Why proper acceleration is $du/dt$ and not $du/d\tau$?

Wikipedia says: > In relativity theory, proper acceleration[1] is the physical acceleration > (i.e., measurable acceleration as by an accelerometer) experienced by an > object. and says: > In the standard inertial coordinates of special relativity, for > unidirectional motion, proper acceleration is the rate of change of proper > velocity with respect to _coordinate time_. Why coordinate time? As far as I know the moving observer itself measures the proper things. It's clock measures proper time, it's speedometer measures proper velocity. So if the observer accelerates with $10m/s^2$ (and feels that proper acceleration), isn't the speed shown by the local speedometer changes $10m/s$ per second on the local clock?


Decoherence when no one is looking?

I understand that in the single-electron-at-a-time double slit experiment, if a detector is placed before the slit, the interference pattern vanishes. Suppose I left the detector on, but put a bag over its screen (I can't tell what state the electron is before it passes the slits), does the interference pattern come back? If so, does that mean the electron "knows" I'm not "looking" and proceeds to interfere with itself? Edit: I had photons passing through the slit initially. I've updated it to electrons to reflect a more common setup.


Why is the Horizontal Force Constant in Deriving the One Dimensional Wave Equation

My textbook in deriving the wave equation for a one dimensional elastic string stated that the horizontal direction force is constant.I understand that the horizontal components of the tensions on either side of the element have to be equal since it is assumed it moves only in the vertical plane but why does it have to be constant? Don't they just have to be equal?


Probabilistic vs Statistical interpretation of Double Slit experiment

Why is it assumed that the results seen in the double slit experiment are probabilistic and not just a statistical result of some unknown variable or set of variables within the system.


Using eigenvalues to determine the stability/behaviour of the system

first time I've been on physics.se but have used the math and cs before... Anyway, here's my question: If we have a damped pendulum described by the equation $$y'' + ay' + b = 0 , a>0$$ Using the conversion $x_0=y$ and $x_1=y'$ we can convert this into a set of first-order ODEs as follows: $$ x_0'=x_1$$ $$x_1'=-ax_1+b$$ or in matrix form : $$\mathbf{x'}=\begin{bmatrix}0 & 1\\\0 & -a\end{bmatrix}\mathbf{x}+\begin{bmatrix}0 \\\ b\end{bmatrix}$$ Now, the exercise where I'm doing this now says we can use the eigenvalues to determine the stability of the system the eigenvalues of the system. The eigenvalues are $\lambda = 0$ and $\lambda = -a$... I know that an eigenvalue of 0 means a neutral equilibrium such that a small change in the variable means it remains at the changed variable and that a negative eigenvalue means a small change will return to its original value, but I'm struggling to see how this relates to the physical situation of the damped oscillator, and what it means about the behaviour of the system.. Any thoughts would be appreciated


Distance at which a 300m asteroid is visible

Using the best available telescopes, and assuming a typical albedo of 10%, at what distance can a 300m asteroid be seen (well enough to start plotting its course) in 1) visible and 2) infrared wavelengths?


Photon on null geodesic

If given an FRW metric $ds^2 = -dt^2 + a^2(t)[dx^2+dy^2+dz^2]$ and for the trajectory followed by a photon (null geodesic; $ds^2=0$) with affine parameter $\lambda$, know that $g_{\mu\nu}\,\frac{dx^{\mu}}{d\lambda}\,\frac{dx^{\nu}}{d\lambda} = 0.$ How does one find $\frac{dt}{d\lambda} = \frac{w0}{a(t)}$? Already found the nonzero Christoffel coeffs and the remaining geodesic equations, $ 0 = \frac{d^2x^i}{d\lambda^2} + \Gamma^i_{ti}\,\frac{dt}{d\lambda}\,\frac{dx^i}{d\lambda} $ and $ 0 = \frac{d^2t}{d\lambda^2} + \Gamma^t_{ii}\,\frac{dx^i}{d\lambda}\,\frac{dx^i}{d\lambda} $ with $ \Gamma^i_{ti} = \frac{1}{a(t)}\,\frac{da(t)}{dt} $ and $ \Gamma^t_{ii} = a(t)\,\frac{da(t)}{dt} $ I expect it to be pretty simple (yet not seeing it) and that the null geodesic in the $x$-direction come from something like (?): $ ds^2 = -(dt^2)+a^2(t)\,dx^2 = 0 \rightarrow \frac{dt}{a(t)} = \pm dx $ I also believe that $dx/d\lambda = c$ if the affine parameter is related by $\lambda = t$. This is where I am confused and need it to move on to look at the cosmological redshift and derive the ratio of emitted and observed energies of a photon at $t1$ and $t2$. Anything to straighten me out would be great!


Are CCDs approaching the sensitivity of photomultiplier tubes?

I still see photomultiplier tubes (PMT) used for even benchtop spectroscopy experiments. How much more sensitive are they to CCDs? I found a post which claims about 1.5 x more sensitivity for PMTs, but any references or more complete understanding of the problem would be really appreciated. I'm also wondering if they need to be cooled or specialized chips or if commodity imaging chips like from a camera has reasonable S/N characteristics for detection?


Equations of graphs

I am doing a practical write up of an experiment I did. Here is a graph using the raw data I got. ![enter image description here](http://i.stack.imgur.com/n1JMW.png) What is the general formula of this graph, and what do I need to do in order to make that into a straight line graph? Take the natural log of the average speed and plot it against mass? Excel gives me the specific equation of the line, but I need to write down a general equation in order to get max marks. Or could it be that my empirical data is wrong with the theory? I'm having trouble understanding the maths here. Edit: Here are attempts at straight line graphs I have made. ln v vs ln M looks the straightest, is that the one I should therefore use? ![enter image description here](http://i.stack.imgur.com/eZnE8.png) Can someone walk me through how I change the general formula of the first graph into the general formula of a straight line graph?


Stresslet coefficient in fluid dynamics

I encountered a parameter called stresslet coefficient in several literatures, which seems related with viscosity and stream function (also influencing the far field displacement). But there's no Wikipedia page for this concept. Can anyone explain what it is? Thanks!


What is the difference between Rayleigh scattering and Thomson scattering?

After reading the wiki articles I know, that both Rayleigh scattering and Thomson scattering are elastic processes. But what is the essential difference between those two processes, their cross sections and energy dependence?


What is the Lagrangian from which the Klein-Gordon equation is derived in QFT?

1. Is there a well-known Lagrangian that, writing the corresponding eq of motion, gives the Klein-Gordon Equation in QFT? If so, what is it?   2. What is the canonical conjugate momentum? I derive the same result as in two sources separately, but with opposite sign, and I am starting to suspect that the error could be in the Lagrangian I am departing from.   3. Is there any difference in the answers to that two questions if you choose (+---) or (-+++)? If so, which one?


What is the 1977 Wow signal?

Can someone explain about the 1977 Wow signal? I know that it is a signal received from space in 1977, in the form of a series of code "6EQUJ5". I would like to know if there is any thing or event which can generate this signal other than an intelligent life form? If yes then can you tell me which things or events can generate it??


why does headlights dim when a starter motor is turned on

okay first of all, what exactly does the starter motor do? does it increase the current flow and decrease voltage? and does starter motor have its own internal resistance? i understand how there is a voltage drop in the battery terminals, so does the starter motor lower the potential difference to become less than the voltage drop? so for example if the voltage is 12V and the voltage drop is 2V. The potential difference is now 10V. what is the voltage when the starter motor is used? does it become less than 10V? but then after the engine starts the starter motor turns off and the lights return to normal. So is normal 10V?


Using centre of mass in gravitational field calculation

When calculating the gravitational field of a continuous mass, like a rod, at some point outside it, why can't we just use the centre of mass of the rod and then directly plug in the distance between the point and the centre of mass in the gravitational field equation. Why do we need to integrate over the whole rod?


Round bulb and Long bulb in Series Circuit

If you have a series circuit with a battery, one round bulb (with thicker filament), and one long bulb (with thinner filament), the round bulb does not light up and only the long bulb lights up. Also, I understand that the voltage drop in the long bulb is greater, but I don't get how the charges distribute themselves like that. For example, if the round bulb comes first, how does the charges passing by the long bulb "know" to only give up a little bit of energy to save it up for the long bulb? Why does the round bulb not light up?


Why do some particles have a greater mass than others?

The property of mass that almost every particle possesses comes from the Higgs Field. It is this field, which permeates all of space, that particles interact with and hence obtain mass. But why do some particles have a higher mass if they are all interacting with the same Higgs Field? One would suspect that they'd all have the same mass, but there must be some property of a particular particle that makes it interact with the field in a different way, thereby giving it mass.


Exact definition of momentarily comoving reference frame (MCRF)

Consider a particle $P$ in the framework of special relativity with position $r(t)=(ct,x(t),y(t),z(t))$ respect to an inertial reference frame $\Sigma=(ct,x,y,z;O)$. I need to know if the following is a correct definition of a MCRF for $P$: > A momentarily comoving reference frame (MCRF) for P is a frame reference > $\Sigma'=(ct',x',y',z';O')$ such that if $r'$ is the position of $P$ respect > to $\Sigma'$, the following two conditions are satisfied: > > 1) $r'(t'_0)=(ct'_0,0,0,0) $ for every $t'_0$ fixed. > > 2) $\frac{dr'}{dt'}(t'_0)=(c,0,0,0)$ for every $t'_0$ fixed. > > Moreover the motion of $P$ is uniformly accelerated if for every choice of a > MCRF $\frac{d^2r'}{{dt'}^2}(t'_0)$ is constant $\forall t'_0$. If the above definition is wrong I would appreciate a **formal** correct definition of a MCRF. Thanks in advance.


What are the the characteristics of a standing wave?

![enter image description here](http://i.stack.imgur.com/s6Lzd.jpg) May someone explain #24 from this picture? B is apparently the correct answer. Can any of the other answers be correct here? Isn't D also correct?


How is gravitational force is compared to flow of water?

I listened to a lecture. The professor said that the gravitational field around the particle (spherical in shape) can be compared to a pond having a constant height and depth and water is constantly pumped in the center. The water flows radially outward which is similar to that of the gravitational field around a spherical object. Can anyone explain me how to compare gravitational field and this flow of water?


How to create unusual sundial?

I am considering small "artsy" project. I would like to create sundial by placing gnomon on the window and painting hour lines on the window facing wall. Since this is to be placed in bedroom I am constrained by my geographic location, wall, window placement and orientation. The esthetic and size (I have one wall only) of the project is gating factor (otherwise known as "the wife" factor). Due to above I am perfectly OK with the fact that this sundial will "work" by limited time of the day and even limited time of the year. However whenever it will work (that is the shadow of the gnomon will be cast on the "said" wall) I would like it to be as accurate as possible. Also all of the above make the calculations for creating hour lines quite challenging (at least for me) and to be honest I do not know where to start. Could you point me the resources that would help in calculating hour lines (software, tutorials, books, math equations)?   Could you describe how would you approach the task of calculating hour lines? (Resources that are little heavy on math side are OK for me. I am also capable of wring software on my own.)


