title: Is sonoluminescence relevant to the behaviour of Navier-Stokes (or converse)?

text: More precisely, could Sonoluminescence be a singularity of Navier-Stokes(NS)? Is there some other connection that might be interesting, or is it completely irrelevant? Wiki page mentions NS, but says it breaks down on small enough scale.


title: (Special Relativity) Points that can be seen by an observer

text: Let the metric be $$ ds^2=(1+gz)^2dt^2-dx^2-dy^2-dz^2 $$ where $g$ is a positive constant. Let an observer be stationary at $x=y=0$ on the surface $z=0$ and look upwards at an angle $\theta$, how to prove that he will see points of the surface with coordinates $(x,y)$ such that $$ \sqrt{x^2+y^2}=2g^{-1}\tan\theta $$ ? There is a hind saying to use the geodesic equations. I have derived them: $$ \frac{d^2t}{d\tau^2}+\frac{2g}{1+gz}\frac{dt}{d\tau}\frac{dz}{d\tau}=0 $$ and $$ \frac{d^2z}{d\tau^2}+(1+gz)g\left(\frac{dt}{d\tau}\right)^2=0 $$ But I cannot figure out how to prove the above statement. Could anyone help?


title: Interpreting Vector fields as Derivations on Physics

text: I have a subtle doubt about the physical interpretation of the mathematical definition of vector field as a derivation. In basic physics we understand a vector quantity as a quantity that needs more than magnitude to be fully specified, in other words, quantities with the notion of direction. This goes very well with the also basic mathematical definition that a vector is an equivalence class of oriented line segments. However, when we go to the study of manifolds, we see that a better definition of vector is to say that a vector at a point is a derivation on the algebra of smooth functions on that point. But then, we represent forces for instance with vectors, what's the interpretation of representing one force acting on a point by a derivation on the smooth functions on the point? I imagine that there must be some interpretation for that, but I didn't find what's it. Sorry if this question seems silly. I'm just trying to bring together those concepts. And thanks you all in advance.


title: What conservation law corresponds to this local $U(1)$ symmetry of the CCR?

text: It is known that canonical commutation relations do not fix the form of momentum operator. That means that if canonical commutation relations (CCR) are given by $$[\hat{x}^i,\hat{p}_j]~=~i\hbar~\delta^i_j~ {\bf 1}$$ they can be satisfied by the following choice of momentum operators: $p_x = -ih\frac{∂}{∂x}+\frac{∂f}{∂x}$ $p_y = -ih\frac{∂}{∂y}+\frac{∂f}{∂y}$ $p_z = -ih\frac{∂}{∂z}+\frac{∂f}{∂z}$ where $f(x,y,z)$ - arbitrary function. On the other hand, for any choice of $f(x,y,z)$ momentum operators can be transformed to their most frequently used form $(-ih\frac{∂}{∂x})$ (etc for $y$ and $z$) by the following transformation of the wave function $\psi$ and operators $p$: $\psi'=e^{-\frac{i}{h}f(x,y,z)}\psi$ $p^{'}_x =e^{-\frac{i}{h}f(x,y,z)}p_x e^{+\frac{i}{h}f(x,y,z)}=-ih\frac{∂}{∂x}$ Hence, we obtain $U(1)$ gauge transformation using only canonical commutation relations for momentum and position operators. Does this mean that $U(1)$ gauge invariance corresponds to conservation of momentum rather than to conservation of electric charge?


title: Combining two finite number fock spaces into one

text: Say I have two separate systems of identical Bosons, one with N Bosons the other with M. System one is described by a state $|\psi_1\rangle$ the other with $|\psi_2 \rangle$ which are expressed in a Fock space like $|\psi_1\rangle = \sum_{n_1,...,n_{max}} \alpha(n_1,..,n_{max}) |n_1,n_2,..,n_{max}\rangle$ $|\psi_2\rangle = \sum_{n_1,...,n_{max}} \beta(n_1,..,n_{max}) |n_1,n_2,..,n_{max}\rangle$ where $|n_1,n_2,..,n_{max}>=\prod_{k=1}^{max} \frac{(a^{\dagger}_k)^{n_k}}{\sqrt{n_k!}} |vac\rangle$ with "max" denoting the maximum occupied mode, $\alpha$ and $\beta$ some constants depending on each of the values (zero if $\sum_{k} n_k$ is not equal to $N$ for $\alpha$ or $M$ for $\beta$) and the wavefunction satisfying all the usual normalisation conditions. At someone point I wish to bring these two subsystems together, this state can be expressed as an $N+M$ body Fock space. $|\psi_{total}\rangle = |\psi_1\rangle \otimes|\psi_2\rangle$ For distinguishable particles this is fairly trivial, however the symmetry makes it somewhat unclear (to me) how to do this and give states with the appropriate amplitudes. Can anyone tell me, or point me to an appropriate book or paper?


title: Extracting heat energy from a material

text: Does it violate any physical laws to take a portion of the energy out of a system and use it? Specifically I'm referring to heat. (Kinetic energy). For example, if you have a material which has a lot of energy and has a high temperature, is it possible to take a portion of that energy and use it for something else, and leave the material cooler? I realise it is impossible to cool it to absolute-zero because of the third law of thermodynamics.


title: Beads in 'perpetual motion'?

text: This page has an interesting video of beads 'syphoning' out of a glass beaker: http://blog.zennioptical.com/weekly-optical-illusion-crazy-beads/ The host has a few explanations for the effect, but none of them sound plausible to me. The beads are in 'perpetual motion'? 'Shock wave'? What could cause an entire bead chain to syphon out of a beaker like water does out of a syphon? It certainly does not look to me that there was enough initial investment in energy to move the chain as is done. On the other hand, considering that only a small portion of the chain is moving at any one moment in time, the whole system _could_ be considered to be moving very slowly thus meeting the initial energy investment requirement (very low velocity).


title: Why is a general Green's function invariant under translations?

text: I am struggling to understand Green's functions, as used in Quantum Field Theory. One of my main problems is that the source I have been reading has a definition which is certainly correct, but involves enough integrals to obscure any intuition I initially had. Could someone explain what a Green function $G^{(n)}(x_1,x_2,\dots,x_n)$ intuitively represents (or provide a reference to a readable explanation)? In particular, I am being told that the Green function enjoys the invariance $$G^{(n)}(x_1+y,x_2+y,\dots,x_n+y) = G^{(n)}(x_1,x_2,\dots,x_n)$$ Supposedly, this is very simple and intuitive, but I can't see it (other than going a few levels down in the definitions, which in not very enlightning). Could someone please explain this property? _(my usual disclaimer: I am not a physicist, but a mathematician who occasionally comes into contact with physics. Please forgive my lack of physical insight.)_


title: Wave pulse on a freely falling rope

text: Consider a rope hanging from the ceiling (massive / massless irrelevant, I suppose). A wave pulse is set up on the rope. Just as the wave pulse starts propagating on this rope, the top of the rope is cut off and the rope is allowed to fall freely. What happens to the wave pulse? I posed this question as a homework problem for a sophomores' class on Waves, but then I realized that this was probably not as simple as the naïve answer I had. My naïve answer was, since there is no tension in the rope once in free fall (no gravity), the velocity for transverse waves on the rope is 0, therefore the pulse will freeze relative to the rope and fall. My colleagues pointed out to energy and momentum conservation. One of my friends concluded that the rope should do some weird sort of spiraling to conserve momentum and angular momentum about the center of the pulse. I'd appreciate if someone could shed some light on this problem.


title: Center of a mass of a hemisphere

text: How can I show that position vector of the center of a mass of a hemisphere is $(0,0,\frac{3a}{8})$ where $a$ is radius of a hemisphere, $x$ and $y$ axis are laying on the base and $z$-axis is perpendicular to the base and is going through the center of the base, $P$? Is there some formula? I calculated tensor of inertia in a point $P$, can that help? Mass of a hemisphere is $M=\frac{2}{3} \pi a^3 \rho$ and $\rho$ is density.


title: Fermi level alignment and electrochemical potential between two metals

text: I'm trying to get a more intuitive/physical grasp of the Fermi level, like I have of electric potential. I know that, for just a single piece of metal in equilibrium, you have to have the electric potential the same at all points, because if you didn't, that would mean there's necessarily an electric field between the two points, which would put a force on electrons, and move them until the field is gone. But I can't understand the electrochemical potential in the same way. I've been looking at this picture to try and understand what happens when two metals with different work functions are put in contact with each other: ![Fermi levels](http://i.stack.imgur.com/9fhtx.png) In (b), the chemical potential in the two metals have to be aligned, so for this to happen, a small number of electrons flow from the metal with the bigger work function to the one with the smaller (could you say this intuitively makes sense, because the WF is a measure of how hard it is for electrons to escape the material, so they'd rather be in a material where it's easier to escape?). This is where my confusion is. First of all, I don't really know of a good intuitive reason why the chemical potentials even have to be the same in the two materials (like the way I mentioned above for the voltage in a regular, single piece of metal. I don't have any sort of sense like that, here). Right now, it kind of seems like the chemical potential has some sort of "priority" -- when these two metals come into contact, why do their chemical potentials match up, and their electric potentials separate, rather than vice versa, or more intuitive to me, some equilibrium in between where they're both off? (but now I'm more confused -- according to wiki, it seems like the latter _does_ happen, but I don't really know why. Is that a thermodynamic potential trying to minimize?) My second point of confusion is the current that passes between and voltage that there now is across the two pieces of metal, from a practical standpoint. Wiki (above) and Ashcroft and Mermin: ![AM](http://i.stack.imgur.com/fm1FE.png) both say that there is this 'contact' potential across the metals now, but that it only drives current momentarily, and a tiny amount. So if I were to apply an increasing voltage across the metals (in either/both biases), would I need to hit a "threshold" voltage before which there would be no current flowing? How would Ohm's law work here, or would it? Naively, without applying any external bias, it seems like you have $V \neq 0$,$R \neq 0$, but $I = 0$. So should you expect anything funky when you apply external potential? Thank you!


title: Is it mathematically possible or topologically allowable for cutouts, or cavities, to exist in a 3-manifold?

text: A few weeks back, I posted a related question, Could metric expansion create holes, or cavities in the fabric of spacetime?, asking if metric stretching could create cutouts in the spacetime manifold. The responses involved a number of issues like ambient dimensions, changes in coordinate systems, intrinsic curvature, intrinsic mass of the spacetime manifold and the inviolability of the manifold. I appreciated the comments but, being somewhat familiar with the various issues, I felt that the question didn't get a very definitive answer. So, if I may, I would like to ask what I hope to be a more focused question; a question about the topology of 3-manifolds in general. Are cutouts or cavities allowed in a 3-manifold or are these manifolds somehow sacrosanct in general and not allowed to be broken? As I noted in the previous discussion, G. Perleman explored singularities in unbounded 3-manifolds and found that certain singularity structures could arise. Surprisingly, their shapes were three-dimensional and limited to simple variations of a sphere stretched out along a line. Three-dimensional _singularities_ , then, can be embedded inside a 3-manifold and the answer to my question seems to depend on whether or not these 3-dimensional singularities are the same things as cutouts in the manifold. I also found the following, which seems to describe what I have in mind. It's a description of an incompressible sphere embedded in a 3-manifold: "... a 2-sphere in a 3-manifold that does not bound a 3-ball ..." Does this not define a spherical, inner boundary of the manifold, i.e., a cutout in the manifold?


