Solving coupled nonlinear ODEs, controlling numerical instability through numerical viscosity

I'm solving a problem numerically that takes the form $ Q_{ij} \ddot{y}_j +S_{ijk}\dot{y}_j\dot{y}_k +V_i=0$, where $(Q_{ij},S_{ijk},V_i) $ are all low order polynomial functions of the dependent variables $y_i$. The dependent variables are all functions of the variable t. The resolution of this spectral model is controlled by the number of $y_i$, denoted N. Now, when these equations become stiff, or I integrate for long times, numerical instability becomes apparent. If I increase the resolution, these spurious growths are attenuated, but it is not feasible to make N large enough so that they don't appear at all. If one looks at the time history of the higher modes, it seems like their large growth corresponds to the onset of the instability, so I want to attenuate these (un-physical) fast growing components of the solutions. To this end, I have added a term to the leading order term, so that the equation takes the form $Q_{ij}( \ddot{y}_j +\nu j^2 \dot{y}_j)+S_{ijk}\dot{y}_j\dot{y}_k +V_i=0$, where $\nu$ is a numerical viscosity to be prescribed later on. This does an OK job at attenuating the instability (e.g. I get accurate solutions for an twice as long), based on how I choose $\nu$, and the constraints of the problem. I have played around with the power of j as well, and found varying success in different contexts. The issue is, for long times this scheme breaks down, and the solution greatly deviates from the expected solution, making me think I'm not doing addressing the issue very rationally. Indeed, I am doing all of this very naively, and am having trouble finding similar types of problems in the literature. Does anyone have experience damping numerical instabilities in equations that take this form? In particular, is there a way to essentially add a dissipation that removes these fast growing modes? References to relevant literature are also greatly appreciated. Thanks! Nick


Does gravity slow the expansion of the universe? I read through the thread http://www.physicsforums.com/showthread.php?t=322633 and I have the same question. I know that the universe is not being stopped by gravity, but is the force of gravity slowing it down in any way? Without the force of gravity, would space expand faster? Help me formulate this question better if you know what I am asking.


What does the Froude number represent?

While reading on Wikipedia, I read the following > _The Froude number is defined as:_ > > $$\mathrm{Fr} = \frac{v}{c}$$ > > _where $v$ is a characteristic velocity, and $c$ is a characteristic water > wave propagation >velocity. The Froude number is thus analogous to the Mach > number. The greater the Froude >number, the greater the resistance._ While reading the following paper on shallow water waves, I could not understand how/why the Froude number was given to be $$\mathrm{F} = \frac{gt_0^2}{L}$$ Where $t_0$ is the time scale, and $L$ is the length scale along the X and Z axes. Note that the scale for the amplitude of the surface above the mean depth is some other $A$ and not $L$. Please explain how the author has defined the Froude number. Is it the same as the one given on Wikipedia? If so, then please provide a step-by-step reasoning to show that they are indeed the same. If not, then please explain what the Froude number mentioned in the paper represents. EDIT: I'd like to add another minor question as it is related to the abovementioned paper. It mentions that the surface tension is accounted for but I do not understand how that is possible given that there is no term containing the surface tension $\gamma$ explicitly.


Does a Force on the block $M_1$ in this diagram have an effect on the block $M_2$?

Suppose we have two blocks: ![enter image description here](http://i.stack.imgur.com/7ZmYU.jpg) There is no friction. Does $F$ has the same effect on $M_2$?


Is the "Great Attractor" an indicator of the "Multiverse"?

I have heard a bit about the Great Attractor (the gravitational anomaly that seems to be "sweeping" our universe in one direction). Someone (and forgive me, I do not recall the specifics) has hypothesized that this may be an indication of an other universe affecting us gravitationally. I am interested in people's opinions on this. Would this be an experimentally provable phenomenon of string-theory (in all its different incarnations)? Or are there more viable hypotheses that would explain this?


From the perspective of an observer inside a black hole's horizon, where does the energy for Hawking radiation come from?

Would energy be seen to "flow" to the outside of the black hole? Through what mechanism?


Have some trouble finding a useful reference to answer the following questions:   * What's a typical electron-hole recombination time of semiconductors? And how does it depend on the temperature and the band gap? If there's no formula, examples will do.   * What's the probability of an electron being kicked into the conduction band merely by thermal motion? Is it $\exp(-\Delta E/T)$, where $\Delta E$ is the band gap and $T$ the temperature? That thermal excitation should lead to noise in the system, is that correct?


Does the unpredictability in the "micro world" means that everything is if we can look at it close enough imperfect? I mean, there is a saying "You will never stand in the same river again" or something like that. So, if nothing in our makro world is the same, if we look close enough (no two people or flowers are the same; and if they seem, it just means you are not looking carefully or cose enough ;) Is this also true in the micro world? If yes, does that mean that e.g no two atoms of hydrogen are the same and the sign $H$ is only some kind of reference for "from-to" or "plus-minus- something-like-that" tolerance? And the same principle can be applied for smaller particles or even strings etc.? Does that mean that there is no such thing as two identical proto particles and everything is slighty different? And my final concern is... Do we live in "The Imperfect Universe"? ;)


Whirlpools and Tornados

This may not be a great question. But whenever you drain water, a small whirlpool happens, obviously. This got me thinking... Can we model tornados with this effect, would it even be beneficial?


Heterotic string as worldvolume theory of two coincident 9-branes in 27 dimensions?

The heterotic string is a combination of right-moving excitations from a D=10 superstring and left-moving excitations from a D=26 bosonic string, with the left-movers behaving as if the extra 16 dimensions are compactified. The heterotic string is also derived from D=11 M-theory, as an open 2-brane stretched between two "end-of-the-world" 9-branes (spatial boundaries; this is M-theory compactified on a line segment, and the 9-branes lie at the ends). So I am led to imagine a 27-dimensional theory, containing branes. We compactify 16 dimensions, and consider the worldvolume theory of two parallel 9-branes. When they are coincident, we get the heterotic string; when they are slightly separate, we get "heterotic M-theory". A 27-dimensional fundamental theory has been discussed before (hep-th/9704158 section 4; hep-th/0012037; arXiV:0807.4899), but I don't see this particular line of thought discussed.


What experiments, other than Hubble Expansion, support the Dark Energy theory?

Dark energy is introduced as a constant inside Einstein's equations. Its primary purpose, from what I understand, is to make Einstein's equations compatible with the _accelerating_ expansion of the universe. As a consequence, of the "predictions" of dark energy is the expansion of the universe according to Hubble's law. I know there are numerous experiments that verify this expansion (and its acceleration), and thus indirectly support the dark energy theory. My question is:   **Are there _other_ factors that give credit to the existence of dark energy?**   Are there any experiments that support this theory, but not only through the verification of Hubble expansion?


combination of 5 identical capacitors

Given 5 identical capacitors, each EMF. Draw the proper combination of all these 5 capacitors.. 1.) 2.5 2.) 4.17 I really do not know how to solve this thing and if anyone of you knows the title of this lesson please comment it here so I can study it further. We will be having our finals tomorrow.


Two ways of looking at weak current?

A weak current of the form: $$j^\mu=\bar u_e \gamma^\mu \frac{1}{2}(1-\gamma^5)u_\nu$$ implies that it automatically selects the left- handed neutrino. This current can also be written as: $$j^\mu=\bar u_e \frac{1}{2}(1+\gamma^5)\gamma^\mu u_\nu$$ which seems to imply that only left- handed electrons take part in weak interactions. Which statement is right?


How can it be that the beginning universe had a high temperature and a low entropy at the same time?

The Big Bang theory assumes that our universe started from a very/infinitely dense and extremely/infinitely hot state. But on the other side, it is often claimed that our universe must have been started in a state with very low or even zero entropy. Now the third law of thermodynamic states that if the entropy of a system approaches a minimum, it's temperature approaches absolut zero. So how can it be that the beginning universe had a high temperature and a low entropy at the same time? Wouldn't such a state be in contradiction to the third law of thermodynamics?


Would you see a rainbow from refraction when the sun is in front of you?

I know how rainbows are formed, and why. Usually it is said that the Sun must be **behind** the observer, in order for its light to be totally reflected inside the droplet and then reach the observer. But surely not all the light is totally reflected? There must be some radiation refracting from inside the droplet to the air: since different wavelengths refract through different angles, shouldn't we also see a rainbow when the Sun is in front of us?


Lever Mechanics - How to formulate an ideal lever launch

Let's say I have a simple lever as shown below, and the lever is massless and the pivot is frictionless and there is no air resistance. I'm thinking the cradle for the projectile would have to have a specific shape to dictate the launch angle, but for the context of a general equation $\theta$ will be the angle the lever makes parallel to the ground, and will have some value where the object will leave the lever. $\theta_{launch}$ will refer to the angle the lever is at when the object 'launches', so the lever will have swept $\theta_0 - \theta_{launch}$. $t_0$ is the initial time the weight is released and $t_{launch}$ = launch time. ![Lever Launching](http://i.stack.imgur.com/dN2fz.png) # Height Functions $h_{weight}(\theta) = h_{pivot} + l_{shortArm}cos(\theta)$ $h_{projectile}(\theta) = h_{pivot} - l_{longArm}cos(\theta)$ # Desired Equations   * Potential Energy of the system at Initial Conditions   * Velocity of Projectile at Launch   * KE of Projectile at Launch   * Efficiency of machine


converting daily global radiation into hourly values

I would like to convert daily global radiation into hourly solar radiation. While I do not know the terms, I tried using below equations I found in the papers but I have negative radiation values range between -75 to 100 j/m2/s. Any suggestions are welcome.               # radi(h) = r * radi(daily)      > r <\- (pi/24) * ( a + b * cos_d (w))* ((cos_d(w) - cos_d(ws)) / > (sin_d(ws)-((pi/180) * ws * cos_d(ws))) ) > > a <\- 0.409 + 0.5016 * sin_d(ws - 60) > > b <\- 0.6609 - 0.4767 * sin_d(ws - 60)                # W = hour angle of the sun      > w <\- (360/24) * (h-12.0) # h = time of the day (hour) > > ws <\- acos_d(-tan_d(phi)*tan_d(sigma))) # ws = sunset hour angle (degree) > > phi= 36.20 # phi = location latitude angle               # sigma= declination angle of the sun [degree]      > sigma = 23.45 * sin_d (360* ((284+day)/365)) # day=DOY


Surviving Free Fall with the Help of Mr. Bernoulli

What is the best strategy to survive a free fall naked out of a jetliner at cruising altitude (ignoring temperature)? For instance, my strategy would be to streamline my fall so that my terminal velocity was very high. Then at some critical distance above the ground I would pitch upward and attempt to use my forward velocity to achieve lift. Presumably this would slow my fall to some survivable speed. (This is roughly what the space shuttle does after all.) Would it work? And idea if it would be survivable?


Atomic nucleus consisting of only neutrons?

Is it true that the nucleus of all atoms (including radioactive isotopes) contain at least one proton? Is there an atomic nucleus consisting entirely of neutrons? (Let's exclude neutron stars for the moment.) If so, how does one name them? (since Periodic Table starts from atomic number 1, not 0.)


