetries by definition. A concise expert answer would be helpful to close the question.</p> | 5,698 | [
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<p>For instance, given a theory and a formulation thereof in terms of a principal bundle with a Lie group $G$ as its fiber and spacetime as its base manifold, would a principle bundle with the Poincaré group as its fiber and $\mathcal{M}$ as its base manifold, where $\mathcal{M}$ is a manifold the group of whose isometries is $G$, lead to an equivalent formulation? Why? Why not?</p>
<p>On a related note, can any Lie group be realized as the group of isometries of some manifold? </p> | 5,699 | [
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<p>Many recent papers study entanglement in eigenstates of fermionic free hamiltonians (normally on a lattice) using the basic assumption that the reduced density matrices are <em>thermal</em> (e.g. <a href="http://arxiv.org/abs/cond-mat/0212631">Peschel 2003</a>). As I understand it, the theorem that we need in order to proceed with the computation of entanglement is the following: </p>
<p>Consider a fermionic state, pure or mixed, which fulfills Wick's theorem (i.e.: all N-point functions can be expressed from the 2-point correlation function). Then, the reduced density matrix for a block can always be expressed as $\rho_B \propto \exp(-\beta H)$, where $H=\sum_k c_k d_k^\dagger d_k$. The $d_k$ and $d^\dagger_k$ are suitable fermionic operators which can be expressed as linear combinations of the original ones.</p>
<p>The reference given is to <a href="http://www.sciencedirect.com/science/article/pii/0029558260902856">Gaudin 1960</a>. But that article only contains the reverse theorem: <em>if</em> a fermionic density matrix is thermal, then Wick's theorem follows.</p>
<p>So, my main question: Does anybody know about a proof of this theorem?</p>
<p>Moreover: I guess this theorem is related to the thermalization theorem of QFT on curved spacetimes. Is this true?</p> | 5,700 | [
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<p>I'm sure many of us are familiar with the following plot showing the running of the inverse of the fine-structure constants of the SM.</p>
<p><img src="http://i.stack.imgur.com/1PZin.png" alt="running of SM couplings">
(I got the picture from google)</p>
<p>At one-loop, the expressions for the fine-structure constants of the hypercharge, weak, and color gauge groups are,</p>
<p>$$\frac{1}{\alpha_1 (\mu)} \approx 59 - \frac{41}{20 \pi}\log[\frac{\mu}{M_Z}]$$
$$\frac{1}{\alpha_2 (\mu)} \approx 30 + \frac{19}{12 \pi}\log[\frac{\mu}{M_Z}],$$
$$\frac{1}{\alpha_3 (\mu)} \approx 8.5 + \frac{7}{2 \pi}\log[\frac{\mu}{M_Z}],$$</p>
<p>where the numerical values of $\alpha$'s in the first terms in RHS are at the mass of the Z boson $M_Z$. Now, if I plot these (from $\mu=100$ GeV to $\mu=2\times 10^16$ GeV) using Mathematica, I get this:</p>
<p><img src="http://i.stack.imgur.com/M3A9X.jpg" alt="enter image description here"></p>
<p>The slopes of the $log$'s in the previous equations do not produce the shape appearing in the first plot. </p>
<p>What am I missing here? How is the first plot, which we usually see in the literature, produced? </p>
<p><strong>Update</strong>: What I was missing is skills with Mathematica :p, Now I get the correct plot:
<img src="http://i.stack.imgur.com/KQcdc.jpg" alt="enter image description here"></p> | 5,701 | [
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<p>This is a question about terminology.</p>
<p>Given two vertex algebras $V_1$ and $V_2$ (= chiral CFTs), there are two kinds of maps $V_1\to V_2$ that one might want to consider.</p>
<p>1) Morphisms of VOAs that send the conformal vector of $V_1$ to the conformal vector of $V_2$. In other words, morphisms that send the copy of the Virasoro algebra in $V_1$ to the copy of the Virasoro algebra in $V_2$. Examples of those come from e.g. from conformal embeddings. This type of morphism forces the central charges of $V_1$ and $V_2$ to be equal.</p>
<p>2) Morphisms of VOAs that are not required to send the conformal vector of $V_1$ to the conformal vector of $V_2$.
