radiation thermodynamics paradox

This question is concerned with a thermodynamic paradox for radiating bodies and radiation in a cavity of a specific shape. Consider two nested shells that are axisymmetric ellipsoids with the same two foci, A and B, as shown in the figure (line AB is the axis of symmetry). Cut the system along the vertical plane of symmetry and remove the right side of the outer shell, and remove the left half of the inner shell. Then connect the two halves with vertical surface, as shown in the figure, to make it a continuous enclosure. The result is a figure of rotation shown by the thick black line in the figure. Next, make the inner surface of it a perfect mirror. The property of such a cavity is that each ray emitted from point B comes to point A; but not each ray emitted from point A comes to point B - some rays emitted from A (shown in blue) come back to A. Now, put two small black bodies (say, two spheres of some small radius) at points A and B. Thermodynamic equilibrium requires that eventually the temperatures of the two spheres equilibrate. However, according to the geometric properties of this cavity, all energy emitted from B comes to A but only a fraction of energy emitted from A comes to B; so the equality of temperatures is not consistent with balance of emitted and absorbed power. How to resolve this paradox? ![enter image description here](http://i.stack.imgur.com/igGAT.jpg)


physical difference between A and B

We all know that when we say A it sounds different than when we say B. I was wondering what exactly can be the difference between saying A and B in terms of physics. I first thought that it may due to difference in the combination of frequencies. Then I realized I can say any alphabet in many different tones. So what exactly in terms of physics is the difference between saying A and B ?


Numerical Computation of Linbald Equation

Can anybody suggest me a good algorithm for the time evolution of the reduced density matrix using Linbald equation. My Hamiltonian is time dependent. I am aware about Qotoolbox and Qutip. I have checked both things but I don't have any clues about the algorithms they are using. I have to find the entanglement dynamics in a bipartite system under dissipation and forcing.


How is free energy built into a Metropolis Monte Carlo simulation of an Ising model?

In the Metropolis algorithm, the change in the energy given by the hamiltonian is compared for flipping a spin. This is not the free energy, but for systems above absolute zero you are trying to minimize the free energy, not the energy. So how is free energy built into this kind of simulation? How is entropy?


how do you destroy magnetic field - demagnetize?

And what happens with the magnetic field of a star that goes supernova? The magnetic radiation is scattered through the cosmos? Each particle will go away with its own magnetic radiation?


A question about BRST current in bosonic string theory

I have a question about Eq. (4.3.3) in Polchinski's string theory book volume I, p. 131. It is said > Replacing the $X^{\mu}$ with a general matter CFT, the BRST transformation > of the matter fields is a conformal transformation with $v(z)=c(z)$, while > $T^m$ replaces $T^X$ in the transformation of $b$. Noether's theorem gives > the BRST current $$ j_B = c T^m + \frac{1}{2} : cT^g : + \frac{3}{2} > \partial^2 c, $$ $$ = c T^m + : bc \partial c : + \frac{3}{2} \partial^2 c, > \tag{4.3.3}$$ My question is, what is the explicit expression of $T^m$? According to this thesis, p 29, $$-\frac{1}{\alpha'}: c \partial X \cdot \partial X = :c T_X:$$ Suppose this expression is correct, I cannot use it to vertify Eq. (4.3.11) $$T(z) j_B(0) \sim \frac{ c^m - 26}{2z^4} c(0) + \frac{1}{z^2} j_B(0) + \frac{1}{z} \partial j_B(0) \tag{4.3.11}$$ if in (4.3.11), $T(z)= -\frac{1}{\alpha'} : \partial X^{\mu} \partial X_{\mu} : \tag{2.4.4}$ and I applied contraction Eq. (2.2.11).


Why is light called an 'electromagnetic wave' if it's neither electric nor magnetic?

How can light be called _electromagnetic_ if it doesn't appear to be _electric_ nor _magnetic?_ If I go out to the sunlight, magnets aren't affected (or don't seem to be). And there is no transfer of electric charge/electrons (as there is in AC/DC current in space). In particular, the photons (which light is supposed to be composed of) have no electric charge (nor do they have magnetic charge). I'm looking for an explanation that can be appreciated by the average non- physicist Joe.


Bandpass of diffraction grating - exit slit width?

I'm supposed to derive a relation for the range of wavelengths that's being transmitted by a spectrometer (bandpass) in terms of the dispersion, focal length and exit slit width. Given grating is $100mm$ wide, slit separation $d=5.6\times 10^{-7} m$, focal length $f=1\space m$, exit slit width $w=100\mu m$ and spectrometer working in first order regime. Then I'm supposed to find value of slit width that corresponds to a bandpass of theoretical resolving power of $\lambda \approx 500 nm$ **Attempt** I found dispersion $\frac{d\theta}{d\lambda} = \frac{p}{d cos \theta} $ and resolving power $Np$. Thus The amount of wavelengths contained in the dispersion is simply $\delta \theta \times \frac{d\lambda}{d\theta} $ But what is the angle of spread? Is it simply $\delta \theta = \frac{w}{f}$. If so, then I find the bandpass $\Delta \lambda = \frac{w}{f} \times \frac{d}{p} = 5.6 \times 10^{-11} $. Last part of the question doesn't make sense, as resolving power is only dependent on $N$.


Exponential of a differential operator

I have a differential operator $L$, $\displaystyle L = i (t\frac{\partial}{\partial z} - z\frac{\partial}{\partial t})$ I can trivially hit this operator to $x,y,z$ and $t$ as $L x$, $L t$, $L y$, $L z$. But I have a problem with _exponential of that operator_. I want to hit this operator to $x,y,z$ and $t$ as well. $\exp(i\eta L)\,\,x$ ($\eta$ is rapidity in this case) The first thing that comes to my mind is to use definition of exponential of an operator: $\displaystyle \exp(A) = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} ...$ But, I don't know why, I don't want to use this infinite sum. There should be a smart way of doing this.. Do you have any suggestions for me?


Conservation of stress-energy and the fluid equation in cosmology

I'm trying to derive the equation for the cosmological fluid: $$\dot \rho + 3 \frac{\dot a}{a}(\rho +P)=0$$ by starting from the conservation of the stress-energy tensor: $$\nabla^\mu T_{\mu \nu} = 0$$ with the stress-energy for a perfect fluid in its own frame being: $$ T_{\mu \nu} = \text{diag} (\rho, a(t)^2 P,a(t)^2 P,a(t)^2 P) $$ in a spatially flat FLRW metric: $$g_{\mu \nu} = \text{diag}(1,-a(t)^2,-a(t)^2,-a(t)^2)$$ But I keep getting a bogus answer! Consider the equation you get from $ \nabla^\mu T_{\mu \nu} = 0$ when $\nu =0$: $$ \begin{align*} \nabla^\mu T_{\mu 0} &= 0 \\\ g^{\mu \alpha}\nabla_\alpha T_{\mu 0} &= 0 \end{align*} $$ $T$ is diagonal, so $\mu$ must be zero, but $g$ is diagonal as well, so if $\mu$ is zero, then so is $\alpha$. This gives: $$ \begin{align*} g^{0 0}\nabla_0 T_{0 0} &= 0 \\\ \nabla_0 \rho &= 0 \\\ \dot \rho &= 0 \end{align*} $$ Because $\rho$ is just a scalar, so the covariant derivative is the partial derivative. Except this answer is wrong.


Positive Mass Theorem

I'm a third year maths undergrad doing a project on minimal surfaces. However I'm really struggling to understand what the PMT is trying to explain? Could anyone help explain this (as simply as possible)


About Efimov States and Halo-Nuclei

I read that > _Halo nuclei could be seen as special Efimov states, depending on the subtle > definitions._ (The last sentence in the second to last paragraph of this > Wikipedia article.) This does not seem trivial to me in the least. Can someone shed some light on this for me?


Why can't we have a wave of particles?

I understand the nature of light can be complex and has extensive theories/experimental data. We hear light can be both a wave and particle, so why can't it be both, a wave of particles?


Identity in quantum operator tutorial

I'm reading this tutorial by Ben Simons entitled _Operator methods in quantum mechanics_ in connection with his course in advanced QM, and I'm a bit puzzled by an identity in page 25, a bit above relation (3.3): With the momentum operator $\textbf{p}=-i\nabla$ and the vector $\textbf{a}$ we have $e^{-i\textbf{a}\cdot\textbf{p}}=e^{\ \textbf{a}\cdot\nabla}=\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}a_{i_1}\cdots a_{i_n}\nabla_{i_1}\cdots\nabla_{i_n}\ ,$ where repeated indices are summed over. What confuses me here is that (in 3D) $\textbf{a}$ and $\nabla$ have 3 components while this expression seems to refer to component $i_n$ where $n\rightarrow\infty$, and I don't recognize the usual expansion of the exponential. What am I missing here?


First variation of the action in relativistic notation - Landau & Lifshitz "Classical theory of fields"

In Landau & Lifshitz's book, _Classical theory of fields,_ the action for a free particle is defined as: $$\tag{8.1} S= \int ^b _a {-mc \ \text d s}=0,$$ where $$\text d s=c\,\text d t\sqrt{1-\frac {v^2}{c^2}}$$ is the the invariant space-time interval beetween points along the particles worldline. From the latter expression of the Lagrangian, it's easy to obtain the expressions of the momentum and the energy of the particle. However, after a "classical" derivation, the author repeats the calculations with a different notation (I'll post the derivation from the "particle in a electromagnetic field" case omitting the 4-potential terms, because I think that my book contains an error in the free particle case): > Since $\text d s= \sqrt {\text dx^i \text dx_i}$: $$\delta S = -mc \int ^b > _a\dfrac{\text dx_i \text d\delta x_i}{\text ds}.$$ Integrating by part, > introducing the 4-velocity $u_i =dx_i/ds$, we get: $$\tag{9.10}\delta S =-mcu_i\delta x^i|^b _a+\int_a ^bmc\dfrac{\text d u_i}{\text d s} \delta x^i \,\text ds.$$ I'm very puzzled about the meaning of these two lines; I have three questions: **1)** What does the notation $\text d \delta x_i$ mean? **2)** How does he obtain the $\delta S$ expression? **3)** How does he pass from the first integral to the second? If someone could explain in detail I'd be very grateful. Note, I have no problem in getting the result, I can obtain it by replacing $\text d s =c\text d t \sqrt {...}$ and doing the variation on $v$.


Is energy since the big-bang still here in the present?

Based on the concept that energy can never be destroyed and is only transferred. Does it mean that energy since the formation of literally everything still here today?


Operator Ordering Ambiguities

I have been told that $$[\hat x^2,\hat p^2]=2i\hbar (\hat x\hat p+\hat p\hat x)$$ illustrates **_operator ordering ambiguity_**. What does that mean? I tried googling but to no avail.


When there is a charged conducting object near an another charged conducting object the charges on these objects accumulate to the sides where the closest points between these objects are. However, since these objects are conductors the charges also have to move on these objects. For a perfect conductor(~superconductors)would the charges still move on the object because to my mind it should accumulate on the sides and be static. I am pretty confused at this point so I hope someone could explain it to me: Would the charges on a superconductor near an another charged conductor move and produce magnetic field?


Finding the least capacitance?