Resources and requirements for amateur spectroscopy

If I wanted to move beyond just looking through my eyepieces and taking photographs of astronomical objects, what can I do do move into astronomical spectroscopy? Are there any good resources for this? Pitfalls? And perhaps most importantly, what kind of initial outlay would I have to make?


How to find time taken for a spinning top to stop?

The angular position of a spinning top is given by $\theta = t^3 - 72t$, where $t$ is in seconds and $\theta$ in "radian".


QM interpretations

I don't fully appreciate what the discovery of the decoherence phenomenon adds to the Copenaghen interpretation of QM. I will be more precise: the Copenaghen interpretation, if I am not wrong, is summarized by the following concepts:   1. QM can, and has to, predict _only the probabilities_ of measurement outcomes ("has to" because descriptions of nature such as the classical one, where every quantity always assumes a certain value is in disagreement with our current experimental observations, such as the violation of Bell's Inequality.).   2. In order to do QM, we use state vectors. However, these must not be considered as a real _physical entity_ , but rather _tools_ in a similar way as probability distributions in classical physics (This means that in an EPR experiment, the change of the vector state describing Bob's qubit after Alice has measured hers in a galaxy far away is not in contrast with relativity.).   3. The probabilities are computed via the Born rule: the state after measurement is an eigenstate of the observable measured, and the outcome is the corresponding eigenvalue. Now the decoherence: if one considers the universe divided into a qubit (i.e. a two level system) system (S), apparatus (A) and the rest (E) can give a QM dynamical description of a measuring process on the qubit. Initially, the universe is in the state $|\psi\rangle = (a |0 \rangle + b |1 \rangle)\ |A_0 \rangle | \text{rest} \rangle$. If $|A_i \rangle $ are such that they can _imprint_ themselves in the environment, i.e. they do not entangle themselves with it, after the unitary evolution the SA system is described by the matrix: $\rho = |a|^2 |0\rangle\langle0||A_1\rangle\langle A_1| + |b|^2 |1\rangle\langle 1||A_2\rangle\langle A_2|$. This analysis explains why at macroscopic scales classical physics works, i.e. we don't see superposition of states and probabilities don't interfere: the environment interaction rapidly evolves the vector states into those states that can _imprint_ in the environment, or in mixture of them. This is surely interesting on its own, but is there anything else to understand from decoherence? The fact that superpositions states turn into mixtures of the eigenstates of the measured observable _seems_ to explain the "measurement problem" (that in the Copenaghen paradigm is not a problem actually, but the most important assumption, as stated in item 1 above) but it does not: after the above evolution the system is a classical mixture, but the probabilities are still there. Explaining the measurement problem means to find an evolution that deterministically bring the system into the outcome state (in the Copenaghen interpretation, this is rejected). So decoherence does not explain the Born reduction rule. Additionally, in order to interpret the mixture above as something analogue to a classical mixture, we must have already adopted the Born probability rule, so decoherence does not explain it either. To sum up: having adopted the Copenaghen interpretation of QM, decoherence explains _in its framework_ the transition from quantum world (superposition always possible, quantum interferences) to classical world (no Schroedinger's cat states, bayes rule), but does not replaces or explain any of its axioms...or does it?


Time Dilation Problem

![Here's the formula...](http://i.stack.imgur.com/K1zGw.gif) I'm having some trouble using the time dilation formula. Say an astronaut leaves Earth for 10 years, at 0.85c. How much time has passed according to an observer on Earth? I tried using the following formula: $$t = \frac{1}{\sqrt{1-(v^2/c^2)}}$$ but couldn't seem to get an answer that made sense. Any help would be much appreciated! This concept is super confusing to me.


What is the relationship between Schrödinger equation and Boltzmann equation?

The Schrödinger equation in its variants for many particle systems gives the full time evolution of the system. Likewise, the Boltzmann equation is often the starting point in classical gas dynamics. > What is the relationship, i.e. the classical limit, which connects these two > first order in time equations of motions? How does one approach this, or is there another way in which one sees the classical time evolution? Where are these considerations relevant?


Why is electric field zero inside a hollow metal sphere ?

The figure below shows a hollow metal sphere with a positive point charge $Q$ sitting outside it. What is the electric field at the center of sphere ? The answer is zero (look at here at the beginning of page 4), but I do not understand why? ![enter image description here](http://i.stack.imgur.com/zYFZC.png)


How to find full energy of field of an arbitrary half-integer spin?

Let's have arbitrary half-integer spin $n + \frac{1}{2}$ representation: $$ \Psi_{\mu_{1}...\mu_{n}} = \begin{pmatrix} \psi_{a, \mu_{1}...\mu_{n}} \\\ \kappa^{\dot {a}}_{\quad \mu_{1}...\mu_{n}}\end{pmatrix}, $$ $$ (i\gamma^{\mu}\partial_{\mu} - m)\Psi = 0, \quad \gamma^{\mu}\Psi_{\mu \mu_{1}...\mu_{n - 1}} = 0, $$ $$ \Psi_{\mu_{1}\mu_{i}...\mu_{j}\mu_{n}} = \Psi_{\mu_{1}\mu_{j}...\mu_{i}\mu_{n}}, \quad \partial^{\mu}\Psi_{\mu ...\mu_{n - 1}} = 0. $$ I need to find the general expression for the full energy of corresponding particles, but I don't know how to do that. I tried to do the following.   1. The solution of equations above may be given as $$ \hat {\Psi}_{\mu_{1}...\mu_{n}} = \sum_{s}\int \left(\hat {a}^{\dagger}_{s}(\mathbf p)u^{s}_{\mu_{1}...\mu_{n}}(\mathbf p)e^{-ipx} + \hat {b}_{s}(\mathbf p)v^{s}_{\mu_{1}...\mu_{n}}(\mathbf p)e^{ipx}\right)\frac{d^{3}\mathbf p}{\sqrt{(2 \pi )^{3}2E_{\mathbf p}}}, $$ where $s$ refers to the possible polarizations, $u, v$ are "spinor-tensor" waves which have some normalization law (as example) $u^{+}u = 2E_{\mathbf p}$.   2. The Dirac equation for each $\mu_{1}...\mu_{n}$ may be rewritten in a form $$ i\partial_{0}\Psi_{\mu_{1}...\mu_{n}} = \hat {H}\Psi_{\mu_{1}...\mu_{n}} = E\Psi_{\mu_{1}...\mu_{n}}. $$ So for finding the full energy I must construct integral like $$ E = \langle \hat {H} \rangle = \int \Psi^{\dagger} \hat {H}\Psi d^{3}\mathbf r = i \int \Psi^{\dagger} \partial_{0}\Psi d^{3}\mathbf r. $$ The expression $$ \int \Psi^{\dagger}\Psi d^{3}\mathbf r $$ must be lorentz invariant. But how to make it invariant? My first idea is convolute $\Psi^{\dagger}\Psi$ as $\Psi^{\dagger}_{\mu_{1}...\mu_{n}}\Psi^{\mu_{1}...\mu_{n}}$, and I think that it is the only possible way to construct the "correct" lorentz invariant object (in vector indices space), but it is not strictly. After that I only use conservation law $\partial_{\alpha}(\bar {\Psi}\gamma^{\alpha}\Psi ) = 0$. But I'm not sure at this.  Can you help? I know that in general the expression looks like $$ E = \sum_{s}\int (\hat {a}^{\dagger}_{s}(\mathbf p)\hat {a}_{s}(\mathbf p) - \hat {b}_{s}(\mathbf p) \hat {b}^{\dagger}_{s}(\mathbf p ))E_{\mathbf p} d^{3}\mathbf p . $$


How can something happen when time does not exist?

I saw this documentary hosted by Stephen Hawkins. And if I didn't get it wrong, it says that there was no time before the big bang, time was created there. So how can anything happen when there is no time (eg: the creation of time and the universe)? Is this what he calls a 'singularity'? Which I could think like this: when there is no time nothing can happen, except during this singularity. Of course I don't know of a law that says 'when there is no time nothing can happen..', I just came up with it, as something intuitive.


Why does this state have a Schmidt rank of 1?

A system is entangled if and only if the Schmidt rank is greater than 1. Why does this $$\left[\frac{1}{\sqrt{2}}\left(\left|0\right\rangle+\left|1\right\rangle\right)\right]\otimes\left[\frac{1}{\sqrt{2}}\left(\left|0\right\rangle+i\,\left|1\right\rangle\right)\right] = \frac{\left|00\right\rangle+i\,\left|01\right\rangle+\left|10\right\rangle+i\,\left|11\right\rangle}{2}$$ have a Schmidt rank equal to one? or equivalently why is this not entangled? ($\otimes$ denotes the tensor product)


How is bound charge and free charge possible?

I am studying Introduction to electrodynamics by Griffiths and I came along a concept I cannot seem to understand properly. The concept of free charge AND bound charge. I do not understand how we can have both. I understand bound is created by the presence of an E field and a dielectric. So where does the free charge come from if we are inside a dielectric ? Here is a quote from the book talking about it > In Sect. 4.2 we found that the effect of polarization is to produce > accumulations of bound charge, Pb = - V . P within the dielectric and CYb = > P . n on the surface. The field due to polarization of the medium is just > the field of this bound charge. We are now rejldy to put it all together: > the field attributable to bound charge plus the field due to everything else > (which, for want of a better term, we call free charge). The free charge > might consist of electrons on a conductor or ions embedded in the dielectric > material or whatever; any charge, in other words, that is not a result of > polarization.


Two-fluid description of superfluidity

I'm trying to teach myself about superfluidity and I'm slightly confused on the ''two-fluid'' description. From what I understand, the superfluid is considered to be a mixture of two fluids, a superfluid component and a normal fluid component. Are these two fluids a true ''mixture'' or are is it just a model to break the behavior of the one _real_ fluid into its two different aspects? Could we separate them from each other as if they were truly separate fluids? For example, if it were truly a mixture, we could make it do something that only a superfluid could do, such as pass through an extremely small opening. This would separate the fluids and leave us with just the superfluid component. Would this remaining ''pure''-superfluid still be described by the two-fluid model?


gamma-gamma collision seen from a relativistic rocket

This is a thought experiment. Consider two photons traveling at right angles and intersecting at some point. As seen from Earth these photons will not interact with each other because their energy is too low for particle creation. Now consider a rocket traveling at high relativistic speed towards the intersection point. People on that rocket will "see" these photons extremely blue shifted. They would appear as gamma rays. Now from the rocket point of view there is a higher probability that particle creation would occur, because these photons appear so energetic. Any produced particles like electrons or positrons would be captured by the rocket and brought back to Earth as a proof. The experiment could be repeated to get good statistics. How many particles would that rocket bring to the Earth. None? Or maybe the amount predicted by observers on the rocket? Disclaimer. I am not a physicist. Probably this "paradox" has a very simple resolution just like other relativity paradoxes. I probably made some wrong assumptions.