An example of the above notion is the inclusion of the trivial one dimensional VOA into some other VOA. </p>
<p><b>Question:</b> is there some standard terminology used in physics to refer to those two kinds of morphisms between chiral CFTs?</p> | 5,702 | [
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<p>I am reading Cheng and Li. On page 9, it is written that the coefficient $\frac{1}{2 \cdot (4!)^2}$ for the second order term of the four point function becomes just $\frac{1}{2}$ for the following diagram:</p>
<p><img src="http://i.stack.imgur.com/hzbgJ.png" alt="enter image description here"></p>
<p>I am not getting that answer. Call the vertices in the diagram $y_1$ and $y_2$. Imagine $y_1$ and $y_2$ as a point with four lines coming out. (As it is $\phi^4$) My reasoning is as follows: $x_1$ can be connected to any one of the four lines emanating, after that $x_4$ has three choices for joining to one of the vertices. Ways of joining $x_1$ and $x_4$ is $4 \times 3$. Similarly for $x_2$ and $x_3$. Now $y_1$ and $y_2$ have two lines left which can be joined in a loop in two ways So total number of ways is $4 \times 3 \times 4
\times 3
\times 2$ . So I am getting the coeffecient as $\frac{4 \times 3 \times 4
\times 3
\times 2}{2 \times (4!)^2}=\frac{1}{4}$. </p>
<p>Where am I going wrong? </p> | 5,703 | [
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<p>Consider an adiabatic system as follows. It consists of a gas in a container and a piston.
Initially, the system is at equilibrium and the gas inside it occupies a volume $V_i$ at a pressure $p_i$ which is equal to the outside pressure. Suddenly, the outside pressure changes and reduces to $p_{atm}$. The piston moves to equalize the pressure and the gas expands isothermally to obtain equilibrium. The gas now occupies a volume $V_f$ at a pressure of $p_f$ which is equal to $p_{atm}$</p>
<p>Now, two textbooks I have define the work done from two different viewpoints.</p>
<p><strong>1: From the viewpoints of the surroundings</strong>: </p>
<p>The work done on the system by the surroundings equals $-p_{atm}\Delta V$. Since $p_{atm}$ is pretty much constant for the whole of the whole of the process, we can say that the work done equals:</p>
<p>$$W_1 = -p_{atm}(V_f - V_i)\tag1$$</p>
<p><strong>2: From the viewpoint of the system:</strong></p>
<p>We can write the internal pressure of the system as a function of its volume: $p_{in}(V)$. As during the expansion, the internal pressure changes, the work done by the system equals </p>
<p>$$W_2 = \int_{V_i}^{V_f}p_{in}(V)dV$$</p>
<p>Now, I don't know which definition to use. The work is done by the system (from definition 2) is done on the surroundings. But what about the negative work that the surroundings did on the system? Where did that energy go? Maybe, the two definitions express the same thing: the work done by the system. The negative sign in definition 1 signifies that the work is done by the system. But that would mean that </p>
<p>$$W_1 = W_2$$</p>
<p>We can simplify $W_2$ as follows. Clearly, </p>
<p>$$p_{in}(V) = \frac{p_iV_i}{V}$$</p>
<p>$$W_2 = \int_{V_i}^{V_f}\frac{p_iV_i}{V}dV = p_iV_i \ln{\frac{V_f}{V_i}} \tag2$$</p>
<p>The internal pressure when the volume is equal to $V_f$ is $p_{atm}$</p>
<p>$$\implies p_{atm} = p_{in}(V_f) = \frac{p_iV_i}{V_f}$$</p>
<p>$$\implies V_f = \frac{p_iV_i}{p_{atm}}$$</p>
<p>Putting this into $(1), (2)$ gives us, </p>
<p>$$W_2 = p_iV_i\ln{\frac{p_{i}}{p_{atm}}}$$</p>
<p>$$W_1 = -p_{atm}\left(\frac{p_iV_i}{p_{atm}} - V_i\right) = -V_i(p_{atm} - p_i)$$</p>
<p>Consider some values. Let $p_i = 5 \ \rm{Pa}, V_i = 1 \ \rm{m^3}, p_{atm} = 1 \ \rm{Pa}$.</p>
<p>$$W_1 = -1\cdot(1-5) = 4$$</p>
<p>$$W_2 = 1\cdot1\cdot\ln{\frac{5}{1}} = 1.609$$</p>
<p>Where am I making a mistake?</p> | 5,704 | [
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] |
<p>Does there exist a single plate capacitor(conductor)? <em>if yes</em></p>
<p>How will you define the <strong>capacitance</strong> and potential(difference)
of such conductor?</p> | 5,705 | [
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<p>Due to the <a href="https://en.wikipedia.org/wiki/Propagator" rel="nofollow">Wiki article</a>, "...In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum...".</p>
<p>Let's have the expression for the propagator of some field $\hat {\Psi}_{l}$ in the free theory:
$$
\tag 1 D_{lm}(x - y) = \langle |\hat{T}\left( \hat {\varphi}_{l}(x)\hat {\varphi}_{m}^{\dagger}(y)\right) |\rangle = -\frac{i}{(2 \pi )^{4}}\int \frac{F_{lm}(p)e^{-ip(x - y)}d^{4}p}{p^{2} - m^{2} - i\varepsilon}.