Infinite Wells and Delta Functions

In considering a delta potential barrier in an infinite well, I can just enforce continuity at the potential barrier-it doesn't have to go to zero. Why then does it need to go to zero at the walls of the infinite well? These two cases seem to be very similar to be, I even feel like the well wall is equivalent to a summation of delta functions... Where is my logic faulty?


What is the meaning of non-compactness in the context of $U(1)$ in gauge theories?

In John Preskill's review of monopoles he states > Nowadays, we have another way of understanding why electric charge is > quantized. Charge is quantized if the electromagnetic U(l)em gauge group is > compact. But U(I)em is automatically compact in a unified gauge theory in > which U(l)em is embedded in a nonabelian semisimple group. [Note that the > standard Weinberg-Salam-Glashow (35) model is not "unified" according to > this criterion.] The implication of the third sentence is that, in some circumstances, the $U(1)_{em}$ gauge group may _not_ be compact. How could this be? Since $U(1)$ as a differentiable manifold is diffeomorphic to $S^1$ isn't it automatically always compact? The following paragraph: > In other words, in a unified gauge theory, the electric charge operator > obeys nontrivial commutation relations with other operators in the theory. > Just as the angular momentum algebra requires the eigenvalues of $J_z$ to be > integer multiples of $\frac{\hbar}{2}$, the commutation relations satisfied > by the electric charge operator require its eigenvalues to be integer > multiples of a fundamental unit. This conclusion holds even if the > symmetries generated by the charges that fail to commute with electric > charge are spontaneously broken. is OK, but I don't follow what that has to do with the compactness of $U(1)$.


How does energy transfer between B and E in an EM standing wave?

I'm trying to understand how an electric field induces a magnetic field and vice versa, its associated energy, as well as relating it to my understanding of waves on a string. Using a standing wave as an example, I came up with the equations $\vec{E}=A\sin(\omega t)\sin(kx)\hat{y}$ $\vec{B}=\frac{Ak}{w}\cos(\omega t)\cos(kx)\hat{z}$ I checked them against Maxwell's equations, and they're self-consistent. At time 0, this reduces to: $\vec{B}=\frac{Ak}{w}\cos(kx)\hat{z}$ Since the electric field is 0, based on the Poynting vector, there's no energy transfer at this time. At this time, at a node where $\vec{B}=0$, there's neither electric field nor magnetic field. If there's no energy transfer, and no energy stored in either field, then how can an electric field exist at this point at some time later? How is the energy stored, or transferred from elsewhere?


Line width of green DPSS laser

Could someone suggest what is the line width/mode stability/coherence length of generic cheapo green (532nm) DPSS laser? I've found lots of nice info here: http://redlum.xohp.pagesperso- orange.fr/laser/modeanalysis.html but there are no any DPSS lasers.


How are light and time related?

So, from what I understand: Special Relativity says that light is always observed moving at the speed of light (c). If some object had a velocity of (3/4)c, and the object had some sort of clock attached to it, it would measure differently from a still clock. 1) What would a clock moving the speed of light measure compared to a still clock? 2) Why does it seem that the speed of light is a focal point of time? EDIT: I may need to rephrase this slightly. What I want to know is, why is it that observed time changes at high velocities?


How do physicists justify using $c^2$ in equations?

Excuse my ignorance but I see physicists use $c^2$ in equations and I was wondering what reasoning they use to justify $c^2$ since, as far as I understand, $c^2$ must be an absurd quantity because there is a speed limit imposed by physicists themselves at $c$. If there is no motion faster than $c$, how do you justify using $c^2$ in an equation?


What routine tasks have to be performed on the outside of spaceships/stations?

What kind of routine tasks have to be performed on the outside of spaceships or space stations? Repairs, I presume. Anything else? Is there anything that needs to be cleaned off from the outside, like debris, or some other substance? Can anything on the surface freeze?


Why is the probability that Kip could go back and kill his grandfather  $10^{-10^{60}}$?

In _The Universe In A Nutshell_ , page 153, Hawking says that > _The probability that Kip could go back and kill his grandfather is > $10^{-10^{60}}$._ How do we get this result?


Dirac equation in curved space-time with Torsion

I am looking for pedagogical references in which Dirac equation in space-time with curvature and torsion were discussed.


What if the 126 GeV particle found at LHC that LOOKS like the Higgs is actually not the Higgs Boson?

I am suspicious of the Higgs announcement. Press mania aside, the scientists seem careful to say only that "we've found something which _looks_ like the Higgs Boson". The difference between Higgs-containing theories and non-Higgs-containing theories is a tiny bump on the energy curve, smaller than the resolution of the detectors, which is why we have to perform statistical gymnastics on large data sets to discover or refute the deviation. In other words, the energy distribution of the reactions is consistent with there being a short-lived particle weighing 126GeV, and that's all the announcements seem to really say. Is there anything else in the results which tells us that the extra element is Higgs and not something else? Is there any identifying feature other than lifespan and mass?


Given two surfaces, one that absorbs light and the other that reflects it, will there be a signficant temperature difference?

My likely wrong understanding of light is this: By some way the molecules are structured, a photon hits an atom or a structure, is absorbed, then re-emitted and it's that photon wavelength that I see as light when my eyes absorb the photon/wave. So if say something is painted black, does it follow that it would absorb light? And if so, when compared to an object that was painted white or not naturally black in color would it have a cooler temperature than the surface painted black?


Is it possible that only one hemisphere of a planet has an atmosphere?

Suppose there is a tidally locked planet orbiting a star. The planet's surface consists of a global ocean, that is, liquid water. At the inner hemisphere the temperature is so high that the water is constantly boiling, creating the atmosphere of water vapor. But the vapor does not reach the dark side of the planet, and precipitates around the terminator line. The further side has no atmosphere and an icy surface. Is this setup possible? Can it be such thing that one side of a planet has liquid (and boiling) water surface while the other side has no atmosphere? Is it possible at all that a part of an ocean was boiling while the other one was icy?


number of microstates associated with two-level quantum systems

this is a very simple question, but apparently one that has no simple answer, at least from standard quantum mechanics theory I'm trying to figure the number of simple quantum states (microstates) of the allowed configuration space of a two-level quantum system, ignoring for now mixed states. So if i have a state like: $$ \Psi = \psi_0 | 0 \rangle + \psi_1 | 1 \rangle $$ removing all redundancies from the representation, we are left $$ \psi_0 = \rho_0 $$ $$ \psi_1 = \rho_1 e^{-i \theta_1}$$ $$ \rho_0^2 + \rho_1^2 = 1$$ Now, this state space looks like $[0,1] \times S_1$. I've seen discussions that quantum states take some volume in phase space, but i want to understand that in detail: so in this case, > **Question:** how many distinct states are in this configuration space? is > this a finite set? infinite but numerable set? nonnumerable? **Following Steve suggestion** , lets write the Shannon entropy of this wave function: $$ S( \Psi ) = - \rho_0 \ln \rho_0 - \rho_1 \ln \rho_1 $$ There is some conflict when i see this entropy; it cannot derive from any counting state scheme, in particular because it completely ignores the relative phase factor between the basis states, and as we all we know by now, states with different relative phase factors are physically different; otherwise there would be no interference terms, and no quantum mechanics to speak of!


How to handle Yukawa contractions in calculating SUSY beta functions?

I'm reading Chapter 6 of Martin's introduction to SUSY http://arxiv.org/abs/hepph/9709356, which is about RGEs in the MSSM. I tried to convince myself of some of the calculations, and I was particularly stuck with anything that has Yukawas in it. I don't know how to deal with them. Take for example the anomalous dimension formula, $$\gamma^i_j =\frac{1}{16 \pi^2} [\frac{1}{2} Y^{imn}Y^*_{jmn} - 2 g_a^2 C_a(i)\delta^i_j]$$ where, $g_a$ is a gauge coupling corresponding to a group $a$, and $C_a(i)$ is the Casimir invariant. And $Y^{imn}$ denotes Yukawas. For example, $$\gamma^{H_u}_{H_u} =\frac{1}{16 \pi^2} [3 Y^*_t Y_t - \dots]$$ $$\gamma^{\bar{d}_3}_{\bar{d}_3} =\frac{1}{16 \pi^2} [2 Y^*_b Y_b - \dots]$$ $$\gamma^{\bar{Q}_3}_{\bar{Q}_3} =\frac{1}{16 \pi^2} [Y^*_t Y_t + Y^*_b Y_b - \dots]$$ the dots are for the second term which I know how to calculate. But I don't know how the results for the Yukawas were obtained! My understanding is that the indices $i,j,k$ are family indices for chiral superfields. Explicitly, $$ Y^{ijk} = Y^{\Phi_i \Phi_j \Phi_k}$$ So if I wanted to calculate, the Yukawa part of $\gamma^{H_u}_{H_u}$, I would try: $$Y^{H_u mn}Y^*_{H_u mn} = Y^{H_u \bar{u}_m Q_n}Y^*_{H_u \bar{u}_m Q_n} + Y^{H_u Q_m \bar{u}_n}Y^*_{H_u Q_m \bar{u}_n} + \text{other permutations in superfields?} $$ If I chose third family approximation like in Martin's notes, then $m=n=3$ But I'm not sure this is correct or where it leads. Could someone help me with this. How should I handle $Y^{imn}Y^*_{jmn}$ to get those results?


"windows of order" in the Bifurcation diagram

When looking at the bifurcation diagram of a chaotic system, one observes "windows of order", namely short intervals where the system briefly leaves its chaotic state and then rapidly returns to chaos. These intervals are quite periodical. Does anyone have a good explanation for this? Thanks!


Special Relativity: Transforming Maxwell's equations

I'm working through Einstein's original 1905 paper*, and I'm having trouble with the section on the transformation of Maxwell's equations from rest to moving frame. The paper proceeds as follows: > Let the Maxwell-Hertz equations for empty space hold good for the stationary > system K, so that we have > > IMAGE: Untransformed Maxwell equations > > where (X, Y, Z) denotes the vector of the electric force, and (L, M, N) that > of the magnetic force. > > If we apply to these equations the transformation developed in § 3, by > referring the electromagnetic processes to the system of co-ordinates there > introduced, moving with the velocity v, we obtain the equations > > IMAGE: Transformed Maxwell equations I do not understand just where this second set of equations comes from. The only derivations of this transformation that I have been able to find involve either potentials, four-vectors, or both. However since four-vectors had not been invented in 1905, and because the statement of the transformation is so blunt, it seems that Einstein is using or appealing to a simpler method for finding the transformation. So my question is: **What is the simplest derivation of the transformation rules for Maxwell's equations in special relativity?** Is Einstein appealing to a pre-existing Lorentz method here, or is there a trick to accomplish all of this quickly? Or does the text here simply belie the true work involved in the derivation?   * I cannot give a proper link to the paper owing to link restrictions, but it's at hxxp://www.fourmilab.ch/etexts/einstein/specrel/www/#SECTION21


Orbital radius of Geo-stationary satellite

Could you please tell, Why all the geo-stationary satellites are to be dropped at same height from earth? Why can't it be closer or away from its regular orbit(ie, 35,000 km)? If all satellites are dropped in the same orbit, then will not those collide one another?


