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<p>A ball rests on a smooth surface. The ball's particles are in constant motion. So are the particles of the floor. Some of the ball's particles collide with the floor's particles and transfer kinetic energy. But overall, the kinetic energy of the floor and the ball is constant. They are at thermal equilibrium. </p>
<p>But, according to probability, there is an infinitesimally small chance, but a chance nonetheless, that the particles of the floor align in such a manner that they bump all the particles in the ball just the right way, so that it bounces of the ground.</p>
<p>Notice, the law of conservation of energy is not violated here. The temperature of the floor will decrease in doing so. But the second law of thermodynamics is violated. Entropy is decreasing. If the ball were smaller (much smaller), the probability of this happening would be much larger. </p>
<p>How is this possible? </p>
| 3,276 |
<p>Do you know the last works about <a href="http://en.wikipedia.org/wiki/Bystander_effect_%28radiobiology%29" rel="nofollow">Bystander Effect</a> on cells caused by radiations? I have found research papers until 2004.</p>
| 3,277 |
<p>Equation for an expectation value $\langle x \rangle$ is known to me: </p>
<p>\begin{align}
\langle x \rangle = \int\limits_{-\infty}^{\infty} \overline{\psi}x\psi\, d x
\end{align}</p>
<p>By the definition we say that expectation value is a sandwich: $\langle \psi|\hat{x}|\psi\rangle$. So: </p>
<p>\begin{align}
\langle \psi|\hat{x}|\psi \rangle = \int\limits_{-\infty}^{\infty} \overline{\psi}x\psi\, d x
\end{align}</p>
<hr>
<p>Can you first confirm that these three lines are correct (I am not sure if I understand <a href="http://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation" rel="nofollow">Dirac's bra-ket notation</a> right). If they are wrong please explain: </p>
<p>\begin{align}
\text{1st:}& & \langle \psi | \hat{x} | \psi \rangle &= | \psi\rangle \cdot \hat{x}|\psi \rangle\\
\text{2nd:}& & \langle \psi | \hat{x} | \psi \rangle &= {\langle \psi|}^\dagger \cdot \hat{x}|\psi \rangle\\
\text{3rd:}& & \langle \psi | \hat{x} | \psi \rangle &= {\langle \psi|}^\dagger \cdot \hat{x} \langle\psi |^\dagger\\
\end{align}</p>
<p>How do i derive relations $\langle\psi|\hat{x}|\psi\rangle = \langle \psi |\hat{x}\psi\rangle$ and $\langle\psi|\hat{x}|\psi\rangle = \langle \hat{x}^\dagger\psi |\psi\rangle$?</p>
| 3,278 |
<p>Sorry if this is the wrong place to post this but it seemed the most appropriate.</p>
<p>I read a riddle earlier today that mentioned something about shadows underwater and it got me thinking.. Would a shadow under water/ in the ocean technically be "wet"?</p>
<p>I understand that light isn't matter, but something. </p>
<p>I'm no good with physics myself but it's just a question that's been playing on my mind and would like to know if anyone has a logical answer to it?</p>
<p>Thank in advance</p>
| 3,279 |
<p>I stumbled across this article <a href="http://blogs.scientificamerican.com/cross-check/2010/12/21/science-faction-is-theoretical-physics-becoming-softer-than-anthropology/" rel="nofollow">http://blogs.scientificamerican.com/cross-check/2010/12/21/science-faction-is-theoretical-physics-becoming-softer-than-anthropology/</a></p>
<p>It got me thinking. Why do we really care to make predictions, if we can't falsify them? </p>
<p>I'm certainly not talking about all of theoretical physics, I'm very aware of it's usefulness, but it seems that we're wasting resources on untestable hypotheses. Does it really matter a great deal about what happens inside a black hole, or about Hawking radiation? These are, in my opinion, ideas that won't be testable for sometime, if at all. I (and other physics students in my classes) question if this is really a constructive way to spend our time. </p>
<p>I'm just a first year (almost second!) year physics student, so perhaps something has evaded my attention? I don't mean to seem confrontational, so please don't start a flame war. </p>
| 3,280 |
<p>Please note, I am only interested in classical mechanics discussion on this. Please do not involve quantum mechanics.</p>
<p>Inspired by this question: <a href="http://physics.stackexchange.com/questions/822/is-angular-momentum-truly-fundamental/4045">Is Angular Momentum truly fundamental?</a></p>
<p>My question is:</p>
<p>Can there be a concept of angular momentum separate from "orbital" angular momentum in classical mechanics? For example, can there be a thing such as "intrinsic" angular momentum in a classical theory that could be distinguished from the limit of shrinking a spinning ball to size zero?</p>
<p><br></p>
<p>It seemed obvious to me that the answer should be no. -- No such distinction could be made in a classical theory. However searching on related topics and reading more brought me to this article in wikipedia:<br>
<a href="http://en.wikipedia.org/wiki/Einstein%E2%80%93Cartan_theory" rel="nofollow">http://en.wikipedia.org/wiki/Einstein%E2%80%93Cartan_theory</a><br>
While I don't follow the math details, the overview comments seem to claim there is a classical notion of intrinsic spin, but GR cannot handle intrinsic angular momentum. And a different classical theory is presented which does, and differs from GR. For instance, some important tensors can be non-symmetric now, which can't happen in GR no matter the size we make a spinning ball.</p>
<p>Therefore there seems to be a real distinction between "intrinsic" angular momentum, and the limit of shrinking a spinning ball to size zero, even in classical mechanics. This blows my mind. So if the answer to the above questions are YES!, can someone help explain what this classical "intrinsic" angular momentum is?</p>
| 3,281 |
<p>I am trying to reconcile data that I have found in <a href="http://www.xn--rheinischesmuseumfrphilologie-2bd.com/fileadmin/content/dokument/archiv/silvaegenetica/18_1969/18-1-2-40.pdf" rel="nofollow">one publication (Allen 1969)</a> with data that I found in <a href="http://www.jstor.org/stable/1514306" rel="nofollow">another publication (George 2003)</a> that synthesized this data. The
data is root respiration rate, it was originally measured at $27\ ^\circ C$.</p>
<h2>Approach</h2>
<p>I am trying to convert a rate of oxygen consumed as volume per mass of root per time to carbon dioxide produced as mass per unit mass per time.</p>
<p>In the appendix table, George 2003 reports the range of root respiration rates, converted to $15\ ^\circ C$ and standard units:</p>
<p>$$[11.26, 22.52] \frac{\mathrm{nmol CO}_2}{\mathrm{g}\ \mathrm{s}}$$</p>
<p>In the original publication Allen (1969), root respiration was measured at $27\ ^\circ $C.
The values can be found in table 3 and figure 2.
The data include a minimum (Group 2 Brunswick, NJ plants) and a maximum (Group 3 Newbery, South Carolina), which I assume are the ones used by George 2003:</p>
<p>$$[27.2, 56.2] \frac{\mu\mathrm{L}\ \mathrm{O}_2}{10\mathrm{mg}\ \mathrm{h}}$$</p>
<h2>Step 1</h2>
<p>Transformed George 2003 measurements back to the measurement temperature using a rearrangement of equation 1 from George, the standardized temperature of $15\ ^\circ $C stated in the Georgeh table legend, and Q$_{10} = 2.075$ from George 2003, and the measurement temperature of $27\ ^\circ $C reported by Allen 1969: </p>
<p>$$R_T = R_{15}[\exp(\ln(Q_{10})(T- 15))/10]$$</p>
<p>$$[11.26, 22.52] * exp(log(2.075)*(27 - 15)/10)$$</p>
<p>Now we have the values that we would have expected to find in the Allen paper, except that the units need to be converted back to the original: </p>
<p>$$[27.03,54.07] \mathrm{nmol CO}_2\ \mathrm{g}^{-1}\mathrm{s}^{-1}$$ </p>
<h2>Step 2: convert the units</h2>
<h3>Required constants:</h3>
<ul>
<li>inverse density of $\mathrm{O}_2$ at $27^\circ C$: $\frac{7.69 \times 10^5\ \mu\mathrm{L}\ \mathrm{O}_2}{\mathrm{g}\ \mathrm{O}_2}$ first assume that Allen converted to sea level pressure (101 kPa), although maybe they were measured at elevation (Allen may have worked at \~{} 900 kPa near Brevard, NC)</li>
<li>molar mass of $\mathrm{O}_2$: $\frac{32\mathrm{g}\ \mathrm{O}_2}{\mathrm{mol}}$</li>
<li>treat 10mg, which is in the unit of root mass used by Allen, as a unit of measurement for simplicity </li>
</ul>
<p>Now convert $$[27.03,54.07] \mathrm{nmol CO}_2\ \mathrm{g}^{-1}\mathrm{s}^{-1}$$ to units of $\frac{\mu\mathrm{L}\ \textrm{O}_2}{10\mathrm{mg}\ \mathrm{root}\ \mathrm{h}}$.
The expected result is the original values reported by Allen: $[27.2, 56.2] \frac{\mu\mathrm{L}\ \mathrm{O}_2}{10\mathrm{mg}\ \mathrm{h}}$</p>
<p>$$[27.03, 54.07]\ \frac{\mathrm{nmol}\ \mathrm{CO}_2}{\mathrm{g}\ \mathrm{root}\ \mathrm{s}} \times \frac{1\ \mathrm{g}}{100\times10\mathrm{mg}} \times \frac{3600\ \mathrm{s}}{\mathrm{h}} \times \frac{3.2 \times 10^{-8}\ \mathrm{g}\ \mathrm{O}_2}{\mathrm{nmol}\ \mathrm{O}_2}\times \frac{7.69\times10^5\ \mu\mathrm{L}\ \mathrm{O}_2}{\mathrm{g}\ \mathrm{O}_2}$$</p>
<h2>Result:</h2>
<p>$$[23.8, 47.8]
\frac{\mu\mathrm{L}\ \textrm{O}_2}{10\mathrm{mg}\ \mathrm{h}}$$</p>
<p>These are the units reported in the Allen paper, but they appear to be underestimates . Since the ratio of observed:expected values are different, it is not likely that Q$_{10}$ or the atmospheric pressure at time of measurement would explain this error. </p>
<h2>Question</h2>
<p>Am I doing something wrong?</p>
<hr>
<ul>
<li><a href="http://www.xn--rheinischesmuseumfrphilologie-2bd.com/fileadmin/content/dokument/archiv/silvaegenetica/18_1969/18-1-2-40.pdf" rel="nofollow">Reference 1</a>: Allen, 1969, Racial variation in physiological characteristics of shortleaf pine roots., Silvics Genetics 18:40-43</li>
<li><a href="http://www.jstor.org/stable/1514306" rel="nofollow">Reference 2</a>: George et al 2003, Fine-Root Respiration in a Loblolly Pine and Sweetgum Forest Growing in Elevated CO2. New Phytologist, 160:511-522</li>
</ul>
<hr>
<p>Footnote 1: The values from reference 2 are adjusted from the $15^\circ C$ reference temperature to the $27^\circ C$ in reference 1 using the Ahhrenius equation, but I am off by an order of magnitude so I do not think that this is relevant:</p>
<p>$$R_T = R_{15}[\exp(\ln(Q_{10})(T- 15))/10]$$</p>
<p>$$[26.9, 54.0] = [11.2, 22.5] * exp(log(2.075)*(27 - 15)/10)$$</p>
<hr>
<p><em>note:</em> I have been updating the equation based errors pointed out by Mark and rcollyer, but the problem remains</p>
| 3,282 |
<p>I have seen in popular media, claims that Hawking does not believe the Higgs boson exists due to microscopic black holes and even made a bet against it. This is based on something published in journal Physical Review D.</p>
<p>I don't have access to journal Physical Review D, and I can't find a clear detailed explanation what his claim is, and what his proposed alternative is. Can someone explain it for this curious layman?</p>
| 3,283 |
<p>Does salt water conduct mostly by the ions travelling through the solution, or by electrons collectively flowing or hopping through the solution like in metals?</p>
| 3,284 |
<p>When I first heard about the <a href="http://en.wikipedia.org/wiki/Black_hole_information_paradox">black hole information paradox</a>, I thought it had no content. At the time, papers about it had been written for numerous years and they keep on coming. Now that the press got wind of Hawking's latest one, I thought I should ask about it:</p>
<p>The information paradox relies on the <a href="http://en.wikipedia.org/wiki/No-hair_theorem">no-hair conjecture</a>. However, all its proofs I'm aware of rely on the fact that we end up with a <em>stationary</em> black hole. So once we introduce Hawking radiation, the theorem evaporates right besides its subject.</p>
<p>Basically, we're wondering why a theorem we have proven for the stationary case does not hold for the non-stationary case. That seems hardly surprising to me, but I may have missed something obvious.</p>
<p>On a related note, I always found the no-hair theorem somewhat suspicious because it means after formation of the black hole, we end up with a result stronger than Gauss's law, whereas before formation of the black hole, the generalizations of Gauss's law to relativistic gravity are (again, as far as I know) generally weaker.</p>
<hr>
<p>To illustrate the argument from a different point of view, let me describe the thermodynamic information paradox:</p>
<p>First, lets start with the no-hair theorem, which states that isolated systems will tend towards a stationary equilibrium state, uniquely described by just a few parameters.</p>
<p>While going forward, instead of looking at completely isolated systems, we now allow interaction via absoption and emission of radiation.</p>
<p>The asumption is that because the system has no hairs, no matter the incoming radiation, the outgoing radiation will obey the totally probabilistic thermal laws.</p>
<p>Let's also assume that we're going to reach $T=0$ after all energy has been radiated.</p>
<p>This is, as far as I can tell, a pretty close analogy to the black hole paradox, and has a simple resolution: Physical equilibrium states fluctuate and thus have hairs. In fact, thermal radiation alone will disrupt equilibrium, and just assuming that it doesn't leads to nonsense.</p>
| 3,285 |
<p><code>Question</code>:</p>
<blockquote>
<p>Prove that $p^2$ and ${\bf r}\cdot {\bf p}$ commute with every component of ${\bf L}$ using the identity
$$[{\bf p},{\bf e}\cdot {\bf L}]=i\hbar\, {\bf e}\times {\bf r} $$
where ${\bf e}$ is a unit vector given by ${\bf e}=a\hat{i}+b\hat{j}+c\hat{k}$. where $\sqrt{a^2+b^2+c^2}=1$</p>
<p>As well, prove that ${\bf L}$ commutes with any function $f(r^2)$</p>
</blockquote>
<p><code>Attempt</code>:
$$ [p^2, {\bf e}\cdot {\bf L}]=[{\bf p}\cdot {\bf p},aL_x+bL_y+cL_z]=$$
$$[{\bf p} \cdot {\bf p},aL_x]+[{\bf p} \cdot {\bf p},aL_y] +[{\bf p} \cdot {\bf p},aL_z]$$