$$
Let's use two examples: spinor and scalar fields: for them $(1)$ takes the form
$$
\tag 2 D_{lm}(x - y) = D(x - y) = -\frac{i}{(2 \pi )^{4}}\int \frac{e^{-ip(x - y)}d^{4}p}{p^{2} - m^{2} - i\varepsilon},
$$
$$
\tag 3 D_{lm}(x - y) = -\frac{i}{(2 \pi )^{4}}\int \frac{(\gamma^{\mu}p_{\mu} + m)_{lm}e^{-ip(x - y)}d^{4}p}{p^{2} - m^{2} - i\varepsilon}
$$
respectively.</p>
<p>How to convert $(2), (3)$ into the probability (which lies at interval $[0, 1]$)? Particularly i don't understand what to do with spinor indices in $(3)$ (the probability must be Lorentz scalar, so I need to sum over the indices (?)). And also, why doesn't probability depends on momentum (the propagator doesn't contain info about momentum)?</p> | 5,706 | [
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<p>Consider a particle raised to height $h$ by applying a constant force of $F$ vertically opposite to the direction of force of gravity on it. The potential energy of the particle is</p>
<p>$$W = mgh$$</p>
<p>But, according to the definition of work, the work done on the particle is </p>
<p>$$W' = Fh$$</p>
<p>According to conservation of energy, $W = W'$. But what if $F>mg$? </p> | 5,707 | [
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<p>In equation 7 of the paper, <a href="http://arxiv.org/abs/1301.2326" rel="nofollow">Feynman's clock, a new variational principle, and parallel-in-time quantum dynamics</a>, the clock register is different from Feynman's <a href="http://www.cs.princeton.edu/courses/archive/fall05/frs119/papers/feynman85_optics_letters.pdf" rel="nofollow">original proposal</a>. According to this paper, for a four clock steps quantum circuit, the sequence of the state of the clock register will be $|00\rangle \to |01\rangle \to |10\rangle \to |11\rangle$. It infers that we will need a two qubit register. But according to Feynman's original proposal, it should be $|000\rangle \to |100\rangle \to |010\rangle \to |001\rangle$ i.e. a three qubit clock register is needed.</p>
<p>I failed to understand if there is any specific reason behind this modification. Any help?</p> | 5,708 | [
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<p>I am creating a data acquisition software, based on <a href="http://www.sparrowcorp.com/products/software" rel="nofollow">Sparrow's Kmax</a>. There I would like to add a feature that will show the system's dead time.</p>
<p>I have already a text field that shows the real time and live. My question has to do, with the dead time.</p>
<p>Is it right to say that the dead time is</p>
<p>$\text{dead time}=\dfrac{\text{total time of recording events}}{\text{total time of acquisition}}=\dfrac{\text{live time}}{\text{real time}}$</p> | 5,709 | [
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<ol>
<li><p>According to theory of time dilation, flow of time slows down significantly at the speed of light.Is there any conditions practically or theoretically when flow of time is reduced to zero means it comes to still?</p></li>
<li><p>Whether time & space are infinite?</p></li>
</ol> | 5,710 | [
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<p>Executive summary: "Collapse of the wave function" is inherently supraluminal.</p>
<p>I suggest an easier thought experiment to demonstrate the apparently supraluminal (or FTL) aspect of a quantum mechanical measurement, without spin and polarizers.</p>
<p>A photon is sent through a beam splitter. The photon traverses both arms of the splitter, and is "everywhere" (or "nowhere") until it is detected at one end or the other. According to quantum mechanics, there are no hidden variables. </p>
<p>The two arms of the splitter extend a significant distance (hundreds of meters) in opposite directions. A single photon is injected at a specified time. A short time later the photon is detected at one or the other of the two detectors, labeled A and B, each a random half of the trials. </p>
<p>Clearly the two detectors do not have time to confer on the outcome. Suppose the photon was detected at A. How did detector B instantly "know" that the photon must not appear there?</p> | 5,711 | [
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<p>We know that there is Adler and Bell-Jackiw(ABJ) type anomalies for fermions. In some case, the ABJ anomaly affecs particle physics pheonomelogy, such as pion decays or kaon decays(in the case of pion, we still have a calculation on left/right chiral fermions running on the 1-loop triangle diagram). In some other case, there is 1+1D QED Schiwinger or axial anomaly for chiral fermions. The commutation of fermionic anomaly is usually done by, either a 1-loop Feynman diagram, or a Fujikawa path integral method.</p>
<p>The above may be some examples of anomalies for fermions.</p>
<p><strong>Is there any example of quantum anomalies for bosons (pure bosonic systems)?</strong></p> | 5,712 | [
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<p>In condensed matter, one usually considers Bloch states inside the first Brilliouin zone, which, for 1d system with lattice constant $a$, is $-\pi/a<k<\pi/a$.</p>
<p>But the basis of this, Bloch theorem, is based on periodic boundary condition, which implies $k=\frac{2\pi n}{Na}$ with $n=0,1,\cdots N-1$ and $N$ is the number of unit cells. This interval of $k$ is not the same as the interval given by first Brilliouin zone, which is fine if one just wants to consider the dispersion because dispersion is periodic. However, there may be some problems. For example, if we do a Fourier transformation numerically, we need to use $k=\frac{2\pi n}{Na}$ with $n=0,1,\cdots N-1$ instead of $-\pi/a<k<\pi/a$, otherwise we are not really using periodic boundary condition.</p>
