title: Have I discovered how to calculate the proton's mass using only integers?

text: Could it be possible that the mass of the proton can be calculated by a series of integer sequences? Or is this just a curiosity? $$\sum_{m=1}^{\infty } \frac{1}{10^{26}(m^2+1)_{2m}}=$$               NSum[1/Pochhammer[m^2+1,2m], {m,1,\[Infinity]}, WorkingPrecision -> 50] /10^26       First seven digits match the proton's mass in kilograms. $1.6726218229590580987863882056891582636342622102204\times10^{-27}$   $1.672621\times10^{-27}$ - from OEIS _revised 11/15/12_   $1.672621777\times10^{-27}$ - from Wikipedia What's to say that sometime in the future, the proton's mass won't be made more accurate by adding $4.5\times10^{-35}$ to the current number? **Edit to explain motivation** Whenever I get a result I don't recognize, I look it up on OEIS. I found this number. I posted on Mathematica.SE with the intention of asking for advice on how to prove that it converges. That would make this number a constant. If this is a "fluke" or the result of "small numbers," it's still worth exploring. **Edit: It does converge.** **Final Thoughts** $f_{p}=0.16726218229590580987863882056891582636342622102204$ is the 0-dimensional value of a fractal know as the Hilbert Curve. To get the minimal 3-dimensional value: $f \times 10^{((dimension+1)!)}$ where $0\le dimension \le 3$. This results in the value for a $1\times 1\times 1$ cube (coincidentally, the definition of the gram.) To get kilograms: $f \times 10^{((dimension+1)!+3)}$. I posit that the fractalness is the stabilizing influence on the proton. **Coda** I agree with everyone that I have been wrong-headed about the importance of this constant. I have posted the constant on OEIS A219733. Thanks for your patience.


title: Andreev reflection graphene - metallic superconductor

text: We have BdG equation, $$\left(\begin{array}{cc} \mathbf{p}\cdot\boldsymbol{\sigma} - V & \Delta_{0}e^{i\phi} \\\ \Delta_{0}e^{-i\phi} & V - \mathbf{p}\cdot\boldsymbol{\sigma} \end{array} \right) \left( \begin{array}{c}u\\\v\end{array}\right) = \mathcal{E}\left( \begin{array}{c}u\\\v\end{array}\right)$$ with definition, $$\mathbf{p}\cdot\boldsymbol{\sigma} = -i\hbar v_{F}(\partial_{x}\sigma_x + \partial_y\sigma_y)$$ We solve the equation through ansatz, $$\Phi = Const. \exp(iqy + ik_{0}x + \kappa x)$$ for $k_{0} > 0$ and $\kappa > 0$. The question is: How to use the assumption that $V >> \Delta_{0}, \mathcal{E}$ in the calculations to get nice looking formulas like in Beenakker (A13)?


title: Inner products with orthonormal bases

text: Probably a stupid question here - I think it's a case of me not having sufficient mathematical background to follow this through. In Leonard Susskind's _Theoretical Minimum_ book, he represents the ket-vector A as such: $$ |A\rangle = \displaystyle\sum_{i} a_{i}|i\rangle $$ Which I follow, as I do when he takes the inner product with $\langle j|$ and collapses $\langle j | i \rangle$ to $\delta_{ij}$. However, could somebody please explain to me how he collapses the summation $$ \langle j|A\rangle = \displaystyle\sum_{i} a_{i} \delta_{ij} $$ to $$ \langle j | A\rangle = a_{j} $$ It's probably a really stupid question, so apologies for that, but is it possible that someone on here could step me through it?


title: How can rising bubbles shrink and disappear?

text: I was recently looking at a Wurlitzer juke box, and noticed something strange. It's decorated with liquid-filled tubes. Gas bubbles are injected at the bottoms of the tubes, and the bubbles naturally rise. What was strange was that, as the bubbles rise, they shrink! Not just a little bit, but very noticeably. At first I assumed it was some kind of optical illusion, but actually many bubbles shrink down to nothing and disappear. How can this be? Where does the gas go? Surely it can't be dissolving into the liquid. It can't happen that fast, and the liquid would be saturated by now. ![Wurlitzer bubbles disappearing](http://i.stack.imgur.com/lD9AP.jpg)


title: fit funtion to the Sun electron fluxes data

text: I'd like to fit a function to the Sun's electrons flux data (blue dots), please note that x,y axis are in the log scale. The green dots are the "best" fit from the gnuplot program. I have taken the a/x as a test function, but in the log scale that is a linear function. Could anyone give me advice which test function should I take? Great thanks for any advice:) ![x - electrons energy \[MeV\], y - flux](http://i.stack.imgur.com/kYhwz.png)


title: Deriving kinetic energy in cylindrical coordinate constraints

text: Consider a mass $m$ which is constrained to move on the frictionless surface of a vertical cone $\rho = cz$ (in cyclindrical polar coordinates $\rho, \theta, z$ with $z>0$) in a uniform gravitational field $g$ vertically down. Set up Hamilton's equations using $z$ and $\theta$ as generalized coordinates. ![enter image description here](http://i.stack.imgur.com/Nik1Z.png) The question is, how does one derive the following kinetic energy? $$T = \frac{1}{2}m[\dot{\rho}^2 + (\rho\dot{\theta})^2 + \dot{z}^2]$$


title: On a principal bundle, why is the horizontal vector space not unique?

text: On a principal bundle, at each point you have a tangent vector space. At a given point, the vectors tangent to the fiber form the vertical vector space. Then the vector space at that point is a direct sum of the vertical vector space and what's called the horizontal vector space. This is all standard stuff in gauge theory. What I don't intuitively understand is why is the horizontal vector space not unique?


title: Energy density and Carnot cycle

text: For example, diesel has about 40 MJ/kg, what does it mean? Is that if we burn it we will get 40 MJ of energy per kg, but we can not get all of it to mechanical work? If that is the case, then why can't we get all of the energy to mechanic like in burning?


title: Is there stringy Morse theory?

text: This question is pretty vague and open. I'm just curious if anyone has considered this. Morse theory has a nice physical formulation: a Morse function can be thought of as a potential, so the gradient flow is the force experienced by a particle. The equilibria are the critical points of the potential. If we take a supersymmetric extension of this as a quantum theory, its ground state structure computes the homology of the space the particle moves in. What can strings moving in a manifold subject to a potential tell us about the topology of the manifold?


title: Twin Paradox: Whose time is slow?

text: What I understand about the twin paradox is that if a person stays at rest with something lets say the earth and a traveller moves with a great velocity with respect to the firsy person, then the time of traveller becomes very much smaller than the time of person who stays back, which led to the conclusion that one would age very much and one would not. But if you see from the frame of the traveller, the person on rest on earth seems to be moving very fast, and according to this the person on earth should be younger than the traveller. So, who is actually going to be younger? One of my friends had asked me this question, and I replied that from the frame of the traveller if you measure the velocity of earth bound person you will have to take the same velocity in negative and you will get the same result. Even though he accepted my answer; I am myself not satisfied with it.


title: How slow is a reversible adiabatic expansion of an ideal gas?

text: A truly reversible thermodynamic process needs to be infinitesimally displaced from equilibrium at all times and therefore takes infinite time to complete. However, if I execute the process slowly, I should be able to get close to reversibility. My question is, "What determines when something is slow?" For definiteness, let's take an insulating cylindrical piston with cross- sectional area $A$ and original length $L_0$. There is an ideal gas inside with $n$ molecules of gas with mass-per-molecule $m$. The temperature is $T_0$, and the adiabatic index is $\gamma$. I plan to expand the piston adiabatically to length $L_1$, taking a time $t$ to do so. If I take $t$ to be long enough, the process will be nearly- reversible. However, $t$ being long does not mean "one minute" or "one year". It means $t >> \tau$ for some $$\tau = f(A, L_0, L, n, m, T, k_B, \gamma)$$ What is $\tau$? From purely dimensional considerations, I guess the relationship is something like $$\tau = \sqrt{\frac{mLL_0}{k_bT}}f(n, L/L_0, A/L^2, \gamma)$$, but I don't have a strong physical explanation. **Edit** A meaningful answer should let me do the following: I take a certain example piston and try expanding it a few times, putting it in a box so I can measure the heat released to the environment. I calculate the entropy change in the universe for the expansions. After doing several expansions, each slower than the last, I finally get $\Delta S$ for the universe down to a number that I think is sufficiently small. Next, I plan to repeat the experiment, but with a new piston that has different dimensions, different initial temperature, etc. Based on my results for the previous piston, how can I figure out how long I should take to expand the new one to achieve a similar degree of reversibility on the first try? For reference, the pressure is $$P = \frac{nk_BT}{V}$$ and the speed of sound is $$v = \sqrt{\frac{\gamma k_b T}{m}}$$, and I'm happy to have an answer in terms of these or other derived quantities. Formulas for entropy and thermodynamic potentials can be found in the Wikipedia article.


title: Why is it necessary to represent Schrodinger's equation as a partial differential equation?

text: The Schrodinger equation governs the possible time evolution of a wave function, expressed as a partial differential equation. Isn't this equivalent to the simpler equation $$\omega = \hbar k^2/2m$$ i.e. any wave function that satisfies this dispersion relation will also satisfy Schrodinger's equation, and the equation above is a lot easier to understand?


title: Do celestial objects experience drag from the near vacuum of space?

text: For instance do the planets around the sun experience drag from the near vacuum of space? Or do the (hydrogen) atoms in interplanetary have a mean velocity near orbital speeds? And if so would (small) celestial objects orbiting the sun in the opposite direction (not sure if these would all me comets, or also might be asteroids) experience a significant drag? When searching for whether if this question has been asked before I also found this question in which the best answer also mentions gravitational waves. Would combining drag and gravitational waves significantly increase the dissipation of the orbital energy, compared to just gravitational waves? And would this mean that every orbit would eventually collide?


title: Where does additional heat energy come from in isothermal step of Carnot cycle

text: According to widipedia: > During step 1 (isothermal heat addition or absorption) the gas is allowed to > expand and it does work on the surroundings. The temperature of the gas does > not change during the process, and thus the expansion is isothermal. The gas > expansion is propelled by absorption of heat energy Q1. The gas temperature is the same as the temperature of the reservoir. But where may heat energy Q1 come from as there is no temperature gradient between gas and reservoir?


title: Why NPT ensemble is used for solid state phase transitions?

text: In Monte Carlo simulations of _solid state_ phase transitions, why often Isobaric Isothermal ensemble (NPT) is used ? Why not NVT ? Here, N is number of atoms, P is pressure, T is temperature and V is volume.