$$a[{\bf p} \cdot {\bf p},L_x]+b[{\bf p} \cdot {\bf p},L_y]+c[{\bf p} \cdot {\bf p},L_x] $$</p>
<p>I know that</p>
<p>$$ [AB,C]=A[B,C]+[B,C]A$$</p>
<p>but does that mean that</p>
<p>$$[{\bf A}\cdot{\bf B},{\bf C}]={\bf A}\cdot [{\bf B},{\bf C}]+[{\bf B}\cdot {\bf C}]\cdot {\bf A}$$</p>
<p>Is this the correct procedure? If it is, then I only need to show via the vector identity</p>
<p>$${\bf A}\cdot {\bf B}\times {\bf C}={\bf B}\cdot {\bf C}\times {\bf A}={\bf C}\cdot {\bf A}\times {\bf B}$$</p>
<p>that the commutator above is zero, but I am unsure if the above identity holds for dot products. Furthermore, how do I show that ${\bf L}$ commutes with any function $f(r^2)$? I'm a little confused how to get started on this part of the question. Any help would be greatly appreciated.</p>
| 3,286 |
<p>Why does the air we blow/exhale out from our mouths change from hot to cold depending on the size of the opening we make with our mouth?</p>
<p>It's not just a subtle difference, but significant in my opinion. I'm inclined to discredit the notion that it's just a matter of speed because I can blow fast with an open mouth and still, it's hot; and blow slow with an almost closed (tighter) mouth and again, it's cold.</p>
| 3,287 |
<p>The <a href="http://mathworld.wolfram.com/BrachistochroneProblem.html">brachistochrone problem</a> asks what shape a hill should be so a ball slides down in the least time. The <a href="http://mathworld.wolfram.com/TautochroneProblem.html">tautochrone problem</a> asks what shape yields an oscillation frequency that is independent of amplitude. The answer to both problems is a cycloid.</p>
<p>Is there an intuitive reason why these problems have the same answer?</p>
<p>Proposed operational definition of "intuitive": Imagine modifying the problem slightly, either to the <a href="http://physics.stackexchange.com/questions/7421/brachistochrone-problem-for-inhomogeneous-potential">brachistochrone tunnel problem</a> (tunnel through Earth), or by taking account of the finite radius of the ball. If the answer to the original question is intuitive, we should easily be able to tell whether these modified situations continue to have the same brachistochrone and tautochrone curves.</p>
| 3,288 |
<p>I am interested in the finite-difference beam propagation method and its applications. I try to solve the Helmholtz equation. At first, i would like to solve numerically it for the easiest case, without nonlinearities. Just to make sure I'm on the right way. But i really don't understand how to wright the boundary condition. I chose the transparent boundary condition and i need to write it properly to solve numerically the equation.</p>
<p>So, for a linear, homogeneous and instantaneous medium the Helmholtz equation is writen (in 3D case, z is the propagation direction)
$$
\frac{\partial^{2} E(x,y,z)}{\partial x^{2}} + \frac{\partial^{2} E(x,y,z)}{\partial y^{2}} + \frac{\partial^{2} E(x,y,z)}{\partial z^{2}} = - (k_{0} n )^{2} E(x,y,z)
$$</p>
<p>It can be solved if the initial condition is known, $E(x,y,0)$.</p>
<p>Introducing operator $\hat{S}$
$$
\hat{S} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + (k_{0} n )^{2}
$$
The equation can be written in the following form
$$
\frac{\partial^{2} E(x,y,z)}{\partial z^{2}} = -\hat{S} \ E(x,y,z)
$$</p>
<p>The solution of this equation is
$$
E(x,y,z) = \exp \left [ - i \sqrt{\hat{S}} z \right ] E^{+}(x,y,0) + \exp \left [ i \sqrt{\hat{S}} z \right ] E^{-}(x,y,0)
$$</p>
<p>Considering only the forward propagating component and introducing the propagation operator $\hat{P}^{+}$ the electric field at $z=\Delta z$ can be written through the value of the field at $z=0$ (initial condition written earlier) and so on.
$$
E(x,y,\Delta z) = \hat{P}^{+}(\Delta z) \ E(x,y,0)
$$
where
$$
\hat{P}^{+}(\Delta z) = \sum \limits_{n=0}^{\infty} \frac{1}{n!}\left[- i \sqrt{\hat{S}} \right]^{n} \Delta z^{n}
$$</p>
<p>Obtained expression can be adopted to the <a href="http://en.wikipedia.org/wiki/Crank%E2%80%93Nicolson_method" rel="nofollow">Crank-Nicholson scheme</a>. But it is also necessary to write the boundary condition. How to write the boundary condition if the medium is confined in the transparent walls ? </p>
| 3,289 |
<p>By comparatively examining the operators</p>
<p><img src="http://i.stack.imgur.com/Ox5jp.png" alt=""></p>
<p>a student concludes that `Energy is actually the momentum in the direction of time.' Is this student right? Could he be wrong? </p>
| 3,290 |
<p>Could someone experienced in the field tell me what the minimal <strong>math knowledge</strong> one must obtain in order to grasp the introductory Quantum Mechanics book/course?</p>
<p>I do have math knowledge but I must say, currently, kind of a poor one. I did a basic introductory course in Calculus, Linear algebra and Probability Theory. Perhaps you could suggest some books I have to go through before I can start with QM? Thx.</p>
| 223 |
<p>Is there a simple mathematical expression for the stopping power of a given thickness of armor, given the thickness of armor plate, the radius of a cannon ball, the density of the cannonball and the armor, the tensile strength and/or toughness of the armor, and the speed of the cannonball? For simplicity assume the cannonball is a solid metal sphere and that the armor plate is homogeneous. I realize that in modern warfare the projectiles are pointed and armor plate isn't a homogeneous slab, but I want to understand the simple case. (My question is inspired in part from reading about Civil War ironclads, but I also saw the question about chain mail and thought that if that question was legitimate this one should be more so.) </p>
<p>In case my question isn't clear, what I'm asking is something like the following. Suppose it took 10 cm of iron armor to stop a 20 cm diameter cannonball moving at 300 meters/sec. How thick would the armor have to be to stop a 40 cm cannonball moving at the same speed? Or what if you doubled the speed? Or what if you doubled the tensile strength of the armor? Etc...</p>
| 3,291 |
<p>I was watching Discovery channel the other night, they were telling that time slows down when you travel at a higher speed. This means there is a difference between the actual speed you travel at, and the perceived speed.</p>
<p>Does anybody know what the perceived speed is, the speed it seems you're travelling at?</p>
<p>What does this imply? Does it have any meaning, in some theories perhaps, that the perceived speed is lower? Does this mean that you can never actually travel at the speed of light?</p>
<p>(All this is if we don't account for the fact that's impossible to reach the speed of light.)</p>
<p>I'm also not into physics, so try to keep it simple...</p>
| 3,292 |
<p>I've had a discussion with my father today, about the fuel usage of a vehicle at the same rpm, but a different gear.</p>
<p>He claims that the following situations have the same fuel usage:</p>
<pre><code>Gear: 2
rpm: 2000
fuel usage: 5.1l/100km
Gear: 5
rpm: 2000
fuel usage: 5.1l/100km
</code></pre>
<p>I say it should be something along the lines of:</p>
<pre><code>Gear: 2
rpm: 2000
fuel usage: 4.1l/100km
Gear: 5
rpm: 2000
fuel usage: 5.4l/100km
</code></pre>
<p>Who is right and why?
He is also claims that the force to maintain the speed will be the same across different gears with the same rpm.</p>
<p>By the way, I'm not sure if it should be on physics, but it fitted best here.</p>
| 3,293 |
<p>I have a <a href="http://en.wikipedia.org/wiki/Uninterruptible_power_supply" rel="nofollow">UPS</a> of 1000 Volts connected with 2 batteries each of 150 Amp. How much time it will take to consume the whole UPS (after fully charged) when a device of 1Amp is getting electricity form that UPS.</p>
<p>Please also explain me the calculation.</p>
| 3,294 |
<p>I am following the first volume of the course of theoretical physics by Landau. So, whatever I say below mainly talks regarding the first 2 chapters of Landau and the approach of deriving Newton's laws from Lagrangian principle supposing Hamilton's principle of extremum action. Please keep this view in mind while reading and answering my queries and kindly neglect the systems to which Action Principle is not applicable:</p>
<p>If we use homogeneity of space in Euler-Lagrange equations, we obtain a remarkable result i.e. the conservation of momentum for a closed system.</p>
<p>Now, this result, using the form of Lagrange for a closed system of particles, transforms into $ \Sigma F = 0 $ . Now, how from this can we conclude that the internal forces that particles exert come in equal and opposite pairs?</p>
<p>Is it because for 2 particles this comes out as $ F_{1} + F_{2} = 0 $ and we take the forces exerted by particles on one other to be independent of other particles (i.e. Superposition Principle) as an experimental fact?</p>
<p>I doubt it as whole of Newtonian Mechanics is derivable from Lagrangian Mechanics and supposed Symmetries. So, according to me, a fact like Newton's Third Law should be derivable from it without using an additional experimental fact.</p>
<p>I have an idea to prove it rigorously. Consider two particles $i$ and $j$. Let the force on $i$ by $j$ be $F_{ij}$ and on $j$ by $i$ be $k_{ij}F_{ij}$. Now the condition becomes $\Sigma (1+k_{ij})F_{ij}=0$ where the terms to be included and rejected in summation understood. As this must be true for any value of $F_{ij}$, we get $k_{ij}=-1$. I don't know if this argument or refinement of such an argument holds or not. I can see many questions arising in this argument and it's not very convincing to me.</p>
<p>I would like to hear from you people as to if it is an experimental result used or not? If not, then is the method given above right or wrong? If wrong, how can we prove it?</p>
<p><strong>Addendum</strong></p>
<p>My method of proof uses the fact of superposition of forces itself, so it is flawed. I have assumed that the coefficients $k_{ij}$ are constants and don't change in the influence of all other particles which is exactly what superposition principle says.</p>
<p>As the superposition of Forces can be derived by superposition of potential energies at a point in space and potential energy being more fundamental in Lagrangian Mechanics, I restate my question as follows:</p>
<p><em>Is the principle of superposition of potential energies by different sources at a point in space derivable from the inside of Lagrangian Mechanics or is it an experimental fact used in Lagrangian Mechanics?</em></p>
<p>I, now, doubt this to be derivable as the fundamental assumption about potential energy is only that it is a function of coordinates of particles and this function may or may not respect superposition.</p>
| 3,295 |
<p>In Peskin and Schröder's QFT, equation 2.56, could anyone give a list of all the arguments necessary in order to make all the transitions mathematically rigorous?</p>
<p>I tried composing such a list myself and I came up with:</p>
<ol>
<li>$\frac{d}{dx}\theta(x)=\delta(x)$</li>
<li>$\theta(x)\delta(x)\equiv0$?</li>
<li>If this holds $\int d^4x(\partial_\mu\partial^\mu\theta(x^0-y^0))<0|[\phi(x),\phi(y)]|0>=-\int d^4x(\partial^\mu\theta(x^0-y^0))(\partial_\mu<0|[\phi(x),\phi(y)]|0>)$ then this holds: $(\partial_\mu\partial^\mu\theta(x^0-y^0))<0|[\phi(x),\phi(y)]|0>=-(\partial^\mu\theta(x^0-y^0))(\partial_\mu<0|[\phi(x),\phi(y)]|0>)$. But why? Is it because of the $e^{-ip_\alpha(x^\alpha-y^\alpha)}$ term in $<0|[\phi(x),\phi(y)]|0>$?</li>
<li>Why is the KG operator applied on the Green's function equal to $-i\delta(x-y)$ and not $\delta(x-y)$? Is it a matter of convention?</li>
</ol>
| 3,296 |
<p>I know Hamiltonian can be energy and be a constant of motion if and only if:</p>
<ol>
<li>Lagrangian be time-independent, </li>
<li>potential be independent of velocity, </li>
<li>coordinate be time independent.</li>
</ol>
<p>Otherwise
$$H\neq E\neq {\rm const},$$ or
$$H=E\neq {\rm const},$$ or
$$H\neq E={\rm const}.$$</p>
<p>I am looking for examples of these three situation.</p>
| 3,297 |
<p>This question is about Godel's theorem, continuity of reality and the Luvenheim-Skolem theorem.</p>
<p>I know that all leading physical theories assume reality is continuous. These are my questions:</p>
<p>1) Is reality still continuous according to string theory? what is the meaning of continuous if there is a finite size for a string? (My understanding of string theory is very limited :)</p>
<p>2) According to the Luvenheim-Skolem theorem, any enumerable first order logic theory (physics?) has an enumerable model. This means we must be able to have a discrete theory of physics which is as good as the continuous ones. Is this accurate?</p>
<p>3) Why haven't we found one? I've heard some people say that Heisenberg's formulation of quantum mechanics doesn't speak of anything continuous... is this accurate?</p>
<p>4) If we can find one / have found one, does this mean that our understanding of physical laws can always be reduced into a Turing machine?</p>
<p>5) If we can find one / have found one, how can we explain the use of real constants in physics? (such as Pi)</p>
<p>6) Does Godel's theorem mean that we could always "notice" that some things are true but not be able to prove them from within the universe?</p>
<p>Thanks.</p>
| 3,298 |
<p>Dark energy is constantly pulling all objects away from each other with increasing speed. This in turn causes a red-shift of the light from the most distant object where this effect is most profound. This red-shift will gradually increase as the objects move away faster. Is there going to be a time where the objects travelling away will emit light of such long wavelength that they will ddisappear from sight?</p>
| 79 |
<p>Newton's work was related to Enlightenment philosophy.</p>
<p>Einstein was influenced by Mach.</p>
