title: Why can't light escape from inside event horizon of Black Holes?

text: The simple answer: Its because Gravity of Black Hole there doesn't allow it. See also this and this Phys.SE posts. Isn't it a classical answer? When we're unable to connect Gravity with Quantum Theory, how can we apply classical force concept on a denizen of Quantum realm? My main concern is Quantum Tunneling. Why can't a photon just break the barrier based on Uncertainty principle? I know about Hawking Radiation, but it also doesn't tell anything about it. Cosmic fluctuations of vacuum at event horizon boundary creates pair of particles; one goes inside and one is released with Quantum Tunneling breaking gravitational barrier at that point. What's the real explanation of light not being escaped from inside Black Hole?


title: Approximations in simple pendulum

text: In the approximation $$-(g/ \ell) \sin \theta \approx -(g/ \ell) \theta $$ we make an error $R$ which is $O(\theta ^3)$. If i did well my calculations it is estimated by $$R\leq|(g / \ell)(\theta^3/3!)|$$ however this isn't essential to my two questions. **Q#1** : We do the estimation of the error for $\ddot \theta$. How does the error reflect on the solution? A first rough idea that comes to my mind is the following: we are negliging a factor $R$ in the second derivative of the angle, so approximating $R$ as constant, the error on $\theta$ must be something like: $$ \int _0 ^t \int _{0} ^{t'} R dt' dt'' = 0.5 R t^2 .$$ **Q#2** After getting to the solution: $$\theta(t) \approx \theta _0 \cos (\sqrt \frac {g}{\ell} t) $$ having set $\dot \theta (0) =0$, our teacher did the following. He reconsiders the equation: $$\ddot \theta =-(g / \ell) \sin \theta \approx -(g / \ell) \theta (1-\theta ^2 /3!)\approx -(g / \ell) \theta (1-\frac{<\theta ^2 >}{3!}),$$ where: $$<\theta ^2>=\dfrac{1}{T} \int _0 ^T \theta ^2 \, dt .$$ Now, to evaluate the integral, _he uses the first solution_ : $$\dfrac{1}{T} \int _0 ^T \theta ^2 \, dt =\dfrac {1}{T}\int _0 ^T \theta _0 ^2 \cos ^2 (\sqrt {.} t)dt=\frac{\theta _0^2}{2}$$ Hence: $$\ddot \theta \approx (-g/\ell)(1-\dfrac{\theta _0 ^2}{12})\theta ,$$ which should be better from the first solution. Now, apart from the lot of approximations that maybe will make you feel uncomfortable like me, second question is: is it logical to use the first rough approximation to calculate $<\theta ^2>$, which we need for the better approximation? Shouldn't the second solution be independent from the first to improve it? Thanks to anyone who wants to help.


title: Is a blackbody real or imagined?

text: In my reading of blackbody radiation I am always asked to imagine this or that body being a perfect absorber or emitter of radiation, and I am always left with the impression that a blackbody exists only as a theoretical construct. But is it? Or can one be constructed and tested?


title: Special relativity, spacetime, velocity and units

text: We all know that space and time are the fundamental units, means no mathematical expression can express their relation to other variable fundamentally. But as we know that moving rod has a contracted length and dilated time which depend on the velocity of that rod w.r.t. an inertial frame in which the motion is happening. So it's a natural question that space and time measurements are not alone to be measured in any inertial frame but one has to specify velocity also. It sounds like 3d space and 1d time have another dimension (velocity) to be included in their totality. So velocity is also a form of space and time. So what is wrong here?


title: william herschel discovering infrared problem

text: when william herschel conducted the experiment of separating white light with a prism and measuring the different colors, he put a thermometer past the red color as a control finding it to pick up the most heat, thus concluding infra red. my problem is... Infra red doesn't pass through glass... so how did he pick it up from the prism? was it not glass (i think it was), did it slow the wave down to make it into infra red, or does very near wave infra red pass through it differently? please can someone shed some light !


title: Studying the logical structure of physics as a mathematical object per se?

text: I was wondering is there a branch of mathematical physics which studies the underlying logical structure of physics as a mathematical object per se? Let me explain what I mean by that. I'm interested in a theory which takes an _axiomatic_ approach to physics, clearly distinguishing between what is postulated or in any other way "swept under the rug" and what is clearly a logical consequence of what is taken to begin with. I imagine something like theory of axiomatic systems in mathematics, which studies both internal consistency of an axiomatic system and relative consistency of systems. Obviously, I imagine it would also be necessary to integrate experimental physics to it with fields like probability theory in an attempt to quantitatively express the certainty of some consequences in physical theory. In an ideal case, that theory would produce results which could perhaps allow us to say how certain or how speculative some physical theory is, but not just qualitatively. I believe you can see where I am going. Is there a field like that out there? Or something similar to it?


title: Is a beta decay possible with out the emission of an electron in the process?

text: Is a beta decay possible with out the emission of an electron in the process ? Beta decay involves the formation of a proton and an electron from a neutron.


title: Information loss

text: First time poster! I just burnt a piece of paper containing a 5 digit number I made up randomly and as far as I am concerned no one else will ever be able to retrieve the information contained on that piece of paper. So to me this information is lost irretrievably. From my reading of the literature (more pop science than technical papers) the consensus is information is never lost in a black hole, but is my simple experiment an example of the fact that information can be lost? How can any time reversal process ever possibly recover that five digit number?


title: Time Contraction

text: This is my first time posting on this site. I am a computer programmer that stumbled across a physics text book and have a question on special relativity. So firstly, I understand that there is no preferential inertial frame of reference. Secondly, as a A body travels faster relative to another body B at rest, body A experiences time dilation, effectively resulting in a forward jump in time. Thirdly, the speed at which a body in relative motion can travel is limited by the speed of light (c). Due to the the first premise (that there is no preferential frame), could we not also argue that the body at rest is actually travelling at a negative velocity relative to the other body? Thus, if we consider the situation in from this reversed perspective, a time contraction is then experienced to the body moving at relative negative velocity. For instance, if we consider a person on Earth as being at rest (0 km/h), then could we think of a person standing on Pluto (which has an orbital velocity of 0.159 relative to Earch) as having a negative velocity relative to that of the person stationary on the surface of Earth, thus experiencing time contraction relative to the time experienced on Earth? This can be confirmed by substituting a negative value in the standard time dilation equation for u^2. This results in a value greater than 1 occurring as the divisor, causing the numerator to 'shrink' in value, thus resulting in a 'time contraction' rather than time dilation. With some searching, I have come across a number of examples of time dilation. However, all of them seem to assume that a body on rest is traveling at 0 km/h. Rather, I argue that such a body would be 'relatively' at rest, as from a greater perspective, this body is subject to the turning of the Earth about its axis, the movement of the Earth along its orbit around the Sun, the Sun around the Milky Way, and finally the Milky Way away from other galaxies due to the expansion of the universe. If my above reasoning is correct, then it seems that if there was a body moving at 0 km/h relative to the expansion of the universe, then the Earth would be travelling faster than that body, which means that the Earth would experience time dilation relative to this body, and that body would experience time contraction relative to the Earth. If time dilation equates to a forward jump in time, then it stands to reason that time contraction would likewise equate to a backwards jump in time. Although, it occurs to me that perhaps it is more useful to not think of the body moving backwards in time, but rather to think of it as being stationary relative to the expansion of space-time, thus time is moving forward past the body experiencing time contraction. I know I am not a physicist, but I was wondering if anyone could tell me if my above speculations are correct, or that I'm an idiot, or if all of this is already well documented somewhere (which I have yet to discover). Any clarification would be greatly appreciated.


title: Any example of lower symmetry in high temperature phase than the low temperature phase?

text: All the phase transition cases I came across so far have this property: the lower temperature phase has lower symmetry than the higher temperature one. But it is nowhere explicitly said that, lower temperature phase always has lower symmetry than the higher temperature phase. So, I was thinking, is there any counter-example to show that higher temperature phase in a phase transition can have lower symmetry too?


title: Do Nuclear Power Plants have shutdown and cooling systems protected from a nuclear generated EMP?

text: Forty years ago, I read a report concerning the difficulties of protecting my companies equipment, which was to be installed in Air Force One, from a Nuclear EMP. An EMP generated by a high altitude nuclear detonation seemed particularly problematic. It raises concern about our power plants ability to safely shutdown and cool a core after a nuclear generated EMP. Adding to the concern is the fact that all civilian equipment would be inoperative and unable to assist.


title: Experimental perspective in understanding the Heisenberg Uncertainty Principle

text: I need to confirm whether or not I understand Heisenberg Uncertainty Principle. So the crucial thing is that you need an **"ensemble" of measurements** : $$\delta x \delta p \ge \frac{\hbar}{2}.$$ If I were to conduct an experiment trying to validate this with particles, then I would first measure, say, the position in the $X$ direction of an "ensemble" of particles right? Because when you measure the position of a particle, you get a single number. Thus, "had I tried to measure the momentum in the $X$ direction at the same time measuring position in the $X$ direction of a particle", I would also get a single number for the momentum corresponding to that particle. It is **until I combine all of my measurements of $x$ and $p$ together** in and plot them in a graph that you would literally "see" the uncertainty relation. This is my understanding. So you would calculate $\delta x$, which is the standard deviation of $x$ measurements and $\delta p$ as the standard deviation of $p$ measurements and multiply them together. You calculate the standard deviation by $\sqrt{\sum(x-x_{avg})^2/n}$, where $n$ is the number of particles and $x_{avg}$ is the average of all the position measurements. I do the same thing with the $p$ values for momentum. Am I correct? Furthermore, I want to solidify my understanding further... So can I think of the nice probability density curve that I typically see for the position of a particle is the ideal plot for an ensemble of a zillion measurements right? Its usually a Gaussian. The distribution depends on the potential that you plug into the wavefunction. Nevertheless, the calculated distribution is "as if" you've done a zillion measurements on an ensemble of identically prepared systems. Am I right here? So, this is why when you have "precise $\delta x$" meaning a very small value for $\delta x$", the spread of the position measurements is very narrow. Following the principle, you MUST end up with a very fat or wide curve for the momentum function because its standard deviation must be large to preserve the principle.


title: Counterexamples in quantum theory

text: I'm looking for counterexamples in quantum theory, in the spirit of books like _Counterexamples in topology_ and _Counterexamples in analysis_. A practically identical post, but for PDEs, can be found here. This request will probably be clearer to people who have read those books: I am **not** looking for claimed 'disproofs' of quantum theory by cranks, but rather examples from quantum theory that defy intuition or expectation, or which 'break' theorems that we would otherwise want to hold, and in the process help clarify some of the theory's more subtle issues. Also note that I am not looking for ways in which quantum theory deviates from classical physics, like the tunneling effect or the blackbody spectrum. A good example might be a strange solution to the Schrodinger equation, or to the Dirac equation in a particular field, or maybe something like the Haag–Lopuszanski–Sohnius theorem, etc.


