Could anti matter collisions be or make dark matter?

I've recently seen that space is or could be a quantum vacuum full of particles like matter and anti matter appearing and possibly colliding causing in theory the same effects that dark energy has. My question is could Dark energy or dark matter be a left over waste product of matter/anti matter collisions?


$M^{+}_4$ Randall-Sundrum Brane Calculation

The basic Randall-Sundrum model is given by the metric, $$\mathrm{d}s^2 = e^{-2|\sigma|}\left[ \mathrm{d}t^2 -\mathrm{d}x^2-\mathrm{d}y^2 - \mathrm{d}z^2 \right]-\mathrm{d}\sigma^2$$ where $\sigma$ denotes the additional fifth dimension. Notice the brane is localized at $\sigma=0$; this 'slice' is precisely Minkowski spacetime. To compute the stress-energy tensor, I define a vielbien, $$\omega^\mu = e^{-|\sigma|}\mathrm{d}x^\mu \qquad \omega^\sigma = \mathrm{d}\sigma$$ where $\mu=0,..,3.$ Taking exterior derivatives and expressing in the orthonormal basis yields, $$\mathrm{d}\omega^\mu = \epsilon(\sigma) \, \omega^\mu \wedge \omega^\sigma$$ where we have defined, $$\epsilon(\sigma)=\theta(\sigma)-\theta(-\sigma)$$ which arises because of the absolute value function in the exponent, and $\theta(\sigma)$ is the Heaviside step function. By Cartan's first equation, the non-vanishing spin connections $\gamma^a_b$ are, $$\gamma^{\mu}_{\sigma}= \epsilon(\sigma)e^{-|\sigma|} \mathrm{d}\omega^\mu$$ Taking exterior derivatives once again, and expressing in terms of the basis yields, $$\mathrm{d}\gamma^\mu_\sigma = \left[ \epsilon^2 (\sigma)-2\delta(\sigma)\right]\, \omega^\mu \wedge \omega^\sigma$$ which arises by applying the product rule, and noting that, $$\frac{\mathrm{d}\epsilon(\sigma)}{\mathrm{d}\sigma} = 2 \delta(\sigma)$$ because the delta function is the first derivative of the step function. From Cartan's second equation, $$R^a_b=\mathrm{d}\gamma^a_b + \gamma^a_c \wedge \gamma^c_b$$ the components of the Ricci tensor are, $$R^\mu_\sigma = \left[ \epsilon^2 (\sigma)-2\delta(\sigma)\right]\, \omega^\mu \wedge \omega^\sigma$$ as the second term vanishes. By the relation, $$R^a_b = \frac{1}{2}R^a_{bcd} \omega^c \wedge \omega^d$$ we may deduce the Riemann tensor components, $$R^\mu_{\sigma \mu \sigma} = 2\epsilon^2 (\sigma)-4\delta(\sigma)$$ I believe, in this case, both tensors in the coordinate basis and orthonormal basis are identical. Therefore we obtain the rank $(0,2)$ Ricci tensor, $$R_{\sigma \sigma}=8\epsilon^2 (\sigma)-16\delta(\sigma)$$ As the only diagonal component, the Ricci scalar is identical to the Ricci tensor at $(\sigma,\sigma)$. Using the Einstein field equations, the stress- energy tensor is given by, $$T_{55} = \frac{1}{8\pi G_5}\left[ 4\epsilon^2 (\sigma)-8\delta(\sigma) + \Lambda\right]$$ where $\Lambda$ is the cosmological constant, and $G_5$ is the five- dimensional gravitational constant. The function $\epsilon^2(\sigma)$ is given by, $$\epsilon^2(\sigma)=\theta^2(\sigma)+\theta^2(-\sigma)-2\theta(\sigma)\theta(-\sigma)$$ The last term appears to be the delta function, as it is zero everywhere, but singular at zero. The first terms are unity everywhere, but undefined at zero, therefore, $$T_{55}= \frac{1}{8\pi G_5}\left[ 4\theta^2(\sigma)+4\theta^2(-\sigma) -16\delta(\sigma) + \Lambda\right]$$ However, this disagrees with Mannheim's _Brane-Localized Gravity_ which states, $$T_{ab}=-\lambda \delta^\mu_a \delta^\nu_b \eta_{\mu\nu}\delta(\sigma)$$ where $\lambda = 12/\kappa^2_5$. In his text, $T_{\mu\nu}\propto -\eta_{\mu\nu}$, but in my calculation the entire purely 4D stress-energy vanishes. I can only assume I've done something wrong.


How can I stabilize an unstable telescope?

I have an 80 mm refractor telescope on a tripod, but it shakes on every touch. It's very hard to see via 6 mm (x120) ocular. Even a little wind causes the image to become too unsteady. How can I make my tripod more steady?


Universe is expanding at enormous speed

I'm not an expert but I've come to understand that the universe is expanding at enormous speed. That means that all of the visible galaxies are moving away from us at great speed. I also came to understand that, eventually (in many many years), all the galaxies and all of the rest of the objects outside our own galaxy will move so far away from us that it will be impossible for us to look (or measure) them. This means that eventually our entire observable universe will be our own galaxy and it will be impossible for us to measure anything else, and scientific measurements at that point in time will not correspond to the actual reality... because data will show that other galaxies don't exist. But I have two questions regarding this phenomena:   1. How is it that the light won't reach us anymore? Is the expansion happening at greater speed than light itself, making it impossible for it to ever reach anything?   2. If scientific measurements, at that point in time, will prove to be wrong, because they will show that galaxies don't exist (while they actually do exist), doesn't that mean that something similar could be happening right now as well? We could be measuring something about the universe that we're _dead_ sure about, but it won't be the actual reality.


What are the anomalies with General Relativity?

If general relativity is the newest model of Gravity which is so far been proven. Does it still have any anomalies such as the problem of Mercury's orbit during Newtonian gravity period? If so are there other types of physics to be discovered?


Food kept in a container in my pressure cooker cooks slower

The pressure cooker is heated at the base to cook the food. The pressure cooker releases steam when the force exerted by pressure exceeds the counterweight, at which point the weight rises up allowing steam to escape. The time to cook is measured by the number of times steam is released. When food(dry lentils) is kept in a container inside the pressure cooker it does not cook as well as if it were dumped in there directly. The same amount of water is added to the food in both cases, though in the former case there is also some water in the base of the cooker outside the container. In both cases the cooker was allowed to release steam 3 times. I thought that based on Pascal's law the same pressure would be exerted on the water in the container as it would if kept there directly. Since the temperature is the same, in both cases the food should be completely cooked. However practical results differ. Please suggest a theory which explains this.


How does gravitational lensing account for Einstein's Cross?

Einstein's Cross has been attributed to gravitational lensing. However, most examples of gravitational lensing are crescents known as Einstein's rings. I can easily understand the rings and crescents, but I struggle to comprehend the explanation that gravitational lensing accounts for Einstein's cross. I found this explanation, but it was not satisfactory. ![enter image description here](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Einstein_cross.jpg/300px- Einstein_cross.jpg) Image from http://upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Einstein_cross.jpg/300px- Einstein_cross.jpg Source of the second image. ![enter image description here](http://i.stack.imgur.com/32yvT.png)


Was Einstein's Cross Predicted by Einstein's Theories?

> **Possible Duplicate:**   >  How does gravitational lensing account for Einstein’s Cross? Einstein's Cross is a fascinating phenomena for which I have asked explanation here. However, I'm also interested to know if Einstein's cross was a specific prediction that people had obtained from Einstein's theories such as General Relativity or if the phenomena was merely named after Einstein. The wiki page on Einstein's Ring explains how the gravitational lensing phenomena of rings was predicted by general relativity, however I have yet to discover evidence that Einstein explained why there are 4 dots instead of a ring/crescent.


N=4 SYM in terms of N=1- SO(6) in the yukawa term

I'm trying to write N=4 SYM in terms of N=1 superfields. I have the lagrangian $$\mathcal{L}=\frac{1}{16 k} \int d^2 \sigma \text{Tr} \big[W^a W_a\big]+c.c+\int d^4\theta \text{Tr}\big[\bar{\Phi}^i e^V \Phi^i e^{-V}\big]+\frac{\sqrt{2}}{3}\int d^2\theta \text{Tr}\big[\phi^i [\phi^j,\phi^k]\big]\epsilon_{ijk}+c.c $$ Where the $\Phi^i$ are chiral superfields and V is a vector superfield. In components, this is all fine except for the Yukawa terms $$\mathcal{L} \supset i\sqrt{2} f^{ABC} Z^{i\dagger}_A \psi^i_B \lambda_C - \sqrt{2}\epsilon_{ijk} Z^i_A \psi^j_B \psi^k_C+c.c $$ Where $A,B,C$, are $SU(N)$ gauge group indices, $i,j,k$ number my 3 chiral superfields, which have an explicit $SU(3)$ symmetry, the $Z^i_A$ are the complex scalars from my chiral superfields, $\psi^i_A$ are the fermions from my chiral superfields, and the $\lambda_A$ is the fermion from my vector superfield. The fermions combine into an fundamental $SU(4)$ multiplet $\chi^I=(\psi^i, \lambda)$, and I decompose my complex scalars into real ones in a fundamental $SO(6)$ (isomorphic to $SU(4)$) multiplet, $Z^i=X^a+iX^{a+3}$. I should be able to write the Yukawa terms as $$\mathcal{L} \supset f^{ABC} X^a_A C^a_{IJ}\chi^I \chi^J +c.c $$ essentially putting the scalars into the antisymmetric matrix representation of $SU(4)$, $X_{IJ}=X_{[IJ]}=X^a_{IJ}X^a$. So I need to show that the $C^a_{IJ}$ that I have are an invariant symbol of $SO(6)=SU(4)$, and thus my Lagrangian has that R symmetry. Not sure how to do that... a reference I found says they should be related to the $SO(6)$ gamma matrices (http://arxiv.org/abs/hep-th/0201253, below equation 3.1), but that hasn't been very helpful.


Traditional Transfer Matrix on the Potts model -- how it grows for strip lattices?

What is the transfer matrix size for a strip lattice of width $n$ vertices, with arbitrary $q$??   I am not sure if it is $q^n$ x $q^n$ or something else. Any reference is also welcome.


Would you be weightless at the center of the Earth?

If you could travel to the center of the Earth (or any planet), would you be weightless there?


gravitational force

why does gravitational force of the earth decrease when we move towards the center of the earth whereas it depends on the radius of the earth.The radius of the earth also decreases when we move towards the center


Does gravity change the closer you are to the Earth's core?

Very simple question that has bugged me for some time, does the acceleration produced by gravitation (e.g. $9.8\: \mathrm{m/s^2}$) would change if we placed an observer closer to the planet's core? I mean, if I were to sit on the center of the Earth, would I still feel the Earth's gravitational field?