<p>The founders of quantum mechanics had strong philosophical opinions.</p>
<p>What is the role of metaphysics in the development of physics?</p>
| 3,299 |
<p>In general, when I think of movement through space, I think of this:</p>
<p>$$\frac{dx}{dt}$$</p>
<p>But in special relativity, we also have a concept of relative duration, which means that $t$ must have a rate of change, but with respect to what?</p>
<p>$$\frac{dt}{d?}$$</p>
| 80 |
<p>If I stand exactly in front of a colorful wall, I imagine the light waves they emit, and they receive should randomly double or erase out each other. </p>
<p>So as a result, I imagine I should see a weird combination of colors, or a full-black/full-white/very lightly perception of the wall, when all the light waves that the wall receives and emits cancel out each other or double each other. </p>
<p>Why doesn't that actually happen? Any time I look into a wall, I never see the wall "cancel out" of my perception. Same for radio waves. Shouldn't radio waves not work at all? There are so many sources where they could reflect and cancel out or annoy each other...</p>
| 3,300 |
<p>I know other people asked the same question time before, but I read a few posts and I didn't find a satisfactory answer to the question, probably because it is a foundational problem of quantum mechanics. </p>
<p>I'm talking about the Hilbert space Separability Axiom of quantum mechanics. I'd like to understand why it was assumed this condition in the set of postulates of QFT. Is there a physical motivation of this, or was it only a way to simplify computation? </p>
<p>Mathematically speaking such an assumption is understandable. I read the argument about superselection sectors, where, even in presence of a non separable Hilbert space in QFT, every sector can be assumed to be separable and one can work inside this one, agreeing in this way with the said axiom. But the trouble remain unsolved, why this sector has to be separable? </p>
<p>If you know some old post or some book where I can find this answer and I didn't see please notify me.</p>
| 3,301 |
<p>Following Arnold's [1] definition of the universe as an affine space $ A ^4$ with the group $\mathbb R ^4$ acting on it, we may define a galilean transformation as <em>an affine map $g:A^4 \to A^4$ which preserves the galilean structure</em>, i.e. which preserves time intervals and spatial distance beetween simultaneous events. “Time” is an application $t:\mathbb R ^4 \to \mathbb R $ and $P,Q\in A^4$ are simultaneous if $t(P-Q)=0$?</p>
<p>Furthermore, to say that a coordinate system $\varphi _1$ (read: a bijection $\varphi _1 :A^4 \to \mathbb R ^4$) is in uniform motion with respect to another $\varphi _2$ means that $\varphi_2 ^{-1}\circ \varphi _1 $ is a galilean transformation of $\mathbb R \times \mathbb R ^3 $ (to give a galilean structure to this space we take $t$ to be the projection on the first coordinate).</p>
<p>On the other hand, a Lorentz transformation beetween two coordinate systems in relative motion is often defined as a <em>linear</em> transformation <em>of the space of coordinates</em>: $\Lambda :\mathbb R \times \mathbb R ^3 \to \mathbb R \times \mathbb R ^3 $.</p>
<p>Question is: is it possible in special relativity to characterize</p>
<p>a) the universe as an affine space $A^4$ (with the group $\mathbb R \times \mathbb R ^3$ acting on it) with some additional structure in a way similar to the galilean universe? I think the answer is yes, the structure is given by the pseudo metric on $\mathbb R \times \mathbb R ^3$, $|x|^2 =c^2t^2 - x_1 ^2 -x _2 ^2 -x_3 ^2$. Is this sufficient to fully characterize the universe $A^4$. </p>
<p>b) Lorentz transformation as transformations of $A^4$ that preserve its structure. Again, is it sufficient to say that Lorentz transformations are the linear ones which preserve the space-time distance given by $|.|^2$? And why linear and not affine?</p>
| 3,302 |
<p>The Hamiltonian is given as</p>
<p>$H=-\frac{\hbar^2}{2 m_e r^2}\displaystyle\sum_{n=1}^N \dfrac{\partial^2}{\partial \theta_n^2}$</p>
<p>In the first part we show that the $\psi_k=\frac{1}{\sqrt{2\pi}}\exp(i k \theta)$ is a solution to $H\psi_k = E_k \psi_k$.</p>
<p>$E_k=\frac{k^2 \hbar^2}{2 m_e r^2}$</p>
<p>All fine so far.</p>
<p>$r$ is given to be $4 \times 10^{-10}\text{m}$ and $N=18$ and it asks for the ground state energy.</p>
<p>I am unsure how the $k=0$ state can be occupied since zero energy is forbidden?</p>
<p>But for each quantum number $k=0,\pm 1,\pm 2\cdots$ we have an increasing energy so the ground state will consist of </p>
<p>$(k=0)^2(k=1)^2(k=-1)^2(k=2)^2(k=-2)^2(k=3)^2(k=-3)^2(k=4)^2(k=-4)^2$</p>
<p>Where $(k=0)^2$ denotes 2 electrons being in the state with $k=0$, one with spin up and one with spin down.</p>
<p>The corresponding energy: $\displaystyle\sum_{k=-4}^4 2E_k = \frac{60 \hbar^2}{m_e r^2} = 4.57 \times 10^{-18} \text{J}$</p>
<p>But in the solution the energy is given as $\frac{30 \hbar^2}{m_e r^2} = 2.29 \times 10^{-18} \text{J}$, half of my value. Is this an error or have I missed something?</p>
| 3,303 |
<p>When I perform parametric modeling, if there is significant multicollinearity between variables I think should be independent, but in fact are not, I run into the case where one or more of the coefficients becomes exceeding small (or large) relative to the others. How is that different than what occurs in fine tuning problems of the standard model?</p>
| 3,304 |
<p>My quantum mechanics textbook says that the following is a representation of a wave traveling in the +$x$ direction:$$\Psi(x,t)=Ae^{i\left(kx-\omega t\right)}\tag1$$</p>
<p>I'm having trouble visualizing this because of the imaginary part. I can see that (1) can be written as:$$\Psi(x,t)=A \left[\cos(kx-\omega t)+i\sin(kx-\omega t)\right]\tag2$$ </p>
<p>Therefore, it looks like the real part is indeed a wave traveling in the +$x$ direction. But what about the imaginary part? The way I think of it, a wave is a physical "thing" but equation (2) doesn't map neatly into my conception of the wave, due to the imaginary part. If anyone could shed some light on this kind of representation, I would appreciate it.</p>
| 3,305 |
<p>Suppose a magnetic dipole $\mathbf{m} = m \hat{z}$ is falling towards a circular loop of radius $b$ under gravity. Assuming the dipole always stays along the $z$-axis of the loop, determine the following:</p>
<p>In terms of the height $z$, loop resistance $R$, and speed $v$, determine</p>
<p>(a) The EMF induced around the loop.</p>
<p>(b) The magnetic field of the loop at the dipole created by the induced current.</p>
<p>(c) The force on the dipole.</p>
<p>So far, I have found that flux through the loop of radius $b$ is $\phi = \frac{\mu_0 m b^2}{2(b^2 + z^2)^{3/2}}$ from treating the dipole as a current loop with radius $\epsilon$, where $b >> \epsilon$.</p>
<p>Thank you for your help!</p>
| 3,306 |
<p><a href="http://www.youtube.com/watch?v=yTeBUpR17Rw" rel="nofollow">This</a> video explains that heat at negative temperatures flows from the negative object to the normal object. If the temperature of the normal object is absolute hot, what happens with the heat? The heat can't be transferred to the absolute hot object, and it apparently does not flow in the other direction, so what happens to it?</p>
| 3,307 |
<blockquote>
<p>The difference between a “timelike” spacetime interval and a “spacelike” spacetime interval can be understood in the following way: If the spacetime interval between two events is timelike, there exists a reference frame which measures the proper time between the two events; i.e. it sees the events occur at the same position. It the spacetime interval between two events is spacelike, there exists a reference frame which measures the proper length between two events; i.e. it sees the events occurring simultaneously</p>
<p>Now, suppose that the S reference frame measures the following spacetime coordinates for three
separate events: </p>
<p>Event 1: ($x_1 = 300\, \mathrm{m};\, t1 = 3.0\, \mathrm{\mu s}$)</p>
<p>Event 2: ($x_2 = 700\, \mathrm{m};\, t2 = 5.0\, \mathrm{\mu s}$)</p>
<p>Event 3: ($x_3 = 1400\, \mathrm{m};\, t3 = 6.0\, \mathrm{\mu s}$)</p>
<p>Find $(\delta s)^2$ between Event 2 and Event 1. Is this a timelike or a spacelike separation? Find the speed (relative to S) of a reference frame S’ that measures either the proper time or the proper length between the two events. What is this proper time or proper length?</p>
</blockquote>
<p>So, finding $(\delta s)^2$:</p>
<p>$$(\delta s)^2 = (c \delta t)^2 - (\delta x)^2$$</p>
<p>$$(\delta s)^2 = [(3 \cdot 10^8\, \mathrm{m/s})(2\, \mathrm{\mu s})]^2 - (400\, \mathrm{m})^2 = 200\, \mathrm{km}$$</p>
<p>So it's a timelike interval and event 1 can affect event 2. How can I begin finding the speed of $S'$?</p>
| 3,308 |
<p>The principle of equivalence - that, locally, you can't distinguish between a uniform gravitational field and a noninertial frame accelerating in the sense opposite to the gravitational field - is dependent on the equality of gravitational and inertial mass. Is there any deeper reason for why this equality of "charge corresponding to gravitation" (that is, the gravitational mass) and the inertial mass (that, in newtonian mechanics, enters the equation F=ma) should hold? While it has been observed to be true to a very high precision, is there any theoretical backing or justification for this? You could, for example (i wonder what physics would look like then, though), have the "charge corresponding to electromagnetic theory" equal to the the inertial mass, but that isn't seen to be the case.</p>
| 3,309 |
<p>Ok, I am stumped by this question:</p>
<p>In the system in the diagram below, block M (15.7 kg) is initially moving to the left. A force F , with magnitude 60.5 N, acts on it directed at an angle of 35.0 degrees above the horizontal as shown. The mass m is 8.2 kg. There is no friction and the pulley and string are massless. </p>
<p><img src="http://i.stack.imgur.com/r3W4t.jpg" alt="diagram"></p>
<ol>
<li>What is the normal force (N) on M ?</li>
<li>What is the tension (N) in the string?</li>
<li>What is the acceleration (m/s2) of M (positive is to the right)?</li>
</ol>
<p>I tried and got the following:</p>
<p>Got the X component of the force F: F * Cos (35) = 49.55869868 N<br>
Got the Y component of the force F: F * Sin (35) = 34.7013744 N<br>
Got the force of m on M: 8.2 * 9.8 = 80.36 N (which is what I assumed the tension on the string was) </p>
<p>I got the acceleration of M caused by m = -5.118471338<br>
Then I got the acceleration of M cause by F = 3.156605011 </p>
<p>Putting it all together I got the following answers:</p>
<ol>
<li>129 N</li>
<li>80.4 N</li>
<li>-1.96 m/s^2</li>
</ol>
<p>I (obviously) got all three wrong. The correct answers are:</p>
<ol>
<li>119 N</li>
<li>69.8 N</li>
<li>-1.29 m/s^2</li>
</ol>
<p>So, I'm stuck. There is something missing that is causing my string tension to be off. If I would have gotten the right string tension, all the other answers would have been correct. I know that is where the problem is, so if someone could point out my mistake, that would be wonderful!</p>
| 3,310 |
<p>Mach's principle says that it is impossible to tell if something is accelerating unless there is something else in the universe to compare that motion to, which seems reasonable. However, if you had one detector in the universe, you seem to be able to tell if it is accelerating because an accelerating detector would record radiation where a non-accelerating detector would not, due to the Unruh effect. So, my question is, does the Unruh effect provide a way to tell if something is accelerating, even if it is the only thing in otherwise "empty" space, thereby violating Mach's principle? (At least Mach's principle in its form stated above.)</p>
| 3,311 |
<p>What does it mean for a field (say, $\phi$) to have a charge (say, $Q$) under the action of a group (say, $U(1)$)?</p>
| 3,312 |
<p>How does it work that gravitational forces can affect time and what usable applications could arise from this?</p>
| 3,313 |
<p>I'm looking for a simple explanation of how a whistle operates. I know <em>that</em> forcing air over a sharp lip can set up a wave in a resonating cavity, but <em>how</em>? "Most whistles operate due to a feedback mechanism between flow instability and acoustics"--yes, but what <em>does</em> that feedback mechanism look like?</p>
<p>I was surprised to be unable to find a basic diagram online demonstrating how a whistle operates. I did find lots of images like this:
<img src="http://i.stack.imgur.com/rHEe6.png" alt="enter image description here"></p>
<p>. . . but such images are unhelpful since they don't show exactly what's producing the oscillation!</p>
| 3,314 |
<p>I would like to support open-access journals by choosing to publish in journals which allow readers free online access. Ideally I would also like to retain copyright instead of signing it over to the journal.</p>
<p>What are some of the better open-access journals in physics? I am particularly interested in journals focused on optics, but let's keep the question general.</p>
| 3,315 |
<p>Inspired by this: <a href="http://physics.stackexchange.com/questions/8937/electrical-neutrality-of-atoms">Electrical neutrality of atoms</a></p>
<p>If I have a wavefunction of the 'reduced mass coordinate' for a hydrogen like atom made from an electron and a positron, what is the spatial charge distribution?</p>