title: Spinning liquid to create a centrifuge effect

text: I'm in the business of purifying used cooking oil. Normally, I heat the oil up and let it settle for a couple of days. Water and solids settle to the bottom and cleaner oil remains on top. I'm trying to accelerate this process. So, the idea is to take a vertical tank 5ft diameter, about 10ft tall and put a paddle mixer on top with vertical blades that would extend to about 3" from the wall. If I turn on the mixer, it will spin at about 60 rpm and spin the liquid with it. According to my calculations this should give me a force of 3.5g and thus decrease settling time by about 3.5 times. Please let me know if this can work or if I'm missing something. Your input will be highly appreciated.


title: Why propagation of uncertainty is linear?

text: I'm in doubt with one thing: let's imagine that we have $n+1$ quantities, $n$ of them being directly measured, and the other one being related to the first $n$ by a function $f : \mathbb{R}^n \to \mathbb{R}$. If the $x_i$ is one of the quantities being directly measured and if $\Delta x_i$ is it's uncertainty, then I've learned that the uncertainty of the quantity $y = f(x_1, \dots x_n)$ is given by: $$\Delta y=\sum_{i=1}^{n}{\frac{\partial f}{\partial x_i}\Delta x_i}$$ My problem in understanding this is: well, if $f$ is such a function, it's derivative is the linear transformation $df : \mathbb{R}^n \to \mathbb{R}$ given by: $$df=\sum_{i=1}^{n}\frac{\partial f}{\partial x_i}dx_i$$ Where $dx_i : \mathbb{R}^n \to \mathbb{R}$ is given by $dx_i(a_1 \dots a_n) = a_i$. Hence, what we are saying, is that $\Delta y = df ( \Delta x_1, \dots \Delta x_n)$, in other words, we are saying that the uncertainty of $y$ is given by a linear function of the vector of uncertainties of the quantities $x_i$. Why should that be true ? I mean, what's the reasoning behind this ? I really didn't get the point, my teacher just gave the equation without justifying or anything like that. Any help and reference is appreciated. Thanks a lot in advance, and sorry if this question is silly.


title: Microsoft Excel not graphing $x = y^{1/2}$

text: The experiment was relating the period of one "bounce" when you hang a weight on a spring and let it bounce. I have this data here, one being mass and one being time. The time is an average of 5 trials, each one being and average of 20 bounces, to minimize human error.               t          0.3049s     0.3982s     0.4838s     0.5572s     0.6219s     0.6804s     0.7362s     0.7811s     0.8328s     0.869s      The mass is the mass that was used in each trial (they aren't going up in exact differences because each weight has a slight difference, nothing is perfect in the real world)               m          50.59g     100.43g     150.25g     200.19g     250.89g     301.16g     351.28g     400.79g     450.43g     499.71g      My problem is that I need to find the relationship between them, I know $m = \frac{k}{4\pi^2}\times T^2$ so I can work out $k$ like that but we need to graph it. In this equation, $k$ is the spring constant, $m$ is the mass hanging on the spring and $T$ is the period. I can assume that the relationship is a `sqrt` relation, not sure on that one. But it appears to be the reverse of a square. Should it be $\frac{1}{x^2}$ then? Either way my problem is still present, I have tried $\frac{1}{x}$, $\frac{1}{x^2}$, $\sqrt{x}$, $x^2$, none of them produce a straight line. The problem for SU is that when I go to graph the data on Excel I set the y axis data (which is the weights) and then when I go to set the x axis (which is the time) it just replaces the y axis with what I want to be the x axis, this is only happening when I have the `sqrt` of $m$ as the y axis and I try to set the x axis as the time. The problem of math/physics is that, am I even using the right thing? To get a straight line it would need to be $x = y^{1/2}$ right? I thought I was doing the right thing, it is what we were told to do. I'm just not getting anything that looks right.


title: How does blow-drying a mirror keep it from steaming up again?

text: After a hot shower, the mirror in my bathroom steams up. When I try to clear it with a towel, it immediately refogs. Yet once I use my hair-dryer, it will clear the fog and the mirror will stay clear. I'd like an answer to the mechanics behind this difference. I suppose it got something to do with the heat, and maybe the refogging is only delayed due to it. But that is just me speculating, and I'd like a concrete answer on this phenomenon.


title: 't Hooft limit of coupling fundamental fermions to Chern-Simons theory

text: This question is in reference to this paper: arXiv:1110.4386 [hep-th].   * I would like to know what is the derivation or a reference to the proof of their crucial equation 2.3 (page 12).    * In their analysis of fundamental fermions coupled to Chern-Simons theory, why have they been able to ignore the ghosts altogether? Does it have something to do with working in the light-cone gauge in which the self-interaction of the gauge field probably disappears and hence that removes the necessity to have ghosts?    * Their Lagrangian is 2.1 (page 11) is massless and they say that they can always tune the bare mass to be $0$ and they can always ignore the mass of the fermion. Can someone elaborate on this? Why was this possible? Isn't this tantamount to assumption of conformality which they want to prove in the 't Hooft limit? (Then isn't the argument becoming circular?)  Similarly if this were a scalar field theory then in the same strain one might want to say that the quadratic and the quartic scalar interactions can always be held at zero - but again wouldn't that be an assumption of conformality in the 't Hooft limit? What would be a genuine proof of conformality for this theory or its scalar version?   * Is the scalar version of this theory somehow uninteresting or known?    * In this paper whatever is claimed as the higher-spin currents seem to have their conservation laws broken in in $1/k$ or $1/N$, then isn't the theory interesting only when $k$ and $N$ are both infinite and then isn't that a trivial theory?  What sense does it make to say the Fermion $2$-point function depends on the 't Hooft coupling (as in equation 2.23 page 16)? Isn't that a non-physical quantity to talk of?


title: How does the size and load of a freezer affect the rate of freezing of an item?

text: I undertsand the size and load of a freezer can affect the rate a freezer will freeze food. I thought the temperature alone was the major factors which determined how fast/slow food freezes in freezer. I am told in a deep freezer -10c may achieve the same rate of freezing for a certain food which on a normal freezer would be achieved at -5c. So how exactly does size and load of a freezer affect the freezing rate, if the air in both freezer is at -5c constant then what excatly else does the freezer do. please explain.


title: Anticommutation relations and bispinor field

text: In a case of free Dirac field we have $$ \hat {H} = \int \epsilon_{\mathbf p}\left( \hat {a}^{+}_{s}(\mathbf p )\hat {a}_{s}(\mathbf p ) - \hat {b}_{s}(\mathbf p )\hat {b}_{s}^{+}(\mathbf p ) \right)d^{3}\mathbf p, $$ $$ \hat {\mathbf P} = \int \mathbf p \left( \hat {a}^{+}_{s}(\mathbf p )\hat {a}_{s}(\mathbf p ) - \hat {b}_{s}(\mathbf p )\hat {b}_{s}^{+}(\mathbf p ) \right)d^{3}\mathbf p, $$ $$ \hat {Q} = \int \left( \hat {a}^{+}_{s}(\mathbf p )\hat{a}_{s}(\mathbf p ) + \hat{b}_{s}(\mathbf p )\hat{b}^{+}_{s}( \mathbf p )\right)d^{3}\mathbf p. $$ So, if an operator $\hat {b}^{+}$ act on energy vector, it will decrease the value of an energy. This is commonly referred to as the fact that it creates an antiparticle with negative energy, because $\hat {b}^{+}$ is interpreted as creation operator (only by analogy with the scalar field, if I understand correctly). The solution to this problem is the postulation of anti- commutation relations between operators (in addition, the summary energy of the field begin to be positive definite quantity). But why don't we call $\hat {b}$ the creation operator, and $\hat {b}^{+}$ the destruction operator? Then, requiring the positivity condition for the integrand (if it is possible), we'll get physically correctly result without postulation of anticommutation relations. Where did I make the mistake?


Does every wavenumber of IR result in a different kind of vibration? If that is true, what if a molecule absorb 2 different wavenumbers (which cause different rocking and symmetrical stretching for example ) in the same time?


title: Mechanical shock resistance as a function of shape

text: I have a system where I'm dropping glass tubes filled with some sample from a certain height, along a track. I can apply a back-pressure of air to push them down faster, and in general the faster they go, the better. Additionally, the thinner the tube the better. I'm looking for a framework for how to analyze the resistance to shock, and the relative strength of either a flat-bottomed glass tube landing on a flat- bottomed surface, or a round-bottomed glass tube landing on a round-bottomed surface. I'd also like to know how resistance to shock scales with tube thickness, because in the end I'm specifically trying to get a sense of how thick do I have to make a flat-bottomed tube before it will be able to survive the same drop speed as a round-bottomed tube. I don't necessarily need an exact answer to this question - I realize there are a lot of parameters in the equation. I more or less just need to know how to approach each element in the problem. Ideally I'd like to set up some equation as a function of the size of the vessel and the force and some mechanical property of the glass, but if that's too complicated, a more general framework for estimating the relative strength of each tube is fine.


title: Is Earth/Luna tidal lock approaching faster?

text: Wikipedia pegs the mass of Earth at $5.972\cdot 10^{24}\,\text{kg}$ or $5.972\cdot 10^{21}\,\text{metric tons}$. Assuming Earth accumulates approximately $30,000\,\text{metric tons}$ annually. To illustrate, since man's first known predecessor walked earth about $4.5\cdot 10^6$ years ago, it has gained about $1.35\cdot 10^{11}\,\text{metric tons}$. Similarly Luna gains a proportional amount of mass too. Does gain in mass of the participating bodies bring Earth/Luna tidal-lock closer ever faster as the masses increase & time goes by? Can the acceleration (if my assumption is valid) be approximated? Further, once tidal lock is achieved, if the bodies continue to gain mass are they likely to eventually pull each other apart and coalesce again?


title: Solving the Schrödinger equation for the double-slit experiment

text: I'm not sure if this is the right place to ask a question about the Schrödinger equation, but I'll take my chances anyway. Basically, I would like to know how one can set up a potential function that represents a double-slit barrier and then solve the Schrödinger equation for this potential. Of course, according to classical optics, we will obtain an interference pattern, but it would be nice to see a solution entirely within the quantum-mechanical framework. I see this as a problem in mathematical physics, so hopefully someone could kindly provide me with some references.


title: A differential equation of Buckling Rod

text: I tried to solve a differential equation, but unfortunately got stuck at some point. The problem is to solve the diff. eq. of hard clamped on both ends rod. And the force compresses the rod at both ends. the solution(v(x)) is the value of bending I need. I assuming, that the differential equation of buckling rod is $$ EI_{x}v''''+Pv''=0$$ where $$P$$ is a force. and $$EI_x$$ is inflexibility. Then I find the solution for the diff. eq: $$v(x) = \frac{(\frac{(c_2 \sin(\sqrt(P) x))}{\sqrt(P)}+\frac{(c_1 \cos(\sqrt(P) x))}{\sqrt(P)})}{\sqrt(P)}+c_4 x+c_3$$ the boundary conditions: $$v(0)=v(l)=0=v'(0)=v'(l)$$ gives the trivial solution for $$c_{1},c_{2},c_{3},c_{4}$$ but I need non-trivial solution. Could you please help me to find the mistake or explain what's wrong in my equation?


