Which "space-time coincidences" are described by a "co-ordinate system in which the gravitational field does not appear"?

In Einstein's exposition of the foundations of General Relativity (cmp. http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity , end of §3) there appears an emphasis on considerations of " _coincidences_ ": > _All our well-substantiated space-time propositions amount to the > determination of space-time coincidences. [...] The results of our > measurements are nothing else than well-proved theorems about such > coincidences of material points_ and their relation to " _co-ordinates_ ": > _The introduction of a system of co-ordinates serves no other purpose than > an easy description of totality of such coincidences._ Subsequently (in §4) one more or less specific " _system of co-ordinates_ " is suggested: > The nature of acceleration of an infinitely small (positional) co-ordinate > system is hereby to be so chosen, that the gravitational field does not > appear; this is possible for an infinitely small region. $X_{1}, X_{2}, > X_{3}$ are the spatial co-ordinates; [...] Question:   What, explicitly, is contained in the corresponding " _totality of such coincidences_ ";   or in some specific subset of this " _totality_ "?,   of which the indicated " _co-ordinate system in which the gravitational field does not appear_ " is a suitably " _easy description_ ". Does this " _totality_ " contain, for instance: (1)   for any three distinct " _material points_ " (${\textbf A}$, ${\textbf B}$, and ${\textbf P}$, who are assigned the corresponding unequal constant " _spatial co-ordinate_ " triples $\\{ \, X_{1}^{\textbf A}, X_{2}^{\textbf A}, X_{3}^{\textbf A} \, \\}$, $\\{ \, X_{1}^{\textbf B}, X_{2}^{\textbf B}, X_{3}^{\textbf B} \, \\}$, and $\\{ \, X_{1}^{\textbf P}, X_{2}^{\textbf P}, X_{3}^{\textbf P} \, \\}$, respectively),   and for any " _coincidence_ " ${\textbf A}_{\mathscr X}$ in which ${\textbf A}$ had been participating,   the " _coincidence_ " that ${\textbf A}\\!$'s indication of having seen that ${\textbf P}$ saw ${\textbf A}\\!$'s indication of having seen that ${\textbf B}$ saw " _coincidence_ " ${\textbf A}_{\mathscr X}$   was coincident to   ${\textbf A}\\!$'s indication of having seen that ${\textbf B}$ saw ${\textbf A}\\!$'s indication of having seen that ${\textbf P}$ saw " _coincidence_ " ${\textbf A}_{\mathscr X}$   ? And is " _an easy description_ " of this " _totality_ " afforded by the indicated system for instance in the sense that: (2)   for any two distinct " _material points_ " (${\textbf A}$ and ${\textbf B}$, who are assigned the corresponding unequal constant " _spatial co-ordinate_ " triples $\\{ \, X_{1}^{\textbf A}, X_{2}^{\textbf A}, X_{3}^{\textbf A} \, \\}$, and $\\{ \, X_{1}^{\textbf B}, X_{2}^{\textbf B}, X_{3}^{\textbf B} \, \\}$, respectively) there is another triple, $\\{ \, X_{1}^{\textbf M}, X_{2}^{\textbf M}, X_{3}^{\textbf M} \, \\}$, reserved as constant " _spatial co-ordinates_ " to denote a " _material point_ " ${\textbf M}$, if it happens to exist, as "middle between ${\textbf A}$ and ${\textbf B}$", described by the " _coincidences_ "   * that for any " _coincidence_ " ${\textbf A}_{\mathscr X}$ in which ${\textbf A}$ had been participating,  ${\textbf A\\!}$'s indication of having seen that ${\textbf M}$ saw ${\textbf A\\!}$'s indication of having seen that ${\textbf M}$ saw " _coincidence_ " ${\textbf A}_{\mathscr X}$   was coincident to   ${\textbf A\\!}$'s indication of having seen that ${\textbf B}$ saw " _coincidence_ " ${\textbf A}_{\mathscr X}$, and   * that for any " _coincidence_ " ${\textbf B}_{\mathscr Y}$ in which ${\textbf B}$ had been participating, ${\textbf B}$'s indication of having seen that ${\textbf M}$ saw ${\textbf B}$'s indication of having seen that ${\textbf M}$ saw " _coincidence_ " ${\textbf B}_{\mathscr Y}$   was coincident to   ${\textbf B}$'s indication of having seen that ${\textbf A}$ saw " _coincidence_ " ${\textbf B}_{\mathscr Y}$, and   * that for any " _coincidence_ " ${\textbf M}_{\mathscr Z}$ in which ${\textbf M}$ had been participating,  ${\textbf M}$'s indication of having seen that ${\textbf A}$ saw " _coincidence_ " ${\textbf M}_{\mathscr Z}$   was coincident to   ${\textbf M}$'s indication of having seen that ${\textbf B}$ saw " _coincidence_ " ${\textbf M}_{\mathscr Z}$   ?


What is the principle of equivalence in thermodynamics?

I've been searching for this for a while. There is a principle of equivalence in general relativity: http://en.wikipedia.org/wiki/Equivalence_principle But I need the principle of equivalence in thermodynamics. Is it the same as the second law of thermodynamics? http://en.wikipedia.org/wiki/Second_law_of_thermodynamics


What transformation is the metric of general relativity invariant under?

My limited understanding of metrics comes from Cartan. From there, I understand that a metric is something invariant under certain transformations, e.g. Lorentz in special relativity. But with the metric varying from point to point (event-to-event), what is the meaning of the metric?


Can anyone explain this demonstration with a helium balloon?

See the YouTube video (click) named "Testing the Magnetic Nature of MagneGas.mp4". A helium balloon is attracted to a metal bar.


Demonstration that vibrating basic particles constitute non-vibrating individuals

I am a dilettante in physics; I ask for pardon for my confusion-causing (if any) terminology usage, and also for my imprecise choice of question tags. I know that basic particles of any individual stopping vibrating contradicts Heisenberg's uncertainty principle, so the particles must whenever vibrate. Then how could basic particles constitute a seemingly non-vibrating individual? Is it a natural limitation of human eyesight or ...? Is there any scientific explanation for it? A demonstration not using analogy or common sense is seeking after.


Questions about entanglement from a laymen/quantum hobbyist

Please note I am not a physicist I just read every article I can on it, I understand a good amount on it though. But no maths. (currently trying to learn the maths) By what means can we as humans entangle two particles? How does (just examples are fine) it happen in nature? And I would guess technically there could be a particle in me entangled with a particle in another person; maybe my future wife? I think the answer to the first is... We make them by shooting a photon at a half silvered mirror and somehow it splits and one bounces off the half silver while the other one leaves and we capture the one, and wait for the other and capture it too. Then measure one and measure the other and see they are the same? EDIT: One more question. From one of the answers two entangled particles have the same wave function. Does this mean every particle with the same wave function is entangled? Can an electron and photon or any two arbitrary particles or groups of particles(molecules) entangle?


Topics in particle cosmology

> **Possible Duplicate:**   >  Topics in particle cosmology I am interested in learning more about this interdisciplinary approach. 1) What are some of the top questions in particle cosmology (e.g nature of dark matter, inflationary structure, topological defects in the universe)? 2) Who are some of the top researchers in this field? 3) Recommended review papers (on arxiv, for instance) will be greatly appreciated.


What is the angular distance between Ptolemaic perigees of Mercury?

In his excellent treatment of the history of the science of astronomical distances and sizes, Albert van Helden says (p.29) that > The complicated [Ptolemaic] model of Mercury has the curious property of > producing two perigees, each about 120° removed from the apogee. But when I try to confirm this using the 88 day period Mercury's epicycle, I get approximately half the expected value. In a given time (in days), $t$, Mercury will travel through an angle $\varepsilon = 2\pi t/88$ along its epicycle which will traveled through an angle $\delta = 2\pi t/365$ along its deferent. In transitioning from apogee to perigee, it must be the case (since Mercury must go half way round the epicycle, and then an additional $\delta$ to "catch up" with the angle traveled by the center of the epicycle) that $$\varepsilon=\delta+\pi$$ solving which yields $$t=[2\cdot(1/88-1/365)]^{-1}\approx58$$ which corresponds to about 57°, roughly half the number expected. What is missing from the above reasoning? Is there something about the definitions of epicyclic period I've missed, perhaps?


Using the area element in derivation of geodesic

In the derivation of the geodesic, one starts with the integral of the line element (arclength): $$L(C)=\int_{\tau_1}^{\tau_2}d\tau\sqrt{g_{\mu \nu}\dot{x}^{\mu} \dot{x}^{\nu}}$$ The integrand is then substituted into the Euler-Lagrange equation, which simplifies to the geodesic equation. My question is would it be possible to start with the area element instead of the line element, and thereby arrive at a surface area geodesic equation, which I guess would be an equation for the surface area of a given patch of manifold?


conservation of momentum when a bullet hits a block

why momentum is conserved when a bullet hit a block horizontally even when force of bullet is acting on it and net external force is not zero ?


Why does this cowbell appear to vibrate at the wrong frequencies?

The cowbell in question: http://www.youtube.com/watch?v=OuU0BzXm-qk In most cowbells, the larger the metal dome, the lower the frequency produced. What would cause a phenomenon where the opposite is true: the larger dome creates a higher-pitched sound, as the one in the video does.


Why is free neutron unstable but free proton is stable?

Aren't they basically made up of the same stuffs (quarks) in almost the same configuration?


Do stars appear to move with uniform motion?

The Ancient Greek astronomers had quite an obsession over uniform circular motion; I was wondering if there was a logical reason for this. Did it develop through actual observations of the stars? Do they appear to move uniformly? If so, why? I've read a few things online that attributes this belief mainly to "Pythagorean mysticism", saying that the Ancient Greeks simply looked for an idealistically satisfying description of celestial motions. This too makes me wonder though: 1.) Why uniform? 2.) Why circular?


Calculate speed of proton at infinity

I have this problem to solve: Two protons are on the x axis at x=-1 and x=0. An alpha particle is placed at x=2, and the proton on the left is released, find its speed at infinity. I know that this involves the change in kinetic and potential energies of the particles, but I dont understand which particles are included in that calculation. Is the proton on the left neglected because it is the one released? An explanation in general would be appreciated. For the purposes of following the rules in asking a homework question properly, the specific physics topics I was asking about were: which particles are included in the kinetic and potential energy calculation for determining the speed at infinity of a proton, and why. My effort in solving this question was finding out that work is the change in kinetic energy + change in potential energy, but not knowing which particles to include in the calculation. This should clear up any confusion that this question is off topic.


