title: Why does the water diffuse at right angles in my coffee?

text: I purchase an espresso shot each morning at my local coffee shop, and I usually water it down slightly to take the edge off as well as to cool it. Recently, I have noticed that when I pour the cold water in, it seems that the surface of my espresso is suddenly disturbed in a lattice-like configuration \begin{align*} \nearrow \searrow \nearrow \searrow \\\ \nwarrow\swarrow\nwarrow\swarrow \end{align*} i.e., water moving up at a $45^\circ$ angle and down at a $45^\circ$ angle along the surface. Googling `water diffusion in coffee` has been fruitless unfortunately. Any thoughts? Regards, Zvpunry.


title: When can i separate spin from the wavefunction?

text: I am currently working on a Tight-Binding model and for the derivation of nearest neighbor spin interactions I have terms like $$ \left<x\uparrow\right|\eta\;\mathbf{l}\cdot\mathbf{s}\left|z\downarrow\right> $$ where $\left|x\uparrow\right>$ is an atomic orbital.For the simplification it is very important that i can assume $$ \left|x\uparrow\right> = \left|x\right>\left|\uparrow\right> $$ My question is:   * is this always possible in a solid body or do i neglect some effects with that? (like spin polarization or something?)


title: Rabi oscillation, electron in a box

text: This page on Rabi oscillation says the Hamiltonian has an interaction term $d \cdot E(t)$, where $d = -$e$r$. (I'm sure what they mean is, that's what the off-diagonal terms of the Hamiltonian look like). What is $r$? Is it the vector pointing from the (average) position of the electron while in the ground state to the (average) position of the electron while in the excited state? Or is it the vector pointing from the average position of the electron (averaged over both the ground and excited state?) and the nucleus? I'd like to understand how Rabi oscillation* would work with an "electron in a box."** However, if the former interpretation of $r$ is correct, the average position for both the ground state and excited state are just the center of the box. If it's the latter, OTOH, then there's no positive charge to point to, just the electron. So it seems like either way, there's no interaction between the particle and the field. Is this actually the case? *What I mean is, if you apply an electric field to an electron the same way one does in the Rabi model. **I know that the particle-in-a-box Hamiltonian has countably many eigenstates, and the Rabi model is supposed to be for a two-state system. I guess I'm assuming if the particle begins in the ground state, the evolution will keep amplitude mostly in the ground state and first excited state, and the higher-energy states can be ignored.


title: Definition and motivation for Twistors

text: What are Twistors? Why are they important? This particular statement in Wikipedia is intriguing > _According to Andrew Hodges, twistor space is useful for conceptualizing the > way photons travel through space, using four complex numbers._


title: Adiabatic filling of a container

text: Suppose a thermally insulated container is filled with atmospheric air until the pressure reaches 5000 psi. This could represent the filling of a diving cylinder, before thermal dissipation becomes significant. Initially, then, some mass of air has atmospheric temperature $T_1$ and $p_1$. When forced in to the tank adiabatically, we expect its final temperature to be \begin{align} T_2 &= T_1\left(\frac{p_2}{p_1}\right)^{1-1/\gamma} \\\ &= (300\,\mathrm{K})\left(\frac{5000\,\mathrm{psi}}{14.5\,\mathrm{psi}}\right)^{1-1/1.4} \\\ &\approx 1590\,\mathrm{K} \end{align} This final temperature seems far too hot. A diving cylinder would likely melt at this temperature. More likely, though, something is wrong with my reasoning. Is this not the correct use of the temperature-pressure relations, assuming an ideal gas? If not, what other information is needed to predict the final temperature of the container?


title: Bigger anti-matter particles

text: I have learned about the existence of positrons as a decay product from uranium fission - if I'm not mistaken. Is there any evidence for higher 'mass' anti-matter, or is that mere speculation or rationalisation? i.e. Is the argument that there anti-matter based planets in the universe, or just a soup of positrons?


title: quantum mechanics. Electromagnetics

text: In electromagnetics the intensity of a wave is calculated taking the squared of its amplitude. What is the reason why in quantum waves this cannot be applied to calculate it?


title: What would we observe from thompson's lamp thought experiment?

text: The thought experiment goes like this: > Say there is some circuit which turns a lamp on/off with just a flick of a > switch. Say its off; you flick it, it turns on; flick it again it turns off, > and so on. > > So say you are conducting the experiment for two minutes. When the remaining > time halves, you flick the switch; and when that remaining time halves, you > flick again...This process goes on infinitely, until two minutes is reached. Mathematics says that the sum of all those remaining half-times will eventually tend to two minutes. And the outcome i.e. the final state is always unpredictable. We can't really know if its on/off. Lets not concern about the final state. What would we actually observe towards the end, few micro seconds towards the two minute mark? Wouldn't the lamp appear turned on? What would happen if the time between two flicks tend to zero?


title: What is the energy of a black hole?

text: This might be a stupid question but given Einstein's general theory of relativity $E = m c^{2} $ what is the energy of a black hole? Isn't the mass of a black hole infinite? Wouldn't that be infinity multiplied by the speed of light squared?


title: A problem on fluid flow

text: I am extremely weak in visualizing physical problems in mathematical context. Please help me in solving the following problem and please give as much details as possible. A fluid flows radially (& isotropically) outward from a faucet of radius a into a 2‐D plane at a volume flow rate of $\dot{V}=\alpha$ and it is confined to a thickness $m$. Find the vector velocity function $(\vec{V}(x,y))$ in the 2‐D plane. Express this vector field in cylindrical‐polar and Cartesian coordinate systems


title: Special Relativity: What differential equation describes an accelerated object from a non-inertial reference frame?

text: I am looking for a set of differential equations (to be solved numerically for an educational program) that would describe the position and apparent time of an accelerated clock relative to a non-inertial reference frame. For now, all movement is along a single spatial dimension. I am _not_ interested in considering the time light takes to travel between observers; assume that they are clever enough to account for this delay. As inputs to this system of equations, I would have:   * An arbitrary, time-dependent acceleration a(t) as experienced by an observer in the non-inertial frame. In practice, this would be a simple list such as a(t=0) = 0; a(t=1) = 0.2; ...   * The initial position x(t=0), clock time t'(t=0), and velocity v(t=0) of the accelerated clock as seen in the non-inertial reference frame   * Another time-dependent acceleration a'(t') which describes the acceleration experienced by an observer traveling with the clock. To make this (hopefully) more clear, I'll give an example: an observer A, initially at rest, sees a relativistic rocket pass by carrying observer B. When A's watch reads t=0, she notes that B's position is x(t=0), B's speed is v(t=0) and B's (prominently displayed) clock reads t'(t=0). A has a list of instructions a(t) which tell her to set her rocket engines to achieve a specific acceleration at specific times. B has a similar set of instructions a'(t'). Both A and B consult their _own_ watches when determining when to change their acceleration. I have seen this question: What is the displacement of an accelerated and relativistic object?, which tells me enough to model the situation for an _inertial_ observer, but I have not figured out how to reapply this for a non- inertial observer. Similarly, I can compute the worldlines of non-accelerated objects from the perspective of a non-inertial observer, but it seems something sneaky happens when both the observer and observed are accelerating. I have also tried computing both worldlines from an inertial reference frame and then using the Lorentz transformation [x' = g (x - vt); y' = g (t - vx/c^2)] to ask what B's worldline would look like from A's perspective, but this did not work (I can describe more about this, but for now it should suffice to say that the worldlines were not remotely correct).


title: what general physical arguments discredit induced emission of nuclei?

text: trying to read a bit about "quantum nucleonics", i've encountered this rebuttal from people at Livermore that induced radiation stimulation of Hafnium nuclei is possible. Something that catches my eye is the phrase "The earlier reports were also very unlikely, in any case, on very general physics arguments". **Question:** what 'very general physics arguments' may have referred the author?


title: Drop a star in a river

text: I (as I am sure you will be able to tell) know very little about nuclear physics, so please understand if something here seems dumb or obvious. I am going to question some Hollywood physics here. At the end of Spiderman 2, Octavius tells Spiderman to drop what essentially amounts to a tiny star into the river in order to extinguish it and prevent it from destroying Manhattan. Ignoring the problems of properly containing the fusion reaction on earth, getting it to begin in the first place, and all the other problems with fusion as an energy source that scientists are currently trying to sort out, I am curious about only this ending. As I understand it (and feel free to correct me if I am wrong), fusion works by, at very high temperatures, taking two hydrogen atoms and fusing the nuclei together, creating one helium atom. There are two ways I can think that submerging a star in water could "put out" the star:   1. cooling the star to the point that the fusion reactions no longer take place   2. cutting off the star's supply of fuel (hydrogen) Since I don't know much about nuclear physics, I'm just curious whether either of these would "extinguish" the star, or whether there is another factor I am not considering. As a follow up, I am also curious what the consequence of extinguishing the star might be. would there be a tiny supernova, would the star collapse, or would there be some other kind of death? You can get an idea of the size of the star from this video. **EDIT** As Ali points out in the comments, a star that small probably could not exist at all. For the sake of discussion, I am amending the question as such: Imagine the star in question is the mass and size of our Sun, and that somehow, somewhere, there is a body of water large enough to submerge the Sun completely.


title: Total scattering rate of light for an atomic transition given a laser field with an intensity distribution?

text: I have a laser with an intensity distribution $I(\lambda)$ over a broad wavelength range (arroun 40 nm), I want to calculate the total scattering rate due to the full spectrum of the laser; the the scattering rate is given by $\gamma_p(\lambda) = \gamma \rho_{ee}(\lambda) = \frac{\gamma}{2}\cdot\frac{A(I(\lambda))}{1+A(I(\lambda))+B(\lambda)}$ where $\gamma$ is the linewidth of the atomic transition and $\rho_{ee}$ is the population in the excited state. I was directly integrating $\int\gamma_p(\lambda)d\lambda$, but the result makes no sense. Any idea?


title: Do brick walls effectively protect against ionizing radiation from a nuclear explosion?

text: Of course it depends on the distance and the amount of radiation, so let's say its about 10,000 Rad. Now, can wall made of brick protect someone behind it from such radiation ?


title: What would be the effect on gravity if space expanded at > $c$?

text: If space were to expand at > $c$ (as in inflation) would that mean gravity would no longer have any effect on the curvature of space, since gravity can only propagate at $c$?