A circuit has a self-induction of 1 H and carries a current of 2 A. To prevent sparking when the circuit is witched off, a capacitor of which can withstand a voltage of 400 V is used. The least capacitance of the capacitor connected across the switch must be? I have no idea how to go about this question.


Does the speed at which sound travel depend on the volume (amplitude) of the sound?

Lets say you have a plank is you hit it once and get t time if you hit is 2x as hard will it travel t/2? will it be the same or will it travel only slightly faster?


Intuitive meaning of the Hilbert Space formalism

> **Possible Duplicate:**   >  Intuitive meaning of Hilbert Space formalism I am totally confused about the Hilbert Space formalism of Quantum Mechanics. Can somebody please elaborate on the following points:   1. the observables are given by self-adjoint operators on the Hilbert Space.   2. Gefland-Naimark Theorem implies a duality between states and observables   3. what's the significance of spectral decomposition theorem in this context?   4. What do the Hilbert Space itself corresponds to and why are states given as functionals on the hilbert space. I need a real picture of this.


Measuring the quantum state of light

**A clarification please** The following scheme to measure **linear** polarization states (a single polarizing beam splitter and two photo counters) orientation (as $arctan \sqrt{\frac{v}{h}}$) of **coherent** light pulses cannot discriminate two states which make the same angle with the measuring bases. $\langle n \rangle$ is the average number os photons per pulse. ![a](http://s14.postimage.org/evo2hbpzl/g4322.png) In the example, the $+45^o$ and $-45^o$ polarized pulses will result on the same output. I'd like to know if it is OK to do the following setup and what would be change in precision of the polarization angle, if any: ![b](http://s2.postimage.org/xsyztz1g9/g4361.png)


Rectifying incomplete popular notions in cosmology

In looking at the answers to this question regarding light from distant galaxies ever being visible to us: Expansion of the Universe, will light from some galaxies never reach us? I came across a few concepts that were quite surprising to me. In particular:   * Movement faster than the speed of light   * The big bang was not an explosion outwards from a single point. Granted I am just a rank beginner and self-studier, yet I did study a QM course from Oxford, have read several sets of notes on SR, and readily went through the first hundred pages of "Student Friendly QFT." Yet I have never encountered these notions. My question is where does one acquire this type of information. Not necessarily the technicalities (of, e.g., GR); but just a correct awareness.


How exactly are the different motions of only one kind of fundamental string assumed to give rise to the spectrum of elementary particles we observe?

In string theory, it is assumed that all particles can be described as quanta corresponding to the excitations of only one kind of fundamental string. How can in principle the different motion patterns of one kind of string give rise to the whole particle zoo in the standard model? How are the different properrties, that characterize an elementary particle such as their   * mass   * spin   * charges encoded in the allowed motion pattterns of the string? (I know that the vacuum corresponding to our standard model can not be uniquely identified at present, but I am interested in the general concepts and ideas that should in principle give rise to the different characteristics of the particles we observe.) **Note** _This is just an attempt to extract and reask the useful part ofthis now a bit too overloaded original question._


What is the electric current $I_{12}$ and voltage $U_{12}$ in following electric circuit? $I_{12}$ and $U_{12}$ are between points 1 and 2. ![picture of electric circuit of the question](http://i.stack.imgur.com/FldEO.jpg). I have used this in constructing this electric circuit and as you can see simulator shows wrong electric current($I_{12}=0,34 A$) between points 1 and 2, but why? I mean, because both bulbs has resistivity of $10 \Omega$ and Cell has voltage as 10 V, then should be $I_{12}=0 A$. ![Another configuration showing another value of $I_{12}$](http://i.stack.imgur.com/nm86W.jpg) Now $ I_{12} \approx 0$ so I think something is wrong with this simulator.


Is it really possible to walk on water?

Is it really possible to walk on water or levitate in air. If not then how do some magician,s like dynemo do it simply while walking on street? And i have asked for possibility of walking on water so please exclude the high velocity concept.


Calculating the amount of dissolved gases in a liquid?

The most basic situation, water at room temperature, has dissolved oxygen, CO2, and more. Why is this? How would one calculate how much gas _should_ be dissolved in any given liquid?


How do I properly write Newton's second law for a particle with drag?

> A heavy particle is projected at speed $U$ at an angle $\alpha$ to the > horizontal. The particle is subject to air resistance which is > experimentally found to vary proportionally to the square of the speed. Show > that > > $$\vec{\dot{v}} = -\frac{g}{V^2}\lvert\vec{v}\rvert\vec{v} - g\vec{j},$$ > > where $V$ is the terminal velocity of the particle. If $\alpha = > \frac{\pi}{2}$ (so that the particle is projected directly upwards), find > the maxium height reached and the time taken to reach it. What is the speed > of the particle when it returns to the horizontal? I'm having troubles solving this question, firstly I have some fundamental mis-understandings,   * if the particle is subject to air resistance which is proportional to the square of the speed then do we model newton second law as: $$\vec{\ddot r} = \vec{g} - k|\vec{v}|^2\dfrac{\vec{v}}{|\vec{v}|}$$ or $$\vec{\ddot r} = \vec{g} - k|\vec{v}|^2$$? I'm assuming the former by looking as what's required but _why_? The question says it's just proportional to speed, why in the direction of the velocity?   * Secondly, why do we have to artificially put a negative sign? Does this mean that $k>0$ from now on? As for my attempt of the question I done it as follows: $$\vec{\ddot r} = \vec{g} - k|\vec{v}|^2\dfrac{\vec{v}}{|\vec{v}|} = -g\vec{j} - k|\vec{v}|\vec{v} $$ so we have, $\vec{\dot v} = -g\vec{j} - k|\vec{v}|\vec{v}$ solving this I get the solution of $$ \vec{v} = \dfrac{-g}{k|\vec{v}|} + \vec{c}e^{-kt|\vec{v}|} $$ where $\vec{c}$ is a constant vector, now from here I get terminal velocity as $\dfrac{-g}{k|\vec{v}|} \vec{j}$ but I am unsure how to get $V^2$ from here. on second thoughts if $|\underline{v}| = V$ then I get the required result, but _why_ would the speed be the same as the terminal speed


Question about two vehicles moving toward each other, but the answer doesn't seem correct, am I missing a key concept?

**The problem statement** Vehicle A and Vehicle B are moving in opposite directions on the NJTP. Vehicle A is heading south toward atlantic city while vehicle B is heading north towards Hoboken. In situation 1 and 2 described below, at t = 0s both vehicles are at a distance of separation of 400m and are moving towards each other. Situation 1: When vehicle A is moving at a constant velocity of 30 m/s and travels a distance of 120 m, the vehicles pass each other on the turnpike. Situation 2: When vehicle A is moving at a constant velocity of 80 m/s and after a time of 3s, the vehicles pass each other on the turnpike. For vehicle A label variables as: Via, Vfa, aa, da,ta For vehicle B label variables as: Vib, Vfb, ab,db, tb In the ORDER INDICATED: (a) Calculate the initial velocity of vehicle B or Vib; and (b) Then calculate the acceleration of vehicle B or ab Show all work in DETAIL and keep all numerical values to the nearest hundredth value. Draw and label all pictures Note: You will receive NO CREDIT if you first calculate part (b) and then part (a) For situations 1 and 2 vib = constant ; vfb ≠ a constant value since its value changes w/ time ab = constant **My Work** Situation 1: Da + Db = 400 Via = 30 m/s Vfa = 30 m/s a = 0 m/s2 da = 120 m ta = ? Vave = d/ t = vf + vi /2 Vave = da/ta ta = 4s db = 280 ta = 4s Situation 2: Via = 80 m/s aa = 0 m/s2 vfa = 80 m/s ta = 3s = tb da = ? Using same equation as in situation 2, da = 240 db=160 tb = 3s Situation 1 db = vibtb + 1/2abtb^2 280 = vib(4) + 1/2 (ab)(4)^2 Situation 2 db = vibtb + 1/2abtb2 160 = vib(3) + 1/2 (ab)(3)^2 Solving for ab I get ab = 11.85 / vib Plugging that back into the equation I get Vib = .34 m/s and plugging that into the 280 = (.34)(4) + 1/2 (ab)(4)^2 = ab = 34.82 m/s^2. I'm sorry if I'm new and breaking the rules, but this problem has really been bothering me for the last week. My friend says it's correct, but it just seems too low of an initial velocity.


Induction on a circuit with switch

I have the following circuit: ![enter image description here](http://i.stack.imgur.com/Ss7Iv.jpg) It is subject to a steady, time-invariant magnetic field which points out of the paper. At $t = 0$, the switch closes. I thought that the magnetic flux would decrease at the moment the switch closes and, by Faraday's law, cause the voltmeter to change its reading. However, my textbook says that the voltmeter won't change. How does one arrive at this conclusion?


How can I integrate in $\mathrm{d}t$ the cube of the harmonic oscillator propagator?

I'm redoing the calculations of "Point Canonical Transformations in the path integral", by Gervais and Jevicki; while doing so I stumbled in integrals like $$ \int \mathrm{d}t \, \Delta_F^3(t) = -\frac{1}{12} \frac{i}{\omega^4}, \\\ \int \mathrm{d}t \, \dot{\Delta}_F^2(t) \Delta_F(t) = \frac{1}{12} \frac{i}{\omega^2} $$ where $$ \Delta_F(t) = \int \frac{\mathrm{d}\nu}{2\pi} \, e^{i\nu t} \frac{i}{\nu^2-\omega^2+i \varepsilon}. $$ I tried various methods, without success. For instance, I integrated explicitly $\Delta_F(t)$ and after some calculations I found $\Delta_F(t) = \frac{1}{2\omega} \cos(\omega t)$; but this expression, inserted in the previous ones, does not make the integrals converge. Maybe in these integrals it's important to integrate in $\mathrm{d}t$ before doing the $\mathrm{d}\nu$ integration; but when trying to do so what I obtain is terribly complicated.


Computational package to find the ground state of a particle in 3D domain

I am developing a numerical algorithm to find the ground state of a Hermitian matrix. Obvious applications are quantum many-body systems and particles in various potentials. I am a little stuck with the comparison for the latter example. Could anyone recommend a numerical package to find a ground state of a particle in a simple 3D domain, say, L-shape? UPDATE: Having read the rules of phys.SE more accurately, I want to emphasise that the question is not about numerical algorithms, but about tools of the trade that people in physics community use and recommend.


Electrostatics - Inserting a brass plate between two charges

The question is: if I were to insert a brass plate between two charges, what will happen to the force between the charges? Would it increase, decrease or stay the same?      Does the brass plate increase the value of permittivity of the medium and therefore the force decreases?      The correct answer is that it will increase. But I do not understand how.


Have I calculated Angular Acceleration correctly?