Two-oscillator system coupled to reservoir

I am implementing the Monte Carlo wave-function approach to dissipation problems. So far, I have simulated the quantum harmonic oscillator coupled to a finite temperature reservoir given in section 5A of the above paper. I must now implement the two-oscillator system, whose Hamiltonian I am told is of the form $$ H=\hbar\omega_aa^\dagger a+\hbar\omega_bb^\dagger b-\hbar ga^\dagger a(b^\dagger+b). $$ My question is this:   1. Can I form my wave-function vector in such a way that, for example, every odd entry represents the first oscillator in the corresponding state and every even entry represents the second? As an example, if both oscillators were in the second state, the wave-function would be $$ \left|\psi\right\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\0\\\1\\\1\\\0\\\0\\\\\vdots\end{pmatrix} $$   2. If this is possible, what sort of form would the ladder operators $a,a^\dagger,b,b^\dagger$ take and would the corresponding number operators for each oscillator still be $a^\dagger a$ and $b^\dagger b$? Sorry if this is a dumb question, I have not yet taken a real QM course. Thanks for the help.


Compton scattering multiple wavelengths?

The formula given for compton scattering shows that when x-ray of one specific wavelength hits carbon or some materials, emitted x-ray will be of one new specific wavelength. However, according to scattered x-ray intensity (y-axis) vs wavelength (x-axis) graph, it shows that there are multiple wavelengths for each scattering angle. So, what is the wrong part of my thoughts? How should compton scattering formula be interpreted? And how can one explain that scattered x-ray frequency can be higher than original x-ray frequency?


Using Moments to influence the thickness of beams in structure?

Are moments relevant to the thicknesses of beams in a structure (ie. the larger the moment the thicker the beam)?


Examples of nearly perfect fluids and gases

When learning for physics (hydrodynamics and gases) I wanted to know what would be examples of nearly perfect fluids (no surface effects and no friction) and perfect gases (only elastic collisions and a simplified interaction potential) that are used in experiments etc. When searching for that in the Internet I only found things like _ultra-hot quark-gluon plasma_ and _optically trapped 6Li atoms_ for perfect fluids. I rather thought of more practicable examples like fluid helium, is it near to a perfect fluid? For gases I think that would be hydrogen and light halogens (because the volume of the molecules is neglected), but at which temperatures?


Using the Scalar Electrostatic Potential to Calculate Transition Probabilities

**transition probabilites** of atomic systems prone to some time-varying electromagnetic field are very often calculated using **perturbation theory** leading to expressions including the vector potential $\mathbf{A}$. This approach seems to be very complicated and I am wondering: > ### Is using the electrostatic potential $U$ to calculate "electric > transitions" (approximately) sufficient? This was the short version of my question; below you will find some further notes that (hopefully) contain all information needed to clearify it. In **relativistic quantum mechanics** , the action of an an **electrodynamic field** on an electron can be realized by minimal coupling using the replacement $\partial_\mu \rightarrow \partial_\mu - \mathrm{i}eA_\mu$. Thus, in a certain sense, (quantum) electrodynamics can be derived from the Dirac equation using the invariance of the free-electron action with respect to a phase change. ### Electromagnetic Fields and the Schrödinger Equation Nevertheless, often one must ignore the beauty of gauge theory to be able to **practically calculate** something under certain simplifications using the non-relativistic Schrödinger-equation. There, one can incorporate an external electromagnetic field applying the replacements $$H\left(\mathbf{p},\mathbf{x}\right) \rightarrow H\left(\mathbf{p}-e\mathbf{A}\left(t,\mathbf{x}\right),\mathbf{x}\right)+eU\left(t,\mathbf{x}\right)$$ to the Hamiltonian of the system where $\mathbf{A}$ is the known vector potential and $U$ is the potential corresponding to the electric field. In the Coulomb gauge $\mathrm{div}\mathbf{A}=0$ one can always achieve $U\equiv 0$ if no sources are present and the overall Hamiltonian will look like $$H = H_0 -\frac{e}{m}\mathbf{A}\left(t,\mathbf{x}\right)\mathbf{p}$$ where $H_0 = \frac{\mathbf{p}^2}{2m} + V(\mathbf{x})$ and the term in $\mathbf{A}^2$, the ponderomotive potential has been neglected. ### Time-Dependent Perturbation Theory The last expression for $H$ has the form $$H = H_0(\mathbf{p},\mathbf{x}) + H_t(\mathbf{p},\mathbf{x},t)\ .$$ Using the Dyson series, one can formally write down the time-dependent solution in the interaction picture as $$|\psi_I(t)> = T\exp\left[\frac{1}{\mathrm{i}\hbar}\int_{t_{0}}^{t}H_{t,I}\left(t^{\prime}\right)dt^{\prime}\right]|\psi_I(t_0)> .$$ Then, one takes only the first two terms into account to find that **transition amplitudes** are approximately given by $$A_{n\rightarrow m}\approx\frac{1}{\mathrm{i}\hbar}\int_{t_{0}}^{t}\left\langle m\right|H_{t,I}\left(t^{\prime}\right)\left|n\right\rangle dt^{\prime}\ \mathrm{with}$$ $$H_{t,I}\left(t\right) = e^{\mathrm{i}\left(t-t_{0}\right)H_{0}/\hbar}H_t\left(t\right)e^{-\mathrm{i}\left(t-t_{0}\right)H_{0}/\hbar}\ .$$ ### Relation to the Question In the given case we have $$H_t = -\frac{e}{m}\mathbf{A}\left(t,\mathbf{x}\right)\mathbf{p}$$ and assuming that $$\mathbf{A}\left(\mathbf{x},t\right) = \mathbf{A}\left(\mathbf{x}\right) T(t)\ ,$$ and further using $$\mathbf{p}=\frac{\mathrm{i}m}{\hbar}\left[H_{0},\mathbf{x}\right]\ ,$$ one will have to solve $$A_{n\rightarrow m}\propto <m|\mathbf{A}\left(\mathbf{x}\right)\cdot \mathbf{x}|n>\int_{t_0}^t e^{-\mathrm{i}\omega_{mn}\tau}T(\tau) d\tau\ +\ \mathrm{h.c.}$$ with $\hbar \omega_{mn} = E_m - E_n$, where the $E_i$ are energy levels of the unperturbed $H_0$. Using here the vector potential $\mathbf{A}$ is the correct way to incorporate the field, of course. For me, it seems **extremely complicated** to do so since to calculate the transition probabilities for an **arbitrary field distribution** one will have to expand $\mathbf{A}$ into vector spherical harmonics. So, the use of a scalar potential like $U$ would greatly simplify computations and my question can now be re-asked as > ### Why can't we just simply use $U(\mathbf{x})\cdot T(t)$ to calculate the > transition amplitudes $A_{n\rightarrow m}$ for "electric excitations"? **Thank you very much** for insights, comments and corrections.


Why electrons get excited?

Why and how are electrons get excited and what happen inside an atom when electrons get excited?


Time Evolution Operator in Interaction Picture (Harmonic Oscillator with Time Dependent Perturbation)

**1\. The problem statement, all variables and given/known data** _Consider a time-dependent harmonic oscillator with Hamiltonian_ $$\hat{H}(t)=\hat{H}_0+\hat{V}(t)$$ $$\hat{H}_0=\hbar \omega \left( \hat{a}^{\dagger}\hat{a}+\frac{1}{2} \right)$$ $$\hat{V}(t)=\lambda \left( e^{i\Omega t}\hat{a}^{\dagger}+e^{-i\Omega t}\hat{a} \right)$$ *(i) Compute $\hat{U}_S(t,0)$ using the interaction representation formula (Equation 1 in next section) to second order perturbation theory. (ii) Compute $\hat{U}_S(t,0)$ using (Equation 2 in next section) to second order perturbation theory. **2\. Relevant equations** EQUATION 1: $$U_I(t,0)=1-\frac{i}{\hbar}\int_0^t dt' V_I(t')+\left( \frac{-i}{\hbar} \right)^2 \int_0^t dt' \int_0^{t'} V_I(t')V_I(t'') + \dots$$ EQUATION 2: $$U(t,0)=1+\sum_{n=1}^{∞}\left( \frac{-i}{\hbar} \right)^n\int_0^t dt_1 \int_0^{t_1} dt_2 \dots \int_0^{t_{n-1}}dt_n H(t_1)H(t_2)\dots H(t_n)$$ **3\. The attempt at a solution** So I know that for the interaction picture the transformation of the operator $\hat{V}_I$ is.. $$\hat{V}_I=e^{\frac{i}{\hbar}\hat{H}_0 t} \hat{V} e^{\frac{-i}{\hbar}\hat{H}_0 t}$$ I also know that both operators and kets evolve in time. So I use the interaction picture equation of motion on the ladder operators so I can obtain an expression for them as a function of time. $$\frac{d\hat{a}}{dt}=\frac{1}{i\hbar}\left[ \hat{a},\hbar \omega \left(\hat{a}^{\dagger}\hat{a} + \frac{1}{2} \right) \right]$$ $$\frac{d\hat{a}^{\dagger}}{dt}=\frac{1}{i\hbar}\left[ \hat{a}^{\dagger},\hbar \omega \left( \hat{a}^{\dagger}\hat{a} + \frac{1}{2} \right) \right]$$ I then got.. $$\hat{a}(t)=\hat{a}(0)e^{-i\omega t}$$ $$\hat{a}^{\dagger}(t)=\hat{a}^{\dagger}(0)e^{i\omega t}$$ I plugged these into the expression for V to get, $$\hat{V}=\lambda \left[ \hat{a}^{\dagger}(0)e^{i(\Omega + \omega)t} + \hat{a}(0)e^{-i(\Omega + \omega)t} \right]$$ So now what needs to be done, is to transform this into the interaction picture and then plug it into Equation 1 from above and integrate. But this seems very messy and I am having doubts if this is the correct way to I also know that both operators and kets evolve in time. So I use the interaction picture equation of motion on the ladder operators so I can obtain an expression for them as a function of time. go about this problem. If anyone can shed some light onto this I would really appreciate it!


exponential potential quantization

What are the energies $E_{n}$ of the Schroedinger operator $$ -\frac{d^{2}}{dx^{2}}y(x)+ae^{bx}y(x)=E_{n}y(x) $$ for some real and positive 'a' and 'b' with the Boundary conditions $ y(0)=0=y(\infty) $? Working a bit I get the quantization condition $$ J_{E_{n}/4}(\sqrt{-a})=0,$$ ($ J_{a}(x) $ is a Bessel function) but unfortunately this equation yields only imaginary roots.