<p>When we solve the hydrogen atom, we change into coordinates of the center of mass, and the separation distance with the reduced mass. Here, the masses of the constituent particles are the same. So the center of mass is equidistant from the positron and electron, and so discussing r and -r is just swapping the particles. Since the probability distribution for all the energy levels of the hydrogen atom are symmetric to inversion (images can be seen here <a href="http://panda.unm.edu/Courses/Finley/P262/Hydrogen/WaveFcns.html" rel="nofollow">http://panda.unm.edu/Courses/Finley/P262/Hydrogen/WaveFcns.html</a> ), this seems to say no matter what energy level positronium is in, the charge distribution is neutral? Since the energy level basis is complete, this seems to say we can't polarize a positronium atom without dissociating it!? This doesn't make sense to me, so I'm probably making a big mistake here.</p>
| 3,316 |
<p>So I was cruising around at YouTube and saw this <a href="http://www.youtube.com/watch?v=im1iNq02Kz0&feature=feedu" rel="nofollow">sweet video</a>, and as I was watching started to wonder: "How is this possible?".</p>
<p>For a little bit of background, in case you decide to not watch the video, what is happening is an individual can hook out a 'pack' that sucks up water and then shoots the water downward:</p>
<p><img src="http://i.stack.imgur.com/tMc9z.jpg" alt="enter image description here"></p>
<p>This water Jet Levitation (JetLev) causes the rider to be able to fly upward into the air and move about forward, backward, side-to-side and even underwater (you have to watch the video for that). So my question is: How exactly is this possible?</p>
| 3,317 |
<p>Cavendish measured the gravitation constant $G$, but actually he measured that constant on the Earth. What’s the proof that the value of the gravitation constant if measured on Neptune would remain the same? What’s the guarantee of its being a constant?</p>
<p>There are many such constants; I just took one example so that you can understand what I intended to ask. Is the value of the speed of light really a constant? Who knows that it wouldn’t change its value on other planets, or, more precisely, what’s the guarantee that the speed of light remains the same even near a black hole? If no one can ever reach a black hole, then how can a scientist claim the value of the speed of light?</p>
| 643 |
<p>IANAP, so feel free to berate me for thinking apocryphal thoughts! Just as magnetism has two charges, in which particles of like-charge repulse and particles of dissimilar charge attract, might gravity have two charges in which particles of like-charge attract and particles of dissimilar charge repulse?</p>
<p>In practice, the state of magnetism means that there is no system composed of many particles in which all particles attract. Rather, there is a net 0 charge if there are equal numbers of each particle type.</p>
<p>My silly theory regarding gravity would mean in practice that there would be two (or more) "clumps" (or universes) in existence, which are racing away from each other. So in our clump (universe) we see only attracting particles, because all the opposing particles have long since separated out and are racing away beyond the boundary of the observable universe. Just like the alien who lands in China and assumes that all humans have slanted eyes, we only observe the attracting particles (or "charge") and disregard the other, unobservable, "charge".</p>
<p>Is there any way to disprove this idea, or like string theory can I go one believing it as it can never be disproved?</p>
<p>Thanks.</p>
| 81 |
<p>I need to send infrared light from its emitter to a distance of about 10-12 feet. Is this possible?</p>
| 3,318 |
<p>What happens microscopically when an electrical current starts to flow? I'd like to understand microscopically what happens in detail when electrons start moving (quasi-classically).</p>
<p>Electrons can have different velocity, they can produce electromagnetic fields, leads have free electrons and rigid atom cores and there exist electromagnetic fields. That's all the ingredients you should need?</p>
<p>Electrons only move due to EM fields, so basically this question boils down to what the EM fields look like and how they build up?! In steady state, what is the electric and magnetic field distribution in/around the lead? And what about the transient state?</p>
<p>What happens when you attack a battery to a lead? Are there EM fields between battery poles or why are electrons pushed? How do the EM field start to push electrons along an arbitraritly shaped long lead?</p>
<p>[EDIT: Ideally an explanation with the Drude model (which partly derives from Fermi model) or an explanation why that model isn't sufficient. Also stating the EM fields consistent with the electron density distribution would be important (i.e. $\vec{E}(r,\theta,z)$ and $\vec{B}(r,\theta,z)$) because otherwise it's hand-wavy arguments.]</p>
<p>(Please consider all remarks in this question. I know common arguments for parts of the question, but I've never seen a full microscopic in detail explanation.)</p>
| 3,319 |
<p>In Ady Stern's review of the Quantum Hall effect, he says of a quantum hall system "The spectrum at $\Phi = \Phi_0$ is the same as the spectrum at $\Phi = 0$..." Can someone explain why this is? It seems like the applied magnetic field certainly changes the hamiltonian, and thus the spectrum, but apparently not when the flux is a single quantum. </p>
<p>Also I apologize for any newbie mistakes or if this is answered elsewhere, I'm pretty new to stackexchange. Thanks. </p>
| 3,320 |
<p>This is freshly ground pepper on water. </p>
<p>Why is there a triangular configuration of the water around the pepper fragment? Surely all these pepper fragments have different shapes? You can clearly see one of these triangles on the lower left edge of the reflection of the kitchen light.</p>
<p><img src="http://owen.maresh.info/pepper.png"></p>
<p>With pepper, this configuration doesn't last long, the fragments clump. </p>
| 3,321 |
<p>The Situation:</p>
<p>A ball is placed in a beaker filled with water and floats. It is also attached to the bottom of the beaker via a string. </p>
<p>The Question:</p>
<p>The ball is attached to the beaker, thus making the ball and beaker (and string included) a system. However, it is floating. Does the ball add to the weight of the beaker? </p>
| 3,322 |
<p>I am stuck with this process of calculating the tree-level scattering amplitude of two positive helicity (+) gluons of momentum say $p_1$ and $p_2$ scattering into two gluons of negative (-) helicity with momentum $p_3$ and $p_4$. </p>
<p>This is apparently $0$ for the diagram where one sees this process as two 3 gluon amplitudes with a propagating gluon (of say momentum $p$) and $p_1$ and $p_2$ are attached one each to the two 3 gluon amplitudes. I want to be able to prove this vanishing. </p>
<p>So let $p_2^+$ be with $p$ and $p_3^-$ and rest on the other 3 gluon vertex. </p>
<p>I am working in the colour stripped formalism. Let the Lorentz indices be $\rho$, $\sigma$ for the propagating gluon. And for the external gluons $p_1^+$, $p_2^+$, $p_3^-$, $p_4^-$ let $\nu, \lambda, \beta, \mu$ respectively be their Lorentz indices. Let the auxiliary vectors chosen to specify the polarizations of these external gluons be $p_4, p_4, p_1, p_1$ respectively. So the "wave-functions" of these four gluons be denoted as, $\epsilon^{+/-}(p,n)$ where $p$ stands for its momentum and $n$ its auxiliary vector and in the spinor-helicity formalism one would write, </p>
<ol>
<li><p>$ \epsilon^{+}_\mu(p,n) = \frac{<n|\gamma_\mu|p]}{\sqrt{2}<n|p>} $</p></li>
<li><p>$\epsilon^{-}_\mu(p,n) = \frac{[n|\gamma_\mu|p>}{\sqrt{2}[p|n]} $</p></li>
</ol>
<p>Hence I would think that this amplitude is given by, </p>
<p>$\epsilon^{-}_{\mu}(p_4,p_1)\epsilon_{\nu}^{+}(p_1,p_4)\epsilon_\lambda^+(p_2,p_4)\epsilon_\beta^-(p_3,p_1)\left( \frac{ig}{\sqrt{2}} \right)^2 \times \{ \eta^{\mu \nu}(p_4-p_1)^\rho + \eta^{\nu \rho}(p_1-p)^\mu + \eta^{\rho \mu}(p - p_4)^\nu\} \left ( \frac{-i\eta_{\rho \sigma}}{p^2}\right)\{ \eta^{\lambda \beta}(p_2-p_3)^\sigma + \eta^{\beta \sigma}(p_3-p)^\lambda + \eta^{\sigma \lambda}(p - p_2)^\beta \} $</p>
<p>One observes the following, </p>
<ol>
<li><p>$\epsilon^{-}_\mu(k_1,n). \epsilon^{- \mu}(k_2,n) = \epsilon^{+}_\mu(k_1,n).\epsilon^{+\mu} (k_2,n) = 0$</p></li>
<li><p>$\epsilon^{+}_\mu(k_1,n_1).\epsilon^{-\mu}(k_2,n_2) \propto (1-\delta_{k_2 n_1})(1-\delta_{k_1,n_2})$</p></li>
</ol>
<p>Using the above one sees that in the given amplitude the only non-vanishing term that remains is (upto some prefactors), </p>
<p>$\epsilon^{-}_{\mu} (p_4,p_1) \epsilon_{\nu}^{+}(p_1,p_4) \epsilon{_\lambda}^{+}(p_2,p_4)\epsilon_{\beta}^{-}(p_3,p_1) \left\{ \eta^{\nu}_{\sigma}(p_1-p)^\mu + \eta_\sigma^\mu(p - p_4)^\nu\right\}\times \{ \eta^{\lambda \beta}(p_2-p_3)^\sigma\}$</p>
<p>(..the one that is the product of the last two terms of the first vertex factor (contracted with the index of the propagator) and the first term from the second vertex factor..}</p>
<ul>
<li>Why is this above term zero? (..the only way the whole diagram can vanish..) </li>
</ul>
| 3,323 |
<p>Specifically the Hamiltonian takes the form of</p>
<p>$$\hat H = \frac{\Delta }{2}{\hat \sigma _z} + {\omega _1}\hat a_1^\dagger {\hat a_1} + {\omega _2}\hat a_2^\dagger {\hat a_2} + {g_1}\left( {{{\hat a}_1}{{\hat \sigma }_ + } + \hat a_1^\dagger {{\hat \sigma }_ - }} \right) + {g_2}\left( {{{\hat a}_2}{{\hat \sigma }_ + } + \hat a_2^\dagger {{\hat \sigma }_ - }} \right),$$</p>
<p>a three body version of Jaynes-Cummings model.</p>
<p>I'm currently trying to diagonalize this Hamiltonian, a first step in our application of quantum Zeno effect to a three-body system. </p>
<p>I guess this Hamiltonian simply has no close-form diagonalization, just like in classical physics there is no closed-form general solution for a three-body system. So my question is: what are several symbolic approximation techniques to diagonalize an Hermitian operator? Better if that techniques particularly suits this Hamiltonian. The values of $\Delta, \omega_1, \omega_2, g_1, g_2$ need not be general; they can be set, say, all equal in order to simplify calculation.</p>
| 3,324 |
<p>Suppose you are given an $n$-qubit quantum channel defined as $\mathcal{E}(\rho) = \sum_{i} p_i X_i \rho X_i^\dagger$, where $X_i$ denotes an $n$-fold tensor product of Pauli matrices and $\{p_i\}$ is a probability distribution. The Holevo-Schumacher-Westmoreland capacity of the channel is defined by
$$
\chi(\mathcal{E}) = \max_{\{q_j, \rho_j\}} \left[S\left(\sum_j q_j \rho_j\right) -\sum_j q_j S\left(\rho_j\right) \right],
$$
where $S$ denotes the von Neumann entropy of a density matrix (see, for example, <a href="http://theory.physics.helsinki.fi/~kvanttilaskenta/Lecture13.pdf" rel="nofollow">http://theory.physics.helsinki.fi/~kvanttilaskenta/Lecture13.pdf</a>). Is it known how to calculate this number as a function of $p_i$ and $n$?</p>
| 3,325 |
<p>It is often stated that in quantum gravity only charges coupled to gauge fields can be conserved. This is because of the no hair theorem. If a charge is coupled to a gauge field then when it falls into a black hole the black hole acquires a corresponding field. However if it is not then the black hole doesn't "remember" it. Apparently it implies we can't have exact global symmetries, only gauge symmetries.</p>
<p>How is this expectation realized in string theory? Is it true string vacuum sectors cannot posses global symmetry? Can we prove it?</p>
| 3,326 |
<p>Given that spacetime is not affine Minkowskispace, it does of course not possess Poincare symmetry. It is still sensible to speak of rotations and translations (parallel transport), but instead of</p>
<p>$$[P_\mu, P_\nu] = 0$$</p>
<p>translations along a small parallelogram will differ by the curvature. I have not thought carefully about rotations and translations, but basically you could look at the induced connection on the frame bundle, to figure out what happens.</p>
<p>This is all to say that spacetime has obviously not <em>exact</em> Poincare symmetry, although the corrections are ordinarily very small. Most QFT textbooks seem to ignore this. Of course it is possible to formulate lagrangians of the standard theories in curved space and develop perturbation theory, too. But since there is no translation invariance, one can not invoke fourier transform.</p>
<p>My questions are: </p>
<ul>
<li>Why is it save to ignore that there is no exact poincare symmetry? Especially the rampant use of fourier transforms bothers me, since they do require exact translation invariance. </li>
<li>How does one treat energy momentum conservation? Presumably one has to (at least) demonstrate that the covariant derivative of the energy momentum tensor is zero.</li>
</ul>
<p>Any references that discuss those issues in more detail are of course appreciated.</p>
| 3,327 |
<p>Let's assume an observer looking at a distant black hole that is created by collapsing star.</p>
<p>In observer frame of reference time near black hole horizon asymptotically slows down and he never see matter crossing event horizon. So black hole is visible as some kind of frozen object - falling matter almost stops near horizon and light emitted or reflected by it becomes more and more redshifted as it approaches horizon.</p>
<p>On the other hand black hole emits thermal radiation that causes it's evaporation. After some time it will finally disappear.</p>
<p>As these two points seems extremely incompatible with each other, my question is how transition from such frozen object to state of no black hole (complete evaporation) looks for distant observer?</p>
<p>What I'm curious about is where this Hawking radiation comes from. I suppose it will be surface of the horizon, but in fact from observer frame of reference there is nothing at this surface yet - matter didn't fall on it yet, it's just approaching it asymptotically.</p>
<p>What is strange from classical point of wiew is how Hawking radiation can be visible being emitted from place where there is nothing?
