Binary Black Hole Solution of General Relativity?

This is rather a technical question for experts in General Relativity. An accessible link would be an accepable answer, although any additional discussion is welcome. GR has well known solutions relating to single Black Holes: Schwarzchild, Rotating & Rotating with Charge. These solutions demonstrate some non-trivial GR behaviour. However do there exist any (corresponding) binary star/black hole solutions? Because of the non-linearity of GR such a solution could well demonstrate additional properties to a "solution" that simply consisted of a pair of "distant" Schwarzchild solutions glued together.


Deriving Planck's radiation law from microscopic considerations?

In the usual derivation of Planck's radiation law, the energies or frequencies $\omega$ of the oscillators depend on the measurements $L$ of the black body. The model is such that the only characteristic energy is given by these oscillator excitations and in terms of the temperature $T$. Also, the oscillation amplitude vanishes on the walls. Specifically, the structure of the atom walls doesn't enter the computation. After all, that's pretty much the definition of a black body. My question: > Are there computations (maybe QED + statistical physics?) for more realistic > systems, which might model the interaction of photons with the wall atoms > and which give results such that one can see the limiting case to an ideal > black body? The question came up when I wondered about the theory behind emission and absorptions of photons on the walls of a black body.


What parameters determine the acceleration a spacecraft can acquire using the gravity of a celestial body?

Contemporary spacecraft frequently rely upon gravity to accelerate/decelerate. Given a spacecraft already in space - What factors determine the acceleration it can acquire from the gravity of another body? Is there an upper limit to such gravitational acceleration?


Does proton make neutron charged by the process of induction/friction/conduction?

If two bodies undergo friction, the one of the bodies which has electrons less tightly bound than the other loses them. Here the protons do also have charged quarks which they could exchange with neutrons to make them charged either by the process of conduction (may not be possible by friction), but it does not happen I think so. If so, Why? Edit: Can the neutron develop induced charge due to protons inside the nucleus? Though we can't expect exchangeable isolated quarks, can we expect them to orient or get settled to one side, so that neutron can develop induced charge? I might have misunderstood sometimes, if so, pardon me and explain.


Find the position of the final image

I was going through Geometrical Optics and came across the following question in my textbook: Q. An object is placed $21$ cm in front of a concave mirror of radius of curvature $20$ cm. A glass slab of thickness $3$ cm and refractive index $1.5$ is placed close to the mirror in the space between the object and the mirror. Find the position of the final image formed. The distance of the nearer surface of the slab from the mirror is $10$ cm. Assume only paraxial rays. Brute force led me to the answer-at the position of the object itself . I have the following questions to ask: $1.$Is my answer correct ? $2.$What can be the best possible way to solve such a problem? $3.$Is the data about the distance of the slab from the mirror really required? Thanks for the help... **NOTE:** I posted this on Math S.E. because Math S.E. **does** have a **physics** tag.


Does the Cosmic Microwave Background obey $\sigma T^4$?

The fact that the CMB has a temperature of 2.7 degrees Kelvin is well publicized, but I have a harder time finding a measure of the wattage per unit area. The Wikipedia article on the CMB is totally silent on the subject. Then when I do find a source for W/m^2 for the CMB, they use the blackbody formula. > A flux of radiation has a Kelvin temperature. We see the temperature of the > CMB as 2.735 Kelvin. We can convert this to watts/meter2 for blackbody > radiation with the Stephan and Boltzmann law. K4*5.5698E-8_W/(m2*K4) = _W/m2 > > 2.735 Kelvin = 3.11E-6_W/m2 > > When one sees something, it is in terms of W/m2 and the inverse square law. That source does mention the fact that _expansion of space decreases photon density_. But I don't actually see a revision of the W/m^2 figure. I can't understand if they mean to say that the combination of temperature and density change balances it out so that it still follows the blackbody radiation law. There's also the matter of whether the gas from which the CMB came was a true blackbody or had some reflectivity to it. Surely we have a measured value for this. Does the value come anywhere close to the Stefan–Boltzmann law prediction? Does the expansion of space predict a deviation from that intensity or not? Do non-blackbody properties of the primordial gas predict a deviation from that?


