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<p><strong>EDIT</strong> - I have included the context of the quote I am interested in, as people seem to be as baffled by Einstein's quote as I am:</p>
<hr>
<p>In a 1920 address Einstein says this: </p>
<blockquote>
<p>Think of waves on the surface of water. Here we can describe two entirely different things. Either we may observe how the undulatory surface forming the boundary between water and air alters in the course of time; or else-with the help of small floats, for instance - we can observe how the position of the separate particles of water alters in the course of time. If the existence of such floats for tracking the motion of the particles of a fluid were a fundamental impossibility in physics - if, in fact nothing else whatever were observable than the shape of the space occupied by the water as it varies in time, we should have no ground for the assumption that water consists of movable particles. But all the same we could characterise it as a medium.</p>
<p><strong>We have something like this in the electromagnetic field. For we may picture the field to ourselves as consisting of lines of force. If we wish to interpret these lines of force to ourselves as something material in the ordinary sense, we are tempted to interpret the dynamic processes as motions of these lines of force, such that each separate line of force is tracked through the course of time. It is well known, however, that this way of regarding the electromagnetic field leads to contradictions.</strong></p>
<p>Generalising we must say this:- There may be supposed to be extended physical objects to which the idea of motion cannot be applied. They may not be thought of as consisting of particles which allow themselves to be separately tracked through time. In Minkowski's idiom this is expressed as follows:- Not every extended conformation in the four-dimensional world can be regarded as composed of world-threads. The special theory of relativity forbids us to assume the ether to consist of particles observable through time, but the hypothesis of ether in itself is not in conflict with the special theory of relativity. Only we must be on our guard against ascribing a state of motion to the ether. (<a href="http://www-groups.dcs.st-and.ac.uk/~history/Extras/Einstein_ether.html" rel="nofollow">http://www-groups.dcs.st-and.ac.uk/~history/Extras/Einstein_ether.html</a>) </p>
</blockquote>
<p>I have seen a lot of animations doing exactly this, depicting the lines of force as changing through time. For example: <a href="https://www.youtube.com/watch?v=VdoL8IOwJw0" rel="nofollow">https://www.youtube.com/watch?v=VdoL8IOwJw0</a> (go to the very end of the video, or look at minutes 1:53-2:08 , 7:31-7:48) </p>
<p>What are the contradictions that Einstein is talking about here? </p>
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<blockquote>
<p>Problem: The marble rolls down the track and around a loop-the-loop of radius R. The marble has mass $m$ and radius $r$. What minimum height $h$ must the track have for the marble to make it around the loop-the-loop without falling off?<br>
Express your answer in terms of the variables $R$ and $r$.</p>
</blockquote>
<p>I found this solution to be very reasonable:</p>
<p>$$mg = m a_c = m \frac{V^2}R $$
which leads to
$$V = \sqrt{g R} $$</p>
<p>The energy at the top of the loop KE = Delta PE</p>
<p>$$\frac12 m V^2 + m g (2R) = m g h \\
\frac12 (g R) + g (2R) = g h \\
\left(\frac12+2\right) R = h $$</p>
<p>so $h = 5/2 R $</p>
<p>However the correct answer is actually $\frac52(R-r)$, I think it's because the radius of the loop is measured from the center of the ball rolling on it, so the they subtracted $R$ from $r$, but how would you derive that? I tried the same steps just using $R-r$ instead of $R$ but I got a different answer.</p>
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<p>From what I've (hopefully) understood from the AdS/CFT correspondence, physical quantities have a dual version. For example, the position in the bulk is the scale size (in renormalization), and waves in a curved gravitational background are the dynamics of quantum criticality. </p>
<p>But the partition functions of both the gravitational and the conformal field theory sides are equal (in the right limits):</p>
<p>\begin{equation}
Z_{QG} = Z_{CFT}
\end{equation}</p>
<p>Which allows things like correlation functions to be calculated on one side, for 'use' on the other. </p>
<p>From this, it would seem like thermodynamic quantities like magnetization and internal energy have the same meaning on each side of the correspondence. Is that correct? Or does, for example internal energy, have a different meaning on the gravitational side? </p>
<p>If it remains an internal energy, what exactly is it an internal energy of? </p>
<p>Similarly, if the correlation functions in the CFT are, for example, static and relate to different positions, what do they correspond to in the gravitational side?</p>
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<p>I am not sure if this question is too naive for this site, but here it goes. In QFT calculations, it seems that everything is rooted in formal power series expansions, i.e. , what dynamical systems people would call <a href="http://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Lindstedt_method" rel="nofollow">Lindstedt series</a>. However, from what I heard, this series (for QFT case) is known to have zero radius of convergence, and it causes tons of difficulties in theory. My question is, are there approaches that start with an iterative process that has a better chance of converging (e.g., a fixed point iteration), and build computational methods for QFT from there?</p>
<p>In other words, when there are so many approaches to approximate the exact solution of, say nonlinear wave (and Klein-Gordon, Yang-Mills-Higgs-Dirac etc) equations on the classical level, why do we choose, when we quantize, only a couple of approaches, such as power series, and lattice regularization (the latter essentially a finite difference method)? Note that it is milder than making QFT completely rigorous, it is just about computing things a bit differently.</p>
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] |
<p>In Ryder Page141, it is written "<em>the electromagnetic field, like any massless field, possesses only two independent components, but is covariantly described by a 4-vector $A_{\mu}$</em>". </p>
<p>Why are there only two independent components? Shouldn't all four components be independent from one another?</p>
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<p>Is there any experiment that would show that radio waves (I am talking about macroscopic wavelength, say between 0.01m - 2m) are made out of individual photons? Sort of an equivalent of the photo-electric effect? Or maybe a photon-multiplier for radio waves? Can a single photon of radio waves be picked up by 2 antennas at the same time? Or is it absolutely senseless to see radio waves as a stream of individual photons?</p>
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<p>As a grad student, I have a single publication, a conference proceeding, to my name. So, my question is what do I need to do to obtain a post-doc position? Obviously, my references are going to be important. But, what other methods are available to me to demonstrate my competence? As I am US based, is it worth my time to apply for an <a href="http://sites.nationalacademies.org/pga/rap/" rel="nofollow">NRC fellowship</a>?</p>
| 4,822 |
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<p>Rather surprised I haven't seen many questions or discussion regarding the <a href="http://www.wired.com/wiredscience/2011/12/higgs-lhc-anticipated/" rel="nofollow">rumored confirmation of the Higgs field</a>. As I understand it, the energies where they saw things were actually quite a bit higher than they had predicted (guessed?). </p>
<ul>
<li>What does it mean that the energies where it was detected are higher than anticipated?</li>
<li>Does it impact the way we understand the standard model to work?</li>
</ul>
<p>EDIT (Moshe, December 13): Now that the announcement is out, these questions and the more general one of potential implications of this for various ideas of BSM physics can be answered here.</p>
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<p>The focus point is an interesting region of the cMSSM parameter space at high $m_0$ and low $m_{1/2}$. Features are high scalar masses (> 1-2 TeV), light charginos / neutralinos (which are higgsino-like), and fairly low fine-tuning. One can also achieve correct dark matter relic densities and the correct higgs mass.</p>
<p>There seems to be a curve in the $m_0, m_{1/2}$ plane in this region, where if you approach it, the $\tilde\chi^0_2 \,\tilde\chi^\pm_1$ cross section goes up radically (maybe even diverges, spectrum generators can be finicky in that region). Why is that so? You can also rephrase the question, why do the chargino and neutralino masses go down there? I'd expect to first order a simple relationship between $m(\chi)$ and $m_{1/2}$, but around the focus point the masses seem very sensitive to $m_0$.</p>
<p>I guess this is due to the renormalization flow of the particle masses. The slope of the running masses seems a bit steeper than usual in focus point SUSY. There seem to be some interesting cancellations going on. It would be great if someone could elaborate on this.</p>
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<p>Modelling the diffusion of a gas dissolved in water in a vertical column of water, several meters deep. Also assuming the water is completely still, so only diffusion plays a role. (Actually a model of methane and oxygen diffusion in the water in peatlands.)</p>
<p><a href="http://en.wikipedia.org/wiki/Fick%27s_laws_of_diffusion" rel="nofollow">Fick's law</a> says that diffusion flux $J$ depends on the concentration gradient as</p>
<p>$J = -D \frac{\partial C}{\partial z}$.</p>
<p>where $C$ is the gas concentration. (And $D$ diffusion coefficient.)</p>
<p>But</p>
<ol>
<li><p>There is also hydrostatic pressure gradient, and in a higher pressure more gas can be dissolved in a volume of water and still be in chemical equilibrium with a volume of lower pressure of water with a somewhat smaller concentration of dissolved gas. With a small enough gas concentration gradient, the diffusion should actually go against the gradient.</p></li>
<li><p>The water can have different temperatures at different depths. And the solubility of the gas is temperature-dependent. So again, if the concentration gradient is small, there could actually be diffusion flux from lower concentration to higher concentration, if the latter is also much colder (higher solubility in cold water).</p></li>
</ol>
<p>How to modify the diffusion equation to account for these effects? As far as I understand, the gradient driving the diffusion should not be calculated from the concentration, but from then chemical potential of the gas dissolved in water.</p>
<p>So in other words, how to calculate the chemical potentials of oxygen or methane dissolved in water at certain pressure and solubility (temperature)?</p>
| 4,825 |
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<p>just a quick question on $F_{\mu\nu}^a$. I'm correct to think $F_{\mu}^{\mu,a}$ vanishes, aren't I? (Just want to make sure...) My reasoning is as follows:</p>
<p>The derivative terms cancel anyways - that's obvious - so the only "critical" term of $F_{\mu}^{\mu,a}$ is $f^{abc}A_{\mu}^b A^{\mu,c}$ but this vanishes because the combination of A's is symmetric but the $f$ totally antisymmetric. Am I right?</p>
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<p>Consider an expanding universe with the following metric in conformal time/co-moving coordinates:</p>
<p>$$ds^2=a^2\left[-c^2\left(1+\frac{2\phi}{c^2}\right)d\eta^2+\left(1-\frac{2\phi}{c^2}\right)\left(dx^2+dy^2+dz^2\right)\right]$$</p>
<p>What would be the formula to compute the redshift $1+z=\frac{\nu_S}{\nu_O}$ of a photon knowing ($\lambda$ is the affine parameter):</p>
<ul>
<li>$[\eta_O, x_O, y_O, z_O]$, $[\frac{d\eta_O}{d\lambda}, \frac{dx_O}{d\lambda}, \frac{dy_O}{d\lambda}, \frac{dz_O}{d\lambda}]$, $a_O$, $\phi_O$ for the observer</li>
<li>$[\eta_S, x_S, y_S, z_S]$, $[\frac{d\eta_S}{d\lambda}, \frac{dx_S}{d\lambda}, \frac{dy_S}{d\lambda}, \frac{dz_S}{d\lambda}]$, $a_S$, $\phi_S$ for the source</li>
</ul>
<p>and considering that neither the observer nor the source have peculiar velocities regarding to the background.</p>
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<p>I hope lab / experimental physics is fair game for this web-site. If not, sorry!</p>
<p>I'm designing a sensor system to perform specialized [astronomy and space-sciences] experiments, and need a "reality check" to support or adjust my theoretical calculations.</p>
<p>What I need is the "counts per second" produced by any modern APD (avalanche photo-diode) sensor through a telescope of any specific "aperture" of a star of any "visual magnitude". I also need the number of "counts per second" of "nothing" (the "dark count") to subtract that from the "counts per second" when illuminated by the star (to determine the "counts per second" generated by the star alone).</p>
<p>This "reality check" will help me assure various "inefficiency allowances" I made are realistic. Examples:</p>
<pre><code>#1: overall detector QE over relevant visual [and near IR] wavelengths.