title: Need some basic help with notation and the Christoffel symbols

text: Apologies in advance if some of the questions below seem overly simple. In an introductory GR book, I find the following expression for the autoparallel of the affine connection (the upper bound of the summation operators should be 3, for brevity's sake I reduce it to 1; also I leave the summation operators explicit because the Einstein convention still freaks me out a little): $$\ddot{x}^{\mu} +\sum_{\rho=0}^{1}\sum_{\sigma=0}^{1} \Gamma_{\rho \sigma}^{\mu}\dot{x}^{\rho}\dot{x}^{\sigma}~=~0\qquad\qquad\qquad Eq. 1$$ Where $$\Gamma_{\rho \sigma}^{\mu}~=~\sum_{\nu=0}^1\frac{\partial x^{\mu}}{\partial x^{\nu}}\frac{\partial^2 x^{\nu}}{\partial x^{\rho} \partial x^{\sigma}} \qquad\qquad\qquad Eq. 2$$ So Eq. 1 written out more explicitly would be: $$\ddot{x}^{\mu} +\left(\frac{\partial x^\mu}{\partial x^0}\frac{\partial^2 x^{0}}{\partial x^{0}\partial x^{0}}+\frac{\partial x^{\mu}}{\partial x^{1}}\frac{\partial^2 x^{1}}{\partial x^{0}\partial x^{0}}\right)\dot{x}^{0}\dot{x}^{0} $$ $$+ \left(\frac{\partial x^\mu}{\partial x^0}\frac{\partial^2 x^{0}}{\partial x^{1}\partial x^{1}}+\frac{\partial x^{\mu}}{\partial x^{1}}\frac{\partial^2 x^{1}}{\partial x^{1}\partial x^{1}}\right)\dot{x}^{1}\dot{x}^{1}~=~0.$$ 1) Did I write this out correctly?   2) Particularly, am I right in assuming the off-diagonal terms (eg. $\frac{\partial x^\mu}{\partial x^\nu}\frac{\partial^2 x^{\nu}}{\partial x^{0}\partial x^{1}}\dot{x}^{0}\dot{x}^{1}$) are all zero, since the dimensions are orthogonal to each other?   3) Could I have saved space by writing $\partial_\nu x^\mu$ instead of $\frac{\partial x^\mu}{\partial x^\nu}$ and $\partial_{\rho \sigma}x^\nu$ instead of $\frac{\partial^2 x^{\nu}}{\partial x^{\rho}\partial x^{\sigma}}$? Then, about fifteen pages later, I find the following expression for the geodesic: $$\ddot{x}^{\mu} +\sum_{\rho=0}^{1}\sum_{\sigma=0}^{1}\Gamma_{\rho \sigma}^{\mu} \dot{x}^{\rho}\dot{x}^{\sigma}~=~\frac{d \ln{L}}{d \tau}\dot{x}^\mu \qquad\qquad\qquad Eq. 3$$ Where $$\Gamma_{\rho \sigma}^{\mu}~=~\frac{1}{2}g^{\mu \nu}\left(\frac{\partial g_{\nu \sigma}}{\partial x^\rho}+\frac{\partial g_{\nu \rho}}{\partial x^\sigma}-\frac{\partial g_{\rho \sigma}}{\partial x^\nu} \right)\qquad\qquad\qquad Eq. 4$$ ($L$ is the Lagrangian and $\tau$ is the parameter, but this is not really important for my present questions) 4) I am having trouble seeing how Eq. 2 and Eq. 4 are the same. Would appreciate it if someone could point me in the right direction. 5) Eqs. 1 and 3 are not single equations but 4 by 4 systems of equations, right?


title: Quantum Hall Effect and Edge States

text: In quantum hall effect we measure the hall conductance (in transverse direction) which is quantized. My question how do they take care of the edge states that are in the longitudinal side?


title: How does sun light after it has passed window's pane still heat me up?

text: I believe it is so because most of photons' energy has successfully passed the glass. But is it so? And how can I roughly estimate part of light's energy which will pass obstacles like glass? And how fast beam will of light will lose energy while traveling from window's pane(What is the rate of heat loss)? Thanks.


title: How exactly are Calabi-Yau compactifications done?

text: To compactify 2 open dimensions to a torus, the method of identification written down for this example as $$ (x,y) \sim (x+2\pi R,y) $$ $$ (x,y) \sim (x, y+2\pi R) $$ can be applied. What are the methods to compactify 6 open dimensions to a Calaby-Yau manifold and how exactly do these methods work?


title: Energy in nuclear reactions

text: I have two questions, been trying to get definite and intuitive answers to them for some time so hopefully you can help me: 1) I understand both the strong force and binding energy but what is the _relationship_ between the two? 2) What actually causes energy to be released when nuclei fuse or split? In my high school textbook it says changing the average binging energy between nucleons causes energy to be released... could you please make this clearer?


title: Dirt energy, possible?

text: Doesn't $E=mc^2$ mean that mass can be converted to energy? But from what I have studied in high school nuclear physics, it seems that the only "$E=mc^2$" we can get is from binding energy between nucleons. This may sound really stupid but is it possible to actually convert matter into energy using a machine on earth? Like get 1 kilogram of dirt and convert it to $c^2$ joules? What would it take to make this happen?


title: How can this formula be analyzed intuitively?

text: Given $$ \bar{U}_{vib}(T) = R \sum_{\text{n.m.}} \left( \frac{\theta_{vib, n.m.}}{2} \right) + \frac{\theta_{vib, n.m.} e^{-\theta_{vib, n.m.}/T}}{1-e^{-\theta_{vib, n.m.}/T}} $$ where $\bar{U}_{vib}$ is the molar vibrational energy of a polyatomic molecule, n.m. refers to "normal modes" $\in \\{1,2,3\\}$ and $\theta_{vib,1}=2290$, $\theta_{vib, 2} = 5160$ and $\theta_{vib, 3}=5360$, all in $K$. Which component of the sum will have the lowest value at $T=3000K$? Can one see that directly somehow by a smart analysis or will I specifically need to compute the various terms of the sum?


title: Does the Higgs boson give mass to ALL other particles?

text: The Higgs Field gave mass to other particles via spontaneous symmetry breaking; does this mean it gives mass to all particles that have mass - such as neutrinos, quarks or weak bosons and strong bosons (gluons)? In the articles I've seen its not clear what the criteria is for the Higgs Field to couple to particles to give mass; Anna v's answer suggest that electo-weak vertexes are required; but the linked article suggests it also couples to quarks.


title: Hydrostatic friction: why do water droplets stay at rest on an inclined glass surface?

text: Tjis is a non-expert question on a (seemingly simple) text-book topic. The question is about **"hydrostatic friction"** , defined as follows. Consider a drop of water resting on a flat surface. If the surface is slightly inclined, then the drop will not run off but just stay in place. > Does this phenonemon have a simple description? "Simple" as in "surface tension is simply described by a constant $\gamma$ which gives energy per unit area, $dE = \gamma \ dA$" or "Coulombic friction force is equal to normal reaction force times the coefficient of static friction $\mu_s$." **EDIT-1:** First answer revived my hope for a simple gravity + surface- tension solution. If the glass plate were horizontal, the droplet is known "chooses" energetically optimal contact area. Now the same with tilt (gravity):   1. impose no-slip condition,   2. minimize total energy with fixed contact area,   3. compare two optimal shapes with slightly different contact areas. I hope there will be a critical angle beyond which the gain of gravitational energy overcomes losses to surface tension. Need more effort to write down an solve the variational problem (in cylindrical geometry for simplicity). **EDIT-2** : Found a recent review article the relevant subject: Rev.Mod.Phys. **81** , 739 (2009); full text available on author's website. If this helps, will post an answer.


title: frames of reference

text: > **Possible Duplicate:**   >  Help Me Gain an Intuitive Understanding of Lorentz Contraction Frames k and k' are inertial frames. Frame k' is moving at a velocity of magnitude v relative to frame k in the direction of the x-axis. there are rigid rods in each frame oriented along the direction of the relative velocity. Someone in frame k measures the rod that is stationary in k as having length L1 and the rod stationary in frame k' having a length L'2. What are the lengths of the rods measured by someone in frame k'?


title: Why does a gas get hot when suddenly compressed? What is happening at the molecular level?

text: My guess is that the molecules of gas all have the same speed as before, but now there are much more collisions per unit area onto the thermometer, thus making the thermometer read a higher temperature. If this is so, then density is directly related to temperature when a substance experiences a change in density. Is this the case?


title: Is the Møller wave operators $\Omega_\pm$ related to $\lim_{t\rightarrow\infty}U(t)$ of field theory?

text: When we want to compute correlation functions $\langle\Omega|\,T\hat{\phi}(x_1)\ldots|\Omega\rangle$ in an interacting quantum field theory, we relate it to the free-field objects $|0\rangle$ and $\hat\phi_I(x)$ using the interaction-picture time-evolution operator in the limit $T\rightarrow\infty$. Eventually, we arrive at an expression like (see Peskin and Schroeder eqn. 4.30) $$\langle\Omega|\,T\hat{\phi}(x)\hat{\phi}(y)|\Omega\rangle=\lim_{T\rightarrow\infty}\mathcal{N}\langle 0|U(T,t_x)\phi_I(x)U(t_x,t_y)\phi_I(y)U(t_y,-T)|0\rangle.$$ The $U(T,t_x)$ and $U(t_y,-T)$ sitting at the end of the correlator look very much like the Møller wave operators of non-relativistic scattering theory $$\Omega_\pm=\lim_{t\rightarrow\mp\infty}U(t)_\text{full}U_0(t),$$ that relate the in-asymptote and out-asymptote states to the actual state at $t=0$. So my question is, are these two like the same thing, with the same properties? _i.e._ they are isometric, etc...


Is there a matching material interior for the Kerr solution of Einstein's equations? I can only find informal and conflicting information about this. Some time ago, I've heard that it was expected to be a "ring of matter". A few days ago, a friend told me that "it is not expected to exist one" but he couldn't say why. Was this subject already settled?


title: Why does adjoint representation matter in some field theories?

text: Recently I am reading a paper about monopoles. In several cases, it seems that writing fields in adjoint representation of the gauge group makes a difference. Once it leads to different group after symmetry breaking when using other representation. And I also noticed statement like this, "An important open question is whether an analogous Bogomolny monopole's mass bound can be obtained if the Higgs field is not in the adjoint representation." Can anyone kindly shed light on this. Thanks! **Update:** I reckon any field (either EM field in real space or Higgs field in internal isotopic space) be in a certain type of representation space of the symmetry group associated with the Lagrangian or action. This space also dictates some constraints on the fields, e.g., specific tensor or spinor structures ( _anything more???_ ). And what representation space you use contains physics as well, that is to say, we have to check it by experiments. Perhaps this question addresses on a particular case. Either does the explicit and concrete 2nd answer. Is this understanding correct?


title: Some doubts about photons

text: **I am reading Berkeley Physics Course vol. 4 (Quantum Mechanics) , chapter 4 (photons).** (1) **Section 46** : book says: consider a typical photon emitted by the source. It can be regarded as a a wave train of finite duration, spreading out in all directions in space and carrying a total energy $\hbar\omega$. If this is true, a photon emitted by a star which is few light years away, would have spread over a sphere of radius of few light years. Then it is not possible to "detect" that photon using a detector. > Then why does the book say like this? In **section 47** , the book gives same example of photon spreading over light years. But in this section, it says that how the people can use this as a paradox; and clarifies that paradox by saying that classical expression for "energy density" refers to a large number of photons, and not for single photon. > So does photon really spread in all directions? * * * (2) In **Section 38** book says: almost monochromatic photons can not be split into 2 photons of the same frequency which carry only a fraction of the energy of the original photon. While in **section 48** , it says (with reference to double-slit diffraction experiment) photon came through BOTH the slits. > Does a photon REALLY goes through both the slits? > > Doesn't it indicate that a single photon can be split into two photons of > same frequency ? * * * Any help will be appreciated. Thanks


title: Unit for expressing energy eigenvalue in infinite potential well problem

text: How are the energy eigenvalues expressed in a infinite potential well problem( Joules/eV)?


