title: Why is the critical density of the Universe non-zero?

text: I'm missing something and I wondered if someone could enlighten me. As I understand it, Einstein's equations say, among other things, that matter curves spacetime. I also understand, the Universe has a 'critical density', defined to be the mass (or energy) density required for space to have zero curvature (found by solving Friedmann's equations). Why does the Universe need a non-zero density (the critical density) for zero curvature? Many thanks.


title: If a theory of everything exists, is it necessarily unique?

text: There is a lot of interesting debate over whether a "theory of everything" (ToE) is allowed to exist in the mathematical sense, see Does Gödel preclude a workable ToE?, Final Theory in Physics: a mathematical existence of proof? and Arguments Against a Theory of Everything. Since as far as I can tell this is still an open question, let's assume for now that formulating a ToE is possible. I'm curious about what can be said about the uniqueness of such a theory. Now the hard part, where I'm pretty sure I'm about to back myself into a logical corner. To clarify what I mean by unique: physical theories are formulated mathematically. It is possible to test whether two mathematical formalisms are equivalent (right? see bullet point 3 below). If so, it is possible to test the mathematical equivalence of various theories. From the Bayesian point of view, any theory which predicts a set of observables is equally valid, but the degree of belief in a given theory is modulated by observations of the observables and their associated errors. So now consider the set of all possible formulations of theories predicting the set of all observables - within this set live subsets of mathematically equivalent formulations. The number of these subsets is the number of unique ToEs. Now the question becomes how many subsets are there? Possibilities:   * It can be proven that if a ToE exists, it is necessarily unique ($1$ subset).   * It can be proven that if a ToE exists, it is necessarily not unique ($>1$ subset).   * It can be proven that it is impossible to say anything about the uniqueness of a ToE, should it exist (it is impossible to test mathematical equivalence of theories).   * We don't know if we can say anything about the uniqueness of a ToE, should it exist. So this is really asking about the ensemble of closed mathematical systems (physical theories) of an arbitrary (infinite?) number of variables (observables). This is honestly a pure math question, but here _strongly_ physically motivated. I suspect the answer is probably the fourth bullet, but surely there has been some research on the topic? Hopefully someone familiar with the ToE literature can shed some light on the question.


title: What is the meaning of following expresion $C=\frac{\delta Q}{dT}$ mathematicly

text: Our professor raised the following question during our lecture in Statistical Physics (even so it's related to Thermodynamics): Many text books (even wikipedia) writes wrong expressions (from mathematical point of view) for Heat Capacity Coefficient, and the right way to wright it is as following: $$C=\frac{\delta Q}{dT}$$ But as we see it is neither usual differential, nor a functional derivative, so the question what is this? I couldn't find the answer in math books, and it is true that many text books writes it in very different ways mixing exact and inexact differentials, so anybody have a clue what is the right expression for c and why from mathematical point of view?


title: commutators in an uncertainty relationship derived from a partition function?

text: The maximum information principle for the discrete case gives rise to a partition function (>>> see details here) $$Z(\lambda_1,\ldots, \lambda_m) = \sum_{i=1}^n \exp\left[\lambda_1 f_1(x_i) + \cdots + \lambda_m f_m(x_i)\right]$$ _**Question:_** _is it possible to identify from this equation commutators in an uncertainty relationship?_


title: Basic Torque: Did TA get it wrong?

text: The question itself: > A modern art sculptor anchors 1 meter of a cast iron pole in the ground and > leaves 3 meters of the pole sticking out of the ground at a 60° angle. The > owner of the metal shop told him that the pole is rated to withstand up to > 9000 Nm of torque before it bends. If the sculptor decides instead to hang a > 100 kg aluminum pheasant from the pole in the previous problem, what is the > net torque on the pole? The "Correct" answer (since it's all multiple choice and online) is **0Nm** , but for the life of me I can't figure out why there isn't any net torque when the pole has something hanging on the end. Am I missing something incredibly basic, or was the wrong answer input as the correct one?


_**Question:_** _Does there exist a commutator to entropy in an uncertainty relationship?_ Similar Energy and time for instance.


title: Etymology of "Renormalisation"

text: Just out of curiosity, does anyone know why "renormalisation" is so named? Who first came up with the term, and why was it used? I did a mathematics undergraduate so to me "normalisation" means rescaling a Hilbert space vector to have unit norm. Heuristically the term renormalisation makes some sense, as it is a systematic process for "normalising" divergent quantities to give well-defined probabilities. But it seems like a bad name to me, because it's patently impossible to truly "normalise" a divergent quantity. As far as I know, the correct perspective is that renormalisation replaces bare parameters by physical ones, thus removing the infinities. Is there something I have misunderstood, or is that another case of historical contingency clouding the terminology. Surely "reparameterization" would be a better name?


title: Why was the universe in a extraordinarily low-entropy state right after the big bang?

text: Let me start by saying that I have no scientific background whatsoever. I am very interested in science though and I'm currently enjoying Brian Greene's _The Fabric of the Cosmos_. I'm at chapter 7 and so far I understand most of general ideas he has talked about. Or at least, I think I understand them :-) There is however one part, at the end of chapter 6 that I can't grasp. It is about entropy and the state of the universe a few minutes after the big bang. On page 171 he says: > Our most refined theories of the origin of the universe -our most refined > _cosmological_ theories- tell us that by the time the universe was a couple > of minutes old, it was filled with a _nearly uniform hot gas_ composed of > roughly 75 percent hydrogen, 23 percent helium, and small amounts of > deuterium and lithium. The essential point is that this gas filling the > universe had extraordinarily _low_ entropy. And on page 173-174: > We have now come to the place where the buck finally stops. The ultimate > source of order, of low entropy, must be the big bang itself. In its > earliest moments, rather than being filled with gargantuan containers of > entropy such as black holes, as we would expect from probabilistic > considerations, for some reason the nascent universe was filled with a hot, > uniform, gaseous mixture of hydrogen and helium. Although this configuration > has high entropy when densities are so low that we can ignore gravity, the > situation is otherwise when gravity can't be ignored; then, such a uniform > gas has extremely low entropy. In comparison with black holes, the diffuse, > nearly uniform gas was in an extraordinarily low-entropy state. In the first part of the chapter Brian Greene explains the concept of entropy with tossing the 693 pages of _War and Peace_ in the air. At first, the pages are ordered. The specific order they are in make sense and are required to recognize the pages as a readable book called _War and Peace_. This is low entropy. It is very highly ordered and there is no chaos. Now, when you throw the pages in the air, let them fall and then pick them up one by one and put them on top of each other, the chances you get the exact same order as the initial state are extremely small. The chance you get another order (no matter what order, just not the one from the beginning) is extremely big. When the pages are in the wrong order, there is high entropy and a high amount of chaos. The pages are not ordered and when they are not ordered you would not notice the difference between one unordered state and another one. However, should you swap two pages in the ordered, low entropy version, you _would_ notice the difference. So I understand low entropy as a highly ordered state with low chaos in which a reordering of the elements would be noticeable. I hope I'm still correct here :-) Now, what I _don't_ understand is how a uniform mixture of hydrogen and helium can by highly ordered? I'd say you wouldn't notice it if some particles traded places. I'd say that a uniform mixture is actually in a state of _high_ entropy because you wouldn't notice it if you swapped some hydrogen atoms. Brian Greene explains this would indeed be the case when gravity plays no important role, but that things change when gravity _does_ play a role; and in the universe right after the big bang, gravity plays a big role. Is that because a reordering of the particles would change the effects of gravity? Or is there something else that I'm missing here?


title: If we know the universe is made up of a relativistic ether, why wouldn't gravity just be a pressure gradient of the ether?

text: According to one of the answers to a previous question ( Can photons be holes in an ether? ), we know a relativistic ether exists. If we are drifting in such a superfluid, why wouldn't gravity simply be understood to be essentially the pressure differences between baryonic matter and the ether at this fundamental level? Imagining a "universal" fluid that surrounds us to infinity for all intents and purposes. Assuming baryonic matter by its (jittery?) nature takes up more space than the equivalent ether it displaces, wouldn't this create a pressure gradient in the ether which would attract other baryonic matter according to the amount of ether displacement? (Similar to the way a hot air balloon rises to where the density of the air inside the balloon matches the air outside?)


title: How could two current flow in the opposite direction?

text: ![enter image description here](http://i.stack.imgur.com/wZfjr.jpg) Could someone explain what the green underlined part means? It doesn't make sense to me. It says the current (induced) flows CCW (that means towards the right). Then it says it flows to the left?


title: Curvilinear coordinate system around body of revolution

text: In _Boundary-Layer Theory_ by Schlichting he gives the boundary-layer equations for a body of revolution according to the paper by Boltze$^1$. Unfortunately, this paper is in German. He apparently uses the following local curvilinear coordinate system to derive the boundary-layer equations: ![Fig. 11.6 from *Boundary-Layer Theory*](http://i.stack.imgur.com/zPgH8.jpg) I'm not specifically interested in deriving the boundary-layer equations for a body of revolution, however, I am working on a problem with a very similar geometry and am having trouble wrapping my head around the coordinate system. **Question:** For the locally orthogonal basis vectors $\mathbf{e}_x$, $\mathbf{e}_y$, and $\mathbf{e}_\theta$, what are the _gradient_ , _divergence_ , and _Laplacian_ operators? This is what I have tried: $$x = \hat{x}$$ $$y = \hat{y}$$ $$r(x)\cos\theta = \hat{z}$$ where $\hat{x}$, $\hat{y}$, and $\hat{z}$ are coordinates in a Cartesian system with the same origin. Lamé's coefficients are then: $$h_x = \left\vert\left(\frac{\partial \hat{x}}{\partial x}, \frac{\partial \hat{y}}{\partial x}, \frac{\partial \hat{z}}{\partial x}\right)\right\vert = 1$$ $$h_y = \left\vert\left(\frac{\partial \hat{x}}{\partial y}, \frac{\partial \hat{y}}{\partial y}, \frac{\partial \hat{z}}{\partial y}\right)\right\vert = 1$$ $$h_\theta = \left\vert\left(\frac{\partial \hat{x}}{\partial \theta}, \frac{\partial \hat{y}}{\partial \theta}, \frac{\partial \hat{z}}{\partial \theta}\right)\right\vert = \left\vert r(x)\sin\theta\right\vert$$ With these coefficients it is straightforward to define the operators since this system is orthogonal (for example here are the formulas). I do not think that the results above are correct though since it compares to a local Cartesian system at each point. What are the correct transformations?   1. Boltze, Ernst. Grenzschichten an Rotationskorpern in Flussigkeiten mit kleiner Reibung. Georg-August-Universitat zu Gottingen., 1908. **UPDATE:** For any future Googlers, the correct transformation is: (based on Fig. 4 of @Floris's answer, $\xi$ is aligned with $U_\infty$ in the picture above) $$\xi = x\cos\phi - y\sin\phi$$ $$\eta = -(r(\xi) + y\cos\phi + x\sin\phi)\sin\theta$$ $$\zeta = (r(\xi) + y\cos\phi + x\sin\phi)\cos\theta$$ where $\phi$ is the local angle of the body with respect to the axis of rotation, i.e. $\phi = arctan(\frac{dr}{dx})$. The paper by Boltze assumes this angle is small. The Jacobian is then: $$J = r(\xi) + y\cos\phi + x\sin\phi$$ which is equivalent to Boltze's value for small $\phi$ and $y \ll r(\xi)$. And the Lamé coefficients are: $$h_x = 1$$ $$h_y = 1$$ $$h_\theta = r(\xi) + y\cos\phi + x\sin\phi$$


title: Relationship between lightlike and spatial compactification

text: The compactification of a spatial dimension, say $x^1$ given by the identification $x \sim x^1 + 2\pi R$ is said to be related to the lightlike compactification by a Lorentz boost : $$ \left( \begin{array}{c} x^0 \\\ x^1 \\\ \end{array} \right) \sim \left( \begin{array}{c} x^0 \\\ x^1 \\\ \end{array} \right) +2\pi \left( \begin{array}{c} -R \\\ R \\\ \end{array} \right) $$ What exactly is the relationship between them? How can I see this? And what does it mean from a physics point of view?