#2: loss of light in atmosphere before entering telescope.
#3: loss of light in telescope optics.
#4: loss of light in fiber (if any).
#5: anything/everything else.
</code></pre>
<p>As implied, I am only interested in the APD operating in "photon counting mode" (not analog).</p>
<p>I've read about 5 dozen articles that discuss APDs for astronomy, but none give a straightforward value. The closest I found was a vague statement that the limited magnitude was 22nd magnitude on a 6-meter telescope based upon observations of the crab nebula pulsar. But this is not specific and the object is highly variable (on a short time frame). They did not say, for example, whether they consider their "limiting magnitude" is where the count rate rises from 200 per second (dark count) to 220 per second (measurement), or 200cps to 400cps, or over what time period, or any other indication of their definition.</p>
<p>All I need is ONE clear statement of cps for any aperture and visual magnitude star. You'd think I could find that in dozens if not hundreds of articles, but... no. Probably a clear statement like I need exists in some article somewhere, but I haven't seen one. Have you? Or better yet, have you made such an observation yourself?</p>
<p>The following detail is not very important (but just to be complete), my primary applications perform fairly high time-resolution measures on fairly bright stars. In other words, the experiments generally need to measure in the range of "counts per microsecond" to "counts per millisecond". Typically APDs max out at around 15 to 50 million counts per second, and most of my experiments will be working at 10K to 10M counts per second to observe the short time-period phenomenon I need to measure.</p>
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<p>Consider a simplest 3D solution of Maxwell's equations:
$$\vec B=\vec e_z \cos\left(\frac{2\pi}{\lambda}(ct-x)\right),$$
$$\vec E=\vec e_y\cos\left(\frac{2\pi}{\lambda}(ct-x)\right),$$</p>
<p>and propagation is in direction of $\vec e_x$.</p>
<p>I'd like to find some vector potential $\vec A$ and scalar potential $\phi$ for such wave. I've tried using known expression for static uniform magnetic field: $\vec A=\vec e_y B x$, which satisfies $\vec B=\nabla\times \vec A$ and multiplying it by the cosine factor:
$$\vec A=\vec e_y B x\cos\left(\frac{2\pi}{\lambda}(ct-x)\right),$$
but despite it does satisfy $\vec B=\nabla\times \vec A$, it appears that I can't have correct result for $\vec E=-\nabla\phi-\frac{\partial\vec A}{\partial t}$ (at least if I use $\phi=const$). Using another expression for static part of $\vec A$, $\vec A=\frac{B}2\left(\vec e_y x-\vec e_x y\right)$, gave even worse result. Seems either I use wrong expression for $\vec A$, or I have to add non-const $\phi$.</p>
<p>What would be the correct way of determining the potentials?</p>
| 4,829 |
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<p>Given the Steane code
$$
\left|0\right\rangle_L \equiv \frac{1}{\sqrt{8}}(\left|0000000\right\rangle + \left|1010101\right\rangle + \left|0110011\right\rangle + \left|1100110\right\rangle + \left|0001111\right\rangle + \left|1011010\right\rangle + \left|0111100\right\rangle + \left|1101001\right\rangle)
$$
$$
\left|1\right\rangle_L \equiv \frac{1}{\sqrt{8}}(\left|1111111\right\rangle + \left|0101010\right\rangle + \left|1001100\right\rangle + \left|0011001\right\rangle + \left|1110000\right\rangle + \left|0100101\right\rangle + \left|1000011\right\rangle + \left|0010110\right\rangle)
$$</p>
<p>and its relative stabilizers:
$$
K^1 = IIIXXXX
$$
$$
K^2 = XIXIXIX
$$
$$
K^3 = IXXIIXX
$$
$$
K^4 = IIIZZZZ
$$
$$
K^5 = ZIZIZIZ
$$
$$
K^6 = IZZIIZZ
$$</p>
<p>The stabilizer set establishes valid codewords for a state if the equation $$s\left|\psi\right\rangle=\left|\psi\right\rangle,\;\;\;\forall s \in S \;\;\;\;\; (1)$$
is satisfied. That means $\left|\psi\right\rangle$ is a +1 eigenstate of $s$.</p>
<p>We then consider a practical example of the usage of these stabilizers
<img src="http://i.stack.imgur.com/kg9F6.png" alt="enter image description here"></p>
<p>The state of the system is represented by:
$$\left|\psi\right\rangle_F={1\over 2}(\left|\psi\right\rangle_I+U\left|\psi\right\rangle_I)\left|0\right\rangle + {1\over 2}(\left|\psi\right\rangle_I-U\left|\psi\right\rangle_I)\left|1\right\rangle$$</p>
<p>where $U \in \left\lbrace K^1,K^2,K^3\right\rbrace$.</p>
<p>We apply $U$ to the input state and we measure the ancilla qubits (syndrome measurement) to verify the integrity of the input (if $\left|\psi\right\rangle_I$ is +1 eigenstate of $K^1,K^2,K^3$). If the equation $(1)$ is not satisfied, then the corrupted qubit is corrected with a $Z$ gate addressed by the syndrome measurement. </p>
<p>This is how does the system work?</p>
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<p>A long time ago I learned that the electrical resistance of a metallic conductor, when surrounded by a gas, varies with the pressure of said gas.
• What is the name (if any) of the law involved?
• Does the resistance increase or decrease with an increase in gas pressure?
• What is the reason for this?
TIA!</p>
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<p>The problem is as follows:</p>
<p>Consider the system consisting of the <em>symmetric</em> dumb-bell (two particles each of mass $m$ connected by a light, inextensible rod of length $l$) moving in the plane. Use as coordinates the angle $\phi$ between the rod and the $x$-direction, and the Cartesian coordinates of the centre of mass of the dumb-bell.</p>
<p>Write down the expression for the kinetic energy of this system in these coordinates and find the mass-inertia matrix for this system, showing that it is non degenerate.</p>
<p><strong>My attempt at solving this problem was as follows:</strong></p>
<p>$$x=\frac{l}{2}\cos{\phi}$$</p>
<p>$$y=\frac{l}{2}\sin{\phi}$$</p>
<p>Then ${}$ $T=\frac{1}{2}(2m)(\dot{x}^2+\dot{y}^2)=m(\dot{x}^2+\dot{y}^2)$
$$\dot{x}=-\frac{l}{2}\sin{\phi}\dot{\phi}$$
$$\dot{y}=\frac{l}{2}\cos{\phi}\dot{\phi}$$</p>
<p>From this we have that</p>
<p>$$T=m\{(-\frac{l}{2}\sin{\phi}\dot{\phi})^2+(\frac{l}{2}\cos{\phi}\dot{\phi})^2\}$$
$$=m(\frac{l^2}{4}\sin^2{\phi}\dot{\phi^2}+\frac{l^2}{4}\cos^2{\phi}\dot{\phi^2})$$
$$=\frac{ml^2}{4}\dot{\phi^2}(\sin^2{\phi}+\cos^2{\phi})$$
$$=\frac{1}{2}m(\frac{l^2}{2}\dot{\phi^2})$$</p>
<p>Which gives the mass-inertia matrix as</p>
<p>$$\mathbb{K}=\left( \begin{matrix} 0&0\\ 0&\frac{ml^2}{2} \end{matrix} \right)$$</p>
<p>Which is clearly a <em>degenerate</em> matrix.</p>
<p>Could anyone point me in the right direction for with this problem? Thank you.</p>
| 4,832 |
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<p>Do <a href="http://en.wikipedia.org/wiki/Black_hole" rel="nofollow">black holes</a> exist from our point of reference? From our point of reference nothing actually goes inside the event horizon right? So is there anything inside the event horizon from our reference? If not then no black holes actually exist but hollow spheres that have the center of mass inside them? So no information paradox :)</p>
| 165 |
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<p><strong>Note</strong>: My question is duplicate of the following </p>
<ol>
<li><p><a href="http://physics.stackexchange.com/q/87976/">Direction of friction when a car turns</a></p></li>
<li><p><a href="http://physics.stackexchange.com/q/79852/">Why does friction cause a car to turn?</a> </p></li>
</ol>
<hr>
<p>I've gone through many related questions especially <a href="http://physics.stackexchange.com/q/87976/">the first</a>. As I understand the static friction is always opposite to the force applied on the object as shown:<br>
<img src="http://www.school-for-champions.com/science/images/friction-slide_kinetic.gif" alt="image"><br>
But in the case when front wheels of a vehichal are turned the force of static friction is not opposite to the applied force. For example consider a car accelerating forward. The net force on the car is in forward direction which is provided from the rear tyres, if eventually break is pressed static friction(assuming tyres aren't skidding) comes into picture. This friction <strong>should be and is</strong> opposite in direction to the direction of force applied by the rear tyres. When the front tyres are turned the direction of static friction is changed(radially inward) means the direction of static friction is not opposite to the direction of applied force as shown:<br>
<img src="http://img.gawkerassets.com/img/18zoykoy35nhfpng/ku-xlarge.png" alt="image 2"> </p>
<p><strong>Question:</strong> Is the force of static friction is always opposite to the applied force ? If not then what determines its direction?</p>
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<p>Can magnetic fields be "blocked"?