title: Chemical Equilibrium - Why do changes in pressure cause a shift in the ratio of products and reactants?

text: I understand Le Chatelier's principle and how every change to an equilibrium system causes an opposing reaction from the system. I also understand how, when pressure is increased, the equilibrium shifts to the side with the lowest number of moles of gas and vice versa. So, why does this happen? How is it more energetically favourable - or is it something else? Also, are these explanations of why the ration changes for the other factors right:   * Temperature - equilibrium moves to the endothermic side of the reaction as temperature increases because the activation enthalpy of the exothermic reaction increases as the overall energy of the system is less   * Concentration - ratio of products/reactants changes immediately after one part of the reaction's concentration increases, it then approaches the same ratio (assuming one whole side of the reaction is added)   * Catalysts do nothing more than decrease the time it takes to reach equilibrium as they catalyse both sides of a reversible reaction and reduce the activation enthalpy


title: Wave diffraction explanation

text: I'm trying to understand wave diffraction and I found this wikipedia article. It's in Czech so I'll explain a bit. I'm interested in the 4 images I couldn't find on english wikipedia. The first one is diffraction at large slit, second on large obstacle, third on tiny slit and last on slit which size is comparable with wavelength - happens with light on diffraction grating. I got few question to those however. I don't know why, but I think I heard that wave can't pass through slit, which size is smaller than wavelength. For example imagine microwave oven. On doors is some kind of texture with those small slits. Microwaves have wavelength from 1 to 0.001 metres, so those slits should be sufficient and block the wave - that's why they are there in first place. But how is then possible that third case with tiny slit? Also, I'm missing cases what happens at obstacle, which size is comparable to wavelength and if I'm wrong in the above situation, also on a tiny obstacle? (And if I'm right with that microwave thing, does the wave pass as there was no obstacle when hitting something tiny? ). On a slit, I can use Huygens principle to create envelope, but what to do on obstacles? And can someone tell me what range of sizes compared to wavelength is considered for comparable obstacle? And if it matters I imagine waves as always coming from source point, the way Kirchhoff described them as shown here, but I don't need to understand those equations and so on. I'd also appreciate if someone got a good website, images, documents,... well whatever that tryes to explain diffraction without too many formulas. I'm not going to compute anything in the end, just trying to get the best possible image of it all for now. Thanks!


title: Negative energy and large-scale spacetime structure

text: I was reading an essay from Stephen Hawking's on the Space and Time warps and I was trying to make sense on some statements referring to the Casimir effect such as: > The energy density of empty space far away from the plates, must be zero. > Otherwise it would warp space-time, and the universe wouldn't be nearly > flat. So the energy density in the region between the plates, must be > negative. Could anyone tell me what's the logic about energy from a far away place from plates being zero? If that was not the case how would the space-time warp? I might lack the knowledge, but I would like to understand the reasoning or have an intuitive explanation about those statements. Thank you very much!


title: What causes compressed air (from "Dust Off" cans) to freeze, but only when turned upside down?

text: I am using some Dust Off brand cans to clean out a computer after an accident with concrete dust... long story. Inside one of these cans is a flurocarbon which quickly turns into a gas at room temperature. It is not ordinary air as that would be too difficult or expensive to store compressed. Anyway, I noticed that when orientated with the spray head upwards, the stuff which exited the nozzle was mostly vaporised immediately with a small amount of liquid residue. However, when orientated downwards some kind of liquid which quickly froze exited the nozzle. In fact such ice is still frozen on my computer case and is slowly melting. What causes the two distinct states? And why does the frozen stuff that sprays out upside down take ages to melt, but the frozen stuff which exits when orientated upwards melts, and subsequently evaporates very quickly?


title: Conversion from natural units to SI

text: I got questions about converting units from natural system of units to SI. To be exact, I'm solving the problem in Heisenber interpretation of quantum mechanics, and I'm using Heisenberg equation of motion $$ \frac{d}{dt}A(t)=i[H,A(t)] $$ instead of $$ \frac{d}{dt}A(t)=\frac{i}{\hbar}[H,A(t)]. $$ And, of course every operator that I'm using in my calculation i.e. (momentum) is used in this convention $\hbar = 1$. Finally, in my calculations I have got the wanted expression for some expectation value for some variable $t$, but the problem now is how to convert that expression in SI, where $\hbar$ is not $1$. Is there some rule of thumb, or dimensional analysis is the right way to go?


title: Which is the right electron mass?

text: When particles mass can be changed by changing the website, then how to calculate with confidence? For example: Google: electron mass = 9.10938 **188** × 10$^-31$ kilograms Wikipedia: electron mass 9.109 38 **2 15(45)** ×10$^−31$ kg 5.485 799 0943(23)×10$^−4$ u 8.187 104 38(41)×10$^−14$ J/c$^2$ 0.510 998 910(13) MeV/c$^2$ The difference is small, but it makes little difference in accuracy.


title: Staying in orbit - but doesn't any perturbation start a positive feedback?

text: I am not a physicist; I am a software engineer. While trying to fall asleep recently, I started thinking about the following. There are many explanations online of how any object stays in orbit. The explanation boil down to a _balance_ of the object's tangential force with centripetal force. But suppose something upsets this balance by a miniscule amount -- say, a meteorite or a spaceship crashing into Earth. Doesn't this start a **positive feedback** process to break Earth out of orbit? Suppose the meteorite crashes such that the Earth is briefly forced toward the sun. (The meteorite contributes to the centripetal force.) Now Earth is just a smidgen closer to the sun. Due to the equation $F=G \frac{m_1 m_2}{d^2}$ that everyone learns in high school physics, the sun's centripetal force acting on Earth in turn increases! That pulls Earth even closer to the sun, increasing centripetal force yet more, and so on. A similar argument applies to briefly forcing Earth _away_ from the sun. Empirically, I want to say that there's a buffer, such that if the balance of forces is disrupted by less than X%, we remain in orbit. But I cannot justify any buffer from the equation above. So, what am I missing? How can objects in orbit suffer minor perturbations in the balance of tangential and centripetal force and yet remain in orbit, when it appears to me that any perturbation starts a positive feedback loop?


title: Picking up audio using high speed video?

text: In the movie _Eagle Eye_, ARIIA (an intelligence-gathering supercomputer/AI) picks up audio using video recording of the vibrations in a coffee cup. How close is this to reality? I have see it done with laser on a window, and in theory I would imagine you could pick up 5 kHz audio by recording the vibrations with a 10,000 fps high speed hi-res camera. Has it been done? How is the quality? What are the theoretical limitations? What are the practical limitations?


title: Microscopic picture of an inductor

text: I have a good understanding of how inductors behave in electrical circuits, and a somewhat rough-and-ready understanding of how this behaviour arises from Maxwell's equations. However, what I don't have a good mental picture of is how electromagnetic induction works on the microscopic level, i.e. in terms of forces experienced by individual electrons in the wire. Ideally I would like to be able to understand the operation of an inductor, at least qualitatively, in terms of the statistical mechanics of the electron gas in the coil and the electron spins in the core. To clarify: in Maxwell's equations there is a term for $I$, the electric current. But current is a macroscopic quantity - it's the expected number of charges passing through a surface, with the expectation taken over an ensemble. So Faraday's law is a macroscopic relation, just like the gas equation. For the gas equation, we can understand how it arises from the microscopic motion of molecules. I want to understand how Faraday's law arises from the microscopic motion of electrons. Would anyone be able to provide an explanation of how induction works in microscopic terms, or point me towards somewhere I can read up on it?


title: Is a macroscopic pair where I observe one of them quantum entanglement?

text: Of what little I know/understand about quantum entanglement can somebody confirm if the following experiment is a good analogy to quantum entanglement of pair of particles? PS: please don't laugh as this could be very very lame! I take an orange and an apple (of similar shape & weight). I put each of them in a non transparent bag separately and seal it off. I put both the bags in a box. I close my eyes and shake it thoroughly such that I no longer know which bag contains what fruit. Then I randomly pick one bag and take a flight to other part of the country. Now given the above setup, I will not know what fruit is in my bag. In other words the fruit in my bag could be both apple/orange at the same time until I open the bag and see what is inside. As soon as I open it it is determined (similar to wave function collapsing) that I have an orange (for example) and there by making the fruit in the bag that was left in the box an apple (or vice versa). Does this in anyway come close to what they are doing with quantum entanglement?


title: Sign crazyness on the stress energy tensor?

text: I would like to know on what depends the sign of the stress energy tensor in the following formula : $T_{\mu\nu}=\pm(\rho c^2+P)u_{\mu}u_{\nu} \pm P g_{\mu\nu}$ In my case the metric is equal to $g_{\mu\nu}=\pmatrix{-c^2 & 0 & 0 & 0\\\0 & 1 & 0 & 0\\\0 & 0 & 1 & 0\\\0 & 0 & 0 & 1}$ and $\rho$ is the mass density. The problem is that we have:   * Wikipedia Stress-energy tensor: $T^{\mu\nu}=(\rho c^2+P)u^{\mu}u^{\nu} + P g^{\mu\nu}$   * Tenseur énergie-impulsion: $T^{\mu\nu}=-(\rho c^2+P)u^{\mu}u^{\nu} + P g^{\mu\nu}$   * Energie-Impuls-Tensor: $T^{\mu\nu}=(\rho c^2+P)u^{\mu}u^{\nu} - P g^{\mu\nu}$   * Fluide parfait: $T^{\mu\nu}=(\rho c^2+P)u^{\mu}u^{\nu} - P g^{\mu\nu}$   * Dérivation des équations de Friedmann: $T_{\mu\nu}=(\rho c^2+P)u_{\mu}u_{\nu} - P g_{\mu\nu}$ So why so many different signs, and what are the right ones in my case ?


title: Impact parameter in scattering process

text: Let's consider classical two body scattering process. Is the impact parameter the same in the laboratory and inertial frame of references?


title: EM Waves Energy Loss

text: Where does the energy go when two photons interfere destructively at a point on a screen in Young's double slit experiment ?


title: What is the reason behind specific values for charge of electrons, protons?

text: Why do things like protons and neutrons have specific values. Also speed of light is a speed in which even if you go towards it, the speed does not vary. But why does light have to travel at speed?


title: What is the relation between extra dimensions and unification of theories?

text: One of the most used methods in unification of theories is the use of higher dimensions. How does it actually work? If these dimensions are extremely small curled up, how does it affect the universe. There many theories giving explanation through mathematics. But what is the physical reality? Is it possible that these dimensions are not curled up but actually include stuff like multiple universes?


title: Computational Fluid Dynamics methods

text: I have read some articles about the finite difference method on a cartesian orthogonal grid. I understand how it works when Dirichlet boundary conditions are used, or when Neumann boundary conditions are used on a simple boundary (for example the rectangular boundary of the simulation). What I don't understand is how to use the Neumann boundary conditions in boundaries with arbitrary shape (in a 2-dimensional cartesian orthogonal grid). Suppose I want to simulate the potential flow about a cylinder (in 2D it becomes a circle): can I use the finite difference method on a circular boundary, where that circle has been drawn on the cartesian grid (so it is approximated by a set of little squares)?   Thank you in advance.