Electrical Potential Energy And forces Experienced By Particles

I am really stuck on a question and would appreciate a helping hand. A particle has a charge of $+3.5$ $\mu$$C$ and moves from point $A$ to point $B$, a distance of $0.5m$. The particle experiences a constant electric force and it's motion is along the line of action of the force. The difference between the particles electrical potential energy at point $A$ and point $B$ is $EPE_{A}$ $-$ $EPE_{B}$ $= +7.0x10^{-3} J$. (a) Find the magnitude and direction and direction of the electric force that acts on the particle. (b) Find the magnitude and direction of the electric field the particle experiences. The fact that the work done is a positive quantity, leads me to think that the electric force acts in the direction from $A$ to $B$ but I don't know how to quantify it. Thank you for any help you can give!


Numerical solution of the BCS gap equation, with Coulomb potential

I'm interested in the excitons condensation, which is described by an equation which is very similar to the standard BCS gap equation. I will be referring to this article: http://arxiv.org/abs/cond-mat/9501011. The only differences w.r.t. the standard BCS theory are:   1. The momentum dependency of the gap cannot be neglected, so we write $\Delta = \Delta(k)$.   2. The potential must be model realistically, in this case after a Coulomb potential. In the end the gap equation reads: $$ \Delta (k) = \int d^2 k' \underbrace{\frac{e^{-d \left| k - k' \right|}}{\left| k - k' \right|}}_{V_{k k'}} \frac{\Delta(k')}{\sqrt{\Delta^2 + (k'^2 - \mu)^2}} $$ Knowing all the the parameters ($d$, $\mu$), clearly this equations needs to be solved numerically for $\Delta (k)$. My first guess (and this approach does work, indeed, in many similar cases) would be of choosing a "trial" $\Delta$, iterating over and over, hoping sooner or later to converge to a solution which is not the banal $\Delta=0$. This approach does not work here, because the potential $V_{k k'}$ has singularities. The article I am referring to is quite cryptical about it. They mention an unpublished paper as far as the numerical calculations are involved, and simply say that "the equation is converted in matrix form", as: $$ \Delta_i = M_{i j} \Delta_j $$ as expected, and then "the logarithmic singularities are treated separately". Clearly the singularities are on the diagonal of the matrix when the equation is discretized. This process is completely obscure to me. I suppose I cannot just separe the divergent diagonal part and the finite part. I tried parametrizing the divergency as $\frac{1}{\epsilon}$ and letting $\epsilon$ go to zero at the end of the calculation but to no avail. Is there some standard techniques to numerically solve my equation? Thank you in advance.


2 torches with the same power - More lumen?

I didn't find an answer for this. A friend and me argued about the brightness-impact of multiple bulbs/torches. I told him, that 2 torches with the same power (the same amount of lumen and everything) hold side by side, won't be a difference in lumen, if you place the measure-device directly in the center of the "beam". So: The brightness is the same. Means: If I point to the lumen-measure device with one torch (maybe ~0.5m distance), adding another would have no impact on the result, if it has exactly the same power. He denied and said, that the more torches/bulbs you add, the more brightness you will get. He said, that if you add another torch, you see 2 rings of light: The center beam contains the power of both, but at the outside (edges), the beams divide and there would be lesser power, where because there is just the beam of one torch. I draw a picture to show the Idea: ![enter image description here](http://i.stack.imgur.com/5pwnT.png) I said, that you don't have more brightness, but you have a better illumination, which let you think that the light is stronger. But if somebody "beams" to your eyes with one torch, adding another with the same attributes won't make a difference in brightness, if you hold them close together. If you would put 1000 torches next to each other, it won't be much brighter, but the illumination is of course bigger and this may let you think, that there is stronger light. Who of us was right? I couldn't get an evidence for my theory, because I lack the required items. I hope you understand what I mean, my english isn't the best. :) I hope there are some of you, who can give me a good answer to this. We "fight" about this point since months. :D I couldn't find an evidence on the internet.