Homework Help - Physics / Calculus III / Force-Work Problem

I have tried to answer the problem correctly, and I have found similar (and even an identical) problem in my textbook, yet I still can't seem to yield the correct answer with this specific problem. The example problem in my textbook: A crate is hauled 8m up a ramp under a constant force of 200 N applied at an angle of 25 degrees to the ramp. Find the work done. W = F * D cos(25) = (200)(8)cos(25) = 1450J. Can anyone give me any advice on how to solve it? A woman exerts a horizontal force of 4 pounds on a box as she pushes it up a ramp that is 5 feet long and inclined at an angle of 30 degrees above the horizontal. Find the work done on the box. I would think that the work would simply be (4)(5)cos(30), but that is not the correct answer. (I've figured out how to accept answers now, so I'll accept the most helpful answer.)


Noether current for the Lagrangian without Lorentz invariance

I am reading an article by Watanabe & Murayama. It gives a proof on the counting of Nambu–Goldstone bosons without Lorentz invariance. I am trying to derive all the equations to get a better understanding. But I have met some trouble with Eqs. (5) and (6). How does the functions $e_i(\pi)$ are determined so that $\delta\mathcal{L}_{\mathrm{eff}}$ does not depend on $\pi^a$ explicitly? Could anyone provide a derivation of (6) from (4) and (5)? Many thanks!


Steps for going from Polyakov action to low-energy effective action (SUGRA?) in String Theory

A lot of string-theory questions have been asked here. This one is, hopefully, different in that this inquiry is specific and the expected answer would be more mathematical than philosophical in character. To give some context, I'm familiar with LQG, QFT etc. I'm a beginner in CFT and string-theory. On reading papers on AdS/CFT and String Theory the following sentence often appears: " **The low-energy effective action for $x$ dimensional ST of type $y$ reduces to that of supergravity (SUGRA) in $z$ dimensions with matter fields of types $a$, $b$ and $c$** ". The $64,000 dollar question any noob, such as me, wants answered is what are the mathematical/technical steps or arguments which lead us from the fundamental action of string-theory (the Polyakov/Nambu-Goto action) to a low- energy effective description. Of course, I might be asking a question for which the only concise answer is: " _go read GSW_ ". Here's hoping otherwise ;)


Function for heating of a planet by its star

I'm looking for an algorithmic model which will gives rough estimate of the average temperature of a planet's atmosphere (good enough to say whether there will be liquid water anywhere, or if metals and plastics will melt/ignite) based on the luminosity of its star, the radius of its orbit, the constituents of its atmosphere, the mass of the planet, and the surface area of the planet. Specifically, I figure I need two mathematical models. The first would simulate the incoming thermal energy from the star, and the second would calculate how much is lost as blackbody radiation by the planet, taking into account the effects of the 'greenhouse effect'. Does anyone know how to model these things?


What are the similarities and differences between speech and music sounds?

From an acoustic perspective, I guess speech sounds are produced by varying/manipulating:   1. resonance (shaping the cavity; something to do with harmony?)   2. f0 (the length of the string/pipe/whatever) However, in music, only (2) can be varied/manipulated. That is, the resonance/cavity doesn't change. As you probably guessed, I'm neither a physicist, a musician, or a speech pathologist. Can you help me understand better the similarities and differences of sound production for speech vs musical instrument?


Weak localization and constructive interference

To my knowledge weak localization is a result of constructive interference between scattering processes and their time reversed. Traveling a given path in a vector potential leads to the phase $q\int d\vec{x}\ \vec{A}(x)$. A particle traveling the time reversed path gets an additional "$-$"sign as the path is inverted. Somehow the spin flip seems to give an additional sign resulting in constructive interference. How is that? How is the phase of a particle in an vector potential connected to the spin?


Key difference between 3D and 4D solutions

In many papers in theoretical physics especially in the more advanced theories, I notice that physicists solve problems in 3 dimensions (2 spatial+ 1 temporal). In some specific papers (I can't seem to recall exactly which ones), some models solved in three dimensions gave finite solutions, whereas 4D (3+1) were either hard to solve or gave divergent results. I don't seem to understand why adding a 3rd spatial dimension should cause problems if 2 spatial dimensions gave no problems. Intuitively, it seems straightforward matter to extrapolate 4D solutions from 3D. Then what is the essential difference between the 2?


What determines the speed of waves in water?

While I was walking my dog this morning, I passed over a canal filled with boats, barges, and kayaks all of different masses and moving at different speeds. I noticed that all of these vessels left behind wakes, and the waves of these wakes moved through the water at different rates, though my ability to make good observations was limited. This made me wonder: what determines the velocity of these waves? I know that sound travels through water at 1,480 m/s or so, but surface waves travel at different rates (tsunamis travel faster than the ripples I make in my bowl of soup when I blow on it). In particular, how is the velocity of wake waves determined?


Superposition of waves hitting at right angle

I'm considering, in general, a spherical wave coming from a pin hole and landing on a wafer, both directly and off of a mirror. The mirror and wafer are at right angle. If it weren't a pin hole, and the light was coherent and moving straight at the mirror/wafer, I read that the period is $\lambda/(2\sin(\theta)$, but I have no idea why. I think the main part of this problem is just, in general, calculating the period of an interference pattern on a plane, of two light sources of the same wavelength, potentially in phase, hitting at two angles-one from the left, one from the right.


What interaction is responsible for the 21 cm Hydrogen line transition?

The 21 cm Hydrogen line is from the transition between the hyperfine levels of the ground state of the hydrogen atom. So, what interaction is coupling the hyperfine levels? I suspect that it is not the usual EM interaction, which causes the 1s-2p transition.


Is it possible that nuclear fission contributes to climate change?

This is probably a really stupid question, please forgive me. Is it possible that the use of nuclear fission on earth contributes to the increased energy in the Earth's system as according with the Laws of Thermodynamics regarding the Conservation of Energy?


Is a magnetic field affected by the presence of air?

Does air affect a magnetic field's ability to attract ferromagnets? If a magnetic field and a ferromagnet were placed in a vacuum would there be a better performance?


Definition of force for a scalar field

I have symmetry-breaking potential V that its reflection symmetry breaks in $\phi_0$. ![enter image description here](http://i.stack.imgur.com/qjjnz.jpg) The action in presence of gravity is: ![enter image description here](http://i.stack.imgur.com/X2VKk.jpg) How we calculate the scalar mediated force $F_\phi$ ? How can I obtain below relation? ![enter image description here](http://i.stack.imgur.com/7gWpX.jpg) Thanks for your Help!


Quartic terms as two particle interaction?

In some lecture notes I found the sentence: > [...] considering two toy models: the classical and quantum harmonic > oscillator with quartic perturbation. The quartic terms $\sim x^4$ and $\sim > \hat x ^4$, respectively, correspond to the two-particle interaction in > fermionic many-body models. Does the author only refer to a formal similarity (as in second quantization interaction terms are quartic) or can a perturbated harmonic oscillator really represent an interacting system? If so, how can one see this?


Geodesic deviation on a unit sphere

Very little interest in the original version of this question so I've rejigged it hoping for a more positive response. I'm trying to use the geodesic deviation equation$$\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}+R_{\phantom{\mu}\beta\alpha\gamma}^{\mu}\xi^{\alpha}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\gamma}}{d\lambda}=0$$ to show that on the surface of a unit sphere two particles separated by initial distance $d$ , starting from the equator and travelling north (ie on lines of constant $\phi$) will have a separation $s$ given by$$s=d\sin\theta.$$ This is similar to Geodesic devation on a two sphere except that question was solved using simple spherical geometry. My plan is find $\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}$ first by using the absolute derivative $$\frac{DV^{\alpha}}{d\lambda}=\frac{dV^{\alpha}}{d\lambda}+V^{\gamma}\Gamma_{\gamma\beta}^{\alpha}\frac{dx^{\beta}}{d\lambda}.$$ Then take the second derivative of this. Next find $\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}$ by calculating the Riemann tensor part$$R_{\phantom{\mu}\beta\alpha\gamma}^{\mu}\xi^{\alpha}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\gamma}}{d\lambda}.$$ And then try to juggle the results to show the separation $s=\xi^{\phi}$ as a function of $\theta$. The line element for spherical coordinates $$l^{2}=dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\phi^{2}$$ for a great circle of constant $\phi$ on a sphere of unit radius reduces to $$dl^{2}=d\theta^{2}$$ giving $\frac{d\theta}{dl}=\frac{d\theta}{d\lambda}=1$ and $\frac{d\phi}{dl}=\frac{d\phi}{d\lambda}=0$. The absolute derivative for $\Gamma_{\theta\phi}^{\phi}=\Gamma_{\phi\theta}^{\phi}=\frac{\cos\theta}{\mathbf{\mathbf{\sin\theta}}}$ is$$\frac{D\xi^{\phi}}{d\lambda}=\frac{d\xi^{\phi}}{d\lambda}+\xi^{\theta}\Gamma_{\theta\phi}^{\phi}\frac{d\phi}{d\lambda}+\xi^{\phi}\Gamma_{\phi\phi}^{\phi}\frac{d\phi}{d\lambda}+\xi^{\theta}\Gamma_{\theta\theta}^{\phi}\frac{d\theta}{d\lambda}+\xi^{\phi}\Gamma_{\phi\theta}^{\phi}\frac{d\theta}{d\lambda}=\frac{d\xi^{\phi}}{d\lambda}+\xi^{\phi}\frac{\cos\theta}{\mathbf{\mathbf{\sin\theta}}}.$$ And for $\Gamma_{\phi\phi}^{\theta}=\sin\theta\cos\theta$ is $$\frac{D\xi^{\theta}}{d\lambda}=\frac{d\xi^{\theta}}{d\lambda}+\xi^{\phi}\Gamma_{\phi\phi}^{\theta}\frac{d\phi}{d\lambda}+\xi^{\phi}\Gamma_{\phi\theta}^{\theta}\frac{d\theta}{d\lambda}+\xi^{\theta}\Gamma_{\theta\phi}^{\theta}\frac{d\phi}{d\lambda}+\xi^{\theta}\Gamma_{\theta\theta}^{\theta}\frac{d\theta}{d\lambda}=\frac{d\xi^{\theta}}{d\lambda}.$$ However, $$\frac{D\xi^{\phi}}{d\lambda}=\frac{d\xi^{\phi}}{d\lambda}+\xi^{\phi}\frac{\cos\theta}{\mathbf{\mathbf{\sin\theta}}}$$ doesn't look right as it blows up when $\theta=0$. Any suggestions where I might be going wrong?


Dyson air blade principle as a propulsion system for an autonomous underwater vehicle?

Any thoughts on applying this principle to an AUV (autonomous underwater vehicle) ? if the power source is a battery, would its efficiency be beneficial to that of a propulsion system?


Why do spherical waves diminish as 1/$r^2$?

Seeing the derivation of a plane wave in Giancoli, I can't understand why a spherical wave would diminish in amplitude. Shouldn't the peaks still be mutually induced, and therefore nondiminishing?


