title: Why can any pair of master coordinates be used to calculate a nonlinear mode of a nonlinear dynamical system?

text: This is a question I have been asking myself for some time since the following technique is often used in the nonlinear dynamics community, but never managed to get an answer why it could be applied. In symplectic mechanics, consider a dynamical system with no rigid body movements (all eigenvalues are non-zero, ie $\lambda_i\neq 0$). The centre manifold theorem states that the dynamics can be written in the following form: $$ \begin{bmatrix} \dot{\mathbf{x}}_\mathrm{s} \\\ \dot{\mathbf{x}}_\mathrm{c} \\\ \dot{\mathbf{x}}_\mathrm{u} \end{bmatrix} =\begin{bmatrix} [ A_\mathrm{s}] & & \\\ & [A_\mathrm{c}] & \\\& &[A_\mathrm{u}] \end{bmatrix}\begin{bmatrix} \mathbf{x_\mathrm{s}} \\\ \mathbf{x_\mathrm{c}} \\\ \mathbf{x_\mathrm{u}} \end{bmatrix} +\text{nonlinear terms}$$ and that there exists a centre manifold (also stable and unstable manifolds, but they are not of interest here) of dimension the number of lines of $A_\mathrm{u}$. Because all eigenvalues corresponding to the centre manifold ($\mathrm{Re}(\lambda_i)=0$) are purely imaginary and non-zero (no rigid body movements), $A_\mathrm{c}$ is composed of $2\times2$ blocks on the diagonal. Now, we are searching for periodic orbits, so we focus and the centre manifold and we name the coordinates of $\mathbf{x}_\mathrm{u}$ such that $\mathbf{x}_\mathrm{u}=[u_1,v_1,\dots,u_n,v_n]^T$ (even dimension, as explained above). In order to calculate the _nonlinear normal mode $i$_ , many people write that any pair $(u_i,v_i)$ can be chosen as _master coordinates_ and that all other pair $(u_j,v_j)$, $i\neq j$, can be written as: $$\left\lbrace \begin{aligned} u_j &=f_j(u_i,v_i) \\\ v_j&=g_j(u_i,v_i)\end{aligned}\right.$$ **Question:** What ensures that choosing a pair of master coordinates is always possible? If not, could you give a counterexample? **EDIT** The presence of an internal resonance in the dynamical system constitutes a counterexample: the system above does not have any solutions, then. Yet, I don't really understand why is it so.


title: Dot product of vector and its derivative with respect to time? How does $L \cdot\frac{dL}{dt} = \frac{1}{2}\frac{d(L^2)}{dt}$?

text: How does: $$L \cdot\frac{dL}{dt} = \frac{1}{2}\frac{d(L^2)}{dt}$$ where **L** is a vector (I dunno how to make it bold in the equation). How do they reach to this right hand side equation? And what is the difference between **||L||** and **|L|** ?


title: Prove the solution of von Neumann equation will never stabilize if Hamiltonian and initial density matrix commutes

text: Given von Neumann equation $$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$ If we know that $[H, \rho(0)] \neq 0$, how do we prove in details the solution of von Neumann equation will never be stabilized, i.e. $$\lim_{t\to\infty}\frac{d}{dt} \rho(t) \neq 0~?$$


title: Why a multilayer OLED is generally more efficient than a single layer OLED?

text: Could anyone give me a brief explanation why multilayer OLEDS as more efficient than single layer OLEDs? What other advantages exist for multilayer OLEDs over single layer OLEDs?


title: Why is gas mileage typically better when traveling on the highway than on country roads or in the city?

text: The gas mileage of my vehicle tends to improve the more I have been driving on the interstate on that tank of gas - if I go through a tank of gas without at any point driving on the interstate, I will typically get 24-26 MPG, but when I have driven almost exclusively on the interstate, I will typically get 26-29 MPG. What confuses me is that, when traveling on the interstate, I am driving great distances at higher RPMs, which I would expect to correlate to more gas usage and thus worse gas mileage. Why does the inverse seem to be true?


title: How is sweating a pipe an example of capillary action?

text: I learned how to sweat a pipe today from my father. If you're not familiar with the process, this might help. One thing that jumps out at me is this line (from the above link, as well as my father's explanation) > Solder, which melts at low temperatures, wicks into a joint by capillary > action and bonds with copper at the molecular level. This doesn't seem to be quite right to me. I always thought of capillary action as something like putting a piece of paper vertically into a dye and watching the dye rise up the paper. I also didn't think the solder bonded with the copper at the molecular level, just that it melted and filled in the gaps really well. Wikipedia seems to agree with my definition of capillary action > Capillary action ... is the ability of a liquid to flow in narrow spaces > without the assistance of, and in opposition to, external forces like > gravity. When sweating a pipe you place the solder above the pipe, letting it drip down (ie with the assistance of gravity) which makes it _not_ capillary action... right? Is there something I'm just not understanding here?


Does an object's kinetic energy increase, decrease, or stay constant when it reaches terminal velocity while falling?


title: What happens when length of antenna >> lambda

text: Length of a dipole antenna according to antenna theory should be lambda/2 for best reception. I am just curious about the outcome when length of dipole antenna >> lambda. Impedance will be zero in such a case and there will be no signal received. Is it correct?


title: Topological Charge. What is it Physically?

text: I have seen the term topological charge defined in an abstract mathematical way as a essentially a labeling scheme for particles which follows certain rules. However I am left guessing when trying to explain what physical properties of a system lead to the need to introduce this new type of "charge." If it makes any difference, I'm interested in studying the statistical properties of Quantum Hall systems (and in particular anyonic interferometry), where the different topological charges contribute to the total number of quantum states.


title: Is real voltage always the real part of complex voltage?

text: If I have a complex voltage $V_z$ is real voltage $V$ (i.e. the voltage used in the normal ohms law and the voltage we normally talk about) always given by $V=Re(V_z)$. Does the same apply to current? and if it is not the case how do we find $V$ from $V_z$?


title: When sliding down steep hill, is it safer to go down back or front against the hill?

text: Assume you are sliding down a steep hill. Is it safer to go down with your back against the hill or the front of your body against the hill? I assumed that since the surface area of your body which is touching the hill is about the the same regardless of which way you are going down, it won't make a difference. Is this the correct answer?


title: Riemann curvature tensor notation in Wald

text: This question is entirely on tensorial notation in Wald's General Relativity. When specifying the properties of the Riemann tensor on pg39, he states: $R_{[abc]}^{\quad \ \ \ d} = 0$ and > For the derivative operator $\nabla_a$ naturally associated with the metric, > $\nabla_a g_{bc}=0$, we have $R_{abcd} = -R_{abdc}$. and > The Bianchi identity holds: $\nabla_{[a}R_{bc]d}^{\quad \ \ e} = 0$ Questions:   1. What do the square brackets around "abc" mean?   2. Why does $R_{abc}^{\quad d}$ become $R_{abcd}$? What is the relation between the two?   3. What does having $R$ in the square brackets mean? Thank you.


title: How is electric field strength related to potential difference?

text: I was looking at the derivation of drift velocity of free electrons in a conductor in terms of relaxation time of electrons. Here a metallic conductor XY of length $\ell$ is considered and having cross sectional area A. A potential difference $V$ is applied across the conductor XY. Due to this potential difference an electric field $E$ is produced. The magnitude of electric field strength is $E = V/\ell$. I dont understand this part. How is electric field strength potential difference divided by length. Where am I lagging in my concepts?


title: Statistics followed by Neutrinos

text: What does the neutrino particles follow- Dirac or Majorana Statistics?


title: How much the probability wave acts like a wave?

text: A superposition of two probability wave creates standing wave. Well, that is convincing. Dose waves described by schrodinger's equation have other properties of wave? like reflection, refraction, polarization ect.


title: What is the precise definition of Gravitational Sphere of Influence (SOI)?

text: I am trying to understand the gravitational sphere of influence (SOI), but all I get by searching is the formula that you can find on Wikipedia, that is $$ r_{SOI} = a \left( \frac{m}{M} \right)^{2/5} $$ where m: mass of orbiting (smaller) body M: mass of central (larger) body a: semi-major axis of smaller body When inputing the Moon numbers in this formula, we get a SOI of 66,183 km for the Moon over the Earth. This is consistent with other sources on the web, for example the Apollo mission transcripts when they talk about entering the Moon SOI. What I don't understand is that when I calculate the gravitational forces between different bodies using Newton's laws, an object placed at this distance between the Earth and the Moon still gets a bigger pull from the Earth. Say for example that we had an object with a mass of 100 kg, these are the gravitational pull (in Newtons) that it would receive from the Earth and the Moon at different distances : Force from Earth on Earth's surface : 979.866 N Force from Earth at 384400 km (Moon dist) : 0.27 N Force from Moon at 66183 km from Moon : 0.112 N Force from Earth at 318216 km (66183 km from Moon) : 0.394 N Force from Moon at 38400 km from Moon : 0.333 N Force from Earth at 346000 km (38400 km from Moon) : 0.333 N As you can see, the pull from Earth and Moon cancel each other at around 38,000 km, not 66,000 km. This is somewhat counterintuitive to me, as I first thought that a spacecraft (for example) would get more pull from the Moon than from the Earth when it entered the Moon's gravitational sphere of influence. I suspect that it has to do with the fact that the Moon is in orbit around the Earth, i.e. it is in constant acceleration in the same direction as the Earth's pull, but I would like a clear explanation if somebody had one.


title: Intuitively, Why is Power Proportional to $I^2R$

text: As the resistance of a circuit goes down, the power increases because the current increases, assuming constant voltage. Why is this? I feel like resistance and current are inversely proportional, so lowering one should just raise the other, and power should stay relatively constant. What is it about power that makes it work like this? Edit: Again, namely I find it confusing that decreasing resistance actually has a net increase in power used. I feel like while the current will increase, it should be countered by the lowered resistance...


title: How far can one hear sound?

text: I was thinking how far can I hear sound coming from a concert. Today I was walking at night and I could hear sound from somewhere very far. I started following the sound but sound used to disappear momentarily and then reappear. I went atleast 2 miles but could not find where it was coming from. Now I am really confused because sound is energy and because of energy losses, as I understand sound cannot go as far as 20 miles. The nearest concert was atleast 20 miles away. I came back and read articles(I cannot proof the reliability) where sound had traveled 200 miles but it was mostly sounds from explosion or volcanoes. Maybe this makes sense because explosions or volcanoes have alot of energy but how can one explain sound being heard from a concert very far away(approx 20 miles).


title: Faraday Effect does light bend or lose energy?

text: I was reading upon Faraday effect when it said > Faraday effect causes a rotation of the plane of polarization That in mind, does this mean the light can be bent around or does the light loose energy when its waves rotate? Can someone explain more about this effect simply as I'm a middle school going kid, make sure the explanation is simple yet precise please. Thanks!


title: Mass into energy

text: The mass of a nucleus if less than the mass of the protons and nucleus. The difference is knows as binding energy of the nucleus. This nuclear binding energy is derived from the strong nuclear force. Now my question is that does the strong nuclear force provide this energy? How does the mass decrease if the force provides the energy?


title: Candidates for holographic QFT of 4D Einstein gravity

text: If we are to believe that holographic principle holds over a wide number of dimensions, and gravitational theories, but specially, those that are relevant to our universe, then there must be some 3D QFT that is the dual description of our current 4D General relativity, over what it seems to be asymptotically flat de-Sitter spacetime. > What QFT theories are candidates for being the holographic dual of > gravitational theory in _our_ universe?


title: What experiment would disprove string theory?

text: I know that there's big controversy between two groups of physicists:   1. those who support string theory (most of them, I think)   2. and those who oppose it. One of the arguments of the second group is that there's no way to disprove the correctness of the string theory. So my question is if there's any defined experiment that would disprove string theory?