title: Why do they consider radioactive matter with long half lives more dangerous than matter with a short half life?

text: The title says it all. For example why is plutonium considered more dangerous than radioactive iodine?


title: The Superconducting Super Collider: what went wrong?

text: The Superconducting Super Collider was famously cancelled in 1993 after running enormously over budget. According to the wikipedia page: > During the design and the first construction stage, a heated debate ensued > about the high cost of the project. In 1987, Congress was told the project > could be completed for $4.4 billion, and it gained the enthusiastic support > of Speaker Jim Wright of nearby Fort Worth. By 1993, the cost projection > exceeded 12 billion dollars. ... What went wrong? Why was it so much over budget? How can such a fate be avoided by future big physics projects in the US? What did we learn?


title: Representing forces as one-forms

text: First of all, sorry if any of those things are silly or nonsense, I'm just trying to understand better how the concepts of forms, exterior derivative and so on can be used in physics. This question arose because of my first question Interpreting Vector fields as Derivations on Physics. Well the point here is: if some force $F$ is conservative, then there's some scalar field $U$ which is the potential so that we can write $F = - \nabla U$. That's fine, it says that force is a covector, but the point is: when we start thinking about curved spaces, in general instead of talking about gradients and covectors we talk about exterior derivatives and one forms. My question then is: if a force is conservative with potential $U$ then it's correct do represent the force by the one-form obtained by the exterior derivative of the potential, in other words the form $F = -dU$ ? In second place, if the force isn't conservative, is it correct to think of it as a one form yet ? But now what's the interpretation ? I tried to give this interpretation: suppose we're dealing with some manifold $M$ and suppose that $(W,x)$ is a coordinate chart. Then $\left\\{dx^i \right\\}$ spans the cotangent space, and so, if we interpret some force at the point $p$ as some one form $F \in T^\ast_pM$ then we'll have $F=F_idx^i$ using the summation convention. Now if i take some vector $v \in T_pM$ we can compute $F(v) = F_idx^i(v)$, however, $dx^i(v)=v^i$ and hence $F(v)=F_iv^i$ and so my conclusion is: if I interpret force at a point as a one-form at the point, then it'll be a form that when given a vector, gives the work done moving a particle in the direction of the given vector. So if a force varies from point to point, I could represent it as a one-form field that can be integrated along some path to find the total work done. Can someone answers those points and tell me if my conclusion is correct? And again, sorry if anything here is silly, I just really don't know.


title: What is the probability of a Brownian path?

text: Suppose I have a Brownian particle with a diffusion constant $D$ starting out from a given position at time $0$ and follow it until time $\tau$. What is the probability (distribution) that it takes the path $x(t)$? Now to my understanding this motion could be represented by the $N$ integrals $$\int_{-\infty}^\infty\cdots\int_{-\infty}^\infty \exp\left(-\frac{1}{4D} \sum_{i=1}^N \frac{(x_i-x_{i-1})^2}{t_i-t_{i-1}} \right)\prod_{i=1}^N \frac{dx_i}{\sqrt{4\pi D (t_i-t_{i-1})}}$$ where $N\to\infty$. We could also write this out in the continuum form $$\int_{-\infty}^\infty\cdots\int_{-\infty}^\infty \exp\left(-\frac{1}{4D}\int_0^\tau \dot{x}^2(t)dt \right)\prod_{t=0}^\tau \frac{dx(t)}{\sqrt{4\pi D dt}}$$ What I would like is to have this in the form $\int p(x(t))\mathcal{D}x$, so I could readily identify $p(x(t))$ as the probablity of a path. Here $\mathcal{D}x$ is supposed to mean integration over all paths, and from what I gather, this usually is the product between the $dx(t)$'s above. This would mean that the $\sqrt{dt}$ term goes into $p(x(t))$, but this doesn't seem right: For example, if I were to compute the entropy using this probability distribution, I think it would always diverge (although if I had a simple discrete Markovian process and thus never take the limit $N\to\infty$ it seems that I'd be safe). In the Wikipedia article on Onsager-Machlup this too seems to be a "problem", but they just take ratios and thus are exempt from these woes. I think this is what usually happens in statistical mechanics, too, as you can kind of disregard the $\sqrt{dt}$ terms because the partition function is going to be getting rid of them anyway. I'm afraid that this all might be some very elementary stuff about path integrals and Lebesgue measures that I have forgotten and cannot seem to be able to find properly explained in any references I have available.


title: What is the difference between Newtonian and Lagrangian mechanics in a nutshell?

text: What is Lagrangian mechanics, and what's the difference compared to Newtonian mechanics? I'm a mathematician/computer scientist, not a physicist, so I'm kind of looking for something like the explanation of the Lagrangian formulation of mechanics you'd give to someone who just finished a semester of college physics. Things I'm hoping to have explained to me:   * What's the overall difference in layman's terms? From what I've read so far, it sounds like Newtonian mechanics takes a more local "cause-and-effect"/"apply a force, get a reaction" view, while Lagrangian mechanics takes a more global "minimize this quantity" view. Or, to put it more axiomatically, Newtonian mechanics starts with Newton's three laws of motion, while Lagrangian mechanics starts with the Principle of Least Action.   * How do the approaches differ mathematically/when you're trying to solve a problem? Kind of similar to above, I'm guessing that Newtonian solutions start with drawing a bunch of force vectors, while Lagrangian solutions start with defining some function (calculating the Lagrangian...?) you want to minimize, but I really have no idea.   * What are the pros/cons of each approach? What questions are more naturally solved in each? For example, I believe Fermat's Principle of Least Time is something that's very naturally explained in Lagrangian mechanics ("minimize the time it takes to get between these two points"), but more difficult to explain in Newtonian mechanics since it requires knowing your endpoint.


title: Do objects rotate around the torque vector or its center?

text: If I have an sphere and I have a torque vector coming out of it at point A. Would the sphere rotate around its center or the axis of the torque vector?


title: Models of higher Chern-Simons type

text: It has long been clear that (the action functional of) Chern-Simons theory has various higher analogs and variations of interest. This includes of course traditional higher dimensional Chern-Simons theory (abelian and non-abelian) as well as the algebroid-version: the Courant-sigma model, but also seemingly more remote systems such as string field theory (and hence essentialy also its effective truncations), a fact that is already somewhat remarkable. In a recent article we claimed that there is a systematic sense in which also all AKSZ sigma-models are special cases of a general abstract notion of "infinity-Chern-Simons theory". These AKSZ models include, in turn, also the Poisson sigma-model (hence also the A-model and the B-model). Also BF-theory coupled to topological Yang-Mills theory fits in. Therefore in a precise sense all these systems are examples of a single underlying basic mechanism. My **question** is: can you point out other models of interest in the literature (or in your drawer) that look like they might be "of generalized Chern-Simons type", along these lines? (I am not just looking, say, for "Chern-Simons term"-summands in higher supergravity actions, even though these are related, but for example new variants of full higher dimensional Chern-Simons (super)gravity.) For instance: has there been a proposal for a nonabelian 7-dimensional Chern- Simons-type model that might be the holographic partner to the self-dual nonabelian 6d (2,0)-superconformal QFT (so that the state spaces of the former are the conformal blocks of the latter)? While we did come across a natural non-abelian 7-dimensional Chern-Simons type TQFT whose fields are string-2-connections (here), I am not sure how to see if this might be the relevant one. Do you?


title: Why is there no double counting of $s$- and $t$-channels in string theory?

text: In string theory for the four particle tree diagram exchange, why is there some mysterious crossing duality between the $s$- and $t$- and $u$-channels? Why isn't there a double counting in the Feynman diagrams? How is this consistent with a quantum field theoretic description?


title: Sub and super multiplicativity of norms for understanding non-locality

text: In relation to various problems in understanding entanglement and non- locality, I have come across the following mathematical problem. It is most concise by far to state in its most mathematical form and not go into the background much. However, I hope people interested in entanglement theory might be able to see how the problem is interesting/useful. Here goes. I have two finite dimension vector spaces $A$ and $B$ and each is equipped with a norm (Banach spaces) such that $||...||: A \rightarrow \mathbb{R}$ and $||...||: B \rightarrow \mathbb{R}$. Both the vector spaces and norms are isomorphic to each other. My question concerns norms on the tensor product of these spaces (for simplicity, lets say just the algebraic tensor product) $A \otimes B$ and the dual norms. First let me state something I know to be true. Lem 1:   If a norm $||...||$ on $A \otimes B$ satisfies:   $||a \otimes b || \leq ||a|| . ||b||$ (sub-multiplicativity)   then the dual norm satisfies   $||a \otimes b ||_{D} \geq ||a||_{D} . ||b||_{D}$ (super-multiplicativity) where we define the dual of a norm in the usual way as   $|| a ||_{D}= \mathrm{sup} \\{ |b^{\dagger}a| ; ||b|| \leq 1 \\}$ This lemma crops up often such as in Horn and Johnson Matrix analysis where it is used to prove the duality theorem (that in finite dimensions the bidual equals the original norm $||..||_{DD}=||...|$ ). I wish to know the status of the converse, which I conjecture will be answered in the affirmative: Conjecture:   If a norm $||...||$ on $A \otimes B$ satisfies:   $||a \otimes b || \geq ||a|| . ||b||$ (super-multiplicativity)   then the dual norm satisfies   $||a \otimes b ||_{D} \leq ||a||_{D} . ||b||_{D}$ (sub-multiplicativity) My question is simply "is my conjecture true or does anyone have a counterexample?". Although I am inclined to think the conjecture is true, it is certainly not as easy to prove as the first stated lemma (which is a 3-4 line proof). The asymmetry enters in the definition of a dual norm, which allows us to "guess" a separable answer at the cost of having underestimated the size of the norm, but we can not so easily overestimate it!


title: A solvable model for the finite rectangular potential well with a bump in the middle

text: A well known example in quantum mechanics is that of a finite rectangular potential well with a rectangular bump in the middle. I guess this closely approximates the "umbrella" effect of the $NH_3$ molecule. But this potential is not solvable analytically.   * I want to know if there is a solvable Hamiltonian known which mimics the effects of this potential - like from which one can exactly see the effect on the energy levels and the wave-functions of the width of the bump or the height of the bump or the well width on either side of the bump.


title: What is a Hilbert space filter?

text: In a recent paper, > _Side-Channel-Free Quantum Key Distribution,_ by Samuel L. Braunstein and > Stefano Pirandola. _Phys. Rev. Lett._ **108** , 130502 (2012). > doi:10.1103/PhysRevLett.108.130502, arXiv:1109.2330, the authors repeatedly claim that > _the dual teleportation channel act(s) as an ideal Hilbert space > filter[...]_ What do they mean by a Hilbert space filter? How does teleportation act as a Hilbert space filter?