Classical models with unbounded particle number

Is there any classical model which deals with the birth, life and death of particles? What application could it have? I am talking about a 'billiard-ball' kind of model, but the kind in which balls keep popping in and out. What would the configuration space of such a system be like if particle number was not conserved? How would the Hamiltonian evolve with time? I am not interested in a statistical models. I can't think of any real world application of such a model. But perhaps there is a mathematical apparatus to deal with the continuous evolution of such systems?


Ship wake height

It is well known that a ship traveling in a deep water produces wake. Far from the aft of the ship, these wake are called a Kelvin wake. I found the mathematical expression of the shape of the wake, but not its height vs. $r$ where $r$ is the distance from the ship's aft.


What are the viable non-anthropic explanations for the tininess of the cosmological constant?

The cosmological constant is 10-120 times its natural value, but it is yet nonzero. Even TeV-scale supersymmetry breaking can't save it. The renormalization group would seem to imply it ought to be at the Planck scale, assuming naturalness, and no fine-tuning. The anthropic principle explains the size of the cosmological constant nicely, but are there any other explanations? Dynamical mechanisms run into problems with the renormalization group. The mechanism has to act in the infrared, but any ultraviolet definition will be modified by the renormalization group.


How to set up a very simple experiment in optics?

This might come across as a very rudimentary question. My fundamentals of Optics are weak. In the optics chapter of my physics text book I saw diagrams each depicting an object on the left and a lens -- typically convex -- in the middle and an inverted image forming on the right side. Let's say I want to perform this experiment, how can I do it? I have always seen such lenses as parts of complex systems like cameras. Do I get such stand-alone lenses from the market off the shelf? And if I keep a candle on the left side, on what screen do I capture the image on the right side? A vertical plate, paper, or the like won't show an image. (Why is this so?)


How are we able to view an object in a room with bulb..?

This is a very basic question on optics. How are we able to view an object kept in a room with a bulb? From what I understand, light rays from bulb will hit the object and some colour will be absorbed and rest will be thrown off. but how does the thrown off light reach eyes? I thought a light ray will reflect with same angle back (as the angle between incident ray and normal) then how is it guaranteed that a reflected ray will reach eyes? ![enter image description here](http://i.stack.imgur.com/X2En8.png)


What causes there to be an offset between the light lensing through the space neighboring galaxy clusters and the galaxy clusters themselves?

http://www.wired.com/wiredscience/2013/01/musket-ball-dark-force/ > The reason this is strange is that dark matter is thought to barely interact > with itself. The dark matter should just coast through itself and move at > the same speed as the hardly interacting galaxies. Instead, it looks like > the dark matter is crashing into something — perhaps itself – and slowing > down faster than the galaxies are. But this would require the dark matter to > be able to interact with itself in a completely new an unexpected way, a > “dark force” that affects only dark matter. Lawrence M. Krauss > If you removed all of the particles, all of the radiation, absolutely > everything from space and all that remained was nothing that nothing would > weigh something." What weighs something has mass. If you remove everything else then all which remains is spacetime which means mass is associated with spacetime, correct? Is it necessary for physicists to conjure a 'new dark force' or is it more likely mass is associated with spacetime and it is the mass associated with spacetime which 'piles up' analogous to the bow waves of two boats which pass by each other very closely?


Levi Civita covariance and contravariance

I read some older posts about this question, but I don't know if I'm getting it. I'm working with a Lagrangian involving some Levi Civita symbols, and when I calculate a term containing $\epsilon^{ijk}$ I obtain the contrary sign using $\epsilon_{ijk}$. I always apply the normal rules: $\epsilon_ {ijk}=\epsilon^{ijk}=1$; $\epsilon_ {jik}=\epsilon^{jik}=-1$ etc. I believed that there is no difference between covariant and contravariant Levi-Civita symbol. What do you know about this?


Interpretation of space time Minkowski diagram

How to interpret the following space-time diagram in the image. I know how to interpret euclidean distance from Euclidean space diagram ![enter image description here](http://i.stack.imgur.com/OdtPO.jpg) **omit the line "whereas for Euclidean space".**


Phase and Group Velocity of Electromagnetic Waves

Moving charges produce oscillating electric and magnetic fields -we have an electromagnetic wave.   1. In terms of moving charges or at the level of charges, what is phase velocity and group velocity of an electromagnetic wave?    2. What is the origin of these velocities?  I am to not able to relate the definitions of phase velocity(how fast the phase of a wave is moving) and group velocity(velocity of the envelope) to moving charges. I am trying to get an intuitive picture of this. Please help me.


Is Feynman talking about the Zeroth Law of Thermodynamics?

In Volume 1 Chapter 39 of the Feynman Lectures on Physics, Feynman derives the ideal gas law from Newton's laws of motion. But then on page 41-1, he puts a caveat to the derivation he has just completed (italics in original): > Incidentally, when we say that the mean kinetic energy of the particle is > $\frac{3}{2}kT$, we claim to have derived this from Newton's laws.... and it > is most interesting that we can apparently get so much from so little ... > How do we get so much out? The answer is that we have been perpetually > making a certain important assumption, which is that if a given system is in > thermal equilibrium at some temperature, it will also be in thermal > equilibrium with _anything else_ at the same temperature. For instance, if > we wanted to see how a particle would move if it was really colliding with > water, we could imagine that there was a gas present, composed of another > kind of particle, little fine pellets that (we suppose) do not interact with > water, but only hit the particle with "hard" collisons. Suppose that the > particle has a prong sticking out of it; all our pellets have to do is hit > the prong. We know all about this imaginary gas of pellets at temperature > $T$ - it is an ideal gas. Water is complicated, but an ideal gas is simple. > Now, _our particle has to be in equilibrium with the gas of pellets_. > Therefore, the mean motion of the particle must be what we get for gaseous > collisions, because if it were not moving at the right speed relative to the > water but, say, was moving faster, that would mean that the pellets would > pick up energy from it and get hotter than the water. But we had started > them at the same temperature, and we assume that if a thing is once in > equilibrium it stays in equilibrium - parts of it do not get hotter and > other parts colder, spontaneously. This proposition is true and can be > proved from the laws of mechanics, but the proof is very complicated and can > be established only by using advanced mechanics. It is much easier to prove > in quantum mechanics than it is in classical mechanics. It was first proved > by Boltzmann, but now we simply take it to be true, and then we can argue > that our particle has to have $\frac{3}{2}kT$ of energy if it is hit with > artificial pellets, so it also must have $\frac{3}{2}kT$ when it is being > hit with water at the same temperature and we take away the pellets; so it > is $\frac{3}{2}kT$. It is a strange line of argument, but perfectly valid. My question is, what is the proposition of Boltzmann that Feynman is referring to? I can think of three possibilities:   1. It could refer to the Equipartition Theorem, which was independently proven by Maxwell and Boltzmann, since the title of the section is "The Equipartition of Energy".   2. It could refer to the Zeroth Law of Thermodynamics, because "it will also be in thermal equilibrium with _anything else_ " (especially when he emphasizes the "anything else") sounds like the fact that thermal equilibrium is an equivalence relation.    3. It could refer to Boltzmann's H-theorem, which was the derivation of the second law of thermodynamics from Newton's laws, because "if a thing is once in equilibrium it stays in equilibrium - parts of it do not get hotter and other parts colder, spontaneously" sounds like some formulations of the second law. So which of the three is it, if it's any of the three, and how would you prove it from Newton's Laws? Any help would be greatly appreciated. Thank You in Advance.


Practical (maybe naive) question on boiling water

I put a glass bottle in a big pot, bottleneck up, I fill both pot and bottle with very hot water. The bottle is submerged except a few cm of the bottleneck and filled to the brim so some water inside the bottle is above the pot water. Then I light the fire. After a while the water in the pot boils but even if I keep it boiling for 30 minutes the water inside the bottle does not boil. All I see in the water inside the bottle are very tiny bubbles. The same happens if I put the bottle upside down (with some air at the top) What is the reason? Maybe the glass acts as insulator and water inside the bottle cannot reach 100 degrees? Or having a few cm of bottle (and water) above the pot surface cools the water inside?


How does radiation shielding using absorbing materials work?

I understand that, for example, a thick enough sheet of lead can absorb gamma radiation, but I want to understand what actually happens at the molecular/atomic/subatomic level. Also, can the same logic be applied to cosmic particles? I have tried Googling for an answer, but to no avail. Can someone enlighten me?


Anyons without fractional spin?

Is it possible to have particles obeying anyonic statistics but not having fractional spin? I am wondering, because while spin in quantum physics arises from the geometry/topology of spacetime, statistics is connected to the geometry/topology of configuration space. What would particles like this be called?


What is the requirements from an optical lab in terms of darkness

I am characterizing an optical lab and would like to know therefore what are the requirement from a standard optical lab in terms of ambient luminescence. In other words how dark should an optical lab be. The lab I am talking about should fit for experiments of photoluminescence and Raman spectroscopy.


Throwing a ball upwards in an accelerating train

If I throw a ball upwards to a certain height in an accelerating train, will it end up in my hand? At the moment I release the ball, it will have a velocity equal to that of the train at that instant. But because the train is accelerating, the train will have a greater velocity and thus the ball will travel lesser distance and should fall behind me. Is my reasoning correct?


Integration of partition-function over many momentum variables

My integral looks like $$Z(\beta) = \frac{1}{h^3}\int d^3p\ \exp{\left(-\frac{\beta}{2m}\sum^{3N}_{i=1}p_i^2\right)}.$$ I'm confused about how to integrate over seemingly 3N variables in only a 3-dimensional integral.


Gravitational effect of the distant universe

When we observe at close to the cosmological horizon (as close as we currently can) we see the universe as a much younger, denser place. As we feel the gravitational forces from these objects distant, will we measure a tidal force increasing the distance between an object less distant than the cosmological horizon and ourselves? Or for that matter between any two objects at different distances from earth? If so is it great enough to account for the accelerating expansion of the universe?


Derive force from the pair-wise potential equation

How can we calculate the force exerted on particle $i$ by particle $j$ given a general potential function $V(d)$? Let's discuss this genenal question with a concrete example below. To simulate the motion of a set of $n$ particles with equal masses, we need to calculate the resultant force for each particle, which is the sum of force vectors. These particles move along a one-dimensional axis and interact via a pair-wise potential, where the $$V(d) = \frac{\exp(-2d^3) - 1}{d},$$ where $d$ is the distance between particle $i$ and particle $j$. I don't have strong math and physics background, and I've been learning partial derivative and potential energy on wikipedia, but it makes me more confused. Simeon Carstens' answer below explains a lot!! It says $$F_i=−\frac{\partial}{\partial x_i}V(x_1,...,x_n).$$ So my understanding is that the force exerted on particle i by particle j can be expanded as $Fi=-\frac{∂}{∂xi}V(d(xi,xj))=-\frac{∂}{∂xi}\frac{exp(-2(xi- xj)^3) - 1}{xi-xj}$. By $(\frac{u}{v})'=\frac{u'v-v'u}{v^2}$, the above epxression can be derived as $Fi=\frac{(6(xi-xj)^3+1)exp(-2(xi-xj)^3) - 1}{(xi-xj)^2}$. Am I correct? Please tell me if I'm wrong and which part is incorrect! Any help is apprieciated!