> **Possible Duplicate:**   >  What experiment would disprove string theory? A hypothesis without hard evidence sounds very much like philosophy or religion to me. All of them tries to establish a functional model explaining how the world works. In my understanding, string theory is an unprovable theory. What differs string theory from philosophy or religion?


title: Regarding string theory: how to refute the argument "if it cannot be tested experimentally then it is not science"?

text: > **Possible Duplicate:**   >  What differs string theory from philosophy or religion? I find a lot of people disbelieve in string theory, saying that since it cannot be tested experimentally then it is not science (quoting Feynman who has said something along that line), and that it is a complete waste of time, and it should not be funded...etc. My question is how to convince those people that string theory is real science and real physics? and how to reply to the argument that if it cannot be tested experimentally then it is not science? where is the flaw in this argument? Note added: Please if your answer is about attacking string theory or saying that it is science fiction and the like, then do not bother answering.


title: What is the 4th dimension?

text: I have heard before that the 4th dimension is time, however, another theory makes a lot more sense to me. This is that the 4th dimension is the third dimension stacked on top of each other in a similar in which 3d objects are just many 2d planes. I have seen many articles related to the 4thdimension being time, but how do we know this?


title: Is the left-handed neutrino, after seesaw, still the left-handed neutrino?

text: Reading this question and its answer Seesaw type-1 and integrating out heavy fields The diagonalization of the see-saw is described as going from " _interaction basis" right (and left) handed neutrinos_ to _"mass basis" right (and left) handed neutrinos_. Now, can chirality refer to a particular basis? Is it not a general property connected with parity? I would think that our "left handed neutrino" after seesaw is not a left neutrino anymore, but that it has a small mixing with the "right handed neutrino". EDIT: and if it is so... is it always so, or there is some see-saw mechanism that preserves chirality?


title: Eternal clocks and 4D spacetime crystals

text: There was a recent article about the creation of 4D spacetime crystals based on recent theory proposed by Frank Wilczek. This theory is based on breaking time translational symmetry which basically is the assumption that the laws of physics are time invariant (e.g. that we can expect our physical laws are the same as we move through time). The concept of spactime crystals is relatively new, and there are classical and quantum versions. Since the time crystal reportedly returns to its ground state periodically, and it reportedly is not in violation of the second law of thermodynamics, the layman might be tempted to look at it as a some sort of pulsating refrigerator. Is there some better description of the time crystal that makes it obvious that this is really a new concept?


title: Operator methods in Quantum Mechanics

text: I'm currently studying the operator methods in Quantum Mechanics, but I can't figure out the general procedure and its relations with the diagonalization problem.


title: Why does time slow down the closer you are to a mass?

text: When ever i look this up all I get is sites saying how its because general relativity says "-" why does it do it though? it is because there is more motion near gravity than further away? Or is it something completely different?


title: Why is metallic hydrogen degenerate matter?

text: Why is metallic hydrogen considered a form of degenerate matter, akin to neutronium and electron-degenerate matter? I can understand that for the other two, degeneracy pressure is the only force countering inward gravity for very massive stars, but how does this concept also apply to super pressurised hydrogen? Furthermore, why would being supported by degeneracy pressure make the hydrogen metallic in nature?


title: What are magnetic field lines?

text: Magnetic field lines depict the strength of the magnetic field, dense lines in a region imply a strong field. Lines are drawn as per our convenience. We needn't draw infinite lines but just relative amounts: where the field is stronger, more lines; where it isn't, less lines. What I wanted to know was what do these lines imply when they are formed in the magnet and iron filings experiment. All this while we find that these lines can be infinite with relative spacings to depict field strength. It seems that these lines are imaginary and only for our convenience. Yet iron filings form them. So what do they imply? Why do the iron filings follow only specific lines around the magnet? What about the spaces in between two adjacent lines? What are those spaces supposed to be? Why wouldn't the iron filings fill those up as well? Are those spaces supposed to imply absence in field?


title: What are magnetic field lines?

text: Does a magnetic field have concentrations of magnetic force lines as seen when putting iron filings over top a bar magnet or are these imaginary? I.e. are they just an artifact of the iron being a 'conductor' of the magnetic field lines making them look like they are concentrated along these path lines but are really continuum of strength around the bar magnet when there are no filings are present? Also I learned that field lines do not cross, yet there are magnet configurations who's forces are explained as the magnetic force vectors are indeed crossing and are additive such as a Halbach Magnet Array. So what is actually happening here?


title: "low-energy effective action" but in what sense?

text: In string theory, consistency with Weyl invariance imposes dynamics on the background fields through the vanishing of the beta functions. Those dynamics can also be derived from the so-called _low energy effective action_ : $$S = \frac{1}{2\kappa_0^2}\int d^{26} X\; \sqrt{-G}\; \mathrm{e}^{-2\Phi}\,(R-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+4 \partial_{\mu}\Phi\partial^{\mu}\Phi)$$ (at least in bosonic string theory) Maybe I shouldn't worry over lexical denomination, but I find this naming of "low-energy" a bit obscure. In what sense is it used? Would it be because the background fields are supposed to emerge from the fundamental strings? Or because we forget the massive excited states of the string (with masses around the Planck scale and irrelevant for low energy phenomenology)?


title: Why energy saving light bulbs contain rare earth?

text: Reading a french article talking about a _recycling unit of rare earth from energy saving light bulbs_ near Lyon - France (here is the official press release of the Salvay company), I was wondering :   * where is rare earth used in energy saving light bulbs ?   * what advantages does it provide ?   * why don't we use them in regular light bulbs ?


title: Physics of homebrewing heat exchangers

text: In homebrewing on of the key steps when brewing with extract is to rapidly chill the wort from boiling temperatures to about 80F in 30 min. This is needed to reduce risk of environmental contamination, oxidation, and control of formation of Dimethyl Sulfide This isn't too hard for small batches of beer, and most people use ice baths for their brew pots, however, as you get to larger sized batches, most people resort to wort chillers run from tap water to reduce temperature. Usually this works great from getting from boiling to 90F, but the temperature gradient usually is too low to allow for rapid chilling from 90F to 80F. In order to optimize this process, one might like a nice set of physical equations for modeling heat exchange rather than some rule of thumb engineering models for heat exchangers. If one were to model this process from a physicists point of view, what would be the resulting set of equations?


title: Show that the Hamiltonian operator commutes with the angular momentum operator

text: Show that the Hamiltonian operator $\hat{H}= (\hat{p}^2/2m)+\hat{V}$ commutes with all three components of $\vec{L}$, provided that $V$ depends only on $r$. (Thus $H$,$L^2$, $L_z$ are mutually compatible observables). ($\vec{L}$ is $\langle L_x, L_y, L_z\rangle$.)


title: How to know electron and muon neutrinos are different except measuring mass

text: Before the discovery of neutrino mass, how did people aware electron and muon neutrinos are different?


title: Are Electromagnetic Waves The Only Means of Transmitting Information?

text: We've been using EMF to transmit energy (information) for over a century. I was wondering is there any other way to send a message on long distances, even faster than EMF waves can travel? For example there are particles that travel faster than C. Or as another example I have heard the Quantum Entanglement can be useful in this case.


title: Does general relativity fail in conditions with very large gravitational forces?

text: It is said in this wikipedia article (in the 7th paragraph) that where there exists huge masses and very large gravitational forces (like around binary pulsars), general relativistic effects can be observed better: > By cosmic standards, gravity throughout the solar system is weak. Since the > differences between the predictions of Einstein's and Newton's theories are > most pronounced when gravity is strong, physicists have long been interested > in testing various relativistic effects in a setting with comparatively > strong gravitational fields. This has become possible thanks to precision > observations of binary pulsars. But here in whystringtheory.com  (in the last paragraph), it is said that > For small spacetime volumes **or large gravitational forces Einstein has > little to offer** I know that in singularities, GR fails and this is a motivation for quantum gravity theories. But the second quote above says in small spacetime volumes **or** large gravitational forces. Is there any problem with general relativity in conditions with very large gravitational forces (in big enough volumes of spacetime)?


title: What is the most natural new physics one can expect at the TeV scale: new (supersymmetric)particles or some new (non-commutative) spacetime structure?

text: Up to now, nothing else than one Standard Model (SM) Higgs boson-like resonance has been found at the LHC while many predictions based on effective theories using supersymmetry require several Higgs scalars and needs an entourage of sparticles close in mass to tame its quantum instabilities (I borrow more or less from James D. Wells). On the other hand, the spectral and almost-commutative extension of the SM by Chamseddine and Connes, expects only one Higgs boson without other particles in the TeV range. In this noncommutative approach spacetime appears as the product (in the sense of ﬁbre bundles) of a continuous manifold by a discrete space and it has been proved by Martinetti and Wulkenhaar that under precise conditions, the metric aspect of ”continuum × discrete” spaces reduces to the simple picture of two copies of the manifold. **Could it be that this picture of a two-sheets spacetime helps to overcome the technical naturalness issue related to the standard model Higgs ( _replacing temptatively a low energy supersymmetry by a new geometric framework_ ) and has to be taken seriously in order to progress in the understanding of physics beyond the SM?** To ask it differently: **Reminding that the Standard model like Higgs boson is a _natural_ consequence of the noncommutative geometric framework, could it be that the _discreteness of space-time_ usually expected at the Planck scale from quantum gravity already shows up _at the electroweak scale_ through the very existence of the already discovered Higgs boson?** (this formulation could require a new, yet to be defined, heuristic meaning for the term: _naturalness_ ) Last but not least, it is worth noting that to postdict the correct mass of the Higgs boson detected at LHC8, the last version of the spectral model relies on a weak coupling with another scalar that shows up "naturally" in the spectral action just like the Higgs. This "big brother" from the Higgs boson is expected to acquire a vev generating a mass scale above $10^{11}GeV$ for right-handed Majorana neutrinos. It could thus be responsible for a type I see-saw mechanism explaining the neutrino phenomenology beyond the minimal SM. _May be noncommutative geometry can help to make effective theories more alive and kicking! In memoriam Ken Wilson_ _To celebrate the 4th of July "IndependentHiggsday", I wish a happy birthday to the lightest scalar field of the Standard Model and I congratulate experimentalists who work hard to prove physics is alive (andnot ordered by theories ;-)!_ EDIT : The title of the question has been changed in an attempt to improve clarity (after reading _The Higgs: so simple yet so unnatural_ ); the former title was: > Doubling the number of elementary particles or "doubling space-time" to > accommodate Higgs boson phenomenology at 8TeV?