For example, in the game, TitanFall, a robot stop bullets with ( presumably ) a magnetic shield.</p>
<p>I wish to calculate the magnetic force required to stop a bullet within a few microseconds. But the problem I have is that my entire electronic system or the vehicle ( most likely even a building ) will be subjected to massive fields most likely destroying it as well.</p>
<p>Does any know of a way to "block" or divert the field around my vehicle? Is it even possible?
My knowledge in magnetic fields are quite limited at this time.</p>
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<p>It it known that the massive spin-2 irreducible representation of the Poincare group is the traceless symmetrical transverse 4-tensor $h_{\mu \nu}$ with rank 2:
$$
(\partial^{2} + m^{2})h_{\mu \nu} = 0, \quad \partial_{\mu}h^{\mu \nu} = 0, \quad h = 0.
$$
These conditions may be united into one equation:
$$
\partial_{\mu }\partial_{\nu}h - g_{\mu \nu}(\partial^{2} + m^{2})h - (\partial_{\mu}\partial^{\alpha }h_{\alpha \nu} + \partial_{\nu}\partial^{\alpha }h_{\alpha \mu}) + \eta_{\mu \nu}\partial^{\alpha} \partial^{\beta}h_{\alpha \beta} + (\partial^{2} + m^{2})h_{\mu \nu} $$
$$= 0, \tag{1} $$
which is called Pauli-Fierz equation.</p>
<p>Also there is the linearized gravity $g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$, and Einstein equations for $h_{\mu \nu}$ takes the form
$$
\partial^{2}h_{\nu \sigma} - (\partial_{\nu}\partial^{\alpha}h_{\alpha \sigma} + \partial_{\sigma}\partial^{\alpha}h_{\alpha \nu} ) + (\eta_{\nu \sigma}\partial_{\mu}\partial_{\alpha}h^{\mu \alpha} + \partial_{\nu} \partial_{\sigma}h) - \eta_{\nu \sigma}\partial_{\mu}\partial^{\mu}h = - \Lambda T_{\nu \sigma},
$$
which is absolutely equal to $(1)$if $m = 0 $ and $T_{\nu \sigma} = 0$ (the second condition refers to the free field).</p>
<p>So I have the question: can I simply set $m$ in $(1)$ to zero? Does it automatically reduce the number of degrees of freedom (by the number of spin projection values) of massive spin-2 field to the number of degrees of freedom (by the number of helicity projection values) of massless spin-2 field?</p>
| 4,835 |
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<p>I'm struck with two competing ideas on the question in the title.</p>
<p>Listing #1: <a href="http://van.physics.illinois.edu/qa/listing.php?id=2009" rel="nofollow">http://van.physics.illinois.edu/qa/listing.php?id=2009</a></p>
<p>Q: "How far can a magnetic field bend light?"</p>
<p>A: "Unfortunately, the path light takes is not affected by the presence of a magnetic field. Light itself is composed of an oscillating electric and magnetic field, and one very important property of electric and magnetic fields is what we call "linearity." That is, if you have two sources of electric and/or magnetic fields, you can predict what the combined field is just by adding the two source fields together. The two fields don’t change each other at all. "</p>
<p>Listing #2(Answer #1): <a href="http://physics.stackexchange.com/questions/34879/does-electric-charge-affect-space-time-fabric/34883#34883">Does electric charge affect space time fabric?</a></p>
<p>Q: "Does electric charge affect the space time fabric? If so, why?"</p>
<p>A: [See link. Rather, see both links if you must.]</p>
<p>I'm more inclined to consider the latter question and answer as the correct interpretation. Anyway, if anyone could help me out with this conceptualization that would be great, thanks.</p>
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<p>I'm doing the variation of a Lagrangian respect to the metric, but I am having problem with a particular terminus. My action is:</p>
<p>$$ S=\int d^4x \sqrt{-g}[ (\nabla_\mu A^\mu)^2]$$</p>
<p>My lagrangian is:</p>
<p>$$\mathcal{L}=(\nabla_\mu A^\mu)^2$$</p>
<p>to get the energy tensor the formula is $$T_{\mu \nu} = -2 \frac{\partial \mathcal{L}}{\partial{g^{\mu \nu }}}-g_{\mu \nu}\mathcal{L} $$</p>
<p>The question is, is correct this procedure?
$$T_{\mu \nu} = -2 \frac{\partial{(g^{\alpha n }\nabla_\alpha A_n)^2}}{\partial{g^{\mu \nu }}}-g_{\mu \nu}\mathcal{L}$$ $$ =-4(g^{\alpha n } \delta^\alpha _\mu \delta^n_\mu \nabla_\alpha A_n \nabla_\alpha A_n) -g_{\mu \nu}\mathcal{L}$$ $$= -4(\nabla_\alpha A^\alpha \nabla_\mu A_\nu)-g_{\mu \nu}\mathcal{L}$$ </p>
<p>$A_\mu$ is like a electromagnetic field and $(\nabla_\mu A^\mu)^2$ is like a kinetic term</p>
| 4,837 |
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<p>I had a very brief introduction to the <a href="http://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect" rel="nofollow">Aharonov-Bohm effect</a> in class. The lecturer introduced the notion that $H(\Phi=\Phi_0)$ and $H(\Phi=0)$ gives identical energy spectrum and that the Hamiltonians are related by a the large gauge unitary transformation. </p>
<p>I did a quick Google on the <a href="http://en.wikipedia.org/wiki/Large_gauge_transformation" rel="nofollow">large gauge transformation</a>, but I did not quite understand much about it aside from the fact that it is a topological related gauge transformation. Can someone explain a bit more about what that gauge is about and how it is performed? </p>
<p>Also, in a many-body system with the following Hamiltonian (1), in a 3D Torus with flux, $\Phi$ piercing through the hole on the torus, how do show that the energy eigenvalues of $H(\Phi=\Phi_0)$ and $H(\Phi=0)$ are indeed identical? $\Phi_0 = \frac{h}{e}$ in this case.</p>
<p>$$H(\Phi) = \Sigma_{j} \frac{1}{2M}\biggl(\vec{p_j}+e\frac{\Phi}{L}\hat{x}\biggr)^2 +\Sigma_j \ U(\vec{r_j}) + \Sigma_{j<k} \ V\Bigl(\vec{r_j}-\vec{r_k}\Bigr)\tag{1}$$</p>
<p>The general idea of how I will go about solving this is by perhaps acting the hamiltonian on a wavefunction to determine the energy eigenvalue though I am not sure how to do it explicitly. </p>
| 4,838 |
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] |
<p>What telescope refractor aperture size would I need to observe Mars's polar caps?
I have a Levenuhk Strike NG 80mm, and I can see red disk of planet but cannot see any details on the surface.</p>
| 4,839 |
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<p>We all know that our universe is inflating from what is known as the <a href="http://en.wikipedia.org/wiki/Big_Bang" rel="nofollow">Big Bang</a>. However, will our universe continue to inflate at the current rate? Or after reaching a maximum size, will it collapse in a <a href="http://en.wikipedia.org/wiki/Big_Crunch" rel="nofollow">Big Crunch</a>?</p>
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<p>I mean besides the obvious "it has to have finite mass or it would suck up the universe." A singularity is a dimensionless point in space with infinite density, if I'm not mistaken. If something is infinitely dense, must it not also be infinitely massive? How does a black hole grow if everything that falls into it merges into the same singularity, which is already infinitely dense?</p>
| 73 |
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<p>Wikipedia defines Wigner D-matrix as an irreducible representation of groups SU(2) and SO(3). What is a good way to visualize this representation? Is there any physical system which can be kept in mind as a simple example of the same? </p>
<p>A general explanation of the idea of irreps, beyond just the Wigner-D matrix, would be appreciated.</p>
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<p>We see waves propagate in air, water, through the cristal of a metal and along a rope.
Isn't a wave a wonder of Nature, or is it just a simple phenomenon?