<p>Is this right? And is there any more subtle point on which one may easily make mistake?</p> | 5,713 | [
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<p>The colloquial explanation is that the spacetime in front of a ship contracts and the spacetime behind expands. I see how one could <em>think</em> that this would bring you closer, but I don't see that it actually does.</p>
<p>Let's look at the front of the bubble. So spacetime "contracts". If you look at a differential volume being run over, I would think it contracts to a minimum, <em>but then must expand again</em> to pass under the ship. If the bubble is moving, but the ship inside is not accelerating (let's assume it's also motionless in its space), then spacetime must be "piling up" in a (compressed) region in front, because you can't throw away the extra spacetime.</p>
<p>To paraphrase <em>A Wrinkle in Time</em>'s analogy: think of some fabric and a bug that wants to get from where it is to somewhere else. You can fold the fabric and jump across the gap, but what I'm thinking of the Alcubierre drive doing is "scrunching up" the fabric: the bug still needs to walk over the same length of material, it's just that it <em>looks like</em> less distance.</p>
<p>Is all the above bogus? I don't really know what I'm doing with the physics here, as I have more of a mathematical background. I'm not seeing a way around this without introducing some discontinuity in space, which (I don't think) was the point. I also read somewhere that the expansion/contraction analogy could be misleading; is this why?</p> | 5,714 | [
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<p><img src="http://i.stack.imgur.com/7NVNH.png" alt="enter image description here"></p>
<blockquote>
<p>A rod of mass $m$ and length $l$ is pivoted at one end to ceiling and free to rotate in the vertical plane. A disc of radius $R$, which is less than $l$, can be fixed at its other end in 2 ways :</p>
<p>$1$ Disc is free to rotate about its centre</p>
<p>$2$ Disc cannot rotate about its centre</p>
<p>Compare moment of inertia of the 2 systems about the pivoted end.</p>
</blockquote>
<p>I have been thinking for several hours and can't think why they would be different. There is no torque on disc(about centre) and hence there should be no difference between 2 cases</p>
<p>I found a similar question here but that uses Lagrangian which I have not studied (I'm in XII standard). Please explain why the moment of inertia will be different in 2 cases.</p> | 5,715 | [
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<p>i read Landau's book recently. In this book p.43</p>
<p>Landau says from (16.1) (16.2)</p>
<p>can be write down $T_10$= $p_0^2$/2$m_1$=($M-m_1$)($E_i-E_1i-E_i'$)/$M$</p>
<p>For me, it is hard to understand the factor ($M-m_1$)/$M$.</p>
<p>Can someone give me a hand to let me understanding master's idea?</p>
<p>If you can give me a easy example, i will appreciate you.</p>
<p>thx for your help.</p> | 5,716 | [
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<p>Minecraft is a game of mining where one digs and finds ore (to put it simply). the world is composed of "blocks" 3 dimensional cubes that have different "pictures" on them, that make them appear like grass or rock or Steele or diamonds...ect. In minecraft there is also TNT which is a block just like all others except it explodes. That does not relate to the question however which is, How do i find the gravitational constant? I mostly need the formula, which my 10th grade physics class has not taught me yet so i can find the neccesary factors to solve it. Aside from just giving me the formula what ways do you think I could find G? Also, just for more info, most block dont fall but sand and gravel do so they could be key to this problem perhaps.</p> | 5,717 | [
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<p>I assume that the reason water freezes is because as you decrease the temperature, the kinetic energy of the water molecules decreases and the dipole bonding potential eventually over comes the escape velocity of the molecules and they form a crystal structure. What would happen if you attempted to freeze water in the presence of a very powerful electric field? Will the water molecules align to the field and make it harder to form a solid or easier to form a solid?</p> | 5,718 | [
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<p>Despite the fact that zinc is more electronegative than copper,why it will lose electrons more readily than copper in the voltaic cell due to sulphate ions.</p> | 5,719 | [
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<p>If quarks and leptons carried flavor charges that differed across generations (as they do in some theories), then could mixing take place?</p> | 5,720 | [
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<p>I have solved the following problem from Griffiths "Introduction to Quantum Mechanics". </p>
<p>Consider the wavefunction:
$\Psi (x,t) = A e^{-\lambda |x|} e^{-i\omega t} $</p>
<p>Normalize $\Psi$. </p>
<p>Now, we want $ \int_{-\infty}^\infty |\Psi (x,t)|^2 dx = 1$</p>