title: Do a 1-dimensional conformal theory exist?

text: can we have in physic or can we speak about 1-d conformal theory in physics ?? for example in this one dimensional theory what would be the generators $ x \partial _{x} $ or $ \partial _{x} $ ??


title: How does air escape from a pneumatic tire?

text: Obviously, it is caused by the difference in pressure between the inside of the tire and its surrounding environment; but how specifically is the air escaping?


title: Forces and acceleration due to an angle that varies over time

text: I'm having difficulty figuring out how to set a problem up. The problem is as follows: A $5\ kg$ block is being pulled along a frictionless floor by a cord that applies a force of constant magnitude $20\ N$ but with an angle $\theta (t)$ that varies with time. When angle $\theta = 34^\circ$, at what rate is the block's acceleration changing if $\theta (t) = (5*10^-2\ deg/s)t$? Specifically, the problem asks you to determine the horizontal acceleration determined by the net horizontal force, then differentiate with respect for time. This is what I'm having difficulty with. Setting the equation up with only variables, I'd have: $$\sum F_{Net,x} \equiv \vec F \cos{\theta} = ma$$ With $\theta$ as a function of time, how do I differentiate the force equation, or am I going about this completely wrong?


title: Finding initial velocity using conservation of momentum and energy

text: A 14 g bullet is ﬁred into a 120 g wooden block initially at rest on a horizontal surface. The acceleration of gravity is 9.8 m/s^2. After impact, the block slides 6.97 m before coming to rest. If the coefficient of friction between block and surface is 0.568, what was the speed of the bullet immediately before impact? Answer in units of m/s I have tried doing this so far:   1. Find Normal force $N = (m_1+m_2)g$   2. Find $F_f = \mu N$   3. Find $A$: $F_{net}=mA$   4. Find speed: $v_f^2 = v_i^2 + 2ad$   5. Find $v_i$: $m_1v_i=(m_1+m_2)v$ My answer was in the range of 740 m/s. I am confused.


title: Heisenberg picture saves locality in QM?

text: David Deutsch recently released a paper titled "The vindication of locality" where he argues that by using the Heisenberg picture rather than the Schroedinger picture, you can "save" locality. The paper is here: http://arxiv.org/abs/1109.6223 Is Deutsch wrong or has he indeed "proven" locality?


title: Higgs Standard Model Parity

text: In the Standard Model, the Higgs boson is expected to have spin 0 and even parity. I know how to get the spin-0 approach, but how do I argue for the even parity? Could you give a simple and a more detailed explanation for this even parity expectation?


title: Radioactive decay law and the exponential model, is it always valid?

text: The law of radioactive decay reads $$ N(t)=N_0e^{-\lambda t}$$ Is it valid when there is less than 1 nucleus or particle to decay? Obviously, it is nonsense to consider that we have 1/2 of nucleus or a number lower than 1 in N(t). Has someone defined the time when 1 nucleus or particle are left, that is, the quantity $$T(1)=\dfrac{\ln N_0}{\lambda}$$? Obviously, T(1) should be related to half-life T(1/2) and mean lifetime $\tau$ , why is it not common that measure? After all, it seems that when less than 1 nucleus is left, there is no "fair" to continue with the exponential model (or yes?). I read somewhere a sentence: "Humans are not able to understand the exponential law"...Of course, that is quite non exact but the exponential laws are tricky...


title: Decomposition of this wave function in eigenfunctions

text: I have this wave function of a system on a central potential: $V(r)$: $$\Phi(x,y,z)=C(x+y+z)e^{-\alpha r^2}.$$ And I'm asked a few things about probabilities. I don't have problems with that, because I understand how it's done. The main part is decomposing as a sum of eigenfunctions of the form. Changing the variables $x,y,z$ by their spherical forms, we have: $$\Phi = Cr(\sin\theta\cos\phi+\sin\theta\sin\phi+\cos\theta)e^{-\alpha r^2}.$$ The part that depends on $\theta,\phi$ can be expressed as: $$\sin\theta\cos\phi+\sin\theta\sin\phi+\cos\theta=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell} a_{\ell m}Y_\ell^m(\theta,\phi).$$ Being $a_{\ell m}$ the scalar product of our function with the eigenfunctions. What I don't understand is that when we did it in class the teacher said that it was necessary to do it only for $\ell=0,1$, and that we didn't need to go any further. I don't remember why this was possible to say before making any calculations. Why is it?


title: Does a magnetic field sustain a current in a loop?

text: A particular current produces a constant particular magnetic field. If the fact holds under scrutiny, why doesn't a magnet held within a loop sustain a particular current? Are the electron spins responsible for the current creating the magnetic field within a magnet pushed back and turned off? So that, that is, the current stops? Magnetic objects are never used as power supplies it seems, so I'm quite at a loss as to where it is I'm off here if at all.


title: WHY does the "order" of a differential equation = number of "energy storage" elements in a system?

text: OK. in all engineering courses there comes a point when they introduce you to systems theory and modeling of systems (for eg. via the impulse response) and then the Laplace transform. The modern approach to this includes the single order "state space" representation of systems, where as a strategy for finding the "state variables", it is recommended that the energy storage elements be targeted as the state variables $x$ and $\dot x$. Magically, all these courses ONLY allude to the fact that "the order" of a differential equation describing the system of interest becomes "the order" of the polynomial in the Laplace s-plane, which all turn out (the magic bit) to match THE NUMBER OF ENERGY STORAGE ELEMENTS in the system of interest. Question: what is the deep phenomenon whereby this holds? [aside: I looked a bit at the Lagrangian framework of physics w Noether's theorem etc (which to be frank I never learned at all about in school) but that doesn't seem to cover this curious symmetry or fact regarding energy storage]. The conclusion that seems to filter up from engineering systems courses is that anytime a derivative is involved in a physical system's physics, that is indicative of an "energy storage" going on! [my reaction: how bizarre...!] Can anyone shed light please?


title: How many kinds of topological degeneracy are there?

text: Here I want to summarize the various kinds of topological ground-state degeneracy in condensed matter physics and want to know whether there exists any other kind of topological degeneracy. For convenience, let's consider a 2D lattice system with $N$ lattice sites, and we may consider the finite system with open boundary condition(OBC) or periodic boundary condition(PBC). Now there are two kinds of topological degeneracy: (1)The topological degeneracy is **approximate** for finite $N$ no matter with OBC or PBC, and it becomes **exact degenerate** only under the thermodynamic limit($N\rightarrow \infty $). **Example:** FQHE. (2)The topological degeneracy(>1) is **exact** for **any finite $N$** with **PBC(on a torus)** , and it's **nondegenerate** for **any finite** $N$ with **OBC**. **Example:** Kitaev's toric code model. Is there any other kind(in the above sense) of topological degeneracy?


title: Free path distribution

text: I'm studying statistical mechanics, and I'm trying to resolve some problem known from my thermodynamics course. So I want to _calculate mean free path_ for particles with a concentration $n$ and scattering cross section $\sigma$ using **only statistical approach**. I understand that for the first step I need to find the probability distribution of the free path of the molecules of gas between successive collisions and then calculate the average value. But here I have some troubles, because I have never solved something like this. I would be grateful if someone would give me some links or give some ideas.


title: Can we show that time is orthogonal to space?

text: It's easy to show that the time we measure is "in a different direction" from the space directions we measure. However, it's not immediately obvious to me that these directions are **orthogonal**. How do we prove that any directions are orthogonal? What I came up with is the following. [For short distances/does this cause trouble?], we can use the pythagorean theorem. If I have two directions in space, I can simply travel 1 meter in each of the two directions, and then measure the distance between the two endpoints. The result is sqrt(2) meters if and only if your two directions were perpendicular. ![Finding whether two directions are orthogonal](http://i.stack.imgur.com/I4rIu.png) Easy! My question is: **can this method be used to prove that our concept of time is orthogonal to our concept of space?** I have an idea of where to start. We would need to move in space by 1 meter, while moving in time by 1 meter (converting time into meters using c). Then we would need to measure whether the distance traveled was sqrt(2) meters. I'm not sure I know what that would mean. Hopefully this question makes sense; thanks for your assistance!


title: What happens to the physical properties of electrons after diffraction?

text: Particle Wave duality shows us that waves and particles are the same thing. Therefore electrons can be viewed as both particles and waves. The wave properties of electrons can be seen in the double split experiment where a diffracted electron interacts with itself. If a single electron was diffracted, what would happen to it's physical properties? How could the mass spread out like the wave? Similarly how would the charge and spin spread out? These properties are usually considered to be contained in particles, yet if a single electron was diffracted, it surely could not become many particles spreading out. Also, what would happen if something interacted with this 'electron wave' causing part of it to act like a particle again? The wave can't be able to turn back into a particle after spreading out so much, yet it must still be able to act like a particle for certain interactions.


title: Rotational inertia of a sphere and a cube?

text: Can someone explain which would have more rotational inertia, a sphere or a cube, and why? Suppose they have the same mass, the side of the cube is equal to the diameter of the sphere, and the cube's rotation axis is perpendicular to two of its faces. Which one would have more rotational inertia about an axis through the center of mass?


title: Reflected waves and phase changes

text: If a wave passes from a lightweight string to a higher density string, we say that the reflected wave has a pi phase change. Can we say that it has minus pi phase change? If yes, why would that not change anything?


title: Rate of change of a normalised wave function

text: I'm reading Weinberg's Lectures on QM. On top of p23 it says $$i\hbar\frac{d}{dt}\int|\psi(x,t)|^2d^3x=i\hbar\int\psi^*(x,t)\frac{\partial}{\partial t}\psi(x)d^3x-i\hbar\int\left(\frac{\partial}{\partial t}\psi^*(x,t)\right)\psi(x)d^3x.$$ I'm confused about why the second term has a minus sign and why $\psi=\psi(x)$ only.