Are homogeneity and isotropy necessary properties for the correct propagation of waves?</p>
<p>Update</p>
<p>are a rope, water and space/EM field elastic in the same way?</p>
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<p>I am reviewing some concepts in statistical mechanics and am becoming confused with how to calculate probabilities when a system has $N$ non-interacting particles. </p>
<p>For instance, let's say we have $N$ electrons with magnetic moment $\vec{\mu} = (g e/2 m)\vec{S}$. If we apply a strong magnetic field parallel to $\vec{S}$, then </p>
<p>$$ E = - \vec{\mu} \cdot \vec{B} = \pm \frac{g e \hbar}{4 m} = E_{\pm}$$</p>
<p>depending on the orientation of the spin of the electron. Therefore, the partition function for one electron is simply</p>
<p>$$ \xi = 2 \cosh \left(\frac{g e \hbar B}{4 m k T}\right) $$</p>
<p>And the probability to find the electron with spin parallel to the magnetic field is simply $e^{-\beta E_{+}}/Z$. So far, so good.</p>
<p>However, what happens when we have $N$ such electrons? Statistical mechanics says that the partition function for the system is now </p>
<p>$$ Z = \frac{\xi^N}{N!} $$</p>
<p>if we assume that the electrons do not interact with each other.</p>
<p>This is where I get confused. Now, if we want to find the probability that 75% of the electrons have energy $E_+$, then the Boltzmann argument doesn't hold anymore: </p>
<p>$$ \frac{0.75 N }{N} \neq \frac{e^{-\beta E_{+}}}{Z} $$</p>
<p>If the Boltzmann ratio doesn't hold, how can one proceed to calculate the aforementioned probability?</p>
| 4,843 |
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] |
<p>I initially thought that it had something to do with the number of slip systems in FCC vs. BCC, but they're both the same.</p>
| 4,844 |
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<blockquote>
<p>Two balls of the same mass $m$ are connected to each other with rope
of length $l$. One of the balls is also connected to the ceiling with
a rope of the same length $l$. The balls are spinning around the axis
which intersects the point of the connection of the rope in the
ceiling. As a result, they create angles $\alpha$ and $\beta$ with the
verticals. Masses of the two ropes can be neglected. What is the angular velocity of the system?</p>
</blockquote>
<p><img src="http://i.stack.imgur.com/XrKYA.png" alt="enter image description here"></p>
<p>So I made a free body diagram:</p>
<p><img src="http://i.stack.imgur.com/F3O0f.png" alt="enter image description here"></p>
<p>And the equations are:</p>
<p>For the top ball:</p>
<p>$T_1 \cos \alpha = mg + T_2 \cos \beta\\
T_1 \sin \alpha - T_2 \sin \beta = m (l \sin \alpha) \omega^2$</p>
<p>For the bottom one:</p>
<p>$T_2 \cos \beta = mg\\
T_2 \sin \beta = m (l \sin \alpha + l \sin \beta ) \omega^2$</p>
<p>The process:</p>
<p>$T_1 \cos \alpha = 2mg\\
T_1 \sin \alpha = m (l \sin \alpha) \omega ^2+T_2 \sin \beta = m (l \sin \alpha) \omega ^2 + m(l \sin \alpha + l \sin \beta) \omega ^2=\omega ^2 m l(2\sin \alpha + \sin \beta)$</p>
<p>$2 mg \tan \alpha =\omega ^2 m l(2\sin \alpha + \sin \beta)$</p>
<p>And so:</p>
<p>$$\omega = \sqrt{\frac{2g \tan \alpha}{l(2\sin \alpha + \sin \beta)}}$$</p>
<p>However, according to the book the answer is:</p>
<p>$$\omega = \sqrt{\frac{g \tan \beta}{l(\sin \alpha + \sin \beta)}}$$</p>
<p>I'm quite stuck on that. Where am I wrong?</p>
| 4,845 |
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] |
<p>A system is described as follows:</p>
<blockquote>
<p>Consider a system consisting of two rotating bars of length $l$ and with
uniform mass density and each with total mass $m$. The bars are attached
to a common axis at one end around which the can rotate. The distance
between the bars on the axis of rotation is $a$.</p>
</blockquote>
<p>My teacher has asked this question:</p>
<blockquote>
<p>Consider the case $V_{0} < 0$. What is the equilibrium configuration of the
system? Determine the frequency of oscillation around this equilibrium
for small $r$.</p>
</blockquote>
<p>I have the Lagrangian for this system, which is:</p>
<p>$$L=\frac{1}{2}I_{tot}\dot{\theta}_{R}^{2}+\frac{1}{2}I_{rel}\dot{\theta}_{r}^{2}-V_{0}\cos(\theta_{r})$$</p>
<p>So my main problem is, I'm not quite sure when it is in equilibrium? Usually it's when the derivative is equal to zero or something, but I can't just do that to the entire Lagrangian, can I?</p>
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] |
<p>Is <a href="http://en.wikipedia.org/wiki/Brownian_motion" rel="nofollow">Brownian motion</a> a deterministic system? I.e the motion of all particles are completely determined or is there an innate indeterminism like quantum systems?</p>
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<p>Is there the effect of sun rising and sun setting, in terms of Rayleigh scattering and visual spectrum and other factors completely similar and symmetric? I mean can one recognise them from a picture taken from the sky?</p>
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<p>To prevent <a href="http://en.wikipedia.org/wiki/Grain_Boundary_Sliding" rel="nofollow">grain boundary sliding</a> so that <a href="http://en.wikipedia.org/wiki/Diffusion_creep" rel="nofollow">creep</a> is less likely to occur, usually engineers would design components of larger grains or have columnar grain structure to prevent grain-boundary sliding. Why can these two methods prevent grain boundary sliding? For columnar grains, would they be more easy to slide against each other since the grains are parallel?</p>
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<p>If as some people <a href="http://physics.stackexchange.com/questions/72753/what-happens-after-the-collapse-of-a-wavefunction">suggest</a>, there is no collapse of the wave function (is there a standard name for this position), then must one rule out the many-worlds interpretation of QM ?</p>
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<p>Many books on special relativity eventually mention that the geometry of spacetime is special because the metric has a signature $(-,+,+,+)$ which is non-Euclidean. I have encountered many ways this makes it different from normal Euclidean geometry, for example, there is more than one null vector. </p>
<p>I want to study the mathematics of this new geometry in order to develop some intuition for it. I understand that the new geometry is called Hyperbolic geometry. Unfortunately, the information I find about that is all about negatively-curved saddles and Poincare disks, etc, which while interesting, seems quite different!</p>
<p>Can someone point to a good resource for learning just the geometry that underlies SR?</p>
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] |
<p>In the cases of nonlinear acoustics, why is shock formation unlikely when the dispersion is strong when compared to the nonlinearity of the wave?</p>
| 4,852 |
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<p>I have a question about deriving variation of metric under Weyl and coordinate transformations in Polchinski's string theory (3.3.16).</p>
<p>Under transformation $$\zeta: g \rightarrow g^{\zeta}, \,\,\, g_{ab}^{\zeta}(\sigma')=\exp[ 2 \omega (\sigma) ] \frac{ \partial \sigma^c }{\partial \sigma'^a} \frac{ \partial \sigma^d}{\partial \sigma'^b} g_{cd}(\sigma) \tag{3.3.10} $$</p>
<p>how to show
$$ \delta g_{ab} = 2 \delta \omega g_{ab} - \nabla_a \delta \sigma_b-\nabla_b \delta \sigma_a ? \tag{3.3.16}$$
The first term in (3.3.16) comes from Weyl transformation. I am unable to derive the second and third terms.</p>
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<p>Alright, so I've been messing around with turning air pressure into thrust, however I'm only a sophomore in high school, so my physics knowledge is fairly limited. (I've studied higher level physics, but have nothing official.)</p>
<p>So, with that being said, I've been messing around with utilizing thrust from air pressure. I've got so far, that force (used as thrust) would be calculated by:</p>
<p>$f = pa$</p>
<p>where</p>
<p>$f =$ force</p>
<p>$p =$ pressure</p>
<p>$a =$ area</p>
<p>Is this the correct formula for the thrust from a gas flowing through a small hole?</p>
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<p>I have tried to search through the internet about the definition of solar storm or what this term mean but i could found what it means. Also, what is the difference between the geomagnetic storm</p>
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<p>How fast is heat transferred by conduction? Is there some simple, but quantitative way that starts from some properties of the material (e.g. its thermal conductivity) and makes rough predictions, for example about how much time is needed for temperature to change at one end of the body when it is placed, at the other end, in thermal contact with another?</p>
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<p>Decoherence explains how a classical state appears once quantum information in a quantum state leaks out. But presumably that environment has its own quantum state which then leaks out to a larger environment.</p>
<p>Does this mean that decoherence can only be understood locally, and not globally? For example if the entire universe is encoded in a quantum state how can it decohere?</p>
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<p>The centers of black holes and quasars often have jets coming out the two poles of an accretion disk, say north and south. Is it known if the two jets spin in the same direction or opposite directions to each other? </p>
| 4,858 |
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<p>What is the importance of dimension six operators in the study of physics beyond the Standard Model? Are these operators more relevant than dimension five operators like $HHFF$ or operators with derivative couplings?</p>
<p>I often see lagrangians with dimension six operators in effective studies of the standard model, but I do not understand this choice. An example is the paper <a href="http://arxiv.org/abs/1304.1151" rel="nofollow">arXiv:1304.1151</a>, where they have defined: $$\mathcal{L}_{\rm eff}= \sum_n \frac{g_n}{\Lambda^2}\mathcal{O_n},$$
whit $g_n$ being the corresponding couplings and $\mathcal{O}_n$ the dimension six operators.</p>
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<p>I have come up with this differential equation for the evolution of $\vert \Psi \vert^2$, the probability density in quantum mechanics.</p>
<p>Is there a name for this equation? Is the logic sound?</p>
<p>So I start from the conservation of the total probability:</p>
<p>$$ \frac{d}{dt} \int _{all space} \vert \Psi \vert^2 d^3\textbf{r} = 0$$</p>