Friction From an Object On-Top of a Sliding Object

Consider a block $A$ lying on a flat and frictionless table, and a block $B$ lying on top of block $A$. A horizontal force $F$ is applied to block $A$. If there is no friction between blocks $A$ and $B$, then block $A$ will move in the direction of $F$ and $B$ will remain stationary until $A$ slides out from under it, at which point it will fall. If the friction between blocks $A$ and $B$ is nonzero, however, then block $B$ will move with block $A$ in the direction of $F$, as expected. While this result makes sense intuitively, and is consistent with out everyday experience, I've been unable to rationalize this behavior in terms of forces. If block $B$ is moving along with block $A$, then there must be some force acting on $B$ (it started out at rest, after all), and my teacher said this force is a frictional which is equal to $F$. This seems strange, considering my previous experience with friction, which I understand to be a force that _opposes motion_ , not one that causes it. How is it possible that there could be a frictional force acting on $B$ if there is no other force acting on it? Perhaps my understanding of friction is too narrow, but I do not see how we can mathematically or logically justify why friction would be causing motion in a body that is otherwise not subject to any forces.


Expansion of the Universe: Conversion of gravitational potential energy to kinetic energy?

Suppose there is an object floating in space which over time begins to fall toward the source of a gravitational field. As it falls, its motion happens to be such that it gets locked in orbit around the source with a greater velocity than it had before it 'began to fall'. So it's gravitational potential has been converted to kinetic energy. According to relativity, this increased speed should increase the gravitational potential of the object (and therefore the object + the original source of the field), correct? Does this mean that as a result of this, the expansion of the Universe should slow down slightly (because there is now a slightly greater gravitational potential in the Universe)? And if so, would that imply that the conversion from gravitational potential to kinetic energy is in a sense a conversion between the bulk kinetic energy of the expanding Universe and the local kinetic energy of a test mass?


Energy required to demagnetize a soft iron?

How much energy is required to demagnetize a soft iron ferromagnetic material, that has a very low coercive force And a small hysteresis area? Also, would it be possible to demagnetize that same soft iron while it's aligned with an exterior magnetic field via vibrations from hammering/dropping? **The hammering is constant so the vibrations should be constant as well.** That's why I assume it can be demagnetized.


How to make a small tokamak?

$\require{mhchem}$I made a fusor once, like the easy science project: deuterium-deuterium ones, but they're really inefficient. I was wondering if it would be possible to make a small tokamak; not one that will make a ton of power, but I just want to make one for the sake of making one, like a tokamak the size of a pizza box. What materials would I need and how would I go about assembling it? Assume it is deuterium-deuterium or $\ce{p-^11B}$ fusion, maybe $\ce{^3He-^3He}$.


Notational(?) Question in Whiting's Paper "Mode Stability of the Kerr Black Hole"

I am a math grad student attempting to read Bernard Whiting's paper "Mode Stability of the Kerr Black Hole." If you are in a university network, the paper should be easily found by a google search. At the beginning of the paper Whiting refers to the "master equation of Teukolsky" from the paper "Rotating Black Holes: Separable Wave Equations for Gravitational and Electromagnetic Perturbations." However, the equation Whiting writes down appears (to me) to be different than Teukolsky's. I assume I am missing something silly and was hoping that someone else has read this paper and can easily point out my mistake. One needs to look at the paper for all the relevant definitions, but for ease of reference: Whiting writes down $\\{\partial_r\Delta\partial_r - (1/\Delta)\\{(r^2+a^2)\partial_t + a\partial_{\phi} - (r-M)s\\}^2 - 4s(r+ia\cos\theta)\partial_t + $ $\partial_{\cos\theta}\sin^2\theta\partial_{\cos\theta} + (1/\sin^2\theta)\\{a\sin^2\theta\partial_t + \partial_{\phi} + i\cos\theta\cdot s\\}^2\\}\psi = 0$ where $\Delta = r^2 - 2Mr + a^2$ while Teukolsky writes down $\\{(\frac{(r^2+a^2)^2}{\Delta} - a^2\sin^2\theta)\partial^2_t + \frac{4Mar}{\Delta}\partial^2_{t,\phi} + (\frac{a^2}{\Delta} - \frac{1}{\sin^2\theta})\partial^2_{\phi} - \Delta^{-s}\partial_r(\Delta^{s+1}\partial_r) - $ $\frac{1}{\sin\theta} \partial_{\theta}(\sin\theta\partial_{\theta}) - 2s(\frac{a(r-M)}{\Delta} + \frac{i\cos\theta}{\sin^2\theta})\partial_{\phi} - 2s(\frac{M(r^2-a^2)}{\Delta}-r-ia\cos\theta)\partial_t$ $ + (s^2\cot^2\theta - s)\\}\psi = 0$ When I expanded out Whiting's equation out I did not get Teukolsky's equation. For example, it is clear that the $\partial_r$ terms are different. I also got that the constant terms (no derivative terms) did not agree.


Is a track in a cloud chamber made by one particle or many particles?

Is the track in a cloud chamber made by one particle interacting with many particles each emiting a photon or is the track made by many particles interacting on the same trajectory? or is it a cascade of interactions?


Implicit Differentiation, A doubt

$v=v_c(\tau, t)$ is a smooth function and suppose we have a relation $y_c(\tau,v_c;t)=0$ when $x_c$ is written in the form $x_c=c+ty_c(\tau,v_c;t)$, $c$ is real constant, $t$ is real number denotes time, could any one tell me how to find implicit differentiation of the relation $y_c(\tau,v_c;t)=0$ and from there how we get $\frac{\partial}{\partial t} v_c(\tau;t)|_{t=0}=-\frac{1}{2} a_c(\tau)$ where $a_c(\tau)$ denote the initial acceleration. page $2726,$ The paper from which I reading this Chillingworth, D. R. J. "Dynamics of an impact oscillator near a degenerate graze." Nonlinearity 23.11 (2010): 2723.


I want the Saturn's position in terms of Declination and Right Ascension?

I want the Saturn's position in terms of Declination and Right Ascension for a couple of month in the interval of 1 hour in a text file to do a simulation. Which site can provide me these data? Or, there may be some software which can give these data in txt file. I know Stellarium 0.12.0 which show data. But does it provide the options to save the data in the text file also?


How to describe heat transfer between two solid materials?

A general equation for dealing with heat transfer between one material and a region of insulating material. I've seen basic heat transfer equations for one material, but I'd love to see an explanation of how to do two.


Compton Scattering

Compton Scattering essentially states that when a photon of a given wavelength hits an electron the energy level of the electron changes and the photon has its wavelength changed. This seems to be implying that it is the same photon that is reflected outwards. Do we know it is the same photon of a changed wavelength, or is it possible that the original photon merely pushed a 2nd photon out of the electron? (This would mean the original photon is not changing wavelength, that is meerly absorbed).


Universal expansion faster than the speed of light

If the universe is expanding faster than the speed of light, and force carrier particles move at the speed of light, wouldn't this cause infinite universal expansion? Since no forces would be acting upon the outer edge of the universe? (This doesn't account for the apparent acceleration of the universe but it does account for the lack of slowing down for the universe).


Light bodies at high speeds behave like heavy objects when recorded by a high speed camera

This is potentially a very dumb question, but recently I was looking at some slow motion footage of bullets hitting stuff and I realized something interesting. One of the clips showed two metal objects (approximately bullet sized) colliding in midair at high speeds. They didn't go directly at each other. In fact their approach was something like this               ------>           7          /         /        /       /      So as they approached eachother "slowly", when they finally connected the contact surface practically exploded. In any case, the deformation was much larger than what I expected considering the difference vector of their velocities was quite small (they were traveling at the same speeds, but slightly different angles). This reminded me of scenes in sci-fy movies where two large spaceships collide in space but at low speeds. Since their mass is very high, the lack of their speed doesn't matter. They rip through each other. Related to this, imagine you grab an ant or some small insect and throw it at the wall. It might grab the wall or fall off, but it won't be hurt. I imagine if you recorded it with a high speed camera you could apply the same principle. Like bullets, it behaves like a heavy object but since you can't throw it nowhere near as fast as a bullet, it's "heavyness" isn't enough because the camera "slows it down" enough so he doesn't get hurt. My question is, what's the correlation between those three factors? The mass, the speed of the object and the speed of the camera. Could we potentially make small models of spaceships, throw them at each other very fast so when you record them in slow motion it looks like big spaceships colliding slowly?


Canonical equal time commutation relations in QED

I understand that to quantize the classical electromagnetic field one needs to impose commutation relations and express the field in terms of creation and annihilation operators. I notice that the canonical equal-time commutation rule comes in to use here. How do you derive this? What is it meant to represent? ![enter image description here](http://i.stack.imgur.com/pEpBe.png) Also, how do you prove the following commutation relation? ![enter image description here](http://i.stack.imgur.com/BLWNa.png)


How to find the amplitude of magnetic field of this plane wave

Because my previous question was short and confusing, it was put on hold. This is my new question, which getes straight to the point. * * * I have an electromagnetic plane wave which has wavelength $λ$ = 1 mm which is traveling through air parallel with $z$ axis (to +$z$) .The amplitude of the electric field is perpendicular to the $z$ axis and is stated $\vec{E}_o=A(\hat x+\hat y)$ where $A$ = 0,5 V/m. How can I find the magnetic field amplitude of $\vec{H}_o $ . I think I must use Maxwell's equation $ ∇×\vec{E} = − j\omega\mu_0 \vec{H} $, from which follows $ \vec{H}_o = \vec{k} \circ \vec{E}_o / \omega \mu_0$ , but where can the propagation constant $k$ can be derived?


Can a difference in the "speed of time" introduce acceleration?