Information about newton's third law

Well I am still small student of physics, but as far as I know, Newton's Third Law states that, > "For every action there is equal and opposite reaction." Now let me tell you how I understood this, Suppose there are two bodies A & B initially at rest (w.r.t the earth), now if A makes collision with B, of course there will be some force acting on A->B (say $F_{ab}$) and according to the third law there would be some reacting force from B->A (say $F_{ba}$). And both would be numerically equal. But now just consider body A, it has some acting force on it i.e. $F_{ba}$ which is being given by body B, so won't reaction produce for that force? Is that reaction is nothing but $F_{ab}$? Now for both bodies, there should be some **static frictional force** (before movement) acting against the force which (tends to) move the body, so whenever two bodies act with each other, are there 4 force produced?


Problem with an electricity / thermodynamics assignment

I've been trying to figure this one out for a while on my own, so I'd like to ask for your help if you could offer some. The task states: > A heater made out of a wire with a diameter $R = 0.2\text{ mm}$, length > $4\pi\text{ m}$ and electrical resistivity of $0.5\times 10^{-6}\ > \Omega\;\mathrm{m}$ is connected to a voltage source of $220\text{ V}$, > sinked in the water. > > Which mass of water will it heat up from $20^{\circ}\mathrm{C}$ to > $50^{\circ}\mathrm{C}$ in the time of 10 minutes? (C of water = $4200\ > \mathrm{J\;kg}/\mathrm{K}$) I know I have the electrical properties of the wire and the thermodynamic properties of the water, but I don't know how to proceed from there. We've been studying electricity and I am not really aware how I can connect it with thermodynamics?


Aurora Effect from Galactic Wind?

The Solar Wind creates auroras due to interactions with Earth's Magnetic Field. Do we witness any similar effects from the Galactic Wind interaction with the Sun's Magnetic Field?


How does this series expand the expression?

How does $$\sqrt{R^2 + |x|^2} = R + \frac{|x|^2}{2R}+\dots$$ when expanded around the point $x=0$? I tried using a Taylor expansion but it didnt work out.


What is a measurement in MWI

I was reviewing the MWI, and couldn't figure what a measurement is in the MWI? Everett claims that split happens when you measure, but what is it for MWI (not for general QM)? Yes, I know about decoherence, but here I am talking specifically about the split, and how MWI considers the measurement. Also, decoherence is just a part of the problem. I am talking about actual collapse or a split. If an apparatus is of an intermediate size (neither a macroscopic nor a microscopic), how do we handle 'half-splits'? Since MWT is an interpretation and not just a postulate, this should be handled somehow. This question is not a duplicate of question about general measurements in QM like this one: What constitutes an observation/measurement in QM?


ket vector with two "entries"

This is a very simple question. I am learning about angular momentum. In my lecture notes, the symbol $|\lambda,m_l \rangle$ was defined as a eigenfunction of a central potential. Two assumptions are introduced: $L^2 = \lambda \hbar^2$ and $L_z = m_l \hbar$. So, $$ \hat{L}^2 |\lambda,m_l \rangle = \lambda \hbar^2|\lambda,m_l \rangle \\\ \hat{L_z}|\lambda,m_l \rangle = m_l \hbar|\lambda,m_l \rangle $$ But what does $|\lambda,m_l \rangle$ mean exactly? I am comfortable with $|\psi \rangle$, but I do not understand what having two variables in the ket means.


How much merit is there in the heuristic argument of bulk-edge relation for topological insulators?

Take 2D quantum hall insulator for example. The typical argument goes like this: We have a Hamiltonian that has translation symmetry in both directions on a infinite lattice, and we assign a integer number(Chern number) to each of its band. The Chern number is a topological invariant(against homotopy, I suppose), hence remains constant if the Hamiltonian is varied continuously and as long as it is well defined. **Attach** a the quantum hall insulator(which has non-zero Chern-numbered bands) to a trivial insulator(of which all bands have 0 Chern number), and the band of the former must be concatenated with the latter. However this means the well-definedness of Chern number has to break down somewhere, or else one band could have two different Chern numbers. The place where the well-definedness breaks down must be a band gap closing, hence we have conducting (edge) states. My question is, the moment we decide to **attach** two insulators together, we must have broken the translation symmetry along at least one direction, so Chern number is already not well defined, how can we proceed with the argument that follows after? I am aware that there exists more serious arguments about bulk-edge relation like the ones by Yasuhiro Hatsugai, but I haven't got around to read it and I will in the near future. But can someone comment on the previous heuristic argument I phrased? Just how hard is it to make it a rigorous argument, if possible at all? Of course if I'm told Hatsugai's paper is exactly formalizing this heuristic argument I'll get to read it as soon as possible.


Slowing of time under gravity

I am not calling this gravitational time dilation because that is a relativistic effect due to the equivalence principle. Now imagine two light clocks (a clock that ticks due to light) are placed above the surface of the earth. One is placed at sea level and one very high. You might be beginning to think that I am copying Einstein but please bear with me. Consider the following image: ![enter image description here](http://i.stack.imgur.com/07OVX.jpg) The left part of the image shows the light clock placed closer to the earth's surface and the right part shows the clock placed very high. We do not need exact height as we are not doing any calculations. One light ray is emitted from the left end of the clock on the left of the image. Now since it is closer to the earth, it will bend more than the light ray in the other clock and take a longer time to reach the receiving end. The above clock (which is the clock on the right part of the image) is away from the earth and experiences less gravity. Therefore the light ray will bend less and reach the receiving end faster. Just imagine that the clocks are fixed in position and are not moving. They are only ticking when the light ray reaches the other end. Even GR predicts that a clock will run slower in high gravity but doesn't actually give a reason. Won't this also lead to the slowing of time in a higher gravitational area (where the force of gravity is more)? Can this be an actual reason for gravitational time dilation or not? **NOTE** The clocks are not falling. The light ray inside them will still bend as it is not being held in position by anything. Only the clocks are fixed. How can a light ray be fixed? So do not think that the clocks are falling.


How classical chaos can be described quantum mechanically?

How can we describe the chaotic properties of classical systems using quantum mechanics when the Schrodinger equation that describes quantum dynamics is linear? How can we use quantum mechanics that basically underlies on such a linear equation to describe highly nonlinear chaotic properties of classical systems?


If we have a current $I$ flowing down a wire, why must the net bound current be zero?

Say we are dealing with a wire that has a current $I$ flowing through it, i.e. $I$ is the free current. Why must it then hold that the net bound current, that is, the bound volume current, $J_b$, and the bound surface current, $K_b$, adds up to zero? I cannot get an intuitive grasp on why this is true, especially considering the fact that if you, on the other hand, have a uniformly magnetized cylinder, the net bound currents do not add up to zero, since there's no bound volume current, but there is a bound surface current. Similarly, in a solenoid you also get a bound surface current, but there's no bound volume current, so the two again do not add up to zero. Any help with this would be greatly appreciated, since I'm just stuck in trying to wrap my head around this. I can calculate the damn things, but I have no idea why this would be the case on a qualitative level.


Playground of Forces

Why is it that the gravitational force acts on large sized objects while the strong and weak nuclear forces act at subatomic levels only? What is that stops each other to enter each others domain?


How can I split a resultant force into its $x$ and $y$ components?

> Point charge 3.5μC is located at x = 0, y = 0.30 m, point charge -3.5μC is > located at x = 0 y = -0.30 m. What are (a)the magnitude and (b)direction of > the total electric force that these charges exert on a third point charge Q > = 5.0μC at x = 0.40 m, y = 0? ![Charges](http://i.stack.imgur.com/4sdvH.gif) I first drew a diagram, showing that it can be broken down into two .3, .4, .5 triangles, leaving me with a theta of 36.87°. I then calculated that because the charges are equal, each individual force enacted on charge Q is .63N. I know I need to split the force into it's x and y components to find the resultant vector using R=√Rx^2 + Ry^2, but I do not know how to split these forces up.


Quantum Yang-Mills Theory and AdS/CFT

I just read the first chapter of Becker-Becker-Schwarz. To quote: > A remarkable discovery made in the late 1990s is the exact equivalence (or > duality) of conformally invariant quantum field theories and superstring > theory or M-theory in special space-time geometries. Can this AdS/CFT duality be used to work in string theory or M-theory instead of quantum Yang-Mills theory, make the corresponding theory rigorous, and then go back to quantum Yang-Mills theory, thereby solving the open Millenium Prize question?


Energy/Momentum required to deflect earth from its orbit

This question occurred to me when thinking about the firepower people have on earth. How hard is it to change the characteristics of the orbit of the earth when an explosion happens on its surface, using conservation of momentum? I know already that the orbit of the earth is not so accurate already and fluctuates a lot, causing the so-called "leap second". So my question is: How could such a thing be measured/quantified? Can we say something like: The explosion energy required to deflect the eccentricity of the earth's orbit by 1%. And how much would that be? Any ideas?


Solving the Schrödinger equation where the initial wave function is an energy eigenfunction

I was watching Allan Adams' lecture on energy eigenfunctions, and there's one part (around 43 minutes into the lecture) that confuses me. Suppose we have the initial wave function $\Psi (x,0)$ such that $\hat{E}\,\Psi (x,0)=E \,\Psi (x,0)$ for some constant $E$. Then, plugging this into the Schrödinger equation, we'd get: \begin{align} i \hbar \frac{\partial}{\partial t} \Psi (x, 0) &= E \, \Psi (x,0) \\\ \frac{\partial}{\partial t} \Psi (x, 0) &= \frac{E}{i \hbar} \, \Psi (x,0) \tag{1}\\\ \therefore \Psi (x, t) &= \exp\left({-i \frac{E\,t}{\hbar}}\right) \Psi(x,0) \tag{2} \end{align} I'm a bit confused about how to go from $(1)$ to $(2)$. Now if we make the additional assumption that $\hat{E}\,\Psi (x,t)=E \,\Psi (x,t)$ for all $t$, then the Schrödinger equation becomes: \begin{align} \frac{\partial}{\partial t} \Psi (x, t) &= \frac{E}{i \hbar} \, \Psi (x,t) \end{align} and I can solve this differential equation easily and get $(2)$. But from watching that part of the lecture, it seems we only need to assume a weaker statement - that the initial wave function is an energy eigenfunction. But then, it's not clear to me how I can get the solution $(2)$ from $(1)$. Am I missing something? Update: Thanks for all the answers. After reading through the accompanying lecture note, we indeed need to assume that the energy operator is a constant over time.


Physical Chemistry: What's the relationship between orbital overlap and barrier shape?