title: Physical interpretation of scalar tadpole loops

text: In Feynman diagrams, fermionic loops are drawn like this: ![enter image description here](http://i.stack.imgur.com/JkPkc.png) While scalar loops are drawn as tadpoles: ![enter image description here](http://i.stack.imgur.com/kOKCX.png) I assume the difference comes from the scalar not having an anti-particle. But how should one imagine such a tadpole diagram? I can imagine the fermionic loop as creation of particle and antiparticle, but I am at loss to only get a faint idea what the scalar loop should illustrate.


title: Linearity of the time evolution operator for the reduced density matrix of an entangled state

text: Suppose to have a system $S$ immersed in an enviroment; the pure states are elements of $H_S \otimes H_E$, where $H_S$ is the hilbert space of the system and $H_E$ is the hilbert space of the enviroment. The density matrix for the total system $\rho_{S+E}$ evolves according to the Von Neumann equation $$ i \frac{d\rho_{S+E}}{dt} = [H,\rho_{S+E}]. $$ If we're interested only in $S$, we can trace over the degrees of freedom of the enviroment and use the reduced density matrix formalism $$ \rho_S = Tr^E[\rho_{S+E}] $$ What we want to do now is to find a time evolution operator $\Gamma_t$ for the reduced density matrices such that $$ \rho_s(t) = \Gamma_t[\rho_S(0)]. $$ Now, during one lesson, one professor said that if we want the evolution operator to be linear, we **MUST** chose a nonentangled state (i.e. $\rho_{S+E} = \rho_S \otimes \rho_E$). While it can be easily proven that if the initial state is not entangled, the operator $\Gamma_t$ is linear, I don't see why, **in general** , the evolution operator for a reduced density matrix of an entangled state cannot be a linear operator.


title: Can a negative charge ionize air?

text: Can a negative charge ionize air? Adding to this question, I have studied that positive charge ionizes air but never studied that negative charge ionize air. If possible, please explain how does it ionize air?


title: Transparency of gypsum and concrete for RF waves

text: Is there a database (or any other source) of graphs of average transparency of various materials (cardboard, concrete, gypsum etc.) as a function of wavelength?


title: Spin-dependence of the directionality of dipole radiation

text: I am interested in understanding how and whether the transformation properties of a (classical or quantum) field under rotations or boosts relate in a simple way to the directional dependence of the radiation from an oscillating dipole. For example, the EM field from an oscillating electric dipole $\mathbf{d}(t) = q x(t) \hat{\mathbf{x}}$ pointing along the $x$-axis vanishes in the x-direction (in the far-field). On the other hand, the sound radiating from a similar acoustic dipole vanishes in the $y-z$ plane (in the far-field). This result makes total sense classically because EM radiation consists of transverse waves while acoustic radiation is carried by longitudinal waves (try drawing a picture if this is not immediately obvious). The same holds true even if the fields and dipoles under consideration are treated quantum mechanically. Now, the acoustic (displacement) field is represented by a scalar field, while the EM field is a (Lorentz) vector field. This leads me to wonder if one can draw some more general conclusions about multipole radiation, based on the transformation properties of the field in question. **Given a tensorial or spinorial field of rank _k_ , and a multipole source of order _l_ , what is the asymptotic angular dependence of the resulting radiation?** **In particular, how does the radiation from a "Dirac" dipole look, assuming such a thing can even make sense mathematically?** By the latter I mean writing the classical Dirac equation $$ (\mathrm{i}\gamma^{\mu}\partial_{\mu} - m)\psi(x) = J(x), $$ for some spinor-valued source field $J(x)$ corresponding to something like an oscillating dipole. (I understand that in general this term violates gauge invariance and so is unphysical, but I am hoping this fact doesn't completely invalidate the mathematical problem.) Apologies to the expert field theorists if this is all nonsense :)


title: Experiment regarding myopic correction by a manipulation of fingers?

text: Here is a small experiment my tutor once told us for just amusement. It works for myopic people at least, and can be a good check to see if you have myopia. > With your naked eye, ("remove the spectacles" for myopic spectacle wearers) > see a considerably distant object (a few meters). It should be a bit blurred > and unclear if you have myopia. Then, using your two thumbs and first > fingers of both hands, create a tiny square in between. Using a single eye, > look through the square. For a large square, the image is still blurred. But > as you purse your fingers tightly and reduce the dimensions of the space > between, you will eventually reach a point when the image comes in perfect > focus. The image is perfectly clear as you would see it with spectacles. For > an object 1-2m away, the square needs to be much smaller than that in the > picture. ![Not my hands](http://i.stack.imgur.com/8Vw2e.jpg)![Not my hands](http://i.stack.imgur.com/rWO8M.jpg) This works for almost any myopic person. Why does it work? Does diffraction come into play? If it does, I don't see how that can adequately refract rays to act as a lens?


title: Lorentz covariance of the Noether charge

text: The invariance under translation leads to the conserved energy-momentum tensor $\Theta_{\mu\nu}$ satisfying $\partial^\mu\Theta_{\mu\nu}=0$, from which we get the conserved quantity$$P^\nu=\int d^3\mathbf x\Theta^{0\nu}(x)$$But I cannot see explicitly **how this quantity is a four-vector** covariant under Lorentz transformation, since $d^3\mathbf x$ is part of the invariant $d^4x$, $\Theta^{0\nu}(x)$ is part of the covariant tensor $\Theta^{\mu\nu}(x)$, neither of which transforms covariantly. So can someone show me how this becomes correct? And generally, how to show that a Noether charge $Q$ corresponding to the Noether current $j^\mu$, $$Q=\int d^3\mathbf x j^0(x)$$ , is a **Lorentz scalar**?


title: Wien's Displacement Law for real bodies

text: It is known that for perfect blackbodies, $$\lambda T= c$$ where $\lambda= \text{peak wavelength}$   $T= \text{Absolute temperature}$   $c= \text{Wien's constant}$ But this is for perfect blackbodies only, which have no theoretical existence. Does a similar formula exist for real bodies, which expresses $\lambda T$ in terms of its emissitivity $\epsilon$? I googled it, but found no relevant results.


title: Some conceptual questions in BEC

text: In Bose-Einstein condensate (BEC), people often say there is a well defined macroscopic phase. What exactly the macroscopic phase is? (a phase factor $\mathrm{e} ^{i\phi}$ in a many-body wavefuction?) Is the macroscopic phase the same as the coherent phase?


title: Would a rocket burn more fuel to get from Earth's surface to LEO, or to get from LEO to GEO?

text: Would a rocket burn more fuel to get from Earth's surface to Low Earth Orbit, or to get from LEO to Geosynchronous Earth Orbit?


title: Finding The Obtuse Angle

text: I am working on a simple force problem involving inclines. I found a reference of $48^\circ$, in the third quadrant; now, to find the obtuse angle, do I subtract that value from $270^\circ$, or do I add it to $180^\circ$?


title: Understanding tension

text: I'm trying to understand tension. So here it goes: I'll start from the beginning. Let's assume I'm in space and can move around and apply forces. Let's say a rope is attached to a body(which is in space). 1) Let's say the body is immovable. Then the force with which the rope is pulled will be the tension in the rope. Right? 2) Okay, now let's say the body has a mass M and I apply a force F. The body will accelerate towards me with a acceleration F/M. Obviously, the rope will slacken, so to keep the tension in the rope constant, I will have to accelerate away from the body, maintaining my distance to still apply the force. (I just mentioned that because no one really talks about how one can apply a constant force without having to 'keep up' with the body to keep applying the force. I just wanted to make sure that that truly is the case.) Now let's say, I pull the rope and then let it slacken i.e. not try to maintain a constant force, then if I was able to measure the acceleration of the body and if I know it's mass, I will be able to find the force I applied and hence, the tension (=force applied) for that period of time on the rope. Is that correct? 3) Okay. Now, let's say I pull the _rope_ with X newtons of force and another person holds the _body_ and pulls it in the opposite direction with Y newtons of force, shouldn't be the tension in the rope now be (X+Y) newtons, even if the body accelerates in one direction? If the person instead _pushes_ the body with Y newtons then shouldn't be the tension (X-Y) newtons in the rope now? 4) Let's say the ends of the rope are attached to two bodies and the rope is currently slackened. I then give both the bodies some initial velocity in the opposite direction to each other. Hence, the rope at some point will become taut. Is it possible to tell the force, the two bodies will experience when the rope becomes taut and hence find the tension the string will experience? Please correct me if I'm wrong in any of these. I would really appreciate the help. Thank you.


title: Heuristic argument for forces with respect to entropic force

text: Here is an argument that I have with regard to entropic forces and how it can be linked to the other forces in nature.   * Gravitational: This is debated: whether gravity is entropic or not. I am going for the paper that says gravity is entropic. Is Gravity an entropic force after all? . Here is my argument for it. Do let me know if it makes any sense at all. If we launch a projectile, then it follows a parabolic path provided the angle of projection is not 90 or 0. According to the entropic theory, the projectile has a random trajectory. However, if the trajectories are averaged out, then the path obtained is parabolic. Therefore, gravity is entropic. (This argument is rather hand-wavy in nature, I agree. Are there **any alternate arguments that you can provide?** )   * Coulombic: Both gravitational and coulombic laws have similar structures and assumptions, the only difference being that mass has no sign whilst charges do. But that is the tendency of matter. Both are central forces, so if we draw parallels, then shouldn't **Coulombic forces also be entropic?** (Again, hand-wavy) As for the other forces, I have some sort of intuition but I am not sure if it makes any sense mathematically. Like nuclear forces, beyond a certain range, they are repulsive in nature. But since it is a much stronger force than gravity, and if gravity is entropic, (and gases collide randomly) then in order for a gaseous system to 'defy' gravity, won't the **nuclear force also have to be entropic ?** And could the repulsion be because if the particles combine, then the **entropy of the system decreases?** I am not sure. * * * The arguments are quite hand-wavy, but I couldn't think of other ways to imagine it. **Are there any mathematical proofs for the same?**


title: Event Horizon fluctuating due to gravitational waves

text: Do the interiors of black-holes create gravitational waves and if so do these waves cause the radius of the event horizon to fluctuate as the waves pass the horizon ?