I am teaching myself basic mechanics from a standing start. I am trying to understand Angular Acceleration and have set myself a problem to solve. My answer 'feels' wrong, so I'd like some help to understand if I've misunderstood, or miscalculated anything. I've taken many liberties with rounding, please ignore, this is more about the basic process/theory than accuracy. Thanks in advance!! **Problem** > _An object is travelling around a circle with a radius of 40m. It's speed at > (A) is calculated as 50mph. 5 seconds later, it's speed at (B) is calculated > as 40mph. Determine the Angular Acceleration._ **Basic conversions** Circumference = $2\pi r = 251\text{ m}$ Velocity (A) = $22\text{ m/s}$ Velocity (B) = $18\text{ m/s}$ **Angular Velocity at (A)** 251 / 22 = 11.4. Therefore one full revolution would take 11.4 seconds. $$\omega = \theta/t$$ $$\omega = 2\pi /t$$ $$\omega = 2\pi/11.4$$ $$\omega = 0.55 \text{ rad/s}$$ **Angular Velocity at (B)** $251/18 = 13.9$. Therefore one full revolution would take $13.9$ seconds. $$\omega = \theta/t$$ $$\omega = 2\pi /t$$ $$\omega = 2\pi/13.9$$ $$\omega = 0.45\text{ rad/s}$$ **Angular Acceleration** $$\alpha = \frac{d\omega}{dt} $$ $$\alpha = \frac{0.45 - 0.55}{5 - 0}$$ $$\alpha = -0.1 / 5$$ **Answer to Problem** $\alpha = -0.02\text{ }\mathrm{rad/s^2}$


What do mathematicians mean by Navier Stokes existence and smoothness problem?

I still don't know what mathematicians mean by Navier-Stokes existence and smoothness. Since there is a reward for proving it, it seems important to them. (in past several months I've read online articles on this topic). Physically, what do we get from such a proof? Is it possible that we _don't_ have existence and uniqueness mathematically, but that our physics still "works" somehow? Also, is there a consensus on whether, based on physical intuition (we are modelling real things after all), there _must_ be existence and uniqueness to the pure math problem?


Unknown Function in the Tolman-Bondi-de Sitter Metric

I've been working with some dust solutions in General Relativity, practicing calculating the Riemann curvature tensor, and I came across an odd metric: the Tolman-Bondi-de Sitter metric. A quick internet search (to supplement the book I'm reading) can tell you that it describes spherical dust, while accounting for a cosmological constant. It's a pretty simple solution, with a line element of the form $$ds^2=dt^2-e^{-2\Psi(t,r) } dr^2-R^2 (t,r)d\theta^2-R^2 (t,r) \sin^2\theta d\phi^2$$ There's one term in there that has me a bit befuddled, and that I can't find an explanation for in a book or on the Internet: $\Psi(t,r)$ At first, I thought it had to be a simple wavefunction, but after looking at it more, I'm not quite sure. What is it, and what is its significance in the metric?


Can quantum annealing be used for factorization?

It is known that there is a famous quantum factorization algorithm by Peter Shor. The algorithm is thought to be suitable only for quantum gate computer. But can a an adiabatic quantum computer especially that which is capable of quantum annealing be used for factorization? I am asking this because it seems that Geordie Rose claims in his blog that they have a quantum factorization algorithm that is somehow "better than Shor". But the details are unavailable as of now.


Relativistic contraction for a wave packet and uncertainty on momentum

Consider an electron described by a wave packet of extension $\Delta x$ for experimentalist A in the lab. Now assume experimentalist B is flying at a very high speed with regard to A and observes the same electron. The extension of the wave packet will appear contracted, and the uncertainty on momentum will increase. What happens when the later become larger than the electron's rest mass?


How does hot water clean better than cold water?

I had a left over coffee cup this morning, and tried to wash it out. I realized I always instinctively use hot water to clean things as it seems to work better. A google search showed other people with similar results, but this yahoo answer is a bit confusing in terms of hot water "exciting" dirt. What is the physical interaction between hot water and oil or a material burnt onto another vs the cold water interaction?


Stability Group of the Poincare Group

The stability group $G_\Sigma$ is a subgroup of the Poincare group $P(1;3)$. Its generators $X$ in the **front form** leave the hypersurface $\Sigma: x^+ = 0$ invariant. Phrased differently they satisfy the condition: $x'^{+} = x^+ + [x^+, X] = x^+ = 0$. So, for the generators $X$ of the stability group we have $$ [x^+, X]=0 . $$ How can I relate the matter of leaving the hypersurface invariant with the condition: $x'^{+} = x^+ + [x^+, X] = x^+ = 0$ ?


To which real densities do carrier densities in the semi-classical model of a crystal correspond?

In the semi-classical model of a crystal in solid state physics, electrons and holes are assigned effective masses that account for their different mobilities. E.g. in silicon, holes have a bigger mass than electrons. This results in different electron/hole densities as the temperature increases. Why do those different densities not violate conservation of charge? In what way do these imaginary electrons/holes correspond to real particles, in what way do they not?


Is a 125 GeV Higgs large (or small) for the MSSM?

As far I as I know, and from naturalness considerations, a 125 GeV Higgs mass is rather large for the MSSM. This is because in the MSSM $$m_h^2 \lesssim M_z^2 \cos^22\beta + \Delta$$ where $\Delta$ represents top/stop loop corrections to the Higgs. It takes the form $$\Delta \sim \ln(\frac{m_\text{stop}}{m_\text{top}}) + \text{mixing}$$ Moreover, the Z mass is determined by the Higgs mass parameter, $m_{H_u},$ and $\mu$ (in the large $\tan\beta$ limit. i.e. $\tan\beta \geq 10$). And $m_{H_u} \sim - m_\text{stop}$, so the larger $m_\text{stop}$ the larger the fine- tuning in the MSSM. Now, only by large $m_\text{stop}$ (and mixing) one can reach a Higgs mass greater than LEP/LHC bounds. But this is already associated with large fine-tuning as I said. So, in this view, a 125 GeV Higgs mass is large for the MSSM. (At least for the cMSSM.) However, I realised that some people think 125 GeV is much less than what the MSSM predicts! Now this I don't understand at all. So could someone please explain what this view is based on?


Why is momentum (instead of something else) the canonical conjugate of position?

Why did nature decide to make conjugate of position to be momentum? Since energy and position do not commute, why not energy? What determines the pairing of time with energy and momentum with position?


How does time relate to mass and velocity

I understand that the larger the mass the greater gravity is and the slower time is, as well the faster an object is traveling the slower time passes. My question is that since the faster an object travels the more mass it has, is the increase in mass the reason for the change in time, or is it the velocity?


A satellite in orbit fires it's engines for a short interval. Is the new orbit closer or further away?

A satellite is in a circular orbit when its engines turn on to exert a small force in the direction of the velocity for a short time interval. Is the new orbit further or closer to the Earth? The solution is that the new orbit is further away (which is also intuitive) and is justified by stating that there is a positive increase in the total energy which is given by the formula: $$\:E_{T}=-\frac{GMm}{2r}$$ And that states that an increase in the total energy would result in a larger radius. Thus, the new orbit is further away. However, the problem arises when I look at this equation which relates the speed with the radius: $$\:v^2=\frac{GM}{r}$$ Since the small force was in the direction of the velocity, an increase in veloctiy should result in a DECREASE in the radius meaning that the new orbit is closer in according to this equation. Why is it that the first equation is correct to use, while the second one is wrong? Why is the second equation not working here? Thanks!


Do apparent event horizons have Hawking radiation?

As I understand it, black holes have an absolute event horizon and an apparent horizon specific an observer. In addition to black holes, an apparent horizon can come from any sustained acceleration. Firstly, there is the "particle horizon" of the universe, which is the furthest co-moving distance from which light can reach us before the acceleration of the expansion of the universe prohibits it, then there is the case of a constantly accelerating observer in just regular flat space. My zeroth question is if the Hawking radiation from a black hole could be said to come from the apparent or absolute horizon - which is it? Obviously the former would be specific to each observer. For the case of the universal acceleration, dark energy, cosmological constant (please help me with terminology as appropriate): One could ask "do we observe radiation from the horizon?", for which we already have an available answer in the form of the cosmic microwave background (CMB). But I am curious if this would match the temperature and intensity you would expect if you treated the edge of the universe as a black hole horizon. Are these two be equivocal in some sense? Finally, I'm most interested to ask what would occur for the case of a constantly accelerating observer. An apparent horizon exists behind the observer beyond which light may never reach. Could it be possible the Hawking radiation is observed as coming from behind? ![Accelerating Observer](http://i.stack.imgur.com/IJRj1.png) **Could P possibly observe Hawking radiation coming from behind?** Granted, if you took the simple version of the temperature of the Hawking radiation ($T = \frac{\hbar c^3}{8 \pi G M k_B}$) and plugged in some super large mass, you will correspondingly get some super small temperature which will not agree with the 2.7 K of the CMB. Of course, the two cases I'm discussing do not have any definable mass that can be used. I would imagine that in the case of a simply accelerating particle, a greater acceleration would correspond with greater Hawking radiation, which is why it's not an issue for normal stuff, but if we accelerated something small with great force, could it observe radiation from that (false) event horizon?


Dipole and multipole bound state anions: Do these bound electrons behave exactly like conventional electrons in the molecular orbitals?

Recently, I read about dipole and multipole bound anions. Dipole bound anions are those, if I understood correctly, when an electron is attached electrostatically on a neutral molecule which is polar. If the molecule is having a quadrupole moment, the anion is said to be a quadrupole bound anion. These bound electrons will have less binding energy compared to the electrons in molecular orbital. My question is, these dipole, quadrupole and other multipole bound electrons behave like normal electrons in orbitals except that it has less binding energy or different? To be more clear, do these electrons have specific orbitals with orbital angular momentum or not? If so, are these orbitals like normal molecular orbitals (S P D F)? Can we assign electronic states to these electrons? Also, is it possible to excite any electron in the molecule (say valance electron) to this bound state (dipole or any multipole bound state)?


Minkowski metric and definition of coordinate differentials?

This is probably a really silly confusion I have about the definition of “coordinate differentials”, which I thought were things like $dx,dy,dz$ etc. The Minkowski line element $$ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}$$ defines the Minkowski metric $$\left[\eta_{\mu\nu}\right]=\left(\begin{array}{cccc} c^2 & 0 & 0 & 0\\\ 0 & -1 & 0 & 0\\\ 0 & 0 & -1 & 0\\\ 0 & 0 & 0 & -1 \end{array}\right).$$ Using index notation, the line element can be written as $ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}$. In textbooks I have seen the terms $dx^{\mu},dx^{\nu}$ called “coordinate differentials”, which seems OK except $dx^{0}=cdt$. I realise this is trivial, but is it correct to call $cdt$ a “coordinate differential”? To me it looks like a coordinate differential $dt$ multiplied by $c$.


reversible cellular automata

Let's suppose a cellular automaton has a value $b(r,t)$ belongs to $Q$ at site $r$ and time $t$, where $Q$ is the set of possible states at each site. Let $N(r, t)$ be the values of the states of all the sites in some (e.g., Von Neumann or Moore) neighborhood of $r$ at time $t$, taken in some canonical order. The cellular automaton rule is then $$b(r, t + 1) = F(N(r, t))$$ where $F$ is the update function. Let us suppose that $Q = \\{0, 1\\}$ so that we have a bit at each site. One way to construct reversible cellular automata (RCA) is to look at rules of the form $b(r, t + 1) = F(N(r, t)) \text{ xor } b(r, t − 1)$ To see why this is reversible, how to solve for $b(r, t −1)$ in terms of $b(r, t)$ and $b(r +1, t)$. Does it mean that this obeys a condition even stronger than reversibility? One apparent disadvantage of this approach seems to be that computation of $b(r, t+1)$ requires knowledge of the state at time $t−1$ in addition to that at time $t$. Is there way to fix this problem by adding extra state at a site so that the state at time $t + 1$ depends only on that at time $t$? Are there other functions besides $\text{xor}$ that could be used to create RCA in the above fashion for $Q = \\{0, 1\\}$?