If nothing is "objectively real" prior to "measurement", what exactly is a "measurement"? Is there any "objective" criteria to demarcate a process as being a "measurement" or not? If "measurements" are "subjective", is "reality" also "subjective"? Does a stray environmental photon scattering off a quantum system thereby decohering it "measure" the system if that photon is promptly absorbed by something else in the environment which promptly thermalizes and scrambles the "measured" information? Is decoherence per se the same thing as a "measurement", or is a "measurement" something (what?!) "extra"? Place Alice in a sealed box destined to fall straight into a black hole. Before crossing the horizon, Alice "measures" a quantum system and "observes" the outcome. Right after, the box passes through the event horizon before Alice even has a chance to leave any message to the outside world. Bob is outside blissfully unaware of what happened inside the box before it crossed the horizon. Has a "measurement" occurred according to Alice? According to Bob? Is this stackexchange question "measured" if it is archived for centuries or even millennia, and then backed up in colonizing space probes if billions of billions from now, the cosmological expansion of the universe causes a big rip and heat death, and the ultimate decay of all storage media say after trillions of years from now with the decay products of the storage media going off in different directions and ending up in mutually unreachable cosmological patches, and the only way to reconstruct this question is to piece together the decay fragments from different patches? Is this string of words making up this paragraph "objectively real"? Are "you" "objectively real"? If "measurement" is time bound, is the which-way path of a photon "temporarily" "measured" when it crosses a Mach-Zehnder interferometer?


title: Experimental data for gas flow through pipelines

text: Maybe some of you could point a good source of experimental data of gas flow through pipelines? What I need is a very simple flow of some gases through some diameter simple linear pipeline. A graph of pressure loss at different diameters would fit or something similar. I need this to compare theoretical estimates and experimental measurements. Thank in advance.


title: What causes an aircraft to roll when rudder is applied

text: When continuous rudder is applied in a typical light aircraft during straight and level flight at "normal" flying speeds and altitudes, the primary effect is that the aircraft will yaw to the left - so it's flying "sideways" to some extent. It is commonly understood amongst pilots that the secondary effect in most aircraft, is that the aircraft will also start rolling in the direction of yaw (i.e. applying left rudder causes left yaw, and left roll). What causes this roll? And does the Dihedral angle have anything to do with it?


title: What's the physical difference between a convective heater and an infrared heater?

text: Could someone please explain why there are 2 types of space heaters-- one that is convective and one that is infrared? Why does the first one not radiate and why does the second one not heat the air? I know also that a convective heater is built with a spiral wire over a ceramic core that act like a resistor that gets heated. How exactly is infrared heater built?


title: Why sound does not heat up the air?

text: Both thermal energy and air are propagated through vibration of particles so why sound does not heat up the air e.g loud musical instrument does not generate much heat ?


title: What is the formal definition of spin-independent vs. spin-dependent scattering?

text: In the search for WIMPs as the dark matter particle, there is an important distinction between spin-independent and spin-dependent scattering. Roughly, WIMPs scattering from nucleons through a spin-independent coupling are not sensitive to the spin (up or down) of the nucleon, while they are sensitive in this way for spin-dependent couplings. In particular, spin-independent scattering can lead to a boost in the total cross-section due to individual nucleons within a nucleus contributing _coherently_ , while the spin-dependent cross-section is (I think) only proportional to the _net_ spin of the nucleus. Is this distinction only useful (or unambiguous) for scattering mediated through through weak interaction, or does it have a model-independent definition within quantum field theory? Also, do the two types correspond to scattering events mediated through a boson with specific properties (e.g. Z vs. W)? For instance, this paper by Barger et al. says that "The spin dependent scattering cross section is largely governed by Z-boson exchange and is sensitive to the Higgsino asymmetry". But, as is suggested by the mention of the Higgsino, this is made in the context of SUSY and it's unclear to me whether this is generally true. Even if this distinction is only used within the narrow field of WIMP scattering, I would be very appreciative of a mathematical precise definition. Thanks!


title: The correspondence between Poisson bracket and Commutators in Quantum Mechanics

text: I don't understand canonical quantization. In passing from classical to quantum, one replaces the Poisson brackets with the commutators. I don't really understand this. How can we generally show that in the classical limit, the quantum theory that we obtained using this operation leads to the classical theory that we started with? Can there be a quantum theory that has the same classical limit but that can't be obtained using this prescription?


title: Scaling: Gravity and Friction

text: I understand how doubling the length of a shape quadrupes it's area and the analog in 3 dimensions. My question however relates to other physical quantities, for example gravitational field strength. Lets consisder some system under constant gravity. How will the be behaviour of the system change if we change the strength of the gravity. Is their a general way to tell? How does friction scale if we change the size? Are any of these questions related to tr units of these quantities? Eg. Acceleration due to gravity $ \frac{m}{s^2} $ does this mean its somehow proportional to some distance? Is there a conceptual to think about this? The only scaling law articles I found related to length area and volume


title: virtual air-surface temperature difference above a water body

text: I'm trying to work through the equations for the following paper: http://onlinelibrary.wiley.com/doi/10.1029/2009JD012839/abstract and am trying to understand the concept of virtual air temperature, virtual temperature of saturated air at the water surface, and virtual air-surface temperature difference. These play a part in calculating the stability of the atmospheric boundary layer. I have a few questions regarding this work so instead of coming back and forth with multiple posts I'll include them all here. This is the Monin Obukhov stability length scale ![enter image description here](http://i.stack.imgur.com/gsTwg.png) where Tv denotes the virtual air temperature which is defined by ![enter image description here](http://i.stack.imgur.com/coRxH.png) where T is the air temperature, and qz is the specific humidity. My first question is, how is the formula for the virtual air temperature derived? i.e. how do you end up at this equation. Secondly, this formula gives the virtual temperature at the water surface: ![enter image description here](http://i.stack.imgur.com/nlu0F.png) where To is the water temperature, and qs is the specific humidity at saturation. Similar to the first question, how do we arrive here? Finally, the ratio of these values are given by: ![enter image description here](http://i.stack.imgur.com/L5Ir4.png) What can this ratio be used for? I realise it must be something along the lines of determining the potential temperature profile above a water body but can it be used to determine the temperature at a given height (i.e. by knowing this profile)? I realize that I am asking a lot but any advice would be appreciated.


title: Why do dark objects radiate thermal electromagnetic energy faster than light objects?

text: Kirchhoff's law of thermal radiation says that: > For a body of any arbitrary material, emitting and absorbing thermal > electromagnetic radiation at every wavelength in thermodynamic equilibrium, > the ratio of its emissive power to its dimensionless coefficient of > absorption is equal to a universal function only of radiative wavelength and > temperature, the perfect black-body emissive power. I can imagine why dark objects have higher absorption of electromagnetic radiation: The darker the object is the less radiation it reflects back. An ideal black body would absorb all the incident electromagnetic radiation. Is there a similarly simple and intuitive explanation of why dark objects emit thermal electromagnetic radiation faster that light objects and why is the Kirchhoff's law valid? For me it is not intuitive at all and I was not able to find any simple explanation.


title: How to cut a stone on on a White Dwarf

text: I've heard that white dwarfs are extremely dense and hard. So, if I had a piece of white dwarf matter, would it be possible to cut it (or otherwise) into a custom shape? How could one do that?


title: Calculating rotor torque out of mass center

text: Suppose I have an engine with propeller. I know there is aerodynamic force pushing the propeller opposite direction the engine is turning it. One component of this force is creating lift, but second is braking the prop. Now these forces are applied at some distance from engine shaft, thus they're creating torque, which is turning the whole engine if it wasn't fit somewhere. And now my problem: how to describe what happens, after fitting this engine out of mass center. For example take a ruler, that has pivot somewhere and at the end of it, there is this engine with propeller. How do I describe momentum at the center of mass - the pivot, instead at the engine (engine has no weight in this example)?


title: String theory landscape and false vacua

text: What is the multiverse of string theory? I always thought it would be from brane cosmology where there's chances for multiple collisions of the branes that can make a universe. So this "multiverse" would be in the "bulk". But when I read up on the string theory landscape it implies a relation to bubble cosmology and false vacua? How does that work? What's the connection? There's no need for strings in bubble cosmology and no need for bubble nucleation in brane cosmology so... what's going on?


title: Bolometer calibration - voltage and intensity

text: I am interested in possible hazards due to high intensy infrared light sources. Therefore I want to use the "PASCO TD-8553 radiation sensor" (website) to measure the intensity of the radiation of an infrared light source. The bolometer outputs voltage values. In the user manual (see website) it says that these voltage values are proportional to the intensity of the radiaton - but I cannot find any equations or proportional constants in this document. How can I calibrate a bolometer myself? Any experiences using this device?


title: Why does my wife's skin buzz when she's using her laptop?

text: When my wife uses her laptop, if I touch her skin, I can feel a buzz. She doesn't feel the buzz, but she can hear it if I touch her ear. So I'm guessing it's a faulty laptop, and she's conducting an electrical current. But why would she not feel anything, and what would it be that she would be hearing when I touch her ear? * * * More info: The effect is only intermitent - it's pretty reliable in a single session on the laptop, but some sessions it won't happen and others it will. I had the same sensation with a desk lamp that I had several years ago, with no moving parts (as far as I could tell) The effect only occurs when I move my finger - if I'm stationary, I don't notice anything. I was playing with my son, and noticed the same buzz. First I thought he was touching the laptop. Then I realised he had skin-to-skin contact with my wife who was using the laptop.


title: Why can't Jupiter be seen over Thanksgiving break?

text: As of these past few days, Jupiter can be seen in the sky, correct? Around Thanksgiving, why can't it be seen?


title: May 20th Annular Eclipse - Duration from Vancouver?

text: On May 20th, there's an annular eclipse. It's technically visible towards the end from Vancouver. In the middle, the duration is just under 6 minutes. What I want to know is what the duration is expected to be for a viewer located in Vancouver?


title: Electrons skip randomly around their orbits

text: I read where the electron (as well as a few other particles) skips around in its orbit randomly rather than move around the orbit smoothly. This effect has been repeatedly observed in the laboratory over the years. I know if we treat this using quantum physics the electron exist in a fuzzy cloud of probability but in the old days before quantum physics what were they seeing in the laboratories when they found this effect?


title: Is it possible to surpass the diffraction limit for telescopes?

text: Telescopes have angular diffraction limit depending on the observed wavelength and aperture diameter. I've read that it's possible to go beyond the limit for microscopes. But is it possible to do the same for telescopes too?