Energy in an EM wave should depend on frequency

I just finished reading Feynman's _Lectures on Physics_ vol.I, §34-9: "The momentum of light". The author explains that there is a relation between the wave 4-vector $k^{\mu}$ and the energy-momentum 4-vector $p^{\mu}$ of an EM wave, namely $$p^{\mu}=\hbar k^{\mu}, $$ or equivalently $$\tag{deB}W=\hbar \omega, \mathbf{p}=\hbar \mathbf{k},$$ and those equations are called _de Broglie relations_. However, as I learned in my classical electromagnetism course, flux of energy in such a wave is quantified by Poynting's vector, yielding formulas such as the following: $$\tag{1} I=\frac{1}{2 \mu_0 c} E_0^2, $$ where $I$ stands for "average intensity" of the wave and $E_0$ for "maximum amplitude of electric field". > **Question** Where is $\omega$? It does not appear in formula (1) nor in any > other formula based on Poynting's vector. But as of equations (deB) it > should do so. Am I wrong? Thank you.


How can the surface of a liquid exist if there are no attracting molecules above it?

Okay this is probably a stupid question but here goes: We know that molecules in a liquid are attracted from all the molecules around it so there is no net attraction. Well, then how do the molecules at the surface exist because there are only molecules below it and not above it? Won't they accelerate downwards? Also, does this have something to do with surface tension? If it does, then isn't surface tension a force that acts along the surface and not into or out of it?


Radial Schrödinger equation

I found a problem that says: > Show by direct substitution that $R_{10}$ is a solution of Schrödinger's > radial equation. AFAIK Schrödinger's radial equation is $\frac{-\hbar^2}{2m}[\frac{d^2}{dr^2}+\frac{2}{r}\frac{d}{dr}-\frac{l(l+1)}{r^2}]R+E_p(r)R=ER$ and $R_{10}$ in this case is: $2(\frac{Z}{a_0})^{3/2}e^{-\rho/2}$ where $\rho$ is $\rho = \frac{2Zr}{na_0}$ and $E_p(r)$ is: $E_p(r) = -\frac{Ze^2}{4\pi\epsilon_0r}$ The thing is I don't understand very well what $E$ is in this case and (I think) I need it to solve the problem. It's the total energy but in the book (Alonso y Finn) they never talk about the kinetic energy. I suppose it depends on $r$, $n$ and $l$ because once I do the substitution on the left side of the equation and I factorize it I get $R_{10}$ multiplied by a term that's full of those variables.


Formula for molar specific heat capacity in polytropic process

I found this formula for a polytropic process, defined by $PV^n = {\rm constant}$, in a book: $$C = \frac R{\gamma-1} + \frac R{1-n} $$ where $C$ is molar specific heat and $\gamma$ is adiabatic exponent. I do not know how it was derived, can someone guide me?


How can you test to see if a dice is weighted?

I was browsing Etsy today and came across this. What tests are there to see if the dice are usable, ie, if one side isn't favored over another, and if all sides are balanced? Would this just be to roll the dice a large nubmer of times and collect the data? Or are there other tests that could be done without doing that?


What lepton neutrino would be in this reaction?

If I have a reaction (shown below), and I'm supposed to work out what the products will be, what would the lepton neutrino $\nu$ be? e.g electron neutrino: $$ \nu + p \to $$


Converting Angular Momentum?

I am aware of three apparently different forms of angular momentum: spin, orbital and macroscopic, like a spinning crystal or baseball. Can angular momentum be transferred or converted from one of these forms to another? If so, how?


Are the rest masses of fundamental particles certainly constants?

In particular I am curious if the values of the rest masses of the electron, up/down quark, neutrino and the corresponding particles over the next two generations can be defined as constant or if there is an intrinsic uncertainty in the numbers. I suppose that since there is no (as of yet) mathematical theory that produces these masses we must instead rely on experimental results that will always be plagued by margins of error. But I wonder does it go deeper than this? In nature do such constant values exist or can they be "smeared" over a distribution like so many other observables prior to measurement? (Are we sampling a distribution albeit a very narrow one?) Does current theory say anything about this? (i/e they must be constant with no wiggle room vs. no comment) I am somewhat familiar with on-shell and off-shell particles- but I must confess I'm not sure if this figures into my question. I'd like to say that as an example I'm talking about the rest mass of the electron as could be determined by charge and charge/mass ratios. But perhaps this very number is itself influenced by off-shell electron masses? Perhaps this makes no sense. Needless to say I'd appreciate any clarification.


Source of potential energy in a capacitor being pulled apart

When the plate spacing is large the capacitor system is storing more energy than when it is small. If one manually changes the spacing, where does the additional energy "come from" as the plate spacing increases?


Can we see light as it 'interferes' with itself and produces the characteristic double-slit pattern?

This TED talk suggests that we can now watch as a beam of light propagates through a bottle filled with water. My question is: can we use this new technology to perhaps 'see' the photon as it makes its way to the detector screen in the double slit experiment? What implications can femto-photography have on the single/double-slit experiment and physics in general?


Is a singularity a real thing?

I've heard the work a few times now, the most recent in the star trek film. Is a singularity a real thing? If so what is it?


Dispersion in a rectangular prism

Why are the colours visible only at the fringes of the light refracted through a rectangular prism?


Creating matrix Hamiltonian for Feynman's CCNOT

I'm trying to read Quantum Mechanical Computer and to implement the CCNOT logical gate with Mathematica. Since i wish to use the SWITCH implementation of the CNOT [Fig.8] i've realized that i need to use 7 cubits (there's no difference with registry cubits and command line cubits). Also, i know that every operation is described by unitary matrix (we can only rotate ours cubit) i was thinking to use Pauli matrices, composed by tensorial product. As Feynman said, I'll obtain as results a $M=2^n * 2^n$. The vector will rapresent every possibile configuration of my system (it's composed by $2^n$) Also, using the observation that the upsite in the command line are invariant for my system (the expected falue of the number of the spinup commute with the hamiltonian of the spin) i can readuce the dimension of my matrix to $s*2^n$, and so the vector will be a unitary vector with only one $1$ in the corrispondence of the atom with the spin up. I have to create an algebra of operation that preserve the invariant of the clock, so not every possible operation is allowed on my chain (ex: NOT is not allowed because i will have 2 or 0 spin up that is an unacceptalbe substate). An example of a matrix that change my vector state from state $|1>$ to $|2>$ is the matrix $|2><1|$


role of pure mathematics in fusion?

Is there any way a pure math guy can get his rigorous math fix while simultaneously moving fusion research forward? Is fusion simply too complicated to take a pure math approach? What open math questions are there pertaining to fusion?


Does inducing a current in a wire have any effect on the strength of a permanent magnet?

Does inducing a current in a wire result in any changes in the strength of a permanent magnet? Specifically, what would the results of the following controlled experiment be? You set up two alternators that each simply consist of a permanent magnet located inside a stationary loop of wire as depicted here. With the first (control) alternator, you simply rotate the magnet at a certain velocity for a short time, once per day, and measure the amount of induced current on a daily basis. With the second alternator, you hook the magnet up to your favorite (external) source of rotational motion and spin it continuously and measure the amount of induced current at the same intervals as with the control alternator. Would the amount of current change over time (in either one)? If it would, in which direction and why? And if this is still not well-specified enough for an definite answer, what would it depend on? (how fast the magnets are rotating? what kind of circuit the wire is hooked up to? what the magnet is made out of? etc.)


Riemann Tensor Calculation trick(number of element)

When we calculate Riemann Tensor for different curvature we have lots of components. However, there are many components that are zero. How can we argue, based on the symmetry of connection , that those elements are zero? For example if I am calculating the Riemann Tensor of $S^2$ sphere, I get only one non zero component i.e. $R_{\phi,r,\phi}^{\theta}$ = $sin^2 {\theta}$ and other components are zero. So, How can I argue, without calculating that all other components are zero. Edit: (Dimension, No. of independent Riemann Components) = (2,1; 3,6 ; 4,20)


The magnitude of spacecraft anomalous velocities?

That Voyager and several other spacecraft have had anomalous changes in speed is known. According to the news reports, the differences are unsettling. But just how large are these anomalies really? Are they 'way down there' at the levels of computational precession? Or, if I understood how to calculate them, could I do so with my home computer in a reasonable period of time? (Say a few days calculation.)


Why is the Legendre transformation an application of the duality relationship between points and lines?

When I read the Wiki about Legendre transformation, there is a statement > The Legendre transformation is an application of the duality relationship > between points and lines. What's the meaning of this statement?


Lethality of sounds and extreme "loudness"

In theory, could pure sound be lethal? How loud would it have to be? Also, which events are the loudest in the universe, and how loud are they? I'm confining attention to events which occur regularly, e.g. starquakes.


Is there a mathematical derivation of potential energy that is *not* rooted in the conservation of energy?

For simplicity I'll consider only gravity, but in general this question only applies to conservative forces. As per my understanding, the way one gets to the equation for gravitational potential energy is this: $(1)$ $\ddot{r}=-\frac{MG}{r^2}$ is given experimentally $(2)$ $KE=\frac{1}{2}m\dot{r}^2$ is derived theoretically and (thought) experimentally $(3)$ Energy is conserved $(4)$ For a set of generic initial conditions,   * Using $(1)$, find $r(t)$ and then derive $\dot{r}(r)$   * Using $(2)$ find $KE(r)$   * Using $(3)$, $PE(r)=E_0-KE(r)$ **Is this a correct method?** If the answer is 'no' my question ends here, if not: surely $(3)$ is _by definition_ correct (circularly so), and so the equation for potential energy is arbitrary? I would be interested in seeing a method that does not do this (or is correct at all, if the answer _was_ 'no').


Wire morphing shape?

Can one morph a wire by electrical current into a shape different than static shape at beginning? As in the beginning shape being in a spiral, then when electrical energy is passed through the wire it becomes straight. When the current is shut off it would return to a spiral shape as in the beginning form. OR vise versa?


Is it possible to calculate chromacity coordinates using absorbance values?

I have absorbance values for blood plasma. Absorbance was measured in range of 380 – 600 nm every 5 nm ending up with absorbance values for 45 wave lengths. I was browsing through literature, and I came across chromacity coordinates (XYZ) and this seems to be a very convenient way of summarizing data about color in just two data points. I have been reading some literature about different colorimetric systems, but since my background is in genetics, it didn’t make much sense for me. Therefore, I would like to ask you if it is possible to calculate these chromacity coordinates using my absorbance values and if so does anyone now some kind of software for this or exact formulas on how to do it?