title: Spin orbit coupling and fine structure of the Hydrogen atom

text: At university we have just derived the expression for the energy split due to spin-orbit coupling in the Hydrogen atom, i.e. what is known as the FINE structure of Hydrogen. To do this, we considered the situation from the frame of reference of the electron: in this frame, the proton is moving and therefore generates a magnetic field ( because it's a moving charged particle) which couples to the electron's spin. FINE. What if I wanted to consider the situation in the nucleus' (proton's) frame of reference? The moving charged particle is the electron and it is generating a magnetic field (which as it happens couples to the spin of the nucleus and results in the hyperfine structure). But the magnetic field of the electron cannot couple to its own spin? Or can it? How do I explain the energy spit in this reference frame? Physics must be the same in all frames of reference right? Thanks


title: Physical meaning of uncertainty principle + question on symmetry

text: A question on the Uncertainty principle. So we know that it says (for position and momentum) that: $$ \Delta x \Delta p \ge \hbar/2 $$. Where $\Delta p = \sqrt {<p^2> \- <p>^2 }$ and $\Delta x = \sqrt {<x^2> \- <x>^2} $; NOW, Potential with a symmetry. Imagine a square well centred at 0, so extending from -A to A. The potential is symmetric so (I would expect) the particle is as likely to be in the LHS $(-A<x<0)$ as in the RHS $(0<x<A)$; This suggests that both $<x>$ and $<p>$ are 0. In that case $\Delta x = \sqrt {x^2}$ and $\Delta p = \sqrt{p^2}$; So effectively the spread is not just a measure of the uncertainty but it is actually telling us the root mean square of both momentum and position. SO my question is: if I confine a particle in a small volume (assuming a symmetric potential such as a square well so as for the argument above to be applicable), it will have a large momentum and therefore a large kinetic energy, whereas if the square well were larger, its kinetic energy would be smaller. Where does this extra energy come from? How can we understand it physically? (If I were to manually change the width the potential well then I could say that the surplus in energy comes from the work that I've done on the system, but here I'm just considering two separate particles, in different well).


title: Why are only linear representations of the Lorentz group considered as fundamental quantum fields?

text: As described in many Q&As around here, fundamental quantum fields are expressed as irreducible representations of the Lorentz group. This argument is entirely clear - we live in a Lorentz-invariant world and those elements of an observed system that do not mix with others any way we transform the system, either actively or just by looking at it from a different viewpoint, are the only candidates for separate physical entities such as quantum particles/fields. The question however is, why do we consider only _linear_ representations of the group? _Peskin & Schroeder_ mention that every non-linear transformation law can be built from linear ones and there is thus "no advantage" in considering non- linear transformations. Nevertheless, they give no reference. Even if this is true, decomposition of the transformation doesn't seem as a counterargument as long as it is irreducible. There might be considerable issues with canonical quantization of non-linear fields, but if we can construct scalars from them (and thus a Lagrangian), the path integral formulation stands. Another argument appeals to the principle of superposition derived from quantum mechanics. However, once interaction comes into game, we have non- linear field equations violating the principle of superposition anyways. Nonetheless, interacting linear and non-linear "point-to-point" representations of the fields in both cases obey the principle of superposition for their _entire quantum states_. So once again, the non-linear representation does not pose a fundamental problem. The last possible argument conceivable by me, the need of the ability to build a perturbation theory, is more of a technical request than a fundamental restriction (not so far from the request of renormalizable interaction terms, though). So, is there a conclusive principle restricting non-linearity of the representations or is it just one of those "it works so nobody cares" physics' moments? * * * EDIT 1: By a "representation" I obviously mean a "realisation" of the Lorentz group, since a representation in the old-fashioned sense is strictly a linear one. We could understand this as a functor from the space of (velocity) vector fields (on which the first "realisation" of the Lorentz group whas formulated) to a space of non-vector (non-linear) fields. A more down-to-earth formulation is the following: Consider a field colection $\phi_a$ and a Lorentz transform $\Lambda$. Then the field transforms as $\phi_a'(x)=M^\Lambda_a(\phi_b(\Lambda^{-1}x))$, with generally ($\alpha \in \mathbb{C}$) $\alpha \phi_a'(x) \neq M^\Lambda_a(\alpha \phi_b(\Lambda^{-1}x))$ and all the other stuff such as addition also generally violated. * * * EDIT 2: To get a taste of how such a theory would be quantized and where the problems may lay, I am also going to comment on the transformation of the quantized field operators. Consider we have field collection $\phi_a$ transforming as specified above. Now we also assume it's transformation to be analytical and with the use of multi-indices(in bold font) the transformation can be written as $$\phi'_a = \sum_{\mathbf b} m_a^{\Lambda, \mathbf b} \phi^{\mathbf b}.$$ I.e. $\phi^{\mathbf b} = \phi_1^{b_1}\phi_2^{b_2}...$ The powers of the field are no problem in quantum mechanics as these are after quantisation multiple applications of the same field operator. I.e. after quantisation for the collection of field operators $\Phi_a$ it must hold that $$U^\dagger(\Lambda) \Phi_a U(\Lambda) = \sum_{\mathbf b} m_a^{\Lambda, \mathbf b} \Phi^{\mathbf b}, \;\;\;\; (*)$$ where $U(\Lambda)$ is the Lorentz transformation of the quantum state of the field $|\Xi'\rangle = U(\Lambda) |\Xi\rangle$. You can for example see that an addition of two operators of this kind _does not_ transform by the given prescription, but this is generally true also in "normal" quantum theories. Consider for example $\hat{X}$ which transforms as $\hat{X} + a$ under a translation by $a$. However, $\hat{Y} = \hat{X} + \hat{X}$ transforms as $\hat{Y} + 2a$. * * * Basically, the only way I see the disqualification of non-linearly transforming representations happening is if their components could be identified as combinations of components of linearly transforming fields to certain powers. For example, four fields $\phi^\mu$ could be actually identified to transform as three components of a vector field, but squared: $$V'^\mu = \Lambda^\mu_\nu V^\nu, \; \phi^\mu = (V^\mu)^2$$ But is this generally possible and how?


title: Work and its nature?

text: Suppose we have a constant force $\overrightarrow{F}$ and in the direction of force we have 2 point $x_1$ and $x_2$ in that order. And we have to calculate work done on moving frok $x_1$ to $x_2$ and $x_2$ to $x_1$. Why and where is the following going wrong ?   1. Moving from $x_1$ to $x_2$  > $$W = \int_{x_1}^{x_2} \overrightarrow{F} \cdot \overrightarrow{dx}$$ $$W = > F [dx]_{x_1}^{x_2}$$ $$W = F (x_2 - x_1)$$   2. Moving from $x_2$ to $x_1$  > $$W = \int_{x_2}^{x_1} \overrightarrow{F} \cdot (-\overrightarrow{dx})$$ $-$ > sign because $dx$ is in opposite direction as that of $F$   >  $$W = \int_{x_2}^{x_1} -F dx$$ $$W = -F [dx]_{x_2}^{x_1}$$ $$W = -F (x_1 > -x_2)$$ $$W = F (x_2 - x_1)$$ Why is the work done in both cases coming out same?


title: Asymtotically flat spacetime applicable for spacetimes which are not diffeomorphic to $\mathbb{R}^4$

text: I wanted to investigate changes on a compact 4-manifold $M$. More specifically it is the K3-surface. I follow a paper by Asselmeyer-Maluga from 2012. The idea there was to make sure that the manifold is Ricci-flat first, then Asselmyer-Maluga applies a mathematical technique that changes the differential structure and cannot be Ricci-flat anymore. Topologically, though, nothing changes. The technique is based on replacing a suitably embedded torus by a complement of a knot. Asselmeyer-Maluga moves on to relate the used knots to possible particles. My idea was to make the change of spacetime and its physical relevance more clear by showing that some suitable definition of mass was $0$ first, but cannot be afterwards. To make it possible to investigate a more physical spacetime, i.e. to introduce a Lorentzian metric that does not contain closed time-like curves, Asselmeyer-Maluga removed a fourball $B^4$. Now I am new to the definitions of mass in General Relativity and do not really know what is possible to apply and calculate when. Are all the masses like ADM and Bondi that need asymtotical flatness impossible to evaluate as they need "infinite space"? Or is this exactly the typical setting of "manifold minus ball" that I have seen in the coordinate free description of asymtotical flatness? My feeling so far is that it is necessary to use semi-local concepts of mass like the one of Brown-York. Kind regards


title: Rotating hoop with fixed body inside of same mass

text: Please help me to solve this problem. I am unable to understand which force will cause the hoop to bounce. > A small body $A$ is fixed to the inside of a thin rigid hoop of radius $R$ > and mass equal to that of the body $A$. The hoop rolls without slipping over > a horizontal plane; at the moments when the body $A$ gets into the lower > position, the center of the hoop moves with velocity $v_0$. At what values > of $v_0$ will the hoop move without bouncing? ![enter image description here](http://i.stack.imgur.com/o5S2m.gif)


title: Hookes Law and Objective Stress Rates

text: Often, in papers presenting updated Lagrangian simulation methods for solid dynamics, the following procedure for updating the (Cauchy) stress tensor is presented: First, the Cauchy stress tensor is split into a hydrostatic and a deviatoric part: $$\sigma^{ij} = -p\delta^{ij} + S^{ij}$$ The pressure is found using an equation of state. Often the following isothermal approach is used: $$p = c_0^2(\rho - \rho_0)$$ $c_0^2$ being the adiabatic sound speed, $\rho$ the density and $\rho_0$ the reference density. Then, it is stated that Hooke's law is assumed and the deviatoric part of the stress tensor evolves as follows: $$\frac{dS^{ij}}{dt} = 2\mu(\dot{\epsilon}^{ij} - \frac{1}{3}\delta^{ij}\dot{\epsilon}^{kk}) + S^{ij}\Omega^{jk} + \Omega^{ik}S^{kj}$$ where $\mu$ is the shear modulus, $$\Omega^{ij} = \frac{1}{2} (\frac{\delta v^i}{\delta x^j} - \frac{\delta v^j}{\delta x^i})$$ is the spin tensor and $$\dot{\epsilon}^{ij} = \frac{1}{2} (\frac{\delta v^i}{\delta x^j} + \frac{\delta v^j}{\delta x^i})$$ is the rate of deformation tensor. Now, since: $$\dot{\sigma} = \overset{\Delta J}{\sigma} + \sigma\cdot\Omega + \Omega\cdot\sigma$$ where $\overset{\Delta J}{\sigma}$ is the Jaumann rate it holds that: $$ \overset{\Delta J}{\sigma}_{ij} = 2\mu(\dot{\epsilon}^{ij} - \frac{1}{3}\delta^{ij}\dot{\epsilon}^{kk})$$ Now on to my questions:   1. How does one come up with the above equation for the Jaumman Rate? Or, particularly, how does the assumption of Hooke's law yield that equation for the Jaumann rate?   2. Is that equation for the Jaumann rate also valid for other objective stress rates? For example for the Truesdell rate, giving a stress update as follows: $$\frac{dS}{dt} = 2\mu(\dot{\epsilon} - \frac{1}{3}\mathbf{1}{\rm Tr}(\dot{\epsilon})) - {\rm Tr}(\dot{\epsilon})S + \dot{\epsilon}\cdot S + S\dot{\epsilon}^T$$ ($\rm Tr(.)$ being the trace)


title: Why do i see bright red hand when i place flashlight behind my hand?