Or what's more interesting how the matter observer see falling on BH will dissappear as evaporation is in progress?</p>
| 3,328 |
<p>Apologies to all if this has been asked before, I searched but was unable to find one similar. </p>
<p>This is a question that has been bugging me for a while that i haven't really been able to find a suitable answer for. </p>
<p>I am aware that the space between an atoms nucleus and its electron cloud is teeming with virtual particles that allow the exchange of energy that give electrons an assigned energy level or 'shell' but what bugs me is about the space in between atoms.</p>
<p>What is in between atoms? is it classifiable as a vacuum where nothing at all exists? </p>
<p>I would find it hard to believe that atoms are pushed right up against each other at all times due to repulsive charges on the nucleus acting upon any other. </p>
<p>I accept that the gap is unbelievably small but on the scale of atoms and electrons, how small are we talking? Is there even a gap at all? Do we know what is in between or is it unknown? is it a similar process to the virtual particles between nucleus and electrons? </p>
<p>It is to my limited understanding that when particles "collide" there is no physical interaction, rather an exchange of energy through virtual photons. Is that what exists in all of these gaps? a constant exchange of virtual energy that acts as a consistent repulsion between all atoms?</p>
| 259 |
<p>If the object in motion gains mass, will it affect the change in mass of Earth if it stops revolving around Sun, since mass is responsible for gravity how will be the gravitational force change? </p>
| 3,329 |
<p>I read that the <a href="http://www.astronautix.com/engines/f1.htm" rel="nofollow">F-1</a> engine from the 1st stage of the Saturn V rocket is the most powerful engine ever created by mankind, delivering ~200 gigawatts of power.</p>
<p>Thus, I have got two questions:</p>
<ul>
<li><p>Will future rockets propelled by nuclear energy be able to surpass this limit, reaching magnitudes of terawatt power or beyond?</p></li>
<li><p>What is the theoretical limit to a rocket/spaceship engine power, if any?</p></li>
</ul>
| 3,330 |
<p>Is it possible to determine which kinds of 10 basic cloud type will form by reading skew-T graph? Also, can I determine the top and the bottom of the clouds by skew-t graph?</p>
| 3,331 |
<p>This is in continuation of my previous question,
<a href="http://physics.stackexchange.com/q/68210/">IR divergence and renormalization scale in dimensional regularization</a>.</p>
<p>Lubos gave a nice answer there but I want to get to a very specific example which is not found in usual QFT - (in QFT all examples I have seen are such that the number of momenta powers are the same in the denominator and the numerator).</p>
<p>Consider the 3 dimensional integrals in A.5 (page 19) of <a href="http://arxiv.org/abs/1301.7182" rel="nofollow">arXiv:1301.7182</a>. I guess one can say that these are UV divergent when 3>2(ν1+ν2) but the gap can be arbitrarily large depending on the values of ν1 and ν2 - right? Then how do the authors justify using d=3+ϵ expansion get the UV divergence? </p>
<ul>
<li><p>Also I guess these are IR divergent if 3<2ν1..right? </p>
<p>How does one define the $\epsilon$ to get this IR divergence in the $\epsilon$-expansion? </p></li>
<li><p>Also by looking at this A.5 is it clear that $d=3$ is somehow the "critical dimension"? (...the terminology is from Lubos's comments in the previous question...) </p></li>
</ul>
| 3,332 |
<p>I was watching a documentary entitled <a href="http://www.bbc.co.uk/bbcfour/documentaries/features/atom.shtml" rel="nofollow">"The Atom"</a> and one of the statements made was that Atoms behave differently when we look at them. I wasn't too sure about the reasoning behind this and i'm hoping someone could explain how or why this happens.</p>
<p>I'm not too knowledgeable about the field of Physics but the theory of Atoms is rather interesting.</p>
| 3,333 |
<p>I found this problem, and so far I am stumped. I was wondering if anyone wanted to solve it with me, or help me calculate eigenvectors, or just give insight on my questions.</p>
<blockquote>
<p><em>Consider a system of two spin-1/2 particles interacting through the Hamitonian
$$H = A(S_x^2 − S_y^2) + BS_z^2,$$
where $A$ and $B$ are constants and $S_x$, $S_y$ and $S_z$ are the three components of the total spin of the system. Find the energy spectrum and the corresponding eigenvectors.</em></p>
</blockquote>
<p>It's been awhile since I did QM, and I <em>know</em> you can't have simultaneous eigenvectors of more than one measurement of spin along an axis, but how does that relate to this problem where there Sx^2 and Sy^2 are in the hamiltonian? I've been trying to work out 4x4 matrices, but I can't find good eigenvectors, but even then I am confused how applying a pauli matrix (say for x or y) transforms the eigenvector to another eigenvector, which I think is another example of not being able to define simultaneous eigenvectors etc. How would one start this problem? I will break it down to S^2 and Sz terms, but there will be remaining terms, will they just not contribute to the energy spectrum? If someone much more knowledgable than me could solve this problem easily and post it, that would clarify so much to me. For now, I'll try to read Feynman's lecture on the subject.</p>
| 3,334 |
<blockquote>
<p><em>A force of 6 Newton and another 8 newton can be applied together to produce the effect of a single force of a) 1 N b) 11 N c) 15 N d) 20 N.</em></p>
</blockquote>
<p>More generally, if I know the magnitude of one force, say $F_1$, and another, say $F_2$, then how do I find the magnitude of the resultant total force of the two vectors? Is it even possible? And if it is, what do I do? Do I add them up together, or something like that?.</p>
| 3,335 |
<h2><strong>Preface</strong></h2>
<p>The definition of <strong>average speed</strong> of an object is defined by the distance travelled over time:
$$v_{avg} = \frac{x_2 - x_1}{t_2 - t1}$$
The interpretation of speed is that when you multiply speed with the time interval, you should get the distance you travelled at this interval. However, it does not measure the local variation so a better definition is the <strong>instantaneous speed</strong>:
$$v = \lim_{\Delta t\rightarrow0} \frac{\Delta x}{\Delta t}$$
The above are very standard stuff for introduction mechanics course.</p>
<h2><strong>Average speed with equal segments</strong></h2>
<p>Now, if we have speed for consecutive paths with equal length, beginner students often commit an error. They may use the following:
$$v_{avg} = \frac{1}{2}(v_1 + v_2)$$
as average speed between the start point and ending point, which is WRONG in general. The correct way to obtain the effective average speed is
$$v_{avg}=\left(\frac{1}{N}\sum_{i}^{n}v_{i}^{-1}\right)^{-1}$$
</p>
<h2><strong>Mean of varying speed measurements for the same path</strong></h2>
<p>Now, suppose that there is a situation that the distance between a starting point and an ending point is fixed. Now, there are experiments to measure the time spent by the traveller (or vehicle, or particle) moving along this path multiple time. Because the time it takes is always different, so we can obtain a list of speed $v_1, v_2, v_3, ..., v_n$ that differ with each other. Presumably there is a corresponding speed distribution $\mathcal{P}(v)$ for the speed travelling along this path. The <strong>mean of the speed distribution</strong> is given by:</p>
<p>$$\left\langle v\right\rangle = \int_0^\infty \mathcal{P}(v) dv \approx \frac{1}{N} \sum_i^n v_i$$
</p>
<p>Note that it is different from the average speed that defined at the first part. It is the mean of the speed distribution function and it is similar to the "wrong $v_{avg}$" discussed in the previous part. Please <strong>do not mix it up</strong> with the average speed defined above.</p>
<p>So my question: I am seeking an interpretation of this quantity $\left\langle v\right\rangle$, that is, I want to know the <em>situations/problems</em> that this idea of $\left\langle v\right\rangle$ can be applied.</p>
| 3,336 |
<p>Suppose I want to solve Nonlinear Schrodinger equation using imaginary time propagation to get the ground state solution. I choose $t = - i t$, and then solve the equation using split step Crank Nicholson method. All the excited states will decay faster than the ground state and leave finally the ground state of the system.
Suppose, I would like to check whether the obtain solution is stable or not.
In this regard, I would add a small perturbation to the obtained solutions and evolve it. If the solution is stable, it should converge else it diverges.
My question is that is it possible to do the second part using imaginary time propagation?
or I have to do it using the real time propagation.
Can somebody tell me where to use the real time propagation and imaginary time propagation and where not to use?</p>
| 3,337 |
<p>So, in one science fiction story, that tries to be realistic as possible apart from few space magics, humanity has contingency plan to blow up Jupiter. As in, totally destroy it in one massive nuclear explosion.</p>
<p>I'd like to know the effects of such event. Would it totally wreck the solar system or would the whole plan be non-issue?</p>
| 3,338 |
<p>I would like to extrapolate my current router wifi radiation from my phone.
If I know that my router is transmitting at 300mW and my phone displays the strength in -dbm (from 0 to -100 scale) if I have -50dbm strength does it mean that the current radiation is 150mW?