Buckyballs in vacuum

I've read about the idea that buckyballs and other nanostructures could be used to hold drugs and things until they reach certain places in the body and then get released. So I was wondering, if you created a buckyball in air, so that some molecules that are in air (such as oxygen and nitrogen) were inside the buckyball. if you then put the buckyball in a vacuum, would the air be able to escape, or would it be trapped in the buckyball. (And vice versa, a buckyball created in a vacuum put into the atmosphere, would any molecules fit through into the buckyball?)


Between a solid and a hollow cylinders of the same mass, which one has the greater rotational kinetic energy?

I know that rotational kinetic energy is $\frac{1}{2}I\omega^2$. Therefore, the rotational kinetic energy will depend on the moment of inertia. I came to the conclusion that since both have the same mass, both should have the same moment of inertia as   $$I = \sum_\text{over all mass} m r^2$$ But the answer says that since the hollow cylinder has greater moment of inertia, it has greater rotational kinetic energy. How can the hollow cylinder have greater moment of inertia even though the masses of both the bodies is same?


Air density equation in stratosphere

Does the density equation for air on Earth change in the outer atmospheres? I am generating a model for the Red Bull Stratos jump and am trying calculate a precise air density function.


Why can't missing mass be photons?

After a star lives and dies, I assume virtually all of it's mass would be photons. If enough stars have already lived and died, couldn't there be enough photon energy out there to account for all the "missing mass" (=dark matter) in the universe? And if there were enough photons to account for all the missing mass, what would it look like to us?


How to find the force of the compression at the core of a planet?

For example the core of the planet Jupiter, how to find out the magnitude of the force that compresses its core? and can that be thought of as the total force that holds the planet together?


Ground state degeneracy of a variation of Toric Code model

We know that the ground state degeneracy of Toric Code model is 4. An easy way of seeing this is the following: Consider a 2D spin model where all the spins live on the links. The Hamiltonian is $H=-\sum_v{A_v}-\sum_p{B_p}$, where $A_v=\sigma_{v1}^z\sigma_{v2}^z\sigma_{v3}^z\sigma_{v4}^z$ is the product of 4 spins around a vertex $v$, and $B_p=\sigma_{p1}^x\sigma_{p2}^x\sigma_{p3}^x\sigma_{p4}^x$ is the product of 4 spins around a plaquette. All the $A_v$ and $B_p$'s commute with each other and they all square to one; thus each $A_v$ and $B_p$'s can take values $\pm1$. To see the ground state degeneracy, we notice that $\prod_vA_v=1$ and $\prod_pB_p=1$, being two constraints on the $A_v$ and $B_p$'s. Thus different $A_v$ and $B_p$'s can now only label $2^{2N}/4$ different configurations ($N$ being the total vertex number) because of the 2 constraints. But the total Hilbert space is still $2^{2N}$-dimensional, thus on average each configuration has a 4-fold degeneracy. Now suppose we add a third term in Toric Code model, making it $H=-\sum_v{A_v}-\sum_p{B_p}-\sum_p{C_p}$, where $C_p=\sigma_{p1}^y\sigma_{p2}^y\sigma_{p3}^y\sigma_{p4}^y$ being the product of 4 $\sigma_y$'s around a plaquette. What happens to the ground state degeneracy? Note that after adding the third term, the Hamiltonian is still a sum of commuting projectors, thus the counting method should still work; I'm just having trouble counting the number of constraints. My guess is that after adding the $C$ term, the ground state degeneracy reduces to 2. Anyone can prove/disprove me?


Do we hear sounds differently on the highest mountains?

Some searching gives that above 6,000 meter altitude the air density is less than half of that at sea level. Speed of sound is about 15-20% slower and "acoustic impedance" seems to change too. Do humans notice these differences in sound? Does the pitch of tones change noticeable?


Does quantum mechanics predict instantaneous action at a distance even without entanglement?

The suggestion that quantum mechanics implies that instantaneous action at a distance occurs is normally based on the contention that this follows from the entanglement of particles that share a common origin. My question is: does quantum mechanics predict that this happens even when there is no entanglement? In particular, does this happen when two quantum systems interact in such a way that the wave function of at least one of them changes drastically? Take, for example, an interaction between a hydrogen atom and a hydrogen-oxide radical: H + HO = H2O. While the hydrogen atom is free, the wave function of its electron allows it to be anywhere in the universe. Or rather, it is everywhere in the universe. Instantly after the interaction has completed, the amplitude of the quantum wave of the same electron is smaller everywhere, except in the very immediate neighbourhood of the water molecule of which it has become a part. In the rest of the universe, the wave function has taken on other values, if not instantaneously – the reaction could take some time – then at least far quicker than light. One could object that the electron is never really at more than one point at any one time, only we don’t know which until we make a measurement. But that amounts to a refusal to accept quantum mechanics. The effect of a wave on its environment is determined by the totality of the wave, including even those places where its amplitude is partly or entirely imaginary. For example, the hydrogen atoms in a water molecule are positioned at an angle of 108 degrees from each other, as seen from the oxygen nucleus, due to some very particular properties of the wave functions which define the hydrogen-oxygen bonds. That results in water having some very peculiar properties, including the property that its solid form – ice – is less dense than the liquid form and therefore floats on it. Even the mere stability of water molecules is due to the wave function of the electrons which the hydrogen atoms share with the oxygen atom: in this wave function the electrons are as much as near the oxygen atom as the hydrogen atom.


Barrier in an infinite double well

I am stuck on a QM homework problem. The setup is this: ![enter image description here](http://i.stack.imgur.com/uRIu3.png) (To be clear, the potential in the left and rightmost regions is $0$ while the potential in the center region is $V_0$, and the wavefunction vanishes when $|x|>b+a/2$.) I'm asked to write the Schrödinger equation for each region, find its solution, set up the BCs, and obtain the transcendental equations for the eigenvalues. **Where I'm at** : I understand the infinite potential well easily and I have done a free particle going over a finite barrier before (which I understood less well, but I can deal with it it).   * The problem asks me to make use of "a symmetry" in the problem, which is a vague hint. Are they trying to get me to make $\psi$ an even function?   * I am supposed the condition for there to be one and only one bound state for $E<V_0$. How do I go about that?


Problem with convergent geodesics at 2D sphere

There is a chapter on general relativity in the book _Spacetime Physics Introduction To Special Relativity_ by Taylor and Wheeler, which qualitatively explains how attractive gravitational force can be understood in terms of space(time) curvature. They give (chapter 9.5, pp. 281-283 in 2nd edition) an example of two travelers on a 2d surface of sphere who start their motion some distance from each other at the equator straight north and move _along geodesic_ (great circle). Because of the surface curvature they _accelerate_ toward each other, and finally meet at the north pole (Fig. 9-4, attached). ![enter image description here](http://i.stack.imgur.com/x2xDm.jpg) My problem begins once I (probably wrongly) extend the idea and imagine that the surface of the sphere represents the entire closed universe (in which case it would be a 3D sphere). In the case of flat Euclidean universe the two observers moving along straight lines are in inertial motion (they are in free fall though isotropic, homogeneous space) and their distance changes linearly with time, that is without accelaration. When the universe is closed and its curvature is positive and constant, they would observe acceleration although the universe is still homogeneous. What exactly is wrong in my speculation (I assume that in reality such two observers in Einstein 1917 spherical universe would not meet at a single point) and what makes the difference? The fact that real space has 3 dimensions, the fact that one has to consider 3D+1 space*time* (with Minkowski rather than Euclidean geometry in the flat case), the fact that Einstein universe is unstable, or something else?


Why exactly does a boomerang return back to the thrower?

I was always intrigued by the phenomena that govern the returning back of a boomerang to the thrower. Even if it is dependent on various factors such as velocity, aerodynamics, wind resistance and so on I was wondering, there must be some concise explanation to it?


Is coherent light required for interference in Young's double slit experiment?

In this Veritasium video, a home experiment is presented which appears to produce a very good double-slit interference pattern with normal sunlight. The experiment is an empty cardboard box with a visor and a placeholder for a microscope slide with two slits on one side. This is arranged with the slits and visor facing the Sun, so the interference forms on the bottom of the box. ![enter image description here](http://i.stack.imgur.com/x2ssa.png) They claim to observe a good interference pattern from the two slits: ![enter image description here](http://i.stack.imgur.com/4ZNX6.png) Discussions of interference in optics textbooks often stress that coherent light is needed to produce such patterns, and that sunlight and other thermal sources of light do not have such coherence. How, then, is this possible?


Reasons for choosing $SU(3)$ as the color group vs. $SO(4)$

What are the reasons that $SU(3)$ is used for QCD? Why wouldn't the simpler & smaller group $SO(4)$ make a better candidate?


Neglecting mass at asymptotic spacelike momenta

What is the rational/reason for neglecting masses at asymptotic non- exceptional space-like momenta. I have come across this as a first fix for being able to extract information from the Callan-symanzik and Gell-Mann-Low methods. Is it purely physical, or is it both physical and mathematically sound reason? All responses are welcome.


Same partition functions, different theories

I am reading the book "Basic Concepts of String Theory" by Blumenhagen, Lust and Theisen and in page 290 they say: "It follows that the $E8\times E8$ and the $SO(32)$ heterotic string theories have the same number of states at every mass level which are however differently organized under the internal gauge symmetries. So, even though the partition functions are identical, the theories are nevertheless different. The differences show up in correlation functions." I am trying to understand how these differences would show up in correlation functions, ie what would be an example of two correlation functions that are different in these two theories. Any references where this is done/explained would be greatly appreciated!


Influence of a Capacitance Matrix of a voltage measurment

I am having a problem in understanding how in a multiconductor system the Capacitance Matrix influences the measure of voltage between two electrodes. In the case that interest me we have 4 electrodes, 2 emitting ones and 2 receiving ones. We transmit a known sinusoidal current into the the emitting electrodes and we measure a voltage on the receiving ones. We are looking for the phase and amplitude difference between the current emitted and the measured voltage. It is my understanding that the measured voltage is influenced by any conductor present in the medium close-by, therefore we have to determine the Capacitance Matrix to obtain the real voltage measured. $$ V_{real} = V_{measured} * C_{ij} $$ What I don't understand is the relation between the measured voltage ($V_{measured}$), the capacitance matrix ($C_{ij}$) and the real voltage ($V_{real}$). If any of you could help understand this I would be very grateful.