How to compute the change in entropy of s system when the temperature is decreasing? Such a question may be too general,then we may ask a special case:How to compute the change in entropy of Annealing system when such a process may be regarded as Markov Chain ?


title: Entropy and Big Bang aren't mutually exclusible?

text: There are some principles in entropy im having trouble understanding. If Entropy requires that the energy in the universe must always been constant, shouldnt in theory 'Heat Death' have occurred since forever ago even before the Big Bang? Why not? Thank you! [UPDATE] Entropy and energy conservation are laws often used to predict the ultimately fate of the universe. Trough achieving thermal equilibrium and dissipation of mechanical energy the universe would become in a 'Heat Death' state where nothing would really occur. The basic principles of this theory, as I understand, is that: (1) Energy cannot be created (nor it is infinite); and (2) the Universe is a closed system; The problem I see is that there must be a hidden number (3) requisite: Time must be finite, for this to work properly. I cant aknowledge time as being finite because then nothing could have started 'time', (you would need to have a external process [and a external time] and acknowledge that the system is not closed). If time is infinite, then there was an infinite time before the Big Bang, then, in my understanding, the Big Bang couldn't possibly exist and have formed. Because in infinite time 'Heat Death' would always be what there is. This wouldn't suppose that one of the 2 basic principles is wrong (or energy can somehow be created or the Universe is not closed), and Entropy really doesnt apply to the universe at all? Another explanation I see is that if the principles are true, then in fully understandment of all Universe laws Entropy itself must be wrong (the system would never achieve thermal equilibrium or a 'final state'). Either way all propositions I see for the problem achieve that Entropy must be somehow flawed in some level. What are your toughts in this? Im not a physicist and if something Im stating is wrong please enlight me!


title: Does effective potential for a gravitational force have a maximum below $E=0$?

text: The relevant figure is below (taken from Goldstein's Classical Mechanics). This figure plots the effective potential for a gravitational force. Does the effective potential $V'$ go flat below $E_2=0$? After finding $r_{flat}$, the point where the effective force $f'=0$ (or equivalently, where $V'$ goes flat), I got $$V'(r_{flat})=-\frac{mk^2}{2l^2}$$ Now, it looks like this is expression is negative since $m$,$k$, and $l$ are all positive. So my question is: Doesn't this result in a possible parabola that could have an energy less than $E_2=0$? I know it isn't possible for a parabola to have negative energy, so where am I going wrong in my reasoning? Thanks.. ![enter image description here](http://i.stack.imgur.com/0EbPo.jpg)


Consider quantum Hamiltonian of free massive scalar particle: $$\hat{H} = \int d^3x \left[\frac{1}{2} \hat{\pi}^2 (t, \vec{x}) + \frac{1}{2} \partial_i \hat{\phi}(t, \vec{x}) \partial_i \hat{\phi}(t, \vec{x}) + \frac{m_0^2}{2} \hat{\phi}^2(t, \vec{x}) \right].$$ I'd like to show that Heisenberg evolution equation for $\hat{\phi}$ holds, i.e.: $$\partial_t \hat{\phi} (t, \vec{x}) = i [\hat{H}, \hat{\phi}(t, \vec{x})].$$ My problem is to compute the following term: $$\int d^3 y \left[\partial_i \hat{\phi}(t, \vec{y}) \partial_i \hat{\phi}(t, \vec{y}), \hat{\phi}(t, \vec{x})\right].$$ I would appreciate some help on how should I rearrange this commutator to see it vanishes.


title: How is a Qubit in two states under Superposition?

text: I have read a little about Quantum computing. From what I understand, Quantum Superposition is when a qubit is in a state $\alpha|0\rangle$ + $\beta|1\rangle$, where $\alpha$ and $\beta$ are probability amplitudes. Now when we try to measure or see the qubit it collapses to a 0 or 1. Why don't we consider that the electron is in either the 0 or 1 state in superposition but we don't know which state because we haven't measured it yet. I'll give you an example. I ask a friend to go inside a closed room and flip a coin. Now I know that it's either heads or tails with equal probability. So, can I say that the coin is in the superposition state $\sqrt0.5|heads\rangle$ + $\sqrt0.5|tails\rangle$ ? I then ask my friend to dope the coin so it has a higher chance (0.75) of falling heads up. So, can I say that the coin is in a superposition $\sqrt0.75|heads\rangle$ + $\sqrt0.25|tails\rangle$ ? Then I go inside the room and suddenly I see the coin. So can I say it's state collapsed to a heads or tails because I measured it?


title: Is Feynman's explanation of how the moon stays in orbit wrong?

text: Yesterday, I understood what it means to say that the moon is constantly falling (from a lecture by Richard Feynman). In the picture below there is the moon in green which is orbiting the earth in grey. Now the moon wants to go at a tangent and travel along the arrow coming out of it. Say after one second it arrives at the red disc. Due to gravity it falls down toward the earth and ends up at the blue disc. The amount that it falls makes it reach the orbital path. So the moon is constantly falling into the orbital path, which is what makes it orbit. ![enter image description here](http://i.stack.imgur.com/rtYG4.png) The trouble I'm having is: shouldn't the amount of "fall" travelled by the moon increase over time? The moon's speed toward the earth accelerates but its tangential velocity is constant. So how can the two velocities stay in balance? This model assumes that the moon will always fall the same distance every second. So is the model wrong or am I missing something? Extra points to whoever explains: how come when you do the calculation that Feynman does in the lecture, to find the acceleration due to gravity on earth's surface, you get half the acceleration you're supposed to get (Feynman says that the acceleration is $16 ~\mathrm{ft}/\mathrm{s}^2$, but it's actually twice that).


title: Conservation of Linear Momentum at the point of collision

text: This is a pretty basic conceptual question about the conservation of linear momentum. Consider an isolated system of 2 fixed-mass particles of masses $m_1$ and $m_2$ moving toward each other with velocities $v_1(t)$ and $v_2(t)$ respectively. Now conservation of momentum says that at any point during the particles' motion the quantity $$m_1v_1(t) + m_2v_2(t) =constant$$ With non-zero velocities and non-zero masses this constant will be **non- zero**. Let us say the particles collide at time $t_0$. At the point of collision, both particles have velocity **zero**. which would mean that the constant above will be zero. Contradiction. I realize I might be going wrong in my reasoning at the point of collision. In fact, I feel defining velocity at that point would not even make sense, since if one considers the displacement functions $x_i(t)$ $i=1,2$ of the particles, then $t_0$ would represent a point of **non-differentiability** of $x_i(t)$ for $i=1,2$. So **assuming** there are no collisions, by following the text-book derivation I can see why $$m_1v_1(t) + m_2v_2(t) = C_1$$ before the collision and $$m_1v_1(t) + m_2v_2(t) = C_2$$ after the collision would hold true, but not why $C_1=C_2$ Can someone help me in clearing this up?


title: Lorentz invariance of the action for free relativistic particle

text: I tried to check the Lorentz invariance of the standard special relativity action for free particle **directly** : ($c=1$) $$ S=\int L dt=-m\int\sqrt{1-v^{2}}dt $$ Lorentz boost: $$ dt=\frac{dt^{'}+udx^{'}}{\sqrt{1-u^{2}}}$$, $$dx=\frac{udt^{'}+dx^{'}}{\sqrt{1-u^{2}}}$$, $$v=\frac{dx}{dt}=\frac{u+v^{'}}{1+uv^{'}}$$ Substitute these expressions into the action: $$ S^{'}=-m\int \sqrt{1-\left(\frac{u+v^{'}}{1+uv^{'}} \right)^{2}} dt=-\int \sqrt{1-\left(\frac{u+v^{'}}{1+uv^{'}} \right)^{2}} \left(\frac{dt^{'}+udx^{'}}{\sqrt{1-u^{2}}}\right) $$ $$ S^{'}=-m\int \left(\frac{\sqrt{1-\left(\frac{u+v^{'}}{1+uv^{'}} \right)^{2}}}{\sqrt{1-u^{2}}} \right)dt^{'}-m\int \left(\frac{u\sqrt{1-\left(\frac{u+v^{'}}{1+uv^{'}} \right)^{2}}}{\sqrt{1-u^{2}}}\right)dx^{'} $$ But I know that action of this type is invariant under lorentz transformations, so we want to get that $$ S^{'}=-m\int \sqrt{(1-v'^{2})} dt^{'} $$ But it doesnt seem to me that these expressions for action are equal. So I'm confused. Maybe there is a mistake in my calculations?


title: Energy in the electric field of a translating charged sphere

text: The energy stored in the electric field of a uniformly charged non-conducting shell moving with a uniform velocity is more than the energy stored in the electric field of the same shell at rest. What is the source of this extra energy?


title: Modern avatar of Englert's solution?

text: Bosonic solutions of eleven-dimensional supergravity were studied in the 1980s in the context of Kaluza-Klein supergravity. The topic received renewed attention in the mid-to-late 1990s as a result of the branes and duality paradigm and the AdS/CFT correspondence. One of the earliest solutions of eleven-dimensional supergravity is the maximally supersymmetric Freund--Rubin background with geometry $\mathrm{AdS}_4 \times S^7$ and 4-form flux proportional to the volume form on $\mathrm{AdS}_4$. The radii of curvatures of the two factors are furthermore in a ratio of 1:2. The modern avatar of this solution is as the near-horizon limit of a stack of M2 branes. Shortly after the original Freund--Rubin solution was discovered, Englert discovered a deformation of this solution where one could turn on flux on the $S^7$; namely, singling out one of the Killing spinors of the solution, a suitable multiple of the 4-form one constructs by squaring the spinor can be added to the volume form in $\mathrm{AdS}_4$ and the resulting 4-form still obeys the supergravity field equations, albeit with a different relation between the radii of curvature of the two factors. The flux breaks the $\mathrm{SO}(8)$ symmetry of the sphere to an $\mathrm{SO}(7)$ subgroup. My question is whether the Englert solution has a modern avatar, perhaps as the near-horizon limit of some solution.


title: Fluctuations of an interface with hammock potential

text: This question is related to that one. I ask it here since comments are too short for the extended discussion that was going on there. I am interested in a very simple interface model. To each $x\in\mathbb{Z}^2$, we associate a random height $h_x\in\mathbb{R}$. Let $\Lambda_N=\\{-N,\ldots,N\\}^2$. Assume $h_x\equiv 0$ outside $\Lambda_N$. To a pair of neighbouring heights, we associate an energy $0$ if $|h_x-h_y| < 1$ and an energy $+\infty$ otherwise. We then consider the corresponding Gibbs measure. In other words, we put the uniform measure on height configurations satisfying $|h_x-h_y| < 1$ for all pairs of neighbouring vertices, and equal to $0$ outside $\Lambda_N$. It is an open problem to prove that the variance of $h_0$ diverges as $\log N$, as $N\to\infty$ (actually, it's even open to prove that it diverges at all!). On the other hand, it is known to hold, if one replaces $+\infty$ by a suitable function diverging outside the interval (fast enough to guarantee existence of the measure, of course). Obviously, one cannot take the limit in the known arguments... **My question:** What are _quantitative_ heuristic arguments implying such a claim. By quantitative, I mean that I don't want something like "by analogy with the discrete massless free field", because I already know that ;) . I'd really like a non-rigorous, but mathematical derivation. **Update (April 27, 2014):** two colleagues have been able to (rigorously) settle this question in a slightly different geometry (periodic boundary conditions, the spin at the origin forced to be 0). Their preprint can be found here: arXiv:1404.5895. Nevertheless, I'm still intertested in good physical heuristics.