How does gravity work underground?

Would the effect of gravity on me change if I were to dig a very deep hole and stand in it? If so, how would it change? Am I more likely to be pulled downwards, or pulled towards the edges of the hole? If there would be no change, why not?


Why gravity decreases as we go down into the earth?

We all know that gravity decreases as the distance b/w the two increases. Hence $$ g = G \frac{Mm}{r^2} $$ hence acceleration due to gravity is inversely proportional to the distance b/w the 2 bodies. Then why the gravity decreases as we go deep into the earth?


Gravitational force of the Earth

Why does gravitational force of the Earth decrease as we move towards the centre of the Earth? Where as inverse square rule says that distance is less than gravitational force is more.


What is the gravity at the center of the Earth?

> **Possible Duplicate:**   >  Would you be weightless at the center of the Earth? Supposing there is a cavity at the center of the Earth, what is the gravity there? What will be its direction and intensity? Will a body be attracted toward the center of mass of the Earth?


The metric Tensor inside a massive shell

Given a fixed shell with the mass of $M$ and a radius $R$ , what would be the metric tensor for $r<R$? I do know that using Birkhoff Theorem the metric for $r>R$ should be schwarzschild. I'm not sure how to solve $G_{\mu\nu}=0$ for the inner part, and I'm not sure if I can demand continuity at $r=R$.


What the effect of gravity in the center of a planet feels like

> **Possible Duplicate:**   >  Would you be weightless at the center of the Earth? The issue of weightlessness at the center of the Earth has obviously already been discussed, however I am curious as to know what will it feel like as a human. Will it be absolute weightlessness as experienced by being in space, or will the mass of the Earth surrounding you pull you apart in all directions in a free floating expansion with your eyeballs popping out, etc.


Is spacetime flat inside a spherical shell?

In a perfectly symmetrical spherical hollow shell, there is a null net gravitational force according to Newton, since in his theory the force is exactly inversely proportional to the square of the distance. What is the result of general theory of relativity? Is the spacetime flat inside (given the fact that orbit of Mercury rotates I don't think so)? How is signal from the cavity redshifted to an observer at infinity?


Gravity in the center of a hollow neutron star

Imagine a hollow 100 metre diameter (for example) sphere made of incredible dense material (ie neutron star dust etc) but is self supporting (ie the central cavity). Assuming that the sphere skin is reletively thick so that the whole object is exceptionally heavy (eg the mass of planet earth. What would someone who was stationed dead center of the sphere feel? IE would they be crushed by the potential gravity or ripped apart and smeared on the inside other sphere cavity (due to the gravity of the surrounding material?


Why gravity decreases as we go under ground?

We all know that gravity decreases as we go upward, we also know that gravity decreases as we go inside the earth? I don't know why gravity decreases as we go downward or inside the Earth? Please explain?


What is the value of $g$ at centre?

What is the value of $g$ at the centre of the Earth? Is it zero or infinity? My attempt: I know it's zero but applying law made me on fused.


Gravitational acceleration at half Earth's radius

What would be the gravitational acceleration at half at Earth's radius? Something tells me it should be proportional to the mass distributed in that part, but I am not sure. Of course, we assume we know what's the acceleration at Earth's surface. Also, for the sake of it let's assume Earth has uniform density and is a perfect sphere.


quarks annihilation process

I am appreciating particle physics and I read about mesons. In quark's model, mesons are pairs of quark-antiquark. Now I think that in general matter- antimatter annihilate and so I don't understand how meson could be possible. Why don't a couple up-antiup annihilates? Thank you!


Besides transmutation, is there any way to "speed up" the decay rate of radioactive material?

I'm well aware of transmutation as a way to effectively make radioactive material decay faster, however that isn't really what I mean. Doing a quick Google search I found references to several theoretical treatises on the subject, and a few references to experiments with, at best, inconclusive results. Among them, there were a few on the quantum anti-zeno effect, which I already knew about, as well as a couple on something called "field enhanced beta decay" and various methods utilizing lasers. Nevertheless, much to my frustration, there was essentially no example I could find with any sort of empirical testing of these effects. So, I suppose I have two questions:   1. How "good" is the physics behind these claims?   2. Have any of these effects ever been _unequivocally_ observed experimentally?


Did spacetime start with the Big bang?

Did spacetime start with the Big Bang? I mean, was there any presence of this spacetime we are experiencing now before big bang? And could there be a presence/existence of any other space-time before the big bang?


Before the Big Bang

I've heard this saying before I don't know about anyone else. It says, "What ever was before the big bang is _something physics can't explain..!_ Is this saying true (accurate)?


Did time exist before the creation of matter in the universe?

Does time stretch all the way back for infinity or was there a point when time appears to start in the universe? I remember reading long ago somewhere that according to one theory time began shortly before the creation of the universe. Does time have a starting point of note?


Big Bang Anybody?

> **Possible Duplicate:**   >  Did time exist before the creation of matter in the universe?   >  on causality and The Big Bang Theory I was recently watching a Discovery special on the Big Bang theory, and after a couple of drinks and a few detailed discussions with others, I have found that I do not believe the Big Bang ever happened. My reasoning is thus: no matter the precision in digressing the causation of material currently in our universe, a point arrives where the scientists give up (as far as I'm concerned) and say that the Big Bang happened out of nothing. So . . . these scientists say that there was nothing, then the Big Bang happened, and then the scientists go on to explain the glories of the universe in detail without discussing how something came from nothing! Here's my critique: if nothing is nothing, then there is no possibility of something coming from it, because there is nothing. I don't know how we came into existence, other than the possibility that the universe never had a beginning and has always been and is infinite, but that's why I'm asking the Mises community. Any ideas?


Is this a great flaw in big bang theory?

Einstein said that, Time & Space cannot exists without one another. Big bang says, time didn't exist before the big bang. So the Primordial ball referred in the Big Bang theory didn't had any space inside it ?? If primordial ball had space in it, then where did the time go ?? Is this a flaw in big bang theory ?? Or Am I missing something ??


Does the Big Bang need a cause?

> **Possible Duplicate:**   >  on causality and The Big Bang Theory Asking here in layman's terms.. When theoretical physicsists discuss the origin of our Universe, the wider consensus appears to be that it originates from a singularity; a position that rests on observations about the apparent expansion of our Universe. However, the question why singularity itself came into being, still remains. But this question takes a different angle: Does the Universe as a whole need a cause to exist at all? If the law of conservation of energy is universally valid, then questions about a beginning become irrelevant since there can't be any by definition and everything boils down to dynamics of interaction. Thoughts?$$\mbox{ }$$


on causality and The Big Bang Theory

With the notion of causality, firmly fixed by GR, we derived the concept of a singular point from where space-time begun. Causality alone gives us the possibility to talk about a known past (i.e. every effect has a cause) and by this we can trace back in time every event. To reach the moment of the Big Bang we only have to extrapolate backwards the Hubble Law. (I see this like a reversal of effect with cause in every point of space-time.) With all this said, my question is about the moment of Big Bang; if even time started _there_ , we have an effect without a cause and causality breaks. In this perspective, when did causality started to play a role in the evolution of the universe? Does this path allow for a pre-Big Bang theory? Like a universe, where all its mass has been turned into energy (and in this case, with no mass present, the ideas of small and large are equivalent. And an infinite universe would be equivalent with a point sized one. Hence, the possibility of point like starting place for the Big Bang, but with preexisting _time_ ). From the point of view of causality we reach a logical nonsense at the moment of the BB. How can we modify the BBT to avoid this problem? (Or is has been done already?)


> **Possible Duplicate:**   >  Did spacetime start with the Big bang?   >  on causality and The Big Bang Theory   >  Before the Big Bang What's a compact scientific answer to question "(Why there is) / (what is before) the Big Bang?" Some fellow physics / science students ask question "what is before the Big Bang?" or alternatively "Why there is a Big Bang" (assuming the time dimension and hence 'before' is ill-defined at the event of Big Bang). We couldn't get any qualitative answer we universally agree on. Is there some compact way to answer this?


Initial conditions of the origin of the universe

I'm not quite sure this question fits the format of this site but I try to word it the best I can to comply the rules. The question is simple: How far can we go talking about the origin of the universe before admitting that the initial conditions cannot be explained without postulating some kind of god-like, physics-unexplainable, force/whatever? I'm interested to know if there's a mainstream physics theory that aims to prove this.


How can a circuit function with two negative battery terminals facing each other?

Here is a drawing of the circuit that is confusing me: ![enter image description here](http://i.stack.imgur.com/Kag3r.png) I don't quite understand how batteries work in this diagram. If a battery has a negative and positive terminal, there must be a barrier preventing them from neutralizing one another, so how can the potential from either negative terminal ever make it through the top half of the circuit without passing through a battery?


Why doesn't one-photon-irreducible function have any pole at $q^2=0$?

I'm reading the QFT textbook by Weinberg. In volume one chapter 10 page 451, at the lower part of the page he says, > Now, because $\Pi^*_{\mu\nu}(q)$ receives contributions only from one- > photon-irreducible graphs, it is expected not to have any pole at $q^2=0$. $\Pi^*_{\mu\nu}(q)$ is the sum of all one-photon-irreducible graphs, with the two external photon propagators omitted. Weinberg states it within one sentence as if it's self-explanatory, but I cannot understand why it is true. Is there something simple I missed? **Update:** I think what Weinberg had in mind was Luboš Motl's answer, that why he's so brief. In addition Peskin & Schroeder used the same reasoning in page 245: > ...the only obvious source of such a pole would be a single-massless- > particle intermediate state, which cannot occur in any 1PI diagram However P&S also put a footnote immediately after: > One can prove that there is no such pole, but the proof is nontrivial. > Schwinger has shown that, in two spacetime dimensions, the singularity in > $\Pi$ due to a pair of massless fermion is a pole rather than a cut; this is > a famous counterexample to our argument. There is no such problem in four > dimensions. Thus my original question stands justified. I'd be grateful if one can give a reference that elaborates P&S's footnote. Of course explanations by any SE user himself/herself are even more welcomed.


Do mankind and manmade activities/constructions have any effect on the rotation of the Earth?

We walk or ride on our vehicles to our destinations daily. Does our movement have any effect on the rotation of the earth according to Newton's law? What will be the effect if we move all the peoples along with their vehicles at their maximum velocity in one line in one direction along the equator? How much effect will it make?


Why can't supersonic planes "just fly higher" to go faster while maintaining cost?

