title: Feynman's explanation of parallel axis theorem

text: In the book Feynman's Lectures on physics volume 1 chapter 19, He explains prallel axis theorem as follows. > Suppose we have an object, and we want to find its moment of inertia around > some axis. That means we want the inertia needed to carry it by rotation > about that axis. Now if we support the object on pivots at the center of > mass, so that the object does not turn as it rotates about the axis (because > there is no torque on it from inertial effects, and therefore it will not > turn when we start moving it), **then the forces needed to swing it around > are the same as though all the mass were concentrated at the center of > mass** , and the moment of inertia would be simply $I_{1}=MR_{cm}^2$, where > $ R_{cm}$ is the distance from the axis to the center of mass. But of course > that is not the right formula for the moment of inertia of an object which > is really being rotated as it revolves, because not only is the center of it > moving in a circle, which would contribute an amount $I_{1}$ to the moment > of inertia, but also **we must turn it about its center of mass**. So it is > not unreasonable that we must add to $I_{1}$ the moment of inertia $I_{c}$ > about the center of mass. I could see the first half of his explanation, but I could not understand last half of it. Why the forces needed to swing it are the same as if all the mass were concentrated at the center of mass and why we should turn the object about its center of mass?


title: How do whisky stones keep your drink cold?

text: From a discussion in the DMZ (security stack exchange's chat room - a place where food and drink are important topics) we began to question the difference between how ice and whisky stones work to cool drinks. Both are frozen, but when ice is placed in a drink it slowly melts, using energy from the drink, thus cooling it. But whisky stones don't melt, so how do they cool the drink?


title: Is my textbook in error about this geometric optics relation?

text: Given a centered optical system (having an axis of rotational symmetry), let $H_1$ be the intersection of the optical axis and the "principal object plane" (I'm studying in French and have no idea how to translate that... it's the plane whose image is a plane with a scaling factor of 1). Let $H_2$ be the intersection of the optical axis and the "principal focal plane", in other words the image of $H_1$. ![](http://i.stack.imgur.com/j9bQC.gif) My book states that the distances between those points and the focal points, $f_1=H_1F_1$ and $f_2=H_2F_2$, verify the following relations: $$n_1f_1=-n_2f_2$$ $$\frac{n_2}{f_2}=-\frac{n_1}{f_1}=V$$ Where V is the vergence. But these seem to contradict eachother. From relation 1, we have $$n_1=-\frac{n_2f_2}{f_1}$$ Substituting into the second relation: $$\frac{n_2}{f_2}=\frac{n_2f_2}{f_1^2}$$ $$1=\frac{f_2^2}{f_1^2}$$ $$f_1=\pm f_2$$ But the book then goes on to say that that's only true when $n_1=n_2$.


title: Monte Carlo for Random Bond Ising ferromagnet

text: **The set-up:** Consider the Ising model on an $L \times L$ square lattice, where the coupling of each bond is chosen to be $+J$ (ferromagnetic) with probability $(1-p)$ and $-J$ (antiferromagnetic) with probability $p$. For $p=0$ we have the standard Ising ferromagnet with two ground states (all up and all down). For $p \lesssim 0.11$ we will still have a ferromagnet, and the ground states still fall into the classification of 'mostly up' and 'mostly down', but there will be regions of opposing spins. These can take various different shapes while not changing the energy, and so introduce a large degeneracy into the ground state space. But the different shapes can be locally moved between by rotations of $O(1)$ spins at an $O(1)$ energy cost, so they are not like the deep wells of a spin glass. **The question:** Suppose we wish to calculate the thermal average of the energy at a temperature below the Curie temperature (and hence within the ferromagnetic phase). How many Monte Carlo steps are required to do this (equivalenlty: what is the autocorrelation time), assuming that we start in an initially random state? How much does it depend on the Monte Carlo used (such as Metropolis or Heat Bath, and with or without parallel tempering)? Since energy only depends of the length of domain walls, and so doesn't care whether the system is in a 'mostly up' or 'mostly down' state, my intuition makes me think that it is not necessary to wait for the $\exp(L)$ required for the system to flip between these. So just local changes should be enough, and hence the autocorrelation should be polynomial at most. Is this true? Another issue is that the initially random state will be around 50% spin up and 50% spin down, and hence needs to break the symmetry to move towards a ground state. What is the time required to do this? **The background** There's a lot of statements out there in the literature about the feasibility of Monte Carlo studies in certain cases. They tell you that frustration (which this model has) is bad and will lead to inefficient Monte Carlo. They tell the same for spin glasses (which this model is at $T=0$ and high $p$, but not in the regime discussed above). But definitions of inefficiency seem to vary over time and between authors. For me it means higher than polynomial wih $L$. But for some people even polynomial scaling seems enough to claim inefficiency. It seems difficult to find a consistent overview of the issue. But this is a model that I have a particular interest in, and would really like to know at least what the autocorrelation time for the energy is.


title: What occupies the space of a molecule in a gas which moves away due to kinetic energy?

text: The gas state means it has lot of kinetic energy and the momentum of molecules transfer the energy on The container wall and is pressure.but during displacement of each molecule, what occupies it's previous position? Will it be occupied by another molecule?


title: Energy conservation and center-of-mass inertial frames

text: For a given set of fast-moving objects in empty space, the maximum mass-energy that can be extracted by using only interactions _between_ objects in the set (e.g., colliding all of them together) is identical to the relativistic mass- energy that an observer would see when residing in the center-of-mass inertial frame for that set. (You can trust me on that or prove it yourself; it's not difficult.) This is why a proton colliding with earth at $\gamma=10$ velocity does not cause a catastrophic release of ten earth's worth of energy. Since the mass of the proton is entirely negligible in comparison to the earth, the mass-energy available by colliding all objects in the set _{earth, proton}_ is very close to the mass of earth plus ten times the mass of the proton. Thus for any set of objects that are isolated in space, the center-of-mass inertial frame is the only special relativity frame for which the _observed_ mass-energy of all parts is identical to the _available_ mass-energy. More energy can of course be extracted, but _only_ by bringing in objects that were not in the original set. The center-of-mass inertial frame thus is both "distinguished" and unique with respect to that set of objects (only). Now here's the fun part: Define the set to include all objects in the universe. Since you have just placed all your eggs in one basket, so to speak, there are no longer any objects to bring in from outside to alter the result, even in principle. Next, calculate for "distinguished" inertial frame for this set. (For a pretty good guess, try the CMB.) My question is this: Doesn't the above argument imply that under the rules of special relativity the universe really does have a single, unique, and non- trivially distinguished inertial frame, that being its center-of-mass inertial frame? * * * **Addendum 2012-05-30.20:220 EST** **Angular momentum** I skipped over angular momentum in defining the set of objects used to define the minimum energy SR frame. Unlike linear momentum, residual angular momentum in a set of object of course cannot be resolved simply by resetting the frame. Always interesting, angular momentum, since if it exists it necessarily "points" to additional matter that exists outside the current set, at least if you believe in absolute universal conservation of it (which I certainly do). Its energy can be quantified locally, though. **"Center of mass" in curved spaces** @LubošMotl and others brought up the excellent point that you can't really do a center of momentum for curved space. If you assume that the curved space can be embedded within a larger Euclidean space (Nash proved that in his last pre- breakdown paper, which was an amazing work), then it's easy to see that the center of mass of the curved piece is not very likely to fall within the piece itself. It's more likely to wind up somewhere "other" than the surface, e.g. for an ordinary balloon it falls at the _center_ of the balloon! So, when I say "center of mass frame," I'm really using a sloppy shorthand for a procedure that would require following along the locally 3D surfaces to construct an overall average. Theorem: For any smooth manifold decorated with components that move locally in accordance to the accepted rules of mass, energy, and momentum conservation, and whose shape can be approximated as being invariant over the time scales considered for those motions, there exists a singular "manifold rest frame" that is motionless with respect both to (a) the overall form of the manifold if it is irregular, and (b) with respect to the SR energy minimum defined for the entire set of components moving within the manifold. (I am dancing lightly over rotations in 4D; there are two orthogonal ones, makes life interesting. I'm also skipping over the issue of the two different forms of 4D holes, the equivalents of 1-spheres (rings) and of 2-spheres (balloons), on the assumption that such topologies are unlikely for a realistic universe.) (All dimensional solids n>3 have more than one type of, each hole type having an equivalence to one of a series of lower dimensional spheres from the 1-sphere (ring) to the (m-2)-sphere. That's not counting erasures (m=n), splits (m=n-1), and internal voids (m=0) (Do they address that in string theory? If not, how do they specify or distinguish between the 7 or 8 hole types of the n=9 or n=10 spaces used in M-theory? Ah.. hmm... maybe I should make that into a real question?)


title: Name of fermionic CFT theory

text: I'm looking for a name or references to theories that include a stress energy tensor of the form $$T(z)=A:\phi^\alpha\partial\phi_\alpha:(z)+B:\prod_{i=1}^{D}\phi^i:(z)$$ $\alpha=1,...,D$. Where the fields anti-commute. In particular I'd like to know what kind sensible operators can be formed from normal ordered products of the $\phi$ **Edit** As commented below, my main interest is $D=4$ and from my calculations it seems that most of the operators than can be formed are only sensible when $B=0$ and there's only one type of operators (of weight 1) that can be formed when $B\neq 0$.


title: Radial quantization and infrared divergences

text: I am reading Ginspard lectures "Applied CFT" http://arxiv.org/abs/hep- th/9108028 which is not my first material on the subject. He tries to motivates radial quantization on the reason that compactifying the spatial direction avoids potential IR divergences, but I previously learned this as motivated by the use of contour integrals and the powerful tools of complex analysis to relate straighforwardly the OPE to the Virasoro algebras. Are these motivations related? Do most CFT suffer from IR divergences? By definition they cure UV divergences but now that I think I bit more they may suffer from strong IR divergences having a spectrum composed only of massless particles (the same way a massless photons brings IR divergences to QED).


title: The Strength Of A Black Hole

text: Does the gravitational strength of a black hole at its singularity change the more it pulls things into it?


title: Breaking ocean waves on other planets possible?

text: I have always wondered if there were an ocean on another planet with stronger or weaker gravity, would it be possible to have breaking waves?


title: What is the force exerted on a body that doesn't move?

text: If I push against a brick wall I'm exerting a force against the brick wall the the brick wall will exert the same force against me. These will cancel each other out and neither myself or the wall will move. Since I am not moving and therefore have no acceleration how do I calculate the force I exert on the wall given $F = ma$?


title: 2 following gates, inverse circuit

text: I have a circuit that has 4 wires and 2 following each other Toffoli gates. The first Toffoli gate occupies 3 wires from above, the following Toffoli gate occupies 3 wires from below.   What will look like the inverse circuit of this entire circuit?      Thanks much!


title: For an object at rest is the potential energy completely zero?

text: It is said every object in this universe tends to be at their lowest state of potential energy. What can be the lowest state of potential energy? And can the potential energy of an object be 0?