<p>It is fairly straightforward, where the modulus is $|\Psi (x,t)|= r = A e^{-\lambda |x|}$. Therefore I square $r$ and integrate. I deal with the absolute value sign by multiplying by $2$ and integrating from 0 to $\infty$, while dropping the absolute value sign, to get:</p>
<p>$ 2\int_0^\infty (A e^{-\lambda x})^2 dx$</p>
<p>This should give me a factor of $A^2$ which I can take outside the integral sign. However, instead of a simple $A^2$, the solution gives an $|A|^2$. I don't understand where the absolute value sign came from. After all, taking the above expression $r$ as being equal to $|\Psi(x,t)$|, the modulus has already been dealt with. </p> | 5,721 | [
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<p>I understand that the concept of an average of a data list means finding a certain value 'x', which ensures that the sum of the deviations of the numbers on the left of 'x' and on the right of 'x' yields the value 0. Assume we are given the values 2,6,9 and 12. I label these values from left to right in the sequence I have provided as: a,b,c and d respectively. The definition of finding these values' average (let's call this average value 'x') in mathematical form is: <br><br>$${(x - a) + (x - b) = -(x - c) - (x - d)}$$<br><br>
So this equation can be reduced to:<br><br>$$\Large x = \frac{(a + b + c + d)}{4}$$<br><br>
This will provide you with the value for the average of any number of values. Similarly there is a formula in physics that calculates the average velocity from a graph by dividing the sum of the initial and final velocities by 2.<br><br>$$\Large\frac{v_f + v_i}{2} = \bar{v}$$<br><br>There is also another formula which calculates the average velocity from a graph by calculating the slope of the line through two points on the graph. This is done by dividing the change in position by the change in time:
<br><br>$$\Large \frac{x_f - x_i}{t_f - t_i} = \bar{v}$$<br><br>
I can understand the third equation from the top of the page, as I can understand this calculation of average velocity in terms of what I explained in the first two equations from the top of the page. I am unable to reconcile the calculation of the average of velocities as seen in the last equation with the method as listed above in the first two equations. Can someone help me solve this dilemma?</p> | 5,722 | [
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<p>I want to check whether my result for the invariant amplitude of the electron-electron scattering (to lowest order in $\alpha$; t+u channels) is correct or not. </p>
<p>I can't find any reference that has the result explicitly. Can someone point out some kind of database of scattering amplitudes? </p>
<p><strong>Edit:</strong> for completeness I post my result (which might have an error)
$$|\mathcal{M}|^2=\frac{2e^4}{u^2t^2}\left((s^2-8m^4)[(t+u)^2+u^4+t^4]+8m^2ut(4m^2-3s)\right)$$</p>
<p><strong>Update:</strong>
I managed to do the calculation using CompHEP+Mathematica. </p>
<p>On CompHEP I selected the QED model and calculated the diagrams for $e^-e^- \rightarrow e^-e^-$ from which I get the two contributions to the process (t+u channels). Then I exported the symbolic computation of the squared diagrams to Mathematica code which gives
$$ \frac{2e^4}{t^2 \left(-4 m^2+s+t\right)^2} (64 m^8+16 m^6 (t-6 s)+4
m^4 \left(13 s^2+3 s t+3 t^2\right)-4 m^2 \left(3
s^3+3 s^2 t+3 s t^2+2 t^3\right)+\left(s^2+s
t+t^2\right)^2)$$
where the denominator is clearly $t^2u^2$ by using $s+t+u=4m^2$ but the rest is not so trivial to put in the same form as my equation for $|\mathcal{M}|^2$. So if I substitute the value of $u$ on my first equation I get something that doesn't look anything like the second equation. Therefore my first equation is wrong. </p>
<p>Note: I compared the second equation with the expression for the differential cross section of the Møller scattering from a book and it is consistent.</p> | 5,723 | [
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<p>I'm slightly confused as to answer this question, someone please help:</p>
<p>Consider a free particle in one dimension, described by the initial wave function
$$\psi(x,0) = e^{ip_{0}x/\hbar}e^{-x^{2}/2\Delta^{2}}(\pi\Delta^2)^{-1/4}.$$</p>
<p>Find the time-evolved wavefunctions $\psi(x,t)$.</p>
<p>Now I know that since it is a free particle we have the hamiltonian operator as $$H = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2},$$ which yields the energy eigenfunctions to be of the form $$\psi_E(x,t) = C_1e^{ikx}+C_2e^{-ikx},$$ where $k=\frac{\sqrt{2mE}}{\hbar}$, and the time evolution of the Schrödinger equation gives $$\psi(x,t)=e^{-\frac{i}{\hbar}Ht}\psi(x,0)$$ but the issue I face is what is the correct method to find the solution so that I can then calculate things such as the probability density $P(x,t)$ and the mean and the uncertainty (all which is straight forward once I know $\psi(x,t)$.</p>
<p>In short - how do I find the initial state in terms of the energy eigenfunctions $\psi_E(x,t)$ so that I can find the time evolved state wavefunction.</p> | 5,724 | [
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<p>Since you would know that light always travels at the constant velocity with respect to all frame of reference ....according to relativity whenever we are traveling at speed of light our time with respect to relative rest observer would become stopped ..if. it means light travels with respect to all frame of reference at light speed, so it implies that light from the sun would never reach us ..but sadly it would reach us within 8 minutes ..How is that possible?</p> | 5,725 | [