title: Depolarizing threshold for CSS codes

text: Many years ago, when CSS codes were first invented, the error threshold of p=0.11 was found when bit and phase flips are independent. Has a threshold yet been found for the case of depolarizing noise?


title: How do we determine the statistics and spin of quasi-particles?

text: I am considering the Heisenberg XXZ model at the moment. In the literature it says that (in the $J\Delta\rightarrow\infty$ limit, i.e. the ferromagnetic Ising regime) one can either view low-energy excitations as magnons (single overturned spins), or as domains of consecutive flipped spins, named spinons. The latter is supposed to have spin $S=1$. My question is two-fold: -How do we easily note the statistics of the introduced particle and what spin it has? -When do we choose what quasi-particle?


title: What is the deepest man-made underground tunnel?

text: ![http://bm.img.com.ua/img/prikol/images/large/4/0/258204_603914.jpg](http://i.stack.imgur.com/Yn0Jn.jpg) What is the deepest tunnel in the Earth's surface? I have meet numbers, around 300km. **What is the main problem to make tunnel more deeper? What is the problem with 1000km?**


title: Hydrogen atom in partial differential equations, show it is is independent of time.

text: For the hydrogen atom, if $\int|u|^2\ \mbox{d}x=1$ show that the same is true at all later times. Hint: Differentiate the integral with respect to t, taking care about the solution being complex valued. assume that $u$ and $\nabla u\rightarrow \infty$ fast enough as $|x|\rightarrow\infty$.


title: Angle needed for object A to intercept with object B

text: Object B is 15 degrees East of North at a distance of 20km/h. Object B is moving at an average speed of 30km/h in the direction 40 degrees East of North. If object A is capable of moving at 100km/h, at what angle does it need to move to intercept object B? ![obligatory sketch](http://i.stack.imgur.com/O2qUC.jpg) I've tried calculating it as vectors, but got stuck. In the end I just tried using good old geomatry, where $$ \frac{\sin A}{a} = \frac{\sin B}{b} $$ So I called the angle I'm searching for A, and the angle from the line between A and B, and the direction of object B, as angle B. 'a' and 'b' are the average speeds of object a and b. $$ \frac{\sin A}{30 \times t} = \frac{\sin(155°)}{100\times t} $$ $$ A = arcsin\left(\frac{30\times t\times \sin(155°)}{100\times t} \right) = 7.284° $$ So the final angle should be $15°+7.284°=\mathbf{22.284°}$ Is this correct? Isn't there a better way of doing this, like using vector products? PS. I've found a very similar question here, but didn't quite understand how i.e. $\alpha + \beta = 40° $, because my calculations show $ =35° $. And trying to read through comments and answers, confuses me more due to updates being made inconsistent to comments about the updates done. And due to me not having enough reputation, I'm not able to ask counter-questions. BONUS Question: I have been searching high and low on how to make a simple graph of my problem. Like this here question, with a graph of the problem hosted at stack.imgur.com which I guess was generated by LaTeX on stackexchange.com, right? EDIT: Added obligatory sketch!


title: Very short physics question (electromagnetic field): Just draw the figure of the problem?

text: Can you please draw me the figure of the problem? I dont want the solution,just the figure? **A straight, stiff, horizontal wire of length 25 cm and mass 50 g is connected to a source of emf by light, flexible leads. A magnetic field of 1 .33 T is horizontal and perpendicular to the wire. Find the current necessary to float the wire; that is, find the current so the magnetic force balances the weight of the wire.** The solution is very simple : ![enter image description here](http://i.stack.imgur.com/0gbS3.png) _I just want the figure that shows the wire and the electromagnetic field_


title: Significance of angle in relativistic mass equation

text: I was playing with $$M=\frac{m_0}{\sqrt{1-v^2/c^2}}$$ and when I put $v=c\sin\theta$, I got $$M=m_0\sec\theta.$$ And this new equation is appearing so beautiful and attracting to me that I was compelled to think whether $\theta$ appearing here has any physical significance or it is just a mathematical artifact? Whether it is directing to somewhere? Can you please help me? N.B. Here by "directing to somewhere" I mean similar to the following one In the conversion of Coulomb's law to Gauss law it was $r^2$ directing us towards surface area and eventually to Flux concept.


title: Angular Momentum Conservation in Gravitational Interaction

text: thanks for any help. I'm trying to show that in a 2body problem, angular momentum is conserved given that $\dfrac{dp}{dt}=\dfrac{-GMm(rv)}{r³}$, where p is momentum, t time, G gravitational constant, M mass of 1 object, m mass of the other, (rv) is the vector between them and r is the magnitude of the vector between them. I've had a lot of attempts but I don't seem to get very far. Where L is angular momentum, $r_1$ is position vector of mass M and $r_2$ is position vector of mass m, $p_1$ is momentum of mass M and $p_2$ is momentum of mass m. L=$r_1Xp_1+r_2Xp_2$ If $\dfrac{dL}{dt}=0$ then dL will be conserved over time, applying product rule $L=\dfrac{dr_1}{dt}Xp_1+r_1X\dfrac{dp_1}{dt}+\dfrac{dr_2}{dt}Xp_2+r_2X\dfrac{dp_2}{dt}$ noticing that $p_1=m\dfrac{dr_1}{dt}$ and $p_2=m\dfrac{dr_2}{dt}$ clearly $p_1$ is parallel to$ \dfrac{dr_1}{dt}$ and $p_2$ is parallel to $\dfrac{dr_2}{dt}$ and hence when cross producted with them, the result is 0. Giving $L=r_1X\dfrac{dp_1}{dt}+r_2X\dfrac{dp_2}{dt}$ I gather I must now show that either these two terms are 0, or they are equal and opposite. I can't see any reason why they would be 0, and (slightly guessing) it seems to make sense to me that they are terms related to how the angular momentum of one body changes with the other body, and I assume they must therefore be equal and opposite. However when I do $r_1X\dfrac{dp_1}{dt}=-r_2X\dfrac{dp_2}{dt}$ I notice $\dfrac{dp_1}{dt}=\dfrac{-GMm(rv)}{r³}$ and $\dfrac{dp_2}{dt}=\dfrac{GMm(rv)}{r³}$ (since (rv)'s direction is flipped) Therefore $r_1X\dfrac{-GMm(rv)}{r³}=-\dfrac{r_2XGMm(rv)}{r³}$ Defining $w=\dfrac{GMm}{r³}$: $r_1X(rv)(-w)=r_2X(rv)(-w)$ Scalar products are commutative so the (-w)'s easily cancel giving $r_1X(rv)=r_2X(rv)$ Which then the only solutions are if all components of $r_1$ are equal to all components of $r_2$, however angular momentum should clearly be conserved in all positions, not just when the masses are ontop of each other. Thanks again for any help.


title: Doppler effect observed in octaves

text: I have a question about the interpretation of the Doppler effect, when you look at the results as a change in octaves. Nothing actually changes when you look at the result in octaves instead of frequency shift, obviously, but it suddenly seems a whole lot less intuitive. This makes me wonder whether I understand things correctly. If we assume a stationary listener and medium and moving sound source ($-c < v_s < c$, where $c \approx 343 \frac{m}{s}$, speed of sound), the observed frequency $f_L$ is: $$f_L = \left(\frac{c}{c - v_s} \right) f_0$$ where $f_0$ is the emitted frequency. For an object moving away from you (negative $v_s$), the maximum observed pitch is $\frac{1}{2} f_0$, which is exactly one octave down. Intuitively, I find that this makes sense: all frequencies are pitched down, but not by a crazy amount. When the source moves away ($v > 0$), however, one octave upis observed at half the speed of sound: $v = \frac{1}{2} c \Rightarrow f_L = 2 f_0$. And at $v = \frac{3}{4}c$ , it's two octaves up, and you can go all the way up to infinity. So in this case, all frequencies are suddenly scaled to a larger and larger range of frequencies, which seems odd. Is this actually a correct way of looking at it? And what happens for the limit of $v_s \rightarrow c$ (just before the sonic boom)?


title: Hamiltonian form of Noether's Theorem

text: I understand that Noether's Theorem has a Hamiltonian form, whereby {X, H} = 0 iff {H, X} = 0. The proof of this is trivial, as it follows from the antisymmetry of the Poisson Brackets. First question: It remains a question on my mind whether every symmetry, which transforms q(t) into q(t) + k(q(t)) truly has a representation as X(q,p) on the phase space. In the Butterfield paper, they rely on recitfication to make q a cyclic coordinate, in which case the problem is trivial. Is there an easier way to see it in the general case? Second question: What about general non-symmetry transformations? Do they always have a representation X(q,p) on phase space? In other words, suppose I have a know L(q, q'), and p(q, q'). Suppose I define a transformation that changes q(t) into q(t) + k(q(t)), and therefore changes q'(t) into q'(t) + q'(t)dk/dq. Does there exist a corresponding X(q, p) on phase space such that dX/dq = -k(q) and dX/dp = q'(t) dk/dq (after changing variables from q' to p)


title: Classically, how can an electron orbiting a proton radiate given its relativistic energy

text: In classical relativistic Electrodynamics, we are often told that any accelerating point charge inherently radiates (Bremstrallung). (This is the basis for the need for a QM conception of electrons.) We are also told that the innate (not due to the fields) energy of a moving particle is based on relativity, and is written as $E = \gamma m c^2$. Thus, the only dynamic dependence that the relativistic energy has is speed (not velocity ... no direction to $\gamma$). This puzzles me. If both of these are true, then the speed (and thus the energy) of a point charge accelerating in a direction perpendicular to its velocity (i.e. turning without speeding up/slowing down) would not change during the acceleration. How then can it radiate away energy? Note: I do realize that this radiation is experimentally confirmed. It is the generally accepted theory behind such observations that is leaving me confused.


title: Different solutions for floating a DIY torii gate deeper than it's original floating level

text: I'm facing a dead end with my physics experience on a new build I am working on. Any help would be appreciated. Basic problem: I am trying to build a floating wooden Torii gate to be placed on water in a way that the structure's original floating level is pulled/pushed downwards in some way - Maybe 15-30 cm below surface. Reasons for a build like this: 1) It can't be there all year due to weather conditions 2) I want to go through it with my kayak 3) It's flipping cool! Questions: A) How can I achieve a STABLEish structure floating under water surface? I guess the best way would be to somehow pull it downwards with weights? How would YOU do it? B) How can this be calculated? C) Tips, tricks? Ask questions too. ![Mockup image](http://i.stack.imgur.com/svOEo.png)


title: point-particle vs rigid-body

text: As pointed out here point-particle-based modeling can lead to very inaccurate predictions. Could you give an example where point-particle-based model describes reality accurately enough and one where it does not work out well? (from any of the areas of physics more specialized than what is in a classical mechanics book/class, e.g. fluid dynamics) What is the widely accepted rigorous definition of a point-particle?