If not how can I extrapolate the radiation from -dbm scale?</p>
| 3,339 |
<p>In $\phi^3$ theory, are there any formula for determining the Symmetry factor as that is found for the $\phi^4$ theory in any standard book of Quantum Field Theory?</p>
| 172 |
<p>The <a href="http://www.google.com/#q=Gutzwiller+trace" rel="nofollow">Gutzwiller trace</a> is about</p>
<p>$$ d(E)=d_{0} (E)+ \sum_{p.o}A_{p}Cos(S(E)\ell_{p}) $$</p>
<p>and $ \ell_{p} $ are the length of the orbit.</p>
<p>However my question is, how can one derive the length of the orbit from the potential inside the quantum Hamiltonian.
$$ H=p^{2}+V(x) $$.</p>
| 3,340 |
<p>Does anybody know of an example were one could derive some important properties of a physical system from a symmetry of said system. </p>
<p>I´m specially looking for simple classical examples, which could serve to illustrate the importance of finding symmetries of a system to non-physicists (high school students or first year undergrads)</p>
| 3,341 |
<p>Following Edwin Hubble, it is widely believed that the universe is expanding, which is based on the red-shift of light from distant objects. Can <a href="http://en.wikipedia.org/wiki/Dark_matter" rel="nofollow">dark matter</a> cause light to be red-shifted and make it look like the universe is expanding while in fact it is not?</p>
| 3,342 |
<p>I understand from some studies in mathematics, that the generator of translations is given by the operator $\frac{d}{dx}$. </p>
<p>Similarly, I know from quantum mechanics that the momentum operator is $-i\hbar\frac{d}{dx}$.</p>
<p>Therefore, we can see that the momentum operator is the generator of translations, multiplied by $-i\hbar$. </p>
<p>I however, am interested in whether an argument can be made along the lines of <em>"since $\frac{d}{dx}$ is the generator of translations, then the momentum operator must be proportional to $\frac{d}{dx}$".</em> If you could outline such an argument, I believe this will help me understand the physical connection between the generator of translations and the momentum operator in quantum mechanics.</p>
| 867 |
<p>When you fill a glass with water, water forms a concave <a href="http://en.wikipedia.org/wiki/Meniscus" rel="nofollow">meniscus</a> with constant contact angle $\theta$ (typically $\theta=20^\circ$ for tap water):</p>
<p><img src="http://i.stack.imgur.com/VV6ik.png" alt="enter image description here"></p>
<p>Once you reach the top of the glass, the water-air interface becomes convex and water rises up to a height $\Delta h$ above the edge of the glass, allowing you to fill the glass beyond the naive capacity $\pi r^2 h$:</p>
<p><img src="http://i.stack.imgur.com/C44BZ.png" alt="enter image description here"></p>
<p>So when getting myself a glass of water, I came to wonder exactly how much this increases the capacity of a glass, and what physical constants are involved.</p>
<p>My intuition would be that for a very large glass, $\Delta h$ converges to a constant so that the effective water capacity of the glass grows like $\pi r^2 (h+\Delta h)$ (to make things simple I'm assuming that the glass is very thin: $\Delta r\ll r$). Perhaps such a constant depends on the precise shape of the rim of the glass. But if not, perhaps it is a constant multiple of the capillary length?</p>
<p>So, what can we say about $\Delta h$, the "rim contact angle" $\alpha$, or the shape of the water-air interface when the glass is filled at maximum capacity?</p>
| 3,343 |
<p>On p. 49 of Polchinski's book, he says: "Incidentally, the free massless scalar in two dimensions has a remarkably large amount of symmetry -- much more than we will have occasion to mention." </p>
<p>Does anyone know what this is referring to? Is he just saying that there are a lot of different choices of stress tensors that generate the conformal symmetry?</p>
| 3,344 |
<p><a href="https://en.wikipedia.org/wiki/Hooke%27s_law">Hooke's Law</a> tells us that $m\ddot{x} = -kx$. We can apply the <a href="https://en.wikipedia.org/wiki/Chain_rule">chain rule</a> to rewrite $\ddot{x}$ as follows:</p>
<p>$$\frac{\operatorname{d}\!^2x}{\operatorname{d}\!t^2} = \frac{\operatorname{d}\!v}{\operatorname{d}\!t}=\frac{\operatorname{d}\!x}{\operatorname{d}\!t}\frac{\operatorname{d}\!v}{\operatorname{d}\!x} = v\frac{\operatorname{d}\!v}{\operatorname{d}\!x}$$</p>
<p>Substituting this into $m\ddot{x} = -kx$, <a href="https://en.wikipedia.org/wiki/Separation_of_variables">separating variables</a>, and then integrating gives:</p>
<p>$$mv^2+kx^2 = c$$</p>
<p>for some constant $c$ to be determined by initial conditions. If we assume that the spring is stretched to a length $\lambda$ to the right before it is released then we have $v=0$ when $x=\lambda$ and so $c=k\lambda^2$. Hence:</p>
<p>$$mv^2+kx^2 = k\lambda^2$$</p>
<p>For all values of $m>0$, $k>0$ and $\lambda>0$, this gives the equation of an ellipse in the $xv$-plane. Ellipses have two special points, named <a href="https://en.wikipedia.org/wiki/Ellipse#Focus">foci</a>. The Earth follows a more-or-less elliptical path around the Sun, which sits at a focus. The Moon follows a more-or-less elliptical path around the Earth, which sits at a focus.My questions is: <strong>What is the physical significance of the foci of $mv^2+kx^2=k\lambda^2$</strong> </p>
<p>One characterisation of the foci, say $\phi_1$ and $\phi_2$, is that the sum of the distances $\operatorname{d}(\phi_1,p) + \operatorname{d}(p,\phi_2)$ is constant for all points $p$ of the ellipse.</p>
| 3,345 |
<p>I am referring to yet another version as the classical twin paradox.</p>
<p>In my version the moving apart of the twins is entirely induced by space expansion between them and they move apart each other at a very high speed, conceivably even higher than the local speed of light.</p>
<p>After some time space itself contracts again so that they meet again.</p>
<p>The question is: Is there an age difference between the two? </p>
<p>Is it correct to say that both stayed in the same inertial frame so that there is not even a paradox and no resulting age difference or is the situation more subtle?</p>
| 3,346 |
<p>I need to get a nice picture about how electron moves around nucleus? I find concept of probability and orbitals quite difficult to understand?</p>
| 579 |
<p>I've got metal coin : <a href="http://www.worldpeacecoin.org/" rel="nofollow">http://www.worldpeacecoin.org/</a></p>
<p>Ruble/dollar, a coin of disarmament with certificate. But, I am very spleeny person, I fear of it's radiance level and I don't know if I can trust it or check it somehow.</p>
<p>That could look weird but my fears feels realistic. I only know that Soviet R-12 (SS-4) nuclear missile is nuclear missile and I'm not sure how do they extract that metal from it? I think that any metal that could be extracted from nuclear missile should be dangerous for my health because of radiance level.</p>
<p>Please tell me if there is a method to check the radiance level of it or why should not I fear it?</p>
| 3,347 |
<blockquote>
<p><em>Show that the hydrogen atomic wavefunction $\psi_{3,1,1}$ is normalized, and that it is orthogonal to $\psi_{3,1,−1}$.</em></p>
</blockquote>
<p>I'm not sure if I'm supposed to consider the radial part. I can show that the spherical part, Y{m=1, l=1}, is normalized and that Y{m = 1, l=1} and Y{m = -1, l=1} are orthogonal... but I'm not sure how to do the radial component if it is supposed to be considered.</p>
| 3,348 |
<p><em>EDIT: Further clarification in the context of answers/comments received to 20 Jan has been appended</em></p>
<p><em>EDIT: 21 Jan - Response to the Lubos Expansion appended [in progress, not yet complete]</em></p>
<p><em>EDIT: 23 Jan - Visser's calculations appended</em></p>
<p><em>EDIT: 26 Jan - Peter Shor's thought experiments rebutted</em></p>
<p><em><strong>Summary to Date (26 Jan)</em></strong></p>
<p>The question is: are the Morris, Thorne, Yurtsever (MTY) and Visser mechanisms for converting a wormhole into a time machine valid? The objection to the former is that the "motion" of a wormhole mouth is treated in an inadmissable manner by the former, and that the <em>valid</em> mathematical treatment of the latter is subsequently misapplied to a case in which a sufficient (and probably necessary) condition does not apply (existence of a temporal discontinuity). It is maintained that extant thought experiments lead to incorrect conclusions because in the former case correct treatment introduces factors that break inertial equivalence between an unaccelerated rocket and co-moving wormhole mouth, and in the latter case especially do not respect the distinction between <em>temporal coordinate values</em> and spacetime <em>separations.</em></p>
<p>Given that the detailed treatment of the Visser case is reproduced below, a valid argument in favour of a wormhole time machine must show how an interval $ds^2=0$ (the condition for a Closed Timelike Curve) obtains in the absence of a temporal discontinuity.</p>
<p>In considering the MTY (1988) paper, careful consideration should be given to whether the authors actually transport a wormhole mouth or just the coordinate frame that is convenient for describing a wormhole mouth if one happened to exist there.</p>
<p>Issues concerning quantum effects, energy conditions, and whether any created time-machine could persist etc. are off topic; the question is solely about the validity of the reasoning and maths concerning time-machine creation from a wormhole.</p>
<p>The original postings in chronological order are below.</p>
<hr>
<p>Over at Cosmic Variance Sean Carroll recommended this place for the quality of the contributions so I thought I would try my unanswered question here; it is most definitely for experts.</p>
<p>The question is simple, and will be stated first, but I'll supplement the question with specific issues concerning standard explanations and why I am unable to reconcile them with what seem to be other important considerations. In other words, I'm not denying the conclusions I'm just saying "I don't get it," and would someone please set me straight by e.g. showing me where the apparent counter-arguments/reanalyses break down.</p>
<p>The question is: <strong>How can a wormhole be converted into a time machine?</strong></p>
<p>Supplementary stuff.</p>
<p>I have no problem with time-travel per se, especially in the context of GR. Godel universes, van Stockum machines, the possibilities of self-consistent histories, etc. etc. are all perfectly acceptable. The question relates specifically to the application of SR and GR to wormholes and the creation of time-differences between the mouths - leading to time-machines, as put forward in (A) the seminal Morris, Thorne, Yurtsever paper (Wormholes, Time Machines, and the Weak Energy Condition, 1987) and explained at some length in (B) Matt Visser's Book (Lorentzian Wormholes From Einstein To Hawking, Springer-Verlag, 1996).</p>
<p>A -- Context. MTY explore the case of an idealised wormhole in which one mouth undertakes a round trip journey (i.e. undergoes accelerated motion, per the standard "Twins Paradox" SR example).</p>
<p>What is unclear to me is how MTY's conclusions are justified given that the moving wormhole mouth is treated as moving against a Minkowskian background: specifically, <strong>can someone explain how the wormhole motion is valid as a diffeomorphism</strong>, which my limited understanding suggests is the only permitted type of manifold transformation in relativity.</p>
<p>Elaborating... wormhole construction is generally described as taking an underlying manifold, excising two spherical regions and identifying the surfaces of those two regions. In the MTY case, if the background space is Minkowski space and remains undistorted, then at times t, and t' the wormhole mouth undergoing "motion" seems to identify different sets of points (i.e. different spheres have to be excised from the underlying manifold) and so there is no single manifold, no diffeomorphism. [Loose physical analogue: bend a piece of paper into a capital Omega shape and let the "heels" touch... while maintaining contact between them the paper can be slid and the point of contact "moves" but different sets of points are in contact]</p>
<p>I'm happy with everything else in the paper but this one point, which seems to me fundamental: moving one wormhole mouth around requires that the metric change so as to stretch/shrink the space between the ends of the wormhole, i.e. the inference of a time-machine is an artefact of the original approach in which the spacetime manifold is treated in two incompatible ways simultaneously.</p>
<p>Corollary: as a way of re-doing it consistently, consider placing the "moving" wormhole mouth in an Alcubierre style warp bubble (practicality irrelevant - it just provides a neat handle on metric changes), although in this case v is less than c (call it an un-warp bubble for subluminal transport, noting in passing that it is in fact mildly more practicable than a super-luminal transport system). As per the usual Alcubierre drive, there is no time dilation within the bubble, and the standard wormhole-mouth-in-a-rocket thought-experiment (Kip Thorne and many others) produces a null result.</p>
<p>B -- Context. (s18.3 p239 onwards) Visser develops a calculation that begins with two separate universes in which time runs at different rates. These are bridged and joined at infinity to make a single wormhole universe with a temporal discontinuity. The assumption of such a discontinuity does indeed lead to the emergence of a time machine, but when a time-machine is to be manufactured within a single universe GR time dilation is invoked (one wormhole mouth is placed in a gravitational potential) to cause time to flow at different rates at the two ends of the wormhole (to recreate the effect described in the two universe case in which time just naturally flowed at different rates). However, in the case of a simple intra-universe wormhole there is no temporal discontinuity (nor can I see how one might be induced) and the application of the previously derived equations produces no net effect.</p>
<p>Thus, as I read the explanations, neither SR nor GR effects create a time-machine out of a (intra-universe) wormhole. </p>
<p>Where have I gone wrong?</p>
<p>Many thanks,</p>
<p>Julian Moore</p>
<p><strong>Edit 1: ADDITIONAL COMMENTS</strong></p>
<p>Lubos' initial answer is informative, but - like several other comments (such as Lawrence B. Crowell's) - the answerer's focus is on the impossibility of an appropriate wormhole per se, and not the reasoning & maths used by Morris, Thorne, Yurtsever & Visser.</p>
<p>I agree (to the extent that I understand them) with the QM issues, however the answer sought should assume that a wormhole can exist and then eliminate the difficulties I have noted with the creation of time differences between the wormhole mouths. Peter Shor's answer <em>assumes</em> that SR effects will apply and merely offers a way to put a wormhole mouth in motion; the question is really, regardless of how motion might be created, does & how does SR (in this case) lead to the claimed effect?</p>
<p>I think the GR case is simplest because the maths in Visser's book is straightforward, and if there is no temporal discontinuity, the equations (which I have no problem with) say no time machine is created by putting a wormhole mouth into a strong gravitational potential. The appropriate answer to the GR part of the question is therefore to show how a time machine arises in the absence of a temporal discontinuity or to show how a temporal <em>discontinuity</em> could be created (break any energy condition you like, as far as I can see the absolute <em>discontinuity</em> required can't be obtained... invoking QM wouldn't help as the boundary would be smeared by QM effects. </p>
<p>Lubos said that "an asymmetry could gradually increase the time delay between the two spacetime points that are connected by the wormhole", I'm saying, "My application of the expert's (Visser's) math says it doesn't - how is my application in error?"</p>
<p>The SR case is much trickier conceptually. I am asserting that the wormhole motion of MTY is <em>in principle</em> impossible because, not to put too fine a point on it, the wormhole mouth doesn't "move". Consider a succession of snapshots showing the "moving" wormhole mouth at different times and then view them quickly; like a film one has an appearance of movement from still images, but in this case the problem is that the wormhole mouth in each frame is a <em>different</em> mouth. If the background is fixed Minkowski space (ie remains undistorted) at times t and t' different regions of the underlying manifold have to be excised to create the wormhole at those times... so the wormhole manifolds are different manifolds. If the background is not fixed Minkowski space, then it can be distorted and a mouth can "move" but this is a global rather than a local effect, and just like the spacetime in an Alcubierre warp bubble nothing is happening locally.</p>
<p>Consider two points A & B and first stretch and then shrink the space between them by suitable metric engineering: is there are time difference between them afterwards? A simple symmetry argument says there can't be, so if the wormhole mouths are treated as features of the manifold rather than objects in the manifold (as it seems MTY treat them) then the only way to change their separation is by metric changes between them and no time machine can arise.</p>
<p>Of course, if a time-machine <em>could</em> be created either way, it would indeed almost certainly destroy itself through feedback... but, to repeat, this is not the issue.</p>
<p>Thanks to Robert Smith for putting the bounty on this question on my behalf, and thanks to all contributors so far.</p>
<p><strong>Edit 2: Re The Lubos Expansion</strong></p>
<p>Lubos gives an example of a wormhole spacetime that appears to be a time-machine, and then offers four ways of getting rid of the prospective or resulting time machine for those who object in principle. Whilst I appreciate the difficulties with time-machines I am neither for nor against them per sunt, so I will concentrate on the creation issue. I have illustrated my interpretation of Lubos' description below.