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<p><a href="http://physics.stackexchange.com/q/98189/">Here</a> is a question about the canonical momentum that I had asked some days ago, but I still have one point that I am not understand.</p>
<p>Considering a particle moves in a magnetic field with charge $q$ and mass $m$, its hamiltonian is $$H=\frac{\vec{P}^2}{2m}=\frac{(\vec{p}+q\vec{A})^2}{2m}$$
where $\vec{p}$ is the momentum of the particle, $\vec{A}$ is the vector potential of the magnetic field and $\vec{P}$ is the canonical momentum of the particle.</p>
<p>I think, because of the expression of the hamiltonian, the canonical momentum $\vec{P}$ is a conserved quantity.</p>
<p>But by the answer in <a href="http://physics.stackexchange.com/q/98189/">the previous link</a>, it seems that the canonical momentum is not conserved even in a simple example that a particle moves in a homogeneous magnetic field.</p>
<p>I am confused about this question. Is the canonical momentum conserved when a particle moves in magnetic field? </p> | 5,726 | [
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<p>I understand fresnel lenses are manufactured using CNC machines. I was wondering, if it would be possible to use sound to vibrate liquid silicon and then fast cool it into the standard form of a fresnel lens? This occurred to me when I noticed that ripples on a pond resemble the lenses in question, and that certain frequencies can cause liquids to vibrate.
So, hypothetically, could it be done, and if so, would it be faster than machining?</p> | 5,727 | [
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<p>Does string theory contradict the theory of <a href="http://en.wikipedia.org/wiki/Preon" rel="nofollow">preons</a>, especially the <a href="http://en.wikipedia.org/wiki/Rishon_model" rel="nofollow">Harari-Shupe</a> one?</p> | 5,728 | [
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<p>I am trying to understand how to combine uncertainties when they are dependent and independent from each other.</p>
<p>Using this formula :</p>
<p>$$\delta z = \sqrt {\Biggl(\dfrac{\partial f}{\partial x} \delta x\Biggr)^2+\Biggl(\dfrac{\partial f}{\partial y} \delta y\Biggr)^2+2\Biggl(\dfrac{\partial f}{\partial x}\cdot \dfrac{\partial f}{\partial y}\Biggr)\text{cov}(x,y)}$$</p>
<p>Intuitively, if the <a href="http://en.wikipedia.org/wiki/Covariance" rel="nofollow">covariance</a> between the two is zero, the last term will disappear and the equation just becomes the square root of sums for combining uncertainty.</p>
<p>My question is does $\delta z$ then need to be divided by 2 to get the final uncertainty value.(i.e. divide by $N$)</p> | 5,729 | [
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<p>I have googled for the "meaning of universe" where I found the following:
"all existing <strong>matter and space</strong> considered as a whole; the cosmos. The universe is believed to be at least 10 billion light years in diameter and contains a vast number of galaxies; it has been expanding since its creation in the Big Bang about 13 billion years ago."</p>
<p>Now,I have heard about big bang theory which tells that universe was initially a point of infinite density .But in the above definition I see space is also included in the universe. so if the universe was initially a point then <strong>what was outside that point?</strong> And <strong>how this point is expanding if there is nothing outside it?</strong> That is why I don't understand the balloon analogy also which is expanding in space. But here the universe itself contains the space according the google definition.</p>
<p>Probably I have this question in my mind because I couldn't visualize how the space meets on itself in 3 dimension like earth's surface does on 2 dimension.</p> | 62 | [
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<p><a href="http://physics.stackexchange.com/a/88984/7665">In this answer to a question about viscosity of water in the presence of solutes</a>, the Falkenhagen relation is given:</p>
<blockquote>
<p>$$\frac{\eta_s}{\eta_0}=1+A\sqrt{c}$$ where $\eta_s$ is the solution
viscosity, $\eta_0$ the solvent viscosity, $A$ a constant that depends
on the electrostatic forces on the ions, and $c$ the concentration of
the solute.</p>
</blockquote>
<p>Now, in what order of magnitude would I assume A? Can one make general statements which group of solutes will be around which order of magnitude, like "expect A to be around $10^{-8}$ for single atom salts and $10^{-5}$ for long chained organics" (numbers plucked out of thin air) or similiar? </p>
<p>p.s. The relation is quoted from a paywalled paper that I can't access.</p> | 5,730 | [
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<p>According to my understanding of black hole thermodynamics, if I observe a black hole from a safe distance I should observe black body radiation emanating from it, with a temperature determined by its mass. The energy from this radiation comes from the black hole's mass itself.</p>
<p>But where (in space-time) does the process of generating the Hawking radiation happen? It seems like it should be at the event horizon itself. However, here is a Penrose diagram of a black hole that forms from a collapsing star and then evaporates, which I've cribbed from <a href="http://motls.blogspot.jp/2008/11/why-can-anything-ever-fall-into-black.html" rel="nofollow">this blog post</a> by Luboš Motl.</p>