title: Electric field a distance $z$ above the center of a circular loop. The Hard way

text: > _Problem 2.5: Find the electric field a distance $z$ above the center of a > circular loop of radius $r$ which carries a uniform line charge $\lambda$._ This problem is in refereced here (with solution): http://teacher.pas.rochester.edu/PHY217/LectureNotes/Chapter2/LectureNotesChapter2.html But it is solved using a trick. The trick is: okay we know there is symmetry so let us just take the magnitude and not use the vectors. The problem I have is I am unable to solve it without using the trick and I would appreciate someones help in doing the problem without the shortcut, using the vectors. When I do it the r compontent doesn't cancel. Using polar coordinates here is the setup: $$dE = \frac{1}{4 \pi \epsilon_0} \frac{\lambda}{\rho^2} ~ \hat{\rho} ~ r d \theta$$ $$\hat{\rho} = \frac{z \hat{k} + r \hat{r}}{\sqrt{z^2 + r^2}} $$ $$ E = \frac{\lambda}{4 \pi \epsilon_0} \int_0^{2 \pi} \frac{r z \hat{z} + r^2 \hat{r} }{(z^2 + r^2)^{3/2}} ~ d \theta$$ $$ E = \frac{\lambda}{4 \pi \epsilon_0} \int_0^{2 \pi} \frac{r z \hat{z} }{(z^2 + r^2)^{3/2}} ~ d \theta + \frac{\lambda}{4 \pi \epsilon_0} \int_0^{2 \pi} \frac{r^2 \hat{r} }{(z^2 + r^2)^{3/2}} ~ d \theta $$ If I now evaluate the $z$ component I get the correct answer. When I evaluate the $r$ component, shown below the $r$ doesn't not cancel the way it should. why not? $$\frac{\lambda}{4 \pi \epsilon_0} \frac{r^2 2 \pi }{(z^2 + r^2)^{3/2}} - \frac{\lambda}{4 \pi \epsilon_0} \frac{r^2 0 }{(z^2 + r^2)^{3/2}} $$ $$ = \frac{\lambda}{4 \pi \epsilon_0} \frac{r^2 2 \pi }{(z^2 + r^2)^{3/2}} \hat{r}$$


title: Why is Neutron Heavier than Proton?

text: This is Neutron decay: $$n^o \to p^+ + e^- + \overline {\nu_e}.$$ and this is proton one: $$p^+ \to n^o + e^+ + \nu_e$$ so when the $e^+ =e^-$ and $\nu_e=\overline {\nu_e}$ why $n \not= p$? **my idea is that:** $$\nu_e \not= \overline {\nu_e}$$


title: Are Electrons Electromagnetic Waves?

text: I encountered a thought provoking article suggesting that electrons are electromagnetic waves. Is this possible? I may not agree with their entire model, but surely there is the possibility that an electromagnetic wave might impersonate a some aspects of a point charge through aligning its electric component with the electric field.


title: Cross Section Peskin vs Srednicki

text: in Peskin Schroeder after the derivation of the differential cross section there is a comment for the central mass system (CMS), which says: In the special case, where all four particles have identical masses (...), this [the general formula] reduces to the formula[...] (p.107): $$ \left(\frac{d\sigma}{d\Omega}\right)_{CM}=\frac{\left|\mathcal{M}\right|^2}{64\pi^2E_{CM}^2}$$ However in Srednicki the general formula for the differential cross section in the CMS is (p.97): $$\left(\frac{d\sigma}{d\Omega}\right)_{CM}=\frac{\left|\mathcal{M}\right|^2}{64\pi^2E_{CM}^2}\frac{\left|\textbf{k}^\prime\right|}{\left|\textbf{k}\right|} $$ where $\left|\textbf{k}^\prime\right|$ is the outgoing three-momentum in the CMS and $\left|\textbf{k}\right|$ the incoming. Now to get from the formula from Srednicki to Peskin's formula I don't need the masses to be all the same but merely the incoming masses equal to the outgoing masses, so to say elastic scattering. I didn't see a further restriction in Srednicki's formula apart from being in the CMS. If I take e.g. Compton scattering I get with Srednicki's formula: $$\left(\frac{d\sigma}{d\Omega}\right)_{CM}=\frac{\left|\mathcal{M}\right|^2}{64\pi^2E_{CM}^2} $$ but Peskin gets on page 164 with his formula for the differential cross section: $$\left(\frac{d\sigma}{d\Omega}\right)_{CM}=\frac{1}{32} \frac{1}{E_1E_2}\frac{\left|\textbf{k}^\prime\right|}{E_{CM}}$$ where $E_1+E_2=E_{CM}$. I don't see, that this is equal? Is it actually equal? Where did I miss something?


title: Deriving Lagrangian density for electromagnetic field

text: In considering the (special) relativistic EM field, I understand that assuming a Lagrangian density of the form $$\mathcal{L} =-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{1}{c}j_\mu A^\mu$$ and following the Euler-Lagrange equations recovers Maxwell's equations. Does there exist a first-principles derivation of this Lagrangian? A reference or explanation would be greatly appreciated!


title: What is the relation between the representation the Higgs field transforms under, the types of couplings in the theory and Higgs/Coulomb branches?

text: When reading about Higgs and Coulomb 'phases' I came across two separate definitions: The first tells us that the Higgs/Coulomb phases are determined by the representation that the Higgs field transforms under, as explained on Wikipedia. The Higgs field breaks the gauge symmetry; If the Higgs is in the adjoint then we're left with U(1)s and we have a Coulomb phase, if the Higgs transforms in any other representation then (typically) no U(1)s are left and we have a Higgs phase. However, the second definition tells us that the these phases are determined by the potential between two electric test charges (as read in Sannino's book). A potential proportional to 1/r is the Coulomb phase and a constant potential gives the Higgs phase. Also, there are other phases: Confining, free-electric, free-magnetic and perhaps others. Question 1) Is there a link between these two definitions? I can see why the Coulomb phase as described in the Wikipedia article might lead to a 1/r potential (due to the surviving U(1) groups). But why would the Wikipedia definition of a Higgs phase lead to a constant potential? Furthermore, if there is a link, what sort of phase (As in the Wikipedia article) would give us confining, free-electric and free-magnetic potentials? Then there are Higgs and Coulomb 'branches'. These are the moduli (scalar fields) of the N=2 hyermultiplet and vector multiplet respectively. Question 2) Do these link in with the phases described above or are the concepts of branches and phases very much distinct? Christian Samann's notes tell us (on page 13) that the Coulomb _branch_ is what we get when the gauge group breaks to U(1)'s. I think this is just an inconsistency in what different authors mean when they talk about branches and phases.


title: Can this ratio be written any better?

text: This topic is closely related to previous topic where we were to calculate ratio $\lambda_e/\lambda_p$ for proton and electron with **same velocities**. This time we I want to know if it is possible to derive a ratio $\lambda_e/\lambda_p$ for proton and electron who have **same kinetic energies** $(E_{ke} = E_{kp})$. So i write this like: \begin{align} \frac{\lambda_e}{\lambda_p} = \frac{\tfrac{h}{p_e}}{\tfrac{h}{p_p}} = \frac{p_p}{p_e} = \frac{\sqrt{{E_{kp}}^2 + 2E_{kp}E_{0p}}}{\sqrt{{E_{ke}}^2 + 2E_{ke}E_{0e}}} \longleftarrow\substack{\text{Here i know that kinetic}\\\\\text{energies are the same so}\\\\\text{i have to use $E_{kp}=E_{ke}$}} \end{align} I want to know if this can be reduced a bit more or is this the best possible result. If anyone has any idea please do tell.


title: Is it possible to separate the poles of a magnet?

text: It might seem common sense that when we split a magnet we get 2 magnets with their own N-S poles. But somehow, I find it hard to accept this fact.(Which I now know is stated by Gauss's Law) I have had this doubt ever since reading about the quantum-field-theory and I know I might sound crazy but is it really impossible to separate the poles of a magnet? **What about black holes**? They attract matter and everything else only inwards into them, right? So then, are they the only such case/exception? Am I being naive? Is there some proof/explanation that can give closure to this question of an independently existing pole?


title: Determine resultant velocity of an elastic particle-particle collision in 3d space

text: So I have two particles that have collided in 3 dimensional space. I want the particles to rebound off of each other in an elastic manner. How do I determine the resultant velocity vector if I know: Both particles' initial velocity vector, their masses, and their initial position vectors relative to the origin. I tried to follow the instructions on this website but I am not sure how to make their equations work without polar coordinates / how to translate my data to polar coordinates. Also, if you answer my question in terms of polar coordinates, can you explain which angle is theta and which is phi? Since there are 2 different conventions, I am not really sure which angle is being referred to in the explanation on the aforementioned website. EDIT: This isn't for a class. I'm writing a physics engine for a gravity simulation I am making using Unity. I haven't taken kinematics since I was in high school, which is why I am so rusty.


title: The Euler equations as a RNG fixed point

text: In this paper at the at the beginning of the last paragraph on p.2 it is said, that the Euler equations, which are an infinite Reynolds number limit of the Navier-Stokes equations, arise as an RNG fixed point. This fixed point is said to be non-unique for a system witn N mixing species, because it can be choosen from an N+1 dimensional parameter space spanned by N-1 dimensionless mass diffusivities, the dimensionless thermal diffusivity (or Prandtl number) and the dimensionless ratio of the anisotropic and isotropic viscosity. About this issue I have several closely related questions:   1. What kind of fixed point is this when considering a classification into Gaussian/interacting, trivial/nontrivial, etc fixed points   2. What is the expected behaviour of the RG flow around this fixed point?   3. Is the non-uniqueness of this fixed point the same kind of non-uniquenes as the one described on p67 if this paper, which explains that the presence of redundant marginal operators can lead to a whole line of physically equivalent fixed points? If this way of analyzing a fixed point can be applied in my example, would the Euler equations fixed point then correspond to some kind of an N+1 dimensional surface of fixed points where the mass diffusivities, the Prandtl number, and the ratio of the anisotropic and isotropic viscosity play the role of such redundant marginal operators?


title: Which parameter determines how much the 'recoil' or force applied by an object hurts?

text: Another question from my textbook, directly following after this question: > 2 identical bullets are fired, on from a lighter rifle and one from a > heavier one, with the same amount of force. Which rifle will hurt the > shoulder more? First off, I'm not even sure if this is an appropriate question for Physics.SE as it deals with what 'hurts' our bodies more. But since we're essentially determining whether acceleration or force matter more, I thought it could be asked. So on with it. (1) Let the lighter rifle be denoted as $l$ and the heavier one as $h$; [Assignment]   (2) The force exerted by the rifle on the bullets is the same; [From Q]   (3) Every force has an equal and opposite reaction; [Axiom]   (4) The bullets exerts an equal force on the rifles; [From (2), (3)] $$ (5)\ F_l = F_h \ \ [From\ (4)] \\\ (6)\ F = M \cdot a \ \ \rightarrow \ \ a = \frac{F}{M} \\\ (7)\ M_l < M_h \\\ (8)\ a_l > a_h $$ That was easy. Both recoil with the same force, but the lighter one is accelerate faster than the heavier one. But the point is, does higher acceleration mean more hurt? Or is it force that matters? If the former, the lighter rifle will hurt more. Else if the latter, then both will hurt equally.