title: Step in derivation of solution to Dirac equation for hydrogen

text: My text, when solving hydrogen in the Dirac equation, makes the claim $\varphi_{j m_j}^{(+)} = \frac{\mathbf{\sigma} \cdot \mathbf{x}}{r} \varphi_{j m_j}^{(-)}$ where $\varphi_{j m_j}^{(\pm)}$ are the Pauli spinors for $j=\ell \pm \frac{1}{2}$: $\varphi_{j m_j}^{(+)} = \pmatrix{\sqrt{\frac{\ell + m_j + 1/2}{2 \ell +1}} Y_{\ell}^{m_j-1/2} \\\ \sqrt{\frac{\ell - m_j + 1/2}{2 \ell +1}} Y_{\ell}^{m_j+1/2}}$ $\varphi_{j m_j}^{(-)} = \pmatrix{\sqrt{\frac{\ell - m_j + 1/2}{2 \ell +1}} Y_{\ell}^{m_j-1/2} \\\ -\sqrt{\frac{\ell + m_j + 1/2}{2 \ell +1}} Y_{\ell}^{m_j+1/2}}$ Leaving the proof as an exercise, the author suggests that one first note that $\varphi_{j m_j}^{(-)}$ is an eigenfunction of $\mathbf{\sigma} \cdot \mathbf{L}$ (indeed, the eigenvalue is in fact $-\ell-1$) and then show that the commutator $[\mathbf{\sigma} \cdot \mathbf{L}, \frac{\mathbf{\sigma} \cdot \mathbf{x}}{r}]$ is $\frac{2}{r} (r^2 \mathbf{\sigma}\cdot \nabla -(\sigma \cdot \mathbf{x}) ( \mathbf{x} \cdot \nabla) - \mathbf{\sigma} \cdot \mathbf{x}))$ This commutator is not hard to compute, but I do not know how to proceed. Could somebody point me in the right direction?


title: Why do some food items leak water on heating in microwave?

text: ![Gulab Jamun](http://i.stack.imgur.com/FTeaW.jpg) As seen in photo (delicious gulab jamun) after heating a certain food item that contains water, simply leaks it out. The sugary syrup seen in photo was not present with the sweets before heating but was obtained as soon as it was microwaved. As the question says, why does this happen ? Why do food items which contain water, after heating in microwave besides heating, leak out the water content too ?


title: Angular position vector?

text: I'm a mathematician, so I like my angular velocities to be vectors. It makes my angular momenta and torques vectors as well, and so they have nice operations I can do on them. Because of that, I pick my definition of $\omega$ to be $\frac{\vec{r} \times \vec{v}}{\vec{r} \cdot \vec{r}}$, instead of $\frac{d\theta}{dt}$. But if I use that definition, what _is_ $\theta$ now? I tried to integrate $\omega$ with respect to $t$, and all I'm getting is $\int \frac{\vec{r}}{\vec{r} \cdot \vec{r}} \times d\vec{r}$, and I can't go any further. Is there some way of interpreting $\theta$ so that I don't have to use this stubborn integral as a definition? EDIT: I've worked out some examples, and it's path dependent. So it won't simplify past an integral. But (at least in the 2D case), it works out to be $2\pi$ times the winding number around $0$. I'm having trouble extending it to three though.


title: Hipothetical universe history with power law distributed matter

text: As every physicist knows that the history of universe has been dominated from light, matter and vacuum. All of three are supposed uniform distributed at very large scale. What would been happened to history of the universe if matter and light would actually power law distributed? Namely if matter and light forms "clusters" with no characteristic scale.


title: Local nature of a surface charge density

text: Boundary S of a cavity in a very large (perfect) conductor is a connected compact (smooth) surface. A positive point charge +q is placed inside this cavity. From Gauss' law we know that the _total_ induced charge on the surface S is -q. However, is the surface charge density on S necessary _locally_ negative (at every point of S)? How to prove (or disprove) this assertion? I'm looking for a precise, robust argument, not hand-waving type of answers.


title: How does Lunar Orbit Injection (LOI) work?

text: Just what the title states. I read that LOI was used by the Apollo program (and possibly others) to achieve escape velocity. How does it work? Is it merely a matter of centrifugal/centripetal force?


title: Is this a correct description of bonding in a metal?

text: I am reading the paper "Twenty five years of Finnis-Sinclair potentials" by Graeme Ackland, Adrian Sutton, and Vasek Vitek, _Philosophical Magazine_ **2009** , _89_ , 3111-3116. It is a review-type article describing the notion that pair potentials are inadequate for describing bonding in metals. But it turns out that if the attractive interaction is "described by the square root of a sum of pair interactions to neighbors, centered on each atom in the solid," metallic bonding can be described reasonably well. The authors point out: > The novelty is entirely in the square root. The square root captures the > dependence of atomic interactions on the local density: **as the number of > neighbors of an atom decreases, the strength of the remaining bonds > increases**. This immediately predicts an inward relaxation at metallic free > surfaces with a tensile surface stress, both of which are widely observed > but not predicted by models in which the cohesive energy is just a sum of > pair potentials. My question is, is it true for metals that **"as the number of neighbors of an atom decreases, the strength of the remaining bonds increases"**? Is there any way that I can understand this, either in a qualitative or quantitative way? Also, when it says "the strength of the remaining bonds increases," is this referring to the energy _per bond_?


title: Is it possible for the entropy in an isolated system to decrease?

text: As far as I can tell, the concept of entropy is a purely statistical one. In my engineering thermodynamics course we were told that the second law of Thermodynamics states that "the entropy of an isolated system never decreases". However, this doesn't make much sense to me. By counter-example: Consider a gas-filled isolated system where the gas has maximum entropy (it is at equilibrium). Since the molecular motion is considered to be random, at some point in the future there will be a pressure gradient formed by pure chance. At this point in time, entropy has decreased. According to Wikipedia, the second law purely states that systems tend toward thermodynamic equilibrium which makes sense. I then ask a) is the second law as we were taught it wrong (in general), and b) what is the use of entropy (as a mathematical value) if it's effectively an arbitrary definition (i.e. what implications can we draw from knowing the change in entropy of a system)? Thanks in advance for your help.


title: Question about heat engine efficiency

text: I would like to know how much thermal energy is converted to kinetic energy in a steam engine, or a more efficient means if available. I have done some research and found out that Carnot made some equations to determine efficiency. It is also mentioned that as part of the Carnot cycle it is important that the cold temperature reservoir be colder, but I don't understand how that would help increase efficiency of the conversion of thermal energy to kinetic energy. Here is a paragraph taken from Wikipedia at http://en.wikipedia.org/wiki/Carnot_cycle "Lowering the temperature of the cold reservoir will have more effect on the ceiling efficiency of a heat engine than raising the temperature of the hot reservoir by the same amount."


title: Expansion of the Universe, will light from some galaxies never reach us?

text: Is it true that the light from some galaxies will never reach us? The explanation for that is that the Universe expanding faster than the speed of light. But, if the speed of light is constant in all reference frames, then in the reference frame of a galaxy, light must move at the speed of light and thus it must reach us (?). Could you also explain how is it possible that the Universe is expanding faster than the speed of light if an object can never be observed to move faster than light? (without Doppler effects).


title: From 1D problem to easier 2D problem

text: Here I describe an example how a 1D problem can be solved easier by considering a mathematically equivalent problem in 2D. Problem: Find the equation of motion of particle in 1D space with the potential $V(x)=\frac{A}{x^2}$   * Normal Solution:  Conservation of energy $$\frac{m\dot x^2}{2}+\frac{A}{x^2}=E$$ then derivative, integral, maths stuffs.   * 2D solution: It's exactly equivalent to the radial motion of a particle in 2D space: Consider a polar coordinate. Conversation of energy: $$\frac{m\dot r^2}{2}+\frac{mr^2\dot\theta^2}{2}=E$$ Conservation of momentum: $$mr^2\dot\theta=L$$ Hence: $$\frac{m\dot r^2}{2}+\frac{L^2}{2mr^2}=E$$ So it's just a uniform motion of a particle in 2D space then projected on the radius line. The interesting part is that if $A<0$ then we must set $L$ is an imaginary number to get the result. This may be consider a mathematical trick. But I want to know if there is a generalization of this method. Maybe it would be related to Lagrange Equations and generalized coordinates.


title: i can't detect what wrong have i done in a kinematics problem

text: a question was given to us in assignment which was as follows > A person moves 20 metres towards north then 30 metres towards east and then > moves 30 x 1.414 metres towards south west. The whole trip takes 5 seconds. > What is the person's average velocity for this whole journey? i just squared every magnitude of velocity in all the three directions and then I took square root of that sum doing so what i got is this number `55.735267112` so this will be the total displacement of that person and dividing this with the given time [5in this case] we get the answer as 11.1 but the answer is 2 m/s can any one suggest me where is my concept going wrong


title: 3D Minimum uncertainty wavepackets

text: Based on the 1D case mentioned in Griffiths, I decided to try looking at the features of 3D Gaussian wavefunctions, i.e. (position basis) wavefunctions of the form $\psi(\mathbf{r}) = Ae^{-\mathbf{r}^\dagger\mathsf\Sigma\mathbf{r}/4}$, where _A_ is a normalization constant, **r** is position, Σ is a positive-definite symmetric matrix (which by a suitable change of coordinate basis can be made diagonal), and † denotes the conjugate transpose. Applying standard results for Gaussian integrals, I was able to get   * $\langle \mathbf{r} \rangle = 0$   * $\langle r^2\rangle = \operatorname{Tr}\mathsf\Sigma$   * $\langle \mathbf{p} \rangle = 0$   * $\langle p^2\rangle = \frac{\hbar^2}{4}\operatorname{Tr}\mathsf\Sigma^{-1}$ So, substituting into Heisenberg's uncertainty principle and rearranging terms, it follows that, in order to get minimum uncertainty with respect to $\mathrm{r}$ and $\mathrm{p}$, we need to have      $(\operatorname{Tr}\mathsf\Sigma)(\operatorname{Tr}\mathsf\Sigma^{-1})=1$. Here's where I'm running into a difficulty. As I mentioned before, the matrix Σ can always be assumed to be diagonal. Then the only possible solution for Σ is $\mathsf\Sigma = \begin{pmatrix} 1 & 0 & 0\\\ 0 &-1 &0\\\ 0 &0 &1\end{pmatrix}\times\mathrm{constant}$ But this contradicts the fact that Σ is positive-definite (the -1 would imply that one of the coordinates has negative uncertainty, an absurdity). Assuming I did all the calculations correctly, this seems to imply that a Gaussian wavefunction is not the minimum uncertainty wavefunction with respect to **r** and **p**. On the other hand, it's comparatively trivial to show that it _is_ the minimum uncertainty wavefunction with respect to _x_ and _p_ _x_ , _y_ and _p_ _y_ , and _z_ and _p_ _z_ individually. Is there a wavefunction which is the minimum unceratinty wavefunction with both respect to the individual coordinates (e.g. _x_ and _p_ _x_ ) _and_ with respect to **r** and **p**? **Edit** It was asked by marek what I meant by "minimum uncertainty with respect to $\mathbf{r}$ and $\mathbf{p}$". To answer this, recall that the generalized uncertainty principle takes the form of $$ \sigma_A\sigma_B \geq \frac{1}{2}\left|\langle[A,B]\rangle\right|.$$ Although I'm not entirely sure it's valid to do so, I assumed that to calculate the commutator $[\mathbf{r},\mathbf{p}]$ I could use the formalism of geometric algebra (see Geometric algebra). Then $$\begin{align*} [\mathbf{r},\mathbf{p}]f &= \frac{\hbar}{i}\mathbf{r}\nabla f - \frac{\hbar}{i}\nabla(f\mathbf{r})\\\ &= \frac{\hbar}{i}\sum_{jk} \left[x^j\hat{\mathbf{e}}_j\frac{\partial f}{\partial x^k}\hat{\mathbf{e}}^k - \frac{\partial}{\partial x^k}\left(fx^j\hat{\mathbf{e}}_j\right)\hat{\mathbf{e}}^k\right]\\\ &= \frac{\hbar}{i}\sum_{jk} \left[ x^j\frac{\partial f}{\partial x^k} \hat{\mathbf{e}}_j\hat{\mathbf{e}}^k - \frac{\partial f}{\partial x^k}x^j\hat{\mathbf{e}}_j\hat{\mathbf{e}}^k - f{\delta^j}_k\hat{\mathbf{e}}_j\hat{\mathbf{e}}^k\right]\\\ &= \frac{\hbar}{i} f, \end{align*}$$ where $f$ is an arbitrary function, $x^1,x^2,x^3$ are the position coordinates, and $\hat{\mathbf{e}}_1,\hat{\mathbf{e}}_2,\hat{\mathbf{e}}_3$ are the standard Cartesian basis vectors. Thus, the uncertainty principle for $\mathbf{r}$ and $\mathbf{p}$ takes the form $$\sigma_\mathbf{r}\sigma_\mathbf{p} \geq \frac{\hbar}{2},$$ which means that the minimum uncertainty wavepacket with respect to $\mathbf{r}$ and $\mathbf{p}$ must satisfy $$\sigma_\mathbf{r}\sigma_\mathbf{p} = \frac{\hbar}{2}.$$


title: Why is tunneling not a classical idea?