Calculate minimum intensity of a point source visible to a camera

I have two cameras and know the specified minimum faceplate sensitivity for both: Camera 1: Minimum faceplate sensitivity 4 x 10-6 Lux   Camera 2: Minimum faceplate sensitivity 2 x 10-7 lux I also know the minimum intensity (1 x 10-7 W m-2) of a point source (470 nm) visible to Camera 2 at a distance of 0.5 m from the front of the camera (ambient light levels: zero). I would like to be able to calculate the minimum intensity of a point source visible Camera 1 through the same medium (seawater). (Camera 1 is no longer functional and so cannot be calibrated) Although there would obviously be a number of factors that would add error into this calculation, is there a way to calculate this (supposing ideal conditions)?


Question regarding the validity of the big bounce

I have several questions regarding the "big bounce" theory. It appears to be popular among LQG researchers. My questions are as as follows. 1) How one reconciles it with the fact that it is now experimentally established that the universe is expanding in an accelerating manner? 2) The entropy of the universe should have been very very low (maybe zero) at the time of the big bang/ big bounce. What was the entropy "before" that event? Was it even lower? If it was lower then doesn't that mean entropy decreases indefinitely? On the other hand if we accept the entropy was higher then doesn't that violate the second law of thermodynamics? 3) What observational evidence can confirm pre-bounce physics? According to GR there are some points near the bang/bounce where one simply can't easily define time in a meaningful way. Even for a quantum gravity theory, is it possible for any theory to extrapolate itself "before" the "bounce"? If there are no observational consequences after the bounce shouldn't one apply Occam's razor? ![enter image description here](http://i.stack.imgur.com/HeKrp.jpg)


Does effective theory have the same meaning in particle and condensed matter physics

I have a naive question about the meaning of effective theory in particle physics and condensed matter physics. In particle physics, from what I know, the effective theory comes from the Wilsonian renormalization. We start from a theory with a fundamental cutoff $\Lambda$, e.g. Planck scale, and consider a smaller cutoff $\Lambda'$, e.g. LHC scale, $\Lambda'<\Lambda$. The Lagrangian then reshuffle into an infinity series including all possible powers of fields. The theory with $\Lambda'$ cutoff is called effective theory (possibly only a few leading terms) In condensed matter theory, I heard the effective theory means neglect some details, like the BCS theory. Roughly speaking, "the effective" in both fields has similar meaning, that we ignore something ($>\Lambda'$ in particle physics or uninteresting detail in condensend matter). My question is, strictly speaking, does the two "effective" in particle and condensed physics have the same meaning? Can we start from QED, lower the cutoff, lead to BCS theory? (perhaps such crazy attempt will never work out...) or any deeper connection than heuristic similarities?


Classical (or semi-classical) interpretation of photoelectric effect?

This site says that "it has recently been proven that the photoelectric effect can be interpreted classically (or at least semi-classically) in non-particle, wavelike terms". Is anyone familiar with this other explanation of the photoelectric effect?


Yan - Drell process

I'm looking for more information about Yan - Drell process but didn't manage to find a good place to learn from (still looking for one). I understood that this process can help us to understand the inner structure of hadrons / mesons but I would love to get more information what is going on. In which accelerator this process is actually being researched?


Root mean square value for DC

The mean value of alternating current comes to be zero because of half of the cycle being positive while the other half negative. so, take the root means square value of Alternating current given by: $I_{rms} = 0.707\times I_{max}$ But why do we take "rms" for direct current? if we take the simple average of direct current, we would come up with a value which is not zero (it is zero in AC).


Why does a cathode have to be heated to emit electrons?

Considering that electrons are highly mobile inside of a metal, why do they have such a tough time getting out at the edge of it and continuing their trip ballistically?