Hypothetically, lets say we have a space divided equally into two adjacent areas where (somehow) in one of the areas time goes by at half the speed as the other area. Or specifically, when a clock in the fast area shows 1 minute having gone by, a clock in the slow area shows only 30 seconds having elapsed.   1. In order for the speed of light to be constant, would lengths in the fast half be compressed to half the size of the slow half? **My guess:** _Yes_   2. (assuming (1) is correct) Let's create two identical objects, one in each area, positioned as close together as possible without passing into the other area, each given an equal velocity in a direction parallel to the boundary between the two spaces. **Q:** Over time would the distance between the objects increase? **My guess:** _No_   3. OK, (2) was all kinds of lame. So to make it more interesting lets put a strip of space between (but not touching) the two objects where time flows at the average rate between the two areas (so 75% as fast as the "fast" area). Then lets connect the two objects with a rigid bar of negligible size and mass that goes right through the "average speed" area. Then at the exact center of the bar (equidistant between the two objects) we give the compound object a specific velocity - again in a direction parallel to the boundaries between the areas. **Q:** Would the compound object rotate? **My guess:** _yes - but it's just a hunch_   4. (if (3) is correct) Depending on the direction of the rotation, since you would constantly get the two objects changing which area they are in, would the compound object just keep spinning faster, accelerate toward the slow area, or accelerate toward the fast area?   5. I've totally been ignoring mass on purpose. I realize that the same things that cause time dilation also cause a change in mass. Would taking mass into account change the results of any of the above? (Edit below) I realize that this is all hypothetical and is a situation that cannot exist naturally (which is why I tagged it as a thought-experiment). What I want to know is whether the time-dilation portion of a space-time curvature introduces an acceleration above and beyond that caused by gravity, or whether it gets cancelled out by the length distortion, or if the curvature of space-time is what causes the acceleration of gravity and it would make no sense to talk about the effects of time-dilation separately. The thought experiment is just my effort to figure out exactly what the interactions between time and gravity are.


A small spherical gas bubble of diameter $d= 4$ μm forms at the bottom of a pond. When the bubble rises to the surface its diameter is $n=1.1$ times bigger. What is the depth in meters of the pond? Note: water's surface tension and density are $σ= 73 \times 10^{-3} \mbox{ N}$ and $ρ= 10^3 \mbox{ kg/m}^3$, respectively. The gas expansion is assumed to be isothermal. My attempts: I used the equation of pressure: $P_1V_1=P_2V_2$ where $P_1$ is the pressure at the top and $V_1$ is the volume of bubble at the top and $P_2$ is the pressure at the bottom, and $V_2$ is the volume of the bubble at the bottom Because the bubble at the bottom, it received the hydrostatic pressure, so the equation became: $P_1V_1$= $(P_0+\rho g d) V_2$ Since $V_1$ and $V_2$ is sphere, we can use the sphere volume. And $P_1$ is same with atmospheric pressure= $10^5 \mbox{ Pa}$ $10^5 \cdot (\frac{4}{3} \pi r_1^3) = (10^5 + 10^3 \cdot 9.8 \cdot d) \cdot (\frac{4}{3} \pi r_2^3)$ Cancel out the $\frac{4}{3}\pi$ and we get: $10^5 \cdot (r_1)^3 = (r_2)^3 \cdot (10^5 + 9800d)$ Substitute $r_1 = 4 \times 1.1= 4.4 \mbox{ μm}= 4.4\times10^{-6} \mbox{ m}$ and $r_2 = 4 \times 10^{-6} \mbox{ m}$ $10^5 \cdot (4.4 \times 10^{-6})^3 = (4 \times 10^{-6})^3 \cdot (10^5 + 9800d)$ $10^{-13} \cdot (4.4)^3 = 64 \times 10^{-18} \times 10^5 + 64 \times 10^{-18} \times 9800d$ $85.184 \times 10^{-13} = 64 \times 10^{-13} + 627200 \times 10^{-18} d$ $21.184 \times 10^{-13} = 627200 \times 10^{-18} d$ $d= 3.37 \mbox{ m}$ So, the depth of the pond is $3.37 \mbox{ m}$. My question: what is the useful of $σ = 73 \times 10^{-3} \mbox{ N}$ ? I really confused about it. Thanks


How do I calculate speed of air on each side of airfoil?

I am doing a project on Bernoulli's principal for my High School physics course. I made an airfoil in accordance with NACA 2412 (same as a C-152). I am going to put it in a wind tunnel where the wind speed is going to be measured. How can I find the wind speed on top of and on the bottom of the airfoil, once I know what the outside wind speed (or velocity of the airfoil) is? My end goal is to find the pressure differences on the outside of the wing, and I can do that very easily once I know the speed on each side of the airfoil.


Is 3+1 spacetime as privileged as is claimed?

I've often heard the argument that having 3 spatial dimensions is very special. Such arguments are invariably based on certain assumptions that do not appear to be justifiable at all, at least to me. There is a summary of arguments on Wikipedia. For example, a common argument for why >3 dimensions is too many is that the gravitational law cannot result in stable orbital motion. A common argument for <3 dimensions being too few is that one cannot have a gastrointestinal tract, or more generally, a hole that doesn't split an organism into two. Am I being overly skeptical in thinking that while the force of _gravity_ may not be able to hold objects in stable orbits, there most certainly exist sets of physical laws in higher dimensions which result in formation of stable structures at all scales? It may be utterly different to our universe, but who said a 4D universe must be the same as ours with one extra dimension? Similarly, isn't it very easy to conceive of a 2D universe in which organisms can feed despite not having any holes, or not falling apart despite having them? For example, being held together by attractive forces, or allowing certain fundamental objects of a universe to interpenetrate, and thus enter a region of the body in which they become utilized. Or, conceive of a universe so incomprehensibly different to ours that feeding is unnecessary, and self- aware structures form through completely different processes. While I realise that this is sort of a metaphysical question, is 3+1 dimensions really widely acknowledged to be particularly privileged by respected physicists?


What is $\hat{p}|x\rangle$?

Trying to solve a QM harmonic oscillator problem I found out I need to calculate $\hat{p}|x\rangle$, where $\hat{p}$ is the momentum operator and $|x\rangle$ is an eigenstate of the position operator, with $\hat{x}|x\rangle = x|x\rangle$. It turns out that $\hat{p}|x\rangle$ should be equal to $-i\hbar\frac{ \partial |x\rangle }{\partial x}$. But what does this even mean? I'm not sure what to do with an expression like that. If I try to find $\langle x | \hat{p} | x_0 \rangle$, I end up with $-i\hbar \frac{\partial}{\partial x} \delta(x-x_0)$, which I'm pretty sure is not allowed even in physics, where we would scoff at phrases like "$\delta$ is really a distribution and not a function". Edit: Motivation. I have solved Heisenberg's equation and found, for a Hamiltonian $H = p^2/2m + m\omega^2x^2/2$, that $\hat{x}(t) = \hat{x}_0 \cos(\omega t) + \frac{\hat{p}_0}{m\omega} \sin(\omega t)$. I am given that the initial state is $|\psi(0)\rangle = |x_0\rangle$, i.e., an eigenstate of position, and I have a hunch that by finding $\hat{x}(t)|x_0\rangle$ I might find a way to get $|\psi(t)\rangle$. But this is not what I need help with; I am now curious as to what $\hat{p}|x\rangle$ might be, never mind my original problem.


Statistical Mechanics - Distribution of Energies

Consider a state space $\mathbb{X}$. The probability density function under a canonical ensemble is given by the Boltzmann distribution $$\pi_{\mathbb{X}}(x)=\frac{e^{-\beta E(x)}}{\mathbb{Z}(\beta)}$$ where $E(x)$ is the energy of state $x\in\mathbb{X}$, and $\mathbb{Z}(\beta)$ the partition function or _Zustandsumme_. I see the marginal density of energy given as $$\pi_{U}(E)=\frac{\Omega(E)e^{-\beta E}}{\mathbb{Z}(\beta)}$$ where $\Omega(E)$ is the density of states. My question is how do you derive the marginal density for energies from the density over the state space $\mathbb{X}$?. $E:\mathbb{X}\rightarrow\mathbb{R}_{+}$ is not a one-to-one function so one cannot apply the change of variables theorem.


meteor explosion, steam or shock/vibration trigger

Is the explosion of a meteor due to forces imparted to the meteor by entry into atmosphere, or is it due to heating of a meteor that contains water with conversion to steam triggering the explosion? Do chondrites explode more readily because of some water content than nickel/iron meteorites?


What is the meaning of phase difference?

What do you mean by phase of a wave? And phase difference? Waves have always confused me as it's too difficult to visualize them. I am no good at waves mechanics, so if anyone could explain in simpler term? Also if anyone could direct me to complete solution to my problem as to study waves from where? How do you visualize it? How did you study and understand? I have been meaning to ask this question for quite a time but always thought it would be too dumb. Thank you and I am eagerly waiting for an answer and your valuable suggestion


What materials focus EM radiation in the 2.4GHz range

If glass and similar materials refract visible light effectively, what materials would be best for focusing lower frequencies of EM radiation, if any? If not, what other methods exist for focusing these ranges? The thought was inspired by wondering how you might build a camera that captures 'light' given off by Wifi routers


"Optically performed" Fourier Transform

This article says that they are only able to achieve such extremely high fiberoptic data rates because the multiplex light and then use a Fourier Transform to split it up again. But they say that they perform the Fourier Transform "optically". Now I understand the math behing a Fourier Transform, but I still can't wrap my mind around how this would be done in analog with light. Any ideas?


Pressure inside a light bulb

It is known that nowadays most light bulbs are filled with gas in order to minimise the evaporation rate of tungsten. Is e.g. argon put into the bulb with reduced pressure, so that the glass doesn't explode while heating the wire?


What is sound in terms of acoustic sources?

Sound is nothing more than small amplitude, unsteady pressure perturbations that propagate as a longitudinal wave from a region in space which created it (called the source region) into a quiescent (still) region where it is observed by hearing. It is at this point where I get slightly confused in that we may distinguish between two types of pressure perturbations: **hydrodynamic pressure perturbations** and **acoustic pressure perturbations**. The acoustic perturbations are what we term sound, as they are characterized by their ability to _propagate_ into the hearing region. The hydrodynamic pressure perturbations could be a consequence of fluid flow simply changing in the source region, and this may not necessarily propagate and become what we term sound. Perhaps evanescent waves can serve as some insight to what I am trying to explain: with reference to this link the pressure fluctations created by the subsonically pulled wavy plate decay exponentially in the upward direction and so do not constitute a sound. Is my interpretation so far correct? If so are there any rigorous methods for determining if pressure fluctations will propagate (thus becoming sound), or is sound simply that which we can hear outside a source region? Are there other cases besides the linked example for which pressure fluctations are created, but do not propagate as sound?


Pendulum in an elevator

![enter image description here](http://i.stack.imgur.com/p5vwp.png) Suppose we have a pendulum tied to the ceiling of an elevator which is at rest. The pendulum is oscillating with a time period $T$, and it has an angular amplitude, say $\beta$. Now at some time when the bob of the pendulum was at the mean position(equilibrium position/center position), the elevator suddenly starts moving up with some acceleration $a$. I need to find out what is the new angular amplitude of the pendulum, in terms of $a$,$\beta$ and if needed $T$, and $l$(length of the string). I'm completely stuck at this concept. I absolutely do not know how to proceed. My ideas are that we could use some conservation, maybe the energy of the bob. Any hints? P.S. I'm sorry if anything is too elementary.