title: Degeneracy in n-dimensional potential well

text: Knowing that degeneracy occurs in n-dimensional infinite potential well where two wave functions correspond to the same energy, can the same be said for the finite potential well?


title: Gluon Fusion in Top Pair Production

text: In my textbook, there is a feynman diagram showing top + anti-top pair production with a highly energetic gluon resulting from the fusion of two gluons. I know that all vector bosons have spin 1, so how is the spin quantum number conserved in this interaction?


title: Which one is correct Dirac equation?

text: For a particle in potential $U(x)$ in 1D which equation is correct $$i\hbar\frac{\partial\psi}{\partial t}=(cp \sigma_x+mc^2\sigma_z+U(x))\psi$$ or $$i\hbar\frac{\partial\psi}{\partial t}=(cp \sigma_x+mc^2\sigma_z+U(x)\sigma_z)\psi$$


title: Is it that electron of an atom can be found anywhere in the space?

text: > Simple pictures showing orbital shapes are intended to describe the angular > forms of regions in space where the electrons occupying the orbital are > likely to be found. The diagrams cannot, however, show the entire region > where an electron can be found, **_since according to quantum mechanics > there is a non-zero probability of finding the electron anywhere in > space_**. Instead the diagrams are approximate representations of boundary > or contour surfaces where the probability density | ψ(r, θ, φ) |2 has a > constant value, chosen so that there is a certain probability (for example > 90%) of finding the electron within the contour. Although | ψ |2 as the > square of an absolute value is everywhere non-negative, the sign of the wave > function ψ(r, θ, φ) is often indicated in each subregion of the orbital > picture.   >  ![enter image description here](http://i.stack.imgur.com/XSJSM.png)   >  [Cross-section of computed hydrogen atom orbital (ψ(r, θ, φ)2) for the 6s > (n = 6, ℓ = 0, m = 0) orbital.] I have a question here, if electrons can be found anywhere in the space with non-zero probability, can we give a definite boundary for the atom? i.e can we determine the radius of atom? My sir has said me that, radius of atom is around $10^{-10}$m (from the $X$-ray experiments), but as we can have non-zero probability of finding the electron even beyond $10^{-10}$m, how can we say specific radius of an atom?


title: Is the Higgs field part of space?

text: So I looked for some explanations, how I understand it so far is that the Higgs field causes particles to have mass by drag, like drag force in water, which prevents objects from moving at the speed of light, is that correct? so is the Higgs field essentially an element that makes up space?


title: Would QM be detectable in a all boson universe

text: If there was a universe with the same laws as this one, but there were only bosons in it, would QM 'do anything'? Would there be any QM effects - such as an energy level (but that would require fermions..).


title: Can colliding gravitational waves create a black hole?

text: Whether gravitational waves are real or just a coordinate freedom was argued in the early days of GR. Eventually the conclusion was that they were real. And if they are 'real' then I'm curious if... Has it ever been shown that in General Relativity, two (or more) colliding gravitational waves can lead to a black hole?


title: Is it easier to learn more about the seasonal changes in an exoplanet's atmosphere when the exoplanet orbits a binary star system?

text: From a recent ScienceDaily article, we have this... > Scientists detected the new planet in the Kepler-16 system, a pair of > orbiting stars that eclipse each other from our vantage point on Earth. When > the smaller star partially blocks the larger star, a primary eclipse occurs, > and a secondary eclipse occurs when the smaller star is occulted, or > completely blocked, by the larger star. Astronomers further observed that > the brightness of the system dipped even when the stars were not eclipsing > one another, hinting at a third body. The additional dimming in brightness > events, called the tertiary and quaternary eclipses, reappeared at irregular > intervals of time, indicating the stars were in different positions in their > orbit each time the third body passed. This showed the third body was > circling, not just one, but both stars, in a wide circumbinary orbit. > > The gravitational tug on the stars, measured by changes in their eclipse > times, was a good indicator of the mass of the third body. Only a very > slight gravitational pull was detected, one that only could be caused by a > small mass. The findings are described in a new study published Sept. 16 in > the journal Science. "Most of what we know about the sizes of stars comes > from such eclipsing binary systems, and most of what we know about the size > of planets comes from transits," said Doyle, who also is the lead author and > a Kepler participating scientist. "Kepler-16 combines the best of both > worlds, with stellar eclipses and planetary transits in one system." Which brings up some interesting questions. Namely - could we learn more about the exoplanet when it can undergo multiple transits in front of each star? With single stellar systems, we often can only see the exoplanet's transit when the exoplanet is in a single consistent position (analogous to seeing the Earth _only_ during Northern Hemisphere winter, and not at any other time). But with multiple stellar systems, we might be able to see the transit during Northern Hemisphere winter and during another season too (we might even be able to see transits for even more seasons)


title: Is it true that a Space Shuttle launch is powerful like an "average" nuclear explosion?

text: Kind of an odd question here. First let's define what an "average" nuclear explosion is, because that's quite a controversial term - "average". Considering that the peak of nuclear tests was at the 1960s, and that the bulk of nuclear or thermonuclear devices developed in that period were fission bombs, and disregarding radical developments like the Tsar Bomba or the B41 nuclear bomb, I'd say it's safe to proclaim an explosion yielding ~800t of TNT is average on these terms. To back up my numbers I cite the statistics for 60s-era nuclear tests, like Operation Nougat, Operation Dominic, Operation Whetstone, Operation Latchkey and Operation Flintlock. The numbers are readily available on Wikipedia. Now to the Space Shuttle. The solid rocket boosters (SRBs) of a Space Shuttle exert 14MN (14x10^6 Newtons) of thrust each shortly after liftoff, and provide 80% of thrust for the whole stack until they are jettisoned at 46km above sea level. Their thrust level is constant until the fuel is burnt and depleted. The SSMEs (Space Shuttle Main Engines each produce ~2MN of thrust at 100% throttle up until the Shuttle reaches orbit at 320km altitude. Considering that work can be measured in Newtons per meters, ie., J = Nm, therefore: SRBs:   1. 14x10^6N * 46,000m = 644x10^9J   2. x 2 SRBs = 1,288x10^9J SSMEs:   1. 2x10^6N * 320,000m = 640x10^9J   2. x 3 SSMEs = 1,920x10^9J Total: SSMEs + SRBs = 3,208x10^9J or 3.2x10^12J or 3.2TJ. 1kt of TNT equals 4.184TJ, therefore, one Space Shuttle launch is equal to around 760 tons of TNT across the whole 320km stretch. Did I get the work/energy/force relationships right?


title: Atmospheric interference and ground based stellar observations

text: Stellar spectra captured from ground based equipment needs corrections to remove atmospheric spectral noise. Is there an Internet site that shows specific amplitude and wavelength differences between an observed spectrum and a corrected stellar spectrum? The Dec, 2011 issue of Astronomy, page 51 shows the interference of Earth's atmospheric spectra, but the figures do not have amplitude and wavelength data.


title: Parallel capacitors without battery. Does charge flowing after a dielectric input?

text: If I charge two capacitors which are connected parallel $[$the minus (-) of the one opposite to the minus (-) of the other and the plus(+) of the one opposite to the plus (+) of the other.$]$, will I get current (charge flowing) after adding a dielectric material to one of them ?


title: Impedance formula for an edge coupled differential stripline in an asymmetrical stackup?

text: I've been searching and have not been able to find a formula for calculating the impedance of differential lines on inner layers where the dielectric above and below it is not symmetrical. I've seen plenty of examples of symmetrical stackups, but that would not help in my case. Does anyone know what the formula would be for this? Thanks! Edited for more information: For those that don't know, a stackup consists of all the different layers of a PCB (Printed Circuit BOard). You can have, for example, a 4 layber PCB which has individual copper material & glass epoxy stacked on top of each other. This allows the board to have more layers for routing the nets of the circuits.


title: Ring height stats?

text: I'm working on a basic simulation system which will include planets with rings, I want to create dust and asteroids within a 3D ring but I'm struggling to find any stats on the sort of height ranges a ring might have (I can find inner and outer radius estimates, but not the height ranges). Does anybody know a good resource on planetary ring stats? Or celestial bodies in general.


title: Photographs of galaxies many light years far from the Earth

text: Using ground-based telescopes or the Hubble Space Telescope, astronomers take photographs of galaxies which are many light years from the Earth. Does it mean those photographs are as old as the galaxies are far from the Earth? As I read that the Universe is growing very radidly, then its position and shape should have changed after many years. Are photographs useless as a result?


title: How much energy would the Human Torch need?

text: I wasn't sure whether to put this in the SciFi section or here, but since I'm interested in a physical quantity,so I decided to ask it here. I was wondering about approximately how much energy would the Human Torch need to reach X temperature. I was wondering if $Q=mC\Delta T$ would be useful, but I'm not sure if there's such thing as a $C$ for human skin?


title: Magnetic force on a magnetic dipole

text: Can someone please help me understand and answer the question of "why are the forces on a magnetic dipole different in a uniform magnetic field and non- uniform magnetic field?" I know in a uniform magnetic field the magnetic dipole material will tend to align with the field lines of the magnetic field it is in but what happens in a non-uniform one? Doesn't it just change movement rapidly whilst trying to align it's poles with the field line of the magnetic field.


title: What are the theoretical prerequisites for experimental work with SEM and TEM?

text: What level of knowledge and in what areas would be required to perform experimental work with an SEM (Scanning Electron Microscope) and a TEM (Transmission Electron Microscope) respectively? E.g. would a strong background in quantum mechanics be necessary for EELS (Electron Energy Loss Spectroscopy) using a TEM? Assume that both the SEM and TEM machines are modern (made within the last 3 years).   Samples to be observed in the microscope would be metal alloys for materials science research.   Imaging methods of particular interest are EBSD and Surface Imaging for the SEM, and BSE, EELS, and Electron Diffraction for the TEM.


title: Why is Helium-3 better than Deuterium

text: I see that many websites and magazines with physics thematic are pretty excited about mining Helium 3 isotope on the Moon. But this seems to be a very hard-to-get resource. For more than one reason:   1. There is not too much of it on the moon. It actually seems to be 0.007% of the moon soil.   2. Transport to earth would be also very VERY complicated in my opinion.   3. People would be required to take care of the harvesters. Deuterium can be found in the seas as "heavy water". No spaceships needed. Now even when we get the Helium-3, I have a strong suspicion that it contains less energy than Deuterium but it will require more energy to start the fusion. ## Why I think it needs more initial energy to start the fusion?   * Deuterium contains one proton and one neutron. That makes a +1e core.   * Helium 3 has 2 protons and one neutron - That's +2 core. Now the worst problem with fusion reaction is, that once you pull the atoms so close, that the electron shells no longer shield the polarity of the protons they start repelling with electric force. Now if you're trying to pull +1 and +1 together it seems much easier than pulling +2 and +2 together. Just think about the standard equation: $F_e = k*\frac{|Q_1*Q_2|}{r^2}$ where $k$ is the permittivity constant and $Q_{1,2}$ are the proton charges. Repelling force must be **4 times as large**. ![enter image description here](http://i.stack.imgur.com/gORNB.png) ## Why I think it contains less energy? This is even simpler to explain. In the fusion process we merge smaller cores to create larger. We normally start with hydrogen and create helium. Now when we already have Helium fused, it means that we **missed one step** in the fusion. Why would we do that?   Can't actually Helium 3 core be one of cores that are created in Hydrogen fusion?