Motivation for Potentials

This is a hypothetical question about "pedagogy". Let's say I am trying to take someone who has just a very small amount of knowledge about Newtonian mechanics and convince them that the Lagrangian formulation of mechanics is a _natural_ thing to try. First of all, to convince them that this is _natural_ , at the very least, I would have to show how one can reproduce Newtonian Mechanics from Lagrangian Mechanics, in particular, Newton's Second Law. However, to do this, I would need to introduce the notion of a potential. However, I am having trouble justifying this. I would like to be able to say "most forces we encounter in nature are conservative, and hence, we are not losing too much by assuming all of our forces are conservative . . .". However, of the two _fundamental_ forces that come to mind in _elementary_ physics, the gravitational force and the electromagnetic force, only one is conservative. I hardly feel that one out of two is a sufficient justification for the introduction of a potential. But if it is not even _natural_ to assume that our forces always come from a potential, then it certainly won't be _natural_ to define the Lagrangian in a way that reproduces Newton's second law. Is there a nice way of convincing such a student that, even though the electromagnetic force is not conservative, there are still ways of deriving it from some sort of potential? In particular, the difficulty here is that I do not want to assume they have any specific knowledge of electrodynamics.


Why doesn't helium start to fuse while there is still hydrogen fuel?

In all the descriptions of the stellar life cycle it seems as though helium doesn't start being fused until all (most?) of the hydrogen is gone. Is this true? Why is this? It seems counter intuitive. Consider a fire consisting of gasoline (hydrogen), and wood (helium). Sure the gasoline would burn up quicker, but the wood would still be consumed while the initial gasoline was being consumed.


Are the higher-order terms in the series for energy really negligible?

To show that energy in special relativity reduces to $E=m+mv^2/2$ for low velocities, if we make a Taylor expansion of $m\gamma$ around $v=0$ we get $$E=m+mv^2/2+3mv^4/8+\cdots$$ But why can we cutoff after the second term? How does one know all the other terms don't add up to something significant? We get $E = \text{rest energy}+\text{kinetic energy}+\frac{3mv^4}{8} + \cdots$ What kind of energy is the $3mv^4/8$ term?


How do you obtain the fluctuation spectrum of a tubular membrane?

I am reading through a paper. A tubular membrane, submitted to tension $\sigma$ acting as a Lagrange multiplier to conserve area, fluctuates around a cylindrical shape of length L and radius R. Parametrisation is as follows, \begin{equation} \mathbf{r}(\phi,\zeta)= (x,y,z) = R([1+u(\phi,\zeta)]\cos{(\phi)}, [1+u(\phi,\zeta)]\sin{(\phi)}, \zeta) \end{equation} where $0 \leq \phi \leq 2\pi$, $0 \leq \zeta \leq L/R$ and $u(\phi,\zeta)$ is the deformation field. Using the Canham-Helfrich Hamiltonian, $F= \int dS\left( \frac{1}{2} \kappa H^2 + \sigma \right)$ one can obtain expressions, to second order, for the excess energy. Long story short, the excess energy can be represented in Fourier modes from which you obtain the mean squared amplitude using the equipartition of energy: \begin{equation} \langle \lvert u_{m,n} \rvert ^2 \rangle = \frac{k_B T}{\kappa} \frac{1}{(m^2 -1)^2 + \bar{q}^2 (\bar{q}^2 + 2m^2)} \end{equation} The correlation function can also be found: \begin{align} C((\phi - \phi '),( \zeta - \zeta')) &= \langle u(\phi,\zeta)u(\phi ', \zeta ') \rangle \\\ &= \frac{k_B T R}{2 \pi \kappa L} \sum_{m,\bar{q}} \frac{\cos(m(\phi - \phi '))cos(\bar{q}(\zeta - \zeta '))}{(m^2 -1)^2 + \bar{q}^2 (\bar{q}^2 + 2m^2)} \end{align} So my questions are: 1) What is the meaning of the mean squared amplitude and the correlation function? 2) How can I obtain a fluctuation spectrum from them? The paper can be found here: http://www.msc.univ-paris-diderot.fr/~jbfournier/publi/critical_tubes.pdf Many thanks in advance.