title: Free Energy and quantum measurement

text: Free Energy must be expended to reset the state of an measurement apparatus. Is this statement valid in all situations? Is there a Definitive mathematical exposition?


title: Could someone jump from the international space station and live?

text: Felix Baumgartner just completed his breathtaking free-fall skydiving jump from $120,000\,\text{feet} = 39\,\text{km}$ above the Earth, breaking the speed of sound during the process. I was wondering if the next step could be jumping from the international space station. The person would have to overcome the orbital velocity of the station, re-enter the Earth's atmosphere, and land on the Earth with their feet, assisted by a parachute. Would such a stunt be survivable by a human? Could the person wear a space suit like the one depicted in the movie "Sunshine"? Is such a space suit possible? ![enter image description here](http://i.stack.imgur.com/W4fSO.jpg) **UPDATE** : I came across an interesting news article about a company that is working on a skydiving suite that would survive freefall from outer space. Read More Here ![enter image description here](http://i.stack.imgur.com/tbqPi.jpg)


1. How does Baumgartner slow his velocity so that the parachute can open up?   2. What is Baumgartner's speed just before his parachute opens?   3. What causes his speed to slow down just before the chute opens?   4. Is it air-resistance or is it altering his position from pencil to delta? The reason for this question is the supposition his body can endure only a maximum of "anti-force`" when the chute opens up.


title: Supersymmetric cancellation of loop contributions in a SUSY gauge theory

text: It is known that in SUSY models, loop contributions are automatically zero leading to a **_technically natural_** solution of the Higgs mass hierarchy problem. In many SUSY books/notes, it is often shown that, with a Lagrangian consisting chiral superfields, $$ {\cal L} = \int \left\\{\Phi_i^{\dagger}\Phi_i+\left[\frac{1}{2} m_{ij}\Phi_i\Phi_j+\frac{1}{3}\lambda_{ijk}\Phi_i\Phi_j\Phi_k\right]\delta^2(\bar{\theta}) +h.c.\right\\}d^2\theta d^2\bar{\theta} $$ All the loop diagrams of the superfields are identically zero. Thus in terms of component fields, the loop contributions are canceled out. Now, the same kind of vanishing loop contributions should be true in SUSY gauge theory as well. Without any direct/explicit calculation of the loop supergraphs, is there a simple argument/way to see how they are zero? Is that the same reason as the chiral model above, each supergraph contains a factor of $\delta^4(\theta)$ or higher order, which is $0$ ? Even if you provide references, your own interpretation/explanation is appreciated.


title: Metric expansion of space and Newton's second law

text: I wanted to ask the following question: If the expansion of the universe is really accelerating, does that mean a certain force applied on the universe? (According to Newton's second law) What kind of force is that?


title: How to compute $L_{\rm eq}$ from temporal pressure data

text: I have pressure vs time data. how can I compute sound equivalence Level $L_{\rm eq}$ in decibels? can this be done from a frequency spectrum assuming that it is constant?


title: Calibrating the relative intensity of a spectrometer with a blackbody source?

text: I am trying to compare the relative brightness of H-Beta and H-Alpha lines using a CCD spectrometer. In order to correct for the different grating efficiency at the two different wavelengths. I took the spectrum of a 100W bulb to use as a blackbody source. I also know the pixel location that correlates to the H-alpha and H-beta line. I am wondering how to translate the blackbody spectrum to a relative intensity calibration. I thought to try to approximate the peak and use Wien's law to get the temperate of the filament and use that temperature in the Planck's law equation. However, I was unsure if there was a standard method of obtaining a relative intensity calibration for a spectrometer.


title: Relativity of simultaneity and the second law of thermodynamics?

text: Has the connection been noted between these two? I have been studying special relativity and in learning Minkowski diagrams I have noticed that as v approaches c, the relativity of simultaneity is so drastic that all of what any observer would call effects happen simultaneously with their causes. In other words, entropy can't be said to increase for that observer. This relationship between the relativity of simultaneity and entropy holds for all other velocities as well and after a little thought it seems obvious: time dilation occurs occurs with the relativity of simultaneity because a change in the order of events (rel of sim) is a change in the increase of entropy (second law of thermodynamics) for any observer, which defines the arrow of time. If entropy increases as events occur for any frame of reference, then if that frame experiences the relativistic effects of changes in the ordering of events, they will experience time dilation due to a lessening of entropy's increase (to the the extent of the relativistic effects of reordering of events). The relativity of simultaneity must cause time dilation due to the definition of the arrow of time in thermodynamics. I would like to express this mathematically, though I am just learning the math of relativity now. I am curious as to why I have not seen this point made yet. I can hardly imagine this connection would not be seen as interesting enough for anyone to mention in teaching these two fundamental ideas...? Edit: because the question/idea has seemed unclear to many people I will state it like this. I believe there is a deep equivalence between relativity and thermodynamics. Viewing our universe as a system evolving toward higher entropy with time, I believe that any perspective which experiences less time (time dilation) must do so because they experience less entropy. Because entropy is defined by the most probable statistical evolution of events, an inertial frame which experiences time dilation (less entropy) must experience a less probable ordering of events, which happens via the relativity of simultaneity. Note that the probability of experiencing an increasingly highly relativistic perspective (for instance, from random natural accelerations) is increasingly rare exactly because it takes energy to do so, exactly as it is increasingly rare to find thermodynamic systems evolving toward states further and further from maximum entropy. I am not saying anything about violating causality, simply that (referring to thermodynamics) there are evolutions which at times momentarily shift away from maximum entropy, but don't violate the second law because they don't decrease entropy, just as there are relativistic frames that experience an ordering of events from the relativity of simultaneity that slow but do not reverse time.


title: Is $E=mc^2$ false?

text: Recently I have found a Youtube video saying that $$E=mc^2$$ is false. It is said that the real one is $$ E^2=m^2 c^4 + p^2 c^2 $$ where $E$=energy, $m$=mass, $c$=speed of light, $p$=momentum. Is there anything true in it? It says that if it was $E=mc^2$ then as light had only momentum it would not have any energy. Youtube link Youtube Search result


title: Calculating sum of heat transfered

text: I have a formula of thermal conductance heat transfer rate. Here it is: $$ Q = \lambda{S (T_1 - T_2) \over L} \Delta t $$ For my calculations I have got some constant values available $$ Q = 0.58{1 (T_1 - T_2) \over 0.01} \Delta t $$ I can express difference of $T_1$(90°C) and $T_2$(0°C) as (note: $c_1$ and $c_2$, $m_1$ and $m_2$ are equal for my values, so I join them) $$ T_1 - T_2 = {Q_1\over c_1m_1} - {Q_2\over c_2m_2} = {Q_1 - Q_2 \over c_1m_1} $$ $$ 90 - 0 = {172431270 - 0\over2090\times 916,7} $$ for beginning values. Now the problem: $T_1 - T_2$ is not constant, because $Q$ is just a rate of energy in one moment. In next moment, the difference of $T_1$ and $T_2$ will be smaller by $Q$ $$ T_1 - T_2 = {(Q_1 - Q) - (Q_2 + Q)\over c_1m_1} = {Q_1 - Q_2 - 2Q\over c_1m_1} = {172431270 - 2Q\over 2090\times 916,7} $$ This happens every following moment. So, completing everything into one formula, $$ Q = {58{172431270 - 2Q_{moved} \over 2090\times 916,7} \Delta t} $$ My question: Is there any way to do sum of all moments in 10 seconds to get heat transmitted ($Q$) between two solids? The only problem I have there is that $T_1$ - $T_2$ is not constant. I can't evaluate it without using $Q$. Please help.


title: Do all planets rotate around the sun with the same acceleration?

text: I have this question in mind. suppose if by any chance, all planet around sun stopped rotating. then as per formula F= M * A. all planet should fall with same acceleration towards sun and ultimately fall into sun. but even when they are rotating around sun in an orbit then they should have same acceleration, as change of direction is also acceleration. So my question is that do scientist have calculated that all planet around sun rotate with same acceleration ? My guess is that off course they would have, but i just want to know if i am thinking it correctly.


title: Why does the blade height goes on decreasing in a multi-stage axial flow compressor?

text: The axial compressors used in jet engines have their blade heights in decreasing order in the direction of the flow.


title: Why doesn't air freeze?

text: I am in no way experienced in the Physics field so this question may seem a bit silly but i'd appreciate an answer :) Why doesn't air freeze?


title: Electron transitions in an infinite square well

text: Consider an electron in an infinite square well. The expectation values of momentum and angular momentum are all zero for energy eigenstates. An electron transition is accompanied by the emission or absorption of photons. And we know the momentum of a photon is hk and the angluar momentum (spin) is 1. The momentum and angular momentum should be conserved in the transition process. Does that mean all electron transitions between energy levels are prohibited in such system? If such transitions are prohibited, how do we explain light-emitting quantum dots? =============================== Thanks for the answers and comments below. Now I realize a quantum dot is more like an atom than a hollow box. But if we consider the theoretical ideal square well, there seems to be no simultaneous eigenstates of both energy and momentum - at least I can't figure it out. Consider a 2D quare well. The energey eigenstates can be shown as: Energy eigenstates in a 2D Box Apparently they are not momentum eigenstates. Since these states form a complete set, we can set a momentum eigenstate to be: $|k_x\rangle = \sum_{n_x,n_y} C_{n_x,n_y} |n_x,n_y\rangle$ where C's are constants, and $P_x |k_x\rangle = \hbar k_x |k_x\rangle$ Keep going on: $P_x |k_x\rangle = -i\hbar\nabla_x \sum_{n_x,n_y} C_{n_x,n_y} |n_x,n_y\rangle = -i\hbar\nabla_x \left( C_{1,1} \sin(k_{1}x)\sin(k_{1}y)+\cdots\right)$ $= -i\hbar\left( C_{1,1} k_1 \cos(k_1 x)\sin(k_1 y)+\cdots\right)$ $=???\; \hbar k_x \left( C_{1,1} \sin(k_{1}x)\sin(k_{1}y)+\cdots\right) = \hbar k_x |k_x\rangle$ I don't know any way to transform cos*sin's to sin*sin's without messing them up. Is there anything wrong with my calculation?


title: Fermi's Golden Rule

text: It is well known that to calculate the probability of transition in the scattering processes, as a first approximation, we use the Fermi golden rule. This rule is obtained considering the initial system in an eigenstate of the unperturbed Hamiltonian $H_0$ and considering the time-dependent perturbation $V(t)=\begin{cases} V(\underline{r}),& \mbox{if } t_i<t<t_f \\\ 0, & \mbox{otherwise} \end{cases}$ where $t_f-t_i$ is the interval of interaction, and then considering the limit for $t_i \rightarrow -\infty$, $t_f \rightarrow + \infty$. Now, what is the range of validity of this approximation?