First post to this site, and I've got at most a high school background in physics - I really appreciate any answer, but I may not be able to follow you if you're too advanced. I suppose this goes for regular planes too, but I'm especially interested in supersonic planes. I read some reports in the news about various people working on commercial supersonic travel, but there were a lot of comments attached to these news posts listing essentially what were physics constraints that would make such travel severely cost ineffective:   1. "skin friction," causing high heat and stress, leading to different metaled (thus more expensively researched/manufactured) and heavier (thus less efficient) airplanes.   2. increased drag, requiring more fuel to overcome.   3. sonic booms. I'll leave alone sonic booms - I understand as well as I can why this could be hard to engineer around, and why various countries have made generating them over land illegal. The other two I don't get. After spending some time on wikipedia this evening, if I've got this right, it seems that, holding the shape of the airplane constant, skin friction, lift, and drag are each equal to a scalar times density times velocity squared. Density drops as the altitude raises, which seems to mean to me that you could keep drag, lift, and skin friction constant when increasing speed by merely increasing altitude. I assume this is right, so I guessed that the "gas mileage" issue had to do with needing to burn too much more gas to achieve thrust required for the higher velocity. And yet, the wikipedia article on jet engines states that the concorde was actually more fuel efficient than some conventional subsonic turbofan engines used in 747's. Given all this, what did I get wrong? Why can't supersonic planes just fly higher to be as cost effective, or more, than conventional subsonic commercial jetliners, using the same construction materials? Relatedly, why do current jets have service ceilings and max speeds (assuming it's not just about the high stress of breaking through the sound barrier)? Thanks!


Does light induce an electric current in a conductor?

I know that electromagnetic waves induce electric currents in conductors and that's the basis for radio, wi-fi etc. I also know that light is also an electromagnetic wave. So, can light induce a current in a conductor (like a metal wire? or a coil?). And, if the answer is yes, is the same visible for other high-frequency waves (X-rays, gamma)? I heard about the photoelectric effect, but it seems related to the particle theory of light (photons transferring their energy to electrons). So, do high-frequency electromagnetic waves generate electric currents? Is it possible to measure them? Is the skin effect relevant here?


3D movie glasses making white light look red and blue

While waiting for a 3D movie to start, I was playing with the glasses they give you. I understand each lens has different polarized filters, so the left and right superimposed images on the screen go to the correct eyes. The first thing that tripped me up was that rotating the glasses didn't affect the light that passes through it. After searching I bit I discovered about _circular_ polarization, which ignores the angle of the filter and seems to be the standard for cinema glasses. The second thing that tripped me up were the wall lamps. When looking through one lens, the light seemed to have a bluish color. From the other, a reddish/orange color. It was subtle, but other people confirmed seeing it. I figured the lamp's white light had blue and red components, which are of different wavelengths (almost opposite in the visible spectrum, if I remember correctly), but what does the wavelength have to do with circularly polarized filters? And if this line of thought is correct, why does it seems to divide the visible spectrum?


Consequences of Compactness in Physics

If we understand spacetime as a $4$-dimensional manifold $M$, from the point of view of physics what are the consquences of a subset of it being compact? My point here is simple: in math we usually think of compactness as some analogue of finiteness because it shares many properties with finite sets, but what are the consequences of this when we deal with physics? Of course, we need not to go into relativity, we can even think about the usual three space $\mathbb{R}^3$. What are again the consequences of a set $A \subset \mathbb{R}^3$ being compact? Are there any cool things we can get out from this, or we simply use compactness in physics to grant the mathematical properties desired without having any direct impact in the way we understand and interpret those sets? Thanks very much in advance.


Gravitational time delay and contraction of matter

How can any matter contract to its Schwarzschild radius if gravitational time dilation clearly states that all clocks stop at that point. So any contraction any movement would stop. If that is so why all this talk about objects which can never form in the first place?


Black hole formation as seen by a distant observer

> **Possible Duplicate:**   >  How can anything ever fall into a black hole as seen from an outside > observer? Is black hole formation observable for a distant observer in finite amount of time? Specifically, let's take uniform non rotating distribution of 10 solar masses within solar radius, assume no outward pressure, and calculate its gravitational collapse. I assume distant observer would never witness complete formation of black hole with event horizon corresponding to entire 10 solar masses. What would be observable in finite amount of time, then? For example, would there be a black hole with smaller radius, while the rest of mass is still falling into its event horizon?


Would dense matter around a black hole event horizon eventually form a secondary black hole?

> **Possible Duplicate:**   >  Black hole formation as seen by a distant observer Given that matter can never cross the event horizon of a black hole (from an external observer point of view), if a black hole is "fed" with a large amount of matter then the new matter will eventually become extremely compressed, and presumably would be compressed below its Schwarzchild radius. Would secondary black holes eventually form near the original black hole? As an alternative one could also imagine that the combined mass of the original black hole and the new mass around the event horizon becomes contained within the Schwarzchild radius of both masses, and so a new event horizon forms, "swallowing" the new mass around the edge of the original black hole. This mechanism would allow black holes to swallow mass in a finite time.   Would this contradict GR predictions?


Can a black hole actually grow, from the point of view of a distant observer?

> **Possible Duplicate:**   >  Black hole formation as seen by a distant observer I've read in several places that from the PoV of a distant observer it will take an infinite amount of time for new matter to fall past the event horizon into a black hole, due to time dilation effects. This seems to imply that once a black hole is formed, its mass cannot actually grow any further as considered by most observers in the universe. Is that right?


Contour for Klein-Gordon field transition amplitude

In calculating transition amplitude for Klein-Gordon real-scalar field, I encountered the integral, $$ \frac{-i}{2(2\pi)^2\Delta x} \int^{\infty}_{-\infty} \,dk \frac{ke^{ik\Delta x}}{\sqrt{k^2+m^2}} $$ I can see here the integrand has branch cuts at $ k= \pm im $ However, later they do a change of variables $ z= -ik $ and then the integral becomes, $$ \frac{1}{2(2\pi)^2\Delta x} \int^{\infty}_{m} \,dz \frac{ze^{-z\Delta x}}{\sqrt{z^2-m^2}} $$ And it is said that they can wrap the contour around the upper branch cut for $ \Delta x > 0 $ ![enter image description here](http://i.stack.imgur.com/9qmWP.jpg) I am not able to see how this transformation happens and how the contour can be wrapped around the upper branch. Thanks for your inputs.


What limits the maximum attainable Fermi Energy for a material experimentally?

Either through doping or gating. What are some good terms to search for if I'm looking for some experimentally obtained values for particular materials? I'm particularly interested in what the limit is for graphene, if anyone knows. For example, the DOS for regular graphene has states between -3t and +3t, but what is the maximum fermi energy we can examine experimentally? Is there a breakdown of the sample at some limit of doping or voltage?


A confusion in the Derivation of Lorentz Transformation

![http://thecuriousastronomer.wordpress.com/](http://i.stack.imgur.com/7xqyI.png) My doubt is in the equation (1) and (2). Aren't x,y and z also the radiuses? **EDIT** Thank you guys for trying to give a wonderful explanation but I figured out the answer myself and it was just my silly interpretation. I thought the the value of ct on x axis would be equal to ct and forgot that the value is given by a perpendicular and not an arc (of the sphere). I was thinking that where the sphere touches x axis, that is the value of ct but it isn't. The value of ct for x is given by a point on the axis that lies perpendicular to the axis.


so this might have some duplicated inquiry that this question or this question had, and while i think i have some of my own opinion about it, i would like to ask the community here for more opinions. so referring to Duff or Tong, one might still beg the question: why is the speed of light 299792458 m/s? don't just say "because it's defined that way by definition of the metre." before it was defined, it was measured against the then-current definition of the metre. why is $c$ in the ballpark of $10^8$ m/s and not in the order of $10^4$ or $10^{12}$ m/s? similar questions can be asked of $G$ and $\hbar$ and $\epsilon_0$. EDIT: to clarify a little regarding $c$. i recognize that the reason that $c\approx10^9$ m/s is that a meter is, by no accident of history, _about_ as big as we are and a second represents a measure of how fast we think (i.e. we don't notice the flashes of black between frames of a movie and we can get pretty bored in a minute). so light appears pretty fast to us because it moves about $10^9$ lengths about as big as us in the time it takes to think a thought. so the reason that $c\approx10^9$ m/s is that there are about $10^{35}$ Planck lengths across a being like us ($10^{25}$ Planck lengths across an atom $10^5$ atoms across a biological cell and $10^5$ biological cells across a being like us). why? and there are about $10^{44}$ Planck times in the time it takes us to think something. why? answer those two questions, and i think we have an answer for why $c\approx 10^9$ in anthropometric units. EDIT #2: the other two questions referred do **not** address this question. Luboš Motl gets closest to the issue (regarding $c$) but does not answer it. i think in the previous EDIT and in the comments, i made it (the question) pretty clear. i was **not** asking so much about the **exact** values which can be attributed to historical accident. but there's a reason that $c \approx 10^9$ m/s, not $10^4$ or $10^{12}$. reworded, i s'pose the question could be " _why are the anthropometric units (which are about as big as we are) as large as they are relative to their corresponding Planck units?_ " (which **is** asking a question about dimensionless values.) if we answer those questions, we have an answer for not just why $c$ is what it is, but also why $\hbar$ or $G$ are what they are.


The origin of the value of speed of light

Meaning, why is it the exact number that it is? Why not 2x10^8 /mps instead of 3? Does it have something to do with the mass, size or behavior of a photon? To be clear, I'm not asking "how we determined the speed of light". I know there isn't a clear answer, I'm really looking for the prevailing theories.


What causes precession or nutation in a spinning object?

1. What causes precession in a spinning object?   2. What causes nutation in a spinning object?    3. What causes a top, gyroscope, and the earth to wobble?  Just because it's a simple question, I'm not expecting a simple answer, but please do summarize whatever you say in laymen terms, thanks.


QCD phase diagram in the large N limit

I am sending a couple of questions which seem a bit more specific than others on this site, partially to probe if there is a point in doing so. Not sure what is the range of expertise here, and no way to find out without trying, so here goes: I am wondering what is known about QCD, or other field theories, in the regime of large density and low temperatures, specifically studied in the large N limit. I know of the qualitative picture at finite N, but lots of the instabilities (e.g. the superconducting ones) are suppressed at large N and replaced by other interesting phenomena. I am only aware at the moment of the DGR instability to form chiral density waves, and I am wondering what else exists in the vast and possibly quite old literature. Any pointers or entry points to that literature will be appreciated


How does Cooper pairing work?

Cooper pairs are one of the models how superconductivity is explained. What still baffles me is how a vibration of the crystal lattice (the so-called phonon) can interact with the electron (an actual particle), in such a way that it then creates a coupled pair with an other electron... What is the explanation for this behaviour? What is the maths behind it?