This is a question for the physical chemists out there. For a given chemical reaction there is a barrier to traverse in order to proceed from reactants to products. Reactants e.g. an organic base abstracting a proton can approach each other at different orientations - attack angles. At times, this orientation will lead to better orbital overlap between the lone pair on the base and the s orbital of the hydrogen. I have heard people talk about better orbital overlap, lowering/narrowing the barrier to reaction. Is this a valid way of conceptualizing this? Does better orbital overlap change the barrier? I personally have always associated a better overlap as starting further along the reaction coordinate/barrier, not the actual barrier itself being altered? Any thoughts from the physchems?


Are dangerous rays emitted during Solar Eclipse?

It is said one should avoid staring at Sun as it can damage the eyes, but it is also said that one should not come out in sun during eclipse as it emits dangerous rays. Is that true? If yes, why?


Finite Square Potential Well Solving

I've got a question about solving the finite square well. Consider a 1D finite, square, potential well where: $$ V(x) = \begin{cases} 0&\,{\rm if} x<0 \\\ -V_0&\,{\rm if}\,x\in\left(0,L\right) \\\ 0&\,{\rm if} x>L \end{cases} $$ When applying the Schrodinger Equation to Zone II ($V(x)=-V_0$), it will be necessary to write a constant $k$, which my book has defined this way: $$k^2= - 2m\frac{E+V}{\hbar^2}$$ My Question is: Can we write this as $$k^2=2m\frac{E+V}{\hbar^2} $$ Shouldn't we get the same solution? If not, why not? The final solution should not depend on which way we write $k$, correct?


B3LYP vs PBE functionals for conjugated organic systems

Two of the most popular (exchange and correlation) functionals for density functional theory are B3LYP and PBE. Out of the people I've worked with / learned from, mostly the computational chemists were using B3LYP, and the physicists were using PBE. Now, different functionals have different strengths and weaknesses, but these two functionals seem to have a lot of overlap in their applicability. Right now I'm interested in ground-state properties of molecules with extended pi- frameworks in the gas-phase. For example, I'd like to compute the geometries, ground-state electron densities, and molecular polarizabilities of the acenes. Does anyone have concrete information or references on the performance of B3LYP or PBE in this situation, or in general?


Integrating equations with units

I was looking through an old copy of Barron's AP Physics and found this problem relating to impulse which I was initially confused about how to integrate. > **Example 6.1** During a collision with a wall lasting from $t=0$ to > $t=2\text{ s}$, the force acting on a $2\text{-kg}$ object is given by the > equation $\mathbf{F} = (4\mathrm{\ kg\ m/s^4})t(2s-t)\hat{i}$ They work out that the integral is equal to: $$\frac{16}{3}\hat{i}\frac{\text{kg m}}{\text{s}}$$ I am confused about the role of the units in the problem. Looking at the answer, it seems that if I were to just ignore all the units and simply integrate $4t\cdot(2-t)$ that would give me $16/3$, and because that is a force I know it to be $\mathrm{kg\,m/s}$ or $\mathrm{N}$. Why it would be OK to ignore the units in the integral though is somewhat unintuitive to me (other than that I know the end result must be a force) and I feel like it may get me into trouble with other problems. Can someone explain how it is that the scalar values of the original unit quantities still give you the correct answer?


Measurement in reciprocal metres

I'm trying to name a measurement that is measured in reciprocal length, which is in a draft document for vehicle risk management. It currently says:               Crash rate = number of crashes / million kilometres driven      where the measurements are taken annually. So the units are in reciprocal length. I'm not happy with the word "rate" because that seems to imply the number of events over time, whereas the measurement is the number of events over distance. Existing names of units of reciprocal length in various disciplines don't seem to work - it's not a measurement of any cyclic phenomenon. Is there an better word for this measurement?


Why does a CD work as a diffraction grating even with light from a light bulb?

I have a really hard time solving the following problem I accidentally came across today. Looking at the back of a usual CD one sees coloured bands. This is explained by the fact that the surface (pits and bumps) is effectively a diffraction grating. But why is this effect already visible when one uses _incoherent_ light, e.g. from a light bulb, whereas the interference pattern of a single or double slit would be missing with such illumination?


Writing a Wavefunction as a Linear Combination of Eigenstates

We have the following wavefunction for the hydrogen atom: $$\psi(r,\theta,\phi)=\frac{1}{\sqrt{4\pi}}\frac{1}{(2a)^{3/2}}\frac{r}{a}e^{-r/2a}\sin(\theta)\sin(\phi)$$ where $a$ is the Bohr radius. `Question`: How can I express the wavefunction above as a linear combination of the eigenstates of the hydrogen atom's Hamiltonian? In other words, we need to express the above in terms of a linear combination of $$\psi_{nlm}=\sqrt{\left(\frac{2}{na}\right)^3 \frac{(n-l-1)!}{2n[(n+l)!]^3}}e^{-r/na}\left(\frac{2r}{na}\right)[L_{n-l-1}^{2l+1}(2r/na)]Y_l^m(\theta,\phi)$$ For some $n$, $l$, and $m$'s. How can I find such a linear combination without just a simple guess and check; i.e., what is a systematic way to find the linear combination?


What laws (formulas) govern forces between atoms?

What laws (formulas) govern the fundamental forces of nature? For example, gravity is governed by the inverse-square law. I am thinking about how particles attract each other, but also repel. All matter attracts via gravity, but also attract and repel each other due to the electromagnetic force, also governed via the inverse-square law, and then there's the strong and weak force. I understand the idea that two hydrogen atoms repel, but when pushed together hard enough will attract and form helium and release energy. So this "upwards slope" of repulsion which is eventually "snaps" into attraction, like trying to roll a ball up a smooth hill with a hole in the center of it, is explained, likely, by varying forces governed by various laws. Can you help me understand it, **on the level of particles**? Where I am stuck is thinking about two particles, each being attracted and repelled by two different forces each governed by the inverse-square law. This doesn't explain the "snapping" together of atoms. Do I need different math in my forces, or is it ultimately explained by more than just two particles? Here is a diagram of what I mean, describing H + H -> H2: ![enter image description here](http://i.stack.imgur.com/3YesO.gif) _Image source and article:http://employees.csbsju.edu/hjakubowski/classes/ch111/olsg- ch111/equilibkinetics/equilbkin.htm_ Please feel free to reword my question if you understand what I am asking but am not making myself clear.


Irreducible tensor representations with "covariant" indices

As a follow-up of my question on the "most general" $\mathrm{SU}(2)$-symmetric interaction of two spin 1/2 particles, I ponder the following question: Consider an operator acting just on one particle of spin 1/2, and just on the spin part. In second quantization, it can be written as $$\sum_{\alpha\beta} \psi^\dagger_\alpha V_{\alpha \beta} \psi_\beta$$ Now, under $SU(2)$, the operators will transform as $$\psi^\dagger_\alpha = \sum_{\alpha'} \psi^\dagger_{\alpha'} D_{\alpha\alpha'}$$ with $D \in SU(2)$. I can move this transformation to the operator and conclude that it transforms like $$V_{\alpha\beta} \rightarrow \sum_{\alpha'\beta'} D_{\alpha\alpha'} V_{\alpha'\beta'} D_{\beta\beta'}^\dagger$$. (I might have gotten some of the daggers and indices backwards, but the important point is that one matrix is the adjunct of the other). Now if I understand correctly, this allows me to conclude that $V_{\alpha\beta}$ transforms "reducibly like a tensor product of two spin 1/2 representations", i.e., like $\frac{1}{2} \otimes \frac{1}{2}$. From that, I conclude that $V$ should decompose into one component that transforms like a spin $0$ particle and one component that transforms like a spin $1$ particle. Indeed, as $V$ is a $2\times 2$ matrix, I can write it as a linear combination of the unit matrix and the pauli matrices. The former transforms like a scalar, i.e., like spin $0$, whereas the latter transform like vectors. However, I have trouble relating this to what I know about combining spin $1/2$ particles using Clebsh-Gordan coefficients, where I have, e.g., for the singlet $$|0 0\rangle = \frac{1}{\sqrt{2}} \left( \uparrow \downarrow - \downarrow \uparrow\right)$$ Because of this conceptual problem, I also have trouble generalizing this to the case of two interacting spin 1/2 particles, which then should lead to an interaction that transforms reducibly as $1/2 \otimes 1/2 \otimes 1/2 \otimes$ and gives rise to two different singlets. I'd appreciate it if someone could disentangle my misconceptions... EDIT: I forgot to specify what I mean with "covariant" in the title: I think an important thing to notice is that the matrix elements $V_{\alpha\beta}$ are **not** the 4 elements of a "cartesian" tensor of rank 2. The entire operator $V$ might be an element of a tensor of higher rank, or even the sum of elements of tensors of different rank. A simple example for such a thing would be an operator "1 + x", which is the sum of a rank zero tensor (scalar) and the element of a cartesian tensor of rank 1. Now, what I mean with covariant is that one of the indices of $V_{\alpha\beta}$ transforms with matrix $D$ and the other with matrix $D^\dagger$. Also important is that the components of the operator transform with the actual $SU(2)$-matrices and not with some rotation matrix $R \in SO(3)$. I guess this is why I have trouble translating the standard literature on tensor operators to my situation...


Does a photon travel through space exactly at the speed of light?

In practice, given that in the vacuum of the universe is not perfectly empty, does the photon actually reach exactly c? Is it even possible? If it travels at exactly c, then from the photon perspective, it arrives at the same time it leaves its point of origin, so for him the distance between the 2 points is zero? Many thanks.


Centrifugal force?

Everyone calls it a pseudo force and that it is not really present, even there are so many machines listed to advantage of centrifugal force ( centrifuge, washing washing drier),and also it has the same magnitude as centripetal force, so that they cancel cancel each other out but i can definitely feel it every time a sharp turn is taken in the car. So can you explain what centrifugal force is exactly, i don't understand it.


How can Drift Velocity of an electron causes change?

Assume we have a conductor that has current flowing and its placed in a magnetic field so that it experiences the Lorentz force, and it gains Kinetic energy due to that force. Would drift velocity change the electron's direction thus the current's total direction changes, therefore, changing the Lorentz force direction applied on the conductor?


How is stable levitation possible?

This question is with reference to the video in this blog post: http://www.universetoday.com/90183/quantum-levitation-and-the-superconductor/ My question is the following: how is the disc stable in its levitated position? Specifically, 25 seconds into the video, the exhibitor turns the entire thing upside down, and the disc doesn't fall. This contradicts two intuitive ideas I have:   1. Right-side-up, gravity is counteracting the repulsive effect of the magnet. Upside down, gravity is working with it. Unless there's some adaptation going on somewhere else, shouldn't "gravity switching sides" be enough to tear the disc away from the magnet?   2. I remembered something about "inverse-square" forces not permitting stable equilibria from college physics - sure enough Wikipedia calls it Earnshaw's theorem. I don't see how this is exempt from that rule. I guess I don't understand diamagnetism.