text: When I place a working flashlight behind my hand, I see my hand bright red because of the light. So my question is, why do I only see red light after transmission even if incoming light from the flashlight is white? Can we explain these phenomenon mathematically? Do the coefficients of absorption and transmission come in picture? If so, can you explain how? And can we measure them? Or going in more details can we explain it by using photon theory?


title: Does a black hole have any kind of mass?

text: Currently in my academics I am studying about the Gravitation. In the chapter I came across a term called the **Escape Velocity** ( _It's the velocity of any celestial body which is required by an object to escape from body's gravitational field without any further propulsion_ ). When I was going through the chapter I came to know that the escape velocity of black holes is greater than $c$ and that's why even light can't escape from it's gravitational field. So proceeding toward my question, From the information I know, > $$v_{es}=\sqrt\frac{2GM}{R}=\sqrt{2gR}$$ where $v_{es}$ is the escape > velocity, $G$ is the universal gravitation constant, $g$ is the acceleration > due to gravityof the celestial body, $M$ is the mass of the celestial body > and $R$ is the distance between the object and center of gravitation of > celestial body. So my question is that if Black Hole is having that enormous value of escape velocity they must be either having exceptionally large value value of $M$ (i.e their Mass) or they must be having very small value of $R$, which I myself didn't know how can it be defined for a black hole. >   * So does black holes have enormous mass which result in very large value > of $v_{es}$ > >   * Also I want to how $R$ can affect the value of $v_{es}$ in case of black > holes? > >


title: Can internet speed be faster than speed of light?

text: We send email in order to transfer information and, over time, the technology used to send email improves to send email at faster rate. Since we use the Internet to send information from one place to another place all over the world, can information speed be faster than light speed via the Internet?


title: Brownian motion and physical meaning

text: I have read Stochastic Differential Equations by Bernt Oksendal   It constructs Brownian motion by Kolmogorov extension theorem by consider $p(t,x,y)=(2\pi t)^{-n/2} e^{- \frac{|x-y|^{2}}{2t}}$   But I can't understand what is the relation to the Brownian motion in physics.


title: why does alternate current attract and direct current repel

text: Does AC current attract and DC repel our body? If the potential difference remains same will still AC and DC will give us same type of electric shock?


title: How to homegrow large, momocrystalline water ice crystals

text: This is a follow up to this question: **Can one get clear ice crystals from a dirty suspension?**. How could one grow a large - meaning visible with the naked eye - water ice crstal with common household tools and substances or little exotic/rare equipment.


title: how datas are physically stored in and read from hard drives?

text: well i think the question is clear enough but I will describe more : how datas are physically stored in and read from hard drives ? are they related with spin of atoms in the drive ? Is that method same in flash drives and SSDs? Thanks


title: paradoxical results in different applications of the 2nd law of thermodynamics

text: Assume two closed systems adjacent to each other together forming one adiabatic system. Both systems are assumed to have their volumes fixed and can therefore communicate with each other through heat transfer only. Now let study this simple example with three different formulation:   1. The first formulation is what I had always thought to be obvious, when I thought I know Classic Thermodynamics! Assume $\delta Q$ is the heat flow from system $2$ into system $1$: \begin{align} &\Delta S=\Delta S_1+\Delta S_2\,,\qquad \Delta S_1\ge\int_1^2\frac{\delta Q}{T_1}\quad\text{&}\quad \Delta S_2\ge\int_1^2\frac{-\delta Q}{T_2}\\\ &\Rightarrow\;\Delta S\ge\int_1^2\delta Q\,\Bigl(\frac{1}{T_1}-\frac{1}{T_2}\Bigr) \end{align} Now again the Clausius statement of the second law says if $\delta Q\ge0$ then $T_2\ge T_1$ and if $\delta Q\le0$ then $T_2\le T_1$. Therefore, eitherway we would have $$\Delta S\ge\int_1^2\delta Q\,\Bigl(\frac{1}{T_1}-\frac{1}{T_2}\Bigr)\ge0$$   2. But no matter if the process is reversible or not, only assuming that the system has no work ( _volume is fixed, but also assume there is no friction or gravity work and etc._ ) we would obtain: \begin{align} &\delta Q-\delta W=dU=T dS - p dV\quad\Rightarrow\quad\delta Q=T dS\\\ &\Rightarrow\quad \Delta S_1=\int_1^2\frac{\delta Q}{T_1}\;\;\text{&}\;\; \Delta S_2=\int_1^2\frac{-\delta Q}{T_2}\\\ &\Rightarrow\quad \Delta S=\Delta S_1+\Delta S_2=\int_1^2\delta Q\,\Bigl(\frac{1}{T_1}-\frac{1}{T_2}\Bigr)\ge0 \end{align} The problem is that we have derived in the first line $\delta Q=T dS$ which is not correct in general and this suggests that even if the volume is fixed but yet another work should exist, but if the containers contain only solid material what work should be considered to resolve this paradoxical result?   3. In the Emanuel's ``Advanced Classical Thermodynamics" it has been suggested to write the entropy change in a closed system as: $$dS=dS_{int}+\frac{\delta Q}{T_{surr}}$$ wherein, _int_ and _surr_ are the abbreviations for internal and the surrounding, respectively. Assuming the irreversibility caused by the flow of heat between a finite temperature difference $T_{surr}-T$ to be considered inside $dS_{int}$ then it concludes the second law of thermodynamics as: $$dS_{int}\ge0$$ Now returning back to our two-system problem we would have: \begin{align} &dS=dS_{int_{total}}+0\quad\text{&}\quad dS_1=dS_{int_1}+\frac{\delta Q}{T_2}\quad\text{&}\quad dS_2=dS_{int_2}-\frac{\delta Q}{T_1}\\\ &dS=dS_1+dS_2\quad\Rightarrow\quad dS_{int_{total}}=dS_{int_1}+dS_{int_2}-\delta Q\,\Bigl(\frac{1}{T_1}-\frac{1}{T_2}\Bigr) \end{align} The last line requires $dS_{int_{total}}\le dS_{int_1}+dS_{int_2}$ which is not correct obviously and it should have been only "equal" instead of "lower or equal". As is already clear, the second formulation of the problem should have a mistake but I just cannot find it, but the third formulation is rather more serious problem, to me either assuming the irreversibility of heat flow between finite temperature differences cannot be removed from it into $dS_{int}$ or if it is done then only one temperature should be used in the formulation, but then the problem rises that which one, $T_1$ or $T_2$? If it is arbitrary and I can write both $dS\ge\frac{\delta Q}{T_{surr}}$ and $dS\ge\frac{\delta Q}{T}$ then the first formulation above would run into problem! Thanks


title: Future of a universe with scalar field

text: > _Assume that $\phi=\phi_0\gg M_{Pl}$, what can you say about the future of a > universe in a model with $V(\phi)=\frac{1}{2}m^2\phi^2+\epsilon \phi$, where > $|\epsilon|\ll m^2M_{Pl}^2$, is a small parameter and $m$ is such that > $V(\phi_0)\approx \rho_c$._ My method is: I used Klein Gordon Equation \begin{equation}\ddot{\phi}+3H\dot{\phi}+V'=0,\end{equation} where $V'=m^2\phi+\epsilon$. After this I can solve for $\phi$ by using the slow roll conditions. But, what am I supposed to do afterwards? What does the future of universe mean? Am I supposed to show whether the universe will expand or contract?


title: Why is the Big Bang the biggest explosion in the universe?

text: 1. What makes the big bang the biggest explosion in the universe?    2. Why even should such an explosion ever happens?   3. Through what process and how does this amount of energy get released as a big bang?   4. Are there there behind the darkness of universe more big bangs?   5. If there are how could we ever know (since the speed of light is constant)?


title: Examples of manifolds (not) being: flat, homogeneous and isotropic

text: I am looking for (at least) one example of the following manifolds:   1. Flat, homogeneous and isotropic   2. Curved, homogeneous and isotropic   3. Flat, non-homogeneous and isotropic   4. Flat, homogeneous and non-isotropic   5. Curved, non-homogeneous and isotropic   6. Curved, homogeneous and non-isotropic   7. Flat, non-homogeneous and non-isotropic   8. Curved, non-homogeneous and non-isotropic By isotropic, I mean isotropic at least about some specific point. This is what comes to (my) mind:   1. Euclidean space   2. (2-)Sphere   3. Euclidean space with one point removed / Vector space   4. Minkowski space   5. Sphere with one point removed?   6. ?   7. Euclidean space with two point removed?   8. A generic manifold I hope you can provide some better examples.


title: The curl of a special cross product

text: When given two vectors $\mathbf{A}$ and $\mathbf{B}$, the curl of the cross product of these two is given by $$\nabla\times(\mathbf{A}\times\mathbf{B})=(\mathbf{B}\cdot\nabla)\mathbf{A}-\mathbf{B}(\nabla\cdot\mathbf{A})-(\mathbf{A}\cdot\nabla)\mathbf{B}+\mathbf{A}(\nabla\cdot\mathbf{B}).$$ Using this relation, we can write $$\nabla\times(\boldsymbol{\mu}_I\times\mathbf{r})=(\mathbf{r}\cdot\nabla)\boldsymbol{\mu}_I-\mathbf{r}(\nabla\cdot\boldsymbol{\mu}_I)-(\boldsymbol{\mu}_I\cdot\nabla)\mathbf{r}+\boldsymbol{\mu}_I(\nabla\cdot\mathbf{r}).$$ In a lecture course that I'm reading, it is stated that this can actually be rewritten as $$\nabla\times(\boldsymbol{\mu}_I\times\mathbf{r})=-(\boldsymbol{\mu}_I\cdot\nabla)\mathbf{r}+\boldsymbol{\mu}_I(\nabla\cdot\mathbf{r}),$$ which means that the first two terms on the right hand side cancel. These can be written out, which results in $$\begin{array}{l@{\;}l} (\mathbf{r}\cdot\nabla)\boldsymbol{\mu}_I-\mathbf{r}(\nabla\cdot\boldsymbol{\mu}_I)&=\left(y\partial_y\mu_x-x\partial_y\mu_y+z\partial_z\mu_x-x\partial_z\mu_z\right)\boldsymbol{\hat{\textbf{i}}}\\\ &+\left(x\partial_x\mu_y-y\partial_x\mu_x+z\partial_z\mu_y-y\partial_z\mu_z\right)\boldsymbol{\hat{\textbf{j}}}\\\ &+\left(x\partial_x\mu_z+y\partial_y\mu_z-z\partial_x\mu_x-z\partial_y\mu_y\right)\boldsymbol{\hat{\textbf{k}}}. \end{array}$$ (Sorry for the ugly ihat, SE doesn't support \imath for some reason.) In order for this to be zero, each term should be zero. Taking for example the $\boldsymbol{\hat{\textbf{i}}}$ component, this means that $$y\partial_y\mu_x-x\partial_y\mu_y+z\partial_z\mu_x-x\partial_z\mu_z=0.$$ In this case, $\boldsymbol{\mu}_I$ is proportional to an angular momentum operator. Is this a special case, or is the above equation always true? And if so, why?