<p>so that $$\int _{all space} \vert \Psi \vert^2 d^3\textbf{r} = {\rm const}$$</p>
<p>This means that
$$\int _{all space} \vert \Psi (\textbf{r},t) \vert^2 d^3\textbf{r} = \int _{all space} \vert \Psi (\textbf{r}+d\textbf{r},t+dt) \vert^2 d^3\textbf{r}={\rm same\,const}$$</p>
<p>So since the integration is over the same volume I can equate the intergrands?</p>
<p>This gives us:</p>
<p>$$\vert \Psi (\textbf{r},t) \vert^2 = \vert \Psi (\textbf{r}+d\textbf{r},t+dt) \vert^2$$</p>
<p>or
$$\Phi (\textbf{r},t) = \Phi (\textbf{r}+d\textbf{r},t+dt) $$
if we call $\vert \Psi (\textbf{r},t) \vert^2 = \Phi (\textbf{r},t)$. A Taylor expansion to first order gives us:</p>
<p>$$ \Phi (\textbf{r}+d\textbf{r},t+dt) =\Phi (\textbf{r},t) + \nabla\Phi\cdot d\textbf{r}+\frac{\partial \Phi}{\partial t} dt$$</p>
<p>Plugging this into the previous equation gives us:</p>
<p>$$\boxed{ \nabla\vert \Psi \vert^2\cdot \frac{d\textbf{r}}{dt} = -\frac{\partial \vert \Psi \vert^2}{\partial t} }$$</p>
<p>What is $\frac{d\textbf{r}}{dt} $ in the context of quantum mechanics? Is this equation correct? What is the physical meaning of this equation?</p>
| 4,860 |
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<p>I want to simulate a circuit similar to the one below in MATLAB. If you have a state matrix describing the state of 3 qubits, I understand that you could apply a CNOT matrix tensored with and identity matrix to $\psi_{0} $ get $\psi_{1}$, but if you want to apply a controlled operation to the 1st and 3rd qubit to get $\psi_2$, how can you do this? It's like you need "remove" the information about the second qubit, apply a CNOT gate, and then somehow integrate the result back with the superposition of the second qubit... I do not understand how to do this.</p>
<p>In general if I have a superposition of N qubits, how do I apply a controlled operation on qubits i and j?</p>
<p><img src="http://i.stack.imgur.com/rQSXw.png" alt="Simple Quantum Circuit"></p>
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<p>I wondered if magnets could be used to hold a drop of molten liquid metal in air (not for any particular reason just because it could be done), but was disappointed when a quick Google search showed the metal would lose its magnetic traits before it melted.</p>
<p>Are there any other forces that could be used to suspend a drop of molten liquid metal in air such as sound waves, high pressure air, electric currents, or anything else?</p>
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<p>I may have confused after thinking too much about Faraday's law. If an emf is induced in a circuit due to some changing magnetic field, the induced current will be in a direction such that the "induced magnetic field" opposes the original magnetic field (Lenz's law).</p>
<p>So wouldn't the induced magnetic field also generate an induced current that opposes it, and that current would also generate a magnetic field and so on... so it seems like there would be infinitely many induced magnetic fields and currents. This is particularly true if the original magnetic field is a sinusoid so it's infinitely differentiable.</p>
<p>Am I misinterpreting Faraday's law? Is Faraday's law just a description of how the electric and magnetic fields can coexist, so we don't have to worry about "infinite induction"?</p>
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<p>Firstly, not sure if this question ought to be in the space SE site. Please let me know if it should. (Posted in both for now)</p>
<p>Secondly, I don't know a whole lot about physics (I'm just inquisitive). So please try to keep answers simple (or at least dummed down to laymans terms).</p>
<p>Inspired by "A Hitchhikers Guide to the Galaxy" movie (haven't read the book yet)</p>
<p>My question is, for a non-spherical object (lets say a cube for now) of a size and mass like that of Earth, how would objects act in it's gravity? </p>
<p>Aspects of this question lead to the following questions</p>
<p>•Would something weigh more/less in certain areas over other areas?</p>
<p>•I'm aware that it's possible to 'slingshot' around massive spherical objects to change their directory. Would it be possible to slingshot objects around something non-spherical to change their velocity?</p>
<p>•If this massive object was orbiting our sun, would it's orbit be as uniform as ours? if not what would that look like and why?</p>
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<p>I'm working on a certain problem in fluid mechanics, which isn't really my strongest area.
The problem is as follows: Curved pipe is partially submerged in a flowing river so that one end is pointing in the same direction as the velocity of river. Level of water inside of the pipe is 7cm higher than the level of river. Determine the speed of river.</p>
<p>Basically what I argued is that this reduces to Torricelli's law: level of water in pipe is constant so the velocity of the surface is zero (or very nearly so, I assume it would oscillate in real life), therefore water should move on the other end with $v_{2}=\sqrt{2gH}$, but it isn't since $v_{2}=v_{river}$ counteracts the movement.</p>
<p>I've tried to be more rigorous so I took pressures at the submerged end of pipe:
$$p_{\text{water in pipe}}=p_{\text{atmosphere}}+\rho g h_{\text{depth}} + \rho g H_{\text{above water}}$$
$$p_{\text{river}}=p_{\text{atmosphere}}+\rho g h_{\text{depth}} + \frac{1}{2} \rho v^{2}$$</p>
<p>Since they have to be in equilibrium, pressures are equal and you get Torricelli's expression above.</p>
<p>Is my reasoning correct? It's somewhat counter-intuitive to me because the water is moving <em>away</em> from the pipe.</p>
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<p>I'm trying to show that the Hamiltonian for a nanowire with proximity-induced superconductivity </p>
<p>$$
\hat{H} = \int dx \text{ } \left[\sum_{\sigma\epsilon\{\uparrow,\downarrow\}}\psi_{\sigma}^{\dagger}\left(\xi_{p} + \alpha p\sigma_{y} + B\sigma_{z}\right)\psi_{\sigma} + \Delta\left(\psi_{\downarrow}^{\dagger}\psi_{\uparrow}^{\dagger} + \psi_{\uparrow}\psi_{\downarrow}\right)\right]\text{,}
$$</p>
<p>can be written as</p>
<p>$$
\hat{H} = \frac{1}{2}\int dx \text{ } \Psi^{\dagger}\mathcal{H}\Psi
$$</p>
<p>with $\mathcal{H} = \xi_{p} 1\otimes \tau_{z} + \alpha p \sigma_{y}\otimes\tau_{z} + B\sigma_{z}\otimes 1 + \Delta 1\otimes\tau_{x}$ (here $\tau_{i}$ are the Pauli matrix for the particle-hole space and $\otimes$ means the Kronecker product), $\Psi^{\dagger} = \left(u^{\dagger}_{\uparrow}, u^{\dagger}_{\downarrow}, v^{\dagger}_{\downarrow}, -v^{\dagger}_{\uparrow}\right)$ and $\Psi = \left(u_{\uparrow}, u_{\downarrow}, v_{\downarrow}, -v{\uparrow}\right)^{T}$(likewise <a href="http://arxiv.org/abs/1002.4033" rel="nofollow">arXiv:1002.4033</a> and <a href="http://arxiv.org/abs/1206.1736" rel="nofollow">arXiv:1206.1736</a>).</p>
<p>My first idea was to calculate brute force the term $\Psi^{\dagger}\mathcal{H}\Psi$, however, I obtain a confusing solution:</p>
<p>$$
1\otimes\tau_{z} = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1
\end{pmatrix}\text{,}$$</p>
<p>$$\sigma_{y}\otimes\tau_{z} = \begin{pmatrix}
0 & 0 & -i & 0 \\
0 & 0 & 0 & i \\
i & 0 & 1 & 0 \\
0 & -i & 0 & 0
\end{pmatrix}\text{,}$$</p>
<p>$$\sigma_{z}\otimes 1 = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{pmatrix}\text{,}$$</p>
<p>$$1\otimes\tau_{x} = \begin{pmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0
\end{pmatrix}\text{.}$$</p>
<p>$$\Rightarrow \Psi^{\dagger}\mathcal{H}\Psi = \left[u^{\dagger}_{\uparrow}\xi_{p}u_{\uparrow} - u^{\dagger}_{\downarrow}\xi_{p}u_{\downarrow} + v^{\dagger}_{\uparrow}\xi_{p}v_{\uparrow} - v^{\dagger}_{\downarrow}\xi_{p}v_{\downarrow}\right] + i\left[-u^{\dagger}_{\uparrow}\alpha p v_{\downarrow} - u^{\dagger}_{\downarrow}\alpha p v_{\uparrow} + v^{\dagger}_{\downarrow}\alpha p u_{\uparrow} + v^{\dagger}_{\uparrow}\alpha p u_{\downarrow}\right] + \left[u^{\dagger}_{\uparrow}B u_{\uparrow} + u^{\dagger}_{\downarrow} B u_{\downarrow} - v^{\dagger}_{\uparrow}B v_{\uparrow} - v^{\dagger}_{\downarrow}B v_{\downarrow}\right] + \Delta\left[u^{\dagger}_{\uparrow}u_{\downarrow} + u^{\dagger}_{\downarrow}u_{\uparrow} - v^{\dagger}_{\uparrow}v_{\downarrow} - v^{\dagger}_{\downarrow}v_{\uparrow}\right]$$</p>
<p>However, now I have no idea how I obtain that both notation are equivalent. Perhaps I made a mistake or my basic idea is wrong? </p>
| 4,866 |
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<p>I want to apply the Biot-Savart law to calculate the magnetic field at a point created by current flowing through a square/rectangular conductor. More specifically, a trace on a printed circuit board. </p>
<p>To me, a trace on a circuit board is the summation of many infinitesimal rectangular conductors.</p>
<p>I have seen lots of examples for a wire (circular conductor), but not rectangular. </p>
<p>How should I go about getting started with this problem?</p>
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<p>Under normal lens operation, a beam is sent through the centre of the lens along the optical axis (ie perpendicular to the lens's plane). What happens when a beam is sent through a lens at an angle to the optical axis? Does it simply exit the lens at the same angle?</p>
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<p>According to <a href="http://en.wikipedia.org/wiki/Terminal_velocity" rel="nofollow">Wikipedia</a>:</p>
<blockquote>
<p>[Terminal Velocity] is the velocity of the object when the sum of the
drag force (Fd) and buoyancy equals the downward force of gravity (FG)
acting on the object. Since the net force on the object is zero, the
object has zero acceleration.</p>
</blockquote>
<p>I was wondering if this can be more generalized. Assume that we replace the force of gravity with another type of force (for example, the force of a theoretical engine on a rocket whose fuel never loses mass). In this case, the force is not one of gravity (along the y-axis) but of the engine (along the x-axis). However, drag would still play a part in causing the acceleration due to the rocket's engine to reach 0.</p>
<p>As far as I'm concerned, these concepts are the same, but whenever one talks about "Terminal Velocity" they always discuss it in context of gravity. Can the term "Terminal Velocity" be used in the "rocket" case as described above, where the force is NOT that of gravity?</p>
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<p>In a old paper,
<a href="http://arxiv.org/abs/hep-th/9509163" rel="nofollow">http://arxiv.org/abs/hep-th/9509163</a>