Hamilton-Jacobi formalism and on-shell actions

My question is essentially how to extract the canonical momentum out of an on- shell action. The Hamilton-Jacobi formalism tells us that Hamilton's principal function is the on-shell action, which depends only on the coordinates $q$. Therefore, if $S$ refers to the off-shell action with Lagrangian $L$, a result for the canonical momentum is $$ \frac{\partial L}{\partial \dot q} = p = \frac{\partial S^\text{on- shell}}{\partial q} $$ In theory, when evaluating the on-shell action, one plugs in the EOM into $S$ and all that's left are boundary terms. If we illustrate this with the harmonic oscillator in 1D, one gets \begin{align} S^\text{on-shell} &= \int_{t_i}^{t_f} L dt \\\ &= \frac{m}{2} \int_{t_i}^{t_f} \left( \dot q^2 - \omega^2 q^2 \right) dt\\\ &= \frac{m}{2} \int_{t_i}^{t_f} \left( \frac{d}{dt} (q \dot q) - q \ddot q - \omega^2 q^2 \right) \\\ &= \frac{m}{2} \left[ q(t_f) \dot q(t_f) - q(t_i) \dot q(t_i) \right] \end{align} Now, this depends on $\dot q$, whereas it should depend only on generalized coordinates. Also, the on-shell action depends on the boundary values of our coordinate $q$, so we can't differentiate w.r.t. $q(t)$ to get a conjugate momentum $p(t)$ for all times, like it is usually done in the Lagrangian formalism. What am I not seeing?


Fierz identity for Weyl spinors in tensor currents

Using Fierz identity I found that certain four-fermion operator with left $l_i$ and right-chiral $r_i$ Weyl spinors vanish $\bar{l}_1\sigma_{\mu\nu} r_2 \bar{r}_3 \sigma^{\mu\nu} l_4 =$ $ -\frac{3}{2} \bar{l}_1 l_4 \bar{r}_3 r_2 - \frac{1}{2}\bar{l}_1\sigma_{\mu\nu} l_4 \bar{r}_3 \sigma^{\mu\nu} r_2$ $ - \frac{3}{2} \bar{l}_1 (-1) l_4 \bar{r}_3 (+1) r_2 =$ $= -\frac{1}{2}\bar{l}_1\sigma_{\mu\nu} l_4 \bar{r}_3 \sigma^{\mu\nu} r_2 = 0$ Why does this operator vanish? Is that true? Why is the product of tensor currents expressed in tensor and scalar currents only, while for such combinations of Weyl spinors $l_1$, $r_2$, $r_3$ and $l_4$ in the Fierz identity all of them vanish?


Does a current carrying wire produce electric field outside?

In the modern texts of electromagnetism in the presence of stationary currents the electric field is assumed conservative $\nabla \times E =0 $. Using this we get $E_{||}^{out}=E_{||}^{in}$ which means we have the same amount of electric field just outside of the wire! Is this correct? Is there any experimental proof?


How to use an accelerometer to correctly quantify the magnitude of vibrations

Consider the following setup, an accelerometer is placed inside a helicopter, measuring "g" values along 3 axes (x,y,z) every 10 ms (100 times a second), during a helicopter flight. If one, wants to obtain some measure of the maximum g force exerted on an object inside the helicopter, what would be the best way to do this? Make the analysis for each axis independently? Or treat x,y,z as components of a vector ? Is it correct to consider the maximum difference between two consecutive measurements as an approximation of the maximum g force exerted on an object inside the helicopter


Naïve relativistic schrodinger equation

> **Possible Duplicate:**   >  Why are higher order Lagrangians called 'non-local'? Bjorken and Drell presents the equation: $$i\hbar\frac{d\psi}{dt}=H\psi=\sqrt{p^2 c^2+m^2 c^4}\psi=\sqrt{-\hbar^2 c^2 \nabla^2+m^2 c^4}\psi$$ The squareroot can be expanded to obtain an equation with all powers of the derivative operator. What do they mean when they say this leads to a non-local theory? And is this equation incorrect or just impractical?


Is there an equation for convective heat transfer?

Is there an equation I can use to calculate the temperature (as a function of time) of an object which is gaining or losing heat by convection? Or equivalently, the rate of energy transfer from the object to the surrounding fluid (or vice-versa)? It can involve constants representing properties of the object and the surrounding fluid. I know about the heat equation for conduction and the Stefan-Boltzmann law for radiation, but I can't think of one for convection.


compact and non-compact gauge theory

From the answer to this question, I see that the gauge theory has to be compact if the charges need to be quantized. I am not sure in what sense these are necessary and sufficient. So here is a question I had: In electromagnetism, the electric charges are quantized in units of electronic charge, so shouldn't the gauge fields $A$ have values on a compact manifold rather than be a real number ?


What happens to orbits at small radii in general relativity?

I know that (most) elliptic orbits precess due to the math of general relativity, like this: ![precession](http://i.stack.imgur.com/H8l5b.png) source: http://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity I also know that something is different for orbits at radii below a certain value. Wikipedia explains this as follows, I'm confused by this and just want to ask for clarification: > _If the particle slips slightly inwards from $r_{inner}$ (where all three > forces are in balance), the third force dominates the other two and draws > the particle inexorably inwards to $r=0$._ What does this mean? If you drew a graph of the path a particle takes in this unstable regime, what would it look like? Why is the stability transition point further than the Schwarzschild radius? Why does this graph show an unstable point beyond the Schwarzschild radius? For elliptical orbits that come very close to the event horizon, is there some kind of orbital decay? How is energy conserved? In short, do all orbits (with GR effects) look like the precession shown above, or is there another shape that we see if it gets closer to the Schwarzschild radius?


Stability of balanced masses with different surface areas

![enter image description here](http://i.stack.imgur.com/Eze3s.png) Say I have this setup. The two round objects have equal mass and their centers of gravity at the same distance from the shaft. The objects only differ in that they have different surface areas (assume they are both flat-faced). If this apparatus is spun, will it fall over? I believe it won't. The centrifugal forces are balanced since they only depend on mass, and the aerodynamic drag will be in the direction of motion, so it won't pull the shaft in any outward direction.


What starts the movement of a photon

Although a photon has no (rest) mass, it does have a measurable speed. Its movement can be altered by gravity. A photon "travels". If I turn on a flashlight, seen by someone at a distance, the photons from that flashlight traveled, at the speed of light, from the flashlight to the observer. But what "force" instantly starts the movement of a photon at the speed of light?


Moving from point A to point B in zero-gravity

Here's an proposal on how to get from point A to point B in zero-gravity without using any propellant and the question why it wouldn't work: A closed tube, filled with water and a round (solid) object. If you need equations, the volume of enclosed water is the same as the round object, but the round object is 10 times lighter. (imagine a glass with water plus a ping- pong ball). here's a picture: ![alt text](http://img210.imageshack.us/img210/6640/sprite143.jpg) On earth the round object will float on the water inside the tube (subject to one G). In zero-gravity the round object has no preferred position. If we accelerate the tube in zero-gravity by one G, the situation is the same as on earth, the round object "floats". In this example we are accelerating the tube from the left side to the right side. The round object will consider floating to the right AS LONG AS THERE IS acceleration. Now consider adding a pipe to the bottom of the tube and connecting it to the top, a loop. Inside the pipe is a small water pump. If we give this apparatus a push, say one G in a zero-g environment, the round object will move "up", but now we start the water pump and spray the water on the round object, we try to submerge it. It will resist and impart a impulse on the water. like trying to hose down a air balloon floating on the pool. The pump will feel a resistance and hence the whole apparatus will move. Just running the pump at constant speed, same volume of water per second, will do nothing. but if we run the pump faster and faster the whole apparatus will start moving: the amount needs to be geometrical. The point is that we need to keep the apparatus feeling an acceleration, since only then will the round-object "float" and resist the incoming water at the top, hence we have something to "push on". Before you blow the "foul whistle": consider the situation if there were no round floating ball in the apparatus. (the pump runs on solar power or pre-charged battery) (disclaimer: i know standing on a sailboat and blowing into the sail will not get me anywhere, action<->reaction) thanks a bunch Sklivvz for the edit. sometimes the idea just needs to get out, never the mind how it looks like : P


Rotation Group and Lorentz Group

It is often stated that rotations in the 3 spatial dimensions are examples of Lorentz transformations. But Lorentz transformations form a group named the Lorentz Group, $O(1,3)$ which is a group a $4 \times 4$ matrices, $\Lambda$ having the following property: $$ \Lambda^T g \Lambda = g$$ where $g$ is the metric tensor. Now rotations matrices for the 3 spatial dimensions are $3 \times 3$ matrices and form $SO(3)$. How can they be in the $O(1,3)$ ?


Are electrons moving? If so relative to whom? and why is there no magnetic field?

electrons on earth **moving** with us, due to rotation of earth, revolution of earth, sun and our galaxy right?   Then, why is there **no** magnetic field around a **piece** of copper wire?


How to apply Andreev reflection formalism to ferromagnet ,normal metal interface?

The traditional formalism for andreev reflection deals with what happens at normal metal, super conductor interface.http://en.wikipedia.org/wiki/Andreev_reflection (i.e when an electron from normal metal is incident on the super conductor interface, a hole is generated in the normal metal which lets the formation of a cooper pair in the SC which propagates in the appropriate direction) How do we apply a similar formalism when an electron of say up-spin is incident from a normal metal to half-metal (magnet) ? what happens when this electron comes spin polarized at an angle to the interface ? note that unlike in superconductivity, there is a direction associated with the half metal (which is basically a magnet with an orientation)


Topological disorder in condensed matter?

What is meant by topological disorder in condensed matter (both crystalline and amorphous)? For example, please see the following two papers from arxiv.org http://arxiv.org/pdf/0906.3848.pdf http://arxiv.org/pdf/cond-mat/0308612v1.pdf Thanks


Angular velocity $\omega$ by $v$

We have two girls, with mass ($M$). They become close to each other in speed of $V$. The distance between them is $3L$. ![](http://img2.timg.co.il/forums/1_159160419.jpg) I was asked to calculate the Angular velocity ($\omega$) of the two girls. So I set the rotation axis in the middle (where the distance between the two girls and the axis is $1.5L$, and I calculate the Angular velocity using this equation: $\omega=v/r$, where $r=1.5L$, and I got that $\omega={2v\over 3L}$. As I understood, the answer is correct, but this is not the correct way. What is my mistake? (sorry about my english)


What are the theoretical explanations behind the "no-boundary condition" in cosmology?

In his book "The Grand Design" on page 167-173 Stephen Hawking explains how one can get rid of the so called "time zero issue", meaning by this the very special initial state needed for inflation to work (?), by applying the so called "no-boundery condition" which says that time behaves as a closed space dimension for times small enough such that the universe is still governed by quantum gravity. To see better what Hawking is talking or writing about I`d like to understand this a bit more "technically" and detailed, so my questions are: 1) How can situations where QG kicks in lead to a transition of the time dimension to a space dimension? I mean what is the physical reason for this "ambiguity" of the signature of the correspondng spacetime metric? 2) What are the triggering causes of the "symmetry breaking" between the 4 space dimensions when starting from the no-boundary condition such that one of the space dimensions becomes an open time dimension at the beginning of time? 3) Are there (quantum gravity) theories or models that can explain 1) and 2) and if so, how? Since I've just read about these issues recently in the "Grand Design" (and nothing else) I`m probably confused and hope that somebody can enlighten me by explaining some details ...