title: Quantum unscrambling

text: This question is similar to the Phys.SE post Retrodiction in Quantum Mechanics, however, it addresses a different issue: how would you design a machine that can measure a simple quantum system and "rewind it"? The machine is allowed to generate as much entropy as it needs to to do so. Lets say two molecules (~10 atoms each) smash into each-other (~30 km/s relative velocity) and spew atoms all over the place. You try to reconstruct the initial state by precisely measuring the position and velocity of each atom in the cloud and rewinding the dynamics. In classical, deterministic physics, this is doable. In quantum mechanics you can't precisely measure the position and momentum/velocity. However, it may be possible to design the detector to measure position precisely, and put the detector walls far enough away so that the prior momentum can be inferred based on time-of flight. If this will work, will the walls have to get exponentially further as the number of atoms increases? The ultimate "quantum scrambler" is a black hole. Even a plank mass hole is 10^19 heavier than a hydrogen atom. The walls may have to be placed farther away than the de-sitter horizon, making it impossible to extract any information from even the tiniest hole.


title: Electrical neutrality of atoms

text: How is it that atoms with equal numbers of protons and electrons are described as "electrically neutral" when the proton is 1,800 times more massive than the electron?


title: Supergravity calculation using computer algebra system in early days

text: I was having a look at the original paper on supergravity by Ferrara, Freedman and van Nieuwenhuizen available here. The abstract has an interesting line saying that > Added note: This term has now been shown to vanish by a computer > calculation, so that the action presented here does possess full local > supersymmetry. But the paper was written in 1976! Do you have any info what kind of computer and computer algebra system did they use? Is it documented anywhere?


title: Can a non-euclidean space be descripted through an euclidean of higher dimension? so why use non-euclidean?

text: If you draw a big triangle in earth 2D surface you will have an aproximated spherical triangle, this will be a non euclidean geometry. but from a 3D perspective, for example the same triangle from space, it could be depicted as euclidean in 3D. then why we talk about non euclidean instead of adding 1 dimension? It is easier the non-euclidean approach? I don't see how! Sorry if the question is perhaps naive, but this is a long time doubt for me, so I will be grateful for an answer, thanks


title: Uniqueness of W,Z bosons

text: What is real reason for considering only ONE unique **W** and **Z** bosons, and not of 3 separate flavors, in the Standard Model?


title: Frame of Reference Question

text: I'm having trouble understanding this frame of reference question. > You are traveling in a car going at a constant speed of 100 km/hr down a > long, straight highway. You pass another car going in the same direction > which is traveling at a constant speed of 80 km/hr. As measured from your > car’s reference frame this other car is traveling at -20 km/hr. What is the > acceleration of your car as measured from the other car’s reference frame? > What is the acceleration of the other car as measured from your car’s > reference frame? Shouldn't they both appear to have an acceleration of zero, because both velocities are constant? I can imagine sitting in the faster car and watching the slower car, its speed would not appear to change, only its position? Thanks


title: Do spin-spin interactions break time reversal symmetry?

text: I'm sure the answer is yes, but how is this shown? Normally for a single spin-1/2 you have a time reversal operator: $-i \sigma_y \hat{K}$ where $\sigma_y$ is the second Pauli matrix and $\hat{K}$ is the conjugation operator. How is this generalized to two spins? I am thinking of whether or not interactions like exchange ($J \hat{S}_1 \cdot \hat{S}_2$) or the hyperfine interaction (contact Fermi: $a \hat{S} \cdot \hat{I}$) break time reversal symmetry.


title: Meaning of conformal field theory

text: Can anyone summarize what conformal field theory is actually about,i.e. 1) what are its goals? (for example, to study such-and-such fields/functions/maps/etc. to see whether they have such-and such properties, and what it means) 2)it is described as "a quantum field theory". How many quantum field theories are there then? Which one does this correspond to. I don't have background in QFT but I thought the division was only between bosonic/fermionic fields. 3) which techniques it uses? (for example, perturbation expansion) I know there are many good books and references, but I was wondering if I could get a self-contained answer here. A few questions have been asked on this website, but none address my two questions above in a straightforward way, without referencing to books. I have a background in quantum mechanics and classical electrodynamics. Any help will be appreciated! Even if it just involves broad analogies!


title: Has non-conservation of baryon number been observed?

text: CP violation (as I understand it) allows for non-conservation of baryon number, and thus can contribute (at least a little) to the baryon asymmetry in the universe today (far more matter than antimatter). But have we ever seen baryon number change in the laboratory? Or had almost- but-not-quite-conclusive evidence of it?


title: Free body diagram and Newton's second law

text: > Two bodies of mass $m_1=50 \text{kg}$ and $m_2 = 10 \text{kg}$ are connected > with a light rope through a pulley (no friction between the rope and the > pulley). $m_2$ moves on surface with angle $37^{\circ}$ with a friction > coefficient $\mu_k=0.5$. Force $\vec F$ acts on $m_1$. The system was moving > with a constant velocity for some time until the force stopped and then the > system started to accelerate. What's the acceleration of the system? ![Free body diagram](http://i.stack.imgur.com/cp4JM.png) I assumed that the system accelerates to the left, therefore the equation for the $m_1$ would be: $ \begin{cases} N_1=m_1 g\\\ T-N_1 \mu=m_1a \end{cases} $ Therefore $T=m_1(a+g \mu)$ For $m_2$ the equation would be: $ \begin{cases} N_2=m_2 g \cos 37^{\circ}\\\ m_2 g \sin 37^{\circ} - T - N_2 \mu=m_2a \end{cases} $ Therefore $m_2 g \sin 37^{\circ} -T- m_2 g \cos 37^{\circ} \mu = m_2a $ The solution would be: $100 \cdot 0.2024 - 50a -250=10a \\\ a \approx -3.829 \frac{\text{m}}{\text{s}^2}$ Which is wrong according to the answers. The answer should be $\approx -5.83\frac{\text{m}}{\text{s}^2}$ which you can get pretty close if instead of $10a$ in the last step it was $-10a$. So I'm pretty sure that my mistake here is with the signs (of the acceleration), but I can't figure out why I'm wrong.


title: Plane Wave expansion method

text: I really don't know if this is the right forum to ask the question...but please help me if you can!! I was going through the Plane Wave Expansion Method Derivation...But to be honest I could not find any step by step proof of this method. (If you provide me one!! I would be thankful) Next coming to the derivation given in the link below, while going through the derivation, I could not understand where come the components $G_i + G_{i'} \rightarrow G_{i'}$ could be written, and similarly the lines which follow it. Similarly ,how do we introduce $G_i,\ G_{i'},\ G_{ii}$... It would be really helpful if anybody could help me out on this one... Lastly I would also appreciate if anyone could just typeset the question properly as I am new here!! PWM Manual


title: Non-conservative electric fields due to changing magnetic flux?

text: What I read in several places, tells me that, the fact Coulomb's Law follows _inverse-square law and gives a force which is radial_ , implies that a static electric field must be _conservative_.(In short, Coulomb's law _is_ conservative!) But the electric field in presence of a time-changing magnetic flux (either time changing fields or a change of area of the loop, Motional EMF) is non-conservative, since closed-loop line integral of electric field is no longer zero but is $-\frac{d\phi}{dt}$ where $\phi$ is the magnetic flux through the surface bound by the loop along which the line integral is carried out. I have the following question:- Consider a situation for the sake of presenting my argument, where there are only steady currents and no accelerated charges, and the changing flux is due to a wire sliding over another U-shaped wire and thus changing the area bound by the loop formed by the U-shaped end and the moving wire.![enter image description here](http://i.stack.imgur.com/B4b5X.png) In this case, all the components and the charges (metallic free electrons and the positive kernels) are unaccelerated (i.e. on an average) and hence, there should be no radiation of electromagnetic waves. Therefore, whatever the electric field produced, which is responsible for the existence of current in the loop, must be produced by a particular non-accelerated charge distribution in the apparatus. But, whatever the field produced by any complicated arrangement of charges, through superposition must follow the inverse-square and the radial dependence of Coulomb's law and must be conservative, since all the superposed fields are conservative in themselves as the individual charges follow the conservative coulomb's law. > **So, how can a non-conservative configuration of the field, result from the > superposition of fields produced by an arrangement of individual charges > which follow a law (Coulomb's) which is conservative , assuming the absence > of electromagnetic waves?**


title: Breaking of a covalent bond

text: When a bond between two atoms is broken, why only one electron is released. Why not two? (as two electrons make up a covalent bond.)


title: Gauss's Law in action

text: ![enter image description here](http://i.stack.imgur.com/gJq0p.jpg) Need someone to tell me if I got this done correctly (a) Draw Gaussuian cylinder inside the black cylinder to find charge enclosed $Q_{en} = Q(\frac{r}{a})^2$ Apply Gauss's Law $E2\pi r \ell = \dfrac{Q\left ( \frac{r}{a} \right )^2}{\varepsilon_0}$ $E = \dfrac{Q\left ( \frac{r}{a} \right )^2}{2 \pi r \ell\varepsilon_0} = \dfrac{Qr}{2\pi \varepsilon_0 a^2\ell}$ $E = \dfrac{\rho r}{2 \varepsilon_0}$ (b) Basically same principle, but the total charge of the black cylinder $E = \dfrac{\rho a^2}{2r \varepsilon_0}$ (c) 0 under static equilibrium (d) The enclosed charge should be a sum of cylinders' charges. $Q_{en} = \rho V + \lambda \ell = \rho \pi a^2 \ell + \lambda \ell = \ell ( \rho \pi a^2 + \lambda)$ Hence $E2\pi r \ell = \dfrac{\ell ( \rho \pi a^2 + \lambda)}{\varepsilon_0} \implies E =\dfrac{ \rho \pi a^2 + \lambda}{2 \pi r \varepsilon_0} $ The term looks very awkward, so it's probably wrong. Thank you for reading


title: Determine radius, speed, radial and tangential component of velocity

text: ![enter image description here](http://i.stack.imgur.com/Qdhv5.png) I need help mainly with the first question. If you can solve also the second problem,I'd greatly appreciate it.


title: What is the mass of a wave?

text: The slide called "QUANTA" here says that "One Quantum has a definite mass" and the picture shows a wave. So, What is meant by the mass of a wave?


title: gravitational convergence of light

text: light has a non-zero energy-stress tensor, so a flux of radiation will slightly affect curvature of spacetime > **Question:** assume a flux of radiation in the $z$ direction, in flat > Minkowski space it will propagate and spread slightly in the $x$ and $y$ > directions. But what happens if we take into account self-gravitational > effects of light onto itself? will the radiation flux self-gravitate and > eventually reconverge?


title: An EM problem of polarized sphere

text: Today, I'm doing an EM problem and my question is exactly the same as this site shows: http://www.physicsforums.com/showthread.php?t=133501 To reiterate my confusion, why we can't use that integral formula at the end V(r)=1/4 pi epislon...+1/4 pi epislon... This formula appears right in the same chapter of the problem. I know we can solve it by applying the Gauss Law, but why this formula won't work. I calculated it by using this formula, and I get a totally different answer.