title: First-order pertubation theory

text: I'm having some trouble figuring this out, so I was hoping someone could help. I need to show that the first-order pertubation of the ground state energy is not changed by the pertubation $H'$, given by a constant electric field in the z-direction: $H' = eEz$ Now, the first-order pertubation for the ground state energy is given by: $\Delta E = E_{1}^{1} = \left< \psi_{100} |H'| \psi_{100} \right > $ Inserting $H'$I get: $\Delta E = eE\left< \psi_{100} |z| \psi_{100} \right > = 0$ But my question is, why is it, that $\left< \psi_{100} |z| \psi_{100} \right >$ equals zero ? I really can't figure that out. Thanks in advance.


title: How can an an ant lift 50 times its own weight and pull 30 times its weight?

text: According to many sites like this one, an ant can apparently lift 50 times its own weight and pull 30 times its weight. Is it true? Can it be proved using physics? Though most sites agree that an ant can lift many times its own weight, not all agree with exactly _how many_ times its weight. The explanations provided if any are usually vague and do not use specific numbers. Can a specific numerical value be calculated? Secondly, how is it possible for the ant to pull 30 times its weight? I find it unbelievable. Can anyone explain this?


title: Does it take significantly more fuel to fly a heavier airplane?

text: I was reading in the papers how some-airline-or-the-other increased their prices for extra luggage, citing increased fuel costs. Now I'm a bit skeptical. Using the (wrong) Bernoulli-effect explanation of lift, I get this: More luggage$\implies$more lift needed $\implies$ more speed needed$\:\:\:\not \\!\\!\\!\\! \implies$more fuel needed. At this point, I'm only analysing the cruise situation. When the plane is accelerating, this will come into effect, but more on that later. Now, I know that the correct description of lift involves the Coanda effect and conservation of momentum, but I don't know it well enough to analyse this. Also, there will be drag forces which I haven't (and don't know how to) factored in. I can see that viscosity must be making a change (otherwise planes wouldn't need engines once they're up there), but I don't know how significant a 1kg increase of weight would be. So, my question is: **Are airlines justified in equating extra baggage to fuel?** Bonus questions:   * If more baggage means more fuel, approximately what should the price be for each extra kilo of baggage?   * What happens when we consider takeoff and landing? Does a heavier plane have to use a _significantly_ large amount of fuel?


title: What makes an Antenna special?

text: The most important property of an antenna, beyond its conductivity(?) seems to be it's length (or more generally, its shape). But the antenna device is connected to the receiver by an arbitrarily-long connector. What is the fundamental difference between an antenna and the connector? Why doesn't the connector's length, straying from some 1/2, 1/4 length of the targeted band, interfere with the reception?


title: Boundary conditions for static electric field

text: Consider a surface that carries surface charge density. In electrostatics, boundary conditions are studied by showing that there is a discontinuity in the normal component of the electric field across the charged surface and that there is no discontinuity in the horizontal component. My question is aren't the electric fields at the surface always perpendicular to the surface? I mean where does the horizontal component come from if wherever we look locally, close enough to the surface, it looks like a flat infinite patch and there is no horizontal components!


title: The nature of time, according to quantum field theory

text: I will try my best to ask the question that best fits something I have been pondering on for a few days. > **Are virtual particles really constantly popping in and out of existence? > Or are they merely a mathematical bookkeeping device forquantum field > theory?** > > Gordon Kane, director of the Michigan Center for Theoretical Physics at the > University of Michigan at Ann Arbor, provides this answer. > > Virtual particles are indeed real particles. Quantum theory predicts that > every particle spends some time as a combination of other particles in all > possible ways. These predictions are very well understood and tested. > http://www.scientificamerican.com/article.cfm?id=are-virtual-particles-rea We define a second by the " _duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom_ " (from wikipedia). Or in other words, we define a second by movement of a certain amount of waves/particles. Since virtual particles pop in and out of existence, could you still claim that these things have a movement on the quantum level? If not, then how do you explain time/spacetime in the eyes of quantum field theory? And how does quantum entanglement fit in to this picture? I am not a scientist, so please forgive me if my understandings of these concepts are flawed.


title: Radiometric Intensity of Non-Monochromatic LED

text: So I have a problem with the LED with the spectrum shown below ![enter image description here](http://i.stack.imgur.com/72jRs.jpg) I am asked to find the radiometric intensity in Watts/steradian given that the LED produces $5.6\times 10^{-3}$ mcd. We are also told to ignore the yellow spectrum. So I know that the luminous intensity is related to radiometric intensity by: $$I_v=\int v_\lambda I_\lambda d\lambda$$ But I am unsure how to numerically find $I_\lambda$ given that I know everything else?


title: Drawing the circuit from a differential equation

text: Can you please help me in modelling a circuit using the differential equation? In the following equation, $u(t)$ is the input voltage and $y(t)$ is the output voltage. $$y(t)=2u(t)+3\frac{du(t)}{dt}+4\int_0^tu(t)dt.$$ How do I draw a circuit such that the input voltage $u(t)$ and the output voltage $y(t)$ are related by this differential equation?


title: How possible is liberal human flight using a flying superconductor bed?

text: I saw a video by Michio Kaku, showcasing levitation of a small object with a superconductor cooled by liquid nitrogen. It's the same technology used by some trains. Hypothetically, how possible is it to achieve free human flight by some way of:   1. Putting a thin, light (say 5kg) bed, under a human in a fly suit. The bed is coated with the magnetic material of one polarity.   2. This bed has some sort of continuous thrust mechanism, with the required thrust to keep only it afloat & to fly it around. (It's very light due to carbon nano tubes, high technlology etc.)   3. On top of this bed is a freely floating human in a suit made of the opposing superconductor (or however this works) So essentially the bed is what's really flying around and has tiny thrust requirements because of it's weight and the human is levitated at essentially minimal cost (due to the superconductor)


title: Where to find information about rocket launches (speed, height, trajectory)?

text: 1. I want to find experimental data from real launches, with some description of a rocket like full mass, fuel mass, thrust and so on. Just trying to model it.    2. Where to read about precise methods of calculating air resistance for rockets.


title: What happens to gravitational waves after arbitrarily long propagation?

text: Given that some systems may radiate energy in the form of gravity waves, and that gravitational waves weaken proportionally to the distance travelled, what would happen to the waves that never hit anything and just propagated infinitely? Do they just become infinitesimally small? If so, would this imply that energy can exist in a non-quantifiable form?


title: How early can scientists detect that a deadly asteroid will hit planet earth?

text: With the advancements in science and technology, how early can scientists of the world detect that a certain asteroid will hit planet earth and cause damage? Are there any impact avoidance strategies or a disaster recovery plan agreed upon by world nations? Edit: This wikipedia article shows some links to few past projects


title: Motivation to introduce von Neumann algebras in addition to $C^*$algebras?

text: Observables are self-adjoint elements of a $C^*$algebra. As such, this structure seems sufficient to describe physics. A theorem by Gelfand and Naimark says that a $C^*$algebra can always be faithfully represented as bounded operators on a Hilbert space, $B(H)$. Then one can introduces different topologies, and one can see von Neumann algebra as a $C^*$subalgebra of $B(H)$ that is moreover complete in one of those topologies. In another question of Physics stack exchange Why are von Neumann Algebras important in quantum physics? someone also talks about the Borel functional calculus, and one also compare von Neumann algebra to "non commutative" measure theory vs "non commutative" for $C^*$algebras. My question is, is the introduction of von Neumann algebra only a technical thing or has it physical consequences?


title: $V = V_1 + V_2$ confusion? Why is my "proof "incorrect?

text: ![enter image description here](http://i.stack.imgur.com/UXh7H.png) Why isn't $V= V_1 + V_2$? $V=V_a - V_c = V_a - V_b + V_b - V_c$, $V_a - V_b= V_1$ and $V_b - V_c = V_2$ Doesn't that prove that $V = V_1 + V_2$? Regardless of $V_3$, If i'm wrong , is there a way to obtain $V$ in terms of $V_1$, $V_2$ and $V_3$


title: Molecule vs Crystal

text: Feynman mentions in his lectures: > ...the concept of a molecule of a substance is only approximate and exists > only for a certain class of substances. It is clear in the case of water > that the three atoms are actually stuck together. It is not so clear in the > case of sodium chloride in the solid. There is just an arrangement of sodium > and chlorine ions in a cubic pattern. There is no natural way to group them > as "molecules of salt". For me that's not clear. A molecule of water also is stuck in a pattern (at least when it's solid). I'd think of any molecule as 'atoms stuck in a pattern'. Thus why a crystal is not a special case of a molecule?


title: Why do I still get static shocks when hands are wet?

text: Quite a few times now I have washed my hands, and while my hands are still wet I will touch something metal and get a static shock. I thought static electricity doesn't really build up when there's a lot of moisture, so that's why you don't get static shocks in the summer when there's more humidity. Even though it's winter now, I figured I wouldn't feel any static shocks if my hands are wet, but I still do.


title: What are the physical limitations on energy density for a useful, safe battery, and are we anywhere near them?

text: Energy storage seems to be a fairly major bottle neck in technological development at the moment, especially for important things like clean-energy powered transport, but also for consumer goods. Are there physical limitations of batteries for these and similar uses? Lithum-ion batteries seem to be the go-to for these applications at the moment, likely because of their high energy to weight ratio. Even they sometimes explode, so I guess safety is somewhat of a grey area, but for the purposes assume standard-ish safety margins. Are Lithium-ion batteries approaching such physical limitations?


title: Probability in moving from state $\lvert1\rangle$ to $\lvert2\rangle $

text: The system is $|1\rangle$ at time $t = 0$, while $|2\rangle$ at time $t$. The base is orthonormal $\left\\{|1\rangle, |2\rangle\right\\}$. Epsilon and delta are positive constants, which have some energy unit. The system is: $$ \hat{H} = \epsilon\left(-1|1\rangle\langle1|+|2\rangle\langle2|\right)+\delta\left(|1\rangle\langle2|+|2\rangle\langle1|\right) $$ What is the probability of moving from state $|1\rangle$ into the state $|2\rangle$?