If $SU(2)_{L} \times U(1)_{Y}$ breaks to $U(1)_{em}$ when a non-zero mass for the Higgs boson is chosen, why do we still have weak interactions?

As I understand it, when we say that the $SU(2)_{L} \times U(1)_{Y}$ is broken via the Higgs mechanism, this is because the symmetry acts on the Higgs mass in a way that would change it's value. If we want to pick a particular model we need to pick a fixed value of the Higgs mass, and this is only possible if we say $SU(2)_{L} \times U(1)_{Y}$ is broken to $U(1)_{em}$. The Lagrangian is always invariant to $SU(2)_{L} \times U(1)_{Y}$ (even if the Higgs mass changes under $SU(2)_{L} \times U(1)_{Y}$ the Lagrangian is such that it remains invariant). The symmetry breaking is just neccesary in choosing a theory with one particular value of the Higgs vev. Real life does appear to have a fixed Higgs mass so we require that $SU(2)_{L} \times U(1)_{Y}$ breaks to $U(1)_{em}$. But then aren't we saying that observable physics is described by $SU(3)_{C} \times U(1)_{em}$, why does the weak interaction still work in our universe that contains a fixed Higgs mass?


What nonlinear deformations will a fast rotating planet exhibit?

It is common knowledge among the educated that the Earth is not exactly spherical, and some of this comes from tidal forces and inhomogeneities but some of it comes from the rotation of the planet itself. The deformation from the rotational effect makes it more oblate spheroid-like, or as I would prefer, "like a pancake". Here is one site illustrating the behavior, and image: ![Earth's shape from rotational distortion from Mathematical Imagery](http://i.stack.imgur.com/gV7Ay.jpg) Literature exists detailing the mathematical expectations for a rotating planet using just hydrostatic forces, for example, see Hydrostatic theory of the earth and its mechanical implications. I like to imagine a ball of water in space held together by its own gravity. I also don't want to deviate from consideration of only hydrostatic (and gravitational) forces because I think it is sufficient for this discussion. It would seem that the solution of the described problem is in terms of a small change in radius as a function of the azimuth angle, or z-coordinate if you take the axis of rotation to be the z-axis. This is using rotational symmetry. In other words, Earth's deformation due to rotation does not depend on longitude. I want to ask about the extreme case. Imagine a planet rotating so fast that it is a very thin pancake. What will occur in this case? I am curious:   * Will the center hollow out, creating a donut shape?    * Will it break up into a multi-body system? It seems to me that it would be logical for the high-rotation case to break up into 2 or more separate bodies. The reason is that a 2 body system is stable an can host a very large angular momentum. But would it be an instability that leads to this case? When would such an instability occur and could a rotating planetary body deform in a different kind of shape from the beginning, such as a dumbbell-like shape, which would transition into a 2-body system more logically than the pancake shape? ![Would THIS happen?](http://i.stack.imgur.com/aHZ9N.jpg) image link To sum up, how would a pancake shape transition into a dumbbell shape? Or would it? What are the possibilities for the described system?


Tunneling and transmission

Lets say we have a tunelling problem in the picture, where $W_p$ is a finite potential step: ![enter image description here](http://i.stack.imgur.com/aFgQ4.png) If particle is comming from the left a general solutions to the Schrödinger equations for sepparate intervals I, II and II are: \begin{align} \text{I:}& & \psi_1 &= \overbrace{A e^{i\mathcal L x}}^{\psi_{in}} + \overbrace{Be^{-i \mathcal L x}}^{\psi_{re}}& \mathcal L &= \sqrt{\tfrac{2mW}{\hbar^2}}\\\ \text{II:}& & \psi_2 &= C e^{\mathcal K x} + De^{-\mathcal K x}& \mathcal K &= \sqrt{-\tfrac{2m(W-W_p)}{\hbar^2}}\\\ \text{III:}& & \psi_3 &= \underbrace{E e^{i \mathcal L x}}_{\psi_{tr}}& &\\\ \end{align} Where $\psi_{in}$ is an incomming wave, $\psi_{re}$ is a reflected wave and $\psi_{tr}$ is transmitted wave. I used the boundary conditions and got a system of 4 equations: \begin{align} {\tiny\text{boundary}}&{\tiny\text{conditions at x=0:}} & {\tiny\text{boundary conditions}}&{\tiny\text{at x=d:}}\\\ A + B &= C + D & Ce^{\mathcal K d} + De^{-\mathcal K d} &= E e^{i \mathcal L d}\\\ i \mathcal L A - i \mathcal L B &= \mathcal KC - \mathcal K D & \mathcal K C e^{\mathcal K d} - \mathcal K D e^{-\mathcal K d}&= i \mathcal L E e^{i \mathcal L d} \end{align} So now i decided to calculate coefficient of transmission $T$: \begin{align} T &= \dfrac{|j_{tr}|}{|j_{in}|} \\!=\\! \Bigg|\dfrac{\dfrac{\hbar }{2mi}\\! \left( \dfrac{d\overline{\psi}_{tr}}{dx}\, \psi_{tr} - \dfrac{d \psi_{tr}}{dx}\, \overline{\psi}_{tr} \right)}{\dfrac{\hbar}{2mi} \\!\left( \dfrac{d\overline{\psi}_{in}}{dx}\, \psi_{in} - \dfrac{d\psi_{in}}{dx}\, \overline{\psi}_{in} \right) }\Bigg| \\!=\\! \Bigg|\dfrac{\frac{d}{dx}\big(\overbrace{Ee^{-i\mathcal L x}}^{\text{konjug.}}\big) Ee^{i\mathcal L x} - \frac{d}{dx} \left( Ee^{i\mathcal L x}\right)\\! \overbrace{Ee^{-i\mathcal L x}}^{\text{konjug.}}}{ \frac{d}{dx}\big(\underbrace{Ae^{-i\mathcal L x}}_{\text{konjug.}}\big) Ae^{i\mathcal L x} - \frac{d}{dx} \left( Ae^{i\mathcal L x}\right)\\! \underbrace{Ae^{-i\mathcal L x}}_{\text{konjug.}}}\Bigg|\\! = \nonumber\\\ &=\Bigg|\dfrac{-i\mathcal L Ee^{-i\mathcal L x} E e^{i \mathcal L x} - i\mathcal L E e^{i \mathcal L x} Ee^{-i \mathcal L x}}{-i \mathcal L A e^{-i\mathcal L x} Ae^{i \mathcal L x} - i \mathcal L A e^{i \mathcal L x}Ae^{-i \mathcal L x} }\Bigg|=\Bigg|\dfrac{-i\mathcal L E^2 - i\mathcal L E^2}{-i \mathcal L A^2 - i \mathcal L A^2}\Bigg|=\Bigg|\dfrac{-2 i \mathcal L E^2}{-2i\mathcal L A^2}\Bigg| = \frac{|E|^2}{|A|^2} \end{align} It accured to me that if out of 4 system equations i can get amplitude ratio $E/A$, i can calculate $T$ quite easy. Could anyone show me how do i get this ratio?


calculating work done by friction

I want to calculate the work done by friction if the length $L$ of uniform rope on the table slides off. There is friction between the cord and the table with coefficient of kinetic friction $\mu_k$. $$ W = \int F \cdot d \vec{s}$$ I think it would be: $$ W_{fr} = \frac M L g \int_{0}^{L} dx$$ But the solutions (which could be mistaken) say: $$ dW_{fr} = \mu_k \frac M L g \, x \, dx$$ which is then integrated. Should there be an $x$ in the integral? I don't think there should be because you are summing up over an infinitesimal displacement $dx$ and the force of friction is not proportional to the displacement at any instant (I think). ![enter image description here](http://i.stack.imgur.com/lKzuO.png)


Is the photon energy required to cause an atomic transition $\Delta E+\Delta KE$, where $\Delta E$ is the "transition energy"?

An atom "at rest" can absorb a photon, and while some of this energy goes into increasing the energy level of the electron, momentum must be conserved, and so some energy must also increase the kinetic energy of the atom. The amount of $KE$ delivered to the atom is easy to calculate. Therefore, is the energy to cause a transition $\Delta E+\Delta KE$, where $\Delta E$ is the "transition energy"? So that no commenters cover old ground, please see this question.


What is the real-world significance of the Bekenstein bound?

The Bekenstein bound sets the maximum amount of information that can be contained in a region of space/energy, and is usually referred to in the same way as computer storage density: > For example, a single hydrogen atom, if it were to code as much information > as permitted by the Bekenstein Bound, would code about 4×106 bits of > information, since the hydrogen atom is about one Ångström in radius, and > has a mass of about 1.67×10−27 kilograms. (source) > Nature permits a surprising amount of information to be encoded before the > Bekenstein bound is reached. For example, a hydrogen atom can encode about 1 > Mb of information — most of a floppy disk. (source) > The "Bekenstein bound" leaves room for a million bits in a hydrogen atom > (source) But what does this really mean? How could _any_ information be stored in a hydrogen atom?


In solar cells, do photons break apart electron-hole pairs, or create them?

Some sources say that when a photon hits the PV cell, it breaks apart electron-hole pairs. Other sources say that photons _create_ electron-hole pairs. Can anyone explain which one is right? I've read several explanations of what goes on in the solar cell, but they don't seem very clear. To me, who has little prior knowledge, a few aspects of the many different explanations seem to contradict each other. I don't know if it's because some of the explanations said things a certain way for simplicity's sake, but this part seems too significant to ignore. For example, http://www.solarenergyexperts.co.uk/buyersguides/photovoltaic- glass-how-does-it-work/: "When photons (light particles) from the sun hit the cell, the energy breaks up the paired particles. The freed electrons go into the n-type layer, while the holes go down into the p-type layer." science.howstuffworks.com/environmental/energy/solar-cell3.htm : "When light, in the form of photons, hits our solar cell, its energy breaks apart electron- hole pairs." solarjourneyusa.com/bandgaps.php has a hole section called " _Generation_ of electron-hole pairs" This one basically seems to be talking about the same thing, but doesn't really mention electron-hole pairs, and the explanation seems more confusing to me, but anyway, http://www2.pv.unsw.edu.au/nsite- files/pdfs/UNSW_Understanding_the_p-n_Junction.pdf: "...broken bonds created by the light act as holes ... and these holes are also free to move throughout the material. Electrons and holes created in this way are physically near each other: for every electron excited by the light there is a corresponding hole generated. These electrons and holes can remain excited only for a short period of time. In a process called recombination, excited electrons stray too close to holes and the two fall back into bonded positions." This one seems to be saying that photons break apart the bonds between the electron and its atom, which creates a free electron and a hole, i.e. another one saying that an electron-hole pair is created. solarcellcentral.com/junction_page.html : "When photons hit the solar cell, free electrons attempt to unite with holes in the p-type layer." www.solarenergy.net/Articles/how-photovoltaic-cells-work.aspx : "When enough photons are absorbed by the negative layer of the photovoltaic cell, electrons are freed from the negative semiconductor material." So what actually happens when light hits a solar cell?