title: Fundamental properties of motion

text: The first paragraph of the Wikipedia article on the angular momentum operator states that > In both classical and quantum mechanical systems, angular momentum (together > with linear momentum and energy) is one of the three fundamental properties > of motion. I guess it makes sense intuitively. Motion can be described as a combination of translation and rotation. Such movements cause those bodies, per definition, to have angular and linear momentum. Those bodies also have kinetic energy. However, "one of the three fundamental properties" sounds like it would be something well known, well used. I have never seen this being said before. Is it possible to elaborate on why these are the so called "fundamental properties"?


title: Does inflation predict a multiverse?

text: Does inflation necessarily predict a multiverse? Or is it the case that "It all depends on the properties of the initial field responsible for inflation, about which we can only speculate"?


title: Is the Universe Past-Eternal?

text: 1. Does the Borde-Guth-Vilenkin theorem definitively demonstrate that the Universe cannot be past-eternal, whatsoever? Does it not assume a classical space-time while the real world requires Quantum mechanics?    2. Are there successful models in Cosmology that are Past-Eternal?    3. Why might Alan Guth say the Universe might be eternal in the past, when he himself wrote a theorem in 2003 saying it most definitely isn't?


title: Is changing angle of grooves in a Fresnel lens important for the lens to work?

text: I'm thinking of making a huge Fresnel lens out of a plastic sheet I found in my storage. As it's for fun, and I'm not very concerned about precision, I thought I could make it using a nail, rope and two blades positioned so as to form a sharp angle between their cutting edges (or simply form a "V" shape). From looking at a flat Fresnel lens condensation diagram and sketching my own "expanded" lens, I figured I would need to account for groove angle changing when cutting grooves to allow the resulting lens to focus light. But as I'm not very knowledgeable in optics, I don't know if it's important. ![enter image description here](http://i.stack.imgur.com/PwxAW.png) Will the lens still focus light in one spot if the grooves are cut at the same angle? If so, which angle should I choose then?


title: What causes drag in a fluid?

text: What causes resistance of an object to motion within a fluid like water? Please explain to me the molecular dynamics of the situation.


title: How does surface tension enable insects to walk on water?

text: I understand that water molecules at the surface have a net inward attraction due to the lack of water molecules above them. I've been reading a bunch of articles, and they say that this inward pull causes water to act as if the surface is covered by a thin, elastic material, but what does that mean? Is surface tension caused only by the attraction between the water molecules on the surface to other water molecules on the surface? If so, why is surface tension said to be caused by the inward pull; if not, then what role does the attraction to the molecules underneath the surface play in surface tension? If an intruder molecule were attempting to puncture the water surface, how would the water molecules around it behave in order to keep it floating?


title: Is "now" or "the present moment" properly defined in GR?

text: My question is about the extent to which "now" is defined in GR. In Minkowski spacetime, it's possible to define a "now" for an inertial observer by finding a spacelike 3-plane such that, in the observer's frame, all 4-vectors in the 3-plane have zero time component (or something like that - apologies, my geometry is a bit rusty - anyway, a 3-plane in which all vectors are orthogonal to the tangent vector of the observer's world line). This plane can be defined globally, so that my "now" is a slice through the whole of Minkowski space. My question is whether it's possible to define such a thing in general relativity. So, for instance, can I meaningfully speak about what the Andromeda Galaxy is doing "right now"? Or is the "present moment" something that can only be locally defined? I remember seeing something in a Roger Penrose book about this, but I can't find the reference (if anyone knows it, please let me know!)


title: Biggest crater in the Solar System

text: It seems that the biggest crater in the Solar System is Borealis Basin on Mars. The Wikipedia entry and this piece of news say it has been formed by an impact with a body of around 1600 - 2000 km. Mars has a diameter of ~7000 km. The impact velocity was estimated as low 6-10 km/sec. Mars has its own momentum and speed, and I assume collisions in space are plastic. Instinctively, I have some doubts about these calculations, and I am curious how this estimates are done. I am quite sure that these guys know a lot of physics, but what I see is one big ball colliding with a ball a third of its size and leaving only a scratch? Can you land something up to 2700 km big and only get a basin? Was the other body a big pillow? Maybe it was a touch and go? How come Mars was not pulverized to bits?


title: What are the easiest ways to get data out of a FITS file without a library?

text: There are FITS libraries for most programming languages (see this list), but FITS is a simple enough format that it should be pretty easy to extract data without the need for such libraries or utilities. The data portion of an an image is a simple list of pixel values, and of a table is just a catenation of the binary values in each row. The headers are simple enough that it's easy for a person to read the header and see what those rows should be, so my intuition says that it shouldn't be too hard for person to look at the header, and write a short bash script to, say print the rows of a table as an ASCII table that can be processed with awk. After trying it, however, it doesn't seem as easy as I first expected; I generally need to resort to writing (admittedly simple) code. I'm really looking for something I can run on a system I am unfamiliar with, and will only be working on for a few minutes, such as when I am helping someone else on their computer. So, I'd like to avoid having to install anything, or write anything long or unmemorable enough that would be annoying to write with someone looking over my shoulder.


title: Code to sample from & integrate light of a cluster?

text: Are there any publicly available codes to sample from an initial mass function (common ones... Kroupa, Chabrier, Salpeter) to construct a cluster, then use stellar models to generate an integrated spectrum? I imagine such codes must be quite common, as they're relatively easy to write, but require access to stellar model databases or very simple stellar models.


title: What happens when astronomical bodies made up of Dark Matter collapse or collide?

text: Does dark matter not produce heat or radiation after a collision etc?   Or, we detect radiations having no generators?


Is a Smithsonian Astrophysical Observatory (SAO) star catalog, 4 volumes, dated 1977, worth keeping in a library or is it too outdated? Can it be used for epoch 200?


title: Mass scales in See-saw mechenism

text: Are **both** types of Majorana masses $$\mathcal{L}^L_M=-\frac{m_L}{2}[\overline{(\psi_L)^c}\psi_L + \overline{\psi_L}(\psi_L)^c]$$ and $$\mathcal{L}^R_M=-\frac{m_R}{2}[\overline{(\psi_R)^c}\psi_R + \overline{\psi_R}(\psi_R)^c]$$ $SU(2)\times U(1)$ invariant? If not, then both the masses must arise by $SU(2)\times U(1)$ symmetry breaking and both the mass scales are fixed by **electroweak symmetry breaking scale**. Right? Then why does one take $m_R>>m_L$ in see-saw mechanism? Why should we assume $m_R$ is around the GUT scale?


title: What's gonna happen if a nuclear bomb explodes nearby?

text: I live in the capital, 10 km from its center on the 9th floor, my windows facing the opposite direction will I still see a flash? Will I hear something? How loud will it be? How fast would I die? If I survive the blast what should do? Should I go underground or climb higher?


title: Why we cannot use Gauss's Law to find the Electric Field of a finite-length charged wire?

text: One of my physics books has a nice example on how to use Gauss's Law to find the electric field of a long (infinite) charged wire. However, at the very end of the example, the author ends by saying Gauss's Law cannot be used to find the electric field of finite-length charged wire. I could not understand why not. Please this is not a homework problem or nothing like that. This is just a question I have and I have not being able to solve it in my mind yet.


title: Entanglement spectrum

text: What does it mean by the _entanglement spectrum_ of a quantum system? A brief introduction and a few key references would be appreciated.


title: Superposition of electromagnetic waves

text: The superposition of two waves is given by $$\sin(\omega_1 t)+\sin(\omega_2 t)=2\cos\left(\frac{\omega_1-\omega_2}{2}t\right)\sin\left(\frac{\omega_1+\omega_2}{2}t\right).$$ For sound waves, this effect is perceived by the human ear as beats. Does the human eye perceive a similar effect for electromagnetic waves? Is it possible to produce visible light through the superposition of two microwaves?


title: Why are electrons and quarks 0-dimensional?

text: Disregarding String Theory, which says that electrons and quarks are 1-dimensional, why are electrons and quarks 0-dimensional? The way I look at it is that they have mass, and if we were the same size as an up quark and looked at another up quark, it would look like a 3-dimensional object. So why do we say particles are 0-dimensional as opposed to 3-dimensional? I hope I'm correcting in saying they are regarded as 0-dimensional, the reason I think it is because of this quote from Wikipedia: > _String theory posits that the electrons and quarks within an atom are not > 0-dimensional objects, but rather 1-dimensional..._


title: Electric dipole transitions/expectation value of position

text: Part of a homework question asks to show that for $\ell=0$ in both $\Psi_i$ and $\Psi_f$, we have $$ \int \Psi_i^\ast \vec{r} \Psi_f \; d\tau = 0 $$ for the position vector $\vec{r}$. (This is for the electron in hydrogen and the integral is over all space.) The physical interpretation of this is that since the expectation value is zero, such a transition is forbidden. I am having trouble showing the above integral is zero. Since we are asked to show this in general, and not for a special case, it seems the only thing to do is use orthogonality of the $\Psi$'s. Is this correct? Can someone nudge me in the right direction?


title: Clarification of rest mass

text: So I have only just been introduced to the concept of rest mass in Special Relativity. Do we assume that the rest mass of a fundamental particle is constant in all inertial reference frames? i.e. is the rest mass of an electron if it is travelling at constant velocity c/2 (relative to the distant stars) the same as the rest mass of the electron if it is travelling at velocity 0 relative to the distant stars? Is the rest mass simply the "mass" of a particle in inertial frame of reference which has the particle at it's origin (assuming the particle is actually in an inertial reference frame)?


title: Autocorrelation and Power density spectrum : Continuous Markov Process

text: I've been reading through the paper from Gillespie on Brownian motion and Johnson Noise (DOI, PDF). He considers $X_s(t)$, a zero-mean stochastic variable, that is stationary in the sense that all of its moments $ \langle X_s(t)^k\rangle$ are time independent. He further defines the autocorrelation function (2.39) $$\langle X_s(t)X_s(t+t')\rangle \equiv C_X(t')$$ which is independent of time since all the moments are time independent. The mean is over all possible values of X. **I cannot interpret this autocorrelation**. I have been told that it measures how much the random variable fluctuates but I cannot convince myself of that. I think this is the relevant question in order to answer my dilemma : on the same page of the definition of the autocorrelation, he goes on showing that its frequency fourrier transform $$ C_X(t) = \int \limits_{0}^\infty S_x(\nu) cos(2\pi \nu t) $$ can be related to the variance of $X_s(t)$ as $$ \langle X_s(t)^2 \rangle = \int \limits_0^\infty S_x(\nu)d\nu$$ Now in the Ornstein Uhlenbeck process, $X_s(t)$ is to be interpreted as the speed of particle, subject to a drag force and a white-noise random force ($\Gamma$) $$\frac{dX_s(t)}{dt} = -\gamma X_s(t) + \sqrt{c}\Gamma(t)$$ One can then compute the dissipated power spectrum which originates from the drag force. Since power is force times speed, one gets $$ \langle P_{diss} \rangle = \gamma\langle X_s(t)^2\rangle \quad \to \quad P_{diss}(\nu) = \gamma S_X(\nu) = \gamma \frac{2c}{\gamma^2+(2\pi\nu)^2} $$ Where the last equality holds for the process at study. Gillespie simply says that this means that only the low frequency regime contributes to the dissipated power, and the high frequency regime doesn't. But frequency of **what** ? My interpretation would be this : the frequency argument ammounts to saying that if the variable is auto-correlated also for long times (low frequency in fourier), then you will have significant dissipated power. Coming back to my original question: do long autocorrelation times mean that the particle fluctuates a lot, and if so, why ?