title: Since cables carry electricity moving at the speed of light, why aren't computer networks much faster?

text: Why can't cables used for computer networking transfer data really fast, say at the speed of light? I ask this because electricity travels at the speed of light. Take Ethernet cables for example, I looked them up on wikipedia.               Propagation speed   0.64        c      Why only 64% What does propagation speed mean? I know there are other variables affecting the latency and perceived speed of computer network connections, but surely this is a bottle neck. **In other words, I'm asking, what is it about a fiber-optics cable that makes it faster than an Ethernet cable?**


title: Resonance in a 1 ft granite box

text: * I have a granite cube made using 6 slabs of granite 1 foot square and 1 inch thick. The top and bottom slabs have a 1 inch margin around the edge. The slabs are just set together, not notched or mortared or anything.   * I also have a concrete block with pipes running through it that generates a sonic pulse/compression wave from a valve closing. If this were a regular water system we'd all call it water hammer. This water hammer comes _out the top of the block_. It is assumed that this block is not going to self destruct from the water hammer. It pulses about 60 times per minute. There is a working example of this block that I have seen online, so I know how to build it. Now what I'm trying to figure out is what would happen if I would set my granite cube on top of the concrete block. Would the sound reverberate around in the granite box until it blew it apart? Or would it somehow go through the block somewhere else? What would happen? Would it resonate at a certain frequency and produce a hum? Would it make a difference if I took away the bottom slab?


title: Crushing a magnetic field

text: What would happen if you crushed a magnetic field to an ever decreasing size? Thanks. EDIT: How small could the field possibly go? Is there a limit on how small it could get? Is there a maximum field density based on the size?


title: A comprehensive reference for RF linear accelerator's operation

text: I'm doing a research on RF linear accelerators (RF Linac), but while studying the material I encountered many problems. I cannot understand the basics of the RF linear accelerators in many ways, for instance   * Why the particles are accelerated in the gaps and not in the drift tubes?    * How they become in phase with the bunches of particles?    * What does bunching particles mean?  Etc. I am reading some books but none of them has explained the principles of working in detail or with some illustrations and comprehensive figures.   * So where can I find a comprehensive resource describing the structure and operation of Rf linacs?


title: Galilean transformation of wave equation

text: I have this general wave equation: \begin{equation} \dfrac{\partial^2 \psi}{\partial x^2}+\dfrac{\partial^2 \psi}{\partial y^2}-\dfrac{1}{c^2}\dfrac{\partial^2 \psi}{\partial t^2}=0 \end{equation} And the following transformation : $t'=t$ ; $x'=x-Vt$ and $y'=y$ The solution to this has to be : $$\dfrac{\partial^2 \psi}{\partial x'^2}\left( 1-\frac{V^2}{c^2}\right)+\dfrac{\partial^2 \psi}{\partial y'^2}-\dfrac{2V}{c^2}\dfrac{\partial^2 \psi}{\partial x' \partial t'^2}-\dfrac{1}{c^2}\dfrac{\partial^2 \psi}{\partial t^{'2}}=0$$ This proves that the velocity of the wave depends on the direction you are looking at. I don't know how to get to this? If you just substitute it in the equation you get $x'+Vt$ in the partial derivative. Is there another way to do this, or which rule do I have to use to solve it? I was thinking about the chain rule or something, but how do I apply it on partial derivatives?


title: Why is general relativity only formulated in continuum terms?

text: So, when we are discussing Newtonian mechanics, we treat particles as point particles. In continuum mechanics, which I understand to be a version in which mass is continuously distributed, we have equivalent formulations. In Special Relativity, we again formulate everything discretely (a particle has its worldline, its four-velocity, etc.). But, we also have the Stress- Energy tensor, and we can use it to formulate some conservation rules (e.g. ${T^{\mu\nu}}_{,\nu}=0$). However, in General Relativity, the Einstein equation is formulated in terms of a continuum, while the geodesic equation deals with a point particle. Is there a point-particle version of the Einstein equation? Is there a continuum version of the geodesic equation? If so, why not?


title: How to calculate magnetic pole strength?

text: I have 2 magnets. I need to know the force between them. In a previous Phys.SE question, conclusion was: we need to use a dipole-dipole interaction equation, which included m, which the magnetic dipole. I searched how to calculate it, and this is what I found: m=pl where p is magnetic pole strength and l is length of the magnet. I searched a lot on how to calculate p and found nothing.


title: What exactly is a coherent state and why is it interesting?

text: Please note that I do not have a background in physics, so if possible please refrain from a bunch of $ |x\rangle $ notations, unless clearly specifying what it symbolically means. So I have been learning about representation theory lately in particular I have studied square integrable irreducible representations, and I'm interested in the applications of these. I have come to understand that given a square integrable irreducible representation $ U $ of a locally compact group $ G $ on a Hilbert space $ \mathcal{H} $ and an admissible vector $ g \in \mathcal{H} $, then the orbit $ \mathscr{O}_g := \\{U(x)g\mid x\in G\\} $ is a coherent state. Furthermore if the group $ G $ is the (Weyl-)Heisenberg-group, then these coherent states are "classical coherent states"(?). So I understand from this that coherent states can be described by these collections of vectors/functions in a Hilbert space and sometimes they constitute frames and possible wavelets(?). How exactly is such a collection of vectors understood in the context of coherent states? What does a coherent state describe? and why are they interesting? If you can refer me to articles or literature explaining these questions in terms understandable by someone who has mostly had basic mechanics then it'd be much appreciated.


title: Movement of a gyroscope with non-fixed axis

text: Assume one has a gyroscope rotating around an axis with both ends leaning on a dedicated semiplane as shown on the picture below. There is no friction either between the rotor and the axis or between the axis and the semiplanes. The only force is the gravity acting on the rotor. How will the gyroscope behave if one instantly removes one of the semiplanes? ![enter image description here](http://i.stack.imgur.com/pn3ZO.png)


title: Understanding the Wave Function and Excited States

text: A wave function is an infinite dimensional vector space, how can it "live" in $\mathbb{R}^3$? Given the equation that is built like: $$\Psi (x,t) = \sum ^{\infty} _{n=1} c_n \psi _n (x) e^{-i E_n t / \hbar}$$ How does one "excite" a quantum particle in the lab? The excited states simply give a different probability distribution for the particle? For the infinite square well, the wave function in excited state $n$ has exactly $n$ bases, all of which are non zero? If a particle is in its ground state, does that mean that all other bases must be zero? making the wave function dependent only on one basis?


title: Meteorite hunters Search for FIREBALLS

text: normally meteorite hunters search for fireballs....but they totally ignore meteor showers . why so? as far as i can think meteors in meteor showers are usually produced by small particles so they burn up quickly in the upper atmosphere about 30-100 km in . but fireballs are caused by larger objects which may not completely burn up in the atmosphere.hence making fireballs a reasonable starting point to search any other reason?


title: Future Computer Performance

text: Moore's law has succesfully predicted up to now that integrated circuit transister density doubles every two years. However, computer performance is dependent on additional factors like architecture, chip design and software. What physics-related factors will enhance or limit ultimate performance of future computers?


title: What happens when a battery is in space?

text: What would happen If you take a battery in a space ? Would the current flow between terminals being it in vacuum of space?


title: What are the alternative theories of dark energy? ($w \neq -1$)

text: There has been a lot of related questions about dark energy around here but these are usually 2-4 years old and the closest question to mine hasn't really been answered, so I am going to proceed. Experts may skip to the last paragraph, but I am going to give a short account for others. * * * **Standard Cosmology run-through** As I understand the matter, assuming Einstein equations and an almost homogeneous and isotropic universe, the current standard theory of cosmology states from multiple kinds of observations (the cosmic microwave background, supernovae, cosmic shear...) that there have to be missing terms on the right hand side of the so-called first Friedmann equation $$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3} (\rho_{radiation} + \rho_{matter} + ???) $$ This equation basically states: the rate of expansion of the universe is proportional to density of matter and radiation ($\rho_{matter},\,\rho_{radiation}$). Even if we account for matter we detect only indirectly (i.e. _dark matter_ ), there still is another $\rho$ which gives nowadays as much as _three times a bigger contribution_ than all the other densities added up. We call this density usually "dark energy density" $\rho_\Lambda$, but it is really just a label for the need a of a formal energy density instead of $???$ in the equation. Normal energy densities however dilute with $a(t)$ which is a measure of expansion/contraction of any _length_ in the universe. Every kind of energy/matter density gets diluted at a different rate. It follows that some types of densities such as radiation density $\rho_{radiation}$ got already diluted and nowadays plays basically a negligible role in the expansion. On the other hand, matter gets diluted slower and manages to persist at least as a minor element that determines the expansion. We describe the rate of dilution by a parameter $w$ different for every kind of density, so it dilutes as: $\rho \propto a^{-3(1+w)}$. The specific form of this equation is because for a perfect fluid we have $$w = \frac{\rho}{P}$$ Where $P$ is the pressure in the fluid. I however consider talking about "pressure of dark energy" as very far-fetched and confusing when just discussing it's formal need in cosmology since we cannot just suddenly blurp out it is a perfect fluid. For dark energy, we vary the parameter $w$ to get different cosmological models (we also have to vary the actual density $\rho_\Lambda$, we deduce the value only for certain points in our cosmic history) and compare the results with actual observations of the universe. The best fits have a value of $w \approx -1$. I.e., dark energy seems almost not to dilute - $\rho \propto a^{\approx 0}$. In the case of completely constant, undiluting $\rho_\Lambda$, dark energy is more of a property of space, fixed on a unit of volume, rather than a density of any kind of a dynamical particle. Then, the best choice to describe it is the cosmological constant, a modification of gravity, or vacuum energy a result of quantum theory (in a theory of everything, these could turn out to be basically the same thing). * * * **THE QUESTION :** Now to the question - observational cosmology tries hard to assess the parameter $w$ to a higher degree of precision by multiple methods of observation - just to be really sure. Some of the astronomers I have met personally seem to be thrilled about the possibility that $w\neq -1$. But my question is - is there a viable theory which would be able to give us the microscopical model of a density with $w\neq -1$ (but $w\approx -1$)? If yes, how does it relate to current standard and beyond-standard theories? And the last, more technical question, what would then be the "loose ends" of the theory such as free parameters? (You may obviously also describe relevant theories which would predict inconsistent $w$ obtained by different observational/fitting methods. But again, consider that all methods have shown consistent results within observational uncertainty.) * * * **EDIT (a review of Dark energy theories):** I have found a very nice comprehensive review of explanations of dark energy here. It basically assesses that Dark energy can be explained by modifying the "right hand side" of Einstein's equations (sources of gravity), "left hand side" (effective laws of gravity) or by modifying the assumptions of the FLRW equations (i.e. _no_ homogeneity on large scales). The left hand side is either modified by a self-interaction dominated scalar field (quintessence), a scalar field with a non-quadratic kinetic energy (k-essence) and combinations of these. Coupling with dark matter can be introduced, or even the single scalar field can be used to explain both dark energy and dark matter. Sources with postulated equations of state such as Chaplygin gas can always be introduced. The right hand side can be modified to a broad class of gravitational theories called $f(R)$ theories, which introduce higher orders of curvature $R$ into the GTR action $S \sim \int R$. These can be effective terms arising from vacuum polarization or can be fundamental modifications of gravity. It can be shown that an $f(R)$ theory can be conformally transformed into Einstein gravity with a scalar field source. As to my observation, the main advantage of $f(R)$ theories is the fact they are not so arbitrary as quintessence as much more observational constraints are placed on them due to the fact of being a universal gravitational theory. The other gravity modifications result from the usual portrayal of our universe as a 4-dimensional brane with at least one "large" extra dimension. The _Dvali, Gabadadze and Porrati model_ described in the linked article accounts for the observation of current acceleration as a transition from a high-matter-density limiting case, where Friedmann equations apply, to a low- density case, where an effective source term of the modified gravitational interaction becomes non-negligible. The last explanation in the article portrays our observed accelerated expansion as an effect arising from being in a very large under-dense region of the universe. The so-called _Lemaitre-Tolman-Bondi_ models are plausible as to explaining the standard candle observations, the location of the first acoustic peak, but have to assume we are very close to the center of the under-dense region (as long as we keep observing isotropy e.g. in the CMB). Nevertheless, the cited article is two years old, most notably not aware of data from the Planck mission. It would be great to have a commentary on the observational plausibility on the mentioned (or new) theories from an expert.