title: Principal value integral

text: I am reading A. Zee, _QFT in a nutshell,_ and in appendix 1 he has: > _Meanwhile the principal value integral is defined by:_ $$\int dx\,{\cal > P}{1\over x}f(x)~=~ \lim_{\epsilon \rightarrow 0} \int dx\, {x\over > x^2+\epsilon^2}f(x)$$ Please can someone explain to me why this is the case? As I understood it the principal value integral is rather defined as $$\int_a^b dx\,{\cal P}{1\over x}f(x)~=~ \lim_{\epsilon \rightarrow 0^+} \int_a^{-\epsilon} dx\, {1\over x}f(x)+\lim_{\epsilon \rightarrow 0^+} \int_{\epsilon}^b dx\, {1\over x}f(x),$$ where $a<0<b$. But as far as I can see these two definitions are not equivalent.


title: Parity Transformation Behavior of (Weyl-)spinors

text: Suppose $u^\pm (p)$ are Weyl-spinors with helicity $h = \pm$, i.e. solutions of the Dirac equation $p_\mu \gamma^\mu u(p) = 0$. How do these spinors behave under parity transformation? In Peskin&Schroeder, chapter 3.6, "Parity", the authors write $$u(p) = \gamma_0 u(\tilde{p})$$ where the tilde is understood to implement the parity on the coordinates: $\tilde{p} = (p_0,-p_1,-p_2,-p_3)$. The helicities are not explicity mentioned, but I guess this means that they are preserved. If I google for "parity spinors", however, the first result[1] suggest they do (scroll down the page). I also found references suggesting that the phase may change. My Question: Is the formula above correct? Is there an additional phase factor? Does it depend on the representation of the Gamma matrices? [1] http://www.ippp.dur.ac.uk/~krauss/Lectures/QuarksLeptons/Dirac/RealDirac_6.html


title: What does the Fermi Energy really signify in a Semiconductor?

text: In understanding the behavior of semiconductors, I'm coming across a description of the Fermi Energy here and at Wikipedia's page (Fermi Energy, Fermi Level). If I understand correctly, the _Fermi Level_ refers to the energy state at which there's a 50% chance of finding an electron. This varies with temperature. The _Fermi Energy_ is the highest occupied energy state of fermions at absolute zero. I'm a little confused as to the relation of the two terms. Additionally, in a semiconductor, the _Fermi Energy_ falls between the valence band and the conduction band. However, my understanding is that electrons cannot exist between the two bands -- so why isn't the _Fermi Energy_ the top of the valence band?


title: Stresses put through joints on a steel climbing wall

text: Being a fan of stackoverflow, I thought I could try this question here, hope its not too off topic. I'm wanting to build a free standing climbing wall in my garden and dont know where to start working out its strength and so construction. It will be made from interlocking (using bolts) 50cmx50cmx50cm mild steel cubes stacked 3m tall with a base of 1mx1m. The 4 base cubes will then be bolted to a concrete foundation. If I then bolt other cubes to the side of the main stack to create an overhang how can I work out the stress put onto the joints between the overhang and the main stack and between the stack and the foundations? The bolts between the cubes will be 3cm long and 2cm diameter. I havent decided what fixings to use between the stack and concrete as I believe this is where the problem will lie. Thanks for any help its much appreciated!


title: Effects on the lift of a Wing in a hermetically sealed torus

text: To help settle an office discussion could someone please explain why this is/is not possible. Assuming that materials could be found that are light and strong enough for its construction. If you were to feed the back-end of a wind tunnel to the front forming a torus and circulating the air back around could you then place a wing in the this tunnel and produce enough lift to fly the torus?


title: Is the fine structure constant actually a constant or does its value depend on the energy scale?

text: The value of the fine structure constant is given as $$ \alpha = \frac{e^2}{4\pi\varepsilon_0\hbar c} = \frac{1}{137.035\,999..} $$ It's value is only dependent on physical constants (the elementary charge $e$, speed of light $c$, Plancks constant $\hbar$), the vacuum permitivvity $\varepsilon_0$) and the mathematical constant $\pi$, which are considered to be constant under all cirumstances. However the Wikipedia article Coupling constant states > In particular, at low energies, α ≈ 1/137, whereas at the scale of the Z > boson, about 90 GeV, one measures α ≈ 1/127. I don't understand how this can be possible, except that one of the physical constants above or even $\pi$ are actually not constant, but dependent on the energy scale. But that seems nonsense. So what do physicists mean when they say that the fine structure constant $\alpha$ increases with energy? Can you perhaps reformulate the quoted sentence above so that it makes more sense?


title: Would you die if you put your hands on a powerline?

text: You know how birds perch on powerlines without getting electrocuted? What if by some chance that I find myself falling and I grab on one of them? Let's say both of my hands are on the same line, would i get electrocuted? I am thinking I won't because the current won't rush through me and I won't be part of the circuit - me - powerline. How does the ground play a role in this? I've heard people say that the ground creates a potential difference, but how? There is only voltage across the powerlines, the pole connecting to the ground is wood, an insulator? Thanks


title: How is the wattage of electrical appliance specified?

text: What does it mean by a bulb is 60 Watt or 100 watt (I know it means it emits 60 or 100 Joules of energy per second, but at what conditions)? Which one glows brighter when connected in series?? What happens in parallel combination?


title: Is the cross section of a relativistic water hose or string always a perfect circle?

text: Given is a very long tube, such as a water hose or a tubular string with finite thickness, that has a constant circular cross section of radius $r$ along the length and that is at rest in an inertial frame. The hose is either   1. straight or    2. curved with radius $R$, or    3. irregularly shaped. Is the shape of the cross section always a circle for a relativistically fast observer of velocity $v$? What is the observed cross-section radius $r(v)$ of the hose? How does $r(v)$ depend on the geometry of the situation, i.e., on the angles between v and the hose? Is it constant along the length of the hose? The overall shape of the hose will vary for each observer. But will its radius $r(v)$ vary or not? We suspected that the cross section remains circular in all cases, and that $r(v)=r$ in all cases, for all observers, along the full length of the hose. The answer is a bit unexpected, though. P.S. This is not homework. It is a question we discussed at lunch yesterday. P.P.S. If the "hose" would be a string with some radius, the result might also apply to string theory?


title: Determining Average Tidal Effects

text: Maximum tidal heights vary widely across the globe, from 16 m in the Bay of Fundy to mere centimeters elsewhere. These variations are due to coastline and shoreline differences. This makes it difficult to determine a global average ocean tidal maximum height. Let's assume the Earth is a landless planet covered throughout by an average one km of ocean. How can we calculate the maximum ocean tidal height from a new moon or full moon syzygy, assuming semi- major Moon & Sun distances?


title: Does the Grand Canonical Ensemble allow for exchange of particles or not?

text: I was doing some reading on wikipedia and found it interesting that one page says the Grand Canonical Ensemble does not allow for exchange of particles, however another page says it does. So I went on google books and tried to look for a more trust worthy source, again the same happens one source says it allows the other says it doesn't so which is it? Book that says it does allow exchange: http://www.scribd.com/doc/52426748/46/Grand-Canonical-Ensemble Another that says it isn't allowed: http://books.google.co.uk/books?id=5sd9SAoRjgQC&pg=PA62&dq=grand+canonical+ensemble+exchange+particles&hl=en&ei=hL6oTZ2THc_p4wbEh-3DCg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC0Q6AEwAA#v=onepage&q=grand%20canonical%20ensemble%20exchange%20particles&f=false What gives?


title: Isn't wave particle duality of light actually cheating?

text: When answering questions about light, I see that we conveniently shift between wave and particle nature of light to match the answer-- isn't this really cheating? Or, is it the principle that the observation changes the nature of the object?


title: How does holographic radar work?

text: I'm trying to figure out how the mechanics of holographic radar work. AFAIK there is a continuous 3d transmission signal (a dome-shaped antenna?) But because there isn't a direction or time-domain, how do we observe the return signal from objects and locate them? Is there any reference material / books on how this works?


title: Assymptotic freedom significance

text: So I have read a bit on this, and get the idea and mathematical machinery leading up to this. I get that it sheds light on the relationship between coupling strengths and length scales. Can someone tell me the "deep insight", or some other consequence of this that commanded a Nobel prize. Are there other physical implications I am missing?


title: Clarifying constraint forces in Lagrangian dynamics

text: In the Lagrangian formulation, the addition of constraint forces that are unknown can be done with Lagrange multipliers, which allows for the forces to be found. Taking $k$ constraints of the form $f_k(q,t)=0$, we have $$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}-\frac{\partial L}{\partial {q}}+\sum{\lambda_k \frac{\partial f_k}{\partial {q}}}=0.$$ So, the questions:   1. Are the $\lambda_k$ the constraint forces (and therefore equal for each $q_i$), or is $\lambda_k \frac{\partial f_k}{\partial {q_i}}$ the $i$th component of the $k$th constraint force?   2. Should the signs of the constraint forces be interpreted as positive along the $\hat{q}$ direction and negative opposing? I have worked out the Lagrangian for a car going around a banked curve at constant speed, radius, etc., to see if I could derive the constraint forces. $$L = \frac{m}{2}(\dot{z}^2+R^2\dot{\theta}^2+\dot{R}^2)-mgz$$ with constraint $f = z-R\tan(\alpha)=0$. I get, from the $z$ equation (after applying $\ddot{z}=0$), $$\lambda = mg$$ and, from the R equation (after applying $\ddot{R}=0$), $$\lambda=-\frac{mR\dot{\theta}^2}{\tan(\alpha)}.$$ From the FBD, it's easily seen that $tan(\alpha) = \frac{m\omega^2R}{mg}$, which is consistent with the idea that what I really had above were $\lambda_z$ and $\lambda_R$, leading to $F_z=\lambda \frac{\partial f}{\partial {z}}$ and $F_R=\lambda \frac{\partial f}{\partial {R}}$.


title: When solving kinematics questions, how to find time in air?

text: I was trying to help a friend with this question: > _A ball is shot horizontally off a cliff. What would happen to the > horizontal distance it travels if the acceleration due to gravity were > doubled and all other factors remain the same?_ My initial reaction was to do $-\frac{v_0}{g} * 2$ to get the total time in air, if $g$ is doubled, the time in air is halved, and therefore the horizontal distance is halved $ (v_x \cdot *t)$ $t$ is halved while $v_x$ stays constant. My friend was using $y=\frac{1}{2}at^2$ to solve for half $t= \sqrt{\frac{2y}{g}}$ and saying that the time would decrease by a factor of $\sqrt 2$. How do you reconcile these two formulas to find the time in air? I have traditionally used my first method and gotten these questions right, but I am having a brain delay this morning on why you can't use the 2nd formula? Any clarifications would be appreciated!


title: How does matter know to obey physical laws?

text: First of all, you may say that, this is a "why" question and in physics we don't ask such a question and instead we ask about "how". However, I want to discuss this with you. We model the behavior of matter by physical laws, for example by quantum mechanics laws. I want to know how the matter, say electron, knows that it must obey such a law. What forces the electron to obey quantum mechanics laws? If you consider a simulation of the universe in a computer, in the view of someone simulated in that computer the physical laws are just there. They may not know what keeps in check the simulated matter to obey physical laws, but in our perspective those simulated physical laws are there because we implemented them as zero and ones in a hard disk. If we wanted, we could change those physical laws and in the eye of the simulated person everything would be still fine. Is there a scientific attempt to discuss this issue? so that I can read about it.