title: At what starting angle will a billiard ball hit a pocket?

text: Let's say you have a square pool table with one pocket. At what angles can you hit the ball to make it hit the pocket? What relation do these angles hare? Constraints:   * Table is square: $\ell=b$.   * Frictionless   * Bouncing of ball of the sides is ideal   * Ball starts in the lower left corner.   * Pocket is in upper left corner.   * Pocket is 1x ball diameter.   * Pocket will accept a ball from any approaching angle.


title: Applying the right-hand rule for magnetic forces

text: I am confused about the right-hand rule for the Lorentz force. I am using same method as Brightstorm.com is: http://www.youtube.com/watch?v=LK7hv4LX3ys But for some reason it does not match op with this example from my book. ![Right hand rule exampld](http://i.stack.imgur.com/IzR8A.jpg) Could someone explain to me what I am doing wrong? I first curl my finger around the wire so as the current is going downwards. But since the charge is moving upwards, I need to flip my hand upwards, but that flips the curl, and thereby the direction of the $B$-field. What am I doing wrong?


title: Bloch wave function orthonormality?

text: there is this text book that is giving me a hard time for a while now: It shows that Bloch wave functions can be written as $$\Psi_{n\vec{k}}\left(\vec{r}\right) = \frac{1}{\sqrt{V}}e^{i\vec k \vec r}u_{n\vec k}\left(\vec r\right),$$ which is fine to me. It also states that the Bloch factors $u_{n\vec k}\left(\vec r\right)$ may be orthonormalized on the (primitive) unit cell volume $V_{UC}$: $$\frac{1}{V_{UC}}\int_{V_{UC}}d^3r u^*_{n\vec k}\left(\vec r\right) u_{n'\vec k}\left(\vec r\right) = \delta_{n n'}$$. However, and here starts my problem, it then concludes that therefore, the Bloch functions $\Psi_{n\vec{k}}\left(\vec{r}\right)$ fulfill $$\int_V d^3r \Psi^*_{n\vec{k}}\left(\vec{r}\right) \Psi_{n'\vec{k'}}\left(\vec{r}\right)=$$ $$=\frac 1 V \sum_\vec{R} \int_{V_{UC}\left(\vec R\right)}d^3r e^{-i\vec k \left(\vec R + \vec r\right)} u^*_{n\vec k}\left(\vec R + \vec r\right) e^{i\vec {k'} \left(\vec R + \vec r\right)} u_{n'\vec{k'}}\left(\vec R + \vec r\right)=$$ $$=\frac 1 N \sum_{\vec R} e^{i\left(\vec{k'}-\vec{k}\right)\vec R} \frac{1}{V_{UC}}\int_{V_{UC}} d^3r u^*_{n\vec{k}}\left(\vec r\right) u_{n'\vec{k'}}\left(\vec r\right)=$$ $$=\delta_{\vec{k'}\vec{k}}\delta_{n'n}$$ with lattice vectors $\vec R$ and crystal volume $V = N V_{UC}$. **But I just don't get the last two lines.** I mean, the integral in the second last line actually should read $\int_{V_{UC}} d^3r e^{i\left(\vec{k'}-\vec{k}\right)\vec r}u^*_{n\vec{k}}\left(\vec r\right) u_{n'\vec{k'}}\left(\vec r\right)$, shouldn't it? And, if that is true and I didn't miss something important already, I can't understand how that would yield these two $\delta$s ... I'm almost sure I missed something, but I just desperately keep fail getting it, so any help would be greatly appreciated! **EDIT** Thank you very much for your reactions! However, it seems I failed to state my problem clear enough, so I figured it might be best to tell you what my approach so far was step by step so may be someone can see where I actually go wrong: Starting with $$\int_V d^3r \Psi^*_{n\vec{k}}\left(\vec{r}\right) \Psi_{n'\vec{k'}}\left(\vec{r}\right),$$ I partitioned the integration domain $V=NV_{CU}$, thus getting a sum of integrations over the unit cell volume, yielding $$\sum_{\vec R}\int_{V_{UC}} d^3r \Psi^*_{n\vec{k}}\left(\vec{r} + \vec R \right) \Psi_{n'\vec{k'}}\left(\vec{r} + \vec R\right),$$ which happens to be exactly the second line when exploiting $\Psi_{n\vec{k}}\left(\vec{r}\right) = \frac{1}{\sqrt{V}}e^{i\vec k \vec r}u_{n\vec k}\left(\vec r\right)$. I then proceeded using $\Psi\left( \vec r + \vec R\right)=e^{i\vec k \vec R}\Psi\left(\vec r\right)$. But this yields $$ \sum_{\vec R} e^{i\left(\vec{k'}-\vec{k}\right)\vec R} \int_{V_{UC}} d^3r \Psi^*_{n\vec{k}}\left(\vec r\right) \Psi_{n'\vec{k'}}\left(\vec r\right)$$ which disagrees with the third line in the book where the integral is over the Bloch factors $u_{n\vec k}\left(r\right)$ only. However even assuming this is just a typo (which I'm not so sure of ...), I would be confronted with the integral $\int_{V_{UC}} d^3r e^{i\left(\vec {k'} - \vec k\right)\vec r} u^*_{n\vec{k}}\left(\vec r\right) u_{n'\vec{k'}}\left(\vec r\right)$ and I can't see how those two $\delta$s would arise from that either. Thank you all again for your reactions and I hope I know actually stated my problem clearly.


title: How to calculate the dependency of speed and weight of a servo?

text: Here is a digital servo: Servo By using the information of details of that thing what is the way to calculate the dependency of the speed of the movement of the car if the load upon the servo is N kg (or gram)? For example I need to know what maximum weight that servo can handle with a maximum speed of M m/s.


title: Why is the mapped universe shaped like an hourglass?

text: I've watched a video from the American National History Museum entitled The Known Universe. The video shows a continuous animation zooming out from earth to the entire known universe. It claims to accurately display every star and galaxy mapped so far. At one point in this video [3:00 - 3:15 minutes] it displays this text: > The galaxies we have mapped so far. > > The empty areas where we have yet to map. At this point, the shape of the "universe" is roughly hourglass, with earth at it's centre. ![The Known Universe - "The galaxies we have mapped so far"](http://i.stack.imgur.com/TEQFs.jpg) I'm having trouble understanding what this represents and why they shape is hourglass. Is this simply because we have chosen to map in those directions first? Is there a reason astronomers would choose to map in this pattern, or is this something more fundamental about the shape of the universe? Or is it to do with the speed of light reaching us from these distant galaxies? Continuing on from the hourglass pattern, the cosmic microwave background radiation is represented as a sphere and labelled "Our cosmic horizon in space and time". This doesn't help clear anything up. If we can map CMB in all directions, why have we only mapped galaxies in this hourglass shape. ![The Known Universe - "Out cosmic horizon in space and time"](http://i.stack.imgur.com/qZnhm.jpg)


title: How far from a spacecraft would it's exhaust cool to BR temperatures?

text: Just a thought that came to mind ... The rockets on interplanetary scouts fire to impart motive/braking force to the craft. This force, I understand, is simply a reaction engine - the exhaust departs out the nozzle, and drives the engine & attached craft in the desired direction. In an atmosphere this exhaust would eventually disperse. In space one would expect the exhaust to continue in the direction it exited the nozzle. But space is cold. Assuming the rockets fired in space far enough for any gravity knot to have negligible effect, how far from the nozzle would the exhaust cool to near CBR temperature? For instance, the hydrazine thrusters on the Voyagers provided necessary motive force at critical times during it's transit through Our solar system... albeit this was necessarily within a gravity knot.


title: Why is it not possible to distinguish left from right by means of a coil?

text: Why is it not possible to explain to an alien "at the phone" which side is left and which is the right side by defining a simple experimental setup using induction? Defining for instance downwards should not be a problem, or is it?


title: Why doesn't dark energy vary with time?

text: Trying to teach myself the basics of cosmology. I can see why, in an expanding universe, radiation and matter density, and radiation and matter pressure vary with time. But why is it assumed that dark energy is independent of time? In my textbook, it's just sort of stated as if it's blindingly obvious. Many thanks


title: Graphene conductor?

text: I was thinking if we had a a busbar that can carry massive amounts of current(since its resistance is $0.9\, \mathrm{\mu \Omega}$), and it's weight is $15\, \mathrm{kg}$, I assume if graphene would be replaced to that copper it would be much much lighter and capable of having the same amount of current flowing within?


title: Explanation for the minus sign in $\Omega_3$ in the Kappa symmetry of the Green - Schwarz formalism for F1 strings

text: > _Just so that there can be more higher - level physics questions here, let > me post this question + answer. Also because I'm a bit sad that there are > almost no questions on the Green-Schwarz formalism._ In the Green-Schwarz formalism, for F1 strings, we have the action $$S=S_1+S_2$$ Where $${S_1} = - T\int_{}^{} {\sqrt { - \det \left( {{\Pi _{\alpha \mu }}\Pi _\beta ^\mu } \right)} {{\text{d}}^2}\sigma } $$ and $S_2$ is the additional action term that arises through Kappa Symmetry. The transformations of kappa symmetry for F1 strings are intuitively related to that for D0 branes, etc. They are given by: $$\begin{gathered} \delta {X^\mu } = {{\bar \Theta }^A}{\gamma ^\mu } \\\ \delta {\Theta ^A} = - \delta {{\bar \Theta }^A}{\gamma ^\mu }{\Theta ^A} \\\ \end{gathered} $$ So that: $$\delta \Pi _\alpha ^\mu = - 2\delta {\bar \Theta ^A}{\gamma ^\mu }{\partial _\alpha }{\Theta ^A}$$ From this transformation, the variation in $S_1$ by kappa symmetry, is almost trivially (not that trivial, thiough...): $$\delta {S_1} = \frac{2}{\pi }\int_{}^{} {\sqrt { - \lambda } {\lambda ^{\alpha \beta }}\Pi _\alpha ^\mu \delta {{\bar \Theta }^A}{\gamma _\mu }{\partial _\beta }{\Theta ^A}{{\text{d}}^2}\sigma } $$ Here, $$\begin{gathered} \lambda = \det {\lambda ^{\alpha \beta }} \\\ {\lambda ^{\alpha \beta }} = {\Pi _{\alpha \mu }}\Pi _\beta ^\mu \\\ \end{gathered} $$ To determine $S_2$, howefver, is not all that trivial since the simple procedure I've mentioned here, for example, is just not practical by any means. We there fore use a 2-form, form $\Omega_2$, such that: $${S_2} = \int_{}^{} {{\Omega _2}} = \int {{\epsilon ^{\alpha \beta }}{\Omega _{\alpha \beta }}{{\text{d}}^2}\sigma } $$ We may also (formally) define a 3-form $\Omega_3=\mbox{d}\Omega_2,$ so that by Stokes' theorem, we have it that: $$ \int_M {{\Omega _2}} = \int_D^{} {{\Omega _3}} $$ $M$ would be the worldsheet whereas $D$ would be it's interior, ; $M=\partial D$. There are 3 supersymmetric 1-forms prominent to us namely, ${\text{d}}{\Theta ^1},{\text{d}}{\Theta ^2},{\Pi ^\mu }$. If ${\Omega _3}$ is to be supersymmetryic, which it better be, then, we shoul'd have it that ${\Omega _3}$ involves just these 3 1-forms. A very sensible choice of 3-form for $\Omega_3$ is: $${\Omega _3} = A\left( {{\text{d}}{{\bar \Theta }^1}{\gamma _\mu }{\text{d}}{\Theta ^1} -{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{d}}{{\bar \Theta }^2}{\gamma _\mu }{\text{d}}{\Theta ^2}} \right){\Pi ^\mu }$$ Now, when I first learnt this, I was pretty confused at the "$-$" si1gn, (probably because of the fact that I was too confused to bother reading the next line of bbs : ) ...). As I guess many others may also be confused at this, let me ask and answer the question: > Is this minus sign, by any chance, related to the fact that Type IIA String > Theory is chiral? Does this mean that the above expression only holds for > the Type IIA, and not for the Type IIB ? .