Becca Asquith argues that it is possible to prove that if the SU(2)xU(1) sector of the standard model is chiral, then the SU(3)xU(1) sector is vectorial, ie, that at least one of them must be of purely vectorial character.</p>
<p>But the argument there was based on building the Standard Model from Connes-Lott models. At that time, I heard some folklore that this was really a general result, probably arguable on the basis of anomalies. But I can not pinpoint a concrete statement. So, is it a mathematical fact, that colour must be a Vector kind of force because SU(2) is chiral? Or just lore?</p>
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] |
<p>If a quantum system interacts with a "big" quantum system, you have dephasing. </p>
<p>The models of decoherence all have this atog aproach to them, about what is to understood of the interaction of the quantum state with a bath. </p>
<blockquote>
<p>At which point does the bath character of the "bigger" system strike? In which way are the systems involved in decoherence fundamentally different? Why is it clear that a bath will behave in such and such way, that the phases vanish?</p>
</blockquote>
| 4,871 |
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<ol>
<li><p>I think <a href="http://en.wikipedia.org/wiki/Many-worlds_interpretation" rel="nofollow">many worlds interpretation</a> is inconsistent with the EPR paradox. Quantum mechanics says that particles are really in more places at the same time and the particle is really only probability wave - otherwise the concept of probability in quantum mechanics doesn't make a sense, because everything have its own specific cause (There is no probability 1/6 that you roll the dice at the six - because this "probability" is caused as your brain controls your muscles in your hand). But the world where everything have only probable location (and everything is on more locations) makes a sense - because cause could be on more locations too (Imagine particle, what collide with another particle - if both of them have specified position, we know where the second particle will fly. But when probability wave collide with another probability wave, we don't know where they will fly, because we don't know one specific direction of this forces.). But in many worlds interpretation is in one world only one particle at the one place - otherwise we describe another interpretation, where Schrödinger's cat alive and died. So how could be many worlds interpretation possible?</p></li>
<li><p>What causes which world will we live in? (This does not imply that Hugh Everett was wrong - only that many worlds interpretation is incomplete.)</p></li>
</ol>
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<p>So I'm having a tough time deciding between courses next semester. I'm a rising 3rd year undergrad math major whose goal is to get a solid understanding of theoretical physics through advanced math (laugh all you want). So in that view, which one should I choose from</p>
<ol>
<li><p>Theoretical Mechanics: This is a graduate level class covering Lagrangian and Hamiltonian Mechanics. I've already had classical mechanics at the undergrad level, but we didn't do Hamiltonians, and at that level a lot of the conceptual details weren't highlighted. I want to do this class because L and H mechanics seem really fundamental to anything you do in theoretical physics and thus a good understanding of them seems essential.</p></li>
<li><p>Electricity & Magnetism II: This is an undergrad class which will probably be based on the second half of Griffiths. I know that this material is pretty important (E&M Waves and Relativistic E&M) and I'll have to learn it eventually, but to be perfectly honest, I think I have had enough classes based on Griffiths for a lifetime (2 semesters of Quantum along with E&M I).</p></li>
</ol>
<p>As a second question, another choice has been bothering me. In the same view which one class should I choose from:</p>
<ol>
<li>Statistical Mechanics. Standard first semester undergrad Stat Mech class using Kittel & Kromer.</li>
<li>Graduate-level Algebra. I've already had an undergrad algebra class but this may be useful if I decide to go to grad school for math.</li>
<li>Mathematical Methods. A graduate level class covering calculus of variations, greens functions and PDEs, real waves and group theory.</li>
</ol>
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<p>The Everett interpretation has memory robots. Copenhagen requires observer memory states. Consistent histories has its IGUSes. Decoherence has its existential interpretation. All of them refer to memory states of observers. What counts as an observer, and which parts of the observer count as memory states? Why isn't there a precise answer to this question? Does a camera count as an observer and the pixels on a photo as memory states? This is a serious question.</p>
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<p>There is this puzzling thing that is called <a href="http://en.wikipedia.org/wiki/Mpemba_effect">Mpemba effect</a>: paradoxically, warm (35°C) water freezes faster than cold (5°C) water. As a physisist, I've been asked about it several times already. And I have no definite answer to that, apart from the standard: "there are many things that can influence it".</p>
<p>So, does anyone knows about the status or progress on that effect? Any recent reviews, publications or other references?</p>
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<p>I am trying to solve for the equations of motion to simulate a pendulum.
I decided to use the spherical coordinates. The Lagrange equation is:</p>
<p><img src="http://i.stack.imgur.com/yXvMs.png" alt="enter image description here"></p>
<p>where </p>
<ul>
<li>L = length of the rope</li>
<li>ϕ= angle of the projection of the rope on x-y plane with x-axis</li>
<li>θ = angle with the z- axis</li>
</ul>
<p>I solved these equations:
<img src="http://i.stack.imgur.com/BJ0Uk.png" alt="enter image description here"> and
<img src="http://i.stack.imgur.com/E0KAx.png" alt="enter image description here"></p>
<p>and I got </p>
<p><img src="http://i.stack.imgur.com/tLx8b.png" alt="enter image description here"></p>
<p>and $$
\frac{ d}{ dt}(mL^2sin^2θ\dot\phi) = 0
$$
This seems like the change in angular momentum is conserved.
but when I solve it more </p>
<p>$$
\ddot\phi = -2\dot\phi\dot\theta cot\theta
$$</p>
<p>This dose not make sense to me because it goes to infinity when θ goes to 0.
Any ideas on what I am doing wrong.</p>
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<p><img src="http://i.stack.imgur.com/sPwNH.jpg" alt="experiment">
sorry for terrible graphical representation, I did an experiment, i took 6 coins fixed 4 of them in one place by placing some real heavy objects on them , then i took a 5th coin placed it in the final position at the last , all these coins were touching each other and only the fifth one was free to move .
Now i took a striker (6th coin) and collided it with this chain of coins and every time the final coin moved as if the force was transmitted through all these coins in the middle to the last one while themselves NOT(assumption) moving at all . How can this happen how can the force be transmitted to the final coin if the coins in the middle didn't moved at all the experiment works with as much as 10 fixed coins in place of 4 . By far the only explanation i can give is that the coins in the middle do move(very little ) but cant prove this theory . </p>
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<p>Consider a simple, circular orifice with an upstream, high gas pressure and downstream, normal atmospheric pressure. Consider also a flat circular plate that can be positioned anywhere along the perpendicular axis of the orifice.</p>
<p>At one extreme the plate occludes the orifice and there is no flow of gas. At this position the force on the plate is just the static pressure times the area of the orifice.</p>
<p>At the other extreme the plate is positioned far from the orifice and gas flows at its maximum rate.</p>
<p>Between each of these extremes the force on the plate decreases as it moves away from the orifice. Computational fluid dynamics can solve for the forces, but is there a simple, even approximate way to express the force as a function of the plate's position, the area of the orifice and plate, and the upstream total pressure?</p>
<p><strong>third party edit:</strong></p>
<p>This is roughly what the situation might look like:</p>
<p><img src="http://i.stack.imgur.com/8I6Ot.png" alt="enter image description here"></p>
<p>original image from <a href="http://www.usbr.gov/pmts/hydraulics_lab/pubs/wmm/fig/F14_04L.GIF" rel="nofollow">http://www.usbr.gov/pmts/hydraulics_lab/pubs/wmm/fig/F14_04L.GIF</a> , but adapted. Question is: what is the force F on the red disk, as a function of distance d from the orifice? Assume that the pipe diameter $D_1$ is very large compared to size of orifice $D_2$ or downstream length of pipe.</p>
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<p>In a non-relativistic quantum mechanical system in an <a href="http://en.wikipedia.org/wiki/Particle_in_a_box" rel="nofollow">infinite potential well</a>. I try to measure the energy and the position of the system simultaneously. Since, the respective operators do commute according to Heisenberg's uncertainty relation I should be able to measure them both with infinite precision. Now, since I know that there is no potential energy in the well I can use $ E=\frac{p^2}{2m} $ since the potential energy is 0 and determine it's momentum provided I know it's mass. But I shouldn't be able know the momentum and position simultaneously with infinite precision! So where am I going wrong? </p>
| 4,879 |
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] |
<p>I have been struggling with this problem:</p>
<blockquote>
<p>The speed of a projectile when it reaches its maximum height is one half of its speed when it is at half its maximum height. What is the initial projection angle of the projectile?</p>
</blockquote>
<p>Please help</p>
| 4,880 |
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<p>Would gravitons follow the same trajectory as photons through a gravitational lense? would all other particles follow the same trajectory?</p>
| 4,881 |
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<p>In the derivation of Galilean transformations the only assumption is that the two frames are moving with some uniform relative velocity $u$. </p>
<p>Suppose with respect to some inertial frame $O$ the two frames $S$ and $S'$ are moving with the same uniform acceleration $a$.</p>
<p>Let $V$ be the velocity of $S$ w.r.t. $O$. Similarly, let $V'$ be the velocity of $S'$ w.r.t. $O$.