title: Detect missing frequencies in a frequency distribution

text: I have photons with a frequency distribution where one frequency is completly cut out. For example a frequency distribution like that: ![enter image description here](http://i.stack.imgur.com/ruoye.png) My question is: How precise can the position (frequency) of this gap be detected? In an experiment this gap is slightly shifted to the left or to the right (depending on the situation). I want to find out if this shifts can be detected (detect if there was a shift or not). To give some numbers say $\omega_0 = 10^{15}\,\text{Hz}$ and the standard deviation of this double-humped distribution is given by $\sigma = 10^{10}\,\text{Hz}$. So the frequency $\omega_0$ is missing but very close to $\omega_0$ the distribution raises to its maximum. Now imagine that $\omega_0$ moves to the right like $\omega_0 \rightarrow \omega_0 + 10^{8}\,\text{Hz}$. Is it possible to detect such shifts?


title: what is the difference between a blackhole and a point particle

text: Theoretically, What is the difference between a black hole and a point particle of certain nonzero mass. Of-course the former exists while its not clear whether the later exists or not, but both have infinite density.


title: Is dimensional analysis always sufficient to establish equivalence of quantities?

text: In dealing with the Biot-Savart law, it was argued that $$ q\frac{d\vec{s}}{dt}\equiv Id\vec{s} $$ using the fact that the units are equal. Does this kind of argument always work? It seems too simple to be true.


title: Effects of gravity on light

text: If gravity can bend light, why can't gravity slow light. At least momentarily? Wouldn't that give the illusion of the universe expansion speeding up?


title: Weak equivalence principle tests

text: on the wikipedia article about the equivalence principle there is a mention about testing the EP against parity-violating masses; > "The equivalence principle is untested against opposite geometric parity > (chirality in all directions) mass distributions. A parity Eötvös experiment > contrasting solid single crystal spheres of identical composition α-quartz > in enantiomorphic space groups P3121 (right-handed screw axis) versus P3221 > (left-handed screw axis) is appropriate. Equivalence principle parity > violation validates a chiral vacuum background forbidden within general > relativity but allowed within Einstein-Cartan theory; affine, teleparallel, > and noncommutative gravitation theories." I don't understand any of this, is this correct? why the crystal layout has anything to do with a parity-violating fields? (AFAIK electro-weak force is the only known force to be parity-violating, but this force is short range) so why the alignment of the by-comparison-macroscopic crystal atomic layout does matter? is this crackpot physics or i just don't understand it?


title: Where does deleted information go?

text: I've heard that, in classical and quantum mechanics, the law of conservation of information holds. I always wonder where my deleted files and folders have gone on my computer. It must be somewhere I think. Can anyone in principle recover it even if I have overwritten my hard drive?


title: Weinberg dimension 5 operator

text: How to prove that the $\Delta L=2,$ dimension=5 Weinberg operator $LLHH$ is the unique operator which violates lepton number by two units, without derivative couplings, etc.??


title: How was this formula in my textbook was derived?

text: How is this formula derived (it is present in one of the hundred pages of my textbook about physics and fundamental physics) ![enter image description here](http://i.stack.imgur.com/l5bZ3.png) In the case of a system of particles $P_i$, $i = 1, …, n$, each with mass _mi_ that are located in space with coordinates $r_i$, $i = 1, …, n$, the coordinates **R** of the center of mass satisfy the condition


title: What can cause a change in wave's shape - One dimensional wave

text: what can cause a change in wave's shape of one dimensional wave moving through a rope? It's velocity ? or the wave's length ? What can cause him change his shape.


title: Tolerance of Natural Frequency & Resonance?

text: I'm writing a report at the moment about natural frequency, driving frequency and resonance - and I was wondering, is there a typical % tolerance inside which the driving frequency will cause resonance (or exhibit resonance-like characteristics)? Or does this tolerance depend with the material and construct involved? (If you have any sources where I can read about this also, it'd be much appreciated). Note- I'm writing about an oscillatory system - torsion pendulum, so it would be resonance in an oscillatory sense).


title: Two axes for rotational motion

text: I understand that angular momentum is a vector, etc.. But, what really happens when some object, say a ball for example, is set to rotate along two axes? What would the resulting motion look like?


title: What is the mass of a photon in non-empty spaces?

text: It is a well known result of the special theory of relativity that the photon has no rest mass, because for a particle to attain the speed of light, it must have zero rest-mass. I will not dig into this more, but the interested reader can see these questions for more information: How can a photon have no mass and still travel at the speed of light? Why can't photons have a mass Also, this Wikipedia link: http://en.wikipedia.org/wiki/Photon#Experimental_checks_on_photon_mass Now my question is what about a photon in non-empty space? It is well known that light in non-empty spaces travel slightly slower than its speed in empty space. But that means that the photon cannot have zero rest mass, as every mass-less particle should always attain the speed of light (see the links above)! How can this be explained? Let us assume for the moment that the answer to this question is that the the photon does indeed have an extremely small mass in non-empty spaces. But let's direct a ray of light towards a box of glass which is completely empty. Now before it reaches the empty space inside the box, it has velocity c - $\epsilon$, where $\epsilon$ is some small positive real number. Once it enters the empty space, it should raise to the the exact speed of light. However, it is a result of special relativity that any particle with a non- zero mass, however small it is, cannot attain the speed of light. Couldn't this mean that the photon does indeed have a mass, but a very very small one, and that it is actually moving in a speed less than the universal maximum limit of speed?!


title: Form Factor in Rutherford Scattering

text: My question relates to Rutherford Scattering of particles. When we calculate the "differential cross-section" expression for a nucleus with finite size, it is said that the expression is almost the same as that if the nucleus was a point only a "Form Factor" multiplies,which is nothing but the fourier transform of the charge density, I want to know the derivation of this expression with the form factor. I cant find it anywhere.


title: Counting the number of propagating degrees of freedom in Lorenz Gauge Electrodynamics

text: How do I definitively show that there are only **two** propagating degrees of freedom in the Lorenz Gauge $\partial_\mu A^\mu=0$ in classical electrodynamics. I need an clear argument that   1. involves the equations of motion for just the potentials $A^0$ and $\mathbf{A}$, and _not_ the electric and magnetic fields.    2. includes sources $\rho$ and $\mathbf{J}$ in the equations of motion. This is to justify the assertion that there are degrees of freedom that decouple from the rest of system.   3. does not critically rely on _quantum_ field theoretic arguments (although any supplementary remarks are welcome). * * * To illustrate the level of clarity I expect, I provide an argument in the Coulomb gauge $\nabla\cdot \mathbf{A} = 0$: Of the four field degrees of freedom, the gauge condition $\nabla\cdot \mathbf{A} = 0$ removes one degree of freedom (the longitudinally polarized EM waves). To show that among the remaining three degrees of freedom only two are propagating, consider the field equations of motion in the Coulomb gauge: \begin{align} \nabla^2 A^0 &= -\rho/\epsilon_0,\\\ \big[\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2\big]\mathbf{A}&= \mu_0 \mathbf{J}-\nabla \frac{1}{c^2}\frac{\partial}{\partial t}A^0. \end{align} The first equation is NOT a wave equation for $A^0$, and thus does not propagate. The final equation IS a wave equation, and describes the propagation of two degrees of freedom (Gauss' law in first eq. can be solved, and then inserted into the second equation to show that $\mathbf{A}$ only couples to the Solenoidal part of the current).


title: Why is mud muddy?

text: How to explain properties of mixture of sand with water? Why is it so coherent and slippery at the same time? Is it due to hydrogen bonds? Why is mud of smaller grains more slippery and incoherent and mud of bigger grains is more coherent and less slippery?


title: Are standard and isotropic forms of Schwarzschild metric truly equivalent?

text: My admittedly rudimentary understanding of physical meaning of conformal flatness - as pertaining to a stationary observer exterior to a spherically symmetric static gravitating mass $M$: Locally Euclidean, in that the proper differential volume $dV$ between two concentric spherical surfaces centered about M, having differential proper radial spacing $dR$, is given by the Euclidean formula for enclosed volume $dV$ = $AdR$, where $A$ can be taken as the mean value of the two proper surface areas, and we take the limit as $dR\rightarrow0$. Correct? And that in GR, SM (standard Schwarzschild metric) is **not** conformally flat since there $dR$ is greater by factor $\sqrt{-g_{rr}}$, as can be determined by inspection of the line element - e.g.:Wikipedia - archived revision That is, $dV = \sqrt{-g_{rr}}AdR$ for SM, $ > dV$ (Euclidean) - for a _given_ proper areal difference between shells. It's evident by inspection of the line element for ISM: here, for the equivalent differentially separated concentric shells arrangement as above, $dV = AdR$ asymptotically applies as per Euclidean formula? In other words, _by virtue of it's construction as spatially isotropic_ , ISM necessarily claims a conformally flat metric, in-principle measurably physically distinct from that of SM? How then is it that the two are claimed to be physically equivalent? Also is there a precise technical term and definition specifying departure from conformal flatness here? While the above concentric spherical shells situation is the one I was introduced to, there is surely no reason preventing it being dimensionally reduced to one of proper radial spacing between concentric great circles, and in fact then further reduced to an arbitrarily small local sector cut from such concentric circles. Meaning it must be an in- principle locally observable quantity rather than only determinable globally?