text: There is no tunneling in the case of infinite potential barrier, but there is when we have a finite well. In the classical analog, in the first case we have a particle bouncing between to infinitely rigid impenetrable walls and there is no tunneling, same as the quantum case. But if we have a finite barrier, means we have walls of finite rigidity, say made of cork or something. Then the particle would just break through some of the cork and it's probability of being found further in the cork wall will decay steadily. I can understand discrete energy levels being a new thing, because they behave like a wave that's confined and not like particles confined, but why tunneling?


title: orbitting around the earth

text: Does a space station need an initial velocity (tangentially) to be able to orbit around the earth or just the gravitational force acting downward on the station at certain height from the ground is enough?


title: Angular momentum matrices (Schiff section 27)

text: On page 203 3rd edition of Schiff we are given the angular momentum matrices ${J}$ for $j=1$. I am curious as to how these relate to orbital angular momentum for $j = 1$. If we take the corresponding 3x3 matrices for spin given on page 198. Lets just use $$S_x = i \hbar \begin{pmatrix} 0 & 0 & 0 \\\ 0 & 0 & -1 \\\ 0 & 1 & 0 \\\ \end{pmatrix}\,\,\,\mathrm{and}\,\,\,S_y = \begin{pmatrix} 0 & 0 & 1 \\\ 0 & 0 & 0 \\\ -1 & 0 & 0 \\\ \end{pmatrix}$$ With $$J_x = \frac{\hbar}{\sqrt{2}} \begin{pmatrix} 0 & 1 & 0 \\\ 1 & 0 & 1 \\\ 0 & 1 & 0 \\\ \end{pmatrix}\,\,\,\mathrm{and}\,\,\,J_y = \frac{\hbar}{\sqrt{2}}\begin{pmatrix} 0 & -i & 1 \\\ i & 0 & -i \\\ 0 & i & 0 \\\ \end{pmatrix}$$ Using $\vec{J} = \vec{L} + \vec{S}$ we can solve for $L_x$ and $L_y$ by subtracting $J$ from $S$. $$L_x = J_x-S_x = \begin{pmatrix} 0 & \frac{\hbar}{\sqrt{2}} & 0 \\\ \frac{\hbar}{\sqrt{2}} & 0 & i\hbar + \frac{\hbar}{\sqrt{2}} \\\ 0 & -i\hbar + \frac{\hbar}{\sqrt{2}} & 0 \\\ \end{pmatrix} $$ $$ L_y = J_y-S_y = \begin{pmatrix} 0 & -\frac{i\hbar}{\sqrt{2}} & -i \hbar \\\ \frac{i \hbar}{\sqrt{2}} & 0 & -\frac{i\hbar}{2} \\\ i \hbar & \frac{i \hbar}{\sqrt{2}} & 0 \\\ \end{pmatrix}$$ $$L_z = J_z - S_z = \begin{pmatrix} \hbar & i \hbar& 0\\\ -i \hbar & 0 & 0 \\\ 0 & 0 & -\hbar \\\ \end{pmatrix}$$ My question is that when I do this, they don't obey the standard commutation relations $[L_i,L_j] = i \hbar \epsilon_{ijk} L_k$. More curious is the 2x2 case for $j=1/2$ on page 203 again, you get null orbital angular momentum matrices. What went wrong here and what am I missing?


title: What does "soft" in "soft symmetry breaking" mean?

text: For example it is stated that if supersymmetry breaking is _soft_ then stability of gauge hierarchy can be still maintained.


title: How distant is the origin of oldest space radiation reaching us?

text: A comment to my question straightens out a misconception. > That places the furthest theoretically observable edge of universe at 13.798 > bln light years away - this is how long light would take to reach us from > there, and there were no light sources before that date. - > _"That places the furthest theoretically observable edge of universe at > 13.798 bln light years away"_ Makes sense in a naive way, but not so. Part > of the travel occurred when space was smaller than it is now. – dmckee♦ Well, let me ask, by how much was I off? How far is currently the point of universe, radiation/light from which, originating in the early times of the universe (say, 150mln years after big bang, when great inflation had long ended and first stars formed, and the speed of expansion got mostly constant) would reach Earth today? How much distance over Hubble length could have light covered over Hubble time thanks to space expansion?


title: Angular deficit

text: If one starts with a flat piece of paper, removes a wedge, and tapes the paper together, you get a cone. The angle of the removed wedge is called the "angular deficit". Now if this is done in 3 spatial dimensions, removing a 'wedge' along an entire line, there is similarly an angular deficit around the line defect. Using cylindrical coordinates we can write the spacetime metric as:   $g^{\mu\nu} = dt^2 - dz^2 - dr^2 - r^2 d\theta^2$   with $0 \leq \theta \leq 2 \pi - \Delta \phi$ and the boundary condition that $\theta=0$ is equivalent to $\theta = 2 \pi - \Delta \phi$, where $\Delta \phi$ is the angular deficit. This line defect is sometimes referred to as a 'cosmic string'. In GR this can also be used to describe the spacetime outside of a cylindrically symmetric object in certain conditions ( http://prd.aps.org/abstract/PRD/v39/i4/p1084_1 ). Besides the change in boundary condition for $\theta$, this looks exactly like flat spacetime. So the spacetime is locally flat (the Riemann curvature vanishes) everywhere except at the line defect where it is undefined. So without observing the defect directly, its gravitational presence can only be seen in a topological sense as it requires measurements of paths going around the defect. To make sure my intuition is okay up to this point, first I'd like to verify that last line:   In case I'm missing something... In such a spacetime, is it possible to somehow measure the angular deficit "locally" or using paths that have a zero winding number around the defect? This intuitive understanding of angular deficit seems to implicitly require that spacetime is locally flat and have the symmetry of an infinite cylinder. For example, if I broke translation symmetry along the line defect by making it finite in length, one can't even use intuitive topology to state whether a path went 'around' the line defect or not. However, physical intuition states that spacetime around the center of the finite line defect, at distances much less than the finite length, should still approximate the infinite line case. Can "angular deficit" along a path be rigorously defined for arbitrary metrics somehow? If not, what symmetries are necessary to do so?


title: Cancel the Earth's magnetic field

text: Is there any reason why Earth's magnetic poles are so close to its axis of rotation, or is it a coincidence? Why does the Earth have a somewhat uniform magnetic field with two poles? If you take a bunch of random particles and put them together, why should a pole form on each side of this collection? Is it in practice possible to create a device capable of canceling the Earth's magnetic field in a region the size of the North Sea? This could help us control the wandering of certain unwanted crabs.


title: Does Earth's Rotation Affect Its Shape?

text: The question I am working on is, "Consider the following. (a) Find the angular speed of Earth's rotation about its axis. rad/s (b) How does this rotation affect the shape of Earth?" I am fully capable of solving part (a); however, I am not sure how to describe the effect earth's rotation on its shape. I tried to search my textbook for the answer, but could not find anything. Is there an actual effect on the shape?


title: Divergent bare parameters/couplings: what is the physical meaning of it? Do this have any relation with wilson's renormalization group approach?

text: I understand that bare parameters in the Lagrangian are different from the physical one that you measure in an experiment. I'm wondering if the fact that they are divergent has any physical meaning? If they weren't the divergencies that arise in loop calculation cannot "be put" anywhere else. I'm OK with them being different from the physical one, but why is it OK for them to be divergent? EDIT: I have some difficulties in understanding if this has any connection with renormalization group approach of Wilson. They seems to be quite different. In fact in the last case you start with an effective field theory valid up to scale Landa (sharp cutoff) and you "integrate out" the high momentum part of the action to see how the theory behaves at low energies. On the other approach you want (after the renormalization ) to let the cutoff go to infinity and find finite results. That means that the theory is not sensitive anymore about the high energy/short scale behavior. The running of the coupling in Wilson's approach has nothing to do with the bare parameters going to infinity when the cutoff is removed right? Is there any reference that tries to unify this 2 different approaches? I read deeply this two books: Quantum and statistical physics - le bellac Field theory, the renormalization group - amit Do you reccomend any other books/articles?


title: Conservation of 4-momentum in special relativity

text: I understand that the inner product of two 4-vectors is conserved under the Lorentz transformations, so that the absolute value of the four momentum is the same in any reference frame. This is what I (most likely mistakenly) thought was meant by the conservation of momentum. I don't understand why equations such as $P_1=P_2+P_3$ ($P_i$ are 4-momentum vectors for different particles in a collision for example) should hold, within a reference frame. I've been told that you can't just add four velocities together on collision of particles, so why should you be able to do this with the momentum vectors?


title: Born rule and unitary evolution

text: Is the Born rule a fundamental postulate of quantum mechanics, or can it be inferred from unitary evolution?


title: Convert pressure to energy

text: I read a little about electricity being created from pressure, Piezoelectricity. I'm still pretty newbie about it though. Let's say I want to charge a battery from that technique, using my hands to press / tap on something. Besides me having all day to do that, what items/technology would be required in order to achieve that? Any specific links or guidance would be great.


title: Why are these specific stress invariants chosen?

text: I've seen these invariants of the Cauchy stress tensor $S$ defined in multiple places: $$J_1 = \lambda_1+\lambda_2+\lambda_3 = Tr(S)$$ $$J_2 = \lambda_1^2+\lambda_2^2+\lambda_3^2$$ $$J_3 = \lambda_1\lambda_2\lambda_3 = \det{S}$$ where $\lambda$ are the eigenvalues of the stress tensor. But I suppose we could use almost any three functions of the eigenvalues. Is there any specific reason to choose these, other than the fact that they can be calculated without first finding the eigenvalues?


title: How do photons 'connect' during wireless connection

text: So wireless router broadcasts a signal and then your device searches. So what actually happens when the photons 'meet' it's kind of like saying, 'ah your one of us, so we will follow you, show us the way' It's so bizarre, how do photons connect during wireless connection?


title: Is there a name for a substance that is a gas at room temperature?

text: Is there a name for a substance that is a gas at room temperature, such as Hydrogen, Helium, Neon, Argon, Krypton, Xenon, Chlorine, Fluorine, Bromine, Nitrogen, and Oxygen. I am writing a paper where I refer to: a “room temperature gas” that has been cooled to a liquid or solid state. The word that I am looking for should replace “room temperature gas”


title: When we do pull-ups , does the bar takes more weight than when we hang down on the bar?