title: seemingly impossible query

text: While on an expedition to a lake, it was noticed that the wind speeds gusted violently in seemingly periodic wakes and the proposed explanation was Von Karman vortex shedding off of nearby mountains that the lake was in the lee of. Therefore I am attempting to calculate the amount of influence that a Karman vortex street would have on some wind speed readings obtained during that time in order to see if I can detect the influence in wind speed readings collected during that time that the observation of intense gusts was made. From some lengthy research into this topic I have found a cadre of equations which I think give me the desired results of the frequency and maximum wind speed within the Karman vortexes all based solely upon wind speed readings from the lake.I however have come across an issue in the calculations which reduces all parameters to either constants or to be exactly directly proportional to the presumed oncoming wind speed, a highly unlikely and illogical occurrence. The problem rests in the fact that the laboratory calculated kinematic viscosity of air doesn't hold up to large scale observations of vortex street formation, thus necessitation a new parameter as stated in this article. This new parameter is β and is defined as $$β=St/Re$$ with $Re$ being the Reynolds number and $St$ being the Strouhal number. Due to the fact that I don't have the shedding frequency but need to calculate it, I cannot calculate St except through the formula $$St = .212(1-\frac{21.2}{Re})$$ as found in this article. However, doing this and substituting into the parameter of $β$ yields a relationship between the constant $β$ and $Re$, which makes $Re$ constant and in turn $St$ constant as $St$ is defined in terms of $Re$ only. This causes all the parameters that I have to be either linear with respect to any oncoming flow velocity or constant, something obviously wrong. Although I have everything that I need after this point in order to obtain maximum wind speed and shedding frequency, this initial step has stopped me dead in my tracks due to the fact that every author whom I have seen on this subject had a picture to work with to calculate the distances between the vortices and use that and an assumed wind speed to calculate shedding frequency, plug that in to find Strouhal number and then use the $β$ parameter to calculate Reynolds number. All those working in lab tests similarly calculated shedding frequency and recorded it by video taping or counting the eddy shedding process. I, however, only have the wind speed to work with and so have problems by reducing out too many variables. My question therefore becomes how do I define a either a variable $β$ parameter or redefine either my $St$ or $Re$ number so that one can be calculated independently with the limited information at my disposal of only the wind speeds and the ability to assume other parameters such as landmass dimensions etc. Thank you for your time reading this.


title: Stable equilibrium given force

text: If a particle moves under the influence of a resistive force proportianal to velocity and a potential $U$, $$F(x,\dot x)=-b\dot x-\frac {\partial U}{\partial x}$$ Where b>0 and $U(x)=(x^2-a^2)^2$ My thoughts were to make $F=0$, which would result in: $$\dot x= \frac 1b(4x^3-4a^2x)$$ This means that the points of equilibrium are a function of time? EDIT:: So I set up as suggested, using $F=m\ddot x$ to get $$0=m\ddot x+b\dot x+4x(x^2-a^2)$$ I am trying to solve for $x(t)$. I know how to solve the equation that has only x, but I do not know how to solve for the x^3 term. I am curious how to combine these to get x(t)


title: Would my fridge magnet drop off on a non-magnetic planet?

text: As far as I understand, the magnetic power of an ordinary permanent ferromagnet comes from and is maintained by the motion of molten iron in the Earth's core. So a fridge magnet would eventually come loose if no longer in contact with this source of replenishment due to the energy expenditure involved in constantly overcoming the force of gravity. Right?


title: How can we define BF theory on a general 4-manifold?

text: (I have rewritten the question some, with new understanding) 4d BF theory is classically presented as the TFT arising from the Lagrangian $B\wedge F$, where $B$ is an abelian 2-connection (locally a real 2-form) and $F$ is the curvature of a connection $A$. People often throw about this theory on general 4-manifolds, but it seems the naive definition from the Lagrangian above is anomalous for most situations. If one restricts to the simpler case of $A$ and $B$ being globally defined forms (not considering them as connections) the phase space is $H^1(\Sigma_4,\mathbb{R}) + H^2(\Sigma_4,\mathbb{R})$ with $A$ as the first coordinate and $B$ as the second. $A$ and $B$ are conjugate from the point of view of the above Lagrangian, but this has no chance of being symplectic if the first and second Betti numbers are not equal. If $A$ and $B$ are connections, however, the only solution to the equations of motion (which not only set $dA=dB=0$ but also their holonomies) is the trivial one, and it is not a problem However, what I'd really like to consider is $nB\wedge F$. With $A$ and $B$ full-fledged connections. Let us first integrate out $B$. $B$ is a (2-)connection, so $dB$ can be any integral 3-form. We can thus write it as $d\beta + \sum_k m^k \lambda_k$, where $\beta$ is an ordinary real 2-form, and $\lambda_k$ form a basis of integral harmonic 3-forms. Then the action becomes $dA \wedge \beta + \sum_k n m^k A\wedge\lambda_k$. The first term sets $dA=0$ after integrating over $\beta$ and the second term sets the holonomies of $A$ to be nth roots of unity. If our 4-manifold doesn't have torsion, there are precisely $n^{b_1}$ such $A$s. If we do the same thing, integrating out $A$ instead, we get Dirac deltas setting $dB=0$ and the holonomies of $B$ to be nth roots of unity. Again, if we have no torsion, there are $n^{b_2}$ such $B$s. It looks like the determinants for these Dirac deltas are all 1, so from the naive calculation above, $Z_n = n^{b_1}$ but also $Z_n = n^{b_2}$. This is a problem on a general 4-manifold, but like the simpler situation above, there is no issue in quantizing on something of the form $\Sigma_3 \times \mathbb{R}$. What I think must be happening is that when I am integrating out $A$ or $B$, I am really only integrating out the non-flat parts. There should still be some more factors that correct the discrepancy above. I think I should be more careful in defining the above integral. Perhaps it is possible to define it sensibly on 4-manifolds which bound a 5-manifold a la Chern- Simons theory. How does one define this theory on general 4-manifolds? Thanks.


title: Utility of gauge four-potential $A_{\mu}$ (as opposed to electric and magnetic fields ${\bf E}$ and ${\bf B}$) in E&M?

text: The action for an electromagnetic field with source charges is given by $$S= \int \left\\{ \frac{1}{4\mu_0}F^{\mu\nu}F_{\mu\nu} - J^\mu A_\mu \right\\}dx$$ By setting $dS=0$ and taking the Lorenz gauge, the resulting field equation is $$-\Box A_\mu = \mu_0 J_\mu$$ I can see that there are a lot of conceptual advantages to thinking about E&M in terms of electromagnetic four-potentials. However, is this approach actually useful for working out problems? That is to say, is there any **computational** advantage to using this equation, or is it usually simpler to solve the standard form of Maxwell's equations?


title: Surface area of glass (or general material) and strength

text: How does the surface area of a piece of glass (or any general material; sapphire isn't technically glass) correlate with strength? Strength as in resistance to pressure concentrated at a small point? Intuitively, something tells me that the larger a piece of glass is, the less resistant it will be to pressure at a small point. Take the extreme example. A "piece" of glass that consists of one molecule. Applying pressure to this one molecule won't fracture the molecule. (But then perhaps this example is too extreme). I ask because of this comparison, done with a sheet of sapphire "glass" that is clearly much larger than the other, competing piece of glass. Would the size of the sapphire give it an inherent disadvantage in this test? ![enter image description here](http://i.stack.imgur.com/ncbVh.png) **Here is a video of the complete test:https://www.youtube.com/watch?v=u6B-2jTvh1w**


title: A large rock falls on your toe. Which concept is most important in determining how much it hurts?

text: A) The mass of the rock B) The weight of the rock C) Both the mass and the weight are important. D) Either the mass or the weight, as they are related by a single multiplicative constant, _g_. Similarly, if the large rock merely sits on your toe, which concept is most important in determining how much it hurts? For a falling rock, to me it seems that mass would be most important as the pain level would be proportional to the change in momentum, or the impulse, on your foot. For a stagnant rock, it seems either weight or mass would be the same. Am I correct?


title: Inductance of loop to a square wire? What is the induced current and the induced $\bar E$?

text: _I am practising for the elementary-physics -course Tfy-0.1064, this puzzle is a simplified version of the problem 3Ahere (sorry not in English). My basic idea here is to understand a scenario with one loop and then understand the solenoid case._ Suppose a circle around the square of 20-cm-side and $\frac{dB}{dt}=35\frac{mT}{s}$ in the circle, please, see the picture below. What is the induced current and the induced $\bar E$ to the square by the circle? ![enter image description here](http://i.stack.imgur.com/CoIpB.jpg)


title: vector-scalar gravity

text: Has anyone developed a vector-scalar theory of gravity using an acceleration field equation that is based on a ratio in which the numerator is the familiar Newtonian vector sum of all masses times the inverse square of their respective distances, and the denominator is the scalar (not vector) sum of the very same formula? As a scalar sum, the denominator would always be larger than the numerator because there is no cancellation from opposite directions, and the quotient could be interpreted as the proportion of "directionality" of the combined contributions. Or has anyone developed a relativistic theory based on the same ratio of vector to scalar sum of the same contributions?


title: What's the right way to calculate the principal moment of inertia?

text: I am writing a program that incorporates calculating the principal moment of inertia for a protein residue based on its component atom XYZ coordinates. I am exceedingly confused about which formulas to use in calculating principal moment of inertia for my situation. Thus far my program does the following:   1. Calculate elements (Ixx, Iyy, Izz, etc.) of symmetric matrix A   2. Find the eigenvalues (and thus the principal moments of inertia) The formulas I originally used to calculate the elements in step 1 were from pg. 5 of this document. However looking at Wikipedia, the formulas for the symmetric matrix elements are different. I have also been scouring the internet and found open source code where the center of mass is subtracted from the x, y, and z coordinates before beginning calculation of elements Ixx, Iyy, Izz, etc. Which formulas/algorithm do I use to calculate principal moment of inertia for my case? I'm not sure where to begin with picking which formulas to use. Is there a source that is accessible to those with a weak physics background that will help me understand which formula/algorithm to use in calculating principal moment of inertia?


title: What are the principles behind foam formation?

text: We know that while pouring water in a glass, there is close to no foam, but if pouring beer, there is much foam (unless pouring "slowly"). So, can the most prominent factors that contribute to foam formation be listed and explained? I've found an article that explains some concepts about foams and has some curious images. It also states that foams in general are not well understood. But I'd like to hear a **qualitative** explanation of why certain properties of liquids such as viscosity and volumetric flux make it foam or prevent foaming.


title: Asymptotic freedom in QCD

text: From renormalization group equation $$ t \frac{d \bar{g}(t , g)}{dt} = \beta (\bar{g}(t , g)), \quad \bar{g}(1 , g) = g $$ (here $t$ is momentum scale factor, $g$ is initial coupling constant and $\bar{g}$ is an effective coupling constant) and $$ \tag 1 \beta (g) = ag^{2} $$ we can get $$ \bar{g}(t , g) = \frac{g}{1 - gln(t)a} . $$ For QCD $$ \tag 2 a = c\left(-11 + \frac{2}{3}n_{f} \right), $$ where $n_{f} = 6$ is the number of quark types. How to get that for QCD we asymptotically have $(1)$ and $(2)$?