title: Higher order derivatives - Equation of motion

text: One possible starting point to create a physical theory is the Lagrangian $L$. There we assume that the variation of the action $\delta S = \delta \int_{-\infty}^\infty dt \ L = 0$. In classical theories we usually only use $q$ and $\dot{q}$ in the Lagrangian. But there are also effects like the Abraham-Lorentz force, which describes a force $F(q) = \alpha \dddot{q}$, where $\alpha$ is a constant and $q$ is the location of a particle. This would require a higher order derivatives in the Lagrangian. Now I wondered if it is even possible to write down a Lagrangian for such a force, that contains a third derivative in time? Maybe a similar question is if it is possible to get a Lagrangian that results in a friction force $m\ddot{q} = - \gamma\dot{q}$?


title: Difference between $\psi_{\alpha}$ and and $u^{\pm}$ in Dirac fields?

text: What is clear difference between say Psi_1,psi_2,....psi_4 and the U+- and V+- matrices in case of dirac fields or are u,v (or some book use U^(1),U^(2)) matrices some rep of the same


title: How was Avogadro's number first determined?

text: I read on Wikipedia how the numerical value of Avogadro's number can be found by doing an experiment, provided you have the numerical value of Faraday's constant; but it seems to me that Faraday's constant could not be known before Avogadro's number was as it's the electric charge per mole. (How could we know the charge of a single electron just by knowing the charge of a mole of electrons, without knowing the ratio of the number of particles in both?) I just want to know the method physically used, and the reasoning and calculations done by the first person who found the number $6.0221417930\times10^{23}$ (or however accurate it was first discovered to be). Note: I see on the Wikipedia page for Avogadro constant that the numerical value was first obtained by "Johann Josef Loschmidt who, in 1865, estimated the average diameter of the molecules in air by a method that is equivalent to calculating the number of particles in a given volume of gas;" but I can't access any of the original sources that are cited. Can somebody explain it to me, or else give an accessible link so I can read about what exactly Loschmidt did?


title: About the energy with the repulsive potential

text: Consider the reduced radial Schrodinger equation: $$-\frac{1}{2}\frac{\text{d}^2}{\text{d}r^2}\phi(r)+V(r)\phi(r)=E\phi(r).$$ We try to find a bound state (i.e. $\phi(0)=\phi(+\infty)=0$). Here $V(r)=1/r$ or $1/r^2,1/r^3$ etc. A numerical approach showed that the energy $E$ tends to zero when $a$ (i.e. the upper bound for $r$, use it to represent $+\infty$ in the numerical method) increases. ![table1](http://i.stack.imgur.com/m3uuU.png) (In this table $V(r)=1/r^{\alpha}$. $E_1$ is the calculated energy.) My question is, how to explain this result physically? Is that the real $E$ should be $0$, or allowed $E$s actually are continuous? I think it is because of the repulsive potential $V(r)$, that $E$ should be $0$. (P.S. I'm interested in the "ground state", if it exists. My numerical procedure tries to find the minimum energy. )


title: Relativistic equation with arbitrary spin

text: can anyone give me some good references on how to obtain the relativistic equation of particles with arbitrary spin?


title: Can additional airfoil enable a commercial liner to reach 100km altitude?

text: Assuming that it's engines are incapable of dying out at 100km altitude, would mere addition of airfoil area enable a commercial liner e.g. B787 to reach that altitude?


title: Is surface of a solid a streamline?

text: In fluid dynamics, streamlines are defined as line where at each point flow velocity is tangential to the line. Is it correct to say surface of a solid a streamline? On the surface the velocity vector is zero, so it does not make sense to define a streamline. Another similar situation is when fluid is at rest (no solid surface involved). Can we "draw" streamlines for such case?


title: Could we use particle colliders as fusion generators?

text: So I know the basic gist is that fusion power's main issue is sustaining the fusion. I also know that there are two methods. The Torus method and the laser method. The torus magnetically contains plasma and heats it with radiation and accelerates the plasma around to make strong enough collisions that protons fuse. The laser method uses 192 lasers and focuses it on tiny frozen hydrogen pellets and aims to initiate fusion each time pellets are dropped. The though struck me when we could sorta combine the two designs together. The torus doesn't have to worry about making fusion happen at a specific location but it has issues in that the plasma is unevenly heated and leaks. On the other hand, the laser design is extremely complicated in the level of precision needed and would have to repeat this for every pellet. This lead me to think to make something precise and contained at the same time. I see that particle colliders are able to direct two beams of protons and have them collide at a specific spot with a very precise energy. Couldn't we tune the energy of the two beams of protons to the energy required for them to fuse? We have the ability to smash them into bits, surely we have the ability to have them fuse. (I'm thinking about the type of collider that circles two beams in opposite directions) It would be at much lower energies than normal colliders and would be very precise and it would be possible to fuse at a specific location that has greater leeway because for protons that missed collision, they'd just circle around again! Thus protons would efficiently be used and very little would be wasted. There wouldn't be problems of plasma leakage because we are focusing them in a thin tight beam. It seems that this idea has girth, or I feel this way at least, can someone back me up by offering some calculations on how to calculate the efficiency? How would I go about calculating the two circling beams of protons and at what specific velocity would be needed? etc.


title: What alternative shapes may a rocket heading into orbit have?

text: It looks like most rockets that head out of Earth, or even into orbit are pencil shaped (or nearly so). I would take this to mean there is some mass of air such vehicles push out of their way. What alternative shapes may a rocket headed into space, or orbit have? **EDIT:** This will probably seem crazy; I was toying with the idea of a tunnel or more extending some length of the body with some mechanism to pull the air out of the way forward and ejecting it backwards (sort of like a jet) to create a little bit of pressure differential. Sort of like striations on the body of a whale and other marines help it slip easily through water.


title: Points in Spacetime

text: Assume there are two points in spacetime $a=(t,x,y,z)$ and $a'=(t',x',y',z')$. Let's say that the first one is in the origin of spacetime i.e. $a=(0,0,0,0)$. The point $a'$ has two possibilities   1. $a'=(0,10cm,0,0)$ i.e. it's $10cm$ right of $a$   2. $a'=(1ns,1cm,0,0)$ i.e. it's $1cm$ right of $a$ but $1ns$ ahead of it as well. Someone want to send a signal from $a$ to $a'$. Is there a way that these two points(the two $a'$) receive the signal instantly either in space or time? What I am trying to understand is that if someone can consider those points($a,a'$) to be identical either in space or in time.


title: Why does rubber ball bounce back while iron ball doesn't?

text: Suppose there are two balls, one of rubber and the other metallic. There are of the same mass and are thrown on a wall with the same velocity. Why does a rubber ball bounce back while a metallic ball simply falls down after striking with the wall? I know it has got to do something with the change in linear momentum and its elasticity but what?


title: Different batteries connected in parallel

text: If we have 2 batteries one of emf x and the other is of emf y and we connect them in series we get an effective emf of x+y. But what if we connect them in parallel, how to calculate the emf now?


title: Non-isolated universe and arrow of space

text: Could you please help me to understand two statements in physics: 1) **Universe by definition is an isolated system.** But universe expands, space pours into universe, space (vacuum) has zero-point energy, so does expansion means more energy in the universe? If yes, how can it be an isolated system? 2) **There is time arrow, but no space arrow.** If universe can only expand, isn't it that space goes only in one direction: from smaller to larger, that is the space arrow? Aren't time and space arrows similar?


title: Is there any place you can safely skydive without a parachute?

text: While trying to explain to someone how a kitten walked away from a fall from a skyscraper I got to wondering if there's any place a human could do that and I find myself out of my depth. The only place that seems possible is Titan but I'm not sure. My basic thoughts: The gravity there is about 1/7th that of Earth. My impression is that should reduce the terminal velocity accordingly. Where I get stumped is dealing with Titan's atmosphere--it's not like Earth's (for which I'm stumped on figuring what will happen) and it's much colder (which I think will increase density and thus drag by the ratio of the absolute temperatures.) Or is there any other place I've missed? (Gas giants don't count as there's no surface to land on.)


title: Why does sound move faster in solids?

text: I know that the molecules are closer together in solids, and I know thicker springs also respond carry waves faster than thinner springs, but for some reasons I can't understand why. The molecules will have a larger distance to move before colliding with another molecule, but in a thicker medium wouldn't that time just be spent relaying the message between multiple atoms? Why is the relaying between a lot of tight knit atoms faster than one molecule moving a farther distance and colliding with another?


title: Work on Ferromagnetic Object Due to Solenoid

text: I've been going through some equations and such trying to determine the work done by a solenoid on a ferromagnetic object. I have the following: Magnetic field due to solenoid: $\vec{B} = \langle0,0,\mu_0nI\rangle$ (Assuming coils are on xy-plane and current is counter-clockwise) Force of magnetic field: $ F = q\vec{v} \times \vec{B} $ Work: $ W = \int F \cdot dl $ Work of Magnetic Field: $ W = \int_c(q\vec{v} \times \langle0,0,\mu_onI\rangle) \cdot d\vec{r} $ For one, this seems to indicate a work of 0 if the object is not charged, which I have seen in some places but just doesn't seem right. Also, this does not take into account the properties of the object, such as relative permeability, which I guess could have some effect with the charge value. I'm trying to calculate the acceleration of a ferromagnetic object from a magnetic field, is there a better way to do this? I've thought about the following: $ \vec{a} = \frac{q\vec{v} \times \vec{B}}{m} $ However, this is where I started running into the charge issue and thought to calculate it from the work done.


title: Finding an equation relating the mass of a blade of a wind turbine to its velocity

text: I'm writing up my physics coursework and I thought I'd try and find an equation described in the title. This is my attempt: ![Image](http://i.stack.imgur.com/i4EEM.png) Is it correct?


title: Is it possible to make statements about bosonic/fermionic systems by taking the limit $\theta\to \pi$ or $\theta\to 0$, of an anyonic system?