Small object in air

What happens with very small spherical objects ($d=1\mu m$, e.g. a bacterium) in air? Do they fall? How quickly? Does it depend on their mass? We often see objects of little mass e.g. leaves falling from trees, but these object ususally have a very large surface, so their behaviour is very different to spherical objects. _Disclaimer: I'm a biologist._


Why do these equations result an incorrect unit for acceleration?

![](http://i.stack.imgur.com/BB7m9.jpg) Hello everyone. Imagine an object moving around a certain point on a circular orbit. Magnitude of the velocity is constant during the motion ($|v|$). The orbit radius is $r$. (I'd better notice that we're just talking about **kinematic** view of this motion.) According to the image I've uploaded, we'll have: $\large v_x(\theta)=|v|\cdot \cos\theta$ $\large v_y(\theta)=|v|\cdot \sin\theta$ Since perimeter of the circular path is $2\pi r$, and magnitude of the velocity is constant, we'll have: $\large\theta (t)=\frac{|v|\cdot t}{2\pi r} \times 2\pi =\frac{|v|\cdot t}{ r}$ Now we can combine these equations: $\large v_x(\theta)=|v|\cdot \cos(\frac{|v|\cdot t}{ r})$ $\large v_y(\theta)=|v|\cdot \sin(\frac{|v|\cdot t}{ r})$ By this point, everything is okay. But the problem occurs here, where we try to get derivative of $v_x(t)$ and $v_y(t)$ in order to find $a_x(t)$ and $a_y(t)$. As we know by differentiation we have: $\cos^{\prime}(x)=-\sin(x)$ $\sin^{\prime}(x)=\cos(x)$ And we know that acceleration(time) function is derivative of velocity(time). So: $\large a_x(t)=(v_x(\theta))'=|v|\cdot -\sin(\frac{|v|\cdot t}{ r})$ $\large a_y(t)=(v_y(\theta))'=|v|\cdot \cos(\frac{|v|\cdot t}{ r})$ Well, now something is wrong: These two equations result a $m/s$ unit (or something like that) for acceleration, but that's wrong. Acceleration unit must be $m/s^2$, (or something like that). The question is that: **Where does this problem come from?** I couldn't figure it out at all. I don't know, maybe some kind of misunderstanding about derivative concepts cause that. So please try to answer simple, clean as much as possible.


Restriction on vector fields

The 2D vector field (x,-y) does not transform like a vector under rotation(Arfken Vol. 1)! Does this mean we cannot have such a vector field physically?


Gravity on flat object

I was wondering how gravity would behave on object of different shapes.   1. If the Earth was squeezed into a thin disk what would the gravitional acceleration be at the center of the flat surface? Would it be really low because the amount of matter beneath me would be small? If I stood on the edge would the gravitational acceleration be enormous because the amount of matter beneath me was huge?   2. If an object is lowered into the Mariana trench will the effect of gravity increase because it gets closer to the center of the Earth?


How does this paper relate to standard QED?

This paper proposes a microscopic mechanism for generating the values of $c, \epsilon_0, \mu_0$. They state that their vacuum is assumed to contain ephemeral (meaning existing within the limits of the HUP) fermion/antifermion pairs. This affects the mechanism of photon propagation as follows: > When a real photon propagates in vacuum, it interacts with and is > temporarily captured by an ephemeral pair. As soon as the pair disappears, > it releases the photon to its initial energy and momentum state. The photon > continues to propagate with an infinite bare velocity. Then the photon > interacts again with another ephemeral pair and so on. The delay on the > photon propagation produced by these successive interactions implies a > renormalisation of this bare velocity to a finite value Now this description of what happens to the photon sounds awfully like a heuristic description you might give to the photon self energy contribution in standard QED perturbation theory, where, although it's used in the renormalization procedure, it most certainly doesn't make any changes to the velocity of light. Can someone explain how what's being proposed here relates to standard QED. I don't see how they fit together.


Can one define an acceleration operator in quantum mechanics?

It seems most books about QM only talk about position and momentum operators. But isn't it also possible to define a acceleration operator? I thought about doing it in the following way, starting from the definition of the momentum operator: $\hat{p} = -i\hbar \frac{\partial }{\partial x}$ Then we define a velocity operator in analogy to classical mechanics by dividing momentum by the mass $m$ $\hat{v} = \frac{-i\hbar}{m} \frac{\partial }{\partial x}$ In classical mechanics acceleration is defined as the time derivative of the velocity, so my guess for an acceleration operator in QM would be $\hat{a} = \frac{-i\hbar}{m} \frac{\partial }{\partial t} \frac{\partial }{\partial x}$ Is that the general correct definition of the acceleration operator in QM? How about relativistic quantum mechanics?


Vectors and motion in a plane

> A particle travels with speed $50 m/s$ from the point $(3,-7)$ in the > direction $7i-24j$ . Find its positional vector after 3 seconds. **My approach** : It has travelled a distance of 150m in the direction given by the unit vector $\frac{7\hat i - 24 \hat j}{25} $ . So, its position is now $210\hat i -720 \hat j$. But it started from $(3, - 7)$, so I have to subtract that to get $197\hat i - 713\hat j$. But I think this answer is awkward, and I may have gone wrong. So, please tell me if I am right, and if yes, can you suggest any shorter way of doing it?


Polchinski equation 4.3.16

I am trying to obtain the polchinski's equation 4.3.16 which is following $ Q_B^2 = \frac{1}{2}\\{ Q_B, Q_B \\} = -\frac{1}{2}g^{K}_{IJ}g^{M}_{KL}c^Ic^Jc^L b_M =0$ Where $ Q_B = C^I(G_I^m +\frac{1}{2}G_I^g)$ and $C^I$, $b^J$ are anticommuting(ghosts) and $[G_I, G_J]=ig^K_{IJ} G_K$,   $G_I^g = -ig^K_{IJ} C^Jb_K $ are ghost parts and $G_I^m$ are matter part and they satisfy above commutation relations What I have done are $\\{ Q_B, Q_B \\} = \\{ C^I(G_I^m +\frac{1}{2}G_I^g), C^J(G_J^m +\frac{1}{2}G_J^g)\\} =\\{C^IG_I^m, C^J G_J^m\\} +\frac{1}{2} \\{C^IG_I^m, C^JG_J^g\\} +\frac{1}{2} \\{C^I G_I^g, C^JG_J^m\\} +\frac{1}{4} \\{C^IG_I^g, C^JG_J^g \\} = C^IC^J [G_I^m, G_J^m ] +\frac{1}{2} C^IC^J [G_I^m, G_J^g] +\frac{1}{2}C^IC^J[G_I^g,G_J^m]+\frac{1}{4}C^IC^J[G_I^g, G_J^g] =C^IC^J [G_I^m, G_J^m ]+\frac{1}{4}C^IC^J[G_I^g, G_J^g] = C^IC^J ig^K_{IJ}G_K^m+\frac{1}{4}C^IC^J ig^{K}_{IJ}G_K^g =C^IC^J ig^K_{IJ}G_K^m+\frac{1}{4}C^IC^J g_{IJ}^K g^M_{KL}C^Lb_M $ compare with the textbook $\\{ Q_B, Q_B \\} = -g^{K}_{IJ}g^{M}_{KL}c^Ic^Jc^L b_M$ My calculation is something wrong. How can I fix it?


Addes mass forces: can a force depend on acceleration?

My friend and I had a little discussion about added mass forces. I always interpreted $F=ma$ as a cause-effect relationship, so I find rather uneasy to accept that the cause can instantaneously depend on the effect. Is it fine to have a force which depends on an acceleration, in classical mechanics? I came up with some possible solutions to this:   1. It's perfectly fine for **F** =m **a** to be an implicit equation with respect to **a**.   2. The time derivative of the velocity appears as the result of an approximation of a time-delay.   3. It arises due to assumptions made on the nature of the fluid (i.e. incompressible).   4. None of them


Photons emitted at the event horizon?

While looking through the questions, a came across a section about black holes. I immediately though; what would happen if an atom is orbiting a black hole and emitted a photon perpendicular to the event horizon, going away from the black hole. How would light going away from a black hole react to the gravity? Photon "stuck" on the event horizon of a black hole actually talks about a photon stopping at the event horizon, but my question is about a photon just outside of the event horizon and if the photon is slowed down.


Why does centre of mass of ice-container system shift in absence of any net external force?

Consider a cube of ice in a flat based container(the base is very broad).The temperature of the system is at first fixed at a minus Celsius temperature, but then the system is left on a table with the top open to atmosphere.   The ice starts melting, and finally there is only water, spread over the container's base(water doesn't touch walls of the container,so it looks like water spilled on the floor)   Here, the centre of mass of the $H_2O$-container system moves downward because of the lowering of height of water molecules.   But,this system experiences no net external force on it,all the time.Hence, the centre of mass of this system shouldn't accelerate at all!   How is this contradiction sorted out?I feel that since melting is very slow, the centre of mass might move only very slowly, but still that doesn't explain things.For example if the room was very hot, melting wouldn't have been slow, right? Edit:Many people are probably getting confused regarding what perspective I'm taking.Let the container and water(solid/liquid) be a single system.We can think of them together as a point mass.This point mass is at equilibrium,and at rest,situated at the position of the centre of mass of ice+container system(let's say,at a height 'h' above the table.   There may be internal forces happening inside the point mass,but the net external force(resultant) is zero,all the time.Hence, according to Newton's first law,the point mass must remain at equilibrium and hence,at rest.But,when ice melts,the position of center of mass of ice+container system has moved down!   Hence the point mass has to move down, in the absence of any resultant external force.The gravitational force on ice+container is cancelled by normal reaction of table on container.


Shape Created by a Pile of Granular Objects Dropped Uniformly

I've observed many times that if you drop a lot of a 'granular' substance in one place and keep the nozzle out of which the substance flows, that the shape of the pile created very much resembles a bell curve. The situation is a bit hard to explain so for example image somebody holding a small pipe vertically above the ground and dropping a large amount of sand through that small pipe. The shape created by the sand on the ground largely resembles a bell curve wrapped around the vertical axis. I come from a math background and am really unsure of how to even start proving or disproving the proposition that the shape formed by the grains is a bell curve. Any help and input would be appreciated.


Calculating the Probability Current of a Travelling Wave

Calculate the probability current density vector $\vec{j}$ for the wave function : $$\psi = Ae^{-i(wt-kx)}.$$ From my very poor and beginner's understanding of probability density current it is : $$\frac{d(\psi \psi^{*})}{dt}=\frac{i\hbar}{2m}[\frac{d\psi}{dx}\psi^{*}-\frac{d\psi^{*}}{dx}\psi]$$ By applying the RHS of the above equation : $$\frac{i\hbar}{2m}[-A^{2}ikxe^{-i(ωt-kx)}e^{i(ωt-kx)}-A^{2}ikxe^{i(ωt- kx)}e^{-i(ωt-kx)}]$$ This gives : $$\frac{-2iA^{2}ik\hbar}{2m}=\frac{k \hbar A^{2}}{m}$$ This is not the correct answer. :( What have I done wrong ? In the model workings instead of A in the complex conjugate of the wave function they have written $A^{*}$. Why is this necessary since $A$ is likely to be a real number anyways ?