Laws of fluid flow in porous medium

What are the equations that describe the flow of a fluid in a porous medium? Is there a variation of the Navier-Stokes equations? I would like to model the flow of air through a sponge-like structure. Instead of modelling the structure of the sponge, I would like to treat it like a homogenous area, where I solve appropriate equations for conservation of momentum, mass and energy.


is gravity always the weakest force

> **Possible Duplicate:**   >  What does it mean to say "Gravity is the weakest of the forces"?   >  Strongest force in nature It is usually taught that the gravitational force is the weakest but I wanted to know if it will be always so. For example, On earth gravity is weaker than electromagnetic force or weak nuclear force, so is it that on black hole also if gravity is strongest than other forces shall be more stronger than it or it's that gravity becomes most strong suppressing others. Because if other forces increase in same proportion than it will have serious implications.


What does it mean to say "Gravity is the weakest of the forces"?

I can understand that on small scales (within an atom/molecule), the other forces are much stronger, but on larger scales, it seems that gravity is a far stronger force; e.g. planets are held to the sun by gravity. So what does it mean to say that "gravity is the weakest of the forces" when in some cases, it seems far stronger?


Why do we say that the electric forces are stronger than gravitational forces?

May be I am asking a stupid question but I am very confused...I understand both forces are inversely proportional to the distances squared, and that the gravitational constant is roughly $10^{-20}$ times the Coulomb's constant. But one involves charges, while the other involves mass. To me this makes as much sense as a saying a second is larger than a meter. Is it related to the energy densities of these fields? If yes, how? I think that it is not the duplicate of the mentioned one because the point asked in that question is why do we consider gravitational fields weaker even if they are more dominant in universe at macroscopic levels but I want to ask that on what criteria we define stronger or weaker fields....What do we exactly mean to say by declaring any one field stronger than the other ? Is it about their energy densities?


Escape Velocity question

I would like some help in understanding the derivation of the formula for escape velocity. Why does $t$ approach infinity? And why is $v(t)$ approaching zero? I know that $r(t)$ is infinite because the gravitational field is infinite. But I don't understand why $t$ (time) and velocity would be zero. I don't think I understand what $v(t)$ is. Is it velocity at a given point in time? And when time increases, you are farther away from the body producing the gravitational field and thus velocity needed for escape is lower? .... Here is a photo of the derivation: ![](http://i.stack.imgur.com/KIH3X.png) From the wikipedia page: http://en.wikipedia.org/wiki/Escape_velocity


Do Baryons have Quantized Vibrational and Rotational States?

Given that baryons are not point particles but are composed of three quarks, are there quantized vibrational and rotational states analogous to those of molecules? If not why not, and if so are there spectroscopically observable transitions between such states?


Tension in a curved charged wire (electrostatic force) - does wire thickness matter?

Consider a conducting wire bent in a circle (alternatively, a perfectly smooth metal ring) with a positive (or negative) electric charge on it. Technically, this shape constitutes a torus. Assume that the radius of the wire is much smaller than the radius of the circle. **Find the tension in the wire due to the repulsive electrostatic force** (in Newtons). For this problem, here are example physical values, (may or may not be necessary to solve the problem) charge, $Q = 1 C$, circle radius, $R = 1 m$, wire radius, $r = 1 mm$. Use the most straightforward notations possible for Coulomb's law, $F=\frac{k q_1 q_2}{r^2}$. This is a question about the mathematics of a set of inverse-square problems, not about theoretical E&M. Other notations that will be of useful to this discussion:   * $\lambda = \frac{Q }{ 2 \pi R } $ = linear charge density    * $\sigma = \frac{\lambda }{ 2 \pi r }$ = surface charge density **What happens to the tension in the wire as the wire radius goes to zero** with a constant circle radius, assuming:   * A constant charge on the entire thing (also constant linear charge density)    * A constant surface charge density    * A constant voltage    * If it really bothers you, consider the tensile stress for the above as opposed to the wire tension, although my question is for the latter The following are some helping equations. Any of them could be wrong. Electric field from an infinite straight line charge $$F(r) = \frac{ 2 \lambda k }{ r }$$ Electric field face-on for a ring of charge, R is radius of ring and d is distance away $$F(d) = \frac{2 \pi \lambda R k d }{ (d^2+r^2)^{3/2} }$$ Electric potential in vicinity of ring of charge, $\rho$ is distance from z-axis with torus in xy-plane, centered at origin. Electric field is simple knowing this.   * $l = \sqrt{ (\rho+R)^2 + z^2 }$ = distance to furthest point on ring    * $K(x) =$ Elliptic integral of the 2nd kind $$E(\rho,z) = \frac{ 4 R k \lambda }{l} K\left( \frac{ 2 \sqrt{ R \rho } }{ l } \right) $$ Now, I think that a valid approach would be to find the net effective field that a differential part of the ring experiences. _If_ you found this, I believe the tension could be written as follows, with T being tension, R the circle radius, and F the electric field. $$T = \lambda R F$$ I have my own thoughts on the answer (and more equations than you would ever want), and I'll return to offer them later, but I certainly don't have a complete picture, and I want to see what approaches people here use. Maybe this could be answered very simply and maybe some people will know intuitively which cases limit to infinity.


How many boats does it take to find an acoustic buoy by Doppler shift?

Inspired by this question on the Doppler shift, suppose there is buoy somewhere on the surface of the ocean emitting a pure frequency. You get to place some boats wherever you want on the surface of the ocean, moving whatever direction you want with whatever speed you want. The boats listen to the pure frequency, which will in general be Doppler shifted due to the relative motion between buoy and boat. If you do not know ahead of time the frequency the buoy is emitting, is it possible to deduce the buoy's location simply by observing the frequency measured by the boats? What is the minimum number of boats necessary, how should they be positioned, and what velocities should they have? This is a toy problem with the mathematics of the Doppler effect, so let's leave out attenuation of sound over distance and assume the surface of the ocean is a plane. Also, if the boats could continuously monitor the Doppler shift, they could collect extra information based on their own changes in position and velocity, so imagine locating the buoy based solely on a single frequency measurement from each boat.


Three-Dimensional Gravity

Does anyone have any references that discuss gravity in three-dimensions? I'm trying to make my way through some papers by Witten relating $SL(2,\mathbb{C})$ Chern-Simons theory and gravity in three spacetime dimensions and would like some explanation about the latter of the two topics. Thanks.


Is position fundamentally different from other observables?

I apologize in advance for what might be a very naive question, and for its science-fictionesque flavor. It's still, I think, a real physics question. Suppose I have a (quantum) particle, whose state is represented by some element $x$ of some Hilbert space $H$. Given an observable $O$, I can express $x$ as a sum of eigenstates of $O$. For example, I could take $O$ to represent the particle's position --- but so far, at least, "position" --- as opposed to any other observable --- seems to play no special role. On the other hand, at the macro level, our brains seem to assign a very special role to position --- namely, when we're deciding whether to think of some collection of particles as an "object", one of our main criteria is (I think) whether these particles occupy nearby positions, and whether they continue to do so over time. Moreover, we ourselves are objects by this criterion. One can (perhaps barely) imagine some other sort of creature that perceives a collection of particles as an object when (and only when) those particles have, say, similar momenta (regardless of their positions) and continue to do so over time --- or similar values of some linear combination of position and momentum, or similar values of some other observable $O$ for which we don't even have a name. Presumably these creatures would themselves be collections of particles that are $O$-localized but not necessarily position-localized. But for us humans, position seems to play this unique role. **Question 1.** Is the unique role of position a matter of physics or of biology? In other words, is there anything in fundamental physics that rules out (or renders unlikely) the sort of $O$-localized (but not position- localized) creatures I'm imagining? **Question 1A.** Let me reword that. It seems clear that (spacetime) position plays a special role in relativity, which quite plausibly rules out my $O$-localized creatures. So let me ask more specifically whether there is anything in **quantum mechanics** (as opposed to all of physics) that explains these creatures' absence. **Question 2.** If the answer to Question 1A is yes, then how does one reconcile this with the fact that any basis of the Hilbert Space $H$ can be transformed via an isomorphism to any other basis, which seems to say, in essence, that there can be nothing special about the position basis, and hence nothing special about the position operator? **Question 3.** If relativity grants a special role to position and quantum mechanics does not, does that all by itself constitute a fundamental incompatibility between relativity and quantum mechanics? And can it be viewed as a significant failure of quantum mechanics that it cannot account for this aspect of the world? **Edited to add:** I'm glad for the answers I've gotten so far, but I infer from some of them that I need to be clearer about what I'm asking. The question is not just "Why can't there be $O$-creatures?". Instead, it is: "If we have good reasons for thinking that $O$-creatures are unlikely, while knowing that position-creatures like us exist, how can we reconcile this with the fact that quantum mechanics sees no essential difference between $O$ and position? (Or am I wrong about the "no essential difference" part?)


Is heat energy from electronics wasted in the winter?

In the winter, are not all electronics heating the house - thus no energy is wasted? **Considering the energy I'm paying for:**   If I leave my computer running while I'm out, how much energy is actually wasted in the winter? Is all of the energy used in the computer expelled as heat into my home? To be even more specific:   **If my computer ran at a constant 500 watts and I had a 500 watt space heater, would they heat my home the same (thus cost the same)?** If so, wouldn't it make sense to leave the computer on in the winter, since the heat produced is not really wasted? That way I always have instant access. (Ignore variables like computer lifespans, gas furnaces, and cost- effectiveness for the sake of the question.)


Is everything 100% efficient at heating?

Everyone always talks about the efficiency of their appliances. I was wondering if everything was 100% efficient at heating its surroundings ?


Are all means of heating equally efficient?

In terms of cost of electricity, does it matter if I heat my room using a space heater, an open oven, a flood light, an ASIC? Assume no energy escapes the room, that the heat quickly becomes uniformly distributed, and ignore the fact that these produce heat at different rates. Consider only the amount of heat produced per unit of electricity.


We went to a planetarium last night last night and watched "Black Holes" narrated by Liam Neeson. If I recall correctly, he said that scientists believe that they have identified a number of Black Holes, and that they think there are White Holes at the bottom of Black Holes and instead of drawing matter/light in, the White Holes spew matter out. I didn't get to ask this question after the session was over: Have scientists been able to find an example of a possible White Hole?


Radiance equation

I am trying to understand the equation of the radiance, but there is one thing i don't understand: $L_r = \frac{d^2\Phi}{d\omega \space dA\cos{\theta}}$ Why is that second exponent there in the numerator? Thanks ahead!


Has hep-th/0312070 forgotten to fix $s_{0} = 1/2$ for the fermionic states in the second table on page 52?