title: An Equality in The Calculus of Many Instantons

text: I am reading the review on instantons. When I tried to derive formula (2.27) on page 17, I always get the different coefficient of $gF_{mn}$ term. My calculation is just directly expanding the first term $-\frac{1}{2} \int d^4x {\rm{tr_N}} F^2_{mn}$ in action to quadratic form, and leaving the second term $i\theta k$ invariant. Under the fluctuation $A_n \to A_n +\delta A_n $, the first term $-\frac{1}{2} \int d^4x {\rm{tr_N}} F^2_{mn}$ becomes $$ -\frac{1}{2} \int d^4x {\rm{tr_N}} F^2_{mn} \\\\-\frac{1}{2} \int d^4x {\rm{tr}_N }\left( 2(D_n\delta A_m)^2 -2D_n\delta A_mD_m\delta A_n +2g F_{mn} [\delta A_m,\delta A_n] \right) +...$$ Integrating by parts and disregarding the surface term, the quadratic term becomes $$ \int d^4x {\rm{tr}_N }\left( \delta A_m D^2 \delta_{mn}\delta A_n -\delta A_mD_nD_m\delta A_n+2g F_{mn} \delta A_n \delta A_m \right)$$ Using the equalities $D_n D_m=D_mD_n-gF_{mn}$, $D_n\delta A_n=0$, the quadratic form becomes $$ \int d^4x {\rm{tr}_N }\delta A_m\left( D^2 \delta_{mn} +g F_{mn}+2g F_{mn} \right)\delta A_n $$ So the coefficient of $gF_{mn}$ term is 3. What is wrong with the above calculation?


title: How much energy would it take to blow up the earth?

text: There is a common statement running around that we as a species has enough nuclear weapons to blow up the earth several times over. What I want to know is: by how many orders of magnitude is that a wrong statement? (Just getting a lower bound would be fine.)


title: Physical interpretation of $\iiint (∇\cdot\vec E)\mbox{d} V$

text: 1. Can anybody explain the physical interpretation of Gauss's law $$\iiint (\nabla\cdot \vec E)~\mbox{d}V~=~\frac{Q}{\epsilon_0}? $$    2. Also, how is the differential form of Gauss's law obtained from the integral form?    3. How can volume multiplied by a gradient equals flux through a closed surface?


title: Spectral radiance unit conversion

text: I have spectral radiance data in SRUs (spectral radiance units), as a function of wavelength: $$a = \mu W cm^{-2} sr^{-1} nm^{-1} $$ However, I am working with software which requires my data in the form: $$b=mWcm^{-2}sr^{-1}\mu m^{-1}$$ Are these two units are equivalent?


title: Finding Interatomic Spacing

text: Here is the problem I am working on > For what temperatures are the atoms in an ideal gas at pressure P quantum > mechanical ? Hint: use the ideal gas law P V = N kT to deduce the > interatomic spacing (the answer is $T < (1/k)(h^2/3m)^{3/5} P^{2/5})$. > Obviously we want m to be as small as possible and P as large as possible > for the gas to show quantum behavior. Put in numbers for helium at > atmospheric pressure. Is hydrogen in outer space (interatomic distance ≈ 1 > cm and temperature ≈ 3K) quantum mechanical? According to the answer key, to find the interatomic spacing, we need to find the size of a single gas particle. One gas particle corresponds to N=1, and the volume is V = d^3. This leads to $d = \left( \frac{kT}{P} \right)^{1/3}$ I have two objections to this, for which I hope you provide correction. Firstly, assigning the volume V=d3 implies that we are assuming that the atoms are square? Secondly, how does finding the size of a single gas particle provide us with the interatomic spacing. It would seem that the most we could deduce from such information is, that closest two gas particles could get. Are we to assume that the gas particles are this closely packed? Wouldn't the gas solidify at this point?


title: Why does nuclear matter tend to maximize pressure?

text: I'm reading a text about equations of state of dense nuclear matter. It is often stated that the phase with maximum pressure is preferred. Why is that?


title: "Conceptualizing" neutrinos

text: Layman here. EE and BS physics. I am "content" in viewing photons/electromagnetic radiation as an "emergent" property of spacetime? due to the electrons ("particles..?") and all their activity jumping around in atoms. [or is this likely too simple?] ... in the "same" manner that sound and water waves are an "emergent" property of air and water particles. OK. So WHAT do neutrinos represent? Can I visualize them as an emergent property of neutrons and protons [and quarks?] and all their activity in the nucleus? Neutrinos are a real mystery to me, as if everything else weren't!


title: Why does the classical Noether charge become the quantum symmetry generator?

text: It is often said that the classical charge $Q$ becomes the quantum generator $X$ after quantization. Indeed this is certainly the case for simple examples of energy and momentum. But why should this be the case mathematically? For clarity let's assume we are doing canonical quantization, so that Poisson brackets become commutators. I assume that the reason has something to do with the relationship between classical Hamiltonian mechanics and Schrodinger's equation. Perhaps there's a simple formulation of Noether's theorem in the classical Hamiltonian setting which makes the quantum analogy precisely clear? Any hints or references would be much appreciated! **Mathematical Background** In classical mechanics a continuous transformation of the Lagrangian which leaves the action invariant is called a symmetry. It yields a conserved charge $Q$ according to Noether's theorem. $Q$ remains unchanged throughout the motion of the system. In quantum mechanics a continuous transformation is effected through a representation of a Lie group $G$ on a Hilbert space of states. We insist that this representation is unitary or antiunitary so that probabilities are conserved. A continuous transformation which preserves solutions of Schrodinger's equation is called a symmetry. It is easy to prove that this is equivalent to $[U,H] = 0$ for all $U$ representing the transformation, where $H$ is the Hamiltonian operator. We can equivalently view a continuous transformation as the conjugation action of a unitary operator on the space of Hermitian observables of the theory $$A \mapsto UAU^\dagger = g.A$$ where $g \in G$. This immediately yields a representation of the Lie algebra on the space of observables $$A \mapsto [X,A] = \delta A$$ $$\textrm{where}\ \ X \in \mathfrak{g}\ \ \textrm{and} \ \ e^{iX} = U \ \ \textrm{and} \ \ e^{i\delta A} = g.A$$ $X$ is typically called a generator. Clearly if $U$ describes a symmetry then $X$ will be a conserved quantity in the time evolution of the quantum system. **Edit** I've had a thought that maybe it's related to the 'Hamiltonian vector fields' for functions on a symplectic manifold. Presumably after quantization these can be associated to the Lie algebra generators, acting on wavefunctions on the manifold. Does this sound right to anyone?


title: Centripetal acceleration for particle that moves in a circle at non-constant speed

text: How is $a_{R}=v_T^{2}/r$ derived for changing tangential speed? ($a_R$-centripetal component $r$-distance from axis $v_T$- tangential speed)


title: Equations instead of psychrometric charts

text: I want to create a program that will accurately simulate a condensor. I want to use the data in psychrometric charts. But I cannot and hence want to use equations that show similar data. Any idea where to start?


title: Reference for the Landau-Lifshitz system

text: I'm interested in understanding the dynamics of the discrete Landau-Lifshitz system. It's solutions to equations like $$\frac{\partial X_n}{\partial t} = X_n\times (X_{n-1}+X_{n+1})$$ where the $X_n$ are unit vectors in spheres that sit on a discrete lattice (parameterized by $n$, and possibly finite with boundary conditions or infinite). This system is Hamiltonian, but apparently it is also very chaotic/non-integrable. I'm not interested in understanding the associated quantum or continuum system (although I'm aware they are related). Does anyone know a good paper that discusses the dynamics of these classical systems? In particular, one that explains their non-integrability and chaotic behaviour? I'm sorry if this question isn't appropriate here.


We know that: **Nambu-Goldstone bosons** come from Goldstone theorem: a spontaneous (continuous)-symmetry breaking of the system leads to massless scalar modes. **quantum anomaly**: is the anomalous phenomena where the (classical Noether) symmetry G respected by the physical system in a classical limit, but this symmetry G is broken by the quantum effect. This is the case where the action $S$ preserves the symmetry, but the path integral partition function $Z=\int [D\Psi][D\Phi]\dots e^{iS}$ and the measures $[D\Psi][D\Phi]\dots$ _do not_ preserves the symmetry. * * *   * **Question** : whether there is any example that Nambu-Goldstone bosons can be derived from a spontaneous symmetry breaking caused by quantum anomaly effect? Alternatively, whether there is a known theorem to prove **``No Nambu-Goldstone bosons from a quantum anomaly symmetry breaking?''** * * * [words of caution & side remark]: I offer some further thoughts if you wish you can skip it. There is a statement learned: **Nambu-Goldstone bosons do not generally appear for a spontaneously broken symmetry if the relevant global symmetry is broken by the effects of the anomaly and instantons. This is regarded as a reason why we do not observe a light pseudo-scalar meson $\eta'$ in the QCD mesons.** 1 among the 9 mesons is this $\eta'$, which stands for axial $U(1)_A$ anomalous symmetry breaking, there are still $SU(N_{flavlor})_A=SU(3)_A$ which is spontaneous broken by dynamical symmetry breaking, which induces 8 among the 9 mesons, such as three $\pi$, three $\kappa,$ and one $\eta$. There are also $SU(N_{flavlor})_V=SU(3)_V$ broken by the nonzero explicit quark masses($m_u\neq m_d \neq m_s$). There is also $U(1)_V$ broken by Sphaleron, such that numebr of baryon $N_{baryon}$ is not conserved, but which only conserved numbers of $N_{baryon}-N_{lepton}$. Anyway, altogether $$SU(N)_V \times SU(N)_A \times U(1)_V \times U(1)_A$$ makes it to be the full $U(N)_V \times U(N)_A$. Simply that **$U(1)_A$ symmetry is anomalous broken by the quantum effect, but we do not see its Goldstone boson $\eta'$.** ps. please one may read full comments below the questions.


title: What is 'past null infinity'?

text: For example, in the sentence "there is no incoming radiation at past null infinity".


title: Right-angle lever paradox in special relativity

text: I remember to have read somewhere an interesting special relativity "paradox" considering two perpendicular rods $A$ and $B$ of equal proper length $L$ fixed at point $O$. ![rest frame](http://i.stack.imgur.com/ULxzd.png) In the "rest" frame equal (in magnitude) forces are applied on the edges of the rods, perpendicular to them, thus giving zero net angular force: $$-F_AL_A + F_BL_B = -FL + FL = 0$$ (assuming $z$-axis is towards us). In the frame moving parallel to rod $B$ with the velocity $v$ it's length is diminished due to Lorentz-contraction $$L_B' = L \sqrt{1-v^2/c^2}$$ and perpendicular component of the force is diminished also $$F_B' = F\sqrt{1-v^2/c^2}$$ while $L_A' = L$ and $F_A'=F$. So the net angular force in this frame is not zero: $$-F_A'L_A' + F_B'L_B' = -FL + FL (1-v^2/c^2) = -\frac{v^2}{c^2} FL$$ and the system should be rotating clockwise with angular acceleration. I don't remember the explaination of the paradox as I did not understand it at that time, but now it seems to be very simple: even in Newtonian Mechanics total angular force would not be zero in "moving" frame given the fact that the net force is not zero. If we take into account that rods are _fixed_ relative to $O$, we should assume that in the junction point $O$ there is an additional force which compensates the forces on the edges of rods. The same argument seems to be valid in the relativistic case too. However, I remember that the explaination was not that simple. Am I wrong with my explaination, or the statement of the paradox is wrong? Does anyone know similar "paradox"?