title: Black hole collision and the event horizon

text: Will the event horizons of a two black holes be perturbed or bent before a collision? What will the shape of the event horizon appear to be immediately after first contact?


title: Elasticity of Space; How does the expansion of Space affect gravity?

text: Does space have an elastic quality? What I was thinking about was if space is expanding, is it being 'stretched', like a balloon being blown up, and if so, is this causing gravity to weaken? Imagine space as a 2 dimensional sheet (got this from one of Brian Greene's books) with planetary bodies resting on it and causing a depression in it, if you were pulling this sheet from all sides over a period of time, you would cause the depression of the planetary body to decrease and eventually become flat, which if we go back to reality, would mean that the gravitational 'constant' had changed to the point where the planetary body had no influence on those objects which were previously orbiting around it (or even residing on it's surface). Is this the case in reality? Or does space not have an elastic quality? If not, can you explain to me what exactly it means for space to be expanding? In case you didn't notice, I'm a layman (hence the Brian Greene books :p), so try to keep your answers/explanations conceptual if possible.


title: How can a string be unidimensional if they can be open ended or close ended?

text: I just don't understand how a object with 2 ends can be unidimensional.


title: Do atoms of a surface excite to reflect the light?

text: How do surfaces reflect certain colours and absorb the others?


title: How should observers determine whether they can be described as being "defined on a Lorentzian manifold"?

text: Consider infinitely many distinguishable observers, no two of whom ever meet; and who generally "keep sight of each other", but not necessarily "each keeping sight of all others".   How should they determine whether or not they can be described as being "defined on a Lorentzian manifold"? [This question refers to terminology of http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity and is meant as follow-up to that question; in the attempt to ask perhaps more originally.] **Edit**   The phrase "defined on a Lorentzian manifold" appears a very general condition. To be more specific consider instead the question:   "How should the given observers determine whether or not some subset of the entirety of events in which they (separately) participated can be described as "open set of a 3+1 dimensional Lorentzian manifold"?


title: Do bullet/gun physics change if the gun is traveling very fast?

text: If I'm running at say $400\: \mathrm{m/s}$ and a bullet travels at $400\: \mathrm{m/s}$ and I fire the gun, will I see the bullet leave the barrel? I either see it stay in the barrel floating because we are moving at the same speed. \-- Or -- The bullet would move and have a speed of $800\: \mathrm{m/s}$. What is the correct answer?


title: Locality in Quantum Mechanics

text: We speak of locality or non-locality of an equation in QM, depending on whether it has no differential operators of order higher than two. My question is, how could one tell from looking at the concrete solutions of the equation whether the equ. was local or not...or, to put it another way, what would it mean to say that a concrete solution was non-local? edit: let me emphasise this grew from a problem in one-particle quantum mechanics. (Klein-Gordon eq.) Let me clarify that I am asking what is the _physical meaning_ of saying a solution, or space of solutions, is non-local. Answers that hinge on the _form_ of the equation are...better than nothing, but I am hoping for a more physical answer, since it is the solutions which are physical, not the way they are written down .... This question, which I had already read, is related but the relation is unclear. Why are higher order Lagrangians called 'non-local'?


title: References for nuclear masses, mass deficits, decay rates and modes

text: Where can I find the base data for computing the energy release of nuclear decays and the spectra of the decay products? * * * My immediate need is to find the energy release by the beta decay of Thorium to Protactinum upon receiving a neutron: $$\mathrm{Th}02 + n \to \mathrm{Th}03 \to \mathrm{Pa} 13 + e^- + \bar{\nu}$$ The estimated amount of energy released from Beta decay is roughly 1eV. Minus the neutrino loss, how much kinetic energy is released as heat?


title: What happened to the work done by friction here?

text: **Problem/Solution** ![](http://img832.imageshack.us/img832/1313/88791065.jpg)! **Question** What happened to the work done by friction BEFORE it touched the spring? Why was that neglected? Also they say there is no physical meaning behind the negative root, so what is the "unphysical" meaning behind the negative root? How are we supposed to know that the speed is constant before and just as it makes contact with the spring? **@atomSmasher** ![](http://img59.imageshack.us/img59/917/friclk.jpg)! I am referring to the green region. Shouldn't it be $E_i = \frac{1}{2}mv_A ^2 - f_k(x_b + x_{green distance})$ $E_f = \frac{1}{2}kx_B ^2$ I realize we would have two unknowns then.


title: What kind of interactions denote 'observation'?

text: I'm quite perplexed by the notion of 'observation' in regards to the collapse of a particle's probability wave. Does a particle's wave only collapse when it is involved in a strong interaction (such as a collision with another particle, like bouncing a photon off another particle to determine the other particle's position) or does any interaction with another particle or field also cause this collapse? For example, I presume traditional detectors such as those mentioned when talking about the double slit experiment are devices that do exactly as I stated above and (in the case of the double slit) have a stream of photons (like a curtain) going from the top of the slit to the bottom which the electron (or particle being shot at the slits) has to pass through and therefore get smashed into by the photons. So given an environment where there was no other gravitational or electromagnetic influence, or where the effects of such have been taken into account, what would happen if you had a region of space within which a massive neutral particle 'is' (ie, it's probability wave fills this volume), and then shot a beam of photons across the bow of this region (so as to pass nearby, but not intersect), and then have a photo-sensitive plate on the other side to see if the photons passed straight by, or were pulled towards one direction slightly by the warping of spacetime caused by the massive neutral particle (gravitational lensing on a mini scale)? Does this even make sense? And if the photons were being pulled slightly towards the massive neutral particle, would the waveform collapse at this point, since if the mass of the neutral particle was known, the amount by which the photons path had been bent could confirm it's position couldn't it? Or perhaps take the above and use an electromagnetically positive particle as the target and an electromagnetically negative particle as the beam you shoot by it to see if it is effected by the electromagnetic field caused by the target particle if that makes more sense. I might be making some silly errors or assumptions here, I'm only a layman with no formal education or in depth knowledge, so be gentle :)


title: Cosmological constant of standard model of cosmology and observational data

text: I am curious whether the current Lambda-CDM model of cosmology matches well with observational data, especially expansion of the universe. How well does Lambda-CDM defend its established status from other models, such as quintessence (quintessence can be said to extend Lambda-CDM, but there are some models against the standard model, I guess.)?


title: How does artificial gravity reduction (weightlessness) work on Earth?

text: Some of us have been to those amusement park/carnival/theme park rides where you enter some capsule/machine, and you will float, etc. There is also some other thing where you go inside a big room with normal gravity, and you are closed in and you start to float (i.e. no gravity), and the amount of gravity can be toggled by a person outside the room, or by some dial/switch, etc. It is like the picture I placed below. ![enter image description here](http://i.stack.imgur.com/y2NbX.jpg) I do not know what those things are called, and I will update accordingly when the names are presented, but what I am asking is how does that work? How can the pull of gravity, result of the spacetime continuum affecting all bodies of mass, be artificially changed? If so, why don't we just alter gravity everywhere with these tools, and move giant, heavy objects easier and such?


title: Equation for relativistic electron and two-component spinor

text: Recently I heard that there is some "alternate" equation for the Dirac one. It can be introduced if we refuse some properties of the theory describes the electron, which Dirac used in his original article. Then we will get theory with spin 1/2-particle and with modified (in compare with Dirac equation) mass. I thought that it is connected with the invariance of theory under discrete Lorentz transformations. If we need it, we must to create theory which describes the sum of representations $\left(\frac{1}{2}, 0\right) + \left( 0, \frac{1}{2}\right)$ (because it is invariant under discrete transformations and as the consequence - under full Lorentz group transformations), so, finally, we will get the Dirac equation. But if we refuse the requirement of invariance under discrete transformations, we may describe two-component wave- function. Some analisys leads to the Klein-Gordon equation for it. But it has problems with propability density, so I think that the way of getting this alternate equation is another. Can you help me?


> Before answering, please see **our policy** on resource recommendation > questions. Please try to give substantial answers that detail the **style, > content, and prerequisites** of the book or paper (or other resource). > Explain what the resource is like as much as you can; that way the reader > can decide which one is most suited for them rather than blindly relying on > the suggestions of others. Answers which just suggest a book or paper may be > deleted. > > And please, note that any answers to this are **community-owned** , which > means that they are subject to heavy editing, especially to make them comply > with the book policy, to avoid deletion. > > This question specifically asks for the level to be that of **Hobson**. Are there any good video lectures for learning general relativity at the level of Hobson? The internet has by L. Susskind, but it is a simple level. I found good one here http://www.youtube.com/watch?v=hcTNxJF_AjU&list=SPB11E80403FB98CFF however, the acoustics is a little bit not so good.


title: Minkowski diagram and length contraction

text: The length contraction means that an object is the longest in the frame in which it is at rest. Lets assume i have a meter stick with length $\Delta x$ in my rest frame which is $x,ct$ and i want to know how long my meter stick seems to an observer moving with a frame $x',ct'$. **1st:** I draw world lines of a meter stick in a rest frame and they are vertical (parallel to $ct$ axis) as meter stick is stationary in this frame. **2nd:** If an observer in a moving frame $x',ct'$ wants to measure my meter stick he measures its edges at the same moment in his time, so i draw a tilted line (parallel to $x'$ axis). **3rd:** If i mesure the length $\Delta x'$ which is a length of a meter stick as observer in frame $x',ct'$ sees it, it seems to me that **he sees a longer distance than me.** This is not correct. Could anyone tell me what am i missing here? ![enter image description here](http://i.stack.imgur.com/r4QzA.png)


title: Matrix representation of state

text: This is a quantum mechanics question, I don't quite understand what it's getting at... > Suppose the we have a state described by $|1\,\,\, m\rangle$. Let its matrix > representation be $\vec u$. The angular momentum measured in $(0, > \sin\theta, \cos\theta)$ direction is $+1$. What are the components of $\vec > u$? I presume one starts by assuming a basis $|1\,\,\, 1\rangle, |1\,\,\, 0\rangle, |1\,\,\, -1\rangle$ or something? I have no idea of what's going on...