<img src="http://i.stack.imgur.com/D1wh7.jpg" alt="Block Universe Wormhole"></p>
<p>As I understand it, there is nothing in GR that in principle prevents one from having a manifold in which two otherwise spacelike surfaces are connected in such a way as to permit some sort of time travel. This is the situation shown in the upper part of the illustration. The question is how can the situation in the upper part of the illustration be obtained from the situation in the lower part?</p>
<p>Now consider the illustration below of a spacelike surface with a simple wormhole (which I think is a valid foliation of e.g. a toroidal universe). As time passes, the two mouths move apart thanks to expansion of space between them, and then close up again by the inverse process (as indicated by the changing separation of the dotted lines, which remain stationary)</p>
<p><img src="http://i.stack.imgur.com/JT5eo.jpg" alt="alt text"></p>
<p><strong>Edit 3: Visser's calculations reproduced for inspection</strong></p>
<p><em>Consider the result in the case where there is no temporal discontuity using the equations derived for the case where there is a discontinuity, given below</em></p>
<p>Visser, section 18.3, p239
The general metric for a spherically symmetric static wormhole</p>
<p>$$
ds^2~=~-e^{2\phi(l)}~dt^2~+~dl^2~+~r^2[d\theta^2~+~sin^2\theta~d\psi^2]~~~~~(18.35)
$$</p>
<p>Note that "there is no particular reason to demand that time run at the same rate on either side of the wormhole. More precisely, it is perfectly acceptable to have $ф(l=+\infty)~\neq~ф(l=-\infty)$"</p>
<p>Reduce to (1+1) dimensions for simplicity and consider
$$
ds^2~=~-e^{2\phi(l)}~dt^2~+~dl^2~~~~~~~~~~(18.36)
$$</p>
<p>The range of l is (-L/2,+L/2) and l=-L/2 is to be identified with l=+L/2.
Define</p>
<p>$$
\phi_\pm\equiv\phi(l=\pm~L/2); ~\Delta\phi\equiv\phi_+~-~\phi_-~~~~~~~(18.37)
$$</p>
<p>at the junction $l=\pm~L/2~~$ the metric has to be smooth, ie. $ds = \sqrt{g_{\mu\nu}{dx^\mu}{dx^\nu}}$ is smooth, implying
$$
d\tau=e^{\phi_-}dt_-=e^{\phi_+}dt_+~~~~~~~~~(18.38)
$$
Define the time coordinate origin by identifying the points
$$
(0,-L/2)\equiv(0,+L/2)~~~~~~(18.39)
$$
then the temporal discontinuity is
$$
t_+=t_-e^{(\phi_-~-~\phi_+)}~=~t_-e^{(-\Delta\phi)}~~~~~~(18.40)
$$
leading to the identification
$$
(t_-,-L/2)\equiv(t_-e^{-\Delta\phi},+L/2)~~~~~~~~~(18.41)
$$
which makes the metric smooth across the junction. Now consider a null geodesic, i.e. ds=0, which is
$$
{dl\over{dt}}=\pm{e^{+\phi(l)}}~~~~~(18.42)
$$
where the different signs correspond to right/left moving rays.
Integrate to evaluate for a right moving ray, with conventions $t_f$ is the final time and $t_i$ is the initial time
$$
[t_f]_+=[t_i]_-+\int_{-L/2}^{+L/2}e^{-\phi(l)}dl~~~~~(18.43)
$$
then apply the coordinate discontinuity matching condition to determine that the ray returns to the starting point at coordinate time
$$
[t_f]_-=[t_f]_+e^{\Delta\phi} = [[t_i]_-+\oint{e^{-\phi(l)}}dl]e^{\Delta\phi}~~~~~(18.44)
$$
A closed right moving null curve exists if $t[_f]_-=[t_i]_-$, i.e.
$$
[t_i]^R_-={{\oint{e^{-\phi(l)}dl}}\over{e^{\Delta\phi}-1}}~~~~~(18.45)
$$</p>
<p><strong>Edit 4: Peter Shor's thought experiments re-viewed</strong></p>
<p>Peter Shor has acknowledged (at the level of "I think I see what you mean...") both the arguments against wormhole time machine creation (the absence of the required temporal discontinuity in spacetime if GR effects are to be used, and that wormhole mouth motion requires metric evolution inconsistent with the Minkowksi space argument of MTY) but still believes that such a wormhole time-machine can be created by either of the standard methods offered a thought experiment. This is a counter to those thought experiments and whilst it does not constitute proof of the contrary (I don't think such thought experiments are rich enough to provide proof either way), I believe it casts serious doubt on their interpretation, thereby undermining the objections.</p>
<p>The counter arguments rely on the key distinction between the <em>values</em> of the time coordinate and the <em>separation</em> of events ($ds^2$). Paragraphs are numbered for ease of reference.</p>
<p>(1) Consider the classic Twins Paradox situation and the associated Minkowksi diagram. When the travelling twin returns she has the same t coordinate ($T_{return}$) as her stay-at-home brother (who said they had to be homozygous? :) ) As we all know, despite appearances to the contrary on such a diagram, the sister's journey is in fact shorter (thanks to the mixed signs in the metric), thus it has taken her "less time" to reach $T_{return}$ than it took her brother. Less time has elapsed, but she is not "in the past".</p>
<p>(2) Now consider the gravity dunking equivalent Twins scenario. This time he sits in a potential well for a while and then returns to his sister who has stayed in flat space. Again their t coordinate is the same, but again there is a difference in separations; this time his is shorter.</p>
<p>(3) Now for the travelling/dunking Twin substitute a wormhole mouth; then the wormhole mouths are brought together they do so at the same value of t. The moving mouths may have "travelled" shorter spacetime distances but they are not "in the past"</p>
<p>(4) Suppose now we up the ante and give the travelling/dunking Twin a wormhole mouth to keep with them...</p>
<p>(5) According to the usual stories, Mr A can watch Ms A receding in her rocket - thereby observing her clock slow down - or he can communicate with her through the wormhole, through which he does <em>not</em> see her clock slow down because there is no relative motion between the wormhole mouth and Ms A. Since this seems a perfectly coherent picture we are inevitably led to the conclusion that a time machine comes into existence in due course.</p>
<p>(6) My objection to this is that there are reasons to doubt what is claimed to be seen through the wormhole, and if the absence of time dilation is not observed <em>through the wormhole</em> we will not be led to the creation of a time machine. So, what <em>would</em> one see through the wormhole, and why?</p>
<p>(7) I return to the question of the allowable transformations of the spacetime manifold. If a wormhole mouth is rushing "through" space, the space around it must be subject to distortion. Now, whilst there are reasons to doubt that one can arrange matter in such a way as to create the required distortion (the various energy condition objections to the original Alcurbierre proposal, for instance), we are less concerned about the <em>how</em> and more concerned with the <em>what if</em> (particularly since, if spacetime cannot "move" to permit the wormhole mouth to "move", the whole question becomes redundant). The very fact that spacetime around the "moving" wormhole mouth is going to be distorted suggests at least the possibility that what is observed through the wormhole is consistent with what is observed the other way, or that effects beyond the scope of the equivalence principle demonstrate that observation through the wormhole is not equivalent to observing from an inertial frame. Unfortunately I don't have the math to peform the required calculations, but insofar as there is a principled objection to the creation of a time machine as commonly described, I would hope that someone would check it out.</p>
<p>(8) What then for the dunking Twin? In this case there is no "motion" of the wormhole mouth, so no compensating effects can be sought from motion. However, I believe that one can apply to the metric for help. Suppose that the wormhole mouth in the gravitational well is actually embedded in a little bit of flat space, then (assuming the wormhole itself is essentially flat) the curvature transition happens outside the mouth and looking through the wormhole should be like looking around it: Mr A seems very slowed down. If, instead, the wormhole mouth is fully embedded in the strongly curved space that Mr A also occupies, then the wormhole cannot be uniformly flat and again looking through the wormhole we see exactly what we see around it (at least as far as the tick of Mr A's watch is concerned.) but the transition from flat to curved space (and hence the change in clock rates) occurs over the interior region of the wormhole.</p>
<p>(9) Taken with the "usual" such thought experiments, we now have contradictory but equally plausible views of the same situations, and they can't both be right. I feel however that no qualitative refinements will resolve the issue, thus I prefer the math, which seems to make it quite plain that the usually supposed effects do not in fact occur. Similarly, if you disagree that the alternative view is plausible, since the "usual" result does not seem plausible to me,maths again provides the only common ground where the disagreement can be resolved. I urge others to calculate the round-trip separations using the equations provided from Visser's work.</p>
<p>(10) I say the MTY paper is in error because it treats spacetime as flat, rigid Minkowskian and then treats the "motion" of a wormhole mouth in a way that is fundamentall incompatible with a flat rigid background.</p>
<p>(11) I say Visser is in error in applying his (correct) inter-universe wormhole result to an intra-universe wormhole where the absence of the temporal discontinuity in the latter nullifies the result.</p>
<p>(12) These objections have been acknowledged but but no equally substantive arguments to undermine them (i.e. to support the extant results) has been forthcoming; they have not been tackled head on.</p>
<p>(14) I am not comfortable with any of the qualitative arguments either for against wormhole-time machines; an unending series of thought experiments is conceivable, each more intricate and ultimately less convincing than the last. I don't want to go there; look at the maths and object rigorously if possible.</p>
| 3,349 |
<p>We start with the general case of $AdS_{p+2}$ i.e AdS space in $p+2$ dimension.