Example in the book: A simple accelerometer

# A simple accelerometer > _You tape one end of a piece of string to the ceiling light of your car and > hang a key with mass m to the other end (Figure 5.7). A protractor taped to > the light allows you to measure the angle the string makes with the > vertical. Your friend drives the car while you make measurements. When the > car has a constant acceleration with magnitude a toward the right, the > string hangs at rest (relative to the car), making an angle $B$ with the > vertical._ > >   * _(a) Derive an expression for the acceleration $a$ in terms of the mass > m and the measured angle $B$._ > >   * _(b) In particular, what is a when $B$ = 45? When $B$ = 0?_ > > I don't care about the answers, the important thing is the following:- The book says The string and the key are at rest with respect to the car, but car, string, and key are all accelerating in the +x direction. Thus, there must be a horizontal component of force acting on the key. That's the reason the book decided to consider a force in the $+x$ direction, but I'm looking for a better explanation: how would I find detect the force in the $+x$ direction in another way? To me, when I draw the free body diagram of the string, there looks to be no force acting on the $+x$ direction! I understand it starts with noticing that the string is attached to the ceiling of the car, and that the car has force causing acceleration in one direction, but I don't know how to go further than that.


How does one interpret the Dirac equation with a self-field potential?

EVERY QFT text I've ever examined states that if there is an external vector potential, $A_\mu$, then one writes the Dirac eq.(or Klein-Gordon eq.) using a covariant derivative to include this U(1) gauge field, $A_\mu$. No problem. Wikipedia states that one must also include the contribution due to the electron self-field, $A_\mu'$. Again, makes perfect sense. Thus if there is no external 4-vector potential, the Dirac equation should exhibit the internal one! However, all texts state the "free" Dirac equation, without an accompanying vector potential. If I now solve the free Dirac equation, the spinorial solutions can be used to construct the Dirac charged current. Using this current as the source, I can then solve the classical D'Alembertian eq. for the corresponding internal self-field $A_\mu'$. If I insert $A_\mu'$ back into the Dirac equation, the solution must be different than those obtained from the free Dirac equation. I am simply wondering what the interpretation of the solutions will be now?


What happens when a supersonic airplane flies through a cloud? Will it punch a hole or is it more like a bullet through water (= hole closes immediately after the aircraft has passed)? Is there some special effect because of the supersonic speed? Or maybe the question should be: Does the airflow around an airplane change when the sound barrier is broken?


Boundary conditions for fields in Kerr/CFT

I am reading a paper by Guica et al. on Kerr/CFT correspondence (arXiv:0809.4266) and I'm not sure if I got this. They choose the boundary conditions, like a deviation of the full metric from the background Near- Horizon Extremal Kerr (NHEK) metric. Let's say that we can write that deviation like $$\delta_\xi g_{\mu \nu}=\mathcal{L}_\xi g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu$$ And the most general diffeomorphism which preserve the boundary conditions given in the text is: $$\xi=[-r\epsilon'(\varphi)+\mathcal{O}(1)]\partial_r+\left[C+\mathcal{O}\left(\frac{1}{r^3}\right)\right]\partial_\tau+\left[\epsilon(\varphi)+\mathcal{O}\left(\frac{1}{r^2}\right)\right]\partial_\varphi+\mathcal{O}\left(\frac{1}{r}\right)\partial_\theta$$ What my mentor told me, while briefly explaining this, is that we basically need to find the most general $\xi$ such that $\mathcal{L}_\xi g_{\mu\nu}$ is within the class of the boundary conditions. But how do I find these $\xi$? How can you find these boundary conditions and diffeomorphisms? Or better jet, how do I find diffeomorphism using those boundary conditions? :\


Non-additivity of energy (thermodynamics)

In page 15, Chapter 1 (Laws of Thermodynamics) of the book "An Introduction to the Study of Stellar Structure" by Nobel laureate S. Chandrasekhar, it is mentioned (in context of total energy of two subsystems brought in contact): > In general when two bodies are brought in contact, the energy is not > additive; it is easy to see however, that the deviation is directly > proportional to common surface area of the bodies, hence, for the large > volumes the deviations from the additive law can be neglected I always thought that energy is additive as a fundamental property (or by definition). I did not find anything further to understand the statement. Please, help me understand this.


Numeric value of the electrons drift velocity in superconductors

Somebody knows the numeric value of electrons drift velocity in superconductors? How this value depends from the used superconductor material? What's about the current? Since the electrical resistance tends to zero, how many electrons can pass an area (a layer) that is occupied e.g. by 100*100 atoms?


Why does the Gallum-Arsenite bandgap narrow for higher temperatures and is this because of unavoidable impurities?

I found this sentence: > GaAs at 300K contains 10^15 acceptor atoms per cubic centimeter. The bandgaps for Gallium Arsenite are: 0 Kelvin : 1.518 eV 300 Kelvin: 1.424 eV Now I am wondering:   1. Why does Gallium Arsenite have this impurity?   2. Is the bandgap at 300 Kelvin is stated for a _pure_ GaAs semiconductor or does it already incorporate the influence of the impurity?   3. If it doesn't what else does then cause the narrowing of the bandgap?


How to derive following thermodynamic relation?

I'd would like to derive the following $$\bigg(\frac{\partial U}{\partial V}\bigg)_T = T \bigg(\frac{\partial p}{\partial T}\bigg)_V - p$$ What I know is that the internal energy $U$ is a function of temperature and volume. Hence, a small change in $U$ can be related to changes in $T$ and $V$ by $$dU =\bigg(\frac{\partial U}{\partial T}\bigg)_V dT + \bigg(\frac{\partial U}{\partial V}\bigg)_T dV$$ But I'm not sure where to go from here.


Feynman diagram for $\overline{K}\,\!^0$ antimeson production on the quark-level

I've recently stumbled upon a physics problem concerning $\overline{K}\,\\!^0$ antimeson production. In this particular example, colliding a $\pi^-$ meson with a stationary proton yields a $K^0$ meson and a $\Lambda^0$ hyperon: $$\pi^-\,[\overline{u}d] + p\,[uud]\rightarrow K^0\,[d\overline{s}] + \Lambda^0\,[uds]$$ This can be expressed in a Feynman diagram by letting the $u$ and $\overline{u}$ quarks annihilate to a gluon, out of which a pair of $s$-$\overline{s}$-quarks is generated. However, if a $\overline{K}\,\\!^0$ particle would be generated by the same method, in order to conserve the baryon number and the strangeness, more than just a particle must be produced. For example, the following reaction could take place, so that every quantum number is conserved: $$\pi^-\,[\overline{u}d] + p\,[uud]\rightarrow \overline{K}\,\\!^0\,[s\overline{d}] + K^0\,[d\overline{s}] + n\,[udd]$$ However, I can't seem to find a corresponding Feynman diagram for the reaction. I am guessing that the $\Lambda^0$ hyperon decays weakly and somehow yields the antikaon and the neutron, but I can't figure out how... Does anyone have a clue what the Feynman diagram could be?


is really an atom stable?

Half filled and fulfilled atomic orbitals are stable because of : high exchange energy. The problem is with exchange energy. We have learnt that the half and fulfilled orbitals have maximum no. of electrons with the same spin so they can exchange their positions and thus release energy. BUT, we have also learnt that the electrons of the same orbitals have the same energy. Furthermore, the electrons of the same sub shell have the same energy in the absence of magnetic field. so, how on earth do they lose energy?


Question concerning Isospin symmetry

I'm currently taking an introductory course to particle physics and I'm now trying to understand the concept of isospin. However I do have some trouble. So let's write the up- and down Quark as a Flavour-Doublet (u,d). We then obtain $\mid u \rangle = \mid \frac{1}{2} \frac{1}{2}\rangle$ and $\mid d \rangle = \mid \frac{1}{2} \frac{-1}{2}\rangle$. Now I want to know what particles I can get by combining these two quarks. So let's write $\frac{1}{2} \otimes \frac{1}{2} = 0 \oplus 1$. So I have a singlett and a triplet. Most books now tell me that the triplet corresponds to the pion-triplet and the singlet to the $\eta$-particle. Now I can for example write by using Clebsch Gordon coefficients that $\mid 1,1 \rangle = \mid \frac{1}{2} \frac{1}{2}\rangle \mid \frac{1}{2} \frac{1}{2}\rangle = \mid u u \rangle$. But this doens't correspond to any pion (They are mostly products of ups and anti-downs etc). However some books interpret this as the product of two proton wave function. This is quite confusing? Can someone help me? I'd me more than happy. Thanks in advance.


Inertia in an empty universe

I was reading a recent article on Mach's Principle. In it, the author talks about inertia in an empty universe. I'll quote some lines from the article: > Imagine a single body in an otherwise empty universe. In the absence of any > forces, (Newton's second law gives): $$m\mathbf{a} = 0$$ What does this > equation imply? Following Newton we would conclude from that $\mathbf{a} = > 0$, that is, the body moves with uniform velocity. But we now no longer have > a background against which to measure velocities. Thus $\mathbf{a} = 0$ has > no operational significance. Rather, the lack of any tangible background for > measuring motion suggests that $\mathbf{a}$ should be completely > indeterminate. And it is not difficult to see that such a conclusion follows > naturally, provided we come to the remarkable conclusion that $$m = 0$$ In > other words, the measure of inertia depends on the existence of the > background in such a way that in the absence of the background the measure > vanishes! I don't see how the argument is complete. For example, in an empty universe, how is it possible to assign a value of 0 to a force? And wouldn't the existence of mathematics in such an empty universe be questionable?


Nature as a calculator

I wondered whether physical experiments can find the outcome of otherwise intractable mathematical problems. For example, solitons or other non-perturbative effects in QFT cannot be seen at any order in perturbation theory, and are difficult to calculate, but could in principle be measured. Is nature doing the whole path integral/the whole perturbative expansion at once? Does that mean a computer could, in principle do it, in a finite time? or can nature solve math problems that are otherwise impossible?


How does quantum jitter "fix" the masses of strings?

So the masses of strings should be huge because of the high energy levels. Which means, in other words, the strings are vibrating with enough energy to have large masses as point particles. But quantum jitter offsets the high energy via negative energy, and makes the masses just right (what we observe). How does this occur? Where does quantum jitter come from? What is negative energy? Is quantum jitter changing the vibration/resonance frequency of the string? If no, how is it acting upon the string?


Can the expansion space time reverse itself and contract the same way?

If there's a mechanism for space-time expanding faster than the speed of light, is there an example of It contracting in the same manner? If whatever mechanism is causing it to expand, can the underlying mechanism reverse itself?


Are there any theories or suggestions for how the multiverse came into existence?

I've just seen a documentary about the multiverse. This provides an explanation for where the big bang came from. But it leaves me wondering: how did the multiverse come into existence? Because this is a step further than something we don't even surely know the existence of, I'm only wondering if there are any theories or suggestions.


Perfect fluids in cosmology?

In cosmology, it is often assumed that the equation of state of a cosmological fluid is of the form $p=w\rho$. Why is this? Is it the equation of a _perfect_ fluid? Why does $w=0$ for matter $1/3$ for radiation and $-1$ for a cosmological constant?