title: Interpreting a Hamiltonian in terms of 'hopping' operators

text: I am having some trouble interpreting a Hamiltonian in terms of "hopping" operators. The Huckel model for nearest neighbour interaction in graphene is given by $$H=-t\sum_\vec{R}|\vec R\rangle\langle \vec R+\vec \tau|+|\vec R\rangle\langle \vec R+\vec a+\vec \tau|+|\vec R\rangle\langle \vec R+\vec b+\vec \tau|+\text{h.c.}$$ where $t>0$ is a constant, $\vec a$ and $\vec b$ are the unit cell vectors for a two-atom unit cell, and $\vec \tau$ is the position of the second atom in the basis, the first atom being at the origin of the unit cell. The position of $\vec\tau$ is $\frac13(a,b)$ generalized over all space. How can I interpret the different terms in the Hamiltonian as hopping processes?


title: Hertzian dipole in time varying electric field

text: Suppose we place a Hertzian dipole (short, ends loaded with capacitance) in a time varying electric field $\vec{E}\left(t\right)$, with magniture $E\left(t\right)$ and direction as shown in this figure. What will be the (steady state) current $I\left(t\right)$ flowing through the resistance $R$? ![Dipole in time-varying electric field](http://i.stack.imgur.com/EM03d.png) I expect that if $E\left(t\right)$ is time invariant, there should be _no_ current? So I was wondering if $\displaystyle \frac {d E\left(t\right)}{d t}$ appears in the expression somewhere? I understand that the condition of a loop inside a time varying magnetic field is handled by Faraday's law, with the EMF given by $\displaystyle\mathscr{E}\left(t\right) = -\iint_A \frac{d\vec{B}\left(t\right)}{dt} \cdot d\vec{a}$. So is there something similar for the electric case?


title: Can the total amount of energy in the entire universe be quantified?

text: Classical thermodynamics leads one to believe that if energy is transferred, and the universe is heading for maximum entropy, then back extrapolating to moments before the big bang, one could find a definitive answer of the total amount of energy of the system(universe) contains. However I can't find this question explored without running into dark matter threads.


title: How trustworthy are numerically-obtained periodic solutions to the three body problem?

text: I recently found out about the discovery of 13 beautiful periodic solutions to the three-body problem, described in the paper > Three Classes of Newtonian Three-Body Planar Periodic Orbits. Milovan > Šuvakov and V. Dmitrašinović. _Phys. Rev. Lett._ **110** no. 11, 114301 > (2013). arXiv:1303.0181. I am particularly impressed by how elaborate the solutions are, and the tantalizing hint of an infinity of other distinct orbits given by the analogy with a free group. The solutions can be viewed in the Three-Body Galery, which has animations of the new orbits in real space and in something called the 'shape sphere', which is described in the paper. ![enter image description here](http://i.stack.imgur.com/2DPJi.jpg) * * * I was aware aware already of the figure-of-eight solution, which is described nicely in > A new solution to the three body problem - and more. Bill Casselman. AMS > Feature Column. and which was discovered numerically by Christopher Moore ( _Phys. Rev. Lett._ **70** , 3675 (1993)). I understand that the figure-of-eight solution has been proven to actually exist as a solution of the ODE problem, in > A Remarkable Periodic Solution of the Three-Body Problem in the Case of > Equal Masses. Alain Chenciner and Richard Montgomery. _Ann. Math_ **152** > no. 3 (2000), pp. 881-901. There is also a large class of solutions called $N$-body choreographies by Carlés Simó, in which a number of bodies - possibly more than three - all follow the same curve. Simó found a large class of them in 2000 (DOI/pdf), though this nice review (DOI) seems to imply that formal theorematic proof that they exist as periodic solutions of the ODE problem is still lacking. * * * So, this brings me to my actual question. For the numerical simulations, however well you do them, in the end you will only have a finite-precision approximation to a solution of a differential equation that is propagated for a finite time. Additionally, you might do a numerical stability analysis that strongly suggests (or rigorously proves?) that you are (or are not) in a stable orbit. However, this is quite a ways away from a rigorous existence theorem for a periodic orbit with that symmetry. With this in mind, then, **in what spirit are these simulations done?** Is it purely a numerical approach, in the hope that good numerics do indicate existence, but with a rigorous proof left to the mathematicians through whatever other means they can manage? Or is there some overarching theorem that indicates the existence of a truly periodic solution after a given threshold? What tools are there for proving existence theorems of periodic solutions?


title: Demagnetization of a permanent magnet and lenz's law

text: This video ( http://www.youtube.com/watch?v=Vs3afgStVy4 ) demonstrates lenz's law, if the current produced by the magnet fall induces an opposing magnetic field , after many tries , can this field demagnetize the magnet? or decrease its magnetization ?


title: Rope tension between two objects

text: ![enter image description here](http://i.stack.imgur.com/ORjPY.png) Let me explain the picture just to make sure. Object with mass 2 kg is being pulled with a force of F = 14.4 N. The whole system has an acceleration of a = 2.8 m/s^2. The friction coefficient between the objects and the ground is k = 0.2 What is the tension in the rope? So the question is simple I believe but I can't understand the concept of tension in the rope (red line). Also if it helps I have a solution for the problem, it says that the tension is 4.8 N. Can someone please elaborate this? Thank you!


title: Variable definition in wave function for scattering particle?

text: For the wave function of a scattered particle when finding the scattering aptitude we have: $$\psi(r)=Ae^{ik_0∙r}+\frac{2\mu}{\hbar^2} ∫G(r-r')V(r')\psi(r')d^3r'$$ I was wondering what the variables $r$ and $r'$ represent. I think r just represents the position of the particle and $r'$ the size of the detector, but this seems wrong. I say this as when we do the Born approximation on it we substitute the $r'$ for e.g. $r_1$ . So what do they actually represent? thanks


title: What exactly is the 'lift' of a sailboat as explained by Bernoulli principle

text: I have heard that when a sailboat is sailing against the wind, it operates on the principle of 'lift'. I am unable to understand the explanation, based on Bernoulli principle, completely. My question is, when it says 'lift', it literally means that the boat is being 'lifted' out of the water? Like when we throw a stone across the lake and it skims and hops on the water?


title: Does measuring the operator of a wave function collapse the wave function to the measured eigenstate?

text: Suppose you have a state described by the wave function $\psi(x) = \phi_1(x)+2\phi_2(x)+3\phi_3(x)$ , where the $\phi$s are normalised eigenfunctions of a Hermitian operator $\hat{O}$ with eigenvalues $\lambda_1 = 1, \lambda_2 = 5, \lambda_3 = 9$. **What happens to the wave function if you measure $\hat{O}$ and you obtain the result $\lambda_2=5$?** My answer is that the wave function immediately collapses to $\psi(x) = 5\phi_2$ _**EDIT** : My answer should be $\psi(x) = 10\phi_2$, I think_ Is that correct? Or should I state the probability with which the wave function will collapse to that state: $<\hat{O}> = \sum|c_n|^2\lambda_n$ where $|c_n|^2$ is the probability of the $n^{th}$ state.


title: Why won't this reactionless drive idea work (motor with a moving weight in space)

text: I know people will say it violates many laws of motion and conservation but could anyone explain why it is so? It's NOT a question about free energy. Imagine a motor in space. It has an arm (rod) attached to it. At the end of the arm there's a ball (weight) attached to it. The motor produces a circular motion. So the motor is the center and the rod is the radius. To the ball there is a rope attached which connects to the rod aswell, but the rod keeps the ball in place. The rope is longer than the radius. The rod has a realease mechanism to let the ball go. When the ball is released, it disconnects from the rod and flies away in a tangent. Then the ball flies for some distance until it is stopped by the rope. Does the sudden stopping of the ball by the rope make the motor move off its place and head in a single direction? Is there any opposite reaction force to the direction of the moving ball on the moment that it is released? Is there an opposite reaction force somewhere that makes this impossible and cancels out any potential movement causing forces all together? Is it possible that this can make the system as a whole (motor, ball, rope and rod) move in a single direction without reaction mass being expelled and without external forces acting upon it? And if so, will this system stop moving when it catches up with its ball? I appreciate help a lot.


title: Does a lack of windows protect against shockwave of a nuclear attack?

text: I know that one of the dangers of a nuclear attack is the shockwave which compresses the air and can cause a pulmonary embolism in creatures in the area of effect. Is the shockwave directional in effect? In other words, lets say your building is on the outskirts of a city and is built so that all the windows face AWAY from the city and only thick, concrete walls face the city side. Will this protect the occupants from the shockwave of an explosion over the city or will the shockwave wash over the building and then go "backwards" through the windows?


title: Is there a connection between the core of the earth and heaviest element produced by stars?

text: I was unable to answer this question from my daughter. Is it just a coincidence or is there a connection between the following two observations: (1) the core of the earth is made of an iron (-nickel alloy) as well as the existence of iron meteorites with the fact that (2) the heaviest element that can be produced by a star (prior to supernova) is iron? Related: Elements of a Planet reveals nearby supernova remnant?


title: Calculating angular momentum

text: A particle of mass $m$ is at a very large distance $p$ from origin $O$ and is moving with velocity $\vec{V}$ which is perpendicular to $\vec{OP}$. I have to calculate angular momentum $L$ of the particle. I know that $\vec{L}=\vec{r}\times m\vec{\dot r}$.   Since $|\vec{r}|=p$ and $|\dot r|=V$ and $\alpha=90^{\circ}$ is angle between $\vec{r}$ and $\vec{\dot r}$, I got that $L=|\vec{r}||m\vec{\dot r}|sin\alpha=pmV\cdot 1=pmV$. In the book, it's written that $L=pV$. What happened with mass?