Can sand falling in a floating hourglass cause it to sink? (Follow up to hourglass question)

This is related to this question:Does the weight of an hourglass change when sands are falling inside? At Brigham Young University, there is a display consisting of a sealed off liter bottle with a sunken sealed off hourglass. When one turns over the bottle, the hourglass floats as the sands flow down. Eventually, the hourglass reaches some transition, and precipitately changes from floating to sinking. It has a sign next to it saying that it's for a fishing lure,and that you should not reveal the answer if you get it. What causes it to change from floating to sinking so quickly?


How general relativity gets to an inverse-square law

I understand that a general interpretation of the $1/r^2$ interactions is that virtual particles are exchanged, and to conserve their flux through spheres of different radii, one must assume the inverse-square law. This fundamentally relies on the 3D nature of space. General relativity does not suppose that zero-mass particles exchanged. What is the interpretation, in GR, of the $1/r^2$ law for gravity? Is it come sort of flux that is conserved as well? Is it a postulate? Note that I am not really interested in a complete derivation (I don't know GR enough). A physical interpretation would be better. Related question: Is Newton's Law of Gravity consistent with General Relativity?


Potential step and its transmission / reflection

Lets say we have a potential step with regions 1 with zero potential $W_p\\!=\\!0$ (this is a free particle) and region 2 with potential $W_p$. Wave functions in this case are: \begin{align} \psi_1&=Ae^{i\mathcal L x} + B e^{-i\mathcal L x} & \mathcal L &\equiv \sqrt{\frac{2mW}{\hbar^2}}\\\ \psi_2&=De^{-\mathcal K x} & \mathcal K &\equiv \sqrt{\frac{2m(W_p-W)}{\hbar^2}} \end{align} Where $A$ is an amplitude of an incomming wave, $B$ is an amplitude of an reflected wave and $D$ is an amplitude of an transmitted wave. I have sucessfuly derived a relations between amplitudes in potential step: \begin{align} \frac{A}{D} &= \frac{i\mathcal L-\mathcal K}{2i\mathcal L} & \frac{A}{B}&=-\frac{i \mathcal L - \mathcal K}{i \mathcal L + \mathcal K} \end{align} I know that if i want to calculate transmittivity coefficient $T$ or reflexifity coefficient $R$ i will have to use these two relations that i know from wave physics: \begin{align} T &= \frac{j_{trans.}}{j_{incom.}} & R &= \frac{j_{trans.}}{j_{incom.}} \end{align} * * * **Question 1:** I know that $j = \frac{dm}{dt} = \frac{d}{dt}\rho V \propto \rho v \propto \rho k$ But what is a density $\rho$ equal to? **Question 2:** I noticed that $\mathcal L$ and $\mathcal K$ are somehow (i dont know how) connected to the wavevector $k$ from the equation in 1st question but how? How can i make it obvious?


equation of motion for the scalar field via variational principle in general relativity

I would like to find the equation of motion for the scalar field $\phi$ by varying the following action in General Relativity. Special Relativity: $$ S = -\tfrac{1}{2}\int d^4\xi\, \eta^{ab} \partial_a \phi\partial_b\phi $$ General Relativity: $$ S = -\tfrac{1}{2}\int d^4x \sqrt{g}\, g^{\mu\nu} \partial_{\mu} \phi\partial_\nu \phi $$ I was able to get the correct equation using the covariant derivatives. Since they are constant with respect to the metric partial integration works and one obtains $ \Box\phi = 0$. But since for the scalar field the covariant and the partial derivates are the same I wanted to vary the action with the partial derivatives. Note: $\phi \rightarrow \phi + \delta\phi$ $$ \begin{align} \delta S &= -\tfrac{1}{2}\int d^4x \sqrt{g}\, (g^{\mu\nu} \partial_{\mu} \delta\phi\partial_\nu\phi + g^{\mu\nu} \partial_{\mu} \phi\partial_{\nu} \delta\phi ) \\\ % &=-\int d^4x \sqrt{g}\, g^{\mu\nu} \partial_{\mu} \delta\phi\partial_\nu\delta\phi \\\ % &=\int d^4x \, \partial_\nu(\sqrt{g}\,g^{\mu\nu}\partial_\mu\phi)\delta\phi \end{align} $$ The third line one obtains after partial integration. But at this point I'm stuck. I know the following definition: $$ \Box\phi = \tfrac{1}{\sqrt{g}}\partial_\mu(\sqrt{g}g^{\mu\nu}\partial_\nu\phi) $$ But I'm not able to convert my obtained solution into this. Thanks for your help!


Why does a temperature increase on a fixed volume increase entropy?

I heard that this statement is correct. However, it seems odd to me. The number of possible microstates is still the same, so isn't the entropy constant?


Towing of asteroid

I recently studied that NASA has planned to tow and place it in the orbit of the moon. My doubt is when asteroid is placed in the orbit near moon.since the gravitational field of earth is very high.what will it revolve around the moon or the earth. Can anyone clarify my doubt ??


4-velocity and 4-acceleration in instantaneous rest frames

I am trying to solve this problem: > _Consider a rocket moving relative to an inertial frame $\mathcal{F}$ , such > that its worldline is given by > $$x^{\mu}=c^2/g(\sinh(g\tau/c),\cosh(g\tau/c)-1,0,0).$$ What are the > components of four acceleration relative to the instantaneous rest frame of > the rocket, $\mathcal{F}'$?_ I (think I) understand how to do this using Lorentz transformation: $dt/d\tau=(1/c) \cdot dx^0/d\tau=\cosh(g\tau/c)$. This is equal to $\gamma$ and it is then straightforward to compute 3-velocity and then use the Lorentz matrix to get $$dx'^\mu(\tau)/d\tau=(c,0,0,0) \quad ; \quad d^2 x'^\mu(\tau)/d\tau^2=(0,g,0,0)$$ However, when I first saw this I immediately thought that by definition the 4-velocity in the instantaneous rest frame of the rocket would be (c,0,0,0) because in $\mathcal{F}'$ the 3-velocity is zero and $\gamma$ would be 1 and I was wondering if this is a valid reasoning. Even if it is, **why is the following wrong**? If $v'^\mu=dx'^\mu(\tau)/d\tau=(c,0,0,0)$, then $a'^\mu=dv'^\mu/d\tau=(0,0,0,0)$ since $c$ is a constant. But this contradicts the calculations given by the Lorentz transformation and I don't understand why and given this I have no idea of **how to interpret 4-acceleration**.


Why do clocks measure arc-length?

Apologies in advance for the long question. My understanding is that in GR, massive observers move along timelike curves $x^\mu(\lambda)$, and if an observer moves from point $x^\mu(\lambda_a)$ to $x^\mu(\lambda_b)$, then his clock will measure that an amount of time $t_{ba}$ given by the curve's arc length; $$ t_{ba} = \int_{\lambda_a}^{\lambda_b}d\lambda \sqrt{-g_{\mu\nu}(x(\lambda))\dot x^\mu(\lambda)\dot x^\nu(\lambda)} $$ will have elapsed where $g_{\mu\nu}$ is a metric on spacetime with signature $(-,+,+,+)$. **Why is this so?** * * * Here is how I would attempt to justify this fact in special relativity with $g_{\mu\nu} = \eta_{\mu\nu}$. Consider an inertial observer $O$ in $\mathbb R^{3,1}$, and suppose that this observer sees a clock, which I'll call observer $O'$ moving around on a curve $x^\mu(\lambda)$. If $O'$ were also an inertial observer, then given any event with coordinates $x^\mu$ as measured by $O$, observer $O'$ would measure the coordinates of the event to be $x'^\mu = \Lambda^\mu_{\phantom\mu\nu} x^\nu + x_0^\mu$ for some Lorentz transformation $\Lambda$. If $O'$ is not inertial, then this is no longer true, and there is some more complicated family of transformations, say $T_\lambda$ between events as seen by both observers. I would argue, however, that if we were to partition the interval $[\lambda_a, \lambda_b]$ into a large number $N$ of intervals $I_1=[\lambda_a, \lambda_i], \dots, I_N=[\lambda_{N-1}, \lambda_b]$ with $\lambda_n = \lambda_a+n\epsilon_N$ and $\epsilon_N=(\lambda_b-\lambda_a)/N$, then on each interval $I_n$, $O'$ is approximately an inertial observer in the sense that $$ T_{\lambda_n} = P_n + \mathcal O(\epsilon_N), \qquad (\star) $$ for some Poincare transformation $P_n$. Then we would note that since $O'$ is stationary in his own reference frame, he measures his worldline to have the property $\dot x'^\mu(\lambda) = (\dot t(\lambda), \mathbf 0)$ so that $$ I_{ba}=\int_{\lambda_a}^{\lambda_b}d\lambda \,\sqrt{-\eta_{\mu\nu}\dot x'^\mu\dot x'^\mu} = \int_{\lambda_a}^{\lambda_b} d\lambda \, \sqrt{\dot t^2} = t(\lambda_b) - t(\lambda_a) = t_{ba} $$ On the other hand the integral on the left can be written as a Riemann sum using the partition above, and we can invoke ($\star$) above to get \begin{align} I_{ba} &= \lim_{N\to\infty}\left[\sum_{n=1}^N \epsilon_N\sqrt{-\eta_{\mu\nu}\dot x'^\mu(\lambda_n)\dot x'^\nu(\lambda_n)}\right] \notag\\\ &= \lim_{N\to\infty}\left[\sum_{n=1}^N \epsilon_N\sqrt{-\eta_{\mu\nu}\dot x^\mu(\lambda_n)\dot x^\nu(\lambda_n)} + \mathcal{O}(\epsilon_N^2)\right] \notag\\\ &= \int_{\lambda_a}^{\lambda_b}d\lambda \,\sqrt{-\eta_{\mu\nu}\dot x^\mu\dot x^\mu} \end{align} Combining these two computations gives the desired result. **How do others feel about this argument?** I'm not completely comfortable with it because of the assumption $(\star)$ I made on $T_\lambda$. I imagine that in GR a similar argument could be made by invoking local flatness of the metric.


Nature of gravity: gravitons, curvature of space-time or both?

General relativity tells us that what we perceive as gravity is curvature of space-time. On the other hand (as I understand it) gravity can be understood as a force between objects which are exchanging (hypothetical) virtual particles called gravitons, similar to the way electromagnetic forces are due to objects exchanging virtual photons? At least at first glance, the two concepts seem mutually exclusive. Is there a description of gravity which includes both, or is this contradiction one of the problems in combining GR with quantum mechanics?


How do gravitons impact on general relativity?

As I'm reading about GR a lot lately, I was wondering: how do gravitons (if they exist ofc) impact the general relativity? Since in GR, when we look at particles moving in space-time, we are only looking from geometric point of view so to say. Since gravity is represented with curvature of space-time via Einstein equations, we don't say gravity is a force that influences on bodies, gravity is just curvature affecting the bodies. So if there is a graviton (gravitons) which would be mediators of gravity as a force within or not within the Standard Model, how would this be reconciled with the view of gravity as a curvature of space and time? I would guess that this kinda question was asked by some scientists and answered, but I never really read anything on it. I don't even remember seeing gravitons mentioned in standard books about GR. Are there any explanations about it?


How do I calculate stopping power?