Is it possible to extract energy from mass rotating in space?

Consider a single isolated rotating mass (for example planet), is it possible to extract energy out of its rotation? If yes, how could that theoretically be achieved?


Physical Explanation of Being Able to "Think"

This may look like a philosophical question, but I'm looking for physical explanations (if there's any), that's is why I'm asking it here. What is the ability of thinking? We are all creatures consist of flesh and bones. Our brains are also nothing but flesh and water? Why are we thinking? What happens to this thinking power when we die? How and why does it disappear? Does it really disappear, or go to somewhere else in a sense we don't know and understand yet? We observe that, different creatures have different capabilities of thinking. The creature which has the most advanced thinking ability is human. Besides human cats can think up to some degree; they know how to hunt, they decide where to hide and where to find food from. Ants think too, even if it is weaker then cats can do; they find food and quickly run away when you try to pick one of them from floor with your hand, because they sense danger and decide to escape from there. Cells can also think in some sense, but it is much less capable that we don't even call it "thinking". Most living creature sense themselves as separate being. What happens if we stop the time, we compile and exact verbatim copy of a normal human being, then start the flow of time once again? Would the second copy think too as the original one? Or would the second copy start his life in vegetative state? Or would it just be a stack of dead flesh? How does physics explain this? Is there any particle that causes us to think? One century ago, we didn't even dream that this many sub atomic particles existed. Can the ability of thinking be a cause of some unknown physical particle of a flow of vector field (like electromagnetic field) which hasn't been discovered yet? Do you expect that in the future some scientist(s) would discover this mystery? Is there any research going on this? What do we know about "thinking" today? (Note: There wasn't appropriate tags for my question. I would appreciate if another user with higher reputation could add some tags for this question and remove this note.)


A brief explanation needed regarding newton's law in particle

So based on the topic, this is the question. "A particle with mass $m$ is moving along x-axis with $v_0$ at $t=0$ and $x=0$. The particle is acted by an opposing force with magnitude proportional to the square of velocity. Find out the a) velocity b) position and c) acceleration of particle at any time ($t>0$)" I have the answers and solution for this but hardly understand them (I dont understand at all actually. Plus where does β come from?) Basically I just need a clarification on question a) as b) and c) are related to the first question. This is a self study for final exam. Answer provided by my friend but she also copied it from a source. So without understand it we do it correctly. But it seems to be useless. So I hope you guys can help me. This is the answer. a) velocity ![Part a](http://i.stack.imgur.com/nU9kp.jpg) b) position ![Part b](http://i.stack.imgur.com/PknJE.jpg) c) acceleration (t>0) ![Part c](http://i.stack.imgur.com/cY0hx.jpg)


Not so simple problem using momentum, energy and angular velocity...?

I have an object in free space (no gravity) with angular momentum $ = \omega_i $, and some velocity vector $=\vec{V_i}$. To simplify we will say it has a mass-less rigid rod length $ = \ell $, connecting two small masses both of mass $ = M $. The masses are small in the sense of a radius equal to the rod radius both much smaller than $ \ell $. Of course for simplicity keep this in the 2D plane. I know precession can play a role, but accounting for it can make the problem easier, I've already done that. Now, we want to change the velocity of the object by $ \Delta V $. We must do this by taking a bit of the matter of mass $ = m $ off the object and get it moving away at a velocity $ = \vec{V_m} $. While keeping $M>>m$ What will be the most energy efficient mechanism to do this? What will be the most momentum efficient mechanism to do this? The final answer, like with the precession related solution, gives some $ \vec{V_m} $ value parallel or perpendicular to $\vec{V_i}$, and some mass "$ m $" proportional to "$ M $". The goal will be to obtain maximum $ \Delta V $ with minimum energy or momentum. While keeping $M>2m$ Right now I am getting different solutions based on minimizing energy vs. momentum, is this logical? Why? This has helped me get to my solution thus far. Thanks JCooper and Maksim Zholudev. **EDIT:** The answer should be at least partially derived using formulas. Some relation between $ \Delta V $ and the input energy or momentum must be shown.


Friction on an object moving with momentum over a surface

I'm familiar with the equations for friction for a static object and an object moving at steady speed over a surface from high school physics. But we never learned how an object moving only due to momentum experiences friction. This is something I've modeled several times while building simple 2D games, but I have no idea if what I have made matches reality in any way. Is the force of friction dependent on the speed of the object across the surface? Or is the force constant as the object decelerates? Is there a simple equation giving the acceleration (deceleration) of the object depending on it's mass and velocity?


Photon number conservation during scattering

I was reading this writeup on the Kompaneets equation and the Sunyaev- Zel'dovich effect. On page 3, section 2 the author states > There is no way to increase the mean energy of a planckian distribution > without changing the particle number. But as far as I understand, photon number is conserved during a scattering process, and therefore isn't it possible that for a given Planckian distribution all the photons gain the same energy through scattering and thus increase the mean energy of the Planckian distribution? Or am I missing something here?


A question about polarization in quantum mechanics

We start our question we a definition A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if \   1. $P$ is Lagrangian   2. P involutive   3. dim$P\cap\bar P \cap TM$ is constant Now, Introduce an hermitian form on $P$ defined by  $$b(X,Y)=i\omega(X,\bar Y)$$ . Note that when $P$ is real then $b$ is vanishes identically on $P$. Cinsequently, $b$ projects onto a non-degenerate form on the quotient $P/{(P\cap \bar P)}$ and we denote it by $\bar b$. $P$ is said to be of type $(r,s)$ if and only if $\bar b$ has signature $(r,s)$ i.e. its matrix is $$diag(\underbrace{1,1...,1}_{r},\underbrace{-1,-1...,-1}_{s} )$$ for $0\le r+s=n-dim_{\mathbf C }P\cap \bar P$ . Then, $P$ is said to be positive if $s=0$. In the case if $r=s=0$ then $P$ is real Philosophically the main goal in defining polarizations in quantum mechanics is to find the wave functions that are covariantly constant along its directions. When the form $b$ is not positive show that there are no globally defined wave functions


Gravitational compression / compression in general

How exactly does gravitational compression, or compression in general, increase temperature? It seems counter-intuitive seen as temperature usually increases from the solid to the gas phase.


Photons and Black holes

How many photons in one Planck volume would it take to form a tiny black hole? A photon doesn't have mass but it does have energy, $1.0101 \times 10^{-37}$ Joule for red $650$ nm wavelength light if I'm correct. A photon is a point- like boson so an infinite number of photons can fit into any given area. So more photons is equal to more energy which would bend space-time even more till they themselves wouldn't be able to escape from their own bend in space- time.


Can modern twistor methods to calculate scattering amplitudes be applied to renormalization group calculations?

As explained for example in this article by Prof. Strassler, modern twistor methods to calculate scattering amplitudes have already been proven immensely helpful to calculate the standard model background in searches for "new physics". If I understand this correct, the "practical" power of these methods lies in their ability to greatly simplify the calculation of scattering processes, which are due to limited computer power for example, not feasable applying conventional Feynman diagrams. Depending on the system considered, a renormalization group transformation involves the calculation or summation of complicated Feynman diagrams too, which usually has to be simplified to obtain renormalization group equations which are numerically solvable in a finite amount of time. So my question is: Could the new twistor methods to calculate scattering amplitudes be applied to simplify investigations of the renormalization group flow, in particular investigations of the whole renormalization group flow field beyond a single fixed point, too? Are such things already going on at present?


Can Mirror box simulate long light travel?

Imagine that there is a cube box that has mirrors all 6 faces in . If we use a strong laser and enter in the box from a small hole on the box. The laser light travels in the box long time that we can detect the laser via a detector on other hole of the box. 1) Is it possible to simulate the long light travel in it (for example a day or week)? 2) Is it possible to proof that there is no ether via that box? If we move the box in a fixed velocity what we can observe about receive time on detector. I ask a question first time in this website If it is asked question or not appropriate for your format sorry for that.


Uniform distribution of charge on a plane and on a bar: conditions to have no motion of the bar

On the plane z=0 there is a superficial charge distribution such that $\sigma$ is constant. Near to the plane, there is a bar, charged uniform with total charge q. At the extremities the bar has two constraints, so it can't turn. If I want to find the constraints force and the force momentum needed to block the bar, can I consider the charge q as a single point charge and put it in the midpoint of the bar?


Is spin just due to uncertainty in angular momentum?

I can't seem to think of any way to envision electron spin. Can it be thought of as the uncertainty in angular momentum?


Conservation of Angular Momentum, as related to a flywheel

Trying to work out some pesky flywheel dynamics for a project I'm working on, would love some for your assistance to better understand the underlying concepts. For a given flywheel (thin-walled cylinder, assume a spoked bicycle wheel) rotating in the x-y plane, I'm trying to calculate the force generated in either direction along the z-axis. It seems to me, in line with Newton's first law of motion extended to rotational dynamics, what forces are physically being generated that prevent a rolling wheel from falling over?


How to calculate fluid(oil / hydrocarbon) loss under pressure

I'm trying to calculate the amount of fluid that would flow through an area dependant on the amount of pressure that there is. I'd also like to know the rate at which it would flow. Essentially I have a very basic model of a well drilling system. At the moment the variables / parameters for each of the objects in question are defined by the user but may be set to constants to enable testing. The assumptions I'm working on are: The viscosity of the oil / hydrocarbons is set to constant. Once drilled the area of the hole will not change (there will be no collapse etc). The well is never ending so constant pressure. If someone could point me in the right direction I'd be very thankful. Some of the assumptions may be way off so guidance in general is also appreciated, many thanks.


Representation of a Signal and its evolution via Quantum System

I am a total beginner in the field of Quantum Mechanics. So, the question I am asking may be a silly one. So kindly give me possible answers or advice for modifications. Recently I am learning the concept of qubit. The quantum theory tells that a $n$-qubit system is represented by a unit vector in $(\mathbb{C^2})^{\bigotimes{n}}$ with some basis set. Now, we can also express a $2^n$ dimensional complex signal as a vector at any point of time, and with suitable normalization it is nothing but a qubit. Now, as far as I know, the evolution of a qubit is always dictated by a unitary operator, whereas, the evolution of a signal can be dictated by any arbitrary operator. So, is there any way, any sort of transformation that allows,given a signal and its evolution, to create its equivalent representation as an evolution of qubit, so that we can solve problems of signals using methods of Quantum Mechanics.


Why does the water-pressure of my shower fall if I hold the shower head high?