Let's have the system of point-like non-interacting particles and it's own angular momentum $$ \mathbf L_{1} = \mathbf L - [\mathbf R_{E} \times \mathbf P], $$ where $\mathbf R_{E}$ - center of energy of the system, $\mathbf P $ - impulse of center of energy, and it's operator, $$ \hat {\mathbf L}_{1} = \hat {\mathbf L} - [\hat {\mathbf R}_{E} \times \hat {\mathbf P}] . $$ What physical value is described by following operator? Does it connect with spin of the system? In my opinion, it consist only orbital component of full spin of the system (i.e., summ of orbital angular momentum of the particles in the center of energy system), but does not contain spin as quantum value. Can you help me?


title: Does the (relativistic) mass change? Why?

text: I learned recently that when an object moves with a velocity comparable to the velocity of light the (relativistic) mass changes. How does this alteration take place?


title: Why does an object with higher speed gain more (relativistic) mass?

text: Today, in my high school physics class, we had an introductory class on electromagnetism. My teacher explained at some point that an object with a very high speed (he said it started to get somewhat clearly noticable when travelling at 10% of the speed of light) will gain mass, and that that's the reason why you can't go faster than light. One of my classmates then asked, why is this so? Why does an object with higher speed gain more mass? This of course is a logical question, since it is not very intuitive that a higher speed leads to a higher mass. My teacher (to my surprise (responded saying that it is a meaningless question, we don't know why, in the same way we don't know why the universe was created and those kind of philosophical questions. I, being interested in physics, couldn't believe this, I was sure that what he said wasn't true. So after a while of thinking I responded saying: _Can't we describe it with Einstein's $E=mc^2?$ If an object gains speed, he gains more (kinetic) energy. With this equality we see that the more energy an object gets, to more massive it becomes._ He then replied saying that this formula is used for different cases, whereupon he gave a vague explanation as to when it is used. He gave me an example to show what I said was incorrect; when a car goes from $10 m/s$ to $40m/s$, according to what I said we would see a big increase in mass, and we don't (this sounded logical to me). So here I am, with the following questions:   * Why does an object with a higher speed have more mass (than the same object with a smaller speed)?   * When is $E=mc^2$ used and why is my argument incorrect in explaining this phenomenon?


title: How to tell if QM is really random?

text: Given a stream of random binary numbers(*) Is there any way to differentiate if they came from a Truly Random or from a formula/algorithm ? how? if there is no way to decide this, then, I can't find any basis, to keep denying that behind the "truly random" of quantum mechanics can be a hidden algorithm. I know I am talking about the posibility of a "hidden variables" theory, but I can't find any other explanation. (*) Is known that is possible to create computable normal numbers from which first is possible to extract an infinite random binary stream, second, it tell us that a finite logic expression (again a binary stream) can contain a rational, and even an irrational number, so there is no big difference within a bit stream and the measurement of a random outcome, (and even less if we consider the accuracy) I say this, because the argument that obtaining bits is not the same as answer the complex question that can be made to quantum experiments, I think that "random" results can be perfectly read from a random stream of bits


title: Do magnets lose their magnetism?

text: I recently bought some buckyballs, considered to be the world's best selling desk toy. Essentially, they are little, spherical magnets that can form interesting shapes when a bunch of them are used together. After playing around with these buckyballs for a while, I wondered: "Can these guys ever lose their magnetism?" Then I went a step further and thought, "How are magnets affected by the 2nd law of thermodynamics?" So, how are magnets affected by the 2nd law of thermodynamics? Do they break down and lose their magnetism over time (like iron rusts over time)?


title: Are negativity of the Wigner function and quantum behaviour equivalent?

text: _I've read the following question:_ Negative probabilities in quantum physics _and I'm not sure I understand all the details about my actual question. I think mine is more direct._ It is known that the Wigner function can become negative in certain region of phase-space. Some people claim that the negativity of this quasi-probability distribution signifies that the system behaves quantum mechanically (as opposed to classical physics, when probabilities are always positive). Apparently, there are still some controversies about this point. Please read the answers from the previously cited post: Negative probabilities in quantum physics **I would like to know whether there is an equivalence between the negativity of the Wigner distribution and some quantum behaviours or not.** Is it still a question under debate / actual research or not ? My main concern is that there are more and more experimental studies of the Wigner function (or other tomography captures) reporting negativity of the Wigner function. I would like to understand what did these studies actually probe. As an extra question (that I could eventually switch to an other question): What is the _quantum behaviour_ the negativity of the Wigner function may probe ? _Having not a lot of time at the moment, I would prefer an explicit answer rather than a bunch of (perhaps contradictory) papers regarding this subject. But I would satisfy myself with what you want to share of course :-)_


title: Reduced graphs and pinch-singular surfaces

text: I am reading a book on perturbative QCD by John Collins. In Chapter 5, the terms **_reduced graph_** and **_pinch-singular surface_** are used for the analysis of mass singularities. However, their meanings are not explainded very clearly. It says (Sect 5.1.4, page 91) > The PSSs (for the physical region, which is all that concerns us) are where > the on-shell propagators and momenta correspond to classically allowed > scattering processes treated in coordinate space. I cannot get the key points from this _definition_. For reduced graphs, it provides some examples. But I still cannot understand why they are important and how they are related to the Feynman diagrams. Could someone help me to clarify these terminologies? Many thanks!


title: Are there formulae for calculating stellar luminosity and effective temperature as a function of age?

text: > **Possible Duplicate:**   >  Are there formulae for calculating stellar luminosity and effective > temperature as a function of age? Is there a manageable formula or set of formulas or simple algorithms that approximate stellar luminosity and effective temperature (or radius) as a function of stellar age? I'm aware that accurate modeling of these attributes is complex and is determined by many factors; what I'm looking for is something that serves as a decent approximation of the sort used in numerous illustrations or applets that show example "paths" taken by representative stars as they trace their evolution across the H-R diagram. (From this original question posted on Physics.)


title: Does the ensemble of effective Lagrangians in the String theory landscape mostly include gauge theories?

text: String theory false vacua can be described by effective Lagrangians at low energy. Is there generally a correspondence between these effective Lagrangians and SU(N) gauge theories? Or do the effective Lagrangians often not respect local invariance with respect to some or any gauge groups?


title: How does a hole's size affect the distance that water will squirt

text: I took a bucket, drilled 2 different sized holes on the side near the bottom and filled it with water. The stream of water the proceeded from the larger hole traveled further than the stream from the smaller one. How does the size of the hole affect the distance that the water travels? See also: Torricelli's Law.


My teacher mentioned that field line density = no. of lines / area and the total area of a sphere is $4\pi r^2$ and so an electric force is inversely proportional to $r^2$. Actually, why can the total area of the sphere be applied to this case and is this true? How does one come up with the Coulomb's law?


title: Ball rolling down an inclined plane going in to a loop

text: I apologize if this question is not up to par. When I was doing exercises in basic mechanics I checked the answers and I can't seem to find what I'm doing wrong. Suppose we have a ball with mass $m$ and radius $r$ on an inclined plane with height $h$. At the end of the inclined plane is a loop with a radius of $R$ and we can assume that $r<<R$. We are asked what the minimum height is the inclined plane should have so as to let the ball complete the loop. Here is my reasoning: We have $U_1\geq K_{rot,2}+K_{trans,2}+U_2$. For the ball rotating we have $I=\frac{1}{2}mr^2$ and $\omega=\frac{v}{r}$. So $mgh\geq \frac{1}{2}I\omega^2+\frac{1}{2}mv^2+mg\cdot2R=\frac{3}{4} mv^2+mg\cdot 2R$. The minimal speed to complete the loop implies $F_{centripetal}=\frac{mv^2}{R}= mg$ So $v^2= Rg$ and we have $gh\geq \frac{3}{4}Rg+g\cdot 2R$ which means $h\geq 2\frac{3}{4}R$, while the book says that the answer should be $h\geq 2.70R$. Can you explain what I am doing wrong? Thank you **EDIT:** Moment of inertia corrected. For the ball rotating we have $I=\frac{2}{5}mr^2$ and $\omega=\frac{v}{r}$. So $mgh\geq \frac{1}{2}I\omega^2+\frac{1}{2}mv^2+mg\cdot2R=\frac{7}{10} mv^2+mg\cdot 2R$. After the edit with correction of the moment of inertia I am getting the right answer. We get $gh\geq\frac{7}{10}Rg+g\cdot 2R$ so $h\geq2\frac{7}{10}R$ in accordance with the book.


title: Rigid body dynamics of tossing of a coin

text: While tossing a coin, it is commonly experienced that you get a head, if you toss it up with the head side up, and a tails if you toss with the tails side up. Is there a mathematical proof of this using classical mechanics? I would like to see a simple model of the coin as a symmetric top, and consider the precision of the body axis of symmetry about the angular momentum.


title: Force on an electron at an arbitrary point inside a cylinder (electron gun)

text: I want to simulate the path of an electron through the anode of an electron gun. I therefore need to calculate the force on the electron due to the electric field from the anode and apply that to its existing velocity. The electron can start at any arbitrary point $p$ with velocity $v$ (both vectors). I guess that I need to integrate the force on the electron from the whole inside surface of the cylinder. I'll be doing a numerical integration. I'm struggling to find the appropriate vector equations. I can see that $F = q\frac{V}{d}$ but all these terms are scalar not vector quantities.


title: Isn't it incorrect for the minimal gauge coupling and related calculations in Prof. Ezawa's book on quantum Hall effect?

text: He is CORRECT. I use $\mathbf{B}=\left(0,0,B_{\perp}\right)$ and he use $\mathbf{B}=\left(0,0,-B_{\perp}\right)$. $B_{\perp}>0$. # Nov.28.2012 Basically I got mad with conventions. 1.Here is the link of the book (second edition): http://books.google.fr/books/about/Quantum_Hall_Effects.html?id=p3JpcdbqBPoC Here is another link for one of his review articles: http://iopscience.iop.org/0034-4885/72/8/086502 2.I am not happy with the negative sign of the commutator $\left[X,Y\right]=-il_{B}^{2}$. Here is my calculation: $\left[X,Y\right]=\left(\left[x,p_{x}\right]+\left[-p_{y},y\right]\right)/eB+\left[-P_{y},P_{x}\right]/e^{2}B^{2}=il_{B}^{2}$ 3.In my calculation, I used some conventions different from Prof.Ezawa's book and article. Here is my convention: $X:=x-P_{y}/eB\;;\; Y:=y+P_{x}/eB$ **However** , Prof.Ezawa use this convention: $X:=x+P_{y}/eB\;;\; Y:=y-P_{x}/eB$ To be prepared for being driven mad, please compare them carefully. 4.I think he must be wrong somewhere, for example, in his book (2nd.ed), (10.2.5) and in his article (2.15) $\left[P_{x},P_{y}\right]=i\hbar^{2}/l_{B}^{2}$ but you know we use minimal coupling $\mathbf{p}\rightarrow\mathbf{p}-\frac{q}{c}\mathbf{A}$ in this problem, $q=-e,e>0,{\mathbf{p}+\frac{e}{c}\mathbf{A}}$ for electrons, as Prof.Ezawa suggested in his book (10.2.3) and his article (2.12). Under this convention, I calculated $\left[P_{x},P_{y}\right]$ as following: $\left[P_{x},P_{y}\right]=-i\hbar e\left(\left[\partial_{x},A_{y}\right]+\left[A_{x},\partial_{y}\right]\right)/c=-i\hbar eB/c=-i\hbar^{2}/l_{B}^{2}$ Oh my god here is a negative sign. 5.To summarize, I think if we take Prof.Ezawa's convention and apply his result for $\left[P_{x},P_{y}\right]=i\hbar^{2}/l_{B}^{2}$ during the calculation of $\left[X,Y\right]$, we will get his result. But his result for $\left[P_{x},P_{y}\right]=i\hbar^{2}/l_{B}^{2}$ seems not correct. 6.Someone save my day... ![No!](http://i.stack.imgur.com/yNnWs.jpg)