Space and time translation of quantum field

In book I am studying, the authors want to calculate the spectral representation of the g-lesser (and g-greater) functions and find a general expression for these. In order to do so they start with the expression for the fields in Wigner coordnates (translated in time and space) $\hat \psi \left( \mathbf R + \frac{\mathbf r}{2} , t + \frac{\tau}{2} \right) $. They then claim that this may be written like $ \hat \psi \left( \mathbf R + \frac{\mathbf r}{2} , t + \frac{\tau}{2} \right) = \exp \left[ \frac{i}{2} \left( \hat H \tau + \hat{\mathbf P} \cdot \mathbf r \right) \right] \hat \psi \left( \mathbf R , t \right) \exp \left[ - \frac{i}{2} \left( \hat H \tau + \hat{\mathbf P} \cdot \mathbf r \right) \right] $ where $H$ is the Hamiltonian and $\hat{\mathbf P}$ is the total momentum operator. What are the underlying assumptions for the equation above to be true? One obvious thing is that the Hamiltonian must not be explicitly time dependent (since the time translation operator then gets a more complicated expression involving the time ordering operator). Is it for example understood that the total momentum operator and the Hamiltonian are commuting? If the Hamiltonian and the momentum operator do not commute then since $\exp \left[ \frac{i}{2} \left( \hat H \tau + \hat{\mathbf P} \cdot \mathbf r \right) \right] \neq \exp \left[ \frac{i}{2} \hat H \tau \right] \exp \left[ \frac{i}{2} \hat{\mathbf P} \cdot \mathbf r \right] $ the order of which the time and space translations are made matter. Perhaps someone could also help me clarify what the meaning of this is...


Is it possible to derive fermi-dirac or bose-einstein statistics using quantum operator formulations?

I've been looking through theory on identical particles to get a better grasp of the uncertainty principle but it would be very interesting if these results could be extracted from the formalism as well. Does anyone have any leads on this?


Motion of elevating airplane

I wonder how the motion of an airplane which will not try stop to elevate should look like. Of course, at some point the plane will no longer be able to go to higher altitudes but is it going to stay on a circular trajectory or things are more involved?


Why is ski jumping not suicidal?

At least on television, ski jumpers seem to fall great vertical distances before they hit the ground - at least a few dozen meters, though I couldn't find exact distances via a quick search. And yet they almost always land on their feet as if they just fell two or three meters. (Here's a whole lot of footage from the Vancouver Olympics if you need to refresh your memory.) Without going into the level of equations (which I wouldn't understand), why are ski jumpers able to fall such great heights without seriously injuring themselves?


Bose-Einstein-statistics out of fermionic many body system

ok, let me try this again. How do I get to atoms obeying Bose Einstein statistics from considering the fermionic many body problem of a bunch of electrons, protons and neutrons forming these (bosonic) atoms. [ORIGINAL QUESTION] this keeps bugging me: Whenever there is a system of bosons (lets not consider photons/phonons) then we use BE statistics to describe it. But shouldn't we also be able to describe it (more accurately) by FD statistics with regard to its constituent electrons, protons and neutrons? How is this consistent with the symmetry requirements on the wave function and phenomenas as Bose Einstein condensation?


How to derive the Karman-Howarth-Monin relation for anisotropic turbulence?

I find the derivation of the Karman-Howarth-Monin relation in the book Turbulence from Frisch (1995) a bit to short. Can someone point me to a more detailed derivation of that relation, if possible in a publicly available source (lecture notes, thesis, arXiv paper)? Or can someone explain to me how Frisch obtains the equation that follows the sentence on page 78: > Starting from the Navier-Stokes equation (6.6), we obtain $$\begin{align} > \partial_t \frac{1}{2}\langle v_i v'_i \rangle =& \- \frac{1}{2} \partial_j > \langle v_i v_j v'_i\rangle \- \frac{1}{2} \partial'_j \langle v'_i v'_j > v_i\rangle \\\ & \- \frac{1}{2} \langle v'_i \partial_i p \rangle \- > \frac{1}{2} \langle v_i \partial'_i p' \rangle\\\ &\+ \frac{1}{2} \langle > v'_i f_i \rangle \+ \frac{1}{2} \langle v_i f'_i \rangle \\\ & \+ > \frac{1}{2} \nu \left( \partial_{jj} + \partial'_{jj} \right) \langle > v_iv'_i\rangle. \qquad (6.11) \end{align} $$


Calculation of a bending moment

I'd like to calculate the bending moment of a cantilever, fixed at its base, and submitted to a certain stress on a specific spot, but I can't find the proper definition of this bending moment (first time I'm encountering this name, and google didn't not help me so far).


How can I interpret negative values of potential evapotranspiration?

If I extract Potential Evaporation (PET, W/m$^2$) from the National Centers for Environmental Prediction (NCEP), climate reanalysis data (downloadable as netCDF files here), there are some negative values, e.g. for summer in southern Manitoba (50N, 100W), I get the attached distribution of PET values.: ![enter image description here](http://i.stack.imgur.com/G6Gct.png) Is this a real physical process, such as condensation (dew formation), or is it an error (e.g. should I average over these values or set them to zero)? * * * **update** requested metadata: The metadata (contained in the files) says that the valid range is -800, 5200; here is further description of the 'pevpr' variable (provided in the data header):   * long_name = "Monthly Mean of Potential Evaporation Rate"   * valid_range = -800,5200 ;   * units = "W/m^2" ;   * add_offset = 0 ;   * scale_factor = 1 ;   * missing_value = -9.96921e+36 ;   * precision = 1 ;   * least_significant_digit = 0 ;   * var_desc = "Potential Evaporation Rate" ;   * dataset = "CDC Derived NCEP Reanalysis Products" ;   * level_desc = "Surface" ;   * statistic = "Mean" ;   * parent_stat = "Individual Obs" ;


Is there a relation between supersymmetry and entropy?

Considering that entropy denotes the level of order/disorder in a system, would it be possible for entropy and supersymmetry to exist at the same time? Or, are they entirely unrelated?


Excitation probability for 2 Level Atom

I currently started learning Quantum Optics using the book "Atom Physics" of Christopher Foot. Currently I am working through Chapter 7 and a question occured concerning the intuitively physical understanding: Assume a two level atom with energy levels $E_{1}$ and $E_{2}$ (with $E_{1}$<$E_{2}$ is given). The eigenstates of the system are given by $|1\rangle$ and $|2\rangle$ respectively. Now assume that we perturbe the system by using a time dependent Electric field of the form \begin{equation} \vec{E} = \vec E_{0} \cos\left(kz - \omega t\right) \end{equation} and expand the solution of the Schroedinger Equation in terms of the unperturbed eigenstates \begin{equation} \Psi(\vec r,t) = c_{1}(t) \ e^{-i (E_{1}/\hbar) t} \ |1\rangle \+ c_{2}(t) \ e^{-i (E_{2}/\hbar) t} \ |2\rangle \end{equation} Let's assume that $c_{1}(0)=1$ and $c_{2}(0)=0$. Using the dipole approximation as well as the rotating wave approximation as discussed for example in www.physics.ox.ac.uk/Users/lucas/BIII/QuantumNotes.pdf, we obtain that \begin{equation} |c_{2}(t)| \sim t^{2} \ \frac{\sin^{2}(x)}{x^{2}} \end{equation} with $x=(\omega-\omega_{0})t/2$ and $\hbar \omega_{0}:=E_{2}-E_{1}$. Plotting this as a function of $\omega-\omega_{0}$ we obtain a "Fraunhofer-diffraction"-like graph. My Question: 1) As the interaction time increases the width of the maxima is decreasing. What is from a physical / intuitive perspective the reason for this. 2) There are also certain minima occuring for certain frequencies of the light. What is the physical interpretation (intuitively) for this? I am looking forward to your responses. Thanks in advance.


Finding the mass of 7cm string

If a string has $120$ g and length $40$ cm. I found that the linear mass density is given by $$120g/40cm=\ linear\ mass\ density$$ How do I use this to find the mass of a $7$cm of the same kind of string? * * * What I have done so far: $$120g/40cm = m/7cm$$ $$m=7cm*120g/40cm$$ $$m = 21$$


Lorentz spinors of $SO(n,1)$ and conformal spinors of $SO(n,2)$

* It would be great if someone can give me a reference (short enough!) which explains the (spinor) representation theory of the groups $SO(n,1)$ and $SO(n,2)$. I have searched through a few standard representation theory books and I couldn't find any.   * More specifically I would like to know how a Lorentz spinor of $SO(n-1,1)$ (say $Q$) is "completed" to a conformal spinor of $SO(n,2)$ (say $V$) by saying,  $V = (Q, C\bar{S})$ where $C$ is a "charge conjugation operator" and $S$ is probably another $SO(n-1,1)$ spinor. Is there some natural Clifford algebra representation ($\Gamma$) lurking around here with respect to which I can define the "charge conjugation operator" as $C$ such that $C^{-1}\Gamma C = - \Gamma ^T$? (...in general a representation of the Clifford algebra also gives a representation of $SO(n,1)$..I would like to know as to how this general idea might be working here...)   * Some of the other aspects of this group theory that I want to know are an explanation for facts like,      * $Sp(4)$ is the same as $SO(3,2)$, and the fundamental of $Sp(4)$ is the spinor of $SO(3,2)$     * $SU(2,2)$ is the same as SO(4,2), and the fundamental of $SU(2,2)$ is the spinor of $SO(4,2)$ (...just two "facts" hoping that people can point me to some literature (hopefully short!) which explains the systematics of which the above are probably two examples...)


Flat space metrics

This question concerns the metric of a flat space: $$ds^2=dr^2+cr^2\,\,d\theta^2$$ where $c$ is a constant. Why is it necessary to set $c=1$ to avoid singularities and to restrict $r\ge 0$? Thanks.


canonical and microcanonical ensemble

What does one mean by **canonical** and **micro canonical ensemble** in statistical mechanics? Can one elaborate on this in a very simple way with examples? Pardon me, if it is a very simple thing; I am a math student.


Tips on teaching Dimensional Analysis?

What's a good way to explain dimensional analysis to a student? Here's a simple question which this method would be useful: Let's say a truck is moving with a speed of 18 m/s to a new speed of 13 m/s over a distance of 48 meters. How long did it take for the truck to break to its new speed, and what was the acceleration of the truck? Edit @TimGoodman and @Laar, Hmm. I was under the impression that you could solve such questions by analyzing their dimensional properties. For example, based on V1 and V0 in the example above I know the person decelerated 5m/s over a distance of 48 meters. If I want the time it took (first question) I can simply look at the units and see if I can eliminate all units except s (seconds). 48m/5m/s allows the meters to cancel out leaving 9.6 seconds. Now I have a new peice of information and I can now solve the second question "What was the acceleration of the truck?" Since I'm solving for acceleration I know my final units will be m/s^2. I know the velocity of the deceleration was 5m/s and it took him 9.6s. Dividing 5m/s/9.6s = .52 m/s^2. My final units match and so by using dimensional analysis I've solved the problem. Maybe I have a misunderstanding of dimensional analysis. Any suggestions?