Link to the original paper: The Gauge/String Correspondence Towards Realistic Gauge Theories (arXiv paper) On page 52 we see that, for a theory of Dp-branes placed at an orbifold (orbifold = $C_{2}$/$Z2$) fixed point, we have a certain spectrum of open string NS and R states. The first thing to note is that the orbifold action has the effect of making the spinor labels behave as follows: $s_{3} = s_{4}$ or $s_{3} = -s_{4}$. This means the 3rd and 4th components are no longer independent. Since $s_{i}$ equals $1/2$ or $-1/2$, this means there are now 16 different combinations of the state $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$ corresponding to the different 16 permutations of the values of $s_{0}, s_{1}, s_{2},s_{3} = -1/2,+1/2$. We would have had 32 different permutations corresponding to $s_{0}, s_{1}, s_{2},s_{3}, s_{4} = -1/2,+1/2$ if $s_{4}$ wasn't fixed by the value of $s_{3}$. However, this isn't the end of it. We must also take a GSO projection which projects out half the states. This leaves 8 states (8 permutations) for each $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$. So far so good; we now have 2 different types of Chan-Paton factor in front of our $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$. Thus, we must consider two types of $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$. This is why each row of the second table on page 52 has 2 states with different Pauli matrices in front. Each of these 2 states contributes 8 permutations so we have 16 fermions for each row of the table. But this is where I threw a spanner into the problem. For the _physical_ Ramond sector states, don't we normally fix $s_{0}$ to equal 1/2. This is the case for flat spacetime and is explained on page 23 of Notes on D-branes. So now we have $s_{0}$ and $s_{4}$ fixed leaving permutations of $s_{1}, s_{2}, s_{3} = 1/2, -1/2$. This would give us 8 states per $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$ and after GSO projecting half the states out we'd be left with 4 per $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$. The result: I'd expect 8 fermions per row of the 2nd table on page 52 of our original source. My interpretation seems to be supported by page 30 of UV/IR Mixing in Noncommutative Field Theories and Open Closed String Duality  which says we have 4 copies of each spinor. Note: It ignores $s_{0}$. I'm guessing they do this as it is fixed by the physicality condition. Sorry for the length of my description and thanks in advance.


What really makes a rainbow happen?

Yes, a rainbow is formed when a raindrop refracts light, but why then does the whole sky not become a huge rainbow when it rains? Would the light not be dispersed into ordinary 'white' light? What forces it to look as if each end is nearly touching the ground?


Is Information a potential or kinetic kind of energy?

It is said that _the law of least action is that nature tries to convert potential energy into kinetic one as fast as possible._ Information can't be thought without a physical realisation, see here. It may be thought as a form of entropy. So information is physical and therefore it's connected to a certain kind of energy. But is it a potential or a kinetic kind of energy?


Ionizing an atom with a proton

What happens when a proton collide with an atom? How does the ionization occurs, assuming the proton has enough kinetic energy? How much energy is required for the proton to enter the atomic nucleus, resulting a higher chemical element ion?


The inertia tensor is "global", how do I make it local?

A solid cuboid has the following moment of inertia: $$I= \begin{pmatrix} \frac{1}{12}h^2+d^2 & 0 & 0 \\\ 0 & \frac{1}{12}w^2+d^2 & 0 \\\ 0 & 0 & \frac{1}{12}w^2+h^2 \end{pmatrix} $$ When integrating the torque to the angular velocity, as far as I know, this holds: $$v_a=\int I^{-1} T dt$$ Where $T$ is the torque and $t$ is the time. However, this seems to only hold when the orientation is the identity orientation. How do I extend this onto an arbitrary orientation? Do I need to make the inertia tensor local in some way? Note that the angular velocity $v_a$ is "global" (!)


Dealing with experimental data

I have some experimental data about a value $n$, now, I am supposed to give, in the ending, a single value with an error: $n=a\pm b $. I have originally 6 values of $n$, each one comes as an indirect measurement from direct measurement, each one with it's systematic errors, so in the ending I have those 6 values, each one with an error. So what I guess I have to do is to mix the systematic error with the random error the way I've been taught $(E_{sys}^2+E_{rand}^2)^{1/2}$. The systematic is already calculated, what do I use for the error? the mean of systematic errors? Another question is that those values, which are by the way moles of a quantity of a gas, have been got from different ways (basically from calculating different isothermic curves and getting the $n$ value that best fits each of them), so they're actually not from the same kind of measurement, but from different ones. This makes me doubt about how this would affect the calculation of the final error, if it does, or if I can just do it the way I said above.


Explanation of energy levels in molecules, atoms, nuclei and their relationship

Why are the energy levels of molecules, the atoms that form them and the nuclei inside the atoms considered separately? Or phrased in a different way- what is it that makes their energy levels so different? What is the relationship between the energy levels of molecules, atoms and nuclei? I'd interested to learn different interpretations on this question.


Is 't Hooft's Determinism based on the holographic principle?

Does 't Hooft's determinism work need the holographic principle in order to work or is it just an extension of his work?


What are torque parameters?

Torque is just the turning force on an object, i.e the force multiplied by the perpendicular distance from the force to the pivot. But when torque parameters are mentioned in mechanical engineering papers, what are they referring to? It's the plural "parameters" thats confusing me. Is it the collection of properties such as maximum torque, starting torque, etc? If so, can somebody please provide me with a list of the all, or some of the "torque parameters", along with what each means? Finally, if looking at the torque parameters of a crankshaft, what would be the important ones to pay attention to, and what are some typical values?


How to write a noisy state as a separable operator

Bipartite operators close enough to the identity are separable. But how does one compute the product operator terms of the separable expansion? In particular, if $\left| \Phi \right> = m^{-1/2} \sum_{i=1}^n \left| i \right> \otimes \left| i \right>$ is the $n$-dimensional Bell state then $I \otimes I - \left| \Phi \right>\left< \Phi \right|$ is separable. But how do I write this as a sum of positive product operators $\sum A_i \otimes B_i$?


fast quantum random number generator + limited decoherence rate => Schrödinger cat state?

Suppose that fast quantum random number generator (QRNG, https://qrng.physik.hu-berlin.de/) is placed in a subsystem which has limited interaction with the rest of the world. What would happen if decoherence rate for this subsystem were not fast enough to extract all the random numbers being created by QRNG? It seems that then the subsystem has to evolve into a state with macroscopic coherent properties (Schrödinger cat state). At least this occurs if one simulate QRNG using measurement of a rotated qubit and copying a result in a vector of qubits representing a (partly) closed subsystem. If so, why the question about feasibility of macroscopic coherent states is still not settled?


Does the slip-stick phenomenon have any application?

The slip-stick phenomenon is present all around us, be it the noise of car breaks or in earthquakes. But does it have any real-life application?


Direct rocky planet formation without protostellar disk?

Inspired by this question, which sort of asked what I'm asking, but emphasized a different point (and the answers accordingly addressed that). Can a rocky planet form directly from the collapse of an interstellar gas cloud? The picture I have of rocky planet formation is that dust and rocky material in a protostellar disk accumulates into a rocky planet. There must be some mechanism to prevent the accretion of significant amounts of (hydrogen/helium) gas, I'm guessing this is something to do with the gas either being unable to cool, reheated if it does cool, or outright removed from some region around the protostar as the stellar wind and/or radiation pressure kick in. My intuition is that even the most metal enriched (astronomer metal, i.e. everything heavier than $\rm{He}$) cloud still by far hydrogen dominated, so that a cloud in isolation must _always_ form a primary star/brown dwarf/gas giant. But has anyone ever heard of a mechanism where a cloud in the right environment (some radiation field that efficiently heats the gas but not the dust/metals perhaps?) could form a rocky body directly, with no associated star? This seems far fetched, but I'm curious to know whether anyone has ever looked at this; literature references appreciated.


Energy conservation in General Relativity

I understand that energy conservation is not a rule in general relativity, but I'd like to know under what circumstances it can still be possible. In other words, when is it possible to associate a potential energy to the gravitational field, so that the energy is constant in the evolution of the system? Here are some examples, is there a convenient way to define energy in these scenarios?   * Just a system of gravitational waves.   * A point mass moving in a static (but otherwise arbitrary) space-time. Equivalent (if I'm not mistaken) to a test mass moving in the field of a second much larger mass, the larger mass wouldn't move.   * Two rotating bodies of similar mass. Overall, I'm trying to understand what keeps us from associating a potential energy to the metric. When we break the time translation symmetry of a system by introducing an electromagnetic field, we can still conserve energy by defining an electromagnetic potential energy. **Why can't we do the same when we break TT symmetry by making space-time curved?**


Calculating coefficient of friction

> Consider a body attached to a horizontal spring and resting on a surface, > inclined at an angle $\theta$ from the ground.![Set- > up](http://i.stack.imgur.com/F2G97.png) > > The spring constant is $k$. Initially the spring was kept in its natural > length while the body was held still by some external agent. When the > external agent was removed, the body slid $x$ units down the inclined plane > to achieve equilibrium.The coeffecient of (kinetic) frictional force acting > on the body is $\mu$ To solve this problem, I can use two methods: 1.Work-Energy Theorem: This yields $$0=mg(x\sin\theta)-\frac12kx^2-\mu (mgx\cos\theta)$$ 2.Equating forces at equilibrium: This yields $$kx+\mu mg\cos\theta- mg\sin\theta=0$$(along the inclined plane) On solving the equations the two methods give different answers, while they should give the same. Is there anything I am missing out here? Please help me solve this problem.


What are Killing spinors?

What are Killing spinors? How can they be motivated? Are they directly related to Killing vectors and Killing tensors and is there an overarching motivation for all three objects? Any answer is greatly appreciated but a less formal one would be preferred.


Discussion of the Rovelli's paper on the black hole entropy in Loop Quantum Gravity

In a recent discussion about black holes, space_cadet provided me with the following paper of Rovelli: Black Hole Entropy from Loop Quantum Gravity which claims to derive the Bekenstein-Hawking formula for the entropy of the black hole. Parts of his derivation seem strange to me, so I hope someone will able to clarify them. All of the computation hangs on the notion of _distinguishable_ (by an outside observer) states. It's not clear to me how does one decide which of the states are distinguishable and which are not. Indeed, Rovelli mentions a different paper that assumes different condition and derives an incorrect formula. It seems to me that the concept of Rovelli's distinctness was arrived at either accidentally or a posteriori to derive the correct entropy formula. > Is the concept of distinguishable states discussed somewhere more carefully? After this assumption is taken, the argument proceeds to count number of ordered partitions of a given number (representing the area of the black hole) and this can easily be seen exponential by combinatorial arguments, leading to the proportionality of the area and entropy. But it turns out that the constant of proportionality is wrong (roughly 12 times smaller than the correct B-H constant). Rovelli says that this is because number of issues were not addressed. The correct computation of area would also need to take the effect of nodes intersecting the horizon. It's not clear to me that addressing this would not spoils the proportionality even further (instead of correcting it). > Has a more proper derivation of the black hole entropy been carried out?