Malcolm Gladwell made a claim in a recent talk that a sling with a stone going at 30m/s has the same stopping power as a .45 calibre handgun. How would I verify whether or not this claim is true - even given some assumptions, like the stone has a weight of 1lb or 2 lbs or whatever. Not quite sure how to work this out. Assume that the bullet is a 185grain 0.45 calibre bullet.


How does Newton's law apply to a person throwing snow onto their snowbank at the side of their driveway?

We know Newton's three laws:   1. A object at rest will remain at rest, and an object in motion will remain in motion unless a external force acts upon it.   2. If an unbalanced force acts on a object, the object will accelerate in the direction of the net force.   3. If an object $A$ exerts a force on object $B$, then object $B$ will exert a equal force to object $A$ in the opposite direction. $F_{a\text{ on }b} = -F_{b\text{ on }a}$ What I think is that as you throw the snow, it begins moving at a constant velocity. However, when the shovel stops moving, the snow will remain in motion, causing it to accelerate/fly toward the snow bank, according to Newton's first law. But is the snow **accelerating** when it leaves the shovel? How does this apply to Newton's second law? Thanks!


Self-adjointness

I know I have posted this question before some time ago. But no one could help so I decided to put my problem in another background. The Schrödinger equation of a free scalar field is given by $i\partial_{t}\Psi[\Phi,t]~=~\underset{A}{\underbrace{\frac{1}{2}\int d^{3}x\left(-\frac{\delta^{2}}{\delta\Phi^{2}(\vec{x})}+|\nabla\Phi|^{2}+m^{2}\Phi^{2}\right)}}\Psi[\Phi,t]$. This is the Schrödinger representation of QFT. Now I want to know, whether the operator $A$ on the r.h.s is essentially self- adjoint? Any idea or advise? My problem is how to handle the functional derivative here.


Strong interaction and the Lagrangian for electromagnetic interaction

The Lagrangian for electromagnetic field has the following expression: $$ L = -\frac{1}{c^{2}}A_{\alpha}j^{\alpha} - \frac{1}{8 \pi c}(\partial_{\alpha} A_{\beta})(\partial^{\alpha}A^{\beta}) $$ (I used Lorentz calibration $\partial_{\alpha} A^{\alpha} = 0 $). If I add the summand $\frac{\mu^{2}}{8 \pi c}A_{\alpha}A^{\alpha}$, I'll get an equations for field (which characterized by some 4-vector $A^{\alpha}$ (not electromagnetic (!!!))) of strong interaction and (for static case) the expression for Yukawa potential. So what is the physical meaning of summand written above? This term is somehow characteristic of the mass of the interaction carriers, but I don't understand the physical meaning of $A_{\alpha}A^{\alpha}$.


A question to the theory of multiverse

Guys I couldn't catch a point of multiverse theory.. Theory: If space-time goes on forever, then it must start repeating at some point, because there are a finite number of ways particles can be arranged in space and time.. Question: It seems like as if multiverse theory stands on a optical argument. We call there is a next universe because we are not seeing that since there is a speed limit of light and our optical universe is 14 billions light yrs in diameter(or whatever it be..since we can not assume our space time a sphere or 2dimensional plane).Doesn't it sound weird? If there is a space between two universe then what is the value of calling it a system of two universe, Isn't it just a single universe? Question:Is it a good idea to call there is a next universe because it is exactly alike? Question: Another question is How does the multiverse theory provides solution to the grandfather paradox? Isn't it like we fold the two dimensional spacetime graph where universe exactly alike to our universe exits??If my above argument is correct then how is it valid?? Note that I am not questioning multiverse theory this time..So please Don't give answer to this question proving my above arguments wrong..Let this be independent question.. I know I am wrong But please Make me clear.


Acceleration, velocity and speed

It is given that acceleration is constant, so can we infer that average speed and velocity are the same?? Moreover, circular motion is out of the question, as the function of x(t) where x=displacement, suggests, that for any t>=0, displacement can not be zero... This is the conceptual problem I am facing in a question: My teacher was reading out the question, and it was asked only to find the avg velocity from the acceleration. She, on her own, added a part to it, asking us to also find avg speed, and then, while discussing solutions, said that a graph must be made in order to solve this...so do you think that it is absolutely necessary? Moreover, if my premise is flawed, then how can the graph even help?? Thanks in advance!


Phys.org Spectral geometry to unite relativity and quantum mechanics, restate in laymens terms?

Lingua Franca links relativity and quantum theories with spectral geometry Could someone give me a short synopsis of this article in laymens terms? What implications does this have in the physics community? Is this work ground breaking or just the start of something that might be? From what I can understand this physicist related two types of maths that we use to model the world around us. Which has been hard to do, because the types of maths are incompatible with each other. But what is this spectral geometry? How does it relate the two types of maths in relativity and quantum theory? What might those maths be? What about the two has made it hard to unite them?


What is the difference between Cramer and Vaidman?

Two very interesting new papers on arXiv last night by Lev Vaidman and friends lead me to ask about the differences between Cramer's transactional interpretation of quantum mechanics (TIQM) and the two state vector formalism (TSVF) advanced by Aharonov, Vaidman and others. At a first look, they both seem very similar to me.   The two papers:   http://arxiv.org/abs/1304.7474 “The past of a quantum particle”, and   http://arxiv.org/abs/1304.7469 “Asking a particle where it has been”.   From these two papers, you can find further references for TSVF, including its origin.   For TIQM, you can start with Wikipedia, and also trace back to the origin. http://en.wikipedia.org/wiki/Transactional_interpretation


Are atoms unique?

Do atoms have any uniquely identifying characteristic besides their history? For example, if we had detailed information about a specific carbon atom from one of Planck's fingerprints, and could time-travel to the cosmic event in which the atom formed, would it contain information with which we could positively identify that they two are the same?


What is it that makes an electron maintain a distance from the positively charged nucleus? Why aren't electrons merely pulled into and absorbed by the nucleus ?


Why do electrons occupy the space around nuclei, and not collide with them?

We all learn in grade school that electrons are negatively-charged particles that inhabit the space around the nucleus of an atom, that protons are positively-charged and are embedded within the nucleus along with neutrons, which have no charge. I have read a little about electron orbitals and some of the quantum mechanics behind why electrons only occupy certain energy levels. However... How does the electromagnetic force work in maintaining the positions of the electrons? Since positive and negative charges attract each other, why is it that the electrons don't collide with the protons in the nucleus? Are there ever instances where electrons and protons _do_ collide, and, if so, what occurs?


What was a second in the early universe?

I have read some popsci articles and documentaries about the early universe and they often explain how various features of the universe came about and at what time. For example hydrogen atoms came about after hundreds of thousands of years. Now, the official SI time unit, the second, is based on the caesium atom's properties. How can we talk about time lengths in seconds or years if there are no atoms yet in the universe? (Or going back further where there aren't even electrons.) Clearly, there must be a way to do it. Then the next question is: why don't we use those field properties or whatever that existed before atoms to define the SI second? If atoms are not needed to get "the beat of time" in the early universe because something else can provide the tick then we could also use those things today instead of relying on the caesium atom for the definition of the second. I hope it's clear what I mean. Thanks!


Quantum Quench Problem

I read about the quantum quench problem in condensed matter physics. But what does really mean? Has anybody a good explanation about the origin of quantum quench problem?


Four-current, Induced Charge and Magnetic Flux

I'm studying Jackiw's "Fractional Charge and Zero Modes for Planar Systems in a Magnetic Field" DOI: 10.1103/PhysRevD.33.2500 but I have difficulties at some points. One of the problems is $$\langle j^0\rangle=\pm\frac{e}{4\pi}B$$ where $j$ is four-current, $B$ is magnetic field. How can I derive this result (and why we used only $j^0$ component)?.


Can a magnet or a magnetic field push gravity?

I have been asking around at my school and at the high school and at EWU but no one can answer this question: can a magnet or a magnetic field push gravity?


What's wrong with Schwarzschild equations?

I don't know much about black holes physics and so I find the Schwarzschild equations with a few contradictions. In particular I am trying to understand this little puzzle. The Schwarzschild Newtonian gravitational field equation is expressed as follows (see http://en.wikipedia.org/wiki/Schwarzschild_radius): $\frac { r^2 }{r_s} \frac {g}{c^2} = \frac {1}{2}$ So once a particle is close to the event horizon such that $r\to r_s$ the equation becomes: $ r \frac {g}{c^2} = \frac {1}{2}$ But then at that point light cannot escape so I would think that $g\to c/t$ where $t=1 sec$. So the equation would now approximate to: $r = \frac {c/t}{2}$ But this implies that $r$ is generalized for all black holes regardless the mass. If $r$ is indeed $r\equiv r_s$ then that contradicts the other Schwarzschild formula where $r$ depends on $m$: $r_s = \frac {2 G M} {c^2}$ Also based on what we know about $r_s$ in massive black holes, a radius of 150,000 km is quite small. What's wrong with this picture?


How is black hole affected by gravity of nearby bodies?

I run into wikipedia articles about ergosphere of rotating black holes. What if some massive body passes nearby some black hole: is something like ergosphere produced, or is the event horizon distorted by the gravity of the passing body?


The most challenging physical phenomena

What are examples of endeavors, in the history of mankind, to understand physical phenomena with models which were proved to be incorrect later, reformed significantly, or are still under development?


Don't heavier objects actually fall faster because they exert their own gravity?

The common understanding is that, setting air resistance aside, all objects dropped to Earth fall at the same rate. This is often demonstrated through the thought experiment of cutting a large object in half, the halves of which clearly can't then fall more slowly just by being sliced in two. However, I believe the answer is that when two objects fall together, attached or not, they do "fall" faster than an object of less mass alone does. This is because not only does the Earth accelerate the objects toward itself but the objects also accelerate the Earth toward themselves. Considering the formula: $F_g = G m_1 m_2/d^2$ We can see that the force of gravity is dependent on **both** the masses, not just that of the more massive object. Of course in everyday situations, we can for all practical purposes treat objects as falling at the same speed. But I'm hoping not for a discussion of practicality or what's measurable or observable, but what we think is actually happening. Am I right or wrong? What really clinched this for me was considering dropping a small Moon-massed object close to the Earth and a small Earth-massed object close to the Earth. This made me realize that falling isn't one object moving toward some fixed frame of reference, but that the Earth is just another object, and falling consists of _multiple objects mutually attracting in space_.


Radial fall in a Newtonian gravitational field

Suppose an object of mass $m$ starts at rest at a radial distance $ r_0$ from a perfectly spherical mass $M$ (where $m << M$), $r_0 > R =$ radius of $M$. Can we analytically determine when $m$ will hit the surface of the $M$? In other words, can we analytically solve this initial value problem: $$ \frac{d^2r}{dt^2} ~=~ - \frac{GM}{r^2} ,$$ $$ \dot{r}(0) ~=~ 0 ,$$ $$ r(0) ~=~ r_0? $$


Do heavier objects fall faster?

This question has been asked multiple times here and all over the internet yet I can't find a conclusive answer:   * Some claim that heavier objects do fall faster: Don't heavier objects actually fall faster because they exert their own gravity?   * Others claim that all objects fall at the same speed regardless of their mass: Free falling of object with no air resistance Which one is the right one?