title: 2nd Law of Thermodynamics

text: I understand that the 2nd law of thermodynamics roughly states that, if you have a body (or a gas in a chamber) that is hot at one end and cold on the other, the heat will always flow from the hot to the cold part and to get the opposite effect, one has to put energy in from the outside (through a machine or something). Now, I don't understand why this fact cannot be explained just through probabilities (of the velocities of the gas molecules, say). It would seem to me that it is simply very, very, very unlikely that faster moving molecules all end up in (approximately) one spot at any time. But from all the fuzz about the 2nd law, I'm led to believe that there has to be more behind it than probability. So where am I wrong? Why is the second law beyond probability? How is the 2nd law tested? (so that one can rule out simple probability?) ps.: I haven't yet had a course on probability theory. So my understanding of it is limited.


title: Current Carrying Wire Loop near Current Carrying Long Wire

text: A rectangular wire loop of (width $a$ and height $b$) carrying clockwise current $I_1$ is a distance $d$ below a horizontal infinitely long wire carrying a current $I_2$ to the right. What is the force on the wire loop? I am not sure what equations I have to use here, it's been quite awhile. I know the magnetic field infinite wire is something like $\frac{\mu_0}{2\pi} \frac{I_2}{r}$ And F=v x B. For the horizontal segment of the loop nearest the wire, does $v=\frac{I_1 a}{e}$? Is that right? does F = $\frac{\mu_0}{2\pi} \frac{I_2 I_1 a}{d e} - \frac{\mu_0}{2\pi} \frac{I_2 I_1 a}{(d+b)e} = \frac{\mu_0}{2\pi e}I_2 I_1 a (\frac{b}{d(d+b)})$ towards the wire. Is that the right answer?


title: Massless limit of the Klein-Gordon propagator

text: I am working with the propagator associated to the Klein-Gordon equation, as derived in "Quantum Physics a functional integral point of view", James Glimm, Arthur Jaffe or as derived here: http://www.wiese.itp.unibe.ch/lectures/fieldtheory.pdf § 5.4 It turns out that the propagator can be evaluated, and a close-form expression for it can be given, namely: $$ C \left( m; \mathbf{x} - \mathbf{y} \right) = \left(\frac{1}{2 \pi}\right)^{-\frac{d}{2}} \left(\frac{m}{\left| \mathbf{x} - \mathbf{y} \right|}\right)^{\frac{d-2}{2}} K_{\frac{d-2}{2}} \left( m \left| \mathbf{x} - \mathbf{y} \right| \right) $$ where $K$ is the modified Bessel function of the second kind. I'd want to take the massless limit in two dimensions; when setting $d=2$ and $m=0$ one of the terms in the r.h.s. of the equation evaluates to $0^0$ while the modified Bessel function goes to infinity. How do I calculate the massless limit for the Klein-Gordon propagator in 2D? Thank you!


title: What do I call the inverse of a propagator?

text: Let's suppose I have a theory described by a Lagrangian as follows: $ \mathcal{L} = A_\mu \underbrace{\left( \partial^2 g^{\mu\nu} - \partial^\mu \partial^\nu + m^2 g^{\mu \nu} \right)}_{K^{\mu \nu}} A_\nu $ (What's the actual theory does not matter.) I know that the inverse of $K^{\mu \nu}$ is called the propagator, I don't know how to call $K^{\mu \nu}$ itself. I would like to avoid something like ''operator'' because I often switch to the path integral formalism...


title: What is a Pseudoscalar particle?

text: 1. Can someone explain to me what is a pseudoscalar particle?    2. And how do experiments figure out that what they're dealing with is a scalar or pseudoscalar?


title: Why is Energy = Voltage x Charge, and how to prove that?

text: As you know the equation $\mathbf{E=V\times Q}$. Where: $\mathbf E$ is the energy measured in joules, $\mathbf V$ is potential difference (Voltage), $\mathbf Q$ is the charge. So my qustion is: How is this equation true or how can I prove that and what is the law of it?


title: How is a second measured? And why is it measured that way?

text: The earth's rotation around the sun isn't exactly 24 hours. It off by some seconds which becomes somewhere around 6 hours per year and 1 day in 4 years(leap year), which brings the question why didn't we modify the measurement of 1 second ever so slightly so as to avoid leap years altogether. Well how is 1 second measured exactly? Wikipedia says > _the duration of 9,192,631,770 periods of the radiation corresponding to the > transition between the two hyperfine levels of the ground state of the > caesium 133 atom. [1]_ Well, why is it measured that way? Is there any technical reason like it is easy to measure and can be standardized easily? Or is it really possible to modify the measurement of time?


title: Time reversal and the solutions to Maxwell equations

text: ![diag](http://i.stack.imgur.com/Cui4i.jpg) The standard Lienard-Wiechert potentials describe the electromagnetic field at a point $P$ at time $t$ due to an arbitrarily moving charge $q$ at the retarded time $t-r/c$. An electromagnetic influence is assumed to travel, with the velocity of light $c$, from the moving charge $q$ at time $t-r/c$ to the field point $P$ at time $t$ while the charge moves along its path. This situation is illustrated on the left-hand side of the diagram. Physically the charge $q$ is in arbitrary motion whereas the field point $P$ is at rest in an inertial frame. But the laws of electromagnetism are assumed to be time reversible. Therefore the time reverse of this process should occur in Nature. This is shown on the right-hand side of the diagram. Have I got this right? Basically all velocities are reversed _including_ that of the electromagnetic influence. Thus we have a picture in which an electromagnetic influence travels from a field point $P$ at time $t$ and intercepts a moving charged particle $q$ at a later time time $t+r/c$. Note that I am _not_ assuming that a signal goes backwards in time. I am "only" assuming that a physical process exists in which cause and effect are switched around in the temporal order. Instead of first moving charge then later field effect we have first field effect then later moving charge. So to what physical situation does the advanced description correspond? I believe that it might describe the situation in which the charge $q$ is at rest in an inertial frame and the point $P$ is in arbitrary motion. Thus, relative to an observer moving with point $P$, charge $q$ has an apparent motion illustrated by the right-hand diagram. The observer, at point $P$ and time $t$, measures a kind of inertial electromagnetic field that is determined by the apparent position of the charge at the advanced time $t+r/c$. Could this be the answer to the meaning of the advanced solutions to Maxwell's equations?


title: Can micro black holes form in the Sun?

text: Wikipedia writes to say               Factor (J)     SI prefix    Value           Item          10E−7            nJ        5.6×10E−7 J     energy per proton in the CERN Large                                                  Hadron Collider in 2011 (3.5 TeV)     10E3             kJ        1.4×10E3 J      total solar radiation received from the                                                 Sun by 1 square meter at the altitude                                                 of Earth's orbit per second (solar                                                 constant)      This leads to the assumption the energy generated at Sol's core is significantly higher. If this assumption is correct, and the earlier reports in media about creation of micro-black-holes in more powerful avatars of LHC are reasonable, **would it be far-fetched to state micro-black-holes may form within a star too?**


title: How do non-transverse photon polarizations cancel in Euclidean QED?

text: First, recall how to write scattering amplitudes in covariant fashion in Minkowskian QED. One starts by considering some process with an external photon whose momentum is chosen to be $k^\mu=(k,0,0,k)$ and let the two transverse polarization vectors be $\epsilon^\mu_1=(0,1,0,0)$ and $\epsilon_2^\mu=(0,0,1,0)$. The Ward identity tell us that if the amplitude for the process is $\mathcal M=\mathcal M^\mu \epsilon_\mu(k)$, where we've factored out the polarization vector for the external photon in consideration, then the amplitude obeys $\mathcal M^\mu k_\mu=0$, on shell. With our setup, this simply tells us $\mathcal M^0=\mathcal M^3$. If we then calculate the square of the amplitude and sum over the physical external polarizations we'd find $|\mathcal M|^2=\sum_{i\in\\{1,2\\}}\epsilon_{i\mu}\epsilon_{i\nu}^*\mathcal M^μ\mathcal M^{ν*}=|\mathcal M^1|^2+|\mathcal M^2|^2$. Due to the Ward identity, this is equal to $−\eta_{\mu\nu}\mathcal M^\mu\mathcal M^\nu$, with $\eta_{\mu\nu}={\rm diag}(1,-1,-1,-1)$, and so we can make the replacement $\sum_{i\in\\{1,2\\}}\epsilon_{i\mu}\epsilon_{i\nu}^*\to-\eta_{\mu\nu}$ and so it's the Ward identity which allows us to write scattering amplitudes covariantly. How does this generalize to the Euclidean case where the metric is $\delta_{\mu\nu}={\rm diag}(1,1,1,1)$? Naively, we need the non-Euclidean signature in order to reproduce the cancellation between the manifestly positive squares of matrix elements as found above so how does this procedure generalize to the Euclidean case? That is, if the true scattering amplitude in the Euclidean case is still $|\mathcal M^1|^2+|\mathcal M^2|^2$ then it seems that this can't be equivalent to $\delta_{\mu\nu}\mathcal M^\mu\mathcal M^{\nu *}$ for any Ward identity type relations between the $\mathcal M^\mu$'s and so it wouldn't be possible to write the scattering amplitude in a manifestly covariant fashion. What's going on? Am I missing something obvious? Do the photon's degrees of freedom change when going to Euclidean space or something?


title: What is the electric charge has the sun and the corona of the sun?

text: What is the net electric charge (in magnitude and sign) of the sun and its corona?


title: Is time ordering defined for a single operator depending of two time variables?

text: The **time ordering** for the purpose of quantum mechanics is e.g. given by $${\mathcal T} \left[A(x) B(y)\right] := \begin{matrix} A(x) B(y) & \textrm{ if } & x_0 > y_0 \\\ \pm B(y)A(x) & \textrm{ if } & x_0 < y_0, \end{matrix}$$ where $x_0$ and $y_0$ denote the time-coordinates of the points x and y. Now strictly speaking, the above definition works by pattern recognition - I can tell the operators $A(x)$ and $B(y)$ appart. I could either imagine one could write down the quantum mechanical theory by repacing all occurences of $\mathcal T \left[A(x) B(y)\right]$ by function notation $\mathcal T \left[A(x), B(y)\right]$ or that there is a computable procedure to make sense of $\mathcal T$ whenever I'm given an operator $F(x,y)$, i.e. $\mathcal T \left[F(x,y)\right]$. Which is the case?


title: Which gets you first when you are falling into a black hole, the black hole singularity or the cosmic background radiation?