title: Do objects have energy because of their charge?

text: My gut feeling tells me things should have energy because of their charge, like they have energy because of their mass. Is this possible? Has it been shown? If not then what is missing to make such an equivalence possible?


title: Statics of Rigid Bodies -- Can there be two possible solutions?

text: I've been working on a question and there seem to be two possible solutions. My own solution does not match the one given in the book. However, after resolving forces and taking moments with both pairs of solutions, they both seem to be consistent, i.e. they both seem to be valid. Here is the question: > A loaded shelf AB is freely hinged to the wall at A, and supported in a > horizontal position by a light strut CD, which is attached to the shelf at > D, 25cm from A, and attached to the wall at C, 25cm below A. The total > weight of the shelf and its load are 80N, and the centre of mass is 20cm > from A. Find the thrust in the strut and the magnitude and direction of the > force exerted by the hinge on the shelf. Everything from this point is my own work, so any of it could be wrong. I've drawn a (possibly incorrect) diagram to represent the situation, and have included it below: ![schematic diagram](http://i.stack.imgur.com/xjyiS.png) The red vector represents the thrust generated by the strut. Since $\|AC\| = \|AD\| = 25\,\tt{cm}$ $\angle CAD = 90^{\circ}$ it follows that $\angle ACD = \angle ADC = 45^{\circ}$. Hence the angle made by $T$ with $DB$ is $45^{\circ}$. Resolving $T$ horizontally and vertically gives $\tfrac{\sqrt{2}}{2}\\!T \, \tt{N}$ and $\tfrac{\sqrt{2}}{2}\\!T \, \tt{N}$ respectively. Taking moments about $A$ gives a clockwise moment of 16 Nm and an anti- clockwise moment of $\tfrac{\sqrt{2}}{8}T \, {\tt Nm}$, meaning that $T= 64\sqrt{2} \, {\tt N} \approx 90.5 \, {\tt N}.$ Let $H$ and $V$ denote the horizontal and vertical components of the reaction at the hinge at $A$. Resolving horizontally and vertically gives: \begin{array}{|c|ccc|c|ccc|} \hline R\left(\rightarrow\right) & H &=& T\cos 45 &\implies& H &=& 64\,{\tt N} \\\ \hline \\\ R\left(\uparrow\right) & V+T\sin 45 &=& 80 & \implies& V &=& 16\,{\tt N} \\\ \hline \end{array} It follows that the magnitude of the reaction force is $16\sqrt{17}\,{\tt N} \approx 66\,{\tt N}$ and the angle with the vertical is about $76^{\circ}$. I've checked the numerical values and all forces and moments seems to balance. **However** , the alternative solution that the thrust is $T \approx 94.3 \, {\tt N}$, the magnitude of the hinge reaction force is about 68 N and the angle of said force is about $78.7^{\circ}$ from the upward vertical are given. I've substituted these in to the problem and all of the forces and the moments again seem to balance. Is it possible to have two possible solutions to such a problem? (There are three unknowns, namely T, H and V, and three linear equations. I would expect a unique solution since no inverse trigonometric functions were used.) If the solution is unique, then who is correct?! Thanks in advance!


title: Super Lie-infinity algebra of closed superstring field theory?

text: Bosonic closed string field theory is famously governed by a Lie n-algebra for $n = \infty$ whose $k$-ary bracket is given by the genus-0 (k+1)-point function in the BRST complex of the string. One might therefore expect that, analogously, closed _superstring_ field theory (in any of its variants) is governed by a lift of that to a super Lie n-algebra for $n = \infty$. The closest to an identification of such that I am aware of is in   * Yuji Okawa, Barton Zwiebach, _Heterotic String Field Theory_ (arXiv:hep-th/0406212) where substructures of the bosonic string field $L_\infty$-algebra are paired with the super-ingredients. This seems to go in the right direction, but does not quite identify a super $L_\infty$-algebra structure. Is there, meanwhile, anything more known that may complete the picture here?


title: What do theoretical physicists need from computer scientists?

text: I recently co-authored a paper (not online yet unfortunately) with some chemists that essentially provided answers to the question, "What do chemists need from computer scientists?" This included the solution of theoretical problems, like combinatorial enumeration and the sampling of certain classes of graphs; and practical programming problems, like open-source implementations of algorithms that are currently only implemented in expensive software packages. This motivates me to ask: what about this field? Are there theoretical issues of combinatorics, algorithm analysis, that physics needs a theoretical computer scientist to solve? Or how about creation of practical tools that would allow a theoretical physicist to do a better job: "If only I had a program that would solve this type of problem for me!" Intended as community wiki.


title: Particle in a 1D box (momentum representation)

text: I have this problem. I want to find the wave function in the momentum space for the particle in a 1D box. We know that the wave function in the position space is: $$Y_n(x) = A\sin{(n\pi x/L)}$$ Well, if I write the Schrödinger equation in the moment space I have: $$\frac{p^2}{2m}Y_n(p) = E_nY_n(p)$$ So, this equation doesn't give me any information about the wave function $Y_n(p)$ I know that I can solve this problem just using the Fourier tranformation, but I'm asking myself if there is another posibility to solve this problem. PD. If I use the Fourier transformation, do I have to integrate just from $O$ to $L$?


title: Status of the little hierarchy problem

text: What is the current thinking on the little hierarchy problem in light of a potential Higgs mass above 120 GeV? A few years ago, at least, I remember various phenomenologists saying that this at least makes life rather difficult for the MSSM.


title: Wilson/Polyakov loops in Weinberg's QFT books

text: I wanted to know if the discussion on Wilson loops and Polyakov loops (and their relationship to confinement and asymptotic freedom) is present in the three volumes of Weinberg's QFT books but in some other name or heading. At least I couldn't naively find these topics in that book. I was hoping to see some detailed discussion about these non-local physical observables and also may be some model calculations and evaluations of them in some gauge theories. In Eduardo Fradkin's lecture notes there exists an evaluation of the Wilson loop in free Maxwell's theory and that too in a certain limit of large time and large separation and thats the only example of such a calculation that I have seen. (Aside: I would be grateful if people can also point to other pedagogic/classic/path-breaking papers/references in this topic of Wilson loops. To start off these are four papers that I came across while searching along this direction: arXiv:hep-th/0003055, arXiv:hep-th/9911088, arXiv:0905.2317, arXiv:hep-th/9803002.)


title: Dimensional Regularization Integral Formula

text: In the formula $$\int \frac {d^{4-2\epsilon} l} {(2\pi)^{4-2\epsilon}} \frac 1 {(l^2-\Delta)^2} = \frac i {(4\pi)^{2-\epsilon}} \Gamma(\epsilon) \left(\frac 1 \Delta\right)^\epsilon,$$ how should I deal with the case when $\Delta<0$?


Follow-up to my other question How does Sol's magnetic field continue to exist at such high temperatures? Assuming Sol's magnetic field is generated by convective currents in it's plasma, how is it that the magnetic field still exists when the star ages to become a white-dwarf? Is the magnetic field of a white-dwarf merely residual?


title: Gentle introduction to twistors

text: When reading about the twistor uprising or trying to follow a corresponding Nima talk, it always annoys me that I have no clue about how twistor space, the twistor formalism, or twistor theory works. First of all, are these three terms some kind of synonyms or what is the relationship between them? Twistors are just a deep black gap in my education. I've read the Road to Reality but I just did not get it from the relevant chapter therein, maybe because I could not understand better the one or two chapters preceding it either ... :-/ So can somebody point me to a gentle, but nevertheless slightly technical source that explains twistors step by step (similar to a demystified book...) such that even I can get it, if something like this exists? Since I think I'd really have to "meditate" about it a bit, I'd prefer something written I can print out, but nevertheless I would appreciate video lectures or talks too.


title: Lumped circuit model of Earth and Sun as a thermal system

text: ![enter image description here](http://i.stack.imgur.com/UqW8O.png) I want to design a thermal model of the Sun heating up the Earth. The circuit below is my own design (I don't know if there is a better way to model it. If there is, please tell me.). I want to simulate this system, by assigning some sinusoidal function to $I_{sun}$. My aim is to monitor temperature change on Earth as a response of the heat delivered from Sun. Next, I am going to assign more complicated functions to $i_{sun}(t)$ in order to model some natural phenomena; like seasons and clouding of the sky. I need some numeric data for completing my model.   So far, I could only find the value of $I_{sun}$ in Wikipedia. If $sin(\omega t) < 0$ then $i_{sun}(t) = 0$ else $i_{sun}(t) = 1.74 \times 10^{17} \times sin(\omega t) \,\,\, W$;   where $\omega = \frac{2 \pi}{24} (rad/hour)$. Can you please give me the other parameters too:   $R_{space}$, $C_{earth}$ and $R_{loss}$ Also; do you have any advises and/or any corrections for me?


title: What is a good introductory text to astronomy

text: What is a good and easy to read introductory text for an adult with limited basic scientific knowledge to astronomy for someone without a telescope and lives in a big city and why do you think that text is good for a beginner?


title: Searching for a specific Feynman quote

text: Pardon that this is strictly speaking not a question about physics but about a quote about physics, but I'm not sure where else to turn to. A while back I read a quote by Feynman in which he says something to the effect that the more different ways one can solve a physics problem the more deeply one understands it. Unfortunately, I don't remember where I read this. I spent significant amount of time googling as well as looking through my Feynman lectures but could not locate it. Does anyone know where to find the quote? Thanks


title: What is the most natural classical polynomial complexity class that includes all of BQP and NP?

text: Since we know that there are some oracle problems which can be solved on a quantum computer, but not on an NP machine with the same oracle, the idea of nondeterministic (i.e. infinitely parallel) machine is not sufficient to describe what is going on in quantum mechanics. The question then arises--- what is? What is the natural classical machine which can simulate a quantum computer efficiently in polynomial time? What is the complexity class of this natural machine?


title: Balmer lines and their positions

text: I have two stars, both of which have Balmer lines at the same wavelength positions. The depths of the Balmer lines on each star are different, and the stars are different as well. Why does this pattern exist?


title: Equation for calculating the Sun's position

text: _SunPosition Calculator_ calculates the Sun's postion based on the location and time. What is the equation used for such a calculation?