Passing a Car With known speeds vs. Time

While driving, it takes 22.0 s to pass a car. If my speed was 64.0 mph when I started to pass, and was 80.0 mph as I finished, how far (in feet) did I travel while passing?


What am I doing wrong finding the eigenvectors of this rotation matrix?

This question is from Principles of Quantum Mechanics by R. Shankar. > Given the operator (matrix) $\Omega$ with eigenvalues $e^{i\theta}$ and > $e^{-i\theta}$ , I am told to find the corresponding eigenvectors. I think these equations are relevant: $$\begin{align}\Omega &= \begin{bmatrix}\cos{\theta} & \sin{\theta} \\\ -\sin{\theta} & \cos{\theta} \end{bmatrix} \\\ \Omega \begin{bmatrix} x_1 \\\ x_2 \end{bmatrix} &= \begin{bmatrix} x_1 \cos{\theta} + x_2 \sin{\theta} \\\ -x_1 \sin{\theta} + x_2 \cos{\theta} \end{bmatrix} \\\ e^{i\theta} \begin{bmatrix} x_1 \\\ x_2 \end{bmatrix} &= \begin{bmatrix} x_1 \cos{\theta} + x_1 i\sin{\theta} \\\ x_2 \cos{\theta} + x_2 i\sin{\theta} \end{bmatrix}\end{align}$$ I let the matrix operate on the generic vector $(x_1, x_2)^T$ and demand that the resulting vector is equal to $(e^{i\theta}x_1, e^{i\theta}x_2)^T$ . From this i get the condition that $x_2 = ix_2$ and $x_1 = -ix_2$ , which implies that $x_1 = x_2 = 0$ . Am i missing something crucial?


Difference between thermodynamics and statistical mechanics

I've just finished a class a few weeks ago which taught thermo and stat mech, and I still don't know the exact difference between the two. Can someone help clear this up for me? (Yeah it's sad, and even sadder that I got an A- in the course)


Negative energies and a partition function

I'm writing down the partition function for a system, for which I know the dispersion relation $$E \left( \mathbf{k} \right) = \sqrt{ \left| \mathbf{k} \right|^2 + m^2 + \cdots }$$ The exact form is not important, what matters is that technically, as the dispersion relation is the solution of a 2nd degree polynomial equation, there should be a $\pm$ sign before the square root. When I write the partition function I get: $$ Z = \sum_{\mathbf{k}} \exp \left( - \frac{E \left( \mathbf{k} \right)}{\beta} \right) $$ but if I wanted to include the negative energies too, I would get: $$ Z = \sum_{\mathbf{k}} \exp \left( - \frac{E \left( \mathbf{k} \right)}{\beta} + \frac{E \left( \mathbf{k} \right)}{\beta} \right) = \sum_{\mathbf{k}} 1$$ which is clearly absurd: my system has dynamics! ;-) Now my question is: are the negative really unphysical? Wouldn't it be more correct to keep track of the two-branch dispersion relation with something along these lines: $$ Z = \sum_{\mathbf{k}} \exp \left( - 2 \frac{E \left( \mathbf{k} \right)}{\beta} \right) $$


Phenomena which are incorrectly declared as resonance phenomena?

In standard college physics text books, high-school books and popular level physics books, the collapse of the Tacoma Narrows Bridge is often taken as an example of _resonance_. However a more detailed analysis shows that the collapse was not due to simple resonance (see the article by Billah and Scanlan and Wikipedia). Now my question is, whether there are further examples from mechanics, which are often incorrectly or oversimplified declared as examples of resonance phenomena (for example in physics text books)? **Edit:** I should add, that the definition usually given in textbooks is the following: Consider a mechanical system which can be described by a differential equation like $\frac{d^2}{d t^2}x(t) + 2 \rho \frac{d}{d t} x(t) + \omega_0^2 x(t) = f(t) $ where $f(t) = f_0 \cos(\omega t)$. Then for example the amplitude or the energy of the system as function of $\omega$ has a maximum near $\omega_0$. In this case one speaks of _resonance_ of the system.


stability of hypothetical lunar atmosphere

Assume that by some means, the moon could be given an atmosphere, of the same density and pressure at the surface as the earth's. Obviously in a stable atmosphere there are temperature variations from pole to equator, day to night, and a lot of interesting dynamics follow from that, but for argument's sake assume an average of something like 0 to 5 C. How long would it last? Clearly there's no air there now (unless you count traces of potassium, or whatever it has), so such at artificial atmosphere would not be permanent, but presumably the moon could hang onto some air, for some time. With the moon's lower gravity, the density or pressure vs altitude curve will be different, and I'd guess that a significantly larger fraction of the mass would be at higher altitudes than on earth, and there's no protection from the solar wind. Someone with the right equations could probably work out how long it would take the pressure at the surface to drop by half.


Why are cgs units the norm in astrophysics?

Other physics communities, e.g. the particle physics one, have their own set of units, custom-tailored to their own needs. Now, the astrophysics community is somewhat similar, in that a lot of measurements are often expressed in terms of corresponding numbers related to the Sun (solar mass, solar luminosity, solar radius, et cetera). However, when it comes to using down and dirty numbers, I've seen cgs units used way more often than SI units. Why is that? Is it just a matter of tradition or is there something more subtle at work?


Observable which dependes on the cutoff

In arXiv:0710.4330v1 Balitsky calculate the eikonal scattering of dipole composed of quark anti-quark, $Tr(U_{x}U^{\dagger}_{y})$, to NLO accuracy. The result he found is: ![enter image description here](http://i.stack.imgur.com/EGe1e.png) Where $\mu$ is the normalization point in the MS bar scheme. I have already read the discussion Why do we expect our theories to be independent of cutoffs? and understood the observable cannot depends on the cutoff. While $Tr(U_{x}U^{\dagger}_{y})$ is observable, taking the derivative according to $\eta$ should produce an equation which is independent of the cutoff (at least naively). Why is it still there? any idea?


Calculating (Glass) thermal conductivity

I read that glass has a thermal coefficient of 0.8-1 W/mK. Given that my window window thickness is around $5\ mm$, then I would calculate my heat loss per area being $160-200\ Wm^{-2}K^{-1}$. Yet standard glass has a U value of $5.6\ Wm^{-2}K^{-1}$. What is going on here? I'm missing something huge...


Question about units of mass, $M = (L^{3})(T^{-2})$?

In section 5 of the "Preliminary: On the measurement of quantities" chapter (page 3) in "A treatise on electricity and magnetism" Maxwell uses, total length, $s=mt^{2}/{2r^{2}}$to show that $m=2sr^{2}/{t^2}$ is in units of Length cubed over Time squared. This is the first I have encountered this way of looking at mass and am wondering:   1. Is this just per-relativity nonsense?   2. Does mass being "length cubed over time squared" have any meaning?


jumping into water

Two questions:   1. Assuming you dive head first or fall straight with your legs first, what is the maximal height you can jump into water from and not get hurt?    In other words, an H meter fall into water is equivalent to how many meters concrete-pavement fall, force wise? (I'm assuming the damage caused will be mainly due to amount of force and not the duration)   2. Assume you jump head first and hold a sharp and strong long object that cuts the water before you arrive, will that make the entrance to the water more smooth and protect you?![enter image description here](http://i.stack.imgur.com/RGzj6.png)


What is the theoretical limit for farthest we can see back in time and distance?

13.2 billion years ago the universe was rather small, having started only half a billion years ago. Today, with the help of Hubble Space Telescope, we are able to capture the light of galaxies emitted at that time. The point at which Earth exists now must have been quite close to those galaxies back then. If so, why is it that it is only now, after 13.2 billion years later the light from those galaxies has reached us? Or in other words, are we sure that the light we are seeing from those galaxies indeed travelled for 13.2 billion years? It looks like as if there was a race between our point running away from those galaxies (with the expansion of universe and space) and the light that was emitted at that time. And only now that light has reached and overtaken us. But if that is so, then wouldn't it put a limit on the oldest light we can see, no matter how powerful the telescope is (even it is more powerful than the James Webb Space Telescope)? This should be expected, because at the time just after the Big Bang, the light emitted by all objects must have already overtaken all other objects, including the location of earth. Therefore we will never see the light that old (close to the time of Big Bang) no matter how powerful the telescope. If this is so, what is the theoretical limit we can see far back in the past?


Why is compressible flow near the choke point so efficient?

Imagine a steady state, one-dimensional, compressible flow in a horizontal pipe of constant cross sectional area. This flow can be isothermal, adiabatic (Fanno), or diabatic (Rayleigh). As an example, the relevant macroscopic energy balance for Fanno flow is: $$\Delta h=-\Delta (KE)$$   In other words, Fanno flow converts enthalpy into kinetic energy. Below are Fanno lines for a real gas (steam) for different flow rates at the same pipe ID and inlet conditions. ![enter image description here](http://i.stack.imgur.com/IFw8A.jpg) The lines clearly indicate the "efficiency" with which enthalpy is converted to kinetic energy. At low velocity (left upper portion of lines), almost no enthalpy is converted while the entropy generation is large (the line is nearly horizontal). Near Mach 1 (the rightmost maximum entropy point for each curve) the enthalpy conversion is high and entropy generation is nearly zero (the line is nearly vertical). This happens whether you are originally subsonic (the upper branch) or supersonic (the lower branch). Another way of saying no entropy generation occurs is that the viscous losses (friction) are zero. This would imply that the molecules no longer impart (as much?) force upon each other near the choke point. This seems absurd. In isothermal flow, you also find that as you near the choke point (interestingly, below Mach 1!), the flow approaches frictionless. As I understand it, these examples are physically realizable flow regimes (if obviously very short in length scale). Why is compressible flow so efficient near the choke point? Is something at the molecular scale really going on or is it simply a poor mathematical model at these velocities? UPDATE: Obviously, in the supersonic regime (lower branch), the reverse process takes place, i.e. kinetic energy is converted into enthalpy. Still, my question holds since this conversion is nearly perfect as you approach Mach 1.