text: If you look up while you are falling into a black hole you see the universe blue shifted, that is, you see the universe moving quickly forward in time compared to your local time. Since this effect increases as you get closer to the singularity, a star aimed at you could blast you with a billion years of radiation in maybe a second. Of course, no star will be perfectly aligned for billions of years. On the other hand, there is lots of cosmic background radiation coming at you from everywhere in the sky and it will be blue shifted too. So, how bad is it? Would this radiation cook you before you hit the singularity?


title: Duality between Euclidean time and finite temperature, QFT and quantum gravity, and AdS/CFT

text: The thoughts below have occurred to me, several years ago (since 200x), again and again, since I learn quantum field theory(QFT) and statistical mechanics, and later AdS/CFT. It is about the duality and the correspondence between Euclidean time and finite temperature, QFT and quantum gravity, and AdS/CFT. So let me summarize the duality between them below as (1)~(5). This can be read from, for example A Zee's book on QFT and for AdS/CFT review, read from any arXiv review of it. > (1). Euclidean QFT in (d+1) -dimensional spacetime > > ∼ Classical statistical mechanics in (d+1) -dimensional space > > ∼ Classical statistical mechanics in (d+2) -dimensional spacetime (where > time does not play much the role) - > (2). Euclidean QFT in (d + 1)-dimensional spacetime, $0 \leq \tau \leq > \beta$ > > ∼ Quantum statistical mechanics in (d)-dimensional space > > ∼ Quantum statistical mechanics in (d+1)-dimensional spacetime - > (3). Euclidean QFT in (d+1)-dimensional spacetime > > ∼ High-temperature quantum statistical mechanics in (d+1)-dimensional space > > ∼ High-temperature quantum statistical mechanics in (d+2)-dimensional > spacetime \- The above (1)~(3) relations can be more precise from considering the partition function of both sides of the duality: $$ Z=\text{tr}[e^{-\beta H}]=\int_{\text{periodic boundary}} D\phi e^{-\int^\beta_0 d\tau_E \int d^\text{d} d x L(\phi)} $$ here $\tau_E$ is the Euclidean time with periodic boundary condition. - In **AdS/CFT correspondence** , or gauge-gravity duality, we learn that > (4). QFT in (d+1)-dimensional spacetime > > ∼ quantum graivty in (d+2)-dimensional spacetime where the bulk radius plays the rule of the renormalization group (RG) energy scale. Such a correspondence has a strong-weak coupling duality, e.g. > (5). QFT in (d+1)-dimensional spacetime at strong coupling > > ∼ classical graivty in (d+2)-dimensional spacetime at weak coupling The duality can be made more precise between: SU($N_c$) $\mathcal{N}=4$ super-Yang-Mills and AdS$_5 \times S_5$ $\frac{R^2}{\alpha'} \sim \sqrt{g_s N_c} \sim \sqrt{\lambda},\;\;\; g_s \sim g_{YM}^2 \sim \frac{\lambda}{N_c},\;\;\; \frac{R^4}{\ell_p^4} \sim \frac{R^4}{\sqrt{G}} \sim N_c$ the **large** $N_c$, number of colors indicates the **small** gravitational constant $G$. * * * **My Question** : Given the relation between the AdS bulk radius as a energy $E$ scale, which is basically related to a time $t$ scale and the temperature via the dimensional analysis $$[E]\sim 1/[t] \sim [T]$$ and given the hinting **relations between gravity and thermodynamics via S Hawking and T Jacobstein works, and perhapsE Verlind, etc.** and Given the suggestive relations between Quantum in (d+1) dim and classical thermo/gravity in (d+2) dim of (1)~(5). How much have we known and have been explored in the literatures about the relations between (1)~(3) and (4)~(5)? e.g. **the relations between:** **the duality between Euclidean time and finite temperature** (e.g. **QFT and thermal-statistical mechanics** ) and **AdS/CFT duality?**


title: Wave or particle in the end

text: In Young's double slit experiment I put a convex lens after double slit and direct interference pattern on to a fibre. Then I expose the other end of fibre to a screen. What will I observe - the interference pattern (slightly jumbled) or a dot? Will photons behave as wave or particle?


title: Really nothing special when falling into a black hole?

text: It has been said time and again, that an observer who falls into a black hole will not notice anything special. Is this really true? There is of course the problem with the tidal forces, but I assume that these can be lowered beyond any limit by making the black hole big enough, right? But what if I take a cigar-shaped spaceship and let it free-fall into a black hole? There will be a moment, where the head is already inside, but the tail is still outside. This of course requires some increase in gravity along the axis of the spaceship, but this increase can be so small, that an observer won't have to worry about it, but still the event horizon will be inside the spaceship. Now the obeserver could send a light beam from the head to the tail of his spaceship and have it reflected there. Since the observer is in free fall and tidal forces are minimal, I would expect that the beam would just bounce back. This would be in line with the statement, that he does not observe anything special. However that cannot be true, because he has just sent a beam of light across the event horizon.


title: QM Question about the Dirac Delta Potential

text: I just wrote down the solution for the bound state of the Dirac delta potential well, for $E<0$, and apparently there is only one specific energy for the bound state, and it is negative. I solved for this just assuming $E<0$. What happens if a particle enters with energy $E<0$ but isn't equal to the bound state energy? I don't understand what the solution is telling me, that there is only one bound state, an exact figure. Does that mean we cannot say what the energy of the impinging particle is until we see what it does? I don't understand why you can't have an $E<0$ that doesn't equal the bound state energy, and if you can, what does that do? In the literature they solve for the bound state energy, then solve for the scattering coefficients for the scattering states. I guess they imply $E$ of the particle can be anything, as long as it's greater than zero. Am I wrong to think they are glossing over this too quickly? It doesn't make sense.


title: Mathematical physics help appreciated

text: Is anyone able to help me answer the following questions? What is the method? a) Monochromatic light is incident on a slit of width 0.200 mm. On a screen 1.43 m beyond the slit, the first-order dark fringe is 4.550 mm from the centre of the primary maximum. What is the wavelength of the light? b) A planet is in orbit a distance of 1.60 × 1011 m from a star which is 2.30 × 1016 m away. What minimum aperture telescope is needed to see the planet separately from the star at a wavelength of 500 nm? c) Monochromatic light of wavelength 525. nm is incident on a diffraction grating with 5415 lines per centimetre. What is the highest order maximum that can be observed?


title: Can we explain Huygen's principle taking into account Maxwell's predictions?

text: Descartes gave the corpuscular model (1637) of light. Corpuscular model was further developed by Issac Newton. Model predicted that if the ray light (on refraction) bends towards the normal then the speed of light would be greater in the second medium. Focault proved it to be false in 1850. Before this, Huygens had put forward the wave theory of light in 1678, this was not readily accepted primarily because of Newton's authority and also because light could travel through vaccum and it was felt that a wave would always require a medium to propagate from one point to the other. However, when Thomas Young performed his famous interference experiment in 1801 ( _click here to watch the video_ ), it was firmly established that light is indeed a wave phenomenon. Huygens explained the interference pattern by considering light to behave as like waves (similar to water waves). It ruled out the corpuscular theory of light as it could not account for the pattern formed. Now, we can assume light to be some what like water waves (strictly speaking at present, it is believed that, light propagates with electric and magnetic field oscillating perpendicular to each other). How are electromagnetic waves produced? Neither stationary charges nor charges in uniform motion (steady current) can be sources of electromagnetic waves. The former produces only electromagnetic fields, while the latter produces magnetic fields that, however, don't vary with time. It is an important result of Maxwell's theory that accelerated charges radiate electromagnetic waves. So, considering that oscillating charge produces electromagnetic radiation. I thought in Sun, vibration or acceleration of charges or ions produces light or electromagnetic radiation. Or else we can say that excited atom emits light. In sun there are infinite number of atoms which emit radiations, and we can't expect all the waves to be in phase as like in water waves (even if we use any monochromatic light source I thought we can expect billions of atoms to be emitting light which may not be in phase). So, I thought considering electromagnetic waves or light to be similar in model to that of water waves over rules from this view of Maxwell's theory. But, Huygen successfully explained that waves will be in phase, and there will be wavefronts as like in water waves. I don't know whether I have misunderstood anywhere, or else _is it that we can't explain Huygen's principle taking into account Maxwell's prediction?_ If any is the case please explain. ![enter image description here](http://i.stack.imgur.com/2RD3Y.gif)   [Water waves with oscillating points in phase] ![enter image description here](http://i.stack.imgur.com/M4TWe.jpg) ![enter image description here](http://i.stack.imgur.com/YLX6n.png) [Huygens model, where he considers light waves to be in phase]


title: Electric impluses inside nerve cells

text: I'm not a biologist, nor physicist, but there is one thing I try to understand about the elecricity inside human body. I wasn't sure whether place this question here, or perhaps on biology portal, but I consider physics to be more descriptive of the processes occuring in nature (than e.g. biology). I always was taught about electricity to be an ordered motion of electrons - and that they are transferred from one atom to another one when electricity "happens" (which means that one atom looses its electron, but acquires a new one from the other atom instead). But after reading some science- popularizating articles on the net on human body, I found out that they mention about whole ions being moved in the human nervous system in order to transfer an electric impulse ("message" inside nervous system). Is it really the case that whole atoms, or even bigger particles, can be involved in the electricity? Even if so, it should have been probably much slower than the electricity with electrons, or even smaller particles, I guess? Whole atoms being "mediums" for electrical charges is something new to me.


title: Surface Water Drainage - Grate Inlet Capcity

text: The situation is a grated drain across a driveway entrance (so a thin long drain) if I know the roughness of the upstream surface (ie mannings roughness), the slope of the surface upstream (which is the same as the slope of the grate) and the effective area of the grate (ie clear opening area of the grate) how do I calculate my approach flow vs my captured flow (in the grated drain)? Ignore the capacity of the channel of the grate I already know how to calculate that.


title: Story about a mathematician, a dinner party, and the three-body problem

text: I remember dimly hearing a story, coincidentally also at a dinner party, and I was trying recently to track the details down with no success. I was hoping someone here might have also heard this story and could help track down a source. The story bears some resemblance to the story about "turtles all the way down". In this story, told as an historical account about a particular mathematician's motivation for studying the three-body problem, the mathematician, perhaps Lagrange, perhaps d'Alembert, perhaps Poincaré, is over at a dinner party where he is the only mathematician. An older woman, perhaps a dean's wife, asks him if he knew about the mysterious planet sharing an orbit with the earth but always diametrically opposed from it, blocked from view by the sun. He tries to convince her that this is preposterous, and that we would know if there was such a planet by the effect on our own orbit. He can't seem to convince her, but--even worse--when he thinks about it harder at home after dinner he can't seem to quite convince himself either. This inspires the mathematician to study the three-body problem in full depth and to show rigorously that the kind of setup described by the woman would be unstable. It is very possible that this account is apocryphal. Nevertheless, I would like to identify a source where it was told, or where it is attributed. Is there such a source?