Classical static equilibrium in atomic physics

Consider a collection of $n+1$ mass weighted points in $\mathbb{R}^3$. Suppose we have one mass located at the point (0,0,0) with mass $m\in\mathbb{N}$ and further suppose we have $n$ masses arranged arbitrarily each with mass -1. Now we want to assign to each mass weighted coordinate a collection of vectors in the following way:   1. There will be a total of $n$ vectors emanating from each weighted point.   2. The assigned vectors $\vec v$ will be of two varieties: $$\vec v\in\\{\frac{\hat r_{ij}}{||r_{ij}||^2},\frac{-m\hat r_{i0}}{||\vec r_{i0}||^2}\\}$$ where $||r_{ij}||$ is the norm of the vector between points $i$ and $j$, $\hat r_{ij}$ denotes a direction away from both points $i,j$ and along the line between $i$ and $j$. $||r_{i0}||$ is the norm of the vector between a point $i$ and the origin and $-\hat r_{i0}$ denotes a direction from point $i$ to the origin. I am interested in whether there exists a 3 dimensional configuration of the points where the sum of all the vectors is zero for each point. If this is not possible I hope someone would give some hints on how to prove it or point me towards a proof in the literature. In essence what I am seeking is whether there exist three dimensional equilibrium configurations for Coulomb's law of electrostatics as you come across the periodic table of the elements. I suspect not but who was the first to prove it?


Why are there equations in physics with factors of 2, 3 and 5, but there aren't any with factors of 7 or 11?

I noticed that there are a lot of equations in physics with factors of 2, 3 and 5 (either in the numerator or in the denominator), but there aren't any with factors of 7 or any prime number greater than 7. It's like the first three prime numbers (2, 3 and 5) are "more important" than the other prime numbers in physics. What is the reason for this?


Relationships between different measure of opacity

I'm reading some papers that compare different values for a materials opacity to a particular particle. The first is given as $\frac{dE}{dX}$, a single particles energy loss per unit column depth (X = $x\rho $) to a continuous process. Makes sense. So then the author goes on to compare his value to that of other authors, who have their opacities for the process expressed in $\kappa $ given in $cm^{2}/g $. They're both presented as measuring the same thing, but I usually think of $ \kappa $ as being an opacity for a large group of particles that are removed discretely from a beam, so that the total energy in the beam dies of exponentially (since energy loss per distance is proportional to the number of particles left in the beam). $\frac{dE}{dX}$ is due to a continuous process acting on all the particles, and so would have a roughly linear effect per distance. So my question is how does one compare the two values. What does $ \kappa $ mean in a context of continuous energy loss. Thanks


Why is scale factor inversely proprotional to temperature?

In all lectures, books and papers I have read about, the scale factor of the universe is inversely proportional to temperature $$a \propto \frac{1}{T}$$ What is the reasoning behind this relation? People did mention stretching of wavelength proportional to scale factor and wavelength being inversely proportional to temperature by Wein's law, but somehow this argument doesn't seem convincing to me. How did we initially come about to this relation?


Is the center of mass in general relativity equal to the center of mass in newtonian gravity?

Consider 2 point masses $A,B$ a distance $d$ away from eachother without velocity or rotation spin. Is the center of mass in general relativity equal to the center of mass in newtonian gravity ? In newtonian gravity the center of mass is where the particles collide , if they collide. Is the center of mass in general relativity also the place where the particles collide , if they collide ? Or maybe the center of mass in general relativity is the place where the particles collide if they had no initial velocity or rotation spin ?


Free energy device - QEG. Is this for real?

I'm not a physicist but I ran across reports on the web of a so called over unity or free energy device that has been created that supposedly works. It's called the QEG. Does this seem legit or is it yet another failed attempt to prove that free energy exists? Of course this goes against the 1st law of thermodynamics. I suppose those who believe in such a device would say they aren't creating energy, they are pulling energy in from the ether?


Electric potential due to a point charge in Gaussian/CGS units

I learned electrostatics in SI units. In SI, the electrostatic potential due to a point charge $q$ located at $\textbf{r}$ is given by $\Phi(\textbf{r}) = \frac{q}{4 \pi \epsilon_0 |\textbf{r}|}$. Now, the Griffiths electrodynamics textbook says, "Converting electrostatic equations from SI to Gaussian units is not difficult: just set $\epsilon_0 \rightarrow \frac{1}{4 \pi}$." So, in Gaussian/CGS units, apparently $\Phi(\textbf{r}) = \frac{q}{|\textbf{r}|}$. However, one textbook (Understanding Molecular Simulation, by Frenkel and Smit) says that the potential due to a point charge is $\Phi(\textbf{r}) = \frac{q}{4 \pi | \textbf{r} |}$. Did I make a mistake, or did Frenkel and Smit? Thank you.


Electric dipole, error in calculation

Currently I am calculating the dipole moment of a metal sphere in a uniform electric field $E_0$ in z-direction. From here I know that the charge density look at page 15 is given by $ 3 \epsilon_0 E_0 cos(\theta)$ Now I wanted to calculate the resulting dipole moment by $ \int_0^R \int_0^{2\pi} \int_0^{\pi} r^3 sin(\theta) cos(\theta) 3 \epsilon_0 E_0 d\phi d\theta dr$ but in this case the integral over $\theta$ is zero and therefore this whole term will be zero which is somewhat strange, since there should be a dipole moment. So what am I doing wrong?


How to understand the Lagrangian of the standard model, effective or "fundamental"

I have a question about understanding the Lagrangian of standard model, should we view it as a "fundamental" or effective theory? The "fundamental" theory here means the theory with physical cutoff (maybe Planck scale) $\Lambda$; the effective theory here means the theory given by lowering the cutoff $\Lambda'<\Lambda$, (Wilsonian renormalization), and we pick up the leading terms as we saw in the action. **The reason to think the standard model Lagrangian is an effective action** We haven't tested the physics up to the physical cutoff (Planck scale?), this is an extra assumption that the standard model is valid up to physical cutoff. To the contrary, in the effective theory, the cutoff is much more reasonable (e.g. LHC scale). **The reason to think the standard model Lagrangian is a "fundamental" action** If we follow the renormalization flow, the coupling constant of QCD will grow in low energy scale, then the perturbation breaks down. We know from lattice calculations of hardron mass, that even in low energy, QCD works. The Wilsonian renormalization group transformation is formulated from perturbative expansion (may be this is just my limited knownledge, correct me if I am wrong). Since QCD agree with experiment beyond the regiom of perturbation, it has to be a "fundamental" action, than effective action. Similar attitude applies to the standard model. It seems the question is rather a mental setup. Since standard model works very well so far at LHC, it seems hard to distinguish these two opinions: whether viewing standard model action as "fundamental" or effective.


Is the Graveyard Really so Serious?

Calculations in relation to black holes are solely in consideration of spacetime curvature and its effects. They are in total alienation with respect to the action of inertial agents[external forces].That world lines do not point outwards from a black hole is consistent with the fact that inertial agents are absent. Can the presence of inertial agents change the picture of the world lines? Let's consider the following situation: _An experimenter [source of "external forces" capable of producing non- geodesics] throws some mass into the black hole. Its surface area and volume increase.This is suggestive of outward world lines from the surface, causing the expansion----points on the surface should move outwards.The presence of "external forces" has modified the entire perspective of the situation._ The black hole picture takes into consideration only the blackhole. _Right from the outset the black hole is considered in isolation from the rest of the universe._ Any type of modification due to the presence of external agents--- artificial or natural ones do not figure in the blackhole calculations. Consideration of such factors can modify the picture of the black hole. Can technology prevent somebody near a black hole from falling into it? Can it remove the apprehensions of what we know to be a graveyard?


Deriving infinitesimal time dilation for arbitrary motion from Lorentz transformations

I'm trying to derive the infinitesimal time dilation relation $dt = \gamma d\tau$, where $\tau$ is the proper time, $t$ the coordinate time, and $\gamma = (1-v(t)^2/c^2)^{-1/2}$ the time dependent Lorentz factor. The derivation is trivial if one starts by considering the invariant interval $ds^2$, but it should be possible to obtain the result considering only Lorentz transformations. So, in my approach I am using two different reference frames $(t,x)$ will denote an intertial laboratory frame while $(t',x')$ will be the set of all inertial frames momentarily coinciding with the observed particle, i.e. the rest frame of the particle. These frames are related by $$t' = \gamma \left(t-\frac{Vx}{c}\right),\quad x' = \gamma \left( x - V t\right),$$ where $V$ is some nonconstant (i.e. time dependent) parameter which is, hopefully, the velocity of the particle in the laboratory frame. Treating $x$, $t$ and $V$ as independent variables (for now) and taking the differential of the above relations, I obtain $$dt' = \gamma \left(dt-\frac{Vdx}{c}\right) - \frac{\gamma^3}{c^2}(x-Vt)dV,$$ and $$dx' = \gamma \left( dx - V dt\right) - \gamma^3 \left(t-\frac{Vx}{c}\right) dV.$$ Imposing either the definition of the rest frame $dx'=0$ or (what should be equivalent) $dx = Vdt$, the only way in which i obtain $dt = \gamma dt'$ is if $dV=0$. So, the derivation breaks badly at some point or I must be wrong in using some of the above equations. Which one is it?


Proof that flux through a surface is independent of the inner objects' arrangement

$$\Phi=\iint_{\partial V}\mathbf{g} \cdot d \mathbf{A}=-4 \pi G M$$ Essentially, why is $\Phi$ independent of the _distribution_ of mass inside the surface $\partial V$, and the shape of surface $\partial V$? That is, I'm looking for a mathematical justification for the characterization of flux as a kind of "flow" of gravitational field through a surface. The reason I worry about that is this. I have encountered the following proof of Gauss' Law for gravity: $$\iint_{\partial V}\frac{-MG}{r^2}\mathbf{e_r} \cdot \mathbf{e_r}d A=\frac{-MG}{r^2}\iint_{\partial V}d A$$ $$=-MG4 \pi$$ However, two assumptions are made in the above proof that I've not convinced myself hold true for the general case:   * The Gaussian surface is a sphere.   * It is assumed that all mass is concentrated at the centre of the sphere. Returning to my original question:   * Why is it true that when the Gaussian surface is deformed (not adding or removing any mass to inside the surface, of course), the amount of flux, $\Phi$, through the surface is equal to the flux in the spherical case?   * Why is it true that the positions of the masses inside the surface doesn't affect $\Phi$? **Edit:** removed reference to Poisson's equation, changed much of question to make it clearer. **Further edit:** I am happy that $$\iint_{\partial V}\mathbf{F}\cdot d \mathbf{A}=\iiint_{ V}( \nabla \cdot \mathbf{F})dV$$ However, my problem is with the following equation: $$\nabla \cdot \mathbf{g}=-\rho G 4 \pi $$ Which can be used to get $$\iint_{\partial V}\mathbf{g}\cdot d \mathbf{A}=\iiint_{ V}(-\rho G 4 \pi)dV=-M G 4 \pi$$


The motion of the particle satisfies $\textbf{v} = \textbf{c}\times \textbf{r}$

Why does a particle with position $\textbf r(t)$ that satisfies $\textbf{v}(t) = \textbf{c}\times \textbf{r}(t)$ follow a circle that lies in a plane perpendicular to $\textbf{c}$ with centre on a line through the origin in the direction of $\textbf{c}$?