title: Feynman Path Integral Formula in Brian Cox' "A Night with the Stars" Lecture

text: The Youtube link keeps breaking, so here is a search on Youtube for Brian Cox' _A Night with the Stars_ lecture. Pause the video on 40.32minutes. What you see he said is called Feynman's Path Integral. $K(q",q',T)=\sum_{paths}Ae^{iS(q",q',T)/h}$ Am I right in thinking this adding all the different paths a particle can take predicting the probability of it landing in a certain position? Once that question is answered how in the world did he simplify it to: $t > \dfrac{x \Delta{x} m}{ h} $--- (42.39minutes) This second equation he got from simplifying was what is used to "predict how long it would take" for his "diamond to jump out of the box", how fascinating. So: what is all this in Feynman's Path Integral? and how did he simplify it to get the other equation? and if you know, what is this second equation called?


title: Two spherical cavities hollowed out from the interior of a conducting sphere. At the center of each cavity a point charge is placed

text: From Griffiths: > Two spherical cavities, of radii a and b, are hollowed out from the interior > of a (neutral) conducting sphere of radius R. At the center of each cavity a > point charge is placed – call these charges $q_a$ and $q_b$.What is the > field within each cavity? By Gauss’s law, fields are $Kq_a/r_a^2,Kq_b/r_b^2$, but thinking in terms of charges, the charge on outer surface cannot affect inside, so contribute nothing to the field inside any cavity. Also does not affects the field in another cavity the charges of one cavity(both present inside and induced). So field inside the cavity must only be due to $q_a$(WLOG), and induced charges $-q_a$. But the result from gauss's law suggests that even induced charges do not affect the field? Is it due to the fact that charges on surface of a solid charged conducting sphere produce no field inside, but these induced charges are not on outside surface [of the surface of cavity but inside the whole sphere], also if both situations are similiar, then there's difference in that the cavity is like a hollow shell. Is my reasoning correct? Somebody please explain. **Edit:** Explain without gauss's law, I know that with gauss law you would eventually explain all( I might also ).


title: Why does change in speed of a wave make it refract?

text: When a light wave enters a medium with a higher refractive index (e.g. from air to standard glass) and its speed decreases, why does that make it refract/bend? I understand that wavelength decreases and frequency stays the same and therefore its speed decreases, but I can't find anywhere whatsoever why the speed decrease cause the wave to refract. So could someone please explain this?


title: Conformal dimensions of the energy-momentum tensor

text: Currently I am reading the Di Francesco-Mathieu-Sénéchal textbook on conformal field theory. Above the equation 5.52, the author argues that the EM tensor should have scaling dimension 2 and spin 2. But why did the author then conclude that both the holomorphic and anti-holomorphic dimension of the tensor are 2? Isn't it that the holomorphic dimension is 2 while the anti- holomorphic one is 0? Another question: why does the EM tensor transform like a primary field?


title: How does one prove Fierz identities?

text: Fierz identities are discussed in the wikipedia article:   http://en.wikipedia.org/wiki/Fierz_identity   but the article doesn't give any derivation. The article implies that they arise from the blade structure of a Clifford algebra. * * * Some notes to be deleted after we have a solution: They seem to arise in QFT calculations and (perhaps as a result) they can always be put into "pure density matrix" form. That is, the spinors can be arranged to appear in an order that makes them natural for a trace operation. For example, with $U,V,W$ as unspecified operators and $\theta,\psi$ as spinors, one might have:   $(\bar{\theta}U\theta)(\bar{\psi} W\theta)(\bar{\theta} V\psi)$   where the unbarred and barred spinors do not always match as they do in "$...\theta)(\bar{\theta}...$". But this can be rearranged to give:   $(\bar{\theta}U\theta)(\bar{\theta} V\psi)(\bar{\psi} W\theta)$   One can then turn the $\psi\bar{\psi}$ type elements into (almost) pure density matrices and treat the problem as one of computing traces:   $\textrm{tr} [ (\theta\bar{\theta}U\theta\bar{\theta} V\psi\bar{\psi} W ]$   And then one can use various properties such as the fact that with pure density matrices such as $|a\rangle\langle a|^2 = |a\rangle\langle a|$ one has that anything that begins and ends with such a thing has to be a complex multiple of the pure density matrix. This would give the Fierz identities without explicitly looking at the blade structure of the Clifford algebra and that is very suspicious to me. I doubt it works. On a previous question: Some Majorana fermion identities I intuitively convinced myself that Fierz identities could be established this way (and some very smart people seemed to agree) but when I tried to solve that problem this way I quickly was buried in unsuccessful calculations. Now it could be that that particular problem couldn't easily be so attacked, or my method was misapplied, etc.; I'm not sure.


title: mSUGRA boundary conditions and the MSSM

text: I read that in the MSSM with mSUGRA boundary conditions, the mass spectrum of the model is determined by five parameters at the GUT scale: $m_0$ (universal scalar mass), $m_{1/2}$ (universal gaugino mass), $A_0$ (universal trilinear coupling), $\tan \beta$ (ratio of Higgs vevs), and sign[$\mu$] (Higgs/Higgsino mass parameter). I have two questions: First, why is it the sign of $\mu$ rather than $\mu$? Second, why $\mu$ can take the two opposite signs?


title: How does General Relativity Emerge from Brans-Dicke Gravity with an Infinite Omega Parameter?

text: The action for the Brans-Dicke-Jordan theory of gravity is $$ \\\S =\int d^4x\sqrt{-g} \; \left(\frac{\phi R - \omega\frac{\partial_a\phi\partial^a\phi}{\phi}}{16\pi} + \mathcal{L}_\mathrm{M}\right). $$ The corresponding equations of motion are $$ \Box\phi = \frac{8\pi}{3+2\omega}T\\\ G_{ab} = \frac{8\pi}{\phi}T_{ab}+\frac{\omega}{\phi^2} (\partial_a\phi\partial_b\phi-\frac{1}{2}g_{ab}\partial_c\phi\partial^c\phi) +\frac{1}{\phi}(\nabla_a\nabla_b\phi-g_{ab}\Box\phi) $$ where $\\\G_{ab}$ is the standard Einstein tensor and $\mathcal{L}_\mathrm{M}$ is the matter Lagrangian density. It's supposed to be obvious that this reduces to general relativity in the limit of infinitely large $\omega$. How so? You may assume that the trace $\\\T$ of the stress energy tensor $\\\T_{ab}$ is not zero.


title: What is the difference between a measurement and any other interaction in quantum mechanics?

text: We've learned that the wave function of a particle collapses when we measure a particle's location. If it is found, it becomes more probable to find it a again in the same area, and if not the probability to finding it in the place that was checked decreases dramatically. My question is about the definition of measurement. What makes a measurement different from any other interaction between two particles (gravity and EM fields for example)? In reality, almost every particle interacts with any other particle, so shouldn't there be constant collapse of the wave function all the time? If this happens we're right back in classical mechanics, aren't we?


title: Monte Carlo use

text: Where is the Monte Carlo method used in physics?


title: The neutrality condition and the (non)-vanishing of the one-point correlator for the bosonic vertex operator

text: Consider the massless scalar field Hamiltonian, \begin{align} H = \frac{1}{2}\int \Pi^2- (\partial_x\phi)^2 dx \end{align} with $\Pi \sim \partial_t\phi$ the conjugate field of $\phi$. This Hamiltonian is treated in a number of texts, in particular Shankar's text on Bosonization on page 1841 and the Conformal Field Theory book by Di Francesco, Mathieu and Sénéchal (chapter 6 and 9). Furthermore, my question also relates to this text by Von Delft and Schoeller on bosonization. Shankar defines the vertex operator as the normal ordered operator $:e^{i\alpha\phi}:$. He states (p1843) that the one-point correlator is unity, $\langle :e^{i\alpha\phi}:\rangle =1$, since the exponential can be expanded and only the first term does not vanish. But just before that he also states that the two-point correlator $$ \langle :e^{i\alpha\phi}: :e^{i\beta\phi}: \rangle $$ vanishes _unless_ $\alpha + \beta = 0$. This is the neutrality condition and it follows from the fact that $\phi\rightarrow \phi + a$ is a symmetry of the Hamiltonian, which needs to be respected by the (two-point) correlators. That sounds convincing enough, but then **why does the one-point correlator not vanish according to that same argument**? To add to (my) confusion the normal ordered form is given by $$ :e^{i\alpha\phi}: = e^{i\alpha\phi_+}e^{i\alpha\phi_-} $$ where $\phi_{\pm}$ contain all creation/annihilation operators. So using some rules of exponentiated operators (see appendix C of Von Delft) one can reorder the operators such that $$ :e^{i\alpha\phi(z_1)}: :e^{i\beta\phi(w_1)}: = f(z-w) :e^{i(\alpha+\beta)\phi(w_1)}: + \cdots $$ with $f(z-w)$ a (singular) c-number. This is of course just the OPE. But that would imply that if the right hand side has a non-vanishing correlator, then the left hand side is non-vanishing as well. **If the one-point correlator does not vanish, why does the two-point correlator need to obey the neutrality condition?** Di Francesco has some other, more elaborate proofs using Ward identities (Chapter 9) that show that indeed $\alpha+\beta = 0$ in order for the correlator to be non-zero. In particular that means that, according to their conventions, any $N$-point correlator vanishes unless the neutrality condition is satisfied. It could be that the different texts have different conventions that I'm missing and everything is fine. Still, a different question arises: **If the exponentials in the normal ordered operators are expanded, and only the first (unit) term is kept, then surely the one-point correlator is just $1$**? Why does this approach fail, according to their conventions? Let me also mention Von Delft and Schoeller. They also state that the one point correlator of the normal ordered operator $:e^{i\alpha\phi}:$ does _not_ vanish. Instead they define the vertex operator (chapter 9) as $$ V_\alpha = \left(\frac{L}{2\pi}\right)^{-\frac{\lambda^2}{2}}:e^{i\alpha\phi}: $$ with $L$ the system size (periodic boundary conditions). They state that "evidently $\langle V_\alpha\rangle =\delta_{alpha,0} $ in the limit of $L\rightarrow\infty$.". **Does the neutrality condition then only hold in the $L\rightarrow\infty$ limit?**. This would invalidate Di Fransesco's treatment, I presume. So, in short, I'm quite confused on how to "merge" these different treatments.