title: Are different interpretations of quantum mechanics empirically distinguishable?

text: Are the different interpretations of Quantum mechanics just different viewpoints of same physical reality? or the experiments can distinguish them? i mean are they empirically distinguishable or not? I have red a paper in which Asher peres states that quantum mechanics needs no interpretation and we can understand it by a minimum number of necessary postulates, here is the last paragraph of the paper: "All this said, we would be the last to claim that the foundations of quantum theory are not worth further scrutiny. For instance, it is interesting to search for minimal sets of physical assumptions that give rise to the theory. Also, it is not yet understood how to combine quantum mechanics with gravitation, and there may well be important insight to be gleaned there. However, to make quantum mechanics a useful guide to the phenomena around us, we need nothing more than the fully consistent theory we already have. Quantum theory needs no interpretation." This is the link to the paper(it is a paper with only two pages) Peres Can you explain what he means when he says "quantum mechanics needs no interpretation"? i didn't get it from the paper!. Does he mean that we shouldn't try to assign any objective meaning to the wave function and consider it as reflection of our knowledge about the system? If yes, isn't it what Copenhagen interpretation says?!! (This is from wiki page of Copenhagen interpretation:"The wave function reflects our knowledge of the system") And finally, has anyone find that minimum number of postulates?


title: How to derive electron number equation of Bogoliubov Hamiltonian using thermodynamic relations.

text: My question arise from this article: Edge superconducting correlation in the attractive-U Kane-Mele-Hubbard model. I will describe my question in detail so that you might not need to look into that article. **What I want to ask for help is an easy way to derive the electron number equation, may be using some thermodynamic relations just as how I derive the gap equation**. The Hamiltonian considered is the mono-layer graphene with intrinsic SO interaction, plus the negative-U Hubbard term. After the mean field approximation with S-wave superconducting order parameter, we obtain the mean- field Hamiltonian: $$ H=\sum_k\phi_k^\dagger H_k\phi_k+E_0 $$ where $\phi_k$ is the Nambu spinor $\phi_k^\dagger=(a_{k\uparrow}^\dagger, b_{k\uparrow}^\dagger, a_{-k\downarrow}, b_{-k\downarrow})$, $E_0=2N\Delta^2/U$, $N$ is the number of unit cell, and $$ H_k=\begin{pmatrix} \lambda_k-\mu & -t\gamma_k & -\Delta & 0 \\\ -t\lambda_k^* & -\lambda_k-\mu & 0 & -\Delta \\\ -\Delta^* & 0 & -\lambda_k+\mu & t\gamma_k \\\ 0 & -\Delta^* & t\gamma_k^* & \lambda_k+\mu \end{pmatrix} $$ The $\mu$ is the chemical potential in the original Hubbard term, $-t\gamma_k$ is the sum of the graphene hopping integral of nearest neighbors, $\gamma_k$ is from SO term. We can just ignore their physical meaning and regard them as some parameters, they are not very important to my question. Diagonalizing $H_k$, we have four eigenvalues, $\omega_{ks\alpha}=\alpha\omega_{ks}=\alpha\sqrt{(\epsilon_k+s \mu)^2+\Delta^2}$ with $\epsilon_k=\sqrt{\lambda_k^2+t^2|\gamma_k|^2}$, where $s,\alpha$ are $\pm 1$. Now we have gap equation and electron number equation: $$ \frac{1}{U}=\frac{1}{4N}\sum_{ks}\frac{\tanh{(\beta\omega_{ks}/2)}}{\omega_{ks}} $$ $$ n_e-1=-\frac{1}{N}\sum_{ks}\frac{s\epsilon_k -\mu}{\omega_{ks}}\tanh(\beta \omega_{ks}/2) $$ where $n_e$ is the average electron number on one sublattice. The following is how I derive the gap equation, the free energy is: $$ F=-\frac{1}{\beta}\sum_{ks\alpha}\ln{(1+\mathrm{e}^{-\beta\omega_{ks\alpha}})}+\frac{2N\Delta^2}{U} $$ the free energy is minimized when $\Delta$ choose to have its true value, i.e. using $\partial F/\partial \Delta=0$ we can derive the gap equation showing above. **How can I derive the electron number equation?** I know in principle I can derive it by representing the original electron operators instead of the diagonalized Bogoliubov quasi-particle operators, but this is much too complicated even one trying to derive them using Mathematica. So just as I said in the beginning of this question: **I need your help to get an easy way to derive the electron number equation, may be using some thermodynamic relations just as how I derive the gap equation**


title: Taylor series: Epsilon not differentiated?

text: Why isn't epsilon differentiated with respect to time? (see my question on the right) ![enter image description here](http://i.stack.imgur.com/AlAgL.png)


title: Can Entropy decrease through genralized measurement

text: Can entropy of a quantum state decrease through the generalized measurement?


title: Can rain evaporate?

text: Are there any atmospheric condition under which a visible raindrop can fail to hit the ground by evaporating first? I imagine this would require a large vertical temperature difference, and possibly the rain forming very high up. Has anything like this been observed experimentally, or if not - is it possible to perform a calculation to show whether this is plausible?


title: Where did I commit a mistake in calculating rotation?

text: ![enter image description here](http://i.stack.imgur.com/8cyNu.png) I did something wrong in my calculation, can somebody tell me what?


title: Are frequency and likelihood the same across the multiverse?

text: My probability text distinguishes between two interpretations of probability values:   1. the _frequency of occurrence_ "as percentage of success in a moderately large number of similar situations" (coin tosses, etc.), and   2. the _likelihood of an occurrence_ of a unique event (whether a skier medals at Sochi, for example), and argues that the latter conveys no information about the former, since it is a "one-time event". If one considers the multiverse as a whole, however, are these interpretations still distinct? In particular, is the _reason_ that a unique event has a likelihood that it occurs with the corresponding frequency across all universes? Does the answer depend on what kind of multiverse structure is being considered? For example, is it the case that under a many-worlds interpretation of quantum mechanics, these are indeed equivalent, because the likelihood of an event is just the quantum mechanical distribution of possible events among "worlds"?


title: Born rule for photons: it works, but it shouldn't?

text: We can observe double-slit diffraction with photons, with light of such low intensity that only one photon is ever in flight at one time. On a sensitive CCD, each photon is observed at exactly one pixel. This all seems like standard quantum mechanics. There is a probability of detecting the photon at any given pixel, and this probability is proportional to the square of the field that you would calculate classically. This smells exactly like the Born rule (probability proportional to the square of the wavefunction), and the psychological experience of doing such an experiment is well described by the Copenhagen interpretation and its wavefunction collapse. As usual in quantum mechanics, we get quantum-mechanical correlations: if pixel A gets hit, pixel B is guaranteed not to be hit. It's highly successful, but Peierls 1979 offers an argument that it's wrong. "...[T]he analogy between light and matter has very severe limitations... [T]here can be no classical field theory for electrons, and no classical particle dynamics for photons." If there were to be a classical particle theory for photons, there would have to be a probability of finding a photon within a given volume element. "Such an expression would have to behave like a density, i.e., it should be the time component of a four-vector." This density would have to come from squaring the fields. But squaring a tensor always gives a tensor of even rank, which can't be a four-vector. At this point I feel like the bumblebee who is told that learned professors of aerodynamics have done the math, and it's impossible for him to fly. If there is such a fundamental objection to applying the Born rule to photons, then why does it work so well when I apply it to examples like double-slit diffraction? By doing so, am I making some approximation that would sometimes be invalid? It's hard to see how it could _not_ give the right answer in such an example, since by the correspondence principle we have to recover a smooth diffraction pattern in the limit of large particle numbers. I might be willing to believe that there is "no classical particle dynamics for photons." After all, I can roll up a bunch of fermions into a ball and play tennis with it, whereas I can't do that with photons. But Peierls seems to be making a much stronger claim that I can't apply the Born rule in order to make the connection with the classical _wave_ theory. [EDIT] I spent some more time tracking down references on this topic. There is a complete and freely accessible review paper on the photon wavefunction, Birula 2005. This is a longer and more polished presentation than Birula 1994, and it does a better job of explaining the physics and laying out the history, which goes back to 1907 (see WP, Riemann-Silberstein vector , and Newton 1949). Basically the way one evades Peierls' no-go theorem is by tinkering with some of the assumptions of quantum mechanics. You give up on having a position operator, accept that localization is frame-dependent, redefine the inner product, and define the position-space probability density in terms of a double integral. Related: What equation describes the wavefunction of a single photon? Amplitude of an electromagnetic wave containing a single photon Iwo Bialynicki-Birula, "On the wave function of the photon," 1994 -- available by googling Iwo Bialynicki-Birula, "Photon wave function," 2005, http://arxiv.org/abs/quant-ph/0508202 Newton T D and Wigner E P 1949 Localized states for elementary systems Rev. Mod. Phys. 21 400 -- available for free at http://rmp.aps.org/abstract/RMP/v21/i3/p400_1 Peierls, Surprises in theoretical physics, 1979, p. 10 -- peep-show possibly available at http://www.amazon.com/Surprises-Theoretical-Physics-Rudolf- Peierls/dp/0691082421/ref=sr_1_1?ie=UTF8&qid=1370287972


How much torque does it take to turn a doorknob? I'm not looking for an exact answer, just a ballpark for someone who doesn't have a sense of everyday amounts of torque. Here's a very ordinary picture of a very ordinary doorknob, the type I'm imagining for this question: ![an ordinary doorknob](http://i.stack.imgur.com/MhY7U.jpg)


title: How to reconcile the two definitions of work? (mechanical and thermodynamical)

text: When studying classical mechanics, work is defined as: $W_M=\int F_{tot} \hspace{2 mm} dx$. However, for thermodynamics, work is defined as: $W_T=\int -F_{ext} \hspace{2 mm} dx$. I'm having trouble reconciling both definitions. I have found the mathematical relationship between both of them (see below), but I wouldn't know how to interpret either this or the other thermodynamical functions in the light of this. For example, $\Delta U$ is usually related to the conservation of energy, but since the definition of $W$ is different, it isn't that direct a relationship now (for me at least). $$ W_M=\int F_{tot} \hspace{3pt} dx \hspace{3pt} =\int F_{int}-F_{ext} \hspace{3pt} dx $$ $$ W_M=\int F_{int}\hspace{3pt} dx+W_T \hspace{3pt} $$ The books I've seen usually just throw the definition and start talking about the state functions, but do not spend any time reflecting on this definition or analyzing why is it different from the classical one. I found this source which tries to deal with this same issue: it is very interesting, but it gets a little confusing towards the end when he tries to relate his mechanical w and q with his thermodynamical W and Q (I don't understand how he combines equations 1 and 3 or 1 and 5 to get his results). Does anyone understand that last step? Has anyone seen or thought about this issue before? Does anyone know of any other source or book where this issue is dealt with more detail? * * * EXAMPLE PROBLEM (to motivate the question): Consider a very big cylinder with a mobile piston in the middle separating two sections filled with gas (A and B). Both the cylinder walls and the piston are adiabatic, so there is no exchange of heat with the exterior or between sections. The pressure of the gas in A is higher than the pressure of the gas in B. The piston is originally being hold, but we let go of it for just a second and then hold it again (we assume the change in volume was small enough so as to not produce a change of pressures). In this situation, I could consider the gas of section A as my system and claim it gave $p_B \Delta V$ of work to its exterior (the gas in section B, since it is the only exterior with which it interacts). But I could also consider the gas in section B as my system, and claim it received $p_A \Delta V$ of work from its exterior (gas in section A). But since $\Delta V$ is equal in both cases (only changes the sign, which is what determines is work is given or received) but the $P_i$ aren't, there is an energy difference I cannot account for. My intuition tells me that the energy difference must come from the potential energy stored in the original pressure difference, but since thermodynamics (or the way I was taught thermodynamics) starts with a different definition of work and does not explain its relationship with the mechanical one, it is not that trivial to me how to incorporate this into a thorough thermodynamical analysis of this process.