title: Travel time in elliptical orbit around the sun

text: Hi I'm calculating a problem, and I think it ought to be correct - but the result I get isn't. The problem is described as follow: > A spacecraft is traveling from Earth to Venus in an elliptical, but NON- > Hohman transfer orbit, starting at the apocentre at 1 AU. The semi-major > axis of this elliptical orbit, a = 0.75 AU. After a few months the orbit of > the spacecraft crosses the orbit of Venus. What is the “exact” travel time > to this crossing point in days? With some numbers (which I used for calculations): $$\mu_s = 1.33 \cdot 10^{11}$$ $$R_{venus} = 0.72 AU$$ $$1 AU = 150 \cdot 10^6 km$$ From the geometrics (elliptical orbit around the sun) the eccentricity can be calculated: $$a(1+e) = R_{earth} \rightarrow e = \tfrac{1}{3}$$ This then can give the true anomaly in a straight forward manner (from the range equation for an ellips): $$r = \frac{a(1-e^2)}{1+e\cos(\theta)}$$ $$\cos(\theta) = \frac{a(1-e^2)}{R_{venus}} - 1 = \tfrac{25}{27}$$ Now I'm wondering a bit if I actually take the correct angle for theta... In the rest of my calculations I used this value ($\theta = 22.2 ^{\circ}$) as the angle between the sun-apocenter & sun-orbit crossing vectors. (A schematic drawing on paper did make this "seem correct"). Now from this (and using simple schematic drawing/geometry) the following can be said (to get the eccentric anomaly): $$R_{venus} \cos(\theta) - ae = a \cos(E)$$ $$E = 0.9818 rad$$ Now the time can be calculated using kepler's equation as following: $$E - e\sin(E) = M = n(t - t_p)$$ $$n = \sqrt{\frac{\mu_{sun}}{a^3}} = 3.05 \cdot 10^{-7} s^{-1}$$ That gives $$t - t_p = t = 2421311 s = 28 d$$ However this is wrong, apparently the correct result is "72 days". What am I doing wrong? Where did I make mistakes?


title: Are the recently observed Dirac monopoles separable?

text: I just went through Observation of Dirac monopoles in a synthetic magnetic field. What exactly has been observed? More importantly, are these monopoles localized inside the apparatus (no stray monopole field lines coming out), or can they be used to create tangible monopoles? (By tangible monopoles I mean an object that is overall a monopole without any external influence. It may contain a condensate or similar inside)


title: Information about fields and superfields

text: I want some explanation of fields and superfields (types and components), and what the relationship between them and representation of a group.


title: Virtual photon description of B and E fields

text: I continue to find it amazing that something as “bulky” and macroscopic as a static magnetic or electric field is actually a manifestation of virtual photons. So putting on your QFT spectacles, look closely at the space near the pole of a powerful magnet – virtual photons! Now look between the plates of a charged capacitor – virtual photons again! But if it’s all virtual photons, how do we get the difference between a magnetic and electric field? [This question needs the tag "magnetostatics" as well].


title: What mechanisms require the use of different polarities in DC welding?

text: In gas metal arc welding, an electric arc forms between the work piece and a consumable wire, heating the work piece and also melting the tip of this consumable wire, which is continually fed into the weld. I recently found out that it is important to use the correct polarity when welding with direct current. Some electrode (i.e. the consumable wire) materials require that the electrode be positive and others that it be negative. Welding machines come with special mechanisms for reversing the polarity as necessary. Searching for why this would be the case explains, at best, that this determines whether the work piece or the wire gets the most heat. Having always perceived any such heating as entirely symmetric under polarity reversal, this puzzles me. Welding tutorials highlight the importance of polarity, as do the instructions on the welding wire and the machines themselves. What are the mechanisms that affect the weld depending on the polarity used?


title: How long does it take for a metal to reach equilibrium?

text: I wonder if there is a measure of how long a piece of metal takes to reach electrostatic equilibrium. Does it depend on piece's size? Does it depend on the amount of imbalance? Lots of websites and textbooks report "after a very short time". But how much short? Thanks a lot


title: Can we pass electricity through air?

text: Electricity: I was wondering, whether we can pass electricity through air over a distance of 100 meters or so as electricity means the flow of electrons and we have seen the discharge or movement of electrons in a cathode ray tube


title: Quantum Mechanics: Time dependence of an expectation value

text: In Griffiths's _Introduction to Quantum Mechanics_ , he says that the time dependence of an expectation value is $$\frac{d\langle Q\rangle}{dt}=\frac{i}{\hbar}\langle [H,Q]\rangle+\langle \frac{\partial Q}{\partial t}\rangle$$ And I also saw a lecture note saying that time dependence of an expectation value is: $$\frac{\partial}{\partial t}\langle \hat{Q}(t)\rangle=-\frac{i}{h}\langle [\hat{Q}, \hat{H}]\rangle$$ I know they are saying the same thing. For me the second one is quite easy to understand. But how can I understand the first one?


title: Calculate thermal conductance of thick walls of a cuboid, given material conductivity and cube size

text: Note 1: This is probably as much a maths question as physics. Note 2: I am not sure if my use of the the term conductance is correct. I want to make a kiln. The walls will be thick. I can come to two figures for the conductivity of a kiln of given dimensions, between which the actual conductivity will lie. For example: Assume: Kiln inner dimensions: 1m x 1m x 1m Kiln outer dimensions: 2m x 2m x 2m Kiln lining material conductivity: 0.5 W/mK Lower bound kiln wall conductance = MaterialConductivity * Area / depth = 0.5 * 1*1*1*6 / 0.5 = 6 W/K Upper bound kiln wall conductance = MaterialConductivity * Area / depth = 0.5 * 2*2*2*6 / 0.5 = 48 W/K **My question** : in the context of a thick-walled cuboid with consistent wall thickness, how do I calculate thermal conductance? Or more generically:   1. Can I split this up into calculations for 6x 3d trapezoids?   2. How would thermal conductance of the trapezoid be calculated? Thanks


title: Wave function collapse in system with many coordinates

text: Consider general state of a system with spin-$1/2$ $$ \psi = \frac{1}{\sqrt{2}}\left[\phi_{+1/2}(x) \left( \begin{array}{cc} 1 \\\ 0 \\\ \end{array} \right) + \phi_{-1/2}(x) \left( \begin{array}{cc} 0 \\\ 1 \\\ \end{array} \right)\right]. $$ If I created an apparatus which measures _only spin_ would that imply that the $x$-dependence of $\psi$ remains uncertain after measurement? E.g. if the system appeared to be in $+1/2$ state then the coordinate part of wave function (w.f.) didn't undergo collapse $$ \psi' = \phi_{+1/2}(x) \left( \begin{array}{cc} 1 \\\ 0 \\\ \end{array} \right) .$$ Maybe such an apparatus cannot be created (why?) or if one "part" of w.f. is measured then all other "parts" necessarily collapse too, but their values go by unnoticed? **UPD** In a more general way: I have a set of "uncoupled" observables $\alpha$, $\beta$ (say spin and isospin). If I measure $\alpha$ does that mean that $\beta$ will also take a definite value (will be measured indirectly).


Are there any hamiltonian systems without a periodic orbit? Can anyone give me an example? If such a system exists, does this fact have any implication on its quantum version?


title: Centripetal force in frame of reference of body moving In a circle

text: Suppose a body is moving in a circle about a fixed point. In the frame of reference of the body, is the centripetal force felt or is only the centrifugal force felt? More generally, does a body only feel the effect of pseudo forces in an accelerated reference frame?


title: Determination of a unique electric and magnetic field produced by a point charge

text: Let's suppose I have a point charge moving arbitrarily in space. As far I know, Maxwell's equations don't determine a unique electric and magnetic field, they only determine a family of electric and magnetic fields. What other information do I need in order to know uniquely the electric and magnetic field produced by a moving (the particle is moving arbitrarily) point charge? I'm not asking for a method for solving it, I just want to know what other information (apart from Maxwell equations) do I need.


title: Optics & Law of Cosines

text: I'm supposed to use law of cosines on $S_1S_2P$ in the following diagram that relates to a lens: ![triangle diagram](http://puu.sh/5zDtv.jpg) To arrive at the following equation: $$ \frac{r_2}{r_1} = [1 - 2(\frac{a}{r_1})sin(\theta) + (\frac{a}{r_1})^2]^\frac{1}{2} $$ I decided to work backwards from this equation, and got to: $$ r_2^2 = r_1^2 + a^2 - 2ar_1cos(\theta) $$ But I'm not sure how to proceed, or if I'm even on the right track, as I have no idea how to turn the sin into a cos. Can anyone help point me in the right direction? Thanks.


title: Is the sum of the amplitudes of an electromagnetic wave always 1?

text: It's been a while, and I'm trying to verify my understanding. I remember reasoning (but never being taught) that the sum of the (normalized) electric and magnetic waves in a single electromagnetic wave at any single point in time is always one. Like so:               Given A of em-wave is normalized (max = 1, min = -1)              abs(sum(A of m-wave, A of e-wave)) = 1      I haven't been able to find anywhere that says this specifically (I probably suck at searching, or am wrong). Am I correct in my understanding?


title: Is the total energy of earth changing with time?

text: Many years ego, Earth was hot. Over time, it has lost energy and has become colder. Is it now in equilibrium or is its total energy changing?


title: Radio antennas that are much shorter than the wavelength

text: From my limited experience with ham radio when I was a kid, I expect transmitting and receiving antennas to have lengths that are on the same order of magnitude as the wavelength, and in fact I recall having to mess around to compensate for the fact that a given antenna wouldn't be properly resonant over an entire frequency band. This also seems to match up with what we see with musical instruments, where, e.g., a saxophone's tube is half a wavelength and a clarinet's is a quarter. For commercial FM radio with a frequency of 100 MHz, the wavelength is about 3 m, so I can believe that some of the receiving antennas I've seen are a half- wave or quarter wave. But for AM radio at 1000 kHz, the wavelength is 300 m, which is obviously not a practical length for a receiving antenna. Can anyone explain this in physics terms, hopefully without making me break out my copy of Jackson and wade through pages of spherical harmonics? Does AM reception suffer from the length mismatch, e.g., by being less efficient? Does it benefit from it because it's so far off resonance that the frequency response is even across the whole band? Is there a dipole approximation that's valid for AM only? For both AM and FM? If the sensitivity is suppressed for the too-short antenna, is there some simple way to estimate the suppression factor, e.g., by assuming a Breit-Wigner shape for a resonance? This question touched on this issue, but only tangentially, and the answers actually seem inconsistent with the observed facts about AM. Also related but not identical: Radio communication and antennas


title: Quantizing the Dirac Field: which commutation relations are more fundamental?