text: One might naïvely write the (anti-)commutation relations for bosonic/fermionic ladder operators as limits $$ \delta_{k,\ell} = \bigl[ \hat{b}_{k}, \hat{b}_{\ell}^\dagger \bigr] = \hat{b}_{k} \hat{b}_{\ell}^\dagger \- \hat{b}_{\ell}^\dagger \hat{b}_{k} = \lim_{\theta\to\pi} \Bigl( \hat{b}_{k} \hat{b}_{\ell}^\dagger \+ e^{i\theta}\cdot\hat{b}_{\ell}^\dagger \hat{b}_{k} \Bigr) $$ $$ \delta_{k,\ell} = \bigl\\{ \hat{c}_{k}, \hat{c}_{\ell}^\dagger \bigr\\} = \hat{c}_{k} \hat{c}_{\ell}^\dagger \+ \hat{c}_{\ell}^\dagger \hat{c}_{k} = \lim_{\theta\to 0} \Bigl( \hat{c}_{k} \hat{c}_{\ell}^\dagger \+ e^{i\theta}\cdot\hat{c}_{\ell}^\dagger \hat{c}_{k} \Bigr). $$ I.e. as limits of Abelian anyonic commutation relations. Assuming now that some system could be solved for anyons with $0 < \theta < \pi$, would taking the limits of e.g. the energy eigenstates for $\theta\to \pi$ yield in general the correct eigenstates of the bosonic system (which might be harder to solve directly)? I'm inclined to think it would work, but after all, the whole Fock space looks different depending on $\theta$, with all kinds of possible topological nontrivialities.


title: Is there an intuitive explanation for the Southward force caused by the Coriolis Effect on rotating spheres?

text: From the Coriolis Effect article on Wikipedia, the following with regard to the Coriolis Effect on a rotating **sphere** : > By setting vn = 0, it can be seen immediately that (for positive φ and ω) a > movement due east results in an acceleration due south. Similarly, setting > ve = 0, it is seen that a movement due north results in an acceleration due > east. In general, observed horizontally, looking along the direction of the > movement causing the acceleration, the acceleration always is turned 90° to > the right and of the same size regardless of the horizontal orientation. My intuitive (but possibly incorrect) understanding is that if there are two points, `Point A` and `Point B`, at different latitudes in the Northern hemisphere, the Eastward velocities of these points are different because they are at different distances from the Earth's axis of rotation, and this causes the Coriolis effect for a rotating sphere. If a projectile is fired **due North** from `Point A` near the Equator towards `Point B` near the North Pole, the projectile will start off with the higher Eastward velocity of `Point A`, and will land to the **East** of `Point B`, which is moving East at a slower velocity than the projectile. Firstly, is that correct? If that IS correct, that brings me to the quote off Wikipedia. The quote implies that, if a projectile is fired **due East** , it will experience a Coriolis force to the **South**. My intuitive explanation based on differences in velocities between origin and destination does not account for a Southwards movement at all, since the `Point A` and `Point B` velocities are identical if the points are on the same latitude. What am I missing? Would a projectile fired **due East** or **due West** experience any North or South drift caused by the Coriolis effect (or anything else for that matter)? Why?


title: Parton showering in Pythia 6 Monte Carlo generator

text: I have Pythia Monte Carlo (MC) samples where I can't understand the parton showering model. If I print out full decay chains from the events, each event contains multiple string objects with pdgId 92. There can be up to 19 strings which are exactly the same and have exactly the same further decay chains, but have different mothers. And this causes a problem that it's impossible to go up in the decay chain from each final state particle and find for example from which quark or gluon the particle is coming. It's possible to identify mothers of each particle up to the string, and then this string has multiple different mothers. So how does this string-related part work in Pythia?


title: Does string theory tells us anything concrete about black hole decay?

text: **EDIT:** I edited the question to reflect Moshe's objections. Please, look at it again. * * * It's apparently a black hole time around here so I decided to ask a question of my own. After a few generic questions about black holes, I am wondering whether string theory is able to provide something beyond the usual semiclassical Hawking radiation talk. Feel free to provide an answer from the standpoint of other theories of quantum gravity but AFAIK none of the other theories has yet come close to dealing with these questions. That's why I focus on string theory. So let's talk about micro black holes. They have extreme temperature, extreme curvature, and I guess they must be exceptional in other senses too. At some point the gravitational description of these objects breaks down and I imagine this kind of black hole could be more properly modeled like a condensate of some stringy stuff. So let's talk about fuzzballs instead of black holes. >   1. What does that microscopic fuzzball model look like? >   2. What does string theory tell us about the evaporation of those > fuzzballs? Is the Hawking radiation still the main effect (as for the > regular black holes) or do other phenomena take over at some point? >   3. Also feel free to add any other established results regarding black > hole decay (as Jeff did with information preservation). >


title: Intelligent(?) Particles

text: Quite a while ago I read about a series of experiments that basically suggested that a certain kind of particle/atom/(something) were "intelligent" and could appear in two places at once, or essentially could "tell the future" when it came to navigating a "maze" ...I think it might have involved lasers or mirrors? Does any one a) know what I'm talking about, and b) have links/further information on it? Really don't have much more recollection than that I'm afraid. This will probably come across as a rather vague question so my apologies but hopefully someone will know what I'm talking about!


title: Area under a $pV$ diagram

text: What does the area under a Pressure volume diagram equal? I read in my textbook it equals 'external' work done, but why is this? First of all, what exactly is external work? Can you get it external work by the simple formula $W= F\cdot s$? Also, why does the area under a $pV$ diagram equal 'external' work? What is the logic behind that?


title: Deriving the "total" Bose Einstein density of states, including the condensate

text: Is is possible to derive the Bose-Einstein density of states containing the delta function representing the BE condensate?


title: Wavefront emitted by bodies at traveling near the velocity of light

text: I studied that no body can travel with the velocity of light. But, assuming that when a body moves nearly velocity of light, will it obey length contraction law of Einstein or will it emit the same wavefront when it is in stationary state? What will happen to its dimensions? Will it appear same or shrink to an stationary observer?


title: Block on cart, equation of motion

text: Consider a rigid block of $b \times h$ having mass $m$ on cart (as depicted below). The cart is given an acceleration $a$, this leads to overturning of the block. The angle of rotation is indicated by $\theta$. ![](https://imagizer.imageshack.us/v2/224x245q90/203/zfuy.png) This is how far I got (not considering the movement of the cart): The Lagrangian $L=T-V$ is calculated using $$T = \frac12 J \dot{\theta}^2$$ $$V = m g \Delta_y = m g \bigg(r \cos(\alpha - \theta) - \frac{h}{2}\bigg) $$ so that $$\frac{\mathrm{d}}{\mathrm{d} t} \bigg( \frac{\mathrm{d} L}{\mathrm{d} \dot{\theta}} \bigg) - \frac{\mathrm{d} L}{\mathrm{d} \theta} = 0$$ yields the EOM $$J\ddot{\theta} + m g r \sin(\alpha-\theta) = 0$$ Now, my question is: how do I add the acceleration of the cart to the RHS? My initial guess would be $$J\ddot{\theta} + m g r \sin(\alpha-\theta) = m a r \cos(\alpha - \theta) $$ where $ma$ is the force and $r \cos(\alpha - \theta)$ the lever arm. But I don't believe this is true since the block does not experience the acceleration $a$ over its full body. Can anyone help or provide some literature? Thanks.


title: What does it mean for the Leggett Inequality to falsify realism in general in Quantum Mechanics?

text: http://en.wikipedia.org/wiki/Leggett_inequality As you can see in the above link, it claims that Bell's inequality ruled out local realism in quantum mechanics, and the violation of Leggett's inequalities is considered to falsify realism in general in quantum mechanics. This would mean there is no material reality or objective reality! This is being said by Bernard Haisch as well, if I'm not mistaken. Is this true, or is this false?


title: Is it true that Macrorealism can be shown false?

text: http://en.wikipedia.org/wiki/Leggett-Garg_inequality As you can see by the link above, it claims that if the violation of the Leggett–Garg inequality can be demonstrated on the macroscopic scale, it would challenge the notion of realism that the moon is there when nobody looks. How is that even possible given that when you get larger than a molecule, the quantum wavelength shrinks to insignificance? According to the link, "they" are even trying to actively falsify macrorealism by employing superconducting quantum interference devices.


title: Doppler Effect, Why Can't We Just Add Speed to Argument?

text: In considering the doppler effect, why can we not just take the distance between the two as a function of time, divide by the wave velocity, and add this as a shift in our time argument of the sinusoid?


title: Is this mental picture of photon correct?

text: What is exactly meant by a statement like "there are about 400 photons per cubic cm in certain region"? Should I mentally picture this as 400 discrete photons enclosed in that volume, each moving at speed of light in that medium (independently of one another)? Also by one photon I should not mean something like a particle (localized at a point) but a changing electric and magnetic field in phase spread over exactly one full wavelength. If I were able to take a snapshot of the changing fields, these would be like sinusoidal variations in space and if I knew the velocity I would know that after one second the entire field variation will be shifted exactly "c" distance away from the previous location of wave (along the wave vector) and effects of field variation at previous location is obliterated. To these one may add characteristics of polarizations etc but is the above picture basically correct? Also, is there any restriction on the number of photons in such a picture (if it is correct at all)? Or is there any condition so that we must consider multiple photons (so that we get continous wave) rather than a single photon?


title: Lagrangian for relativistic massless particle

text: For relativistic massive particle, the action is $$S ~=~ -m_0 \int ds ~=~ -m_0 \int d\lambda ~(\dot x ^\mu \dot x_\mu)^{\frac{1}{2}} ~=~ \int d\lambda \ L,$$ where $ds$ is the proper time of the particle; $\lambda$ is the parameter of the trajectory; and we used Minkowski signature $(+,-,-,-)$. So what is the action for a massless particle?


title: What does the data in various stages of analysis from a particle collision look like?

text: I've been following the news around the work they are doing at the LHC particle accelerator in CERN. I am wondering what the raw data that is used to visualize the collisions looks like. Maybe someone can provide a sample csv or txt? Edit: In addition to the raw data, it also seems that I should be interested in the data used at the point where a physicist might begin their analysis, possibly at the "tuple" stage of the data transformation. I'm familiar with RDF tuples, are there any parallels between the two tuples?


title: Direct exposure to the vacuum of space

text: I was watching a few sci-fi movies and was wondering the real science explaining what would happen if you were to be subject to the conditions of outerspace. I read the wikipedia article on space exposure, but was still confused. If a person was about the same distance from the sun as earth is, would they still freeze to death? (as shown in the movie Sunshine) I'm reading from all sorts of sites with conflicting information about what would actually happen when a person is exposed to the vacuum of space...


title: Equivalence between Differential Geometry and Mechanics?

text: Given a metric $$ ds^{2}~=~ g_{a,b}dx^{a}dx^{b}. $$ Here Einstein's summation convention is assumed for $a$ and $b$. Then given the Laplacian over that metric, can then we find a metric $ g_{a,b} $ so $$ \Delta _{g}f ~=~ - \nabla^2 f +V(x)f. $$   * the first term on the left is the _Laplacian_ operator in curvilinear coordinates.   * the term on the right is just the ${\rm div}({\rm grad})f$ in Euclidean coordinates.   * $V(x)$ is a potential in $n$-dimensions. The idea, if this is possible, is to turn physical problems into geometrical problems, for example solving the Schroedinger equation would be equivalent to solve a Dirichlet problem for some Laplacian.