title: Period of pendulum in falling lift

text: A simple pendulum suspended from the ceiling of a stationary lift has period $T_0$. When the lift descends at steady speed, the period is $T_1$. When the lift descends with constant downward acceleration, the period is $T_2$. Explain why $T_0 = T_1 < T_2$. I know that for small angles of oscillations, $T \approx 2\pi\sqrt\frac{l}{g}$ where $l$ is the length of the pendulum. For the first two cases, since $\sum F = 0$, the value of $g$ is apparently the same. However, I am unable to explain why the period is largest for $T_2$. If $g$ is the same for all cases, shouldn't their periods be the same?


title: Is the second law of thermodynamics a real law?

text: Everytime I ask people why the second law of thermodynamics is true, they say because the probability of entropy increasing is very large. This for some reason doesn't seem right for me. Just because something has a high probability of happening doesn't mean that it is a fundamental law of nature. Can someone say that because it is very unlikely to win the lottery that it is a LAW that you won't win?


title: Special Relativity moving in space

text: Given that time is 'just another' dimension and people get hung up on the fact that we cannot go back and forth in time like the other dimensions. Is there any proof that the corner of 8th Avenue and 14th Street at 3pm is the same place as as 8th and 14th at 4pm? Consider that in a 3 dimensional space, coordinates (8, 14, 3) isn't the same spot as (8, 14, 4) why can we assume that in a 2D place with Time that (8, 14, 3pm) is the same place as (8, 14, 4pm) only that the time has changed? Could it be that we cannot move back and forth in the other dimensions either, i.e. if we walk forward 10 meters and back 10 meters, it isn't really the same place since the time component has changed.


title: How to move from Special to General Relativity

text: I have understood special relativity nicely, and right now I am trying to learn general relativity from D'Inverno's book. I an finding it rather difficult to understand the pre-requisite math (i.e. tensors) for general relativity. Please suggest some source from where I could learn the general relativity math, as if it's being taught to a kid. Also note, that I have a 5 year undergraduate Degree in Geo-physics, so I believe I have a good enough background. PS: Apology, if this is a duplicate question


title: DC Josephson effect

text: I was taught that currents are always associated with some nonequilibrium situtations. Now I encountered the DC Josephson effect where **equilibrium** current can be calculated from this formula: $I = \frac{2e}{\hbar}\frac{dF}{d\phi}$ where $F$ is just the free energy. 1) First of all, where does this formula come from? 2) This macroscopic phase of the superconductor $\phi$ should somehow be included in internal energy, or am I wrong? 3) What about long junction limits - is this a general formula? Maybe I can handle this effect by Green function method or other machinery? I am trying to understand where this current comes from and what is the general transport scheme in that situations? If you stick together two superconductors and make a loop, the current will also flow? How can you define a current through thermodynamic quantities (I know the classical nonequilibrium thermodynamics, but thats the point). Any references are welcome.


title: Mixing of Ideal gas - Thermodynamic equilibrium

text: I always get confused what exactly happens when two ideal gases mix. Consider the initial situation where two gases are in a box, separated by a rigid and adiabatic wall between them. Now when the wall between them is removed, they come to equilibrium (of course assuming the process is done quasi-statically). Initially the thermodynamic quantities of the gas be $$ U_i,\: S_i,\:T_i,\:P_i,V_i \:\:\: \:\:\: \:\:\: \:\:\: \:\:\: \:\:\: \:\:\:i=1,2 $$ Now after the wall is removed, the condition for equilibrium is attained by the condition $$ dS = 0 $$ I am not able to apply this and find out exactly the final values of the thermodynamic quantities for an arbitrary ideal gas situation (I am not particularly able convince the situation physicaly) (for instance if we consider and the two ideal gases to obey equation, $$ P_iV_i = c_iN_iRT_i \:\:\: \:\:\: \:\:\:i=1,2 $$ where $ c_i $ is the degrees of freedom of the gas, like $ c_i = \frac{3}{2},\:\frac{5}{2} $ for monatomic and diatomic respectively.


title: Why does resistance increase with length?

text: This is for the case of wires only (pipes are easy). I see that if I place a 9-Volt (constant) battery across any wire, the current is proportional to resistance. I don't understand why, for a constant potential difference, a current would be slower for certain lengths of wire, aren't all the electrons experiencing the same potential difference as before and therefore moving at the same rate?


title: Maximum magnetic strength achievable

text: What will be the maximum magnetic strength to which a piece of ferromagnetic substance can be magnetised? What will the maximum strength possible for a piece of iron? How would the shape of the ferromagnetic substance affect this strength? What would be the current needed in the solenoid if this strength is to be achieved?


title: Capacitor physics problem

text: I read in a book that if both the plates of a parallel plate capacitor are given equal positive charges q, then the charges on the facing surfaces will be zero and the charge on the outer surfaces is q for each plate. How is this possible? Shouldn't the charge distribute uniformly on both the surfaces of each plate?


In k$\cdot$p theory, how does one calculate the bulk inversion asymmetry coefficients given in table 6.3 in Winkler? Winkler's book on spin-orbit coupling effects is available free online. In chapter 6 (section 6.2.1) he discusses BIA invariants and gives the formulas for computing them in Equation 6.2, while in table 6.3 he lists a few of them computed for some materials. He states that Band parameters were taken from Landolt-Bornstein which does indeed list the various energy gaps, and maybe the Kane parameter P. My question is how does one compute these coefficients in table 6.3 for a different material? I have searched for weeks and tried to use equations 6.2, however I have not found one reference that lists the values of the other Kane parameters P' and Q, and therefore I cannot use them accurately. Winkler quotes Landolt-Bornstein as the reference for table 6.3, however in LB the values of P' and Q are never given, only energy gaps (E0, E0', Delta ... etc) Any help would be greatly appreciated!


title: Gaussian surface and and Gauss law

text: Can we consider a cube as a Gaussian surface, for a point charge located at its center.since,Gaussian surface is a closed surface which has a constant electric field but in this case the both the magnitude and direction of electric field is changing at different points on the surface of cuboid. Can we use Gauss law for this surface in this condition.


title: Why can't you see meteors, but you can see comets from the moon?

text: Why can't you see meteors from the moon and why can you see comets?


title: Why is M42 red in the pictures, but green when observed at the eyepiece?

text: I saw magnificent images of the Orion Nebula (M42) in pictures from Wikipedia, ![M42 \(Orion nebula\)](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Orion_Nebula_- _Hubble_2006_mosaic_18000.jpg/600px-Orion_Nebula_- _Hubble_2006_mosaic_18000.jpg) However, when observed with a telescope, the nebula appears green hued, and I can't see any of the characteristic red color of the picture. Where's the difference?


title: Computing linearized gravitational wave emission from point-like masses

text: I'm trying to compute the gravitational wavefront created from a set of moving masses. I'm trying to apply the equation $$ h_{jk} = \frac{2}{r} \frac{d^2 Q_{jk}}{dt^2}$$ Where $h$ is the linearized perturbation of the metric and $Q$ is the quadrupole moment. $Q_{jk} = \int{\rho(x) x^j x^k dx^3}$ My question is, would these equations be still valid for a bunch of point-like masses? the $\rho(x)$ would become a $\delta(x)$, and because of that $h$ would become zero everywhere except on the masses, so I wouldn't get a meaningful far field from these quantities What would be the first-order approximation of GW emission from some moving point-like masses?


title: distance of electron from proton

text: An electron is projected, with an initial speed of $1.10 \times 10^5 \text{m/s}$, directly towards a proton that is essentially at rest. If the electron is initially a great distance from the proton, at what distance from the proton is its speed instantaneously equal to twice its initial value? i know it can be solved by equating the total energy as K.E+P.E(electron)=K.E+P.E(Proton) kinetic energy is $ \frac{1}{2}mv^2$. how can i find out potential energy of each particle?


title: Born–Oppenheimer approximation

text: Reading few sources on Born–Oppenheimer approximation I don't understand one particular thing. If you look for example here (PDF, 70 KB) and focus attention on equations 14 and 15 than it is clear that $$ \nabla_{A}^2 \left( \psi_k(\mathbf{r}; \mathbf{R}) \chi_k(\mathbf{R}) \right) = \psi_k(\mathbf{r}; \mathbf{R}) \nabla_{A}^2 \chi_k(\mathbf{r}; \mathbf{R}) \+ 2 \nabla_{A} \psi_k(\mathbf{r}; \mathbf{R}) \nabla_{A} \chi_k(\mathbf{r}; \mathbf{R}) \+ \chi_k(\mathbf{R}) \nabla_{A}^2 \psi_k(\mathbf{r}; \mathbf{R}) $$ where $$ \nabla_A^2 = \frac{\partial^2}{\partial X_A^2} + \frac{\partial^2}{\partial Y_A^2} + \frac{\partial^2}{\partial Z_A^2} $$ and $$ \mathbf{R} = \\{ \mathbf{R_i} \\}_{i=1}^N = \\{ (X_i, Y_i,Z_i) \\}_{i=1}^N $$ is a set of all nuclear coordinates. But honestly I don't undestand why is that so. The fact is that $\psi_k$ depends _implicitly_ only on $\mathbf{r}$ and _parametrically_ on $\mathbf{R}$ (that's why I think they are delimited by $;$ and not just $,$). As far as I know this parametric dependence means that for each set of nuclear coordinates $\mathbf{R}$ there is a complete set of electronic wave functions $\\{ \psi_k(\mathbf{r}) \\}_{k}$ which are functions of electronic coordinates only. And then of course when you differentiate $\psi_k(\mathbf{r}) \chi_k(\mathbf{R})$ twice with respect to $\mathbf{R_A}$ you get just $\psi_k(\mathbf{r}) \nabla_{A}^2 \chi_k(\mathbf{R})$ because $\psi_k(\mathbf{r})$ is constant with respect to $\mathbf{R}$. And one more thing - the linked resource (and many others) claimed that the _chain rule_ is used going from 14 to 15. I don't see any usage of chain rule but I see a usage of _product rule_. Seems like I don't understand what's going on here but this is critical step because non-adiabatic coupling terms come from this expansion.