Furthermore, let $V_0' - V_0 = u$ (const.). Then</p>
<p>$$V = V_0 + at$$
$$V' = V_0' + at$$</p>
<p>Then the relative velocity is $V' - V = u$.</p>
<p>This is the only result required in deriving the Galilean transformation. So why do people assume that the reference frames be inertial. (I know the point is so that Newton's laws would be valid, but exclusively in the derivation of the transformation equation is this assumption needed?) The same applies in the derivation of Lorentz transformation.</p>
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<p>1, If I understand correctly, people talk about <a href="http://en.wikipedia.org/wiki/BQP" rel="nofollow">BQP</a>, <a href="http://en.wikipedia.org/wiki/QMA" rel="nofollow">QMA</a>, etc are usually referring to digital quantum computer/Turing machine and not about analog quantum computer. Based on the papers <a href="http://arxiv.org/abs/1208.3334" rel="nofollow">http://arxiv.org/abs/1208.3334</a> and <a href="http://arxiv.org/abs/0712.0483" rel="nofollow">http://arxiv.org/abs/0712.0483</a> we know that almost all the quantum chemistry methods are QMA-complete/hard. However, like Hartree-Fock, DFT methods, etc still QMA-complete/hard on analog quantum computer? Are there any papers prove that digital and analog quantum computer are equal in computational complexity theory?</p>
<p>2, Assume there is a material/molecule, its Hamiltonian just exact the same as a Hartree-Fock/DFT equation, then we measure the ground state energy of this material/molecule, can we say this material/molecule act as an analog quantum computer and solve this QMA-complete/hard problem? If not, why?</p>
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<p>Let $G$ be the group of the permutation of $N$ particles, $P\in G$. Therefore, there are $N!$ elements in $G$. For its subgroup, e.g., even permutation, we can calculate $\text{sign}(P)$ and get $\text{sign}(P)=+1$. Could you please explain the meaning of the the function $\text{sign}(P)$ and the formula as follows</p>
<p>$$P'' = P P' \to \text{sign}(P'') = \text{sign}(P)\text{sign}(P') $$? </p>
<p>PS: The sign function, in which the argument is a operator,i.e., an element of a group) is hard to me. As I know the sign function $\text{sign}(x)$ is defined as a function of numbers. That is: the sign function of a real number $x$ is defined as </p>
<p>\begin{align}
\text{sign}(x) = \left\{
\begin{array}{lr}
1 & : 0 < x < \infty\\
0 & : x=0 \\
-1 & : -\infty <x<0
\end{array}
\right.
\end{align}</p>
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<p>I am trying to create an exhaustive list of all assumptions which work as the base of the CHSH inequality.</p>
<ol>
<li>Locality - this means an object can be influenced only by its surroundings. So, the events taken place at Alice and Bob's ends cannot influence each other.</li>
<li>Realism - the value of the observed quantity is independent of observation.</li>
<li>The quantity being observed is a discrete random variable.</li>
<li>Repeated rounds of experiments are independent of each other and evenly distributed.</li>
<li>A measurement will always produce a result (a photon will always be detected).</li>
<li>No enhancement assumption - it means when a measuring setup is placed between the source of the entangled particle and a detector the probability of measurement doesn't increase.</li>
</ol>
<p>Am I missing anything?</p>
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] |
<p>In inelastic collisions, kinetic energy changes, so the velocities of the objects also change.</p>
<p>So how is momentum conserved in inelastic collisions?</p>
| 213 |
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<p>Do gravitational lenses have a focus point? Could I burn space ants?</p>
| 4,886 |
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<p>Hello I wanted to know where does the integral of following picture come from and what are the alternatives in it? How and where can i find information i need to know to understand this text?<br>
thank you in advance!
<img src="http://i.stack.imgur.com/LBBlE.png" alt="Detailed balance formulation"></p>
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<p>After being excited by a photon, an electron of a photoactive molecule jumps to a higher electronic state. When it relaxes, the molecule emits a photon (in simple terms). How is this photon "generated"? Photons are particle/waves, right? So somehow this particle has to form. I picture it in a way similar to when you blow through a ring of soap-water and a bubble forms.</p>
<p>How does the photon really form?</p>
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<p>Heisenberg said that we can't tell precisely both the location of an electron and the momentum of it in the same instant. If we observe one thing, the other is changed. How he concluded this principle? </p>
<p>I want to say if we want to locate an electron we'll send photons to it, but after hitting the electrons, will the photons ever come back? It'll lose its energy and transfer it to electron changing its momentum right? How he concluded this principle?</p>
<p>Also what will happen if by some means we're able to observe both things accurately? What will be the benefit?</p>
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<p>If $ \lbrace f,g \rbrace $ is Poisson bracket
and $\epsilon_{ijk}$ is Levi-Civita symbol, how to show that</p>
<p>$$ \epsilon_{iab}\epsilon_{jcd}(x_ap_d\lbrace x_c,p_b \rbrace+x_cp_b\lbrace x_a,p_d \rbrace) = x_ip_j-x_jp_i$$</p>
<p>where $x_i$ are generalized coordinates and $p_i$ are momentums ?</p>
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<p>if I transform the time-dependent Schrödinger equation without a potential I get: </p>
<p>$$ - \hbar \omega \psi(\omega,x) = \frac{- \hbar^2}{2m} \frac{\partial^2 \psi(\omega,x)}{\partial x^2}$$</p>
<p>The solutions is clealy: $$\psi(\omega,x)={\it C1}\,{{\rm e}^{{\frac {\sqrt {2\omega m}x}{\sqrt {\hbar}}}}
}+{\it C2}\,{{\rm e}^{{-\frac {\sqrt {2\omega m}x}{\sqrt {\hbar}}}}
}
$$</p>
<p>I don't really understand this result. The problem is, that if I want to transform back, the Fourier-transform will be divergent, so what does this mean regarding my solution? Is there a work-around to get rid of this divergence? Why did this Fourier-transform fail?</p>
<p>(Should I have used the Laplace-transform?)</p>
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] |
<p>What are some good condensed matter physics books that can fill the gap between Ashcroft & Mermin and research papers? Suggestions for any specialized topics (such as superconductivity, CFT, topological insulators) are welcomed.</p>
| 453 |
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<p>I have another question on the notation in Shankar. I think it's sloppy, but I also may just be misunderstanding it. Again, this is at the very beginning of the math intro.</p>
<p>He has:</p>
<blockquote>
<p>$$a\left| V \right\rangle \to \left[ {\begin{array}{*{20}{c}}
{a{v_1}}\\ {a{v_2}}\\ \vdots \\ {a{v_n}} \end{array}} \right] \to
\left[ {\begin{array}{*{20}{c}} {{a^*}v_1^*,}&{{a^*}v_2^*,}& \cdots,
&{{a^*}v_n^*} \end{array}} \right] \to \left\langle V \right|{a^*}$$
It is customary to write $aV\rangle $ as $a\left| V
> \right\rangle$ and the corresponding bra as $\left\langle aV \right|$.