title: Anti-symmetric forms on Dirac spinors

text: In order to describe invariant forms on Dirac spinors $S$ one can find trivial subrepresentations in $S \otimes S$. If we use $S \cong (1/2, 0) \oplus (0, 1/2)$ then \begin{multline} [(1/2, 0) \oplus (0, 1/2)] \otimes [(1/2, 0) \oplus (0, 1/2)] =\\\ (0, 0) \oplus (1, 0) \oplus (1/2, 1/2) \oplus (1/2, 1/2) \oplus (0, 1) \oplus (0, 0) \end{multline} Therefore representation theory predicts existence of two invariant forms. It is usually claimed that this two forms are $$ D_1(\chi, \psi)=\bar{\chi}\psi=\chi^T\gamma_0\psi=\chi_R^T\psi_L+\chi_L^T\psi_R $$ and $$ D_2(\chi, \psi)=\bar{\chi}\gamma_5\psi=\chi^T\gamma_0 \gamma_5\psi=\chi_R^T\psi_L-\chi_L^T\psi_R $$ The form $D_1$ is symmetric and it's quadratic form (with complex conjugation on the first argument) is usually used to construct Dirac's Lagrangian. From the other hand, it is known that on Weyl spinors one can also find an antisymmetric invariant forms given as $$ \chi^T_L\sigma_2\psi_L . $$ Let me use this to construct one more anti-symmetric invariant form on Dirac spinors as a sum of two forms on Weyl spinors $$ D_3(\chi, \psi)=\chi^T_L\sigma_2\psi_L+\chi^T_R\sigma_2\psi_R $$ Form $D_3$ is not a linear combination of $D_1$ and $D_2$ and thus I get a contradiction with representation theory prediction. Where I made a mistake?


title: When you breath in a helium balloon, do you lose weight?

text: We all know the trick of breathing in a helium balloon to get a funny squeaky voice! When you do this, would it be correct to say you lose weight? On the one hand, the helium should make you more buoyant in the surrounding air, similar to how taking a big lungful of air when swimming helps you to float better, so it would seem that you are decreasing in weight. On the other hand, you are taking more matter inside your body so you must obviously increase in mass.


title: What form of energy is released in $E = mc^2$?

text: In the famous mass energy equivalence $ E = mc^2$ whenever any form of mass, lets say mass defect in nuclear reactions or any other example of mass is converted to Energy, the released energy is in what form ? Heat ? Light ? Other ?


title: Ricci scalar higher dimensions

text: I was wondering if there is a straightforward way to compute the Ricci curvature of a metric that has the form (à la Kaluza-Klein): $g_{MM}\equiv\begin{pmatrix}g_{\mu\nu}&g_{\mu i}\\\g_{i\nu}&g_{ij}\end{pmatrix}$ Ie, finding an expression of the form: $\mathcal{R}(G_{MN})=R(g_{\mu\nu})+R(g_{ij})+(\text{scalar depending on both metrics})$ I tried using the usual formula $R^M_{NPQ}=2\partial_{[Q}\Gamma^M_{P]N}+2\Gamma^S_{M[P}\Gamma^M_{Q]S}$ where $A_{[MN]}=\frac{1}{2}(A_{MN}-A_{NM})$. But this becomes really quickly a mess due to the different indices. Does anyone has an idea how to find the result quickly? For a kaluza klein metric where $g_{ij}$ is a scalar and $g_{\mu j}$ is a vector this is straightforward, but I can't find a nice solution.


title: What is going on with star's energy?

text: The sun radiates $3.8 × 10^{26} J$ energy each second, this energy is absorbed by various celestial objects but most of it is reflected or emitted back in the space, similarly there must be energy pouring into empty space from various star systems, what is happening to all this energy ? In what form is it thriving if it is thriving ? Does it ultimately get absorbed by various objects ? Is it bringing some kind of change to the motion of galaxies etc ? Is there some sort of energy sink somewhere ( maybe getting sucked in by Black holes) ? Does it somehow combine and form new bodies via energy mass equivalence ? **what is happening ?** Addendum : This maybe a on the edge of site's mainstream physics discussion policy, but till now all I understood was that no definitive proof of where the energy is going is available, @ John rennie says that due to red shift the wave's are losing energy, he says that energy density is tending to zero, but there is no account for this energy. At the same time, @Thriveth mathematically asserts that the universe is losing energy, I find it a bit of a controversial topic regarding energy conservation and can't shake off the idea that maybe this energy could account for dark energy/matter. I add, millions of stars have died/ emptied their energy into the space since the beginning of time, by the logic of either answer, red shift or violation of energy conservation we can say that this energy is somewhere we can not account for, similarly we till now can't account reasons for dark matter/energy can't we explain both of these with each other ?


title: Beam power and electric field after a beam splitter

text: Consider a beam with power $P_1$ and electric field amplitude $E_{01}$. It is sent through a 50/50 beam splitter that produces beams with power $P_2=P_3=P_1/2$. What are the electric field amplitudes of the split beams, $E_{02}$ and $E_{03}$? From what I understand, $P=KE_0^2$ where $K$ is a constant. Therefore, $E_{02}= \sqrt{P_2/K}=\frac{1}{\sqrt{2}}\sqrt{P_1/K}$, $E_{03}= \sqrt{P_3/K}= \frac{1}{\sqrt{2}}\sqrt{P_1/K}$ Now say I can recombine the beams in phase and without any losses, then $E_{04}=E_{02}+E_{03}$. The power of the recombined beam is $P_4=KE_{04}^2 $ $= K[E_{02}^2 + E_{03}^2 + 2E_{02}E_{03}]$ $=\frac{1}{2}P_1+\frac{1}{2}P_1 + P_1$ $=2P_1$ So $P_4 >P_1$ and I've created power from nowhere! What is wrong with this picture?


title: Why aren't we Boltzmann brains in an infinite universe?

text: Either space is finite or it is infinite. a) - If space is infinite in extent, either it is thermal over an infinite volume, or it is in the vacuum state for most of it. If it is thermal, infinity is a large place, and random statistical fluctuations alone will ensure there are infinitely many places in the universe just like the Earth (or maybe just your room, or your brain), but not necessarily beyond. This is Max Tegmark's level I multiverse. Only a doubly exponentially small fraction of the universe will be like that. Doubly because human length scales are already exponential compared to the microscopic scale, so it's exponential of exponential. But infinity is a large place. If most of the spatial extent of the universe is in the vacuum state, in quantum field theory, we have the Reeh-Schlieder theorem for any bounded region, and so, there is still a doubly exponentially small probability that the region will be just like your brain. b) - If space is finite: either it remains bounded in size over all time, or it expands indefinitely, or it will end in a big crunch. b1) -If it is finite and bounded, it will undergo Poincare recurrences, and there will be infinitely many brains just like yours spread out over time. b2) - If it expands without bound, eventually the volume of space will exceed doubly exponential, and there will be many replicas of your brain spread out over space in the future. So why aren't we Boltzmann brains? In general relativity, two locally identical Boltzmann brains can be related by a diffeomorphism. There is no absolute position or absolute time distinguishing them. So we have to identify all these Boltzmann brain replicas _relationally_. Are they all the **same** brain, all those infinitely many of them? Infinitely many identical brains, all identified. Quantum evolution is nondeterministic, so they don't all evolve the same. So figure out the fraction of identical brains evolving in a certain manner, and we get frequentist probabilities. This requires a measure over these brains. There is no invariant measure in eternal inflation, the measure problem.


title: Explanation of the Graetz circuit

text: My knowledge of circuits is pretty rudimentary and I've never really understood circuits, so I'm having trouble with the concept of **Graetz circuits** : ![enter image description here](http://i.stack.imgur.com/wVX2z.jpg) When you register the voltage on the resistor R on a screen of an oscilloscope, you get: ![enter image description here](http://i.stack.imgur.com/RKpxg.jpg) Why is this exactly? As I said, my knowledge of circuits is very limited. My idea: The sign at the bottom of the circuit means it is an AC current. If the current goes from left to right it will be stopped by the diode above and let through by the diode below (to the left of the resistor) and vice versa if it goes from the right to the left. Is this correct?


title: Vacuum-ether and spacetime

text: in the past you could not give an explanation for various phenomena in which was an action at a distance like magnetism or gravity that occurred in a vacuum; For this reason, it was hypothesized ether as something that would by connecting to bring the information of the positive charge to the negative charge (remaining in the field of magnetism), in short, was inconceivable that this attractive force could exist in empty. At a later stage it was thought to .......... Then with General Relativity about gravity is thought to a deformation of the space-time, for which (for example) the Earth has a fixed course in its turn around the sun Now excuse me if my question to the introduction I wrote inaccuracies, but did not know how to explain otherwise ........ I come to the question whether in the past had broken into the ether because it was inconceivable that a force could be transmitted in a vacuum, now why it can deform according to the General Relativity? If it is empty what deform? Then I have to re-introduce the ether for something to deform? I hope that no one gets a laugh for this my silly question, but instead to help me understand a little better as far as possible. Luca


title: Greens function in EM with boundary conditions confusion

text: So I thought I was understanding Green's functions, but now I am unsure. I'll start by explaining (briefly) what I think I know then ask the question. * * * **Background** Greens are a way of solving inhomogeneous differential equations by first solving the response to a unit impulse. I am aware they are a _tool_ that can sometimes be given a physical meaning. I fear those two realms may be crossing for me where they shouldn't. $\hat{L} f(x) =s (x)$ $\hat{L} G(x) =\delta (x)$ Using these we can obtain the following general expression for the solution: $f(x)=\int G(x-x')s(x')dx'$ Physically, in EM, the Greens function appears to be in the form of a potential due to a single point charge. I.e. we can construct our solution for an arbitrary charge distribution by 'summing' over everything to create our known charge distribution. $\nabla^2\Phi=-\frac{\rho}{\epsilon}$ $\nabla^2G(x-x')=-\frac{\delta(x-x')}{\epsilon}$ $\Phi(x)=\int G(x-x')\rho(x')dx'$ Where $G(x-x')=\frac{1}{|x-x'|}$ When we start to talk about boundary conditions there is an ambiguity in the definition of our Greens function such that we can have a form as follows: $G(x-x')=\frac{1}{|x-x'|}+F(x-x')$ so that $\nabla^2F(x-x')=0$ I guess this is where it starts to get blurred for me. * * * **Question:** Jackson 2.7 a) Consider the half plane z>0 with drichlet boundary conditions on the z=0 plane. We wish to write down the greens function for this situation. Many solutions I have found jump right to the greens function of: $G(x,x')=\frac{1}{|x-x'|}-\frac{1}{|x-x''|}$ But I do not understand how they got here - they say it is obvious- but given what I understand I must be missing something. How is this the greens function if there is no charge? They appear to allude to the existence of a charge and an image charge. It seems like we are trying to solve the laplace equation which is homogeneous (so I don't understand the use of greens here). I am guessing the boundary conditions are the critical thing I am not understanding fully. Feel free to correct any of my misinterpretations and mistakes in general.