title: What was the first ionization radiation?

text: While trying to make somewhat of a timeline of the history of ionizatig radiation, i am wondering about the following questions: The first photoelectrical effect was found 1839 by Alexandre Becquerell - the photoelectrical effect and therefore UV-light, would be the first ionization radiation. Is that right? While i am quite sure about it for electromagnetical radiation, i am not sure about particle radiation. Was there any ionizating particle radiation before the discovery of uran-radiation by Antoine Becquerell? A source for proof would be quite a nice extra, thanks!


title: The Dual Nature of Matter

text: I can't seem to understand the dual nature of matter completely. If electrons have a wave nature, then if two electrons were to collide, wouldn't they undergo interference and form an electron wave of larger or smaller amplitude? Does that mean the electrons merge and turn into a "bigger" electron? Then again, this would mean that the electrons would have a phase difference, so what determines the phase of matter? Also, if we were to consider two photons colliding, would they also undergo interference and form a "bigger" photon? Why don't normal everyday objects undergo interference? Just because the wavelength (according to the de Broglie equation) of something is small doesn't mean that it doesn't undergo interference (take for example gamma rays which have a very small wavelength but are the most powerful form of radiation).


title: Weakly Guiding Approximation

text: I was reading a chapter on Fiber Optics and I encountered `Weakly Guiding Approximation`. I am reading John M. Senior and it says `Optical fibers used in communication purposes follow weakly guiding approximation where the relative refractive index (delta) is much less than 1.`I tried searching different sources but with no success. Kindly help me understand it. Thanks.


title: Representation of operators in quantum mechanics

text: For which systems we represent the Hamiltonian as a differential operator and for which system we represent it by a matrix? Can the momentum be represented by a matrix operator?


title: Why does a star collapse under its own gravity when the gravity at its centre is zero?

text: The gravity at the centre of a star is zero as in the case of any uniform solid sphere with some mass. When a massive star dies, why does it give rise to a black hole at it's centre? I know how to derive the field equations for gravity inside a star assuming the star as a uniform solid sphere of mass M and radius R. I need to know how to find the expression for the total pressure due to gravity at the centre.


title: Generating artificial gravity by using rotation

text: If I want to design for futuristic space stations and I want to use rotation to produce artificial gravity. One of such designs consists of a giant ring that is rotated about its centre. If it were rotating fast enough, inhabitants in the ring will feel a ‘gravitational’ force just like on Earth. Suppose there are two clocks in the space station, one at the centre and the other attached to the rotating ring. Which clock would be slower?


title: Graphene with a disclination and the spin-orbit coupling

text: I am trying to follow the methods used in this paper (http://arxiv.org/pdf/1208.3023.pdf) to construct the Hamiltonian of a graphene cone, but taking into account the spin-orbit coupling. The paper constructs the Hamiltonian for a graphene cone with a Haldane mass term of the form $m \tau_z \sigma_z$. I wish to take into account the spin. So, instead of the mass term having the form $m \tau_z \sigma_z$, it should look like $m \tau_z \sigma_z s_c$, where $s_c = \vec{n} \cdot \vec{s}$ accounts for the spin projection on the surface of the cone. The spin degree of freedom can easily be added to the massless term of the Hamiltonian by putting $s_0$ at the end (since it does not affect it). (If this were flat graphene, the mass term would be $m \tau_z \sigma_zs_z$ like in the Kane-Mele model.) The problem is that when I construct my Hamiltonian, I don't know "when" to add the mass term. Do I start from the standard 2D graphene Hamiltonian (with spin), add this mass term, and then use the required formalism to take into account the conical topology? Or can I already start from the cone's massless Hamiltonian (which is already in block-diagonal form) and simply add the mass term to it? Ideally I would like to obtain a Hamiltonian which is in block-diagonal form, which is attainable if we neglect the spin (as shown in the paper).


title: Gravitationally bound systems in an expanding universe

text: This isn't yet a complete question; rather, I'm looking for a qual-level question and answer describing a gravitationally bound system in an expanding universe. Since it's qual level, this needs a vastly simplified model, preferably one in a Newtonian framework (if this is even possible) that at least shows the spirit of what happens. If it isn't possible, and there is some important principle that any Newtonian model will miss, that is probably the first thing to point out. Otherwise, ... I'm thinking of a setup that starts something like this: _A body of mass $m$ orbits another of mass $M \gg m$ (taken to be the origin) in a circle of radius $R$ (and consequently with a period $T = 2 \pi \sqrt{ R^3 / GM }$). At time $t = 0$, space begins to slowly expand at a constant Hubble rate $H$ (where "slowly" means $ HT \ll 1 $, so that the expansion in one period is negligible compared to the orbital radius)._ Now here is where I'm not sure how to phrase the problem. I'm thinking instead of something like the "bead on a pole" problems. Then the bead has generalized Lagrangian coordinates relative to a fixed position on the pole, but the pole is expanding according to some external function $a(t)$, a la FLRW metric (For a constant Hubble rate, $H = \dot a / a$ and so $ a(t) = e^{Ht} $). So, we can call the bead's distance from the origin in "co-moving units" the Lagrangian coordinate $q$. Then the distance of the bead from the origin is $d(t) = q(t) a(t)$, giving us a time dependent Lagrangian. Basically, I need a 3D version of this that will work for the gravitationally bound system described. I also hope that the problem can be solved as an adiabatic process, which might mean changing coordinates to be around the original position relative to the more massive body at the origin, rather than co-moving coordinates. EDIT 1: I won't claim that I fully understood the details of Schirmer's work, but I think one of the big takeaway points is that the cosmological expansion damages the stability of circular orbits and causes them to decay. I finished my very hand-waivey "Newtonian" model of a solar system in an expanding universe, and I don't think it captures the essence of this part of the GR solution. The model does have several sensical properties:   * There is a distance at which bound solutions are impossible and the two bodies are guaranteed to expand away from each other   * There is still a circular orbit (I haven't checked that it is stable) whose radius is modified by a term involving the Hubble constant.   * Using solar parameters, this shift is entirely negligible; it is larger (though still small) on a galactic scale. Here is the model: Without this scaling factor, the Lagrangian would be $$ \mathcal{L} = \frac{1}{2} m \left( \dot r^2 + r^2 \dot \theta^2 + r^2 \sin^2 \theta \, \dot \phi^2 \right) + \frac{GMm}{r} $$ Adding in the scale factor, all the kinetic terms will pick up a factor $e^{2Ht}$ and the potential a factor $e^{-Ht}$: $$ \mathcal{L} = \frac{1}{2} m \left(\dot r^2 + r^2 \dot \theta^2 + r^2 \sin^2 \theta \, \dot \phi^2 \right) e^{2Ht}+ \frac{GMm}{r} e^{-Ht} $$ (I have not added in any derivatives of the scaling factor; this would mean I was just changing coordinates and not adding in an externally controlled expansion of space). Since the problem is spherically symmetric, the total angular momentum is conserved and the motion lies in a plane (the expansion of space does not change this) at $\theta = \pi/2$. This reduces the effective Lagrangian to $$ \mathcal{L} = \frac{1}{2} m \left( \dot r^2 + r^2 \dot \phi^2 \right) e^{2Ht} + \frac{GMm}{r} e^{-Ht} $$ Now let's change coordinates to $r = \eta e^{-Ht}$. This represents a point that is stationary with respect to the origin (i.e. the massive body that the smaller body orbits). Then $\dot r e^{Ht}= \dot \eta - H \eta $, so the Lagrangian becomes $$ \mathcal{L} = \frac{1}{2} m \left((\dot \eta - H \eta)^2 + \eta^2 \dot \phi^2 \right) + \frac{GMm}{\eta} $$ This looks like a Lagrangian that has been modified slightly by the parameter $H$. After these manipulations, $\phi$ is still a cyclic coordinate, and its conjugate momentum $p_\phi = m\eta^2 \dot \phi = L$ is still conserved. The momentum conjugate to $\eta$ is $$ p_{\eta} = m (\dot \eta - H \eta)$$ So, the equation of motion for $\eta$ is $$ m \frac{d}{dt} (\dot \eta - H \eta) = m (\ddot \eta - H \dot \eta) = - mH(\dot \eta - H \eta) + m \eta \dot \phi^2 - \frac{GMm}{\eta^2} $$ Canceling terms and substituting in the equation of motion for $\phi$, this becomes $$ \ddot \eta = H^2 \eta + \frac{L^2}{m^2 \eta^3} - \frac{GM}{\eta^2} $$ So, the expansion of space shows up as a force term proportional to $\eta$. For large $\eta_0$ at time $t = 0$, there is a solution $\eta(t) = \eta_0 e^{Ht}$, where $\dot \eta = H \eta$ agrees with Hubble's law (the corresponding comoving coordinate is $r(t) = \eta_0$). There is also an a highly unstable orbital solution for large $\eta$, where gravitational force just balances cosmological expansion. This new term also shifts the location of the "stable orbit" slightly (I haven't checked that it actually is stable in this situation). The location of the stable circular orbit when $H = 0$ is $$ \eta_0 = \frac{L^2}{GMm^2} $$ Then let $$ \eta = \eta_0 + \delta \eta = \eta_0 \left( 1 + \frac{\delta \eta}{\eta_0} \right)$$ To lowest order, the shift in location of the stable orbit is $$ \frac{\delta \eta}{\eta_0} = \frac{H^2}{GM/\eta_0^3 - H^2} $$ For the orbit of the earth around the sun, this shift would be entirely negligible- about 15 pm. Taking the mass to be the mass of the Milky way and the distance to the distance of the Sun from the center, the shift would be a bit larger - a fractional amount of about 2e-7, which is several hundred AU - but still largely small.


title: Light Focus Guider

text: How can I easily guide a focused light source to a different location without losing focused light intensity? If there is an item, where can I buy this? I heard about fiber cable, but I do not know how can I focus light into this fiber.


title: The consistency conditions of constrained Hamiltonian systems

text: I am studying the Hamiltonian description of a constrained system. There are some questions puzzled me for days, which I have been stuck on it. From the lagrangian, we can obtain the primary constraints, and for the consistency conditions, we have to make thse primary constraints preserved in time, which lead us to some further constraints. It is clear that $$\\{\phi^{(1)}_i,H_c\\}+u^{(1)}_j\\{\phi^{(1)}_i, \phi^{(1)}_j\\}=0 \qquad (1)$$ yeilds the secondary constraints. But what equation would yeild the tertiary constraint? Is it $$\\{\phi^{(2)}_a,H_c\\}+u^{(1)}_j\\{\phi^{(2)}_a, \phi^{(1)}_j\\}+u^{(2)}_b\\{\phi^{(2)}_a, \phi^{(2)}_b\\}=0 \qquad (2)$$ or just $$\\{\phi^{(2)}_a,H_c\\}+u^{(1)}_j\\{\phi^{(2)}_a, \phi^{(1)}_j\\}=0 \qquad ? \quad(3)$$ Namely, should we take the secondary constraint into account, when move from the second stage to the third stage. (Here $i,j$ denote the all the dimensions of primary constraints and $a,b$ denote the dimensions of secondary constraints.) And when should this procedure be terminated? Should we stop or move on, when some constraints don't commute with each other? If we move on, what are the next stage constraints? Is it the parts without the terms involving undetermined $u$s? For example, in Eq.(2), if   $\\{\phi^{(2)}_a, \phi^{(1)}_j\\}\neq 0$, can it still yeild the tertiary constraints, $\\{\phi^{(2)}_a,H_c\\}=0$ ? And which Hamiltonian should we use to derive the Hamilton equations, the Hamiltonian with only primary constraints or the one with all constraints? I am trying to make my questions clearer, so the description is lengthy. Thanks!