text: When I do pull-ups, I feel I push down to the bar. But does the bar really take more weight than just hang down? For people who don't know pull-ups and hang down, here is an illustration. Left: Hang Down-----------------------Right: Pull ups ![Pull-Ups](http://i.stack.imgur.com/AY9Xl.png) So, does in right picture the bar take more weight than the left one? Thanks in advance.


title: Salt water evaporation

text: Pardon my no-knowledge of the topic but I am curious about following characteristics of sea water evaporation:   1. is it more efficient to have thinner ponds to which water is re-added or deeper ponds?   2. as the salinity of water dictates the evaporation is it more efficient to arrange ponds based on salinity and move water around, or just keep a pond until it evaporate?   3. when sea water evaporates, does crystal salt fall down or remain on top?


title: Correlation functions and connection to ward identities

text: I have the following definition of a general correlation function $$ \langle \Phi(x_1)\dots \Phi(x_n)\rangle = \frac{1}{Z} \int [d\Phi] \Phi(x_1)\dots\Phi(x_n)e^{-S[\Phi]} $$ I have only just started to learn about these functions, so could someone explain what this equation actually means? I see that parts are reminiscent of what you find in statistical mechanics, such as $Z$ denoting the partition function and $\exp(-S[\Phi])$ denoting the weight function or Boltzmann factor, and I think $[d\Phi]$ stands for the integration measure over a set or family of fields, so the bracketed notation emphasizes that, rather than an integration over points. But I can't make sense of all the pieces collectively. My other question is to do with a derivation in the book by Di Francesco 'Conformal Field Theory' P.43. He defines the quantity $$ \langle X \rangle = \frac{1}{Z} \int [d\Phi'] (X + \delta X) e^{-S[\Phi] - \int d x \partial_{\mu}j^{\mu}_a \omega_a(x)}, $$ where $X$ is a collection of fields and $\delta X$ is its variation under the transformation. He then expands this result to first order in $\omega(x)$ to obtain $$\langle \delta X \rangle = \int d x \partial_{\mu}\langle j^{\mu}_a(x)X\rangle\omega_a(x)$$ and then identifies $$\delta X = -i\sum_{i=1}^{n} (\Phi(x_1)\dots G_a \Phi(x_i)\dots \Phi(x_n))\omega_a (x_i)$$ but I am not sure how he obtained these two equations. Any help would be great, many thanks.


title: What is the difference between "accelerating a particle to light speed" and "turning on a flashlight"?

text: According to how we understand the universe it would take an infinite amount of energy to accelerate the ambiguous "particle" to light speed. Yet there are all kinds of reactions that cause light, seemingly without providing the infinite amount of energy needed to accelerate particles to light speed. Burning wood in a campfire or heating tungsten in a vacuum both cause particles that were not traveling at light speed to somehow become traveling at light speed, otherwise both would be totally useless. What am I doing when I light my cigarette that the guys at CERN can't figure out?


title: Is the presence of the phase velocity of De-Broglie wave in the Lorentz Transformations a mere coincidence?

text: We have the term $c^2/v$ in the lorentz transformation for time: $t' = \gamma (t - x v/c^2)$. Now, taking into account of the fact that $c^2/v$ is the phase velocity of the De-Broglie wave for a particle with speed v, this reduces to $t' = \gamma (t-t_1)$, and I could explain this term $t_1$ as > the time taken by a de-broglie wave excited at $x=0$ in the frame $xyzt$ at > time $t=0$ to reach the point $x=x$ in the same frame. But then, during the time special relativity was published, quantum mechanics and the whole idea of de-broglie wave itself didn't exist. (and of course I know de-broglie waves don't really travel thorough real space...) So my question is how did this rather unusual term turn up in special relativity, whose meaning was understood only 20 years later? I've gone through some textbook derivations of lorentz transformations, but couldn't gather any clue on this seemingly weird connection between relativity and QM. Thanks in advance for your help :)


title: Can we glue femto-cameras to photons?

text: I know its not as easy as saying to glue nano-cameras to photons. Please consider the following extract (related to Heisenberg's Uncertainty principle) from the Modern's abc of Chemistry: > ...to observe the position of the electron accurately, light of > appropriately small wavelength should be used. But the photons of radiations > of smaller wavelength will have higher momentum ($p=h/\lambda$). _When even > a single photon of this light used to observe the position of electron > strikes against it, a large amount of momentum will be transferred to the > electron at the time of collision_. This will change the velocity of the > electron and consequently will result into greater uncertainty in velocity > or momentum...... If a single photon of light can transfer a large amount of momentum to a particle of mass like electron, can we create sticky femto-cameras with electron like properties (if possible in future) which can stick to photon? If it is not possible to create sticky nano-cameras or for any other reasons the glue idea doesn't work, atleast can we spread those non-sticky cameras (of electron mass) over the moving photons, so that they can carry them to long distance? It helps us to know what universe is at the far distance. I have no idea w.r.t particle physics or relativity or even what photon is, what ever I know is that passage. I know there are more difficulties w.r.t this idea, it may be related to creation of those femto-cameras or associating a sticky property to it or any other. I want this concept to be discussed w.r.t whether it is possible that photons can do the job, if we _have_ such femto-cameras. Sometimes I might have misunderstood, if so pardon me and explain.


title: Can pure maths create new theories in physics or does the "idea" ALWAYS come before the math?

text: I am in a debate with a friend about the value of string theory in physics. He is concerned that we are wasting valuable intellectual and financial resources on a path that is fanciful and can't ever hope to be verified by experiment and evidence (11 - 20 dimensions etc). My point is not to agree with string theory but to argue that maths is powerful and capable of produce new ideas that can be verified. His question to me is "Can math can provide new and viable ideas without an original idea based on the natural world or through observation" I can't think of any good examples where the study of maths has resulted in new theory about physics. Can anyone?


title: Photo of reflection in magnifying mirror shows normal magnification. Why?

text: I just took a picture of myself in one of those magnifying shaving mirrors and in the photo I am unmagnified! Why is this?


title: In coordinate-free relativity, how do we define a vector?

text: Relativity can be developed without coordinates: Laurent 1994 (SR), Winitzski 2007 (GR). I would normally define a vector by its transformation properties: it's something whose components change according to a Lorentz transformation when we do a boost from one frame of reference to another. But in a coordinate-free approach, we don't talk about components, and vectors are thought of as immutable. For example, Laurent describes an observer using a timelike unit vector $U$, and then for any other vector $v$, he defines $t$ and $r$ uniquely by $v=tU+r$, where $r$ is orthogonal to $U$. The $(t,r)$ pair is what we would normally think of as the coordinate representation of $v$. In these approaches, how do you define a vector, and how do you differentiate it from things like scalars, pseudovectors, rank-2 tensors, or random objects taken from something that has the structure of a vector space but that in coordinate-dependent descriptions would clearly not transform according to the Lorentz transformation? It seems vacuous to say that a vector is something that lives in the tangent space, since what we mean by that is that it lives in a vector space isomorphic to the tangent space, and any vector space of the same dimension is isomorphic to it. [EDIT] I'm not asking for a definition of a tangent vector. I'm asking what criterion you can use to decide whether a certain object can be described as a tangent vector. For example, how do we know in this coordinate-free context that the four-momentum can be described as a vector, but the magnetic field can't? My normal answer would have been that the magnetic field doesn't transform like a vector, it transforms like a piece of a tensor. But if we can't appeal to that definition, how do we know that the magnetic field doesn't live in the tangent vector space? Bertel Laurent, Introduction to spacetime: a first course on relativity Sergei Winitzki, Topics in general relativity, https://sites.google.com/site/winitzki/index/topics-in-general-relativity


title: Momentum, Impulse and Newton's Second Law of Motion

text: Newton formulated his Second Law as such: $$\sum{\vec{F}} = \frac{\delta \vec{p}}{\delta t}$$ and of course, $\vec{p} = m \vec{v}$. Why is it that if the net force $\sum \vec{F}$ is constant (which implies that the rate of change of momentum is constant), then $$\frac{\delta \vec{p}}{\delta t} = \Delta \vec{p}$$ In other words, why does the rate of change of momentum equal to the total change in momentum? If there were previous answers on this, please inform me! Thanks :)


title: Typical time scales for spin dynamics and lattice vibrations in magnetic solids

text: In a paper from the 1990s ([1]) on magnetovolume effects in ferromagnets, it is written that _in most real situations, the moment (or spin) autocorrelation time is much larger than the period for typical atomic vibrations. That is, in the average time between successive spin fluctuations on a given moment, the moment- bearing atom experiences many (10^2-10^5) atomic vibrations cycles._ They only give two references in this connection. One is [2], which seems to be very specific for solitons in 1D Ising chains (and whose reading is proving quite challenging to me), the other is the beautiful book on Solid State theory by Ashcroft and Mermin [3], where I cannot find this piece of information. Is there a general consensus on whether spin time scales are longer than those of vibrational degrees of freedom/phonons in solids? Is there any theoretical argument supporting that claim? Is it a well established experimental fact? [1] B. Grossmann and D. G. Rancourt, Phys. Rev. B, 54, 12 294 (1996) [2] D. G. Rancourt, Solid State Commun, 58, 433 (1986) [3] N. W. Ashcroft and N. D. Mermin, _Solid State Physics_ (Saunders College, Philadelphia, 1976)


title: About partially polarized light and the degree of polariztion

text: When I was taking Optics course, I found there were several questions about polarization of light. I use the textbook of Hecht.   1. It seems that the definition of degree of polarization may be not so well-defined if $V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}$. For a elliptical polarized light, there is no natural polarized part, but still $V\neq 0$.   2. I found it hard to deal with partially polarized light. First the definition. What is the definition of partially polarized light? Light with $0<V<1$?   3. Is all partially polarized light can be presented by the superposition of a plane polarized light and a natural light? I suppose it is true but why? Is there any formal explanation?   4. Consider a real problem. Suppose there is a beam of light incident on an air-glass interface with $n_{ti}=1.5$ at a certain degree, say $30°$. Then how to characterize the reflected light or the transmitted light since they are all partially polarized. What is $V$ for these lights? I gauss the answer may be $V=\frac{|r_p|^2-|r_s|^2}{|r_p|^2+|r_s|^2}$. But I can't convince my self why this correspondes to the definition above. (This is essentially a problem in Hecht.) Thanks a lot! I am really confused with that.


title: Vertical uniform circular motion.. can it really be uniform?

text: This is the picture in my mind: ![enter image description here](http://i.stack.imgur.com/VTbX7.png) For centripetal force, I learned that: $T-mg\cos\theta= \frac{mv^2}{r}$ In vertical circular motion, the velocity is naturally going to decrease as kinetic energy is converted to potential energy as the particle moves up the circle- and thus resulting into a non-uniform circular motion. However, I was told that if tension changes accordingly, the velocity will remain constant(as from the equation). What I do not get about this part of the explanation is that which force will balance mgsinθ (tangential force in the diagram )?


title: Higgs mechanism in QED

text: I'm trying to understand the Higgs mechanics. For that matter, I'm exploring the possibility of giving mass to the photon in a gauge-invariant way. So, if we introduce a complex scalar field: $$ \phi=\frac{1}{\sqrt{2}}(\phi_{1}+i\phi_{2}) $$ with the following Lagrangian density (from now on, just Lagrangian) $$ \mathcal{L}=(\partial_{\mu} \phi)^{\star}(\partial^{\mu} \phi)-\mu^2(\phi^{\star}\phi)+\lambda(\phi^{\star}\phi)^2$$ and $\mu^{2}<0$. We note that the potential for the scalar particle has an infinity of vacuums all of them in a circle of radius $v$ around (0,0). We introduce two auxiliary fields $\eta,\xi$ to express the perturbations around the vacuum $$ \phi_0=\frac{1}{\sqrt{2}}[(v+\eta)+i \xi ]$$ Introducing the covariant derivative and the photon field, I have to compute the following thing $$(D^{\mu} \phi)^{\dagger}(D_{\mu} \phi) $$ The derivatives included in $(D^{\mu}\phi)^{\dagger}$ are supposed to act upon the $(D_{\mu} \phi)$?


title: Mass in terms of energy

text: Given the relationships between mass and energy in relativity, and given that particles with mass can be created given energy over the threshold energy, and vice-versa, can we say that mass is simply an extremely dense form of energy? Or is there a deceptive parallel between the two?


title: The cooling curve of tin during solidification

text: I'm going to measure the freezing point of tin by recording the cooling curve. It sounds like a dull experiment because all I have to do is to heat up the tin, wait for it to cool, and the computer does all the job. But in the lab manual it says:"keep a heating voltage of 30V so that the tin won't cool down too fast". What will happen if I just cut off the power source after the tin is melted? * * * Follow-up: I measured the two cooling curves. The curve without the 30V heating drops faster, which is expected. And the peak due to the latent heat released by supercooled tin is lower. But the melting points of tin calculated from the two curves are almost equal, with a difference of 0.04K. So I guess nothing special happened when I turn off the heater. And the error caused by temperature gradient is not significant in this particular experiment.