Gravity question

> **Possible Duplicate:**   >  Don’t heavier objects actually fall faster because they exert their own > gravity? When true: The force pulling the "heavy" object down is greater BUT it also takes more force to accelerate a heavy object. These two effects cancel out. As I heard, the bigger mass/size an object the bigger attraction force between each other, then, why the big mass object did not accelerate faster due to the attraction force is greater than light object.


Falling from Earth's sphere of influence

How can I make a graph of an object falling from say, the Earth's sphere of influence, where acceleration(,a) is the real force of gravity at the objects radius? ie, using $$s=ut+\frac{1}{2}at^2$$ I can find the position s after a time t due to acceleration, a. This does not account for any gravity gradient, which is what I want to account for. Basically, $$ s=ut+\frac{1}{2}(\frac{GM}{r^2})t^2 $$ But there is a problem, firstly, I don't think that would work as it still only uses a constant acceleration, also there is a circular dependancy with r and s basically having the same meaning. (s + r would be the original radius at t=0) I'm not sure how helpful this will be, but recall (Hopefully accurately) that the area under an acceleration time graph is the distance travelled. Currently I have a function for the force at any radius [f(x)=GM/r^2], and a function for the position with the acceleration for a point r in it. In my case, it is [s(x)=ux+1/2 f(r) x^2] **Edit:** What I have realised is that when I calculate the position it returns s=ut+1/2 at^2 (duh...) So every point on the graph is the distance travelled had the object been accelerating for time,x all the way from the starting distance, rather than accelerating from the last point. Hope that makes it clearer, is there a way to get around this? P.S. In the graph, the actual force is not what I want and I am assuming that the object has a mass of 1kg, thus the force is the acceleration. Thanks in advance, Leo


Distance traveled in a simple two body problem

I'm trying to program an $N$-body simulation and I'd like to be able to test it with a known solution to a simple, two-body problem. I've looked at multiple sources, but I just don't know how to apply it to my simple test case. Two objects at rest placed 10 meters apart with mass of 1. The force between them is a modified gravitational force of F = 10 * m1 * m2 / r^2. How long will it take for each object to travel 4 meters?


If an object fell from the moon

Ignoring the moons gravity, if an object sitting still (relative to the Earth, i.e. not in orbit) was dropped from the moon. How long would it take to hit the Earth?


One dimensional motion with changing acceleration. Falling to a large body from a great distance

> **Possible Duplicate:**   >  Radial fall in a Newtonian gravitational field My math and physics are rusty. I am trying to calculate the time an object takes to fall to a large body. Before you answer $1/2at^2$, the conditions where $1/2at^2$ apply is where $a$ is a constant. Constant acceleration is a fine approximation when the distance the object falls is $<<$ than the distance to the center of the large body. i.e. 100 meters on the surface of the Earth. I'm looking at problems where distance fallen is 50% or more of the distance between objects. Gravity varies with the inverse square of distance. Given two bodies, $m_s$ and $m_l$. $$ F = m_sa = m_la $$ $$ F = Gm_sm_l/r^2 $$ $$ m_sa = Gm_sm_l/r^2 $$ $$ a = Gm_l/r^2 $$ Thus acceleration of $m_s$ varies with the inverse square of the distance to the center of $m_l$, and versa visa. Given R is the radius of the $m_l$, defining the surface of $m_l$. Given $h$ is the distance $m_s$ is above $m_l$'s surface, subsitute $R+h$ for $r$. $$ a = Gm_l/(R+h)^2 $$ This is as discussed at stackexchange 35878. Given the initial velocity of $m_s$ is zero, what is the time, $t$, it takes $m_s$ to fall distance $h$? An alternative question is give a $t$, how far with $m_s$ fall? This is where I'm rusty. I don't believe one can't simply substitute $Gm_l/(R+h)^2$ for $a$ in $1/2at^2$, but I don't know what else to do. The stackexchange articles, 15587 and 41741, discuss $a(x)$ where $a$ changes linearly with distance. Here $a$ changes with the inverse square of the distance. I did a lot of web searching and did not find this topic discussed. The discussions are with $a$ being constant or changing in a linear manner. Thanks for the help.


What's an equation for two astronomical entities both of 4000 tonnes in weight, colliding?

I have next to no knowledge of any physics, but would be happy if you could answer my question... I want to know an equation for two astronomical entities such as the star Sirius (2.02 solar mass) colliding.


Basic question about law of gravitation

> **Possible Duplicate:**   >  Radial fall in a Newtonian gravitational field This is how Wikipedia defines Newton's law of Gravitation: > Every point mass attracts every single other point mass by a force pointing > along the line intersecting both points. The force is proportional to the > product of the two masses and inversely proportional to the square of the > distance between them: > > $F=G\frac{m1m2}{r^2}$ > > where > >   * F is the force between the masses, >   * G is the gravitational constant, >   * m1 is the first mass, >   * m2 is the second mass, and >   * r is the distance between the centers of the masses. > Now, say 2 spheres, one the size of the earth and the other the size of a ping pong ball are placed say 10 km apart. There are no other forces acting on the system other than gravitation. If I've understood rightly, then _both_ the ball & the earth sized sphere will be pulled towards each other with the same huge force. My question is- how can we calculate when the 2 spheres will meet? And while calculating that shouldn't we consider the fact that the force is changing every moment because the distance between them $r$ changes every moment? How can I include this factor into my calculations? Further, since the force is changing every moment, are the spheres undergoing acceleration or _accelerated_ acceleration?? (Forgive me if my terminology is improper. I'm a beginner.) Is there any name for such forces & accelerations which change at a predictable rate as in this question?


Determining impact velocity over long distances

Everything I have found concerning impact velocity from a given height deals with constant acceleration due to gravity. I would like to know how to work in varying acceleration into the equation. So, time for the question... How does one go about determining the velocity at which an object with negligible mass $m$ and radius $a$ will hit the surface of a much larger object with mass $M$ and radius $r$ from a initial distance $R$? To be a bit more specific, let's say we have a closed system with two objects, one comparable to the moon and another with a small mass, let's say a basketball. The problem I have with this situation is the acceleration due to gravity is not constant at any given distance. Newton's equation for this is: $$g = GM / R^2.$$ Ideas?


Is there any problem a quantum finite state machine can do faster than a classical finite state machine?

All of the quantum algorithms I've seen so far require a turing-complete quantum computer, at least as far as I can tell. Are there any quantum algorithms that require only a quantum finite automaton? If so, how does their asymptotic complexity compare to the classical versions of those algorithms?


How do you calculate heat flux (Kw/m2) at the focal point of a mirror?

can anyone help me to determine the heat flux (Kw/m2) on a focal point of a parabolic dish having a diameter of 1.5 meter and a focal length 60 cm ??? please awaiting your soonest reply for my senior project :( Regards


Generalized Coulomb's Law

This question is about the generalization of Coulomb's law to continuous bodies of charge. The basic statement of Coulomb's Law involves two discrete charges $q_1$ an $q_2$: $$\vec{F}_i = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r_{12}} \hat{r}_i $$ Here $i$ represents the charge on which the force is exerted, and $\hat{r}_i$ represents the unit displacement vector between the other charge and the charge $i$. Many treatment of electrostatics extend this law to the case that one charge is not discrete, but rather a continuous body. The force on the discrete charge $Q$ is then: $$\vec{F} = \frac{Q}{4 \pi \epsilon_0} \int \frac{dq}{r^2} \hat{r} $$ Here $dq$ is the infinitesimal charge element of the continuous body, while $r$ and $\hat{r}$ represent the distance and displacement vectors between $dq$ and $Q$. Continuing this way, we could probably propose an expression for force between two continuous bodies of charge, like so: $$\vec{F} = \frac{1}{4 \pi \epsilon_0} \int \int \frac{dq_1 dq_2}{r^2} \hat{r}$$ However, I have not really seen this expression in the literature/treatments of electrostatics. Does anyone know why this is the case? Is the expression not useful, or are there no applications demanding the above expression?


Stability of the vacuum state of interacting quantum fields

"Stability" is generally taken to be the justification for requiring that the spectrum of the Hamiltonian should be bounded below. The spectrum of the Hamiltonian is _not_ bounded below for thermal sectors, however, but thermal states are nonetheless taken to be stable because they satisfy thermodynamic constraints. In classical Physics, we would say that the thermal state has lowest _free energy_ , which is a thermodynamic concept distinct from Hamiltonian operators that generate time-like translations. The entropy component of free energy, meanwhile, is a _nonlinear_ functional of the quantum state (presuming that the definition of entropy in quantum field theory would be at least this much like von Neumann's definition in terms of density operators), so we can reasonably expect the sum of the energy and entropy components to have a minimum in the state of greatest symmetry, as we see for thermal states. [It seems particularly notable in this context that the entropy is not an observable in the usual quantum mechanical sense of a linear functional of the quantum state.] The presence of irreducible randomness in quantum mechanics presumably puts quantum field theory as much in the conceptual space of thermodynamics as in the conceptual space of classical mechanics, despite the quasi-functorial relationship of "quantization", so _perhaps_ we should expect there to be some relevance of thermodynamic concepts. Given this background (assuming, indeed, that no part of it is _too_ tendentious), **why should we think that requiring the Hamiltonian to have a spectrum that is bounded below should have anything to do with stability in the case of an interacting field?** The fact that we can construct a vacuum sector for free fields in which the spectrum of the Hamiltonian is bounded below does not seem enough justification for interacting fields that introduce nontrivial biases towards statistically more complex states. This question is partly motivated by John Baez' discussion of "quantropy" on Azimuth. I am also interested in the idea that if we release ourselves from the requirement that the Hamiltonian of interacting fields must have a spectrum that is bounded below, then we will have to look for analogues of the KMS condition for thermal states that restore some kind of analytic structure for interacting fields. I asked a related Question here, almost a year ago. I don't see an answer to the present Question in the citations given in Tim van Beek's Answer there.


Relation between decay probability and the energy of particle

Is there any way to find the energy of a particle through its decay probability?


Why there's a whirl when you drain the bathtub?

At first I thought it's because of Coriolis, but then someone told me that at the bathtub scale that's not the predominant force in this phenomenon.


What causes a whirlpool to form in a bathtub?

I mean, when you have a full bathtub and then let it empty, under what conditions does a whirlpool form? Could we devise an experiment and measure this effect? When it comes to fluids, I don't know why I can't think of anything to test a hypothesis.


Why can glass absorb ultraviolet light?

Please see the following photos. (I cannot post them...) http://i1163.photobucket.com/albums/q554/startanewww/CIMG4545.jpg http://i1163.photobucket.com/albums/q554/startanewww/CIMG4546.jpg From the first photo, the book mentioned _Don't cover the paper with glass because glass absorbs enough ultraviolet light to slow the damage process._ (line 8) My questions:   * Why can glass absorb ultraviolet light?   * Will papers with dark colours fade fastest? Why is that the case? (see the second last paragraph)


Slowdown rate of rotating body due to friction force

This isn't a homework question, but it might as well be. The problem I have been pondering is: > If a disc (or children's roundabout if you like), of radius r, mass m, is > spun around it's center with an initial force F, and thereafter there is the > friction force (of either the axle or air resistance or both) of f, then how > long will it take to come to a stop? I have thought about it and have come up with not much. My first way is thus: $F = ma$, so $a = F/m$, the initial acceleration ( or should that be $(F-f)/m$ ?). And then the deceleration is $a = -f/m$. I'm not sure how to calculate the initial linear velocity, but assuming I have it, $u$, say, then I could say that after time $t$ the velocity is $v = u -at = u -ft/m$, where f is the friction force. So then the disc would stop spinning when $ t= um/f$. I am aware that this is wrong (well, it might work if we were dealing with linear motion). Straight away it seems wrong because it doesn't take into account the radius of the disc and also the slow down seems linear, when from observation it seems rotating discs slow down and taper off to a standstill. But that is as far as I got. I have tried to use angular motion equations (well $\omega r = v$) but I am stuck at this point, and of course, finding the initial velocity. Any help is appreciated.