I live in a very old house - build 1902 - in the 4th floor in the city of Karlsruhe (Germany). I have a shower and the gas-heater for the water is in it: ![enter image description here](http://i.stack.imgur.com/tpiBK.jpg) If I hold the shower head over some height (see image), the water pressure is suddenly reduced. Whats the reason for that? (I guess the way the pump works might be the reason.) ![enter image description here](http://i.stack.imgur.com/rvn6k.jpg)


What were Feynman's objection(s) to a cubic lattice universe?

In this video of Feynman discussing the scientific method, starting at around eight minutes and 30 seconds, Feynman describes the proposition that space consists of a cubic lattice of points (as opposed to space being continuous). Feynman says "then we can prove that [the proposition] immediately is wrong," but he doesn't elaborate on the proof. > My question is: What are the objection(s) to a cubic lattice universe that > Feynman refers to in this video? I'm not looking for every possible objection to a cubic lattice universe. I'm wondering what in particular the (apparently immediate and obvious) objections are that made Feynman so dismissive of the idea.


Conserved quantities in generalized n-body problem

Given a collection of point-particles, interacting through an attractive force $\sim \frac{1}{r^2}$. Knowing only $m_1a=\sum_i \frac{Gm_1m_i}{r^2}$ and initial conditions we can deduce the motion of the system. Consequently we can observe that three quantities remains constant A) center of mass of the system B) total energy C) angular momentum How can we derive these 3 facts directly from $m_1a=\frac{Gm_1m_2}{r^2}$ ? Are these quantities conserved for any attractive force $\sim\frac{1}{r^n}$ ? Given any monotonically decreasing force for $r$ in $(0,\infty)$, which are the conserved quantities?


X-ray and neutron powder diffraction: peak broadening due to crystal size

When analysing powder diffraction patterns, the broadening of peaks can be used to estimate crystal sizes. Smaller crystal size gives larger broadening according to the Scherrer equation: $$ \beta = {{K\cdot\lambda}\over{D\cdot\cos \theta}}.$$ What is the physical origin of this effect? Edit: Added homework tag as this is related to my thesis on soil studies using diffraction methods.


Torque definition and right hand rule not arbitrary

I have read the following: http://www.feynmanlectures.caltech.edu/I_20.html#Ch20-S1 The formula for $\tau_{xy}$ is derived in this chapter: http://www.feynmanlectures.caltech.edu/I_18.html#Ch18-S2. In this derivation until equation 18.11 ($\Delta W=(xF_y-yF_x)\Delta\theta$) the term $xF_y-yF_x$ does not seem to be arbitrary. It would not make any sense the other way around like: $yF_x-xF_y$, because of the equations 18.6 ($\Delta x=-PQ\sin\theta=-r\,\Delta\theta\cdot(y/r)=-y\,\Delta\theta$) and 18.7 ($\Delta y=+x\,\Delta\theta$). Then $\tau_{yz}$ and $\tau_{zx}$ are derived (in the first link) by symmetry. $$\begin{alignedat}{6} &\tau_{xy}~&&=x&&F_y&&-y&&F_x&&,\\\\[.5ex] &\tau_{yz}~&&=y&&F_z&&-z&&F_y&&,\\\\[.5ex] &\tau_{zx}~&&=z&&F_x&&-x&&F_z&&. \end{alignedat}$$ * * * I have some ideas and wonder whether or not they are true: I have got the feeling that they are wrong but I have no idea why they should be wrong.   1. _So, since the formula for the torque in the $xy$-plane ($\tau_{xy}$) is not arbitrarily-derived, there is to torque $\tau_{yx}$, since $yF_x-xF_y$ would not be correct in equation 18.11._   2. In my first link, below equation 20.1 there are 2 pictures which shall demonstrate how the letters $x$, $y$ and $z$ can be interchanged. _How would that make sense?_ Since equations 18.6 and 18.7 are true, we "live" in a right-handed coordinate system and $\tau_{yx}$ would not make any sense.   3. Since $\tau_{yx}$ would not make any sense, the arguments in my first link down to equation 20.9 show that the right-hand rule is not arbitrary. What is wrong with my arguments? What does $\tau_{yx}$ mean?


1D Acoustical Relations beyond nearest neighbor couplings

Consider some 1D Lattice of atoms with nth neighbor coupling of strength k_{n}. I'm looking for the dispersion relation for acoustical phonons under these conditions. I start with the Lagrangian, $$L = K- V$$ $$L = \sum^{\infty}_{n} \frac{1}{2}m \dot{x}_{n}^{2} - \sum^{\infty}_{p=1} \frac{1}{2}k_{p} \\{(x_n-x_{n+p})^2 + (x_n - x_{n-p})^2\\}$$ Mass is the same for each atom. The Lagrange equation should be $$m \ddot{x}_{n}=\sum_{p=1} k_p(x_{n-p}+x_{x+p}-2x_n)$$ Now, if I use a travelling wave solution as an ansatz, I should get my dispersion relation as some infinite series. Is this correct? If so, help me out because I can't make it work. Thanks!


Absorption cross section and absorption coefficient

What is the absorption cross section, how is it measured? How to convert it to the absorption coefficient (measured in cm$^{-1}$)?


characterization of potential

So i have a force field $F(x,y)$ and i have to find out wether it is a potential or not. My first idea was to calculate : $dU=Fdr$ (where $r$ is the radius vector) , to integrate on both sides and hence to see if $U$ is path dependent or not. That turned out to be right. I was thinking about another approach: we know that , if a potential exists then $F=-\nabla U$ So if $F$ is a potential it must be possible to find such a $U$ , wich in my case wasn't possible, hence $F$ is not a potential. My question: (1) Is the second approach right? (2) are there othere ways for solving this problem? Thanks in advance.


What are local electrons in a crystal?

I am reading Pekar's "Research in Electron Theory of Crystals" and I came across a passage I find a bit unclear: > The theory developed below takes into account the dielectric polarization of > a an ionic crystal by the electric field of the conduction electron. The > local polarization that results from this is related with the displacement > of the ions and consequently is inertial. It cannot follow the relatively > rapidly moving electron and therefore forms a potential well for the > electron. The depth of this potential well turns out to be sufficient for > discrete energy levels of the electron to exist in it. The electron, being > in a local state on one of these levels, can maintain with it sown field the > aforementioned local polarization of the crystal. Because of their inertia, > the ions are sensitive not to the instantaneous value of the electron field, > but to the average field. The latter can be calculated as the static field > of the $|\psi|^2$ cloud of the electron; it produces a static polarization > potential well, which in turn maintains the electron stationarily in a local > state. Such states of the crystal with the polarization potential well, > which in turn maintains the electron stationarily in the local state. Such > states of the crystal with a polarization potential well, in which the > electron is localized, were called by the author polarons Now, what exactly does he mean by local vs. conduction electrons? Are local electrons those that are not moving and are in the crystal? That doesn't seem right. What does it mean for an electron to be in a local state? (Also what does he mean by "inertial"?) IN fact, it would be nice if one explains this passage in understandable terms so that I can have some intuitive picture in mind.


Zigzag flow of water along a vertical glass window

I've observed this behavior many times. When it rains, the rainwater will form vertical channels along a glass window. The flow of water is mostly confined within these vertical channels and the channels are (more or less) stable. But sometimes - and I suspect this happens when the flow intensity in one of the channels increases - the channel will switch from a vertical configuration into a zig-zag configuration. The zig-zag is composed of short segments running horizontally that are connected by semicircular (vertical) segments. The zig-zag is unstable and lasts only for 0.1 second or so. Then the channel reverts to its vertical configuration. I have made photographs of this behavior but I cannot find them now. I have seen similar patterns in the book "The self made tapestry" by Philip Ball, page 145. This shows growth instabilities in glass cracks. Is says "at higher speeds the crack becomes oscilatorry with a constant wavelength". This is what I see in the water flow. It feels counterintuitive. There must be a good explanation for this behavior. Can you point me to it? EDIT Here is a video .


What's the most fundamental definition of temperature? Is it the definition concern about average energy, number of micro states, or what? By "fundamental", I mean "to be applied" in such general cases as Black Hole's Temperature, Accelerated Frame's Radiation,...


Determine stationary angular velocity of wheel with circuit in magnetic field

I have a wheel (free to spin around the $z-$axis) with four spokes that is connected by sliding contacts to a circuit with $U_0 = 0,72V$. Also, there is a B-Field parallel to the $z-$axis ![Imgur](http://i.imgur.com/aLF3N7S.jpg) For the induced electric potential I have: $$U_i = - \frac{1}{2}(R_{outer}^2 - R_{inner}^2)\omega B$$ (with $\omega$ = angular velocity) I'm now asked to find out the constant $\omega_0$ after the system is in a stationary state (moves with constant speed). The assignment points out to look at one mesh (with one spoke) on the wheel and to determine if Kirchhoffs second rule applies and if there is a current flowing in a mesh. The most obvious way I can think of would be (since $U_0$ should be equal in any spoke): $$U_0 - U_i = 0$$ and to solve for $\omega$. But elsewhere I was told that Kirchhoffs rules don't apply in systems with changing magnetic fields. Also I'm not sure if there still would be an emf induced in stationary conditions since the flow wouldn't change anymore then.


Instantaneous angular momentum of a disc

> _Suppose we have a disk of radius $r$ and mass $m$ travelling at velocity > $v$. I want to calculate the instantaneous angular momentum with axis > through the edge of the disc (on the circumference)._ Angular momentum $= I \omega$. $I = \frac{1}{2}mr^2 + mr^2 = \frac{3}{2}mr^2$ by the parallel axis theorem. $\omega = \frac{v}{r}$. Therefore, angular momentum $= \frac{3mrv}{2}$. Alternatively, angular momentum $=p\times r= m r \times v = mrv$. Why do these two methods differ? Which, if any, are correct?


Why does space expansion not expand matter?

REFORMULATED: I have looked at the other questions (ie "why does space expansion affect matter") but can't find the answer I am looking for. My question: There is always mention of space expanding when we talk about the speed of galaxies relative to ours. Why, if space is expanding, does matter not also expand? If a circle is drawn on balloon (2d plane), and the balloon expands, then the circle also expands. If matter is an object with 3 spatial dimensions, then when those 3 dimensions expand, so should the object. If that was the case, we wouldn't see the universe as expanding at all, because we would be expanding (spatially) with it. I have a few potential answers for this, which raise their own problems:   1. Fundamental particles are 'point sized' objects. They cannot expand because they do not have spatial dimension to begin with. The problem with this is that while the particles would not expand, the space between them would, leading to a point where the 3 non-gravity forces would no longer hold matter together due to distance   2. Fundamental particles are curled up in additional dimensions ala string theory. These dimensions are not expanding. Same problems as 1, with the added problem of being a bit unsatisfying.   3. The answer seems to be (from Marek in the previous question) that the gravitational force is so much weaker than the other forces that large (macro) objects move apart, but small (micro) objects stay together. However, this simple explanation seems to imply that expansion of space is a 'force' that can be overcome by a greater one. That doesn't sound right to me. I think some of the problems in this question verge into metaphysics, but I think from his (?) previous answer, Marek can probably explain the physical side of things a bit more thoroughly. I will leave it there cos anything else I write sounds rambling!