title: Materials with a high compressive strength to weight ratio

text: We know about extremely strong materials such as carbon nanotubes. However, this is only in tension. What are some high strength-to-weight materials (both available and hypothetical) in uniaxial compression? Such a material would be useful in supertall structures and vacuum airships. Of my research, PVC pipe seems to be a good bet at 100MPa and only 1.3 g/cm^3. However, this is still inadequate for a vacuum airship. Beryllium is also a likely candidate as it is 1400MPa and only 1.85 c/cm^3, making it doable in a vacuum airship. However, it is very toxic to work with (I don't know if bulk beryllium is safe) and expensive.


title: Why Quantum correlation is not uniform in this diagram?

text: Following diagram is from a Wikipedia article which shows Quantum Correlation for local hidden variables and Quantum Mechanics and experiments confirm Quantum Mechanics predictions. ![enter image description here](http://i.stack.imgur.com/VOnL7.png) My question is why the curve at left side (0-180 degree) is not symmetric with the curve on right side (180-360 degree) although on each side the curve is symmetric with horizontal line?


title: Is it possible to detect subjective decoherence? If yes, how?

text: In his paper from 1994 Thomas Breuer describes a phenomenon of subjective decoherence (p. 43). I wonder whether it can be measured, and if yes, how. I also wonder whether it would allow to create an all-win lottery that can make everybody happy. Suppose a lottery is conducted. The winner is awarded a billion dollars plus the title of eternal Earth dictator. The winner is determined in the following way.   * There is a thoroughly isolated black box inside which a pretender is placed.   * Inside the box an apparatus measures subjective decoherence.   * If subjective decoherence is found, the pretender declared the winner. It follows that any person, when he or she is observing the lottery from outside will see all other pretenders to loose. At the same time any participant that tries the lottery himself will find himself a winner. This is a fascinating lottery which will make all the participation pretenders the winners in their own eyes. Each person who goes through this machine will see other people to congratulate them, to proclaim them the eternal emperor and so on.


title: Is the Lorentz group compact (and if not, is U(1)?)

text: A common statement in any quantum field theory text is that only compact groups have finite-dimensional representations, and that the Lorentz group is not compact, since it is parameterised by $0\leq (v/c)<1$. Fair enough, but $U(1)$ is parametrised by $0\leq \theta <2\pi$, so why is $U(1)$ compact? Should they be described as closed instead, or am I getting lost?


title: Does $E$ really equal $mc^2$?

text: I'm currently in a debate with a co-worker. If mass is sped up to the speed of light, does the mass become energy?


title: Is it possible to have annealing without creep?

text: Annealing can repair a material by allowing atoms to find the minimum energy state; since solids have a surface tension this process will allow cracks to fuse and reverse fatigue. However, annealing comes with it's evil twin: creep. If the atoms have enough thermal energy to remove defects form the crystal they likely will also slowly move in the direction of applied force; the material behaves like a very viscous liquid. But is it possible, given the right material and temperature, to have annealing without creep? If the energy barrier to remove a dislocation is _less_ than the energy barrier to pull an atom out of the crystal lattice, there could be a temperature sweet-spot in which materials wouldn't fatigue but wouldn't creep either?


title: where the proper time is invariant why $d\tau$ is not zero?

text: where the proper time is invariant why change (differential) in proper time $d\tau$ is not zero? $\Delta \tau=\tau_f-\tau_i$ as i know. $d(invariant)=0$ note to comment: action $S=-m_oc^2\int_C d\tau$ > Invariance always applies between some set of conditions. Things can be > invariant in time, invariant over changes in position or over changes of > coordinate systems and so on. In relativity things described as "invariant" > without a descriptive clause are general things that all observers can agree > upon. **The claim that _"proper time is invariant"_ , means that the proper > time between two events (four points in space time) is something that all > observers can agree upon**, not that the measured proper time is zero. something that all observers can agree upon like speed of light but this does not means that the measured in speed of light is not zero $dc\not= 0$, ok guys i got the point: **invariance is not a synonym for being a constant**. thanks


title: Range of forces from mass of force carrier?

text: Why is $\frac{\hbar}{mc}$ a good estimate of the range of the four forces, where $m$ is the mass of the carrier particle of the force? Inputting the pion mass gives $1.4\ \mathrm{fm}$ for the hadronic force. I have one book that tries to justify the formula by the uncertainty principle and another book that says that the argument is bogus. So does this follow from the uncertainty principle?


title: Will a violin string keep vibrating for longer time in vacuum than in air?

text: Hitting a string of a violin or a guitar will cause that string to vibrate, but after short time the amplitude of the vibration will decay, consequently the produced sound will die out. I suppose this decay happens because of friction with air. If this is true then how longer will the string keep vibrating in a vacuumed room? Any way to estimate this? Are there other effects that cause the damping?


title: Metric to describe an expanding spacetime from coordinates reflecting the perspective of a local observer

text: The FLRW metric describes the metric expansion of spacetime from the perspective of comoving coordinates. Given the way this metric is usually formulated, comoving distances stay constant, and the size of the unit ball "shrinks" as you translate it along the time axis (but not the spatial axes). I'm interested in seeing how to reformulate this metric given a choice of coordinates that lines up with the perspective of an observer who is in the space. From the observer's perspective, everything is falling along timelike geodesics that are moving away from it, and the further objects are from the observer, the faster they seem to be falling, and the size of the unit ball around the observer remains constant with time. To keep it simple, I only care about the case where curvature is 0 and we set $c=1$. So the FLRW metric in those circumstances is $$\mathrm{d}s^2 = -\mathrm{d}t^2 + a(t)^2 \cdot (\mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2)$$ Where $a(t)$ is the time-dependent scale factor. I thought that to flip this on its head, you'd want a metric that looks like this: $$\mathrm{d}s^2 = -b(x,y,z)^2 \cdot \mathrm{d}t^2 + \mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2$$ Where we now have the space-dependent scale factor of $b(x,y,z)$, and it affects the time coordinate. I was thinking that for some such suitable function $b$, which is presumably monotonic, you'd end up with a metric which stays constant in time rather than in space, and timelike geodesics that are all constantly moving away from one another. (They're also moving away from the line $(t,0,0,0)$, representing the observer I was talking about.) Am I on the right track here? I'm looking for something which is basically exactly the same as FLRW, but just reflects the different choice of coordinates.


title: Does nonlocal theory violate causality?

text: Let's talk about two kinds of nonlocal theories. The first one frequently derives from integrating out part of the degrees of freedom to obtain a kind of effective theory. Probably, we get an integral potential or something more complex. The second one is the case where we have operators like $\sqrt{-\hbar^2c^2\nabla^2+m^2c^2}$ acting on fields. In some sense, people say they are problematic, at least not fundamental. I wonder if one issue is violation of causality per se. And as far as I know, some researchers also distinguish causality and locality, e.g. E.P.R. paradox.


title: Is there any connection between "Lagrangian and Eulerian formalism of fluid" and "Heisenberg and Shrodinger picture"

text: Is there any connection between "Lagrangian and Eulerian formalism of fluid" and "Heisenberg and Shrodinger picture of Quantum mechanics"? Thanks!


title: On Group Theory: Symmetry Groups and Our Interest

text: Over the past few years, I've been doing a lot of self education in the Quantum Mechanics and General Relativity, and of course, there are mathematical elements of both doctrines that are matrices. More specifically, linear operators and transformations. Most of my texts introduced these ideas as fundamental, and not part of some larger mathematical structure--Group Theory. One thing throughout quantum mechanics that I've noticed is that there is a particular interest in matrices $ A $, such that $Ax = \lambda x$. The Eigenvalue Equation. This linear operator can be said to exhibit symmetry, since it leaves the operand the same up to a constant. For $ \lambda = 1 $, the transformation does indeed leave the operand the same. After some digging, this particular element of quantum mechanics that represents physical measurement, is an element of the Symmetry Group ( call it $ SG(n) $ where $n$ is the dimensionality). Another example is for the Lorentz transformations in special relativity. Each transformation leaves the reference frame the same no matter where you are. I've usually seen it explained that the Lorentz transformation is just the result of a rotation, very similar to that of the Rotation Group $ O(3) $, but since we have a time dimension and we rotate spacetime and suppress the two other spatial variables, we get the Lorentz Group: $ SO(3,1) $, equipped with 3 space + 1 time, dimensions. Both of these groups have been shown to me to be elements of $ SG(n)$ as well. At present it seems that Relativity and Quantum Mechanics are the two pillars that our ultimate goal of Cosmology rests upon. From the lens of Group Theory and by the relation of the elements in the theory aforementioned, I am compelled to conclude that the physical laws and principles that are being discovered present day forward are mathematical elements in this group theory that have meaningful physical and mathematical interpretation down at the physical doctrine level, ie the Energy Operator and the Hamiltonian which brought us to the Schrodinger equation, and the Lorentz Transformation that defined special relativity. This also compels me to believe that further advancement can be found if we have our mathematicians race to discover more and more elements in this larger Group Theory, and have Physicist determine and interpret any Physical meaning in the discovered constructs. My question is then a question of how valid my conclusion is, and how sound does this in turn make my belief.


title: How fart is transported through air?

text: As we know diffussivity of gas is really slow (e.g. $\require{mhchem}\ce{O2}$ and $\ce{H2O}$ in the air are respectively $0.176$ and $0.282 ~\text{cm}^2/\text{s}$). here is a example to specify how slow it is, to evaporate water filled in a $1~\text{m}$ testing tube without external heating, one requires almost $50$ years!! (Note that the vapor is saturated at the water-air interface so the energy for phase conversion is not a issue here.) However, if one farts in a room without sensible air flow, the smell can be noticed by him within $10$ seconds. This is definitely a speed that diffusion can not achieve. Does it mean that the air flow in a 'normal' room can be around $1~\text{m/s}$, or the initial far speed from anus gives a strong momentum which I don't think it could be that strong?


title: Why don't planets have Circular orbits?

text: This might be a completely wrong question, but this is bothering me since many days ago. Given the mass (Sun) curves the space around it, gravitation is the result of such curved space (Correct me if I am wrong, source: A Documentary film). Given any point on a circle with center same as center of the mass, curvature in the space should be equal (Intuition). Planets rotate around the Sun because of the curve in the space they should follow a circular path and the distance between planet and Sun should be at a distance. Given the fact that earth has a elliptical orbit around the sun, and the distance between Earth and Sun varies according to position of the earth. Why do we have a elliptical orbit not a circular orbit.