How do I create a direct current with a magnet? The magnet is not to be moved in the direction of the wire. In fact I'm looking for the contrary to the drawing from https://commons.wikimedia.org/wiki/File:RechteHand.png ![magnetic field](http://i.stack.imgur.com/GVfh0.png) Update 1st: The background is that in the German wiki under Magnetismus you find the sentence "Any movement of electric charges produces a magnetic field." For a wire this is proved by experiment, even a direct electrical current produces a magnetic field. This was illustrated in the drawing above. My first doubt is, do a number of free flowing electrons in the absence of any external fields produce a magnetic field in the same strength as in the wire. In this case I believe only experimental facts. 2nd: If a DC in a straight wire produces a magnetic field the reason for this must be the EM induction. But there are two strange facts. The EM induction connects always three components, two of them produces the third. This are the electrical current, the magnetic field and the movement of the wire or the magnetic field. But the straight wire in the wiki definition under point one does not moves. To proof the wiki definition I remembered that the vector components of the EM induction are a cross product. This is the reason I ask help me to find out what is the reverse process of the above linked drawing.


title: Why defects moderate (reduce) electric field?

text: I am running a simulation with drift-diffusion equation, Poisson equation and Fourier heat equation with a metal/insulator/metal structure. In my simulation, initially electric field is at the top of the asperity and defect generation is high at that point. Note that the initial field enhancement is higher (50%) for sharp asperity but breakdown voltage reduces about 3%. This seems that defect generation moderates/reduces initial electric field. Could any one explain this this behaviour.


Can a charged particle pass through a superconductor? I think it is impossible, because the superconductor will induce an infinite electric field which keeps the particle away.


title: Proof involving the fine-structure Hamiltonian of the Hydrogen atom

text: Given the perturbed Hamiltonian of the Hydrogen atom: $$ H=H_0+H_f=\frac{\bar{p}^2}{2m}+V(r)-\frac{\bar{p}^4}{8m^3c^2}+\frac{1}{2m^2c^2}\frac{1}{r}\frac{dV(r)}{dr}\bar{L}\cdot\bar{S}+\frac{1}{8m^2c^2}\nabla^2V(r), $$ where the last three terms are the relativistic perturbation, the spin- orbit coupling and the Darwin term and they make up the fine structure Hamiltonian $H_f$; assuming we are in the case $n=2$, I'm asked to prove that $H_f$ doesn't link the eigenvectors $| n=2,l=0,m_l=0,m_s=\pm1/2\rangle$ with the eigenvectors $| n=2,l=1,m_l=-1,0,+1,m_s=\pm1/2\rangle$. The only thing that comes to mind is to apply $H_f$ to each of the first two vectors and check that the output kets are only linear combinations of those two. I applied the separate parts of $H_f$ to those two vectors and found immediately that the relativistic part (the one with $\bar{p}^4$) doesn't mix the subspaces. But I'm stuck with the spin-orbit term: my guess is that I should use the Clebsch-Gordan coefficients to change to the $|n,l,j,m\rangle$ basis, but I'm not sure how to do this since I've never calculated CG coefficients. My questions are:   * Am I on the right track? If so, could somebody point out how to calculate the CG coefficients? If I'm not on the right track, what could I do?   * Also, the Darwin Hamiltonian ends up being proportional to $\delta(\bar{r})$. Is the proof that it doesn't link the aforementioned kets, as trivial as it looks? It just multiplies the kets and is thus its own eigenvalue? PD: I forgot to mention earlier that potential is Coulombian: $V(r)=-e^2/r$, which gives the delta in the Darwin term.


title: Example of space-like intervals in spacetime

text: From wikipedia: > When a space-like interval separates two events, not enough time passes > between their occurrences for there to exist a causal relationship crossing > the spatial distance between the two events at the speed of light or slower. > Generally, the events are considered not to occur in each other's future or > past. There exists a reference frame such that the two events are observed > to occur at the same time, but there is no reference frame in which the two > events can occur in the same spatial location. Can someone give me an example of this?


title: Muon 3 Body Decay

text: I'm trying to calculate the maximum energy of the electron in the decay muon >electron + electronantineutrino +muonneutrino in the reference frame of the muon having no kinetic energy. $m_m$=mass of muon $m_n$=mass of neutrino $p_n$=neutrino momentum $p_n1$=electron anti neutrino momentum $p_n2$=muon neutrino momentum $E_e$=Electron energy Using energy conservation E²before=E²after $m_m^2c^4=m_e^2c^4+2m_n^2c^4+p_e^2c²+p_n1^2c^2+p_n2^2c^2$ The mass of neutrino's is negligible $m_m^2c^4=m_e^2c^4+p_e^2c^2+p_n1^2c^2+p_n2^2c^2$ Max energy of electron occurs when the least amount of energy is given to the neutrino's, this will happen when both electrons move together in the opposite direction to the electron, so that all their momentum is used to cancel out the electron's momentum, with the neutrino's sharing the momentum. $p_n1=p_n2=p_n$ $2p_n=-p_e$ $p_e²=4p_n²$ $p_n²=p_e²/4$ $m_m^2c^4=m_e^2c^4+p_e^2c^2+p_e^2c^2/4+p_e^2c^2/4$ $m_m^2c^4=m_e^2c^4+p_e^2c^2+p_e²c²/2$ $m_m^2c^4=m_e^2c^4+3p_e^2c^2/2$ $2m_m^2c^4/3=2m_e^2c^4/3 +p_e^2c^2$ $2m_m^2c^4/3+m_e^2c^4/3=m_e^2c^4+p_e^2c^2$ $2m_m^2c^4/3+m_e^2c^4/3=E_e^2$ $m_e^2c^4/3$ is negligible. $2m_m^2c^4/3=Ee^2$ $sqrt(2)m_mc^2/sqrt(3)=Ee$ $E_e = sqrt(2/3)$ the energy of the muon, however I know this is wrong as the energy should be 0.5*energy of muon. Could someone people point out where I've gone wrong?


title: Has the spin and parity of the Higgs boson been experimentally confirmed?

text: I read in a newspaper that the Higgs boson might be the new boson but that this was not confirmed, because we don't know its properties, e.g. its spin or parity. Now I see it confirmed that it is the Higgs boson, but, really, is it true? I don't have access to the journal Physical Review D, and I can't find a clear detailed explanation of the claim or the alternative possibilities. Could someone explain it for this curious layman?


title: Relating the power to weather variables

text: Could you please help me in this problem? I have 3 independent variables, (T,H,t) , as inputs and one output P ( I have all data for these inputs and the output, done experimentally measured every hour during one year). I want to find a formula of this form: P=f(T,H,t) where t is the time in hours and it is always in the x -axis (index), T is temperature, H is humidity, and P is power. I have all the data, and when I draw them in the same graph during one year, meaning that P , T , and H vs. hours. I found that the behavior of P is oscillating, making a sinusoidal shape over the entire year: ![](http://www10.0zz0.com/2010/12/27/18/159867792.jpg) So, if I make a zoom view to this figure, for example from the 2000 th hour of the year to the 3000 th hour, it is clear that it almost has the same shape but it is oscillating. So, it keeps oscillating and increasing up to the peak point and then it starts decreasing till the end of the year. So, how can I predict the structure of the formula that relates P with T , H , and t ? What is the effect of T,H and t on P . Is there any approach that you advise me to follow? Sorry for this long question and any help from you is highly appreciated. I read many papers but I could not know how to solve the problem.


title: Complete vs General Integral of first order PDE

text: The following is an excerpt from Landau's Course on Theoretical Physics Vol.1 Mechanics: > ... we should recall the fact that every first-order partial differential > equation has a solution depending on an arbitrary function; such a solution > is called the general integral of the equation. In mechanical applications, > the general integral of the Hamilton-Jacobi equation is less important than > a complete integral, which contains as many independent arbitrary constants > as there are independent variables. Can someone clarify what's a complete integral and what's a general integral of a first order partial differential equation? Thanks.


title: Measuring the spring constant in a system with hydrostatic buoyancy

text: I have a physics lab tomorrow and I would appreciate it if you could give me your opinion on whether my reasoning is correct for this situation or not: http://i.stack.imgur.com/WMzA7.jpg The title of the lab is hydrostatic and is roughly described in the attached pic. We will take data using different weights to measure the deformation of the spring and the increase in the level of the water (yes, the fluid is water) and the thing that we have to find is the spring constant "k" but (and my professor was very clear about it) for the whole system. Now I am having some doubts, but I think that he means that we have to find some function of F that uses d (deformation of the spring) and h (increase in water height) and then we'll use least squares (done this in previous labs) to find the constant k, what I'm currently thinking is F = restitution force of the spring m = mass of the submerged body   d = deformation of the spring h = increase in water level   W= weight of the submerged body   B = buoyant force   V= volume of the displaced liquid / small body F=−k*d   B=d(water density)*V*g W=m*g Taking upwards as positive F+B−W=0 but now I have to find either a) the volume of the displaced fluid or the volume of the body, both of these are not data (or can't be found without the dimensions of the container or the small body as far as I know) at all, the Buoyant force has to be smaller than the weight since it's at equilibrium only after the spring force begins acting Is my current train of thought correct? 1) since the difference in water level changes with different masses I think it's safe to assume that the body to be submerged is not completely inside from the beginning, since if it were that way after adding more masses the total displaced volume would not increase II) Still having troubles with the F = W - B part, since I really can't figure how to calculate the Buoyant force without the volume of the object or the water displaced and I can't find the one of the water displaced since I do not know the dimensions of the container. III) the most smart course of action seems to be to find F using only B and W and then use that F with d (spring deformation) to find ''k" IV) I can't grasp what would happen if the mass were to get so big that the body would be completely submerged and "h" would stop growing but it's probably not going to happen in the laboratory I have a feeling I'm missing something really obvious and even a small push in the right direction would be nice


title: What is more efficient: Add milk, and then heat up coffee in the microwave, or microwave than milk?