\begin{equation}
X_{0}^{2}+X_{p+2}^{2}-\sum_{i=1}^{p+1}X_{i}^{2} = R^2
\end{equation}
This space has an isometry $SO(2,p+1)$ and is homogeneous and isotropic. The Poincare Patch is given by
\begin{equation}
ds^2 = R^{2}\left(\frac{du^2}{u^2}+u^2(-dt^2 +d\mathbf{x}^{2})\right)
\end{equation}
According to Equation (2.27) of the article <a href="http://arxiv.org/abs/hep-th/9905111" rel="nofollow">http://arxiv.org/abs/hep-th/9905111</a>, The second metric covers only half of the hyperboloid. Firstly, how do I show this. Secondly, when I go to the asymptotic limit (small radial distance), should the topology of the two spaces be different?</p>
| 3,350 |
<p>An spherical spaceship moving in two dimensions is at position $(x, y)$ and has a velocity $(v_x, v_y)$. It also has a maximum acceleration $a_{max}$. Its goal is to be at position $(x', y')$ with a velocity of $(v'_x, y'_x)$. How long does the optimal path take?</p>
<p>I see that the problem can be reduced to a spaceship at $(0, 0)$ with a velocity of $(0, 0)$, trying to intercept a object currently at $(x'-x, y'-y)$ with a velocity of $(v'_x - v_x, y'_x - y_x)$.</p>
<p>I have a hunch that the optimal path will always be constant acceleration in one direction, possibly with a reversal somewhere along the way.</p>
<p>I'm curious because I believe the total time will be a consistent and admissable heuristic for a Newtonian pathing algorithm that takes velocity into account. </p>
| 3,351 |
<p>Let me start by saying that my knowledge of physics is almost nil (only high school level and I pretty much forgot everything) so bear with me, if I am asking naive questions the answers of which may be obvious to you. </p>
<p>My question is twofold: </p>
<ol>
<li><p>Is it possible to use the motion of the planet or its magnetic field to extract an electric current efficiently?</p></li>
<li><p>If yes, how much electricity could be generated? Enough to power a toaster? A house? A city?</p></li>
</ol>
| 82 |
<p>For positive definite infinite dimensional Hilbert spaces, there is the standard Cauchy norm topology. What if this state space has an indefinite norm or a positive semidefinite one, as in gauge theories or Faddeev-Popov ghosts? Which infinite sums are valid, and which aren't?</p>
<p>Similarly, for the algebra of operators, which norm topology do we choose? Not the W*-one? The C* one?</p>
| 3,352 |
<p>Which kind of truss would produce the best result (aka. best distribution of force):</p>
<p>i) An equilateral, isosceles or a scalene triangle<br/>
ii) 3 small triangles or 2 big triangles (the overall masses of both are equal)</p>
<p>The truces are made of fettucini and need to hold a mass of 6 kg.</p>
| 3,353 |
<p>I know that the Coulomb potential is logarithmic is two dimensions, and that (see for instance this paper: <a href="http://pil.phys.uniroma1.it/~satlongrange/abstracts/samaj.pdf">http://pil.phys.uniroma1.it/~satlongrange/abstracts/samaj.pdf</a>) a length scale naturally arises:</p>
<p>$$ V(\mathbf{x}) = - \ln \left( \frac{\left| \mathbf{x} \right|}{L} \right) $$</p>
<p>I can't see what's the physical meaning of this length scale, and, most of all, I can't see how this length scale can come up while deriving the 2D Coulomb potential by means of a Fourier transform:</p>
<p>$$ V(\mathbf{x}) = \int \frac{\mathrm{d^2 k}}{\left( 2 \pi \right)^2} \frac{e^{\mathrm{i} \mathbf{k} \cdot \mathbf{x}}}{\left| \mathbf{k} \right|^2} $$</p>
<p>I would appreciate some references where the two-dimensional Fourier transform is carried out explicitly and some insight about the physical meaning of L and how can it arise from the aforementioned integral.</p>
<p>Thanks in advance!</p>
| 3,354 |
<p>I was just starting a barbecue fire by blowing on the smouldering coals when I realised I had no idea what the sound was actually caused by. I can make the sound by blowing at almost any flame I can think of, and I guess it is perhaps related to the increased oxygen consumption and a turbulent flow. Why does a disturbed flame make a sound?</p>
| 3,355 |
<p>For a physics issues investigation I chose to investigate what effects lightning could have on an aeroplane while in flight if it was struck and then go on to discuss some possible implications of engineers not taking into account the power of positive lightning.</p>
<p>Just in-case you don't know what positive lightning is, my understanding of it at least is that when charges accumulate in clouds (I won't go into how) in most cases the underside of the cloud is negatively charged and the top of the cloud is positively charged. Basically positive lightning is a lot more powerful than negative lightning as it has a higher voltage and current.</p>
<p>Q1. How would you determine the potential difference between the underside of the cloud (given an overall charge) and the ground (given the overall charge) and hence the electric field strength. $E = V/d$ ? But how would I calculate the voltage?</p>
<p>Q2. I understand that $V = IR$. And this is why the voltage of a positive lightning strike is higher than a negative strike as the resistance for the positive strike is higher (it has to go out to the side of the cloud and THEN down). But why is the current higher? If $I = V/R$ and the resistance is higher, wouldn't the current be lower?</p>
<p>(This question probably isn't as high a level as many of the other questions on this site so you should find it quite easy to answer.)</p>
| 3,356 |
<p>I'm looking for a formula that will return the number of <a href="http://en.wikipedia.org/wiki/Day_length" rel="nofollow">hours per day</a> given a specific location. I was thinking that can be calculated as a difference of sunrise and sunset, but I see that there are some other ways, like in this <a href="http://mathforum.org/library/drmath/view/56478.html" rel="nofollow">topic</a>.</p>
<p>What is the best, fast and correct way to calculate this?</p>
| 28 |
<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/14968/superluminal-neutrinos">Superluminal neutrinos</a> </p>
</blockquote>
<p>I remember not too long ago hearing very much speculation about a discovery that perhaps neutrinos are faster than the speed of light.
I've heard nothing since, not even anything to confirm that they're not, although I strongly assume they're not due to the fact that I've heard nothing since.</p>
<p>In my assumption that they aren't, can someone explain to me why we thought they were and what happened to cause us to abandon the theory? Technical problems?</p>
| 83 |
<p>What is each mathematical step (in detail) that one would take to get from:</p>
<p>$E^2 - p^2c^2 = m^2c^4$ </p>
<p>to</p>
<p>$E = \gamma mc^2$,</p>
<p>where $\gamma$ is the relativistic dilation factor.</p>
<p>This is for an object in motion.</p>
<p>NOTE: in the answer, I would like full explanation. E.g. when explaining how to derive $x$ from $\frac{x+2}{2}=4$, rather than giving an answer of "$\frac{x+2}{2}=4$, $x+2 = 8$, $x = 6$" give one where you describe each step, like "times 2 both sides, -2 both sides" but of course still with the numbers on display. (You'd be surprised at how people would assume not to describe in this detail).</p>
| 3,357 |
<p>I'm teaching myself mechanics, and set out to solve a problem determining the optimum angle to throw a projectile when standing on a hill, for maximum range. My answer seems <em>almost</em> plausible, except for one term, which, to be plausible, needs to have its sign switched. But I can find no hole in my reasoning.</p>
<p>Problem: I am standing on a straight, downward sloped hill, and wish to throw a rock for maximum range. The hill is sloped down from horizontal by $\varphi$. What angle $\theta$ above the horizontal should I throw it at?</p>
<p>My solution:</p>
<ol>
<li><p>Use coordinates so that $x$ is parallel to the hill </p></li>
<li><p>Let $\alpha = \varphi + \theta$ (that is, the angle above the ground that I'm throwing at)</p></li>
<li><p>Then initial $v$ is $v_x = \cos \alpha$, $v_y = \sin \alpha$ (normalizing the units to remove any constants)</p></li>
<li><p>Acceleration due to gravity is then $a_x = -k \sin \varphi$, $a_y = - k \cos \varphi$ (gravity is in the y direction). To make the calculations simpler, assume $k = 2$ (answer holds for any value of gravity, so is same on Moon as on Earth)</p></li>
<li><p>We want to find the alpha which maximizes $s_x$ at the time that makes $s_y = 0$. First, find the time which makes $s_y = 0$; call it t.</p></li>
<li><p>$s_y = t \sin \alpha - t^2 \cos \varphi$. Using the quadratic formula, $s_y = 0$ at $t = 0$ or $t = \frac{ \sin \alpha }{\cos \varphi}$.</p></li>
<li><p>Now, find $s_x$ at this $t$. Substituting in and using basic algebra and trig, we get $s_x = \sin \alpha \, \cos \alpha - \sin ^2 \alpha\, \tan \varphi$. (This makes sense; the first term maxes at $\pi/4$, like we'd expect from symmetry. The second term tells us that if the ground banks down significantly, we should lower our angle of throwing. Very plausible.)</p></li>
<li><p>Taking phi as a constant, we wish to maximize this expression. A little calculus and trig identities gets the derivative equal to $\cos(2\alpha)- \sin(2\alpha) \,\tan \varphi$, which has a zero at $\alpha = \pi/4 - \varphi/2$, or $\theta = \pi/4 - 3\varphi/2$. <strong>Here's where things break down.</strong> The first term, $\pi/4$, seems correct. <strong>But the second term gives ludicrous results.</strong></p></li>
<li><p>Switching the sign of the second term in the alpha equation ends up with $\theta = \pi/4 - \varphi/2$, which gives completely plausible results. But I can't find any error in my reasoning or calculations!</p></li>
</ol>
<p>Can anyone find the missing link?</p>
<hr>
<p>Answer explanation:</p>
<p>As Pygmalion determined, step 4 is wrong. The $a_y$ value is correct, <em>but</em>, $a_x$ should be <em>positive</em>: pointing down the hill.</p>
<p>The answer is independent of the <em>magnitude</em> of gravity; but it depends on the <em>direction</em>.</p>
<p>Revising the derivation:</p>
<p>7. $s_x = \sin \alpha \cos \alpha + \sin ^2 \alpha \tan \phi$</p>
<p>8. Derivative is $\cos(2 \alpha) + \sin(2 \alpha) \tan \phi$, with zero at $\alpha = \pi/4 + \phi/2$, thus $\theta = \pi/4 - \phi/2$. <strong>QED.</strong></p>
| 3,358 |
<p>I would like to know how come if <a href="http://en.wikipedia.org/wiki/Dark_matter">dark matter</a> was electrically charged it would reflect light. What are the equations or the logic behind it? </p>
| 3,359 |
<p>The Dirac adjoint for Dirac spinors is defined as,
$$
\bar{u} = u^{\dagger} \gamma^{0} \, .
$$
However I have come across this,
$$
\overline{\gamma^{\mu}} = \gamma^{\mu} \, , \tag{1}
$$
(where $\gamma^{\mu}$ are the $4\times4$ gamma matrices). Naively applying the same rules as for the Dirac spinor clearly does not get us anywhere,
$$
\overline{\gamma^{\mu}} = \gamma^{\mu \dagger} \gamma^{0} = \gamma^{0} \gamma^{\mu} \gamma^{0} \gamma^{0} = \gamma^{0} \gamma^{\mu} \neq \gamma^{\mu} \, .
$$
So it seems that the Dirac adjoint for a matrix is defined differently, so in trying to figure this out I make the following reasoning, let $A$ be a $4 \times 4$ matrix and $u$ a Dirac spinor so that $Au$ is again a Dirac spinor. Taking the Dirac conjugate (which is defined) gives,
$$
\overline{A u} = (A u)^{\dagger} \gamma^{0} = u^{\dagger} A^{\dagger} \gamma^{0} = u^{\dagger} \gamma^{0} \gamma^{0} A^{\dagger} \gamma^{0} = \bar{u} \;\underbrace{\gamma^{0} A^{\dagger} \gamma^{0}}_{ = \bar{A} ? } \, .
$$
So my guess is that $\bar{A} = \gamma^{0} A^{\dagger} \gamma^{0}$. If this is the case it is straightforward to show that $\overline{\gamma^{\mu}} = \gamma^{\mu} $.</p>
<p><strong>My question</strong> is the following, is the above statement correct? Is it so that the Dirac adjoint is actually only defined for Dirac spinors but it can be sort of extended to $4 \times 4$ matrices as above (allowing one to write $\overline{A u} = \bar{u} \bar{A}$)?</p>
<hr>
<p><a href="http://epx.phys.tohoku.ac.jp/~yhitoshi/particleweb/ptest-3.pdf" rel="nofollow">Link</a> where I found eq. (1) (page 93, eq. 3.249)</p>
<p><a href="http://www.itp.phys.ethz.ch/research/qftstrings/archive/12HSQFT1/Chapter05.pdf" rel="nofollow">Link</a> where I found eq. (1) and the claim $ \overline{X} = \gamma^{0} X \gamma^{0} $ which appears to be missing a "$^{\dagger}$"? (page 9, eq. 5.54)</p>
| 3,360 |
<p>A friend of mine seems to think that wind affects the ground-speed of a ground-based vehicle in the same way airspeed affects an airborne aircraft. i.e. If faced with a 10mph headwind, your car isn't actually traveling at 60mph, it's traveling at 50mph.</p>
<p>I have tried to explain why that is incorrect, but I'm not having any luck. I wondered if any of you smart folks might be able to explain in a convincing, but in relative layman's terms, why wind does not affect the speed of a car like that.</p>
| 3,361 |
<p>Light bulbs, <a href="http://en.wikipedia.org/wiki/Electric_power" rel="nofollow">Wattage</a> meaning?</p>
<p>Two incandescent bulbs (120 V, 25 <a href="http://en.wikipedia.org/wiki/Watt#Confusion_of_watts.2C_watt-hours_and_watts_per_hour" rel="nofollow">Watt</a>) and (120 V, 500 Watt) connected to the same batteries.</p>
<p>Which one shines brighter? And why?</p>
| 3,362 |
<p>In a simply connected container containing a superfluid and rotating, there is a net circulation of superfluid. This is found due to the vortices formed, around which the superfluid rotates. These vortices have been found to generally form a triangular lattice arrangement. Why is the triangular lattice arrangement preferred by the vortices formed ?</p>
| 3,363 |
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