Why is there no current flow when i connect two opposite terminals of two separate batteries

I don't understand this..maybe there is something wrong with my understanding of battery or electricity. I am learning about electricity more deeply this time. So please spare with me... So lets begin with what i know:   * follow of current occurs when there is voltage drop.   * voltage drop happens when there is imbalance in charge.   * a battery is basically two separated chemicals. One side with electrons to spare and other side that needs electron to balance itself. which is imbalance in charge.   * When both sides are connected **\+ side** snatches electrons from **\- side**. (And if there is a medium, conductor, then first electrons are taken from conductor that in turn snatches from **\- side** , which not being able to keep electrons lose them and give up those electrons, we label it as excessive.) The analogy i was taught with used marbles in pipe... When one side has excessive marbles And this pipe is connected to other side that has no marbles the marbles will instantaneously start falling out so even though marbles from container with excessive marbles haven't traveled there yet, they still start falling because filled side marbles are pushing marbles in pipe and which in turn push the last marbles in pipe out. 1st, Aren't electrons **pulled** by atoms that can use more electrons, and not move because one side is overfilled with electrons so they can't **wait to push** some out. Questions:   1. if i connect a conductor to positive side of a battery, shouldn't it make all electrons/marbles fall into empty container as empty container will pull marbles as atoms need electrons?   2. if i connect two batteries' **only opposite sides** to each other shouldn't they drain (balance the each other out, cancelling all **-** & **+** charges)?


What is opposite of electric field?

A negatively charged molecule will generate a field, an electric field, and it will repel all negative ions. So what does positively charged molecule generates that causes other positive molecules to be repelled?


Question regarding inverse relation of resistance with area of cross section

It is said that resistance is inversely proportional to area of cross section. But greater area will have greater electric flux, and greater electric flux will have greater magnetic flux, and greater the magnetic flux will have greater eddy current, which is opposite to the current flowing. So this will oppose the current hence resistance increases. Am I right?


Can light be canceled by merging with an inverted wave?

Can light waves be canceled by merging them with their inverted waves? Seems like it would violate conservation of energy but waves are added together when they overlap, right? Where is the flaw in this logic? I'm thinking polarized laser light, added to its opposite, might becomes dark again. \-- Appreciating your collective wisdom.


What creates black streams in fountain-type fireworks?

For the New Year celebration of 2014, my nephews set off a series of fountain- type fireworks (as shown below). ![](http://i.stack.imgur.com/BgQYd.jpg) When one looks carefully into the shower of light, one notices that a fair number of "long, black and stringy streams of light" are coming out of the fountain. Note: I do not see any black streams in this picture! After reading this online article from the American Chemical Society, the colors of light are, of course, created from electrons jumping orbitals. However, how are these “black streams” created? I assume that somehow visible light is being absorbed to create these "black streams of light". What mechanism is responsible for this to occur? Are there small bits of matter that are large enough and cooled enough to absorb visible light? I really don’t know. Any help would be much appreciated.


How can we model intrinsic curvature?

Can it only be done in Euclidean space? Doesn't Euclidean space only model extrinsic curvature?


How does an object regains its neutrality after being charged by rubbing?

Objects (like comb) can be charged by rubbing as charged particles, particularly electrons, are transferred from one object to other. This can be seen as the object (comb) attracts small bits of paper. After some time, the charge on the body seems to disappear. How does this charge disappear without any external influence?


Post Problem Analysis - Rod in a cylindrical time varying magnetic field?

**A metal rod of length $l$ is placed (as shown in figure) in a cylindrical time varying magnetic field. Find the potential difference across it.** ![enter image description here](http://i.stack.imgur.com/XgIjQ.png) My answer: As electric fields as forming circles around the centre with magnitude $$\frac12r\frac{dB}{dt}$$ whose magnitude is same along any chord at a fix distance from centre: $$\Delta V=E.l=\frac12r\frac32at^{\frac12}.l$$ $$\Delta V=\frac34arlt^{\frac12}$$ **_My question:_ Is $\Delta V =V_M-V_N$ or $\Delta V=V_N-V_M$ $\bf OR$ equivalently which side is at high potential?** Possible   * $(i)$ As Field lines are circular from $M$ to $N$, $M$ must be high potential.   * $(ii)$ As this field line apply a force on electrons- in the rod- towards left, then $M$ must have excess $e^-$ and $N$ must have deficiency; so $N$ would be high potential as $M$ is $(+)$ and $N$ is $(-)$.


How to distinguish high-energetic muons and electrons in the CMS and ATLAS muon detectors?

At a typical energy of about 100 GeV, a muon has a Lorentz factor of about 1000, an electron about 200.000. The flight time to the detector should be around 30 ns (assuming d= 10m from the collision point). While e- would practically have the speed c, the muon's speed is about 0.9999995 c, causing a delay of about 15 fs, barely measurable I think. Is there a way to distiguihs e- from mu, unless we assume prior knowledge (that they have passed the electric calorimeter, about the penetrating power etc). Or, to put it in another way: what is the energy threshold below which they can be distinguished directly?


Electric field outside a hollow cylindrical conducting shell

Imagine that a point charge $+Q$ is located at radius $R$ and height $z=0$ inside of an infinitely long conducting cylinder shell of inner radius $a>R$ and outer radius $b$. From what we know of perfect conductors, we know that $Q_{induced}(r=a) = -Q$, $Q_{induced}(r=b) = +Q$, and the electric field inside the shell is $E(a<r<b)=0$. Is there a way to figure out the magnitude and direction of the electric field outside of the conducting shell as a function of position?


Geometry in diagonal matrix and inertia tensor

For this problem, can anyone explain to me why when $x_1$ axis is aligned with the diagonal of the cube, the resulting inertia tensor will become diagonal? How to interpret this result geometrically? I mean, is there some tricky way for some geometric shape that I don't have to compute the eigenvalue, eigenvectors to find the principal axises, but by observation, I can make a guess first. ![enter image description here](http://i.stack.imgur.com/HyEC3.png) ![enter image description here](http://i.stack.imgur.com/Rkpfz.png)


What is the need to consider a singular spacetime?

To have a consistent superstring theory (which is to avoid the conformal anomaly on the worldsheet CFT) we are forced to build our theory on the critical dimension $n=10$. However, the Standard Model, seen as an effective low energy theory lives on $n=4$. So we are forced to compactify the background spacetime for the string theory, as to be of the form $\mathbb{R}^4\times M$ where $M$ is a compact and "very small" riemannian manifold. $M$ could be a torus, for example. To save some of the supersymmetry after compactification we are furtheremore forced to take $M$ to be Kahler. Given this, my question is: From a physical viewpoint, why do we usually consider nowadays $M$ to admit singularities? In other words why should $M$ be an orbifold and not just a smooth manifold?


Entropy of a naked singularity

According to the wikipedia article http://en.wikipedia.org/wiki/Naked_singularity: "Some research has suggested that if loop quantum gravity is correct, then naked singularities could exist in nature, implying that the cosmic censorship hypothesis does not hold. Numerical calculations and some other arguments have also hinted at this possibility." The entropy of a black hole is proportional to its horizon area. What is the entropy of the black hole if the cosmic censorship hypothesis fails to hold? Do we assume this hypothesis for the expression of the entropy?


Special relativity and quantum mechanics

I know how the Dirac's equation about the relativistic quantum mechanics works. Can anyone tell me how can one combine special relativity and quantum mechanics as a whole - special relativity is valid only under constant speed while the speeds of electrons in quantum mechanics vary with positions.


How many particles are created in the strong electromagnetic field?

Consider a vacuum of charged massless scalar field. Then the uniform and isotropic electric field $E$ is turned on for a time interval $\tau$. The question is, how many scalar particles are created?


Elegant approaches to quantum field theory

I have been reading _Quantum Mechanics: A Modern Development_ by L. Ballentine. I like the way everything is deduced starting from symmetry principles. I was wondering if anyone familiar with the book knows any equally elegant presentation for quantum field theory. Weinberg's books start off nice with the irreducible representations of the Poincare group/algebra but the later chapters lose me with the notation. Also, most books I've read on QFT (Srednicki, Peskin and Schroeder, Mandl and Shaw) make a valiant initial attempt at a nice consistent framework but end up being a big collection of mathematical recipes and intuitive insights that seem to work but the overall structure of the theory seems to be sewn up. The relativistic equations crop out of rather flimsy arguments, the canonical anti/commutation relations are imposed out of density indeterminate air, functional methods are developed cause we know no better, infinities come about with renormalization theory to the rescue but it seems very alien from the initial context. Is there any approach that ties all these things together in an elegant mathematical framework which accounts for all the patch up work that is needed? I am not talking about axiomatization just a global point of view that encompasses all the issues.


How to use Legendre polynomials in order to determine the (an)isotropy of an on-lattice cluster aggregate?

I am currently testing various models of on-lattice (square lattice in two dimensions) cluster growth for anisotropy. I end up with a cluster, the boundary of which, in case of a truly isotropic model, should look like a circle. In anisotropic cases the cluster boundary will deviate from the circular shape. I've been told that the Legendre polynomials can be used in order to determine the anisotropy of a cluster, but I'm not exactly sure of how I can use these Legendre polynomials in order to say something about the isotropy or otherwise of the clusters. I gather that one can write the angular functional dependency of the cluster boundary in terms of a Legendre series due to the orthogonal nature of the Legendre polynomials, but once I've calculated the coefficients of the series I'm not sure how to interpret them in terms of anisotropy of the cluster or even if this is the correct thing to be doing. Any hints, tips or further reading would be appreciated. I am a humble mathematician, so please go easy on the physics jargon as I may not be well versed in it. Thanks for your time and help.


Pure state - density matrix - real life example of boxes in warehouse

So if there are a billion boxes in a warehouse, I would like to know conceptually how to tell if it is in a pure state. I know that if it is in a pure state (not mixed) that the density matrix has only one nonzero diagonal element and that the trace is 1, whereas a mixed state would not be idempotent and the trace would be less than 1. I'm thinking that the warehouse resembles a microcanonical ensemble or a canonical ensemble since the box count is fixed. I also think it wise to imagine it is well-insulated (isolated) so that there is no entanglement with the environment. Would a good way to tell if it is in a pure state would be to measure each box's temperature? (Too many, right) Or instead measure a few times several boxes to make sure the measurement doesn't change and is the same for all in the sample? Or would one just look at the thermostate? ha ha please forget that last remark.


Charging an object by induction

when charging a sphere by induction using a (-) charged object , and we put it to the right side of the sphere, electrons are pushed to the left side , so we ground the left side and the excess electrons escape. but what if we grounded the right side ( which has less electrons than usual ) wouldn't electrons flow from the ground to neutralise this side?


How was the sinusoidal model for propagation developed?

It's a little difficult to explain this question .. but I'll try anyway. To the best of my knowledge propagation models - audio, RF - are modelled as travelling in a sinusoidal form. Surely if a model was constructed based on observation, and then verified by experiment the experiment would more often yield the desired result than not. Is the model for sinusoidal motion/waves deemed true because the equations are proved true by experiment? How do we know these waves are actually sinusoidal? How was this model developed?