text: Adding milk first increases the volume to heat and lowers average temerature, but adding it afterwards seems to have similar effects. How can you compare the two?


title: Creating the proper suction

text: I'm a bit rusty on my fluid dynamics. I'm experimenting with creating my own leaf vacuum. How would I determine how much suction a fan creates?   Bernoulli's equation would apply correct?   I would also need a fan curve right?   Does CFM matter much when creating a vacuum? Ultimately what I'm looking for is the Force @ inlet (P x Ainlet) > Force on leaves (m x g)...?


title: Confusion regarding field operators

text: Second quantisation of the scalar field leads to an algebra of quantum field operators $$ [\phi(x),\phi(y)] = 0, \ \ [\pi(x), \pi(y)] = 0, \ \ [\phi(x),\pi(y)] = i\hbar \delta(x-y). $$ Where the field operators are given by $$ \phi(x) = \int \frac{\text{d}^3 k}{(2\pi)^3 2 \omega_{\mathbf{k}}}\left( a_k e^{i k x} +a^{\dagger}_k e^{-ikx}\right) , \qquad\pi(x)=\partial_0 \phi(x) $$ These are made of two counter rotating terms. However in many body theory I have also seen field operators defined as $$ \phi(\mathbf{x}) = \frac1{\sqrt{V}} \sum_\mathbf{k} a_\mathbf{k} e^{i \mathbf{k}. \mathbf{x}} $$ I can understand why outside of the context of relativistic physics we drop the requirement for covariance, but the counter rotating term has also been dropped and so this definition has a different algebra altogether. The former definition seems like a generalisation of the $\hat{x}$-operator of a 1D harmonic oscillator, whereas the latter seems like a generalisation of the creation operator $\hat{a}$. This would be fine, and I could accept it as a choice of convention were it not for the fact that I have seen Greens functions defined using both. The greens functions as I understand them allow for the calculation of time ordered operator products by giving a general form for the amplitude of measuring different field values at different times. However this does not seem to enlighten me to why this understanding of the Greens function is conflated with the latter definition. I have clearly missed something important, but going through the relevant textbooks has failed to uncover what that is. If anyone can give an explanation of what a Green's function is and also explain how these two definitions can be reconciled I would be greatly appreciative.


title: Intuitive explanation of moments as they relate to center of mass

text: I would appreciate it if someone could give me an intuitive explanation of moments. I understand that that the center of mass could be thought of as the point which an object would balance on a fulcrum. But how do moments relate to this idea? My calculus book connects the ideas with equations and formulas, but I can't seem to get an intuition of what is actually happening. If someone could offer up a useful way of thinking of moments it would be most helpful. Thanks.


title: existence of other forces obeying inv square law

text: Is there any restriction in what we know of physics to the existence of other type of forces that obey the inverse square law in 3 dimensions. I mean other than electromagnetic and gravitational. Also if there were some other kind of force ..would it follow that it would have a dual like electric and magnetic forces are duals of each other.


title: How can my water cool down more quickly?

text: I have a cup and I can only pour hot water inside, I wanna know whether the heat will dissipate more quickly with more water or less water? How about the occasion when my cup is well covered?


title: Tensor manipulation

text: Having a bit of trouble applying what I know about tensor manipulation, given, $T^{\mu \nu} = \left( g^{\mu \nu} - \frac{p^\mu n^\nu + p^\nu n^\mu}{p \cdot n} \right)$, I need to compute quantities such as (i) $T^{\mu \nu}T_{\mu \nu}$, (ii) $T^\mu{}_\nu$ and (iii) $ T^\mu{}_\mu $. Start with (iii) $ T^\mu{}_\mu = g_{\mu\nu}T^{\mu\nu}$ I don't think this can be correct because both indices appear twice. Just by inspection I would guess that $T^\mu{}_\mu = \left( g^{\mu}{}_\mu - \frac{p^\mu n_\mu + p_\mu n^\mu}{p \cdot n} \right)$ $g^\mu{}_\mu =2$ here I summed all the diagonal terms $\frac{p^\mu n_\mu + p_\mu n^\mu}{p^{\mu} n_\mu} $= $\frac{2 p\cdot n}{p \cdot n} = 2$ $T^\mu{}_\mu = 0$? If this is the case then I get the feeling that if the indices are doing things like that, i.e., if you have repeated indices on the tensor, then you will end up with a scalar - and maybe that that scalar will be zero. (ii) $T^\mu{}_\nu = g_{\nu\sigma}T^{\mu\sigma}=g_{\nu\sigma}\left(g^{\mu\sigma} - \frac{p^\mu n^\sigma + p^\sigma n^\mu}{p\cdot n}\right)=\left(g_{\nu\sigma}g^{\mu\sigma} - \frac{p^\mu n_\nu + p_\nu n^\mu}{p \cdot n}\right)$ I think that $g_{\nu\sigma}g^{\mu\sigma} = \delta^\nu_\mu$ and $\frac{p^\mu n_\nu + p_\nu n^\mu}{p \cdot n}$ I think, is simplified as far as it can go. (i) I wanted to start with $T_{\mu\nu}T^{\mu\nu}=T^{\mu\nu}g_{\mu\sigma}g_{\nu\rho}T^{\sigma\rho}$ but I can sort of see that this will, firstly, immediately be: $\left( g^{\mu \nu} - \frac{p^\mu n^\nu + p^\nu n^\mu}{p \cdot n} \right)\left( g_{\mu \nu} - \frac{p_\mu n_\nu + p_\nu n_\mu}{p \cdot n} \right)$. And also that there are tricks involved in getting the answer quickly, but if I multiply out the terms I get $\left( g^{\mu \nu}g_{\mu \nu} - g_{\mu \nu}\frac{p^\mu n^\nu + p^\nu n^\mu}{p \cdot n} -g^{\mu \nu}\frac{p_\mu n_\nu + p_\nu n_\mu}{p \cdot n} + \frac{p^\mu n^\nu + p^\nu n^\mu}{p \cdot n}\frac{p_\mu n_\nu + p_\nu n_\mu}{p \cdot n}\right)$ Here's where my understanding breaks down, how do I evaluate $g_{\mu\nu} p^\mu n^\nu$ in terms of raising and lowering, which index do I choose? It must be a scalar, since it is a double sum. $g_{\mu\nu} p^\mu n^\nu = p \cdot n$. $\left( g^{\mu \nu}g_{\mu \nu} - g_{\mu \nu}\frac{p^\mu n^\nu + p^\nu n^\mu}{p \cdot n} -g^{\mu \nu}\frac{p_\mu n_\nu + p_\nu n_\mu}{p \cdot n} + \frac{p^\mu n^\nu + p^\nu n^\mu}{p \cdot n}\frac{p_\mu n_\nu + p_\nu n_\mu}{p \cdot n}\right)=\delta_\mu^\nu -2-2+ 4 \frac{(p\cdot n)^2}{(p \cdot n)^2} = \delta_\mu^\nu $ Which seems like it could be the right answer, but is it?


title: Force between two charged rods?

text: Suppose that we have two rods of length $l_1, l_2$ connected at one end but free to rotate. These rods have charge density $\lambda$ uniformly distributed, so the total charge of rod $i$ is $\lambda l_i$. Given an interior angle between them $\theta$, what is the force between the two rods? What is the torque? Idea for a solution: find the force on a particle with charge $d\lambda$ from a finite-length rod then integrate over the whole rod. However, it's been a while since I last took an E&M class, so I'm not sure how to do that. I'm also not sure what the torque would be; $k\sin \theta$ makes the most sense until you realize that that would also mean small torque for small angles, which is wrong. This indicates the torque must be discontinuous at $\theta = 0$. Something proportional to $\cot \frac{\theta}{2}$ is my best bet. Edit: this is not homework, though it appears to be. I just thought of it and couldn't easily solve it.


title: Ж ("zhe") in string theory?

text: I was just recently watching a TED talk about string theory, by Thad Roberts, and at around 11:10 into the video he mentions a constant for maximum spacial curvature called "zhe" (the Cyrillic symbol Ж). I had never heard of this before, and I know the use of Cyrillic characters is rare in mathematics, so I decided to Google to to find more information on it and I ended up dry. Does anyone know what the deal with this Ж constant is, and where I could learn more about it? Is this based off some new, obscure research in string theory?


title: Why doesn't the magnetic field of a coil "cancel out"?

text: In a coil, we can see that the current moves right, then left, then right, then left, and so on as it travels down the coil. ![enter image description here](http://i.stack.imgur.com/6fw9C.png) According to the right-hand grip rule, isn't the magnetic field going in one direction (downwards towards me) when the current moves left, and going in another direction (downwards away from me) when the current moves right? Why doesn't the magnetic field "cancel out" as such?


title: What does "action of a gauge group on a particle" mean?

text: I have come across this phrase > _left handed fermions transform under $SU(3)\times{}SU(2)\times{}U(1)$ > differently from the way right handed fermions do._ I am just beginning to learn about how the particles of the standard model are related to irreps of the gauge group. So, I would like to know what the phrase transforming under a gauge group exactly means, and what's the meaning of an action of a gauge group on whatever it acts on.


title: Electron indistintinguishability and handedness

text: I just learned that right ad left handed electrons behave in a remarkably different way under the weak interaction. Up till now I have been told that all the electrons are exact copies of one another and all this fermi-dirac statistics story. Nonetheless, since right handed and left handed electrons DO have fundamental differences, shouldn't this remove their indistinguishability?


title: What coefficients may be used for evaluation the heat keeping?

text: The final goal is to compare two objects (similar to boxes) from the standpoint of keeping the heat inside? May be there is numerical coefficient or any approaches?


The question is simple as stated in the title. How relative humidity and temperature affect rainfalls? I can't find a satisfatory answer in the web. Who can help me to understand me this relation?


title: Energy spectrum of a Dirac electron

text: How do you explain easily "The spectrum of an electron in a repulsive potential " and hence "bound state of charge conjugation" in Dirac hole theory ?


title: Stress-energy Trace of Massless Klein Gordon Field

text: I've calculated the trace of the stress-energy for a massless KG field and I keep getting $T = - (\partial \phi)^2$ in 3+1 dimensions. I'm using $$T_{\mu\nu} = \partial_\mu \phi \partial_\nu \phi - \frac{1}{2} g_{\mu\nu} (\partial \phi)^2$$ Since $g_{\mu\nu}g^{\mu\nu} = 4$, the second term is double the first. On the other hand, David Tong's notes suggest that massless KG is a conformal field theory and is thus has trace = 0. He shows this in 2D euclidean space, but then says "This is the key feature of a conformal ﬁeld theory in any dimension. Many theories have this feature at the classical level" What am I missing?


title: 7 sphere, is there any physical interpretation of exotic spheres?

text: Basically an exotic sphere is topologically a sphere, but doesn't look like a one. Or more accurately: > homeomorphic but not diffeomorphic to the standard Euclidean n-sphere The first exotic spheres were constructed by John Milnor (1956) in dimension n = 7 as S3-bundles over S4. It looks like an SU(2) gauge field on S4? But I don't know more about this? I guess there is lots of expertise here. Could you give a physical interpretation of this 7 sphere? Just to make it more intuitive. http://en.wikipedia.org/wiki/Exotic_sphere Thank you!


title: How thin a filter to filter particles out of salt water to make salt for human consumption?