Why does the sky suddenly look gray through this window?

I am looking to the windowed roof of my building, and I notice that the sky, which has few white clouds, sometimes looks completely gray, as if there was a huge cover of gray clouds, but there are not. By observing it and waiting for the change to happen, I notice it takes a matter of seconds : when the sun shines directly on the building (but not directly on me), the blue sky turns gray. I wonder if this is just an optical illusion because my vision of colour could be relative to its environment, or if there is serious glass/sun/refraction effect or something. I haven't found anything on that matter on the internet or on Stack Exchange, so I'm asking you what you think about it. Additional information : The top window is a very peculiar glass, it's basically a strong glass window (because it's almost horizontal, it has to resist hits) and looks like it is covered by a layer of opaque black, with big holes (surface of holes is greater than surface of black, from a distance you don't see the holes). Also, I'm taking the pictures from a windowed office, you see the sky through two glass windows, but the one closest to me looks basic and I didn't notice a difference when looking through one layer. You will also notice there is a blue light, some neon-like directed at the roof, but during the day its effect is negligible. Here are the pictures, first the sky through top window without sun hitting the window, then with the sun (The colours/ exposition differences don't seem to make a great difference compared to my naked eye observation. The pictures were taken only a few seconds apart, you can see the same cloud in both pictures. The gray sky isn't the most grayish I've observed, it gets even farther away from blue sometimes) : ![Sky through top window without sun hitting the window](http://i.stack.imgur.com/fbIyO.jpg) ![Sky through top window with sun hitting the window](http://i.stack.imgur.com/hBmW9.jpg)


Is the universe expanding faster than we could catch up with it?

Like the title says, could we ever catch up with the expanding universe?


Parallel-plate capacitor

I'm trying to grasp parallel-plate capacitors in a class of electromagnetic field theory. My book leaves out a lot and I couldn't figure it out from what I found on Google.   1. The book's procedure is to place a charge Q on the surface of the one conductor plate and -Q on the other. Now would this charge only be distributed over the surfaces of the plates that face each other, or on all four surfaces?   2. Is it possible to prove that any kind of source of potential difference connected to a capacitor can be represented as two opposite charges like that?   3. The book treats one of the charged surfaces as a conductor surface (or this is my understanding) and this yields, with $\Delta S$ being area, $E_n=Q/\epsilon\Delta S=\rho/\epsilon$ when using Gauss' law (if you place say a thin box around the charged surface because the electric field within the conductor is zero when in equilibrium). I find this way of doing it very odd because first you place Q on the one surface and -Q on the other, and then disregard that you've put -Q on the other surface. And I actually don't understand how come the -Q does not matter. How can this work?   4. A different I think more intuitive way of doing it that I've found online is to treat each conductor as a sheet of charge. Then you have twice the area to spread out the charge over, so you get $E_n=\rho/2\epsilon$ directed inwards from each surface but with different signs, the sum which is $\rho/\epsilon$ in between the plates, just what we want. I like this because it uses the setup with Q and -Q, plus it's easy to see how the fields cancel each other out. But why is it OK to consider a conductor to be a sheet of charge? Is it because the electric field will not be changed by passing through a conductor even though it will temporarily be zero within it? Thank you in advance.


Maximum speed higher than the speed of light

1. I wanted to ask if higher speed than the speed of light will be discovered, will scientists be able to adjust the special relativity to the new situation?   2. I read that informations transmitted faster then the speed of light, but According to special relativity, $c$ is the maximum speed at which all matter and information in the universe can travel. Is there a contradiction here?   3. How the fact that the speed of light is the maximum speed for all matter, affects special relativity?


Gravitational Length Contraction

General Relativity predicts that a clock at rest in a gravitational field will run slower than a clock in free fall. Similarly, will a vertical ruler on the earth's surface be shorter than a ruler in free fall? Why or why not and by how much?


Why is it more convenient to consider space or time as a continuum?

I often find that phisicists and cosmologists make use of Planck's units. I have read propositions that sound like > " _...at the level of Planck's units many law of physics break down_ " > > " _...Planck time, the smallest observable unit of time...before which > science is unable do describe the universe_ " > > " _... it would become impossible to determine the difference between two > locations less than one Planck length apart_ " even in string theory: > "* Planck length is the order of magnitude of the oscillating strings that > form elementary particles, and shorter lengths do not make physical sense*". Notwithstanding this and the fact that in QM (almost) everything is quantizied (discrete) I read that in mainstream they are still considered not discrete, cf. Phys.SE link. I am not particularly aware of any pros, I see only cons; can someone tell me what are the compelling reasons to consider space and time not discrete? Is it a requisite of relativity? As a corollary, I suppose that they must be both discrete or non discrete, right?


Readable books on advanced topics

I realise that there are already a few questions looking for general book recommendations, but the motivation and type of book I'm looking for here is a little different, so I hope you can indulge me. I enjoy reading quite a lot, and tend to prefer books that teach me something new, rather than straight up fiction. When I was younger I used to read a lot of popular science books, and at the time I thought these were teaching me something, but then I got older and went to university and studied theoretical physics, and that killed the genre for me. Now, I know that I can learn about a topic simply by seeking out review papers on it, or finding an appropriate textbook. That's what I do when I need to do something for work. However, most of the time this material does not make for light reading, and requires a significant amount of effort to work through, and is not necessarily the kind of thing I'd want to use as a way to relax. However, occasionally one comes across a very readable book on some aspect of physics. Nielsen and Chuang's Quantum Computation and Quantum Information strikes me as an example of this. There are probably better examples of this, but what are they? This brings me to my question: > Beyond the standard undergraduate topics, which areas of physics or > mathematics have books which both provide fairly comprehensive introductions > and which are actually enjoyable to read? To be clear, by "enjoyable" I mean something more than that they simply be accessible or well written. I mean that it should be something that I could read for relaxation rather than work. To some people this will probably sound like an insane thing to want to do. However modern physics is huge, and it bothers me that I don't know very much about, say, string theory beyond bosonic string theory, and if it is something I can learn more about in my spare time, then that would be great. * * * UPDATE: There appears to be some confusion in the answers over the type of book I am looking for. I am **not** looking for popularisations. Rather I am looking for something at the level of a graduate text, but only those which are particularly readable.


Speed of sound at temperatures below 0 °C

How can the speed of sound be calculated for temperatures below 0 °C (down to -40 °C)? Does the calculation $v=331\ \frac{m}{s} + 0.6 \frac{m}{s°C} \times T$ still hold (where T's unit is °C)?


What is a "Trojan Moon"?

Today I read an article about Saturn's moon Helene, and it was described as a "Trojan Moon" but no further explanation was given. It was difficult to even get any context about the term from the article. What is a Trojan moon and how did the term "Trojan" come about?


Do glass panes become thicker at the bottom over time?

Is it true that a glass window, that has been placed in a wall for about 10 years or more, is thicker on the bottom than on the top? I can vaguely remember my physics teacher saying that this was true. So if this is true, how is this possible?


Wave equation in the case of non-monochromatic light

monochromatic light refers to electromagnetic radiation of a single frequency, $\omega$ The wave equation in this case turns into the well known helmholtz equation.. How about for non-monochromatic light, how can we simplify the wave equation


Product of exponential of operators

in the context of non-relativistic quantum mechanics I want to show that, for any $A$ and $B$ operators $$e^{A}e^{B}=e^{A+B} $$ if and only if $$[A,B]=0$$ I remember my professor told use about looking for a differential equation but I don't remember the details, and I want to be able to prove it. Brute force the series doesn't seem to be a good idea. Any hint will be appreciated thanks.


Power to suspend a mass in air

How much power is needed to suspend a mass in the air? The four parts below deal with the same problem. I post them all here, should there be some subtle things I didn't see. ## The textbook problem A 9th edition college physics textbook has this as an example problem in its energy chapter (details cleaned-up): An elevator car ("a lift") and its passengers has a mass of 1800 kg and negligible friction force in all its moving part. How much power must a motor deliver to lift the elevator car and its passengers at a constant speed of 3.00 m/s? _Answer_ : Speed is constant, so $a=0$. Now let $F$ be the total force and $T$ be the force exerted by the motor. Then, $$ \sum F = T-Mg = 0 \qquad \Rightarrow \qquad T = Mg $$ with power = force x velocity, we then have $$ P = T\cdot v = Mg \cdot v $$ $$ P = 1800 \,kg\cdot9.8\,m/{s^2}\cdot 3.00\,m/s=5.29\times10^4\,{\textrm W} $$ And, what power must the motor deliver at the instant the speed of the elevator is $v$ if the motor is designed to provide the elevator car with an upward acceleration of 1.00 m/s per second? _Answer_ : Now $a=1.00 \,m/{s^2}$, and then $$ \sum F = T-Mg = Ma \qquad \Rightarrow \qquad T = M(a+g). $$ Proceeding as before, we have $$ P = T\cdot v = M(a+g) \cdot v $$ $$ P = 1800 \,kg\cdot (1.00+9.8)\,m/{s^2}\cdot 3.00\,m/s= 5.83 \times10^4\,{\textrm W}. $$ That's the end of the example. **My questions**. We might as well ask these:   1. How much power must the motor deliver to suspend the elevator and its passengers so it's kept at the constant height? That is, zero velocity? Certainly it can't be zero right? Because otherwise the elevator car will free-fall. But how much power is required?   2. What power must the motor deliver at the instant the speed of the elevator is $v=3.00$ (as before) if the motor is designed to provide the elevator car with a _downward_ acceleration of 1.00 m/s per second? I've been looking for an answer but to no avail. ## The helicopter problem This is _the_ real question that motivates this post. I've been asked by a friend who's going to build a model-helicopter. "Ok, I have a 2000 W engine for my 50 kg model-helicopter. Assuming 100% efficiency, will it able to float 1 m above ground for at least 1 minute?" I can't answer that. "Ok then, what is minimum power required for the engine if I want it to float 1 m above ground, assuming 100% efficiency?" I'll ask physics.se ## The pulley problem This is a simpler version of the above two problems. Using a rope and a pulley, what is the minimum power required for a man to suspend a 1 kg mass in the air 1 m above ground for 1 second? ## The table or string problem There's not much problem here. Suspending a 1 kg mass 1 m above the ground? Simple. Put it on a table, or hang it with a string. No movement. No power required. What's happening here?