<p><img src="http://i.stack.imgur.com/Qtjrx.png" alt="enter image description here"></p>
<p>On the diagram I've drawn the world-lines of the star's surface (orange) and an observer who remains at a safe distance and eventually escapes to infinity (green). From the diagram I can see how the observer can see photons from the star itself and any other infalling matter (orange light rays). These will become red-shifted to undetectably low frequencies. But it seems as if any photons emitted from the horizon itself will only be observed at a single instant in time (blue light ray), which looks like it should be observed as the collapse of the black hole.</p>
<p>So it seems that if I observe photons from a black hole at any time before its eventual evaporation, they must have originated from a time before the horizon actually formed. Is this correct? It seems very much at odds with the way the subject of Hawking radiation is usually summarised. How is it possible for the photons to be emitted <em>before</em> the formation of the horizon? Does the energy-time uncertainty relation play a role here?</p>
<p>One reason I'm interested in this is because I'd like to know whether Hawking radiation interacts with the matter that falls in to the black hole. There seem to be three possibilities:</p>
<ol>
<li>Hawking radiation is generated in the space-time in between the black hole and the observer, and so doesn't interact (much, or at all) with the infalling matter;</li>
<li>Hawking radiation is generated near to the centre of the black hole, at a time before the horizon forms, and consequently it does interact with the matter.</li>
<li>The Hawking radiation is actually <em>emitted</em> by the infalling matter, which for some reason is heated to a very high temperature as it approaches the event horizon.</li>
<li>(With thanks to pjcamp) you can't think of them as coming from a particular point, because they are quantum particles and never have a well-defined location.</li>
</ol>
<p>All these possibilities have quite different implications for how one should think of the information content of the radiation that eventually reaches the observer, so I'd like to know which (if any) is correct. </p>
<p>The fourth possibility <em>does</em> sound like the most reasonable, but if it's the case I'd like some more details, because what I'm really trying to understand is whether the Hawking photons can interact with the infalling matter or not. Ordinarily, if I observe a photon I expect it to have been emitted by something. If I observe one coming from a black hole, it doesn't seem unreasonable to try and trace its trajectory back in time and work out when and where it came from, and if I do that it will still <em>appear</em> to have come from a time before the horizon formed, and in fact will <em>appear</em> to be originating from surface of the original collapsing star, just before it passed the horizon. I understand the argument that the infalling matter will not experience any Hawking radiation, but I would like to understand whether, <em>from the perspective of the outside observer</em>, the Hawking radiation appears to interact with the matter falling into the black hole. Clearly it <em>does</em> interact with objects that are sufficiently far from the black hole, even if they're free-falling towards it, so if it doesn't interact with the surface of the collapsing star then where is the cutoff point, and why?</p>
<p>In an answer below, Ron Maimon mentions "a microscopic point right where the black hole first formed," but in this diagram it looks like no radiation from that point will be observed until the hole's collapse. Everything I've read about black holes suggests that Hawking radiation is observed to emanate from the black hole continuously, and not just at the moment of collapse, so I'm still very confused about this.</p>
<p>If the radiation <em>is</em> all emitted from this point in space-time, it seems like it should interact very strongly with the in-falling matter:</p>
<p><img src="http://i.stack.imgur.com/OaYEI.png" alt="enter image description here"></p>
<p>In this case, crossing the event horizon would not be an uneventful non-experience after all, since it would involve colliding with a large proportion of the Hawking photons all at once. (Is this related to the idea of a "firewall" that I've heard about?)</p>
<p>Finally, I realise it's possible that I'm just thinking about it in the wrong way. I know that the existence of photons is not observer-independent, so I guess it could just be that the question of where the photons originate is not a meaningful one. But even in this case I'd really like to have a clearer physical picture of the situation. If there is a good reason why "where and when do the photons originate?" is not the right question, I'd really appreciate an answer that explains it. <sub>(pjcamp's answer to the original version of the question goes some way towards this, but it doesn't address the time-related aspect of the current version, and it also doesn't give any insight as to whether the Hawking radiation interacts with the infalling matter, from the observer's perspective.)</sub></p>