title: Atomic weight in respect to the binding energy?

text: My book says that the weight of helium (with the nucleon number of 4 and proton of 2) is that of $6,6447*10^{-27}$ kg. Earlier the book stated that if the proton number is left out it means that the atom is neutral included the electrons. Does this imply that the weight of the electrons are included in tables if the proton number is excluded?   http://en.wikipedia.org/wiki/Isotopes_of_helium#Table   I can't find the helium isotype with the proton number written out on the list. If I take the one with nucleon number 4 the weight would be   $4.00260325415 \times 1,66 \times 10^{-27} = 6,644321402 \times 10^{-27} $ kg   Is this correct? If I do subtract for two electrons it doesn't add up either. It's just off by a tiny bit. I need to find the weight correctly because I'm using this to estimate the binding energy. Because that is suppose to be equal to the difference.


title: Possibility for radiation in dark matter that is not interactive with regular matter?

text: **Definition:** Radiation in this case does not refer to electromagnetic radiation. It refers to any kind of emission of energy, even energy that does not interact with regular matter. Just like dark matter does not interact with electromagnetic radiation, could regular matter not interact with "dark matter radiation" (I'm not talking about the usual "really high wavelength radiation" kind)?


title: The Faddeev-Popov Lagrangian

text: This is a non-abelian continuation of this QED question. The Lagrangian for a non-abelian gauge theory with gauge group $G$, and with fermion fields and ghost fields included is given by $$ \mathcal{L}=\overline{\psi}(i\gamma ^\mu D_\mu -m)\psi+\sum _{k=1}^{\dim (G)}\left[ \frac{-1}{4}F_{\mu\nu}^kF^{\mu \nu k}+\frac{1}{2\xi}(\partial _\mu A^{\mu k})^2+\sum _{i=1}^{\dim (G)}\overline{c}^k(-\partial _\mu D^{\mu ki})c^i\right] . $$ How does the third term (the one that contains $\xi$) come into the picture? The first terms is the standard fermion Lagrangian interacting with a gauge field, the second term is the standard term for gauge bosons, and the fourth arises because of the introduction of the ghost fields . . . but what about the third?


title: Will a stone thrown in space move forever?

text: If I throw a stone on space, in a place where gravity is equal zero, will the stone move forward forever, because no air, so no friction?


title: Paper in physics - calculations; rounding or not?

text: I'm currently a high schooler, and I'm writing my first scientific paper. The result is fairly simple, and it is nothing too special, but I see it as a nice way to prepare myself for the academic world. The paper involves a lot of calculations and I'm actually wondering how I should approach rounding to make it as scientific as possible. My own guess would be scientific rounding in the intermediate steps, but using the 'correct' value in further calculations. Is this correct?


title: Why can I split white light into separate colors with my eyes

text: I was sitting in a meeting at work today. When I looked up there was a projector facing the rear wall letting out a very bright white light (the projection). When I look right at it, all I could see was white light. But as I illustrated below when I shifted my gaze away I saw the white light split into each individual color. I was able to reproduce this repeatedly just by shifting my gaze. Any explanation? Gama rays? Do I have a super power? ![enter image description here](http://i.stack.imgur.com/2ob10.jpg)


title: Working with a Routhian for a specific system

text: I asked a more general question earlier about the Routhian, but I'm still having trouble working with it. Here's my specific case. Given the following Lagrangian: $$L=(1/2)m(\dot{r}^{2}+r^{2}\dot{\phi}^{2}sin^{2}\theta_{0})-mgrcos(\theta_{0}),$$ I obtained the corresponding Routhian: $$R=(1/2)p_{\phi}\dot{\phi}-(1/2)m\dot{r}^{2}+mgrcos\theta_{0}.$$ I'm being asked to do the following: obtain the Hamilton equation of motion for $p_{\phi}$, show that it's constant in time and find its initial value, obtain the Lagrange equation of motion for $r$ from the Routhian, and obtain the Hamiltonian When I try to do the first part (obtain the Hamilton e.o.m.) I get $∂R/∂p_{\phi}=(1/2)\dot{\phi}$ which doesn't show that it's constant in time or that angular momentum is conserved, which I think is what we're trying to show here. What am I doing wrong? Also, how can I obtain the Lagrange e.o.m. for $r$ using the Routhian? Do I just take $$\frac{d}{dt}\frac{∂R}{∂\dot{r}}-\frac{∂R}{∂r}=0,$$ since $r$ appears in the Routhian and plug in $r=r_{0}$ to find the initial value? Similarly, how do I go about obtaining the Hamiltonian from the Routhian?


title: Where does the heat flow in the Earth's crust switch from primarily solar to geothermal?

text: Ok, maybe more of a geology question than physics, but maybe somebody has been involved in modeling these heat flows? Essentially I'm asking if we know what sort of depth the heat source becomes primarily from below rather than from above. I've heard that the fully solar domain is down to about 5m, and that temperatures below that are fairly stable, but crucially the temperature at the bottom of a 100m borehole is supposed to be around the mean temperature form that latitude, which suggests to me that solar is at least a large input even to that depth. I'm struggling to find any reference to this exact point in the literature.


title: Why and how is sound produced when two objects hit each other?

text: When two objects collide and undergo a partially inelastic collision (so every one we experience in every-day life), they rebound to a certain degree, but kinetic energy is not conserved. Thus, the energy of motion must be converted to another type of energy. Thus, when two object collide, the most common forms of energy that kinetic energy is converted to are sound energy and thermal energy. Some collisions are louder than others just because they cause a bigger local variation of pressure (which then propagates through air until finally gets heard). So, why some objects, even if the velocity before and after the collision seems to be the same, are louder than others? I mean, how do the different material properties enter in the phenomenon? How and why, by means of solid and fluid mechanics or with an heuristical microscopical reasoning, can this process be explained?


title: Why is electric potential positive?

text: If there is a positive charge $q$ at the origin of a coordinate system, the electric potential $\phi$ at a distance $r$ from $q$ is (by definition, if we take the point of zero potential at infinity): $$\phi=-\int_{\infty}^r \vec{E}\cdot d\vec{r}$$ The dot product of $\vec{E}$ and $d\vec{r}$ is $-E\text{ }dr$ because they point in opposite directions, so $$\phi=\int_{\infty}^r E\text{ } dr$$ For a positive point charge $q$, we have that: $$\phi=\frac{q}{4\pi\epsilon_0}\int_{\infty}^r \frac{dr}{r^2}$$ And evaluating the integral we arrive at: $$\phi=-\frac{q}{4\pi\epsilon_0 r}$$ However, the result should be positive according to Halliday-Resnick (fifth edition, page 608). They have the same derivation essentially, except that after evaluating the integral for some reason they get a positive potential. What's up?


title: Buoyancy in gases

text: If buoyancy is caused by pressure gradient and if the pressure of a gas is same everywhere in its container, which means there's no difference of pressures on surfaces of the lighter-than-air object, how come we observe such an effect?


title: scaling laws for density and temperature of high-energy explosions

text: I'm wondering if there are heuristic ways to derive how the peak density and temperature of nuclear explosions scale with the amount of fissile/fusible material. Does it matter what the explosion energy is made of? what if the source of the energy is matter antimatter annihilation? how would that scale? any intuitive way to derive those as well?


title: Why is the canonical ($NVT$) ensemble often used for (classical) molecular dynamics (MD) simulations?

text: Molecular dynamics (MD) simulation is a common approach to the (classical) many-body problem. It relies on integration of Newton's equations of motion to simulate the trajectories of many (e.g., ~1,000-100,000) particles. In my limited exposure to the MD literature, a recurring theme that I think I see (at least on the chemical physics end of things) is that MD simulations are often performed in the canonical ($NVT$) ensemble. Why does this seem to be the case? * * * Here are the common thermodynamic ensembles:   * **Microcanonical ensemble** ($NVE$)     * The system is isolated.     * The total energy $E$ is fixed.     * Every accessible microstate has equal probability. That is, if $\Omega$ is the number of accessible microstates, the probability that a system is in a particular microstate is $\frac{1}{\Omega}$.     * Please correct me if I am wrong, but I think that the microcanonical ensemble satisfies ergodicity -- time averages can be replaced with ensemble averages.   * **Canonical ensemble** ($NVT$)     * The system is _not_ isolated. The system can exchange energy with a heat bath. The total energy of the system + bath is fixed. The _average_ or _equilibrium_ energy of the system is constant.     * The absolute temperature $T$ is well-defined. (Is $T$ fixed? I think so.) $T$ is given by the temperature of the heat bath.     * The probability of finding the system in some microstate $i$ with energy $E_i$ is given by the Boltzmann distribution: $$p_i = \frac{e^{-\frac{E_i}{k_B T}}}{\sum_i e^{-\frac{E_i}{k_B T}}}$$   * **Isothermal-isobaric ensemble** ($NPT$)     * The absolute temperature $T$ and the pressure $P$ are fixed. * * * Looking at this non-exhaustive list of choices, it seems that we can eliminate the $NVE$ ensemble from consideration because "real world" chemistry involves energy exchange with the environment. MD simulations typically do not model chemical reactions, but still, I would say that most chemistry in the "real world" occurs at nearly constant pressure (e.g., atmospheric pressure). So the $NPT$ ensemble seems like a reasonable candidate. What about the $NVT$ ensemble? Constant temperature perhaps seems reasonable for equilibrium, "real world" chemistry, but I am not so sure about constant volume. Now let's jump back to my very rudimentary of MD simulations in the literature. In MD simulations, molecules sit in a simulation box to which periodic boundary conditions are applied. From reading some literature articles it seems that the $NPT$ ensemble is used for equilibration -- to obtain the simulation box size that gives an average pressure of, for example, 1 atm. **Then** , the system is simulated in the $NVT$ ensemble -- that is, the simulation box's dimensions are held fixed, hence fixing the system volume. It is from _this_ simulation in the $NVT$ ensemble that ensemble averages are computed and the system's chemistry is analyzed. Why is the $NVT$ ensemble used for MD simulation production runs?


title: Could the center of the combined mass of the entire galaxy change if there were no external forces acting on that galaxy?

text: Everything in the galaxy orbits the center of the combined mass of the entire galaxy. So could the center of the combined mass of the entire galaxy change if there were no external forces acting on that galaxy ? (By external I mean forces outside of the galaxy)


title: Why conserved quantities in quantum mechanics are experimentally interesting?