text: When quantizing a system, what is the more (physically) fundamental commutation relation, $[q,p]$ or $[a,a^\dagger]$? (or are they completely equivalent?) For instance, in Peskin & Schroeder's QFT, section 3.5, when trying to quantize the Dirac field, they first say what commutation relation they _expect_ to get for $[\Psi(\vec{x}),\Psi^\dagger(\vec{y})]$ (where $i\Psi^\dagger$ is the conjugate momentum to $\Psi$), in analogy to the Klein- Gordon field, then they _postulate_ a commutation relation between $[a^r_{\vec{p}},a^{s\dagger}_{\vec{q}}]$ etc., and then verify that they indeed get what they expected for $[\Psi(\vec{x}),\Psi^\dagger(\vec{y})]$. Why did we need to postulate the value of $[a^r_{\vec{p}},a^{s\dagger}_{\vec{q}}]$? Couldn't we have just computed it off of $[\Psi(\vec{x}),\Psi^\dagger(\vec{y})]$? (which, by expecting to get it, we could have just as well already _postulated_ it). I suppose that would entail explicitly writing something like: $$ a_{\vec{p}}^r = \frac{1}{\sqrt{2E_{\vec{p}}}} u^{r\dagger}(\vec{p})\int_{\mathbb{R}^3}d^3\vec{x}\,e^{-i\vec{p}\cdot\vec{x}}\Psi(\vec{x})$$ and a similar expression for $b$.


title: Can we build a synthetic event horizon?

text: If we imagine ourselves to be a civilization capable of manipulating very heavy masses in arbitrary spatial and momentum configurations (because we have access to large amounts of motive force, for example), then we can imagine building ponderous rings such that light, pointed in a particular direction, will be gravitationally bent around it and will arrive back precisely where it began. This mimics the behaviour of light at the event horizon of a black hole - there is no escape. To what extent can models like this simulate a real black hole and yield real physical insight as to what lies inside?


title: The gravitational random walk

text: When we shoot a single photon out into space, the chance that it will eventually return to our vicinity from a different direction is vanishingly small, even though spatial curvature exists due to the gravitational influence of the distribution of mass energy throughout the universe. It is hard to imagine any regularity of motion for all possible photon directions in this quasi-random mass-energy structure. So lets flip this around, and, assuming a strictly random mass-energy distribution, the question can be asked as to whether any discernable common patterns exist for all possible photon trips? Such patterns need not necessarily exist in 3D space, but may exist in something like lattice k-space, or in Fourier space, or indeed any conceptual space, given the assumption of randomness of the mass-energy distribution throughout the cosmos. Another way to ask this question is to enquire as to what kind of distribution (in any space of our choosing) is to be expected?


title: When (Exactly) Does Bremsstrahlung Radiation Occur?

text: Regarding the Bremsstrahlung Radiation emitted when a charged particle curves inside a uniform magnetic field; Is there a way of calculating the angle through which the charged particle will precess before emitting the first quantum of radiation? Also, is there a way of determining the angle through which the charged particle will precess between the first and second emissions of radiation? My investigations revealed only what I think are the angles of emission relative to the tangent of the curve here.


title: Impulse on adjacent objects

text: Let's say two objects are sitting adjacent (in contact) to each other. If we start pushing one of them, we know that both the objects move, remaining in contact to each other. But let's now consider imparting an impulse to one of the objects, instead of a steady force. I expect the two objects to start moving, but this time with different velocities, and thus developing a separation between them. As a test case, let's say that we push the object which has mass $M$, and the other object has mass $m$; where $M > m$. I expect the smaller mass to move off with a higher velocity. But I can't find a mathematical description for it. Assuming that the force is large enough, we have $F dt$ as the impulse on the combination. On the mass $M$, the impulse is $(F-N) dt$, where $N$ is the contact force between the two masses. Also, $N dt$ is the impulse on the object of mass $m$. How can I calculate the velocities of the two masses, knowing that both the masses were at rest before the impulse.


title: "Lack of inversion symmetry" in crystal?

text: Apparently (first paragraph of this article) the lack of _inversion symmetry_ is some crystals allows all sort of nonlinear optic phenomena. Now. Does anyone know of an intuitive or just physical explanation as to _why_ this is the case? What does _inversion symmetry_ mean and what is so special about it?


title: Strings and their masses

text: How do strings present in particles give mass to them? Is it only by vibrating? I have been trying to find the answer but could not find it anywhere, can this question be answered?


title: Two masses in deep space - collide or orbit?

text: If two identical masses are somehow "released" into deep space (that is, they're subject to no other gravitation forces but their own, and are initially at rest to each other). What decides whether they collide or orbit each other? I'm imagining for example two 1 kg masses initially at rest, say 10 m apart. What happens next?


title: calculating of g force in fun ride

text: we vistited yesterday a fun park and saw a ride which was a big centrifuge only in vertical. this one had a speed meter but it was broken and i also wanted to know how much g-force is on there... so my first thought was to calculate this with the centrifugal-accerleration. I have $\omega = \frac {0.5*2\pi} 1 [\frac {rad} s]$ and $r= 25[m]$, so the accerleration is $a_z = \omega^2r = \pi^2*25 = ~246$ then i can calculate the g force with $g_{force} = \frac {a_z} g$ which gives me about $24g$ that seems fairly too high! what am i missing?


title: What role does electrical charge play in black holes?

text: Not having studied General Relativity, I have sometimes been puzzled by references to the behaviour for "classic" black holes -- as they are popularly portrayed -- as being true for black holes which are not rotating and which have no charge. I don't understand the role rotation _does_ play, but at least because it has to do with the motion of massive bodies, I can understand why it _could_ play an important role. But I have no such intuition for electrical charge. How does _charge_ change the behaviour of a black hole, aside from the obvious role of electrostatic force? (It will preferentially attract particles of the opposite charge, obviously.) But it would seem that charge plays a more intriguing role than just this. The current status of the Wikipedia page on black holes claims that you can theoretically avoid the singularity of a charged black hole. It also describes there that there is a theoretical upper bound on the charge/mass ratio of a black hole: that any would-be black hole exceeding it (which is generally thought to be impossible -- see this related question on trying to force saturation of the charge/mass ratio or black holes) would lack an event horizon (and therefore presumably not be a black hole). Why should that be? Furthermore: from this other related question on repulsion of pairs of charged black holes (and from Willie Wong's comment, below), ir seems that the size of the event horizon may change depending on how close it is to being extremal! Why would the event horizon of a _highly charged_ black hole be different than the event horizon of a _neutral_ black hole, of similar mass? Is there a clear reason for such an interplay is there between electrodynamics (aside from local Lorentz invariance) and general relativity that bring these things about?


title: Definite Parity of Solutions to a Schrödinger Equation with even Potential?

text: I am reading up on the Schrödinger equation and I quote: > _Because the potential is symmetric under $x\to-x$, we expect that there > will be solutions of definite parity._ Could someone kindly explain why this is true? And perhaps also what it means physically?


title: Angle of deflection of an atomic nucleus

text: when firing a proton (for example) to an atomic nucleus, from a distance $D$, the deflection angle of the proton $\alpha $ to the type of changes atomic nuclei? or always constant?


title: Gauss' law question

text: It's actually a teaching conflict at my school. They said that $$\text{Flux}=\frac{q}{\varepsilon_0}.$$ Say for a point charge at the centre of the sphere and let's say we not put water into the sphere, so now $E/ \varepsilon_0=K$. Thus $E= \varepsilon_0 K$. So flux becomes $q/\varepsilon_0 K$. That means that any change in medium changes the flux. However, in FIITJEE I was told that the addition of a medium won't change the flux since the number of electric field lines remains constant for any medium. So changing medium won't change the formula, $\text{flux}=q/\varepsilon_0$. So, does flux actually change with change in medium or not? $K$ is the dielectric constant.


title: Waves produced by a mass of water on the calm sea surface

text: Suppose we have a parallelepiped shaped box full of water on the surface of the sea. Suddenly the box disappears. What is the shape of the waves vs. time caused by the fall of the water contained in the box?


title: Mathematical derivation of interference pattern for electrons?

text: One of the most famous experiments in quantum mechanics in the context of wave-particle duality is certainly passing a beam of electrons through two slits, which results in an interference pattern being formed on a screen positioned after them. Now, **starting from the Schrödinger equation** (which we know is obeyed by electrons), how can we derive the formula describing the interference pattern? I am pretty sure we already know the answer, i.e. it is the same pattern generated by light going through two slits (so a sine squared modulated by a sinc envelope), but how do we get it **_from quantum mechanics_** ?


title: How would Einstein's later years have been different with modern computers?

text: This is a historical question partly, and maybe too broad for this site, but would require some familiarity with modern physics research practice so hopefully appropriate here. Einstein's later years, after general relativity, were (as I understand it) taken up predominantly by his effort to find a unified theory that explains both gravitation and electromagnetism and their interrelation. And from what I understand his approach was mainly mathematical, trying variations of functions and so forth, looking for something that fit - and doing his work predominantly on paper. Had he access to modern computers - programming languages, mathematica, etc. - how might that have changed his progress, and willingness to give up perhaps? And in what ways have computers in general changed / revolutionized theoretical & mathematical physics?


title: Stationary points of the action functional

text: In QFT the principle of stationary action states that we choose fields that will make the action stationary but what if the action has many stationary points? What's the significance of these other solutions?


title: What's the relationship between the energy density of a black-body and its radiant exitance?

text: Through a bit calculation we can derive that in a cavity, the energy density $$u(f,T)=\overline{E(f)}\times G(f)=\frac{8\pi h}{c^3}\frac{f^3}{e^{h\nu /kT}-1}$$ If we take the integral over all frequency, we get $$U(T)=\frac{8πh}{c^3}\frac{(kT)^4}{h^3}{\frac{π^4}{15}}=C_{onst}T^4$$ And Stefan-Boltzmann Law claims that for a perfect black-body $$j^*=\sigma T^4$$ where $j^*$ is the radiant exitance, which is defined as the total energy radiated per unit surface area of a black body across all wavelengths per unit time. And it just so happens that $\frac{\sigma}{C_{onst}}=\frac{c}{4}$, why is that? P.S. The professor told me to refer to some thermodynamics book, where a more general case is discussed. But we don't have that book in our library and the professor's now out of town xD.