Toward the establishment of non-equilibrium (quasi-equilibrium) magnon BEC theory

In 2006, Demokritov et al have reported that they have achieved the observation of quasi-equilibrium magnon Bose-Einstein condensation (BEC) in YIG at finite (room) temperature by using the method called "microwave pumping" http://www.nature.com/nature/journal/v443/n7110/abs/nature05117.html. I know there are various (sometimes controversial) discussion on their experiments and the understanding of the microscopic mechanism; about spontaneous coherence (spontaneous symmetry breaking), spontaneous condensation[A] or driven condensation, thermodynamics properties, the novel dispersion relation of YIG, and thermalization effects. From the view point of theoretical issues (to the best of my knowledge), Prof. Bunkov and Volovik http://arxiv.org/abs/1003.4889 has generalized (extended) the idea of magnon condensation and have proposed a non-equilibrium magnon BEC theory[B], in which they have viewed the magnon annihilation operator associating with (you call) the off-diagonal long-range order, $a_0=\sqrt{N_0} \exp(i\mu t-i\alpha)$, as the macroscopic condensate order paramater. Taking the experimental result by Demokritov et al into account, we can easily expect that the magnitude of the condensate order parameter decreases due to thermalization effects[C] (i.e. The dissipation into environments, phonons) and it takes the following form; $a_0= \sqrt{N_0} \exp(i\mu t-i\alpha -t/\tau)$, [D] where the variable $\tau$ reflects the dissipation. Thus although we can understand an aspect of the above experiment by Demokritov et al, I wonder whether we can regard it as Quasi-equilibrium or non-equilibrium magnon BEC (until $t<\tau$) also in the case where the magnitude of the condensate order paramater is time-dependent; $a_0= \sqrt{N_0} \exp(i\mu t-i\alpha-t/\tau)$ ? [A] In the standard or traditional equilibrium magnon BEC theory treating only equilibrium situations, the idea of "the spontaneous $U(1)$-symmetry breaking" plays the key role. [B] Magnon BEC corresponds to coherent precessions in the language of original spins. [C] By tracing out the enviromental degrees of freedoms, the time-development of the system we focus on becomes a non-unitary form. As an eaxample, please see the famous Caldeira-Leggett model or quantum master equation. [D] When we neglect the thermalization effects or the dissipation, we can easily show and see that the magnitude of the macroscopic condensation order parameter takes a time-independent constant value after switching off microwaves because the $U(1)$-symmetry of the system is recovered. I guess although this theoretically means the stable magnon BEC, it does not reflect the experimental result by Demokritov et al.


Photobombing without app used by Harley Davidson

Here we see a photobombing trick, used by Harley Davidson. They achieve to incrust a logo on a photo you take. The logo is invisible with naked eyes and only revealed on the photo you take. How does it work?


Is a single particle transition allowed between two superconductors having Majorana modes?

Sorry maybe I ask a very naive and stupid question. The motivation to ask it is as follows: I am considering the parity protection of Majorana qubit based on Kitaev's 1D model. As we know, in the Kitaev's chain model there are two free MFs on the end of the chain and the information in these two free MFs is protected by the parity of the superconductor substrate under the chains. My question is as follows: > How about the parity protection if we consider a Kitaev junction (A nanowire > hang on two superconductors and the middle of the nanowire is break up by a > gate voltage)? > > There are four MFs in the junction (from left to right named as 1,2,3,4), of > course the combined parity of 12+34 is protected if we ignore the > quasiparticle poisoning from environment, but how about the parity > protection if we only consider 23? > > Can 12 and 34 also have parity protection respectively?


Where does the change in energy come from when trapping a photon between mirrors?

You have a photon traveling with E=hf and you trap it between two perfectly reflecting mirrors (like a QM particle in a box). The photon has to make a standing wave between the mirrors and its spacial frequency is dependant on the distance between the mirrors, L. Its time frequency, and hence energy are derived from the spacial frequency; Therefore E is dependant on L. Where does the change in energy come from?


Why does the heaviest point on a spinning ball tend to become the topmost pole?

I have a mostly hollow, small clear plastic ball. Inside the ball is a weight stuck to the edge but the weight is smaller than the ball so the ball will always come to rest with the weight at the bottom when placed on a smooth surface. I've been spinning it on my desk with the weight at the top, it spins well despite it being top heavy, I assume like a gyroscope. If I spin it with the weight at the bottom however, it loses stability and it very quickly slows down until it regains stability with the weight at the top again, with further experimentation I found that wherever the weight is, once the ball starts spinning it stabilises with the weight at the top. What causes the rotational energy to transform into gravitational potential energy? Any why is the ball more stable with the weight at the top axis than with it at the bottom axis?


Do wormhole exits require matter? If not, would they change the edge of the universe?

As a layperson who watches way too many theoretical physics and astronomy documentaries, I've seen the folded-piece-of-paper analogy of wormholes a bajillion times. Usually they explain that black holes might connect to other parts of the universe, but they don't really explain what would have to be on the other end for any kind of information to travel through it. When they do, they usually suggest white holes -- but I'm not entirely convinced. I don't have too much trouble understanding how a black hole forms, and to some extent, how that could lead to a distortion of space-time sufficient to causing the "fold". But it's hard to imagine what would cause a white hole to form. Even if there were such thing as negative mass, there doesn't seem to be any way that an object can come to have a negative density sufficient to becoming a white hole, so there couldn't possibly be a process that causes them to form. So if black holes could be wormholes at all, it doesn't seem to be the case that the exit would require matter to be present before it forms. If anything, a black hole wormhole would have to "perforate" a white hole or some other kind of exit into existence -- somewhere. But there doesn't seem to be anything that determines where that exit would form, so one could exist outside the boundaries of the matter that defines the edges of the universe. That would mean that the universe is infinite in size, since white holes could (and probably would, if their position were completely random) appear very, very far away. So then either black holes can't be wormholes, the universe is infinite in size, or I'm missing something. Neither of the former two conclusions have been reached to the best of my knowledge -- so what am I missing?


The position of a particle at any time $t$ is given by $S = V0/a [1-e^{-at}]$. What are the dimensions of $a$ and $V_0$?

To find the dimensions of and V0, I must know the dimension of S and e. So I want to know it.


Find Trace of Dirac Matrix

The matrices present in the Dirac equation must have the following properties:   1. $\\{a^j,a^k\\}_{ab} = 2\delta^{jk}\delta_{ab}$      2. $\\{a^j,\beta\\} = 0$      3. $(\beta^2)_{ab} = \delta_{ab}$ How can one show from this that the trace of $\beta$ must be $0$?


What was the first discovery of the delta baryon $\Delta^{++}$?

The delta baryons (also called delta resonances) are a family of subatomic hadron particles which have the symbols $\Delta^{++}$, $\Delta^{+}$, $\Delta^{0}$, and $\Delta^{−}$ and electric charges +2, +1, 0 and −1 elementary charge respectively.   1. i want first reference claims delta baryon discovery   2. why the mass of all delta particles are the same!?   3. proton and neutron quarks content are the same with $\Delta^{+}$, $\Delta^{0}$, So why they have so different mass!?


How can electrostatic forces provide lift to the gossamer spider?

I was reading this article entitled "“Ballooning” Spiders Use Electrostatic Forces To Generate Lift". What I would like to know is how can a electrostatically charged object generate lift within the atmosphere? If I have a charged piece of paper or silk, I don't notice that it can lift off my hand and fly up 100's of meters, yet a spider can do so to the hight of 4 kilometres according to the article. Clearly my silk or paper is much heavier, but therefore, shouldn't I be able to construct a suitably charged object which could do the same as the spider up, lift up to 4 kilometers? If we assume that spider is using electrostatic charges solely as the means of lift, how can it work? How could we develop a model which can do the same thing?


Are the perpetual machines in this video really perpetual?

This video lists a number of perpetual motion machines that have been proposed throughout history:   * Robert Boyle's Self-Flowing Flask   * A "perpetual train" using rolling cones   * F. G. Woodward's hoop wheel machine   * Jacob Leupold’s Overbalanced Wheel   * Bob Schadewald’s Wheel   * Villard's wheel These are all mechanical machines, which use gravity to operate, i.e, they are a way to transform the force of gravity into energy, and do not need batteries or motors. Since I think that those machines are not really perpetual motion ones, can they turn for hundreds of years or will it stop just after some minutes?


Comparing sensitivity of flat spiral coils with regular solenoids

I'm attempting to compare the sensitivity that can be gained by using flat spiral coils against regular solenoid coils. A nice paper here gives reasonably accurate expressions for the inductance of such coils. More precisely, for a given solenoid of N turns and area A, we can write (note that $d/dt \rightarrow j\omega\times$ when taking Fourier transforms) $V_{induced}(\omega) = j\omega NAB$ and we can define sensitivity as $\left|\frac{V_{induced}}{\omega B}\right| = NA$ to indicate just how much amplitude can be picked up by the coil at a given frequency. This is a good way of comparing solenoid-type coils. For the spiral coil, how would you define an equivalent sensitivity metric? I suppose what this really means is finding the expression for the flux of such a coil, or somehow using its inductance to define pickup sensitivity.


Kirchhoff current loop in Resistor Diode Ladder network

I am looking for an approach on how to apply Kirchhoff current / voltage law in the infinitely long diode ladder network. Can anyone help me with this ? I am looking for 1D differential equation or an implicit equation.![enter image description here](http://i.stack.imgur.com/qMNhd.jpg)


QED Laser Reflection Mirror Expirment

Feynman in his famous lectures in NZ, talked about an experiment with a reflection of a light from a "back" mirror with stripes on it I tried to find a video of such an experiment, but with no success. Can someone help me find such a video? or something very related to it that delivers the same point?


Chern-Simons on a lattice and the framing anomaly

Can someone make or refer me to the argument for why $U(1)$ Chern-Simons theory in three dimensions cannot be defined by a lattice action? (Unlike Dijkgraaf-Witten theories, which are defined on the lattice.) Possibly related: what is the "framing anomaly"?


Red Shift Effects

I dong know much about physics, but I was curious about the red shift effect. I think I have the basic idea of it down in that as an object 1 accelerates from object 2 the light from object 1 will appear more red to object 2 because it stretches the light between them. My question is how fast is the red shift effect? If object 1 and 2 are viewing each other and are stationary with say one light year between them and object 2 suddenly accelerates away from object 1, how soon could object 1 see the light from object 2 shift red? My thinking was that if there was no lag that you could potentially use it to transmit information over vast distances with a sort of red shift mores code.


Lorentz force helix

I have a few questions about the Lorentz force.   1. Is it possible to create a Lorentz force helix in a cloud chamber and if so how would this be achieved?   2. How would the pressure inside the helix be measured in comparison to the pressure surrounding the helix?   3. Is there a limit on how much pressure the helix could produce? Ultimately what I am asking is, whether it is possible to create mini tornadoes and for those tornadoes to support a payload. Thank You