Do Photons interfere when it passes through a slit (one)?

When a light (photons) goes through two slits it creates interference patterns. if the light goes through a "single" slit, does it create interference patterns or does it behave like particles (photons)? is the pattern same for both particles and light waves if it is a "single slit"? In the double slit experiment, if you close one slit (or observe) it is said that light behave as particles (bullets) which means that through one slit light exactly behaves as stream of particles??


Is this really how a capacitor works? Why doesn't it behave like a resistor?

My book says a capacitor is two conducts being connected by an insulator. Now let's take a parallel plate capacitor to simplify the problem I have. Suppose I got two parallel plate capacitor in series and I hook the circuit up with a battery. ![enter image description here](http://i.stack.imgur.com/YWnG8.jpg) As soon as I hook it up, electrons flow (forget conventional current for now) into on the right plate and builds up on that capacitor (and spreads on the surface of conducting plate) and remains stuck there because there is an insulator that blocks the electrons from going anywhere. Now here is my confusion, how does the left plate of $\ C_1$ even build the positive charges and how do the current even run through the circuit if there an insulator blocking the electrons from moving? Is there even current through the circuit?


Gravitational field with cavity removed

I'm just struggling a little with this question: A uniform sphere, of radius $R$, contains a spherical cavity of radius $R/4$, whose centre is $3R/8$ from the surface. The diameter passing through the centres of the sphere and cavity meets the surface at points $X$ and $Y$. Find the ratio of the gravitational field at $X$ and $Y$. My attempt at the solution goes something like this: Using the superposition principle, the gravitation field due to the whole mass is equal to the sum of the gravitational fields due to the remaining mass and the removed mass. The gravitational field due to a uniform solid sphere is zero at its centre. Therefore, the gravitational field due to the removed mass is zero at its centre. The gravitational field due to the solid sphere is equal to the gravitational field due to the remaining mass. Now we know g acts towards the centre of the sphere. As such, both the gravitational field of the combination of the sphere and removed mass and the gravitational field of the sphere only act in the same direction, so we can use the scalar form of the equation. Therefore the gravitational field is given by $g=GMr/R^2$. Then insert $r=-R$ and $R$ for the gravitational field at $X$ and $Y$. But this doesn't seem to be correct as it is just the same as if the removed mass wasn't there..... have I gone wrong in my logic somewhere?


Optical laser pumping and reflectors

Let's say we are building Nd:YAG laser. It is optically pumped by some linear xenon flash lamps, it absorbs light around 750nm and 800nm, and emitted light is at 1064nm. The question is why doesn't 1064nm emission from the flash lamps interfere with laser operation? Why doesn't 1064nm photons emitted in the Nd:YAG at 'wrong' directions (not coaxially to the resonator path) and reflected back and forth from the cylindrical/oval reflector (for the flashlights) interfere with the laser emission? As far as I see it, both these factors should consume precious atoms in excited state, and probably require some 1064nm filter around the Nd:YAG rod...


At CMB recombination (z=1090), what is the radial extent of the last scattering "shell"? a) Delta(z) = .... b) Delta(comoving angular distance)= ....Mpc The WMAP first-year parameters give Delta(z) = 195. Is this still correct?


Flow of liquid among branches

If water is flowing through big pipe is branched into 4 branches of small pipe. Lets say the flow is around 4 m/sec. I have the following questions:     1. What will be the flow rate in each of the pipes? I would be knowing diameter, height of the pipes. Considering all pipes are rigid.     2. What will be the flow rate in each of the pipes, if I close one pipe of the 4 branches? Will water flow speed increase in other 3 pipes? If yes, how we can calculate the gain of speed of flow?     3. What if the pipes are non-rigid, will that have any effect?


Practical method to weigh human limbs with common household items?

What methods could be used to determine ( _or_ estimate within a _reasonable_ margin of error) the mass of a living human's limbs, short of cutting them off? And more interestingly, how can this be done without any high tech equipment, just with the means commonly found in households? A scale for example is allowed. An MRI isn't ;)


Has anyone else thought about gravity in this way?

Picture yourself standing on a ball that is expanding at such a rate that it makes you stick to the ball. Everything in the universe is expanding at this same rate. To escape the earths gravitational pull we would need to jet upward faster than the expansion of the earth. Each object expands at a different rate on its surface according to its size. Thus different gravity affects for different size planets. When in space we are subject to being affected by the most distant body if we stand in its way. I just can not explain the reaction of our tides with our moon. Have any scientists seriously considered an idea like this? Follow up July 23 I am no scientist, but I think someone with more knowledge might explore this idea a little further. At the very lease the idea that every thing in the total universe is expanding, including all parts of the atom can be used as a simple way to see formulas and the same results to the effects of gravity of anything on the surface of a sphere planet or a donut shaped planet. The area of mass will grow but the density will remain the same. The idea can be cross referenced by light shifting etc, to see if it falls in line with the known action of planet gravity and the known expansion affect of the whole universe. Maybe the gravity affect of a planet on its surface dweller is a completely different force than is the force that maintains the orbits of planets. keplers law I believe. What happens when we have an eclipse of the moon?, does the earths orbit around the sun change for time of this eclipse? My summary is that if all scientists can not explain gravity totally, then maybe the common thought for all these years is not completely a correct one.


Define Pressure at A point. Why is it a Scalar?

I have a final exam tomorrow for fluid mechanics and I was just looking over the practice exam questions. They do not provide solutions. But pretty much I have to define pressure at a point and also say why pressure is scalar instead of a vector. I am thinking pressure at a point is $P=\lim_{\delta A \to 0} \frac{\delta F}{\delta A}$. Please let me know if I am wrong. But I do not know at all why pressure is a scalar instead of a vector. I know it has something to do with $d \mathbf{F}=-Pd \mathbf{A}$


Intercept a moving object

Object A can move at 50km/h, wants to intercept object B (currently $15^{\circ}$, east of north from A) moving at 26km/h, $40^{\circ}$ east of north. What angle should A take to intercept B? AB is 20km apart ![enter image description here](http://i.stack.imgur.com/Kcm8I.png) The provided answer looks like: * * * Choose x axis along 20km distance. $26t \sin{(40-15)} = 50t \sin{\theta}$ $\theta = \sin^{-1}{\frac{11}{50}} = 12.7$ $15 + 12.7 = 27.7$ * * * I took a different approach and used $\cos$ and got a different answer ... why is that? $26t \cos{(40-15)} = 50t \cos{\theta}$


Buoyancy fluxes in a stratififed fluid and units

I am calculating the buoyancy flux ($B$) for a stratified fluid as follows: $$ B=\frac{g\alpha S}{C_{pw}\rho_0} $$ where $g = 9.81$ $m/s$; $\alpha = 1.6 t\times10^{-5} + 9.6\times10^{-6} \times (20 \text{ degC})$; $S = 100\text{ }Wm^{-2}$, $\rho_0 = 1000$ $kg/m^3$, and $C_{pw}$ is the specific heat of water. The question I have is: Is $C{pw} = 4200$ or $4.2$ I've seen it used both ways, and I am unsure which I would use with the units of the other terms in the equation.


Treatment of boundary terms when applying the variational principle

One of the main sources of subtlety in the AdS/CFT correspondence is the role played by boundary terms in the action. For example, for a scalar field in AdS there is range of masses just above the Breitenlohner-Freedman bound where there are two possible quantizations and which one you get depends on what boundary terms you add to the action. Boundary terms are also essential in the treatment of first-order Lagrangians for fermions and self-dual tensor fields. These all involve the "UV" boundary as $z \rightarrow 0$ in Poincare coordinates. Then there are dual models of QCD like the hard-wall model where one imposes an IR cutoff and imposes boundary conditions at the IR boundary and/or adds IR boundary terms to the action. My question is a bit vague, but basically I would like references to reviews, books or papers that give a good general treatment of the variational principle when one has to be careful about boundary terms. It would help if they clearly distinguish the requirements that follow from mathematical consistency from those that are imposed because of a desire to model the physics in a certain way.


Confront Order Of magnitudes

Is it correct to say that 9.0 is one order of magnitude smaller than 10.0? Has anyone a link/source about confronting order of magnitudes, apart from wikipedia?


Isn't the uncertainty principle just non-fundamental limitations in our current technology that could be removed in a more advanced civilization?

From what I understand, the uncertainty principle states that there is a fundamental natural limit to how accurately we can measure velocity and momentum at the same time. It's not a limit on equipment but just a natural phenomenon. However, isn't this just an observational limit? There is a definite velocity and momentum, we just don't know it. As in, we can only know so much about the universe, but the universe still has definite characteristics. Considering this, how do a wide range of quantum mechanical phenomena work? For example, quantum tunneling - its based on the fact that the position of the object is indefinite. But the position is definite, we just don't know it definitely. Or the famous light slot experiment? The creation of more light slots due to uncertainty of the photon's positions? What I am basically asking is why is a limit on the observer, affecting the phenomenon he is observing? Isn't that equivalent to saying because we haven't seen Star X, it doesn't exist? It's limiting the definition of the universe to the limits of our observation!


Is the uncertainty principle a property of elementary particles or a result of our measurement tools?

In many physics divulgation books I've read, this seems to be a commonly accepted point of view (I'm making this quote up, as I don't remember the exact words, but this should give you an idea): > Heisenberg's uncertainty principle is not a result of our lack of proper > measurement tools. The fact that we can't precisely know both the position > and momentum of an elementary particle is, indeed, a property of the > particle itself. It is an intrinsic property of the Universe we live in. Then this video came out: Heisenberg's Microscope - Sixty Symbols (skip to 2:38, if you're already familiar with the uncertainty principle). So, correct me if I'm wrong, what we may claim according to the video is: > the only way to measure an elementary particle is to make it interact with > another elementary particle: it is therefore **incorrect** to say that an > elementary particle **doesn't have** a well defined momentum/position before > we make our measurement. We cannot access this data (momentum/position) > without changing it, therefore it is **correct** to say that our ignorance > about this data is not an intrinsic property of the Universe (but, rather, > an important limit of how we can measure it). Please tell me **how** can both of the highlighted paragraphs be true or how they should be corrected.


Uncertainty principle implies the non-deterministic universe?

Does the uncertainty principle imply the non-deterministic universe, or just the fact that our model of the universe, the one based on observation, can be at most non-deterministic, since we will not be able to measure with perfect accuracy, even if such would exist (and therefore we will have to provide probability distributions rather than an event which should happen with overwhelming probability (term used by Terence Tao))?


About Heisenberg uncertainty principle

What would happen if someone invented a way to measure both position and momentum precisely? If it is impossible why?