Expansion of the space-time metric

If the space-time metric is expanding with the expansion of the universe, if I could travel back in time, would I be less dense than the matter in that previous era?


Expansion of space

> **Possible Duplicate:**   >  spacetime expansion and universe expansion? So I've heard that space is expanding very quickly and that the redshift we see when we look at other galaxies is evidence of this. But why doesn't this expansion affect orbits of the planets in our solar system, for instance? I would think that the expansion of space is uniform through the universe...why doesn't it tear our bodies apart?


does space-time expansion affect on fundamental particle?

does space-time expansion affect on fundamental particle or point particle?


Expanding Universe and more

We all are pretty familiar with the Friedmann models of the expanding universe, but at present I would like to concentrate on the model where the universe goes on expanding with the lapse of time. We know that if we mark spots on a balloon and inflate it, distance between these spots increases gradually(with time, of course). We may draw an analogy that these spots represent the galaxies. Now imagine a space within one of these spots, whose dimensions is indeed less than that of the spot concerned. With time, the dimensions of this space goes on increasing with time. If we compare this space to the interstellar(as well as interplanetary) space, and by the process stated above, go on imagining spaces within the one imagined before, we arrive at a limit where the rate of dimension variation of the space with time becomes very small in magnitude. Here I ask my question: could it happen so that objects in the universe(even solids) are expanding at a rate so very feeble(much much smaller than even the rate at which the universe expands), because all objects, be they solid or any other physical form, constitute some space within this universe, on account of its existence, and can be designated as a space within the spot considered in our balloon model. It is indeed a mere whim of mine to compare a universe model with that of a balloon-spot model! Nonetheless, please remove my doubts upon this trifling matter, that has perplexed me a good deal.


Effect of Cosmological Expansion at the Atomic Level

> **Possible Duplicate:**   >  Why does space expansion not expand matter? Does cosmological expansion have any effects at the atomic level?


How does the gravity well change as space expands? If we assume that the Earth's gravitational field curves flat space to create a gravity well then how does the gravity well change as space expands also is the change in gravity well measurable.


Why space expansion affects matter?

If space itself is expanding, then why would it have any effect on matter (separates distant galaxies)?   * Space is "nothing", and if "nothing" becomes bigger "nothing" it's still a "nothing" that shouldn't interact with matter in any way (it doesn't have mass, energy, etc).   * Gravity doesn't have a cutoff distance afaik, so even the most distant galaxies should be attracted to each other. Gravity force would be very very tiny, but it would still dominate "nothing" from space expansion.   * Lets take inertia from Big Bang into account. Inertia would be the primary force that moves galaxies away comparing to their tiny gravity and even more tiny, if any, force of our "nothing" that is still expanding inbetween. Wouldn't expansion decelerate if driven mostly by inertia?


If the universe is expanding then why does the distance between Sun and the Earth is not expanding?

It has been proved that universe is expanding in the accelerating way. If that is true than the space between the Sun and the Earth must also be expanding. But it is not so, why?


How do we know space is expanding when we are part of space?

From what I understand space itself is expanding, and the Big Bang attempts to describe this expansion at the very early stages of the universe. This is usually described in a visual way as 2 dots on the surface of a balloon as the balloon is being inflated. We exist in space, so as space expands we are expanding too aren't we? If this is correct how do we know space is expanding at all? I'll attempt to explain further... 2 dots on a balloon with a ruler drawn between them measuring 1 cm. As the balloon expands, so do the dots, and the ruler, and it always says that the distance is 1 cm. From an observers point of view (somewhere away from the balloon) the dots, ruler, and ruler are all getting larger - but from the point of view on the balloons surface nothing has changed. Another way to describe this would be a man in a room, the room and everything in it (including the man) are getting bigger at the same rate. From the man's perspective nothing is changing but from outside the room - you can see everything is getting bigger. I guess what I am trying to say is that we are in the universe, and expanding at the same rate as it (aren't we?) so how do we know it's getting bigger?


What exactly is expanding when they say universe is expanding?

We know that universe is expanding and galaxies are moving away from each other. Does this mean the galaxies are also expanding in itself and therefore I guess growing larger in volume? Depending on the answer of that, does the solar system also expands?


Why doesn't everything expand when the universe expands?

Everybody has been taught at one point, "oh the universe expands, but that doesn't mean that everything is expanding uniformly, since that means we can't detect the expansion, but only that huge galaxies are moving away from each other". But I'm rather confused. Can't the expanding universe simply be thought as a coordinate axis that expands? The definition of "1" on the coordinate axis constantly expands? After all, spacetime itself is expanding, and not only "average distance of galaxies". So our Planck length would expand, light wavelengths expand, we expand, etc. But clearly this isn't happening. Of course, the handwavy explanation is "electromagnetism/gravity overcomes expansion at small scales", but the electromagnetic and gravity forces are defined in terms of constants relative to our "coordinate intervals", so if spacetime itself is expanding "under the feet" of gravity, shouldn't gravity not be able to do anything about it? We would also be unable to measure the expansion of spacetime since all of our measurement benchmarks and tools are also expanding. Clearly, this isn't happening. What is the _precise_ reason?


Are atoms getting weaker?

If the universe is expanding, it would make sense that the spaces between particles are getting bigger. If this is so, then the particles which make up atoms are also affected. Does that imply the spaces between the components of an atom will become large for the subatomic forces to hold? Are atoms getting weaker?


Cosmic Expansion - Why aren't we ripped off yet?

According to Hubble's law, the universe expands exponentially ever since the big bang.   1. If the space-time expands, what effects does it have upon us, (1.Earth, 2.solar system and 3. Milky Way).    2. What would be the fate of Gravitational force and other forces of nature if the expansion continues.    3. What happens to the dark energy and dark matter then?    4. Won't all the dark energy be used up in the acceleration process?


Expansion of Universe

> **Possible Duplicate:**   >  Why space expansion affects matter? If the Expansion is prevalent, i.e. it is observable and true then shouldn't that result in the expansion of the Milky Way galaxy, eventually our solar system would expand resulting in increase in the planetary distances (from the sun). Could the escape of Pluto be an example to this phenomena ? And is the Expansion resulting in increase in the sizes of the galactic objects ?


Spacial curvature and expanding space

If we take the analogy that in an empty space the space is just a flat sheet then if there is a single planet or a star then the flat sheet will curve below the planet leaving a curvature shaped like a hemisphere below the object, my question is, Does this curvature change as space expands?


Why doesn't Brooklyn expand? (Or "Is the expansion of the universe kinematic?")

> **Possible Duplicate:**   >  Why space expansion affects matter? Imagine two tiny spacecrafts that are moving with the Hubble flow and so are moving away from each other. Let's assume that they've been that way since the very early universe, never firing their engines, just drifting along their Hubble flow geodesics in a homogenous isotropic universe. They then momentarily fire their engines so that they "cancel" the Hubble flow and have a fixed proper separation. Will they now start drifting apart again (presumably due to expansion)? Or will they stay at fixed proper separation, and maybe very slowly move towards each other due to their mutual gravitational attraction? I guess this is another way of asking whether expansion is kinematic and hence can be forgotten (so that "Brooklyn isn't expanding" because gravitational collapse and structure formation have erased memory of the expansion). Or maybe someone will help me refine this question and make me realize I just haven't thought things through completely?


Space within galaxies. Is it stuck by the gravity of the galaxy or expanding and "slipping past"?

I understand that the inter-galactic space is expanding but galaxies themselves are not. What is happening to the space within a galaxy? Is it fixed by the gravity of the galaxy or is it expanding and moving out around it? Using the coins on a balloon analogy, is the balloon surface under the coins (galaxies) stuck to the coin and constrained or is it free to expand out, slipping underneath the coins leaving them in the same place (and the same size)?


spacetime expansion and universe expansion?

First of all, does the expansion of spacetime solely cause the expansion of universe? Secondly, if spacetime is the sole cause, do objects(matter with mass) themselves expand? Thirdly, by spacetime expansion, does time also expand? (at least in general relativity) Thanks.


How is the universe expanding?

> **Possible Duplicate:**   >  spacetime expansion and universe expansion? Is the space between planets is growing or the space between stars is growing or the space between galaxies is growing?


Expanding metre sticks

Given the Universe is expanding. Therefore everything within the U is expanding, in all dimensions, subatomic to cosmic. Then all metre sticks are expanding. The question: How can we measure U expansion with metre sticks that are also expanding? Alice only knew she had grown ten feet tall by comparing herself to her surroundings.


Do atoms expand with universe?

> **Possible Duplicate:**   >  Why space expansion affects matter?   >  Why does space expansion not expand matter? As we know, the universe is expanding, galaxies are away from each other. But what about atoms? Do they also in expanding? What's more, Bohr radius is $$a_0=\frac{\hbar}{m_e c \alpha}$$, if it is increasing, does it means $m_e$ is decreasing due to the density of Higgs field is getting thinner. or $c$ is decreasing or $\hbar$ is increasing?


How is it possible to measure expansion of universe

> **Possible Duplicate:**   >  Why does space expansion not expand matter? If according to Hubble's law space the expanding then shouldn't everything else along with the space expand and we should not be able to measure this expansion as our tools/measuring sticks would also expand in same proportion


What is the difference between matter & spacetime?

If the universe is expanding why doesn't the matter in it expand proportionally making it seem as if the universe is static? Alternatively, as spacetime expands why does it not just slide past matter leaving matter unmoved? What anchors the matter to a particular point in spacetime?


How is tritium illumination possible without negative health effects?

Turns out there's tritium illumination \- a tiny very strong plastic tube will be covered in phosphor and filled with tritium. Tritium will undergo beta decay and a flow of electrons will cause the phosphor to glow. This gives enough light for illuminating hours marks on a wristwatch dial and the hands of the wristwatch for many years and is claimed to not pose health hazard. Now how it is possible to have energetic enough radioactive decay and no health hazard at the same time?


What do subatomic particles look like for a 'speed of light - observer'?

Let's say that an observer is moving with the speed of light relatively to an atom that he wants to look into. He has equipment that precise that he can observe the atom and what is inside. From Einstein's theory we know that for light particles, everything else that moves with velocity smaller than the speed of light, 'looks like frozen, no move'. How would the elements inside the atom look?