text: Part of the QM lecture notes of Prof. Biswas, available online, on the harmonic oscillator reads: "Now it is easy to see from theorem 4.2 that neither P nor X is conserved. The only conserved quantity is H. Direct position measurements, like in scattering experiments, are not possible as that would mean directly measuring interatomic distances in molecules. This makes the measurement of P or X experimentally uninteresting. Hence, we shall discuss the measurement of H alone. These measurements are actually made indirectly in molecular spectra." 1-I do not see why it is impossible to measure interactomic distances based on the fact that x does not commute with the hamiltonian (i.e being not a conserved quantity) 2-The text gives the feeling also that only conserved quantities are experimentally interesting, why is that? Could someone elaborate on 1 and 2 please?


title: Why can't gauge bosons have mass?

text: Clearly, a mass term for a vector field would render the Lagrangian not gauge- invariant, but what are the consequences of this? Gauge invariance is supposed to be crucial for the renormalisation of a vector field theory, though I have to say I'm not entirely sure why. As far as removing unphysical degrees of freedom - why isn't the time-like mode $A_0$ a problem for massive vector bosons (and how does gauge invariance of the Lagrangian ensure that this mode is unphysical for gauge bosons)?


title: Shouldn't LHC have used $p\bar{p}$ collisions, instead of $pp$ collisions, to study baryogenesis?

text: Baryogenesis is the physical process(es) that produced baryon antibaryon asymmetry in the early universe. That means, the laws that governed the bigbang was baryon-antibaryon symmetric. On the other hand it is usually said that LHC is probing the early universe by creating similar conditions of the early universe. My question is, by using $p\bar{p}$ collisions, instead of $pp$ collisions, we are not exactly reproducing the early conditions of the universe because the conditions we created are not baryon antibaryon symmetric, are we?


title: Calculating angle of projectile motion

text: I realize that similar questions have been asked but I'd like to put my problem here and what I have to make sure everything is right. So my problem is: I've got a rock $2.5$ meters above the ground and I shoot it out of a slingshot. It hits another rock which is $3.6$ meters above the ground, $14$ meters away from it's initial position $x$. The initial velocity of the rock is $15.9$ m/s. So the formula I know is: $ \frac{1}{2} \cdot g \cdot t^2 - (v_o \cdot \sin ( \theta ) ) + y $ I also know: $x = (v_o \cdot \cos ( \theta ) ) \cdot t$ But these don't seem to help me solve my problem as I don't have $t$ nor $ \theta$.


title: How to decide convergence of the ensemble average in a Monte Carlo process

text: I am simulating electromagnetic scattering off a rough surface. The usual process is to do a Monte Carlo simulation, which is briefly described as follows.   * Generate a randomly rough surface, and compute the scattered far-field intensity in a particular direction of interest.   * Repeat this process for several different instances of rough surfaces, and get an ensemble average of the far-field intensity.  This ensemble average converges after sufficient instances have been taken. So, my question is that how does one determine in a rigourous manner if convergence has happened? In the electromagnetic scattering community, this question is not addressed, and heuristic estimates are used (e.g. "we used 100 instances") instead. Thanks apriori.


title: R charge of the chiral multiplet in $2+1$ dimensions

text: These are two examples that I am puzzled by,   * One can see in this paper on page 16 that for ${\cal N} =2$ theory on $2+1$ the R-charge of the $\phi$ and the $\psi$ is determined to be $\frac{1}{2}$ and $-\frac{1}{2}$. How?  Further in this case since the supercharges are in a vectorial (self- conjugate!) representation of the R-symmetry group why is the R-charge of $\psi$ and $\psi^*$ different? Since post radial quantization $\bar{Q}=S$ it follows that $Q$ and $\bar{Q}$ transform in exactly the same way under $SO(2)$ and hence shouldn't the R-charges $\psi$ and $\bar{\psi}$ been the same?   * One can see in this paper at the bottom of page 7 a statement about how the scalar component of the ${\cal N}=3$ chiral multiplet in $2+1$ dimensions is in a $SU(2)_R$ triplet. So does this (what?) determine the R-charge of the scalar field and other components in the multiplet?  What is puzzling is why did the authors in the second paper think of the R-symmetry group as $SU(2)$ for ${\cal N}=3$ rather than the $SO(3)$ that would be gotten by definition? I would have thought that the R-charges of a field are the same as the eigenvalue under the Cartan subalgebra of the R-symmetry group of the supercharge(s) which acted on the lowest helicity state state of the multiplet. But isn't the R-charge of the lowest helicity state of a multiplet merely a convention? Can't the scalar field be always chosen to be uncharged (R=0) under the $SO({\cal N})$ R-symmetry of $2+1$?


title: Looking for a list of possible subatomic particle collisions

text: This is going to be a strange question, but here we go. I'm working on a computer puzzle game that will simulate subatomic particle collisions. I am not a physicist by training, but I do dabble. I would like the game to be loosely based on reality, and I've been trying very hard to find a resource that could outline, simply put, what happens when: `proton <-> proton` collides   `electron <-> positron` collides   `electron <-> electron` collides   ...etc. It would be even more interesting to see what happens at varying energy levels (as energy, in the game, will be a resource). If you think think this question is a waste of time, I apologize in advance.


title: Explanation for $E~$ not falling off at $1/r^2$ for infinite line and sheet charges?

text: For an infinite line charge, $E$ falls off with $1/r$; for an infinite sheet of charge it's independent of r! The infinitesimal contributions to $E$ fall off with $1/r^2$, so why doesn't the total $E$ fall off the same way for the infinite line and sheet charges?


title: nature of glass transition

text: I am reading in some book: "The glass transition is similar in appearance to a second-order phase transition, but it is not a true thermodynamic phase transition. This is because the transition temperature depends on the **rate** at which we do the experiment." What's the matter with that **rate**? Is it only a question of mathematical definition of thermodynamic phase or is there any physical reason to exclude rate-dependent transitions?


title: Are there microscopic theories, which work, but which wrongly predict macroscopic behaviour?

text: Motivated by this question (and the P. W. Anderson article linked in that question, which I came across here somewhere today and just read) I wonder about something, which is somewhat bordering an inverse idea of effective field theories: > Can a theory of a microscopic scale accurately describe microscopic effects, > if it is known that the macroscopic behaviours predicted by it don't > correspond with observation? Maybe the answer is a trivial _no_. After all, the "classical limit" is ofter a guiding principle. But I can't really show it. * * * I can think of possibilities where such a scenario might be realized. For example I could imagine a microscopic theory A which has been introduced to describe a certain phenomenon, but which neglects some things on the same scale level as the one of the property of interest (in which the physicist is not interested though) and where these second effect determine more global or macroscopic behaviour. The second effect would be unforseen by the theory A. Are there examples for this? And if so, is there a guiding principle, which says, that a parallel or even more detailed and so called fundamental theory B must become that theory A, if the degrees of freedoms in B (maybe e.g. some vector field components), which the first theory A never contained, are then neglegted?


title: Is boson sampling a problem in 'continuous variable' quantum information?

text: When people generally speak of quantum information in the context of continuous variables, what is generally meant is that observables, like position/momentum or the field quadratures of quantum optics, can take a continuous range of values and that information processing tasks are performed using operators on these variables. However, most work in this area restricts attention to a countable number of modes of the radiation field (in quantum optics). In this case, the state of the system can be described using a countably infinite number of variables, like the number eigenstates in the field. But this is still called 'continuous variable' quantum information because the relevant observables still take continuous values. Is that so? If so, is the boson sampling problem in the domain of continuous variables? I don't know what exactly the relevant variables are in this case. A preliminary reading seems to suggest to me that the observable is the photon number, since what is observed is the probability after measuring in some basis. This would now mean that in the sense I think of continuous variables, boson sampling is not a problem in this domain. Am I right/wrong? Which of my statements here are false?


title: What is meant by the phrase "the mass is protected by a symmetry"?

text: In a particle physics context I've heard this phrase used. I guess it means that the mass of a particle is less than you'd naively expect from $E=mc^2$ after computing the momentum uncertainty corresponding to some appropriate length scale. Assuming my interpretation of the phrase is right, how does a symmetry let you get away with a smaller mass ?


title: Free particle propagation amplitude calculation

text: I have a quick calculational question. In Peskin and Schroeder, Chapter 2, they want to look at the amplitude for a particle to propagate between two arbitrary points, $x$ and $x_0$, in an arbitrary amount of time t given the Hamiltonian $\sqrt{p^2c^2+m^2c^4}$. To do this, we look at the inner product of the time-evolved particle that was at $x_0$ with the $x$ eigenket: $\langle \vec{x}|\exp(-i\hat{H}t/\hbar)|\vec{x_0}\rangle$, correct? If we insert two complete sets of momenta, the integral becomes (give or take factors of $\hbar$): $\frac 1 {(2\pi)^3}\int d^3p\; \exp(-it\sqrt{p^2c^2+m^2c^4}/\hbar)\exp(i\vec{p}\cdot(\vec{x}-\vec{x_0}))$ I'm not quite sure where to proceed from here... Peskin and Schroeder somehow reach: $\frac 1 {2\pi^2 |x-x_0|}\int_0^\infty dp\;p\sin(p|x-x_0|)\exp((-it\sqrt{p^2c^2+m^2c^4}/\hbar)$. To get here, however, it seems that you would have to assume that $p$ points in the same direction as $x-x_0$ to get spherical symmetry or something. Why can we do this? Aren't we summing over all possible momenta? Thanks!


title: Computer cooling with dry ice, ideas and question; thermodynamics

text: I am designing a cooling system for my computer and had a few questions. So I have a computer water cooling radiator and I want to cool it as much as I can. My first two ideas were an old window A/C unit and blowing air over the cold radiator in there onto my radiator. Or put the whole radiator again in water and fill the water with ice indefinetely... My next idea, which I am going to pursue, is to set dry ice on the radiator. Or to submerge the radiator in some sort of liquid with a low freezing point (acetone?) and get that liquid as cold as I can by sublimation the dry ice in it. Using the acetone is a good idea because it can grab all the cold from the dry ice (surface area is covered by liquid; opposed to it just sitting on the rad and 3/4 of the dry ice is incontact with the air doing nothing) But I am not sure how well my radiator would stand up to being submerged in acetone (not to mention explosions...) So what would be my best bet to using the dry ice to chill the water? How do I calculate how much I need for the amount of watts in the watter? I know my water cooling loop must remove about 625watts at max computing power, so how much dry ice do I need to remove that amount of heat? I think setting dry ice on it works well (have done it before) but I am scared of local freezing inside the radiator, and that ice hitting my pump impellor then (no-no). Also if I get the water too cold I will start to get condensation on some parts (no-no).


title: Parachute in vacuum

text: Consider that the earth is vacuum. Consider a person of weight 100 kg is falling from sky with an parachute . He is free falling at height of 3000 m. When will he reach the ground? What would happen if he opens the parachute in diving? Will it work ?