<p><strong>Editorial note:</strong> this question has been changed quite a bit since the version that pjcamp and Ron Maimon answered. The old version was based on a time-symmetry argument, which is correct for a Schwartzchild black hole, but not for a transient one that forms from a collapsing star and then evaporates. I think the exposition in terms of Penrose diagrams is much clearer.</p> | 5,731 | [
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<p>How large in the night sky would Saturn look from Titan's surface? I believe they are tidally locked.</p> | 5,732 | [
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<p>Why 1 <a href="http://en.wikipedia.org/wiki/Astronomical_unit" rel="nofollow">AU</a> is defined as the distance between the Sun and the Earth? (approximately if you like to be precise) </p>
<blockquote>
<p>An astronomical unit (abbreviated as AU, au, a.u., or ua) is a unit of length equal to about 149,597,870.7 kilometres (92,955,807.3 mi) or approximately the mean Earth–Sun distance.</p>
</blockquote>
<p>Shouldn't <a href="http://en.wikipedia.org/wiki/Astronomical_unit" rel="nofollow">astronomical units</a> be defined within metric units (that is, $10^x$), so we can understand massive distances a little easier?</p> | 5,733 | [
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<p>With the recent lunar eclipse, for some reason this question came to me: The reason the Moon turns red is that the only appreciable sunlight hitting it when it's in Earth's umbra is refracted through Earth's atmosphere and shorter wavelengths are scattered out. I couldn't think of any reason why this shouldn't happen with, say, the space shuttle, ISS, or another satellite, but I've never seen any photograph nor discussion of it.</p>
<p>It <em>should</em> happen, right? And if so, have there been any photos of it?</p> | 5,734 | [
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<p>The effect of gravitational waves in transverse traceless gauge on matter is represented by the expansion and contraction of a ring of test particles in the direction of polarization of the wave. </p>
<p>This result is obtained by choosing a gauge which in GR is a choice of a coordinates.</p>
<p>Does that mean that choosing another coordinate system(moving observers) would experiment time dilatation plus the effects of streching and contracting?</p>
<p>EDIT:</p>
<p>When the weak field approximation is used and the metric is split into $g_{ab}=\eta_{ab}+h_{ab}$ the gauge transformations are no longer coordinate transformations and instead define equivalence classes of symmetric tensors in the flat Minkowski background.</p>
<p>Then two Lorentzian frames are related by $h_{ab}=\Lambda_{a}^{\mu}\Lambda_{b}^{\nu}h_{\mu\nu}$, so uniform moving observers will measure time dilatations different from that of Minkoswki when the wave is passing. </p>
<p>Is this correct?</p>
<p>What about accelerated observers? Are there any extra effects in the full non-linear case?</p> | 5,735 | [
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<p>I'm programming a game where different types of objects will be sliding over different types of terrains (Top-down in two dimensions). At my current level of physics education we are given the coefficient of friction between any pairing of surfaces, but that's not practical for the amount of objects and terrains this game (will) have.</p>
<p>So my questions are, </p>
<ul>
<li>How do I find the force of friction between two objects, when I have their surface area and any other constants I need to add? </li>
<li>How can I somewhat accurately simplify that to something on the order of $z=x*y+a*b$ ? </li>
<li>I'd like to account for static vs kinetic friction, is there an easy way to do that?</li>
</ul>
<p>Edit: failing any established approximations, if I make up some abitrary "grippiness" constant for all of the surfaces, what's the best way to combine them into a coefficient of friction?</p> | 5,736 | [
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<p>Can a photon/EM-field-excitation redshifted by spatial expansion be completely dissipated? Does the energy reach a minimum value (Planck's constant) and continue on as normal? Does expansion also cause energy to be lost from travelling massive particles, as they too have a "wavy" aspect? From an informational point of view, where does the info go?</p> | 5,737 | [
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<p>A conductor is placed in a magnetic field, current starts to flow, and the wire experiences the Lorentz force of $x$ Newtons. However, what confuses me is that $x$ is a <strong>value</strong> that is calculated from $F = IL \times B$. If $-EMF$ is induced due to the change in exterior magnetic field(due to the current now flowing in the conductor), shouldn't $I$ be reduced thus total $F$ (which is $x$) should be less than initially calculated? </p>
<p>Also,if two wires have current flowing within them, as they initially "start" when current "beings" to flow, shouldn't there be $EMF$ induced? </p>
<p>And what occurs first? The magnetic force or $EMF$ or do they occur simultaneously?</p>
<p>I myself am confused about this notion, so this might sound a bit absurd please do state if it is unclear. I will review this and try to make it as clear as possible.</p> | 5,738 | [
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0.029960954561829567,
0.004270059056580067
] |
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