Does the friction force change directions with a change in reference frame?

In a basic friction problem with Block A sliding on top of Block B, the direction of the friction force is usually explained as being simply the opposite of the direction of motion. So if Block A is sliding to the right, the friction force is pointing to the left. But this reasoning implicitly assumes that we are calculating friction force from the reference frame of Block B. What if we instead look at the problem from the reference frame of Block A? To Block A, it looks as if Block B is sliding to the left, so an observer on Block A would say that there is a friction force which, to oppose the direction of motion, points to the right. It seems counterintuitive and probably wrong for the direction of friction force to depend on reference frame like this. Where is the flaw in the reasoning above? Are the two reference frames described above not exactly equivalent in a way that leads to the force changing directions?


Books for linear operator and spectral theory

I need some books to learn the basis of linear operator theory and the spectral theory with, if it's possible, physics application to quantum mechanics. Can somebody help me?


1. What's the motivation behind the action principle?   2. Why does the action principle lead to Newtonian law?    3. If Newton's law of motion is more fundamental so why doesn't one derive Lagrangians and Hamilton principle from it?    4. Also does all Lagrangians obey $L=T-V$?    5. I think that it's related to the fact that the kinetic energy of the particle at all points on the path or it's travel time is as small as possible?    6. If so, How can we derive the principle of least action from this fact in detail?


Charge of an electrolytic capacitors

I can't understand the electrolytic capacitors, when a capacitor has a capacitance of 100 microfarads, does that mean that when it is charged with 100 volts will the charge of the plate be 0.01 coulomb? If there is a part of the plate with no isolation, then I touch it, I will be shocked with a charge of 0.01 coulomb and 100 volts?


I read that the number of atoms in the entire observable universe is estimated to be within the range of $10^{78}$ to $10^{82}$. Does the Universe have finite number of particles? If so, how could it be determined?


Positive Mass Theorem and Geodesic Deviation

This is a thought I had a while ago, and I was wondering if it was satisfactory as a physicist's proof of the positive mass theorem. The positive mass theorem was proven by Schoen and Yau using complicated methods that don't work in 8 dimensions or more, and by Witten using other complicated methods that don't work for non-spin manifolds. Recently Choquet- Bruhat published a proof for all dimensions, which I did not read in detail. To see that you can't get zero mass or negative mass, view the space-time in the ADM rest frame, and consider viewing the spacetime from a slowly accelerated frame going to the right. This introduces a Rindler horizon somewhere far to the left. As you continue accelerating, the whole thing falls into your horizon. If you like, you can imagine that the horizon is an enormous black hole far, far away from everything else. The horizon starts out flat and far away before the thing falls in, and ends up flat and far away after. If the total mass is negative, it is easy to see that the total geodesic flow on the outer boundary brings area in, meaning that the horizon scrunched up a little bit. This is even easier to see if you have a black hole far away, it just gets smaller because it absorbed the negative mass. But this contradicts the area theorem. There is an argument for the positive mass theorem in a recent paper by Penrose which is similar. Questions:   1. Does this argument prove positive mass?   2. Does this mean that the positive mass theorem holds assuming only the weak energy condition?


Normalizing the free particle wave function

One way to normalize the free particle wave function > "is to replace the the boundary condition $\psi(\pm{\frac{a}{2}}) = 0$ [for > the infinite well] by periodic boundary conditions expressed in the form > $\psi(x)=\psi(x+a)$" \-- _Quantum Physics_ , S. Gasiorowicz How does this work? What does this mean physically? Or more precisely, why does this approximation suffice? I understand that this makes the wavefunction square-integrable (when integrated from $x=0$ to $x=a$) hence normalizable. Thanks.


Hydrostatics: Log floating in water near a dam

So I am studying for a final and can't seem to solve this. There is a log floating in water and I need to find its weight. The question I have is what parts of the volume of the log count when summing the forces in the Y axis. ![enter image description here](http://i.stack.imgur.com/108N2.png) What i have now for AREA alone is $r^2 - .25\pi r^2$ pushes down and $.5\pi r^2 + r^2$ pushes up. however this does not take into account the air above the log in the upper right quadrant. is the force pushing up taking into account the half circle below the water and also the $2r^2$ above the horizontal of the log? The actual question for this image says: A log is stuck against a dam as shown in the diagram. given the radius of the log of 1.4 m and the length of the log into the page, 10 m find the weight of the log in kN.


How to solve Poisson equation in electrodynamics?

I learned electrodynamics. According to the vector potential determination, $$ \mathbf B = [\nabla \times \mathbf A ], $$ Coulomb gauge, $$ \nabla \mathbf A = 0, $$ and one of Maxwell's equations, $$ [\nabla \times \mathbf B ] = \frac{1}{c}4\pi \mathbf j, $$ I can assume, that $$ [\nabla \times \mathbf B ] = \nabla (\nabla \mathbf A) - \Delta \mathbf A = -\Delta \mathbf A = \frac{1}{c}4 \pi \mathbf j. $$ How to prove that the one of the solutions of this equation is solution like newtonian potential, $$ \mathbf A = \frac{1}{c}\int \limits_{V} \frac{\mathbf j (r) d^{3}\mathbf r}{|\mathbf r - \mathbf r_{0}|}? $$


Sliding force less or equal?

Why is the force required to slide a magnet off a steel plate A LOT less than the force required to directly pull it off? The force required to pull the magnet can be: 20lb While the force required to slide the magnet can be: 1lb more/less. Why is that?


Does the termination point(level) influence water flow from a pipe

I get water to my home from a nearby Tank A at a certain height above ground level. I have a 1" pipe through which I get this water to my home.. I leave this water into my well by connecting a 1" tube to this pipe. Reason for question: I have seen water pressure vary(lower- more flow) depending upon the height I hold my pipe(on the outlet side) when I empty my fish tank, keeping how deep the pipe is immersed on the other side a constant. Now, if I elongate and leave the pipe from Tank A at deeper level below ground level should the flow rate increase?(will i get more water in the same time) as apposed to a shorter pipe which still goes into my well. or simply: Will I empty my overhead tank on 2nd floor quicker if I use the tap on the ground floor instead of the tap on the second floor. The taps being same size.


Reflection, transmission, absorption...how to calculate them?

I was wondering whether there is an equation that enables me to calculate the reflection, transmission, absorption and polarization, when the electric field everywhere is given? Consider this: You have solved the full Mie scattering process, so incident field, the field in the sphere and the scattered field are known. How can one calculate those quantities then?


Why does the car clock dim down so much when the headlight is turned on?

Even on a current new 2012 car, when the green LED clock inside the car is quite bright, but when the car's headlight is turned on, the clock dims down to only about 1/4 of its brightness, which makes the time hard to see. I thought the clock requires merely a watt or even less for its brightness, and the car stereo which output at least 30 watt of music (if comparing a 30 watt speaker for the PC to the car stereo), won't be less loud when the headlight is turned on. Why does the clock dim down so much? Is it explained by Ohm's law? Also, can't it be "parallel" instead of "in series" with the headlight, so that the brightness is not affected?


Inclined conveyor belt mystery

**The situation** An inclined conveyor belt with topmost point $h$ height above the ground receives sand at a constant rate $X$ from a container at a negligible height above the lowermost point of the conveyor belt. There is sufficient friction on the conveyor belt so that the sand stops almost immediately after coming in contact with the belt. **The question** On trying to find the minimum force required to maintain this situation, miraculously (after making some appropriate assumptions) the force needed at the bottom most point for the sand to start moving and the force required to maintain the motion of the sand on the conveyor belt above that point have to be same for the total force exerted by the conveyor belt to be minimum. Mathematically it is easy to reach to this conclusion but I can't understand why this happens physically.


What is the relation between General Relativity and Newtonian Mechanics?

What is the relationship of General Relativity and Newtonian Mechanics? Namely, which laws does GR replace of Newtonian Mechanics, and which laws of Newtonian Mechanics are incorporated into it. Or is GR a complete replacement and overhaul?


Step in a proof that $\textrm{div} \ \mathbf{B} = 0$ from Biot-Savart's law

Notation: The magnetic field $\mathbf{B}$ generated by a point charge $e$ moving with velocity $\mathbf{v}$ is given by Biot-Savart's law $$\mathbf{B} = \frac{\mu_0 e\ \mathbf{v} \wedge \mathbf{r}}{4\pi r^3}$$ where $\mathbf{r}$ is the vector from the charge to the point at which the field is measured, $r = \left| \mathbf{r} \right|$, and $\wedge$ denotes vector product. Question: According to my book: Since $\mathbf{r}\, / \, r^3 = - \,\textrm{grad} \left(1\, /\, r \right)$, we have $$\textrm{div} \left( \mathbf{v} \wedge \frac{\mathbf{r}}{r}\right) = \mathbf{v} \wedge \textrm{curl} \left( \textrm{grad} \frac{1}{r}\right) = 0.$$ What I have is $$\textrm{div} \left( \mathbf{v} \wedge \frac{\mathbf{r}}{r}\right) = \mathbf{v} \, . \, \textrm{curl} \left( \textrm{grad} \frac{1}{r}\right) - \left( \textrm{grad} \frac{1}{r} \right) \, . \, \textrm{curl} \ \mathbf{v},$$ so I think there is a typo in the book ($ \, \wedge$ should be $\, . \,)$. However, I still don't know how to go from my equation to the correct one. Is it because $\textrm{curl} \ \mathbf{v} = 0$? If so, why? I understand that the curl of a gradient vanishes identically.


If a planet were cut in half but stayed hemispherical, how hot would the ocean on the flat side be?

Some time ago I asked a question about gravity on a hemispherical planet. What would gravity be like on a hemispherical planet? Would the water all boil away at first, quickly cooling the core of the planet? Would the ocean boil for centuries?


Electric Field: distributed uniformly in one infinity tape of length

> _One charge density surface is distributed uniformly in one infinity tape of > length with $2a$ width from distance $d$. Determine the Electric Field in > the point perpendicular from the distance $d$ of the centre of tape._ Answer: > $\frac{\rho_s}{\pi\epsilon_0}\tan^{-1}(a/d)$ PS.: I tried every ways (Coumlomb Law and Gauss Theorem) Gauss Theorem (I guess closer) Consider cylindrical surface: $Q = \int_v \rho_v dv = \rho_s.2a.L$ (Call L as infinity length) $Q = \oint_s \vec{D}.d\vec{S} = D_r . 2\pi.d.L$ $\Rightarrow E = \large\frac{\rho_s}{\pi\epsilon_0}\frac{a}{d}$ Coulomb http://i.stack.imgur.com/Ks4gC.jpg The image woth more than thousand words. Please I'd like some advise to solve this problem. http://i.stack.imgur.com/Ks4gC.jpg