What we have found is that $\left\langle aV \right|=\left\langle V
\right|{a^*}$.</p>
</blockquote>
<p>The * means conjugate. This doesn't look correct to me unless I'm missing something. First it would seem that the LHS of the final equation should be a ket not a bra. Then it also seems that it's not really "equals". From what I've seen if it is a bra on the LHS, commuting the scalar shouldn't cause it to result in taking its conjugate. Is the text correct or am I not understanding something?</p>
| 4,892 |
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] |
<p>BEC cold atoms occupy the same ground state. But what about the electrons or other fermions of the BEC atoms? Are they in the same state? Do electrons of one atom interact with those of another?</p>
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<p>Is the magnetic field propagated by photons or by virtual photons? If it is by photons, then doesn't that mean that magnets lose energy and eventually become non magnets?</p>
| 4,894 |
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<p>We are all familiar with the version of Quantum Mechanics based on state space, operators, Schrodinger equation etc. This allows us to successfully compute relevant physical quantities such as expectation values of operators in certain states and then compare with experiment.</p>
<p>However, it is often claimed that the path integral is an "equivalent" way to do all of this. To me, the "equivalent" part is a bit vague. I understand that the Feynman path integral allows you compute the propagator $\langle x | e^{-iHt} |x' \rangle $ by just using the classical Lagrangian of the system. Then any expectation value in a state can be computed by two resolutions of the identity to get an integral over this propagator. This shows that the path integral is a way to compute a quantity that's very useful, but not much more than that, since we still need the concept of operators, and their representation in position space, as well as position space wave functions, all of these objects along with their usual interpretations.</p>
<p>Ultimately, regardless of how you compute things, QM will still be based on probabilities and thus wave functions, however my question is, is there anything analogous to the axioms of Quantum mechanics usually mentioned in textbooks that are based on the path integral?</p>
<p>The path integral if seen as an independent object gives us the propagator, correlation functions, and the partition function (and maybe other objects which I'm not aware of). Are all these sufficient to give us the same information that quantum mechanics based on Hilbert space and operators give us? I would really appreciate if someone can make these connections precise.</p>
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<p>If I place a mercury thermometer in hot water, heat energy will transfer from the water to the mercury inside the thermometer. Will this continue until thermal equilibrium is reached and thus the mercury will show the temperature of the water?</p>
<p>However, if this is so, will the thermometer show the right temperature as some of the heat energy is transferred to the thermometer and this in turn will cause original temperature of water to fall?</p>
<p>Please correct me if I am wrong.</p>
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<p>As Weinberg exposited in his QFT Vol1, there are two equivalent ways of arriving at the same quantum field theories:</p>
<p><strong>(1).</strong> Start with single-particle representations of Poincare group, and then make a multiparticle theory out of it, while preserving principles of causality etc. I would call this approach the second quantization of particles, since second quantization is usually used to emphasize the many-body nature of a theory.</p>
<p><strong>(2).</strong> Start with field representations of Poincare group, canonically quantize it, while preserving principles of causality, positive definiteness of energies etc. I would call this approach the quantization of fields, just as everyone else would call it.</p>
<p>Weinberg showed the proof of the equivalence between the above two approaches using some, though not hard, but let's say nontrivial, mathematics. The equivalence seems like a sheer miracle to me, or a complete coincidence. I do not feel that I understand the equivalence with the current state of mind. Is there a way to trivialize the equivalence? Or putting it another way, is there an a priori reasoning to argue, given the two sets of starting points of (1)(2), we have to get the same theory in the end?</p>
<p>Just as a side remark, many have suggested the term "second quantization" should be totally dumped, because it is really just the first quantization of fields. To me however, it still serves some purposes since the equivalence is not transparent. </p>
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<h2>[I have updated the below description to clarify that the mechanism does not depend on a truly frictionless implementation. Also, since 2 commenters apparently assumed that I was proposing a mechanism to create energy from nothing, it seems necessary to state that this is <strong>not</strong> what I am proposing.]</h2>
<p>I will describe a Maxwell's Demon like mechanism that seems to extract useful work from random motion at a human scale. The questions that I then have about this mechanism are:
1) Does it work? If not, in what way does it fail?
2) If we scale this mechanism down (towards the scale at which the balls in the example can be replaced with atmospheric molecules), at what scale does it first stop working, and why?</p>
<p>MECHANISM: A rigid cylinder (say, 2m in diameter and 10m in length) floats in space. Inside are 10^5 1-cm elastic massive balls colliding at random. There is a piston in the cylinder, consisting of a rigid circular disk which just fits inside the cylinder, welded to a thin rigid rod. The rod runs along the cylinder's axis, through the disk, and extends through holes in the center of the cylinder's ends. This piston can slide with low friction along the axis of the cylinder. The disk is perforated by many 5-cm holes, each of which is covered with a light, rigid hinged trap door, weakly spring-loaded and damped so as to close after being opened. Each trap door is one-way: When initially closed, it is easily pushed aside allowing Northbound balls to pass through, but it reflects Southbound balls. The disk is initially positioned midway along the cylinder, with <strong>half of the balls on each side of the disk</strong>.</p>
<p>ANALYSIS: The piston will move under the influence of a net Southward force due to the randomly bouncing balls, because <strong>Southbound</strong> balls encountering a hole will typically rebound from its trap door transferring considerable momentum to the disk, while <strong>Northbound</strong> balls will typically pass through, transferring little momentum to the disk. This motion is transferred outside the cylinder by the rod, where it can perform useful work.</p>
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<p>I've seen OPEs commonly used in 2d CFT, it's quite apparent to me that, in that case, it dresses a bridge between the algebraic and the operator formalism especially when combined with radial ordering and the use of contour integral. Even more powerful in the minimal models where it leads to the bootstrap equations and the resolution of the 3 pt functions. I also heard that OPE are sometimes used in other circumstances, for instance in the QCD chapter of Peskin & Schroeder's book but I don't recall the motive. I'd be curious to know what generally it is relevant to decompose the products of operators into an "operator basis" ie. associate an algebra to the space of operators.</p>
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<p>I am trying to learn Hamiltonian mechanics. While many textbooks treat closed systems, I have a hard time finding references for forced systems.</p>
<p>For example, if I consider simple systems of masses ($m_i$ connected to $m_{i+1}$ with a spring) it is easy to write down the Hamiltonian. But I'm not so sure how to directly write down the Hamiltonian if say there is an external force that moves for example $m_1$. </p>
<p>Is there a good textbook that treats more general cases like this?</p>
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<p>I came across and interesting effect today, I have a dozen of Neodymium magnets around my house. And, they are very strong. Anyhow, one of them got attracted to my large steel plate table(use for cutting and building). It was almost impossible to take it apart, till I contacted the manufactured of those magnets and they proposed to slid it off. Amazingly, it worked. But what shocked me is the level of huge force decrease. It's like I'm going against friction alone... Only at the edge of the plate I felt a force that is strong, but half way through... The magnet as almost "off" only the edge of the magnet was attracted strongly to the edge of the plate, what explains this?</p>
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] |
<p>The compression spring equations are generally given for helical coil. What are the equivalent equations for alternative coil shapes, like oval?</p>
| 4,902 |
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<p>Spin, position, and velocity are observables which are QM for quantum particles. My question is, what determines whether an observable is QM or not?</p>
<p>For example, why is electric charge not QM? That is, why don't (or can't) particles exist in a superposition of being positive and negative?</p>
<p>What is the underlying mechanism in all of this?</p>
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<p>Does the Higgs Mechanism contradict <a href="http://en.wikipedia.org/wiki/Entropic_gravity" rel="nofollow">Entropic Gravity</a>?</p>
<p>It seems like it probably does. But then again, one is a microscopic theory and the other is macroscopic. Can they live together in harmony? or is the recent CERN stuff empirical evidence against EG?</p>
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] |
<p>Is the dimensionality of spacetime in all usual models constant?</p>
| 4,905 |
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<p>Can somebody explain a detailed procedure, of producing orientation maps? I need to implement this into a software. Right now im able to transform EBSD pattern to probabilistic Hough space, get lines parameters - unfortunetly with some noised/wrong parameters too (any advices?), and autoindex this lines. What is the next step? I read lot of pdf's in the internet, but none explains this in details.
Sorry for my poor english.</p>
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<p>I had a question regarding cosmic event horizon. Let's say that a far away solar system has just crossed the cosmic event horizon due to the expansion of the universe. In that solar system let us say there is a planet orbiting the solar system's star, for half of the planet's orbit, it will be traveling towards us, and the other half it travels away from our planet. So, if the planet crosses the cosmic event horizon while the orbit is going away from us, what happens when the planet travels back towards us so that its relative velocity is less than light? Does the planet re-appear crossing back over the event horizon? And what happens to the entropy of the planet as it crosses back on our side of the horizon? Does entropy of the planet decrease in this case? Or, because the relative velocity is so high, does the planet from our perspective not orbit because time slows down to the point that the planet appears stationary?</p>
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<blockquote>
<p><em>How long does it take a baseball with velocity $(30, 20, 25) m/s$ to
travel from location $r_1 = (3, 7,−9) m$ to location $r_2 = (18, 17, 3.5)m$?</em></p>
</blockquote>
<p>I am thinking that it should be the displacement vector divided by velocity. but velocity is a vector and my text is adamant on not putting a vector in the denominator. I tried it anyway just to see what happens and I get $(.5, .5,.5)s$ .The units do cancel out to give me seconds. The answer is listed in the book as .5 seconds. But how do I arrive there without breaking any rules? (I am assuming delta time is final time - 0.) </p>
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<p>Pretend you are throwing darts at a dart board. You throw dart $d_1$ at time $t_1$. After you throw your first dart, you throw your second dart $d_2$ at time $t_2$. Given that $t_2 > t_1$ in a stationary frame of reference, is it possible for a frame of reference to exist that will make an observer $O_1$ in that frame of reference see $d_2$ hit the dart board before $d_1$?</p>
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<p>Most string theory compactifications analyzed so far have as backgrounds a conformal field theory corresponding to a nonlinear sigma model with a Calabi-Yau target space, or some relatively classical background, possibly with orbifolding. But as long as the central charge of the Virasoro algebra is right, and worldsheet superconformal symmetry is respected, any other superconformal field theory will do just as well. Why haven't string theorists looked at such models more often?</p>
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<p>For <strong>S</strong> and <strong>S'</strong> in standard configuration, the Galilean transformations are:</p>
<p><strong>x' = x - vt, y' = y, z' = z, t' = t</strong></p>
<p>From the Lorentz transformations for v << c:</p>
<p><strong>x' = x - vt, y' = y, z' = z, t' = t - vx/c^2</strong></p>
<p>So it looks as if the Galilean transformations become increasingly accurate for:</p>
<p><strong>vx -> 0, v << c</strong></p>
<p>And exact for v = 0 for all x. </p>
<p>Yet, all text books I've come across state that the Galilean transformatons become more accurate for the condition <strong>v << c</strong> only.</p>
<p>So what are the conditions under which the Galilean transformations become more accurate and why?</p>
| 4,911 |
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