title: Two capacitors sharing charge

text: My two capacitors: ![](http://i.stack.imgur.com/t7Rmy.png) I have these two capacitors ($C_1 = 3\mu F, C_2 = 4\mu F$) both initially under 19 volts. Then, I added a dielectric with $k=4$ at $C_1$ and entire air gap got filled. How can I calculate the potential between the two capacitors, or the electric field of each one of them or the charge of each one of them? The distance between the two of plates of both capacitors is $0.003\ m$. Why the new voltage is gonna be $q_{total}/(4\mu\times4+3\mu)$ and not $q_{total}/(4\mu+3\mu\times4)$? I'm inserting the dielectric in the $C_1$ not in $C_2$...


title: Predicting solar storms?

text: Quote from BBC "A strong solar storm is expected to hit Earth shortly, and experts warn it could disrupt power grids, satellite navigation and plane routes." How is it that we are aware of a massive solar activity that will hit us before it actually does? Isn't that activity all electromagnetic and therefore travelling at the same speed towards us (light speed)? Or are there separate elements to a solar storm, some light speed waves and some physical particles slower than light speed? A http://www.bbc.co.uk/news/science-environment-17295337


title: Impact location that created the moon

text: I was reading an article today about the 1000th orbit of the Lunar Reconnaissance Orbiter, and as many of you know NASA created an animation that simulates the history of the moon. It is speculated that the moon was formed after a planet the size of mars slammed into the earth and some of that debris and debris from the earth eventually coalesced into the moon. What I am wondering is, is there any speculation as to where this collision with the earth happened, and are there any remnants of that collision?


title: Is the Apophis asteroid a concern?

text: I was watching the TED lecture _How to defend Earth from asteroids_ , and the speaker Phil Plait spoke of the possibility of this asteroid hitting Earth. He also said something about a Mayan legend. He says it's not a concern, is there a consensus on that? And what is the source for this Mayan legend?


title: Polar ice deposits on the Moon - dispersed in permafrost loose rubble, or also massive/layered?

text: Water has been discovered on the Moon with remote sensing methods. Read the press release _Diviner results indicate presence of widespread ice on the Moon_ on the "Diviner" infrared instrument on the Lunar Reconnaissance Orbiter (LRO) spacecraft that is orbiting the Moon. Independently, H2O was also found in the spectrum of a dust cloud created by that LCROSS probe fragment when it crashed into the Moon, and had the main LCROSS probe follow it, which analyzed the impact. To my understanding, this was very solid evidence that there is some water in the high-latitude areas on the Moon. What else is known about the polar ice on the Moon? Have there been identified any ice deposits that are more distinct? Say forming tiny ice caps on central peaks on certain craters on the moon, or at crater walls. Or are there ice deposits covered under a well-insulating layer of "soil" and rubble (regolith), and dispersed in this regolith (kind of permafrost)? Or is it just crystal-water bound in minerals? I can dig for science papers myself, but maybe someone has done this already and can recommend some good articles or new findings (e.g. fresh from the AGU 2011 conference).


title: Why don't stars in globular clusters all orbit in the same plane?

text: Globular clusters like Omega Centauri certainly don't seem to be very coplanar at all. In other words, why doesn't the explanation at Why are our planets in the solar system all on the same disc/plane/layer? (quoted below) apply here? > We haven't ironed out all the details about how planets form, but they > almost certainly form from a disk of material around a young star. Because > the disk lies in a single plane, the planets are broadly in that plane too. > > But I'm just deferring the question. Why should a disk form around a young > star? While the star is forming, there's a lot of gas and dust falling onto > it. This material has angular momentum, so it swirls around the central > object (i.e. the star) and the flow collides with itself. The collisions > cancel out the angular momentum in what becomes the vertical direction and > smear the material out in the horizontal direction, leading to a disk. > Eventually, this disk fragments and forms planets. Like I said, the details > aren't well understood, but we're pretty sure about the disk part, and > that's why the planets are co-planar.


title: What is the effect of gravity on gamma rays?

text: I read an article about a Gamma Ray burst linked to a black hole. How does high gravity fields affect gamma rays?


title: Minimum viscosity of liquids

text: In a lecture by Purcell he mentions that he notices that there aren't any liquids with viscosities much less than that of water, even though they go up seemingly unbounded. In an endnote (endnote 1 in that copy), he mentioned that Weisskopf found a reason, but I haven't been able to find what that reason actually was. Note that this is _not_ about the AdS-CFT lower bound.


title: Estimate the damping coefficient of my car

text: I was wondering how I can estimate the damping coefficient of my car by doing the hand bouncing the car body and watching the motion of the car? I just need a rough estimate of the damping coefficient for simulation in matlab of my car with a quarter-car model. The spring constant I found by jacking the car up until the tire barely lifted off the ground and measured the difference between a reference and the new height from the reference. Used 1/4 of the weight of the car with the assumption that the CG is in the middle splitting the weight of the car 50/50 front and rear Mechanics usually bounce the car by applying hand force and watching how the car moves in harmonic motion. Is there a way to estimate the damping coefficient by using this technique? I was thinking Mythbusters type stuff like camera, but I have an IPHONE 4 if I know the FPS of the camera I might be able to measure it frame by frame, if I have a measurement stick and reference point. Not sure if theres a practical way of doing this, I know the motion will be simple harmonic damped motion, so if you count the half-cycles maybe you can get the damping, not sure of the forumula to use? Or is there another technique like dropping the car a few inches or something to estimate it? 24 minutes ago - 4 da


title: So where is mistake in the formula of wave number (magnitude of wave vector)?

text: I have following form for wave vector $k_2=n_2 \omega/c_0$. Now because $\omega=2 \pi c/ \lambda$, then $k_2=n_2 \omega/c_0=\frac{n_2 2 \pi c_0}{c_0 \lambda}=\frac{n_2 2 \pi}{\lambda}$. But problem is that $k$ (is magnitude of wave vector~ wave number) and according to wikipedia $k=\frac{2 \pi}{\lambda}$. So where is mistake?


title: Building some measurement appratus that distinguish between two mixtures

text: We have a measurement $M$ that distinguishs between $\rho_1$ and $\rho_0$, if it has three possible answers 1,2,3 and whenever it answers something different than 3 it's correct. $M$ succeeds with probability $p$ if for every $\rho s$ above it answer the correct answer with probability at least $p$. Now I found the most efficient value of $p$ that $M$ succeeds to disntiguish between $|0\rangle\langle0|$ and $|+\rangle\langle+|$, but I am not sure how to contruct a three dimensional measurement that achieves this value of $p$. Any hints, or references? Thanks.


title: Lagrangian mechanics vs Hamiltonian mechanics

text: First of all, what are the differences between these two: Lagrangian mechanics and Hamiltonian mechanics? And secondly, do I need to learn both in order to study quantum mechanics and quantum field theory?


title: Dilation invariant lattice

text: In one dimension, we can define the lattice $ f(x+a)=f(x) $ so the lattice is translation invariant. But, can we give a lattice so $ f(p^{k}x)=f(x) $ for ALL the primes $ P= 2,3,5,7,11,....$ and prime powers $ k \in Z $ so if we increase the 'size' of the lattice, the system remains invariant? Is this possible? In this case, should we look for the Schrödinger equation, $ -\hbar ^{2} \Theta f(x)+ V(x) $ with $ \Theta = x \frac{d}{dx} $ and $ V(p^{k}x)=V(x)$?


title: solution of schrodinger equation - infinite solutions?

text: In Griffiths's introductory quantum mechanics book, it states that if $\Psi (x,t)$ is a solution to the Schrodinger's equation, then $A\Psi (x,t)$ must also be a solution, where A is any complex constant. This seems to suggest that for any particle with defined mass and kinetic/potential energy, there are many possible wave functions, as $A$ can be any complex constant - therefore, it gets impossible to figure out anything from the wave function. Can anyone show me how I am mistaken? Thank you very much.


title: Does "Air caterpillar" exist or even possible?

text: Air-Elf Aircraft Concept Is Able to Take-off or Land Vertically > Air-Elf aircraft concept is to solve road congestion. The core of the > concept is a new type of wing, “Air Caterpillar”. It can be transformed in > order to meet the needs of flexibility and speed of the aircraft. Air-Elf > aircraft not only can take off and land vertically, but also is able to > perform high-speed cruising.... > > “Air Caterpillar” wings consist of a main structure and many small wings. > These small wings can move horizontally and circularly to provide lifting > force constantly. “Air caterpillar” wings are highly transformable to meet > the parking, vertical takeoff and landing, high-speed cruising requirements.


title: The quickest 6-3 play in baseball: to bounce or not to bounce?

text: What's the quickest way the get the ball (say from shortstop) into the first- baseman's glove, given some fixed initial (throwing) speed? Directly or with one bounce? I'm fairly sure that the answer depends on the initial throwing speed, but where's the cut-off point (roughly)? * * * PS 1: Numbers, _if needed_ : > Under the current rules, a major league baseball weighs between $5$ and > $5\frac1 4$ ounces ($142$ and $149\,\mathrm g$), and is $9$ to $9\frac1 4$ > inches ($229–235\,\mathrm{mm}$) in circumference ($2\frac7 8–3$ in or > $73–76\,\mathrm{mm}$ in diameter). The horizontal distance to be travelled is $36\,\mathrm m$. PS 2: The surface is "clay" (around first base). The ball is covered in leather. PS 3: Let's assume that throwing accuracy and catching abilities are not issues.


title: Visualization of 1D spin chain wave fuction

text: What are the known methods for visualizing quantum states of one-dimensional spin chains? They can be based either on their wave functions or density matrices. My particular interest is in plotting eigenstates of 1D spin chains (e.g. Heisenberg model) in a way in which correlations between particles are related to the self-similarity of the image. **As a side note** , I found a Wikipedia Commons image with its description on the topic (a thumbnail below). However, it is not supported neither by a peer reviewed publication nor by a decent preprint. The only text related is extremely drafty. However, I'm the most interested in published results, and even more when they relate properties of the image to the properties of the state. ![enter image description here](http://i.stack.imgur.com/MM0hN.png)


title: How is information related to energy in physics?

text: I recently attended a talk by Dr. Ravi Gomatam on 'quantum reality', where the speaker suggested, that conservation of energy is not a fundamental law, and is conditional, but the conservation of information is fundamental. What exactly is the meaning of information? Can it be quantified? How is it related to energy?


title: Neutron electric dipole moment and $T$ symmetry violation

text: Our textbook (and other sources I have found) says that non-zero electric dipole moment of neutron would violate $T$ symmetry. They prove this statement by first assuming $\boldsymbol{D}=\beta\boldsymbol{J}$, where $\boldsymbol{D}$ is the dipole moment, $\boldsymbol{J}$ is the angular momentum, and $\beta$ is a constant. But why? Why is $\boldsymbol{D}$ proportional to $\boldsymbol{J}$? Why is $\boldsymbol{D}$ related to $\boldsymbol{J}$ at all? And how can't this argument be applied to other composite particles such as atoms and molecules, thereby breaking T symmetry for most of the world?