text: I have gathered some salt in the Death Valley and for the fun of it I would like to clean it and make it edible. The process I plan to follow is simple:   1. Dissolve the salt in water.   2. Boil water for 30 min to kill bacteria.   3. Filter water to remove foreign particles.   4. Evaporate water to get clean salt. At step 3, I'm planning to buy paper filters with pores of 3 µm to filter fine particles. Is it sufficient for the purpose of human consumption? I'm mostly wondering if there is a key pore size threshold beyond which it is significantly less useful to filter, or if it's just a matter of degree, the smaller the better at an consistent size/benefit ratio.


title: What is the minimum wavelength of electromangetic radiation?

text: As a first approximation, I don't see how a wavelength of less than 2 Planck distances could exist. The question is: are there any other limits that would come into play before that? For example:   * Would the energy density cause the photon to turn into a black hole or something like that?   * Would the energy of the photon exceed the total mass+energy of the universe?


title: What is a gauge transformation of the metric in GR?

text: I'm confused by this question: Why can't General Relativity be written in terms of physical variables? I can't quite see how you can make any change to the metric without either: (a) changing the physics; or (b) changing which point in your manifold represents which physical point in spacetime. Either of these would, it seems to me, disqualify the change as being a gauge transformation, at least in the usual sense. So is there some cunning way of changing the metric that avoids either (a) or (b), or is there some sense in which (b) can still be thought of as a gauge transformation?


title: How did Enrico Fermi calculate the classical Fermi Problem?

text: From Wikipedia: Fermi was known for his ability to make good approximate calculations with little or no actual data, hence the name. One example is his estimate of the strength of the atomic bomb detonated at the Trinity test, based on the distance travelled by pieces of paper dropped from his hand during the blast. Fermi's estimate of 10 kilotons of TNT was remarkably close to the now-accepted value of around 20 kilotons, a difference of less than one order of magnitude. I have not been able to find any references explaining how he made this calculation. Please provide a reference or an example calculation. See also: http://en.wikipedia.org/wiki/Fermi_problem


title: impulse problem

text: ![enter image description here](http://i.stack.imgur.com/Vg1aG.png) The figure above shows a plot of the time-dependent force $F_x(t)$ acting on a particle in motion along the x-axis. What is the total impulse delivered to the particle?               (A) 0     (B) 1 kg⋅m/s     (C) 2 kg⋅m/s     (D) 3 kg⋅m/s     (E) 4 kg⋅m/s      > Is the solution (A)? I'm saying this because impulse is the change of > momentum is zero between $ 0$ and $2$ sec.


title: When a variation of a tensor is not a tensor?

text: In a comment about variation of metric tensor it was shown that $$\delta g_{\mu\nu}=-g_{\mu\rho}g_{\nu\,\sigma}\delta g^{\rho\,\sigma}$$ which is contrary to the usual rule of lowering indeces of a tensor. That means $\delta g^{\mu\nu}$ is not a tensor, while $g^{\mu\nu}$ is a tensor quantity, so taking a variation leads to loss of tensor nature here. How often does that happen to other tensor quantities? Are there general rules or guidelines to know if their variations are tensors or not? **Update:** Misunderstanding clarified. I should not interpret $\delta T^{\mu\nu}$ as $(\delta T)^{\mu\nu}$ and try to apply tensor rules to indices immediately. Instead, I should take it as $\delta(T^{\mu\nu})$, and if I wish to lower indices, I should write $\delta(g_{\mu\rho}T^{\rho\nu})$ and then apply Leibniz rule to the variation as $\delta(AB)=(\delta A)B+A(\delta B)$. That would generally add some terms in case $\delta g^{\mu\nu}\ne 0$.


title: Without Electroweak symmetry breaking

text: How would the present universe be different from one if there was no electroweak symmetry breaking?


title: Potential at a point

text: What is the electric dipole moment of the charge distribution with $q$ at $(0,0,1)$, $q$ at $(0,0,-1)$ and $-2q$ at $(0,0,0)$? I would think that it is $\vec{0}$ by the definition $\vec{p}=\sum\limits_i \vec{r_i}q_i$. So would it follow that the potential field due to it be $0$? Since $V=k {\vec{p}\cdot \hat{r}\over r^2}$ where $\vec{r}$ is the position vector of a point at which we wish to evaluate the potential.


title: Solving Schrödinger's equation for a specific potential

text: I am trying to solve this differential equation: $$-\chi''(\epsilon)+\Big[\epsilon^2+\frac{2F}{hw}\sqrt{\frac{h}{hw}}\epsilon \Big]\chi(\epsilon)=\mu\chi(\epsilon) \tag1$$ This was found using Schrödinger's equation, modelling a particle in potential $V(x)=\frac{1}{2}mw^2x^2+Fx$. I have changed variables and got to this point. Here I then look for solutions in the form $$\chi(\epsilon)=f(\epsilon)e^{-\epsilon^2/2}$$ I then substiution into $(1)$ giving $$f''-2\epsilon f'+f(\mu-1-\frac{2F\epsilon}{w\sqrt{hwm}})=0$$ Here I seek a power series solution for $f$. This gives a recurrence relation for the terms in the power series. This power series must terminate for the solution to make physical sense. If it terminates there exists a $n$ such that $a_n=0$. We can use this to get information about $\mu$, as this term describes the energy levels of the particles, this is the goal. However the recurrence relation I get is: $$a_{k+3}=\frac{a_{k+1}(2k+3-\mu)+\frac{2Fa_k}{w\sqrt{hwm}}}{(k+3)(k+2)}$$ If we set this equal to zero the idea is to eliminate the remaining terms in the series, as they are non-zero, to leave useful information. I can't see how to do this! any help would be greatly appreciated!


title: How is the apparent significance of (length) scales in physics explained?

text: From what I understand, especially from reading arguments on Physics.SE, different (length) scales of a system are extremely important. It's clear that if there are two scales $\delta,d,D,\Delta$ with say $$\delta < d \ll D < \Delta,$$ then effect, which happen on a length scale $d$ might be neglected if one is interested in effects of the length scale $D$. However, I don't see which these points justify arguments involving the length scales per se. As far as I can see, the existence of a quantity itself doesn't imply that physical effects are of that size. If I have a natural length scale $l$, and problems of geometries involving a length $L$, then things could end up with depending on $$l\propto\frac{1}{16^{\pi^2}}L$$ or $$2^{\text{dim}}L\gg L.$$ For example, often people argue that if some characteristic scale $H$ is close to $\hbar$ then things will get problematic. I don't see why this would be a priori justified at all. Conversely, why does a model with some small quantifity automatically have to be suppressed at one point? And if I have a small and a big scale, why are the scales in between relevant, if their value is obviously just some real value times one other values. So how is the apparent predominance in reasoning with length scales to be explained? I'm specifically thinking of field theories here, but not only.


title: Could LEP II have discovered a 125GeV Higgs?

text: LEP II eliminated the Higgs up to 114.5GeV. If it had been run for longer could it have detected a Higgs at 125GeV? I Googled for this without any luck, though I did find a comment that LEP II topped out at 209GeV collision energy, so it seems as though production of a 125GeV Higgs would have been possible. If so, how much longer would it have had to run?


title: Why does aircraft tail not project down from the fuselage?

text: Just what the title states; I was browsing photographs and it struck me how aircraft tails typically 'stand' atop the fuselage. A few aircraft (think it dates to the 40s-50s) have a tail that projects both up-and-down from the fuselage. Why do aircraft not have tail projecting downwards? Is it merely because a bad landing would render the aircraft incapable of flight until repairs were effected, or is there a scientific reason other than monetary?


title: Atmospheric pressure inside a rigid vessel

text: On a macroscopic scale one can explain the atmospheric pressure by the weight of the column of air over a given small area. If you enclose this air in rigid vessel (for example by pressing together two Magdeburger hemispheres without evacuating them), the pressure of the air inside of the vessel remains the same as outside the vessel though the air column is not present anymore. How can one explain this sticking only to the macroscopic/phenemological scale, i.e. avoiding microscopic kinetic gas theory?


title: Optimal selection of generalized coordinates in Lagrangian system

text: **EDITED** : The number of bonds is actually 2, not 1 (look at edit history). Fixed for archiving purposes. **Problem** : The edge A of an homogeneous rod (of length $\ell$ and mass $m$) is performing a smooth circular motion around the center $O$ with a constant angular speed $\omega$. The motion takes place on the $Oxy$ plane without friction under the action of an attractive force $\vec{F}=-k \vec{r}_k$ (k>0), where $\vec{r}_k$ is the vector position of the center of mass. Note: In the original image from my textbook, it appears as if $F_k$ acts towards $(x,y)=(0,1)$ and not towards the origin $(0,0)$. There is no gravity. **Goal** : To calculate the Lagrangian of the system. **Attempted solution** : The rod is a solid body moving on a 2-dimensional plane. Therefore there are $3n-k$ degrees of freedom, where $n$ is the number of the bodies (here $n=1$) and $k$ is the number of bonds (here $k=2$), since the edge A is confined to move along the circle _and_ the angular speed is constant. Subsequently, one needs **1 generalized coordinate** to describe the system. $\phi$ is angle between $(OA)$ and $Ox$ for which $|\dot{\phi}|=|\omega|$, but what would the generalized coordinate be ? ![enter image description here](http://i.stack.imgur.com/wjujc.jpg) * * * Based on the answer from @pppqqq, here is an updated diagram of the problem: ![enter image description here](http://i.stack.imgur.com/rgrrD.jpg) ~~**Pending issues** : ~~ (These have been answered by @pppqqq)   1. Are we assuming that the length of the rod is $\ell'=2\ell$ ? If not, shouldn't it be $x_{CM} = R\cos\phi + (\ell/2) \cos\theta$ ?   2. Shouldn't $y_{CM}$ be $y_{CM} = R sin\phi−\ell\sin\theta$ (assuming that the length of the rod is $2\ell$)?


title: Is the de Broglie wavelength of a photon equal to the EM wavelength of the radiation?

text: Is the de Broglie (matter) wavelength $\lambda=\frac{h}{p}$ of a photon equal to the electromagnetic wavelength of the radiation? I guess yes, but how come that photons have both a matter wave and an electromagnetic wave but other particles not and that matter and electromagnetic wavelength miraculously match?


title: How do you determine the heat transfer from a P-V diagram?

text: I doubt this question has been addressed properly before, but if there are similar answers, do direct them to me. I am currently studying the First Law of Thermodynamics, which includes the p-V diagram and of course, $\Delta U = Q_{to} + W_{on}$. My question is: how do I determine the heat supplied to the system from a given p-V diagram? I know $W_{on} = -p(V_2-V_1)$, but is there a way to find $Q_{to}$ for a given path? Thanks in advance.


title: Why don't black holes within a galaxy pull in the stars of the galaxy

text: visit http://www.nasa.gov/audience/forstudents/k-4/stories/what-is-a-black- hole-k4.html If black holes can pull even light, why cant they pull the stars in the galaxy?


title: Error in Sear's and Zemansky's University Physics with Modern Physics 13th Edition (Young and Freeman)?

text: I was reading up on the Ideal Gas Equation in University Physics with Modern Physics by Young and Freeman when I chanced upon a seemingly illogical mathematical equation. Can anyone rectify this error? Or is it misunderstanding on my part? Here is the portion (Page 600, Chapter 18, Equation 18.12): > $$pV = \frac{1}{2}Nm(v^2)_{av} = > \frac{1}{3}N\biggl[\frac{1}{2}m(v^2)_{av}\biggr]$$ It should be clear that $\frac{1}{3} \neq (\frac{1}{3} \times \frac{1}{2})$.