Atwood machine with spring

I'm just beginning to learn about Lagrangian mechanics, and I am asked to find the kinetic energy of this Atwood machine (See figure). ![Atwood machine](http://i.stack.imgur.com/SZIt4.png) I am told, that the kinetic energy should be: $$T=2m\dot{x}^{2}+\frac{1}{2}m\dot{y}^{2}-m\dot{x}\dot{y}$$ I am also told, that the movement is so slow, that the spring is always stretched, and $x$ and $y$ are my generalized coordinates. So what to do? My first though was, that since the spring is always stretched, all masses are dependent of each other, and I could maybe look at it as a whole string with one mass on the left, and two on the right. But that didn't give me the right answer. So I was hoping someone could give me a hint or something?


Mathematical model of a fluid: which viewpoint is the standard one?

I've been studying fluid mechanics and I have some doubts on the mathematical model of a fluid. First we let $D\subset \mathbb{R}^3$ be a region in three- space filled with a fluid. This region has as many points as the real line. Since in Classical Mechanics when studying systems with a finite number of particles we consider each particle as a point in $\mathbb{R}^3$ I thought that _each_ point of this region $D$ should be considered a particle of fluid. In that setting the fluid would be composed of uncountably many particles. With this logic, I thought on labeling each particle by it's position at time $t = 0$, so that a state of the fluid would be a bijection from $D$ onto $D$. The time evolution would be then $t \mapsto \varphi_t : D \to D$ and we could summarize this with a map $\varphi : D\times \mathbb{R}\to D$ given by $\varphi(a,t) = \varphi_t(a)$. This is exactly the Lagrangian description of the fluid, so I thought I was right in considering each particle of the fluid as a point in $D$. Now, I've been told it's wrong to think it this way, because if it were so, each particle would have zero mass. I've also been told that the right way to think is to imagine each particle as a infinitesimal volume $dV$ containing _some_ real molecules of fluid with mass density $\rho : D\times \mathbb{R}\to \mathbb{R}$ so that each particle would have mass $\rho dV$. Since each piece of volume is infinitesimal we could still identify them with points. My doubt is, when modeling a fluid with math which viewpoint is the standard one and why? The view point as in Newtonian and Lagrangian Mechanics of considering the fluid particles as points of $D$ or the view point of considering the fluid particles as infinitesimal chunks of fluid with volume $dV$ one at each point of $D$?


Artificial planetary magnetic field

I wonder how difficult it is to create an artificial planetary magnetic field with generators? What power they would need? The question is inspired by thinking about possible colonization of Jupiter's moons Io and Europa which are located inside the Jovian radiation belt. Is it possible to create with easy means an artificial magnetic field such that it to shield the surfaces of these moons from radiation? Or it would require astronomical amount of power? By easy means I mean a device that would not require power greater that normal industrial power plant, best of all, solar-powered or based on once-charged superconductor coils.


Is a capacitor in an open circuit charged?

Say I have a circuit consisting of a battery, a wire, an open switch, and a capacitor. The circuit is open since the switch is open. My book says that the capacitor will only be charged when the switch is closed, but I don't see why this is true. I would expect the capacitor to be charged a little - not as much as if the circuit is closed, but still charged none the less. To further illustrate my point consider this: If the circuit is open, the current must be zero. Consequently the field must be zero. For the field to be zero, the capacitor's field must cancel out the battery's field. Therefore the capacitor _must_ be charged. Generalizing this concept, shouldn't capacitors be charged (to a lesser degree) in open circuits? EDIT: In other words, if the field is zero, the capacitor _must_ be charged to cancel out the field of the battery.


Why is radiative forcing from CO2 logarithimic and not a decreasing exponential?

I have a question about climate science, but this is relatively straight forward and (I hope) easy. I've been reading that the radiative forcing caused by CO2 in the atmosphere (or any gas) is proportional to the logarithm of the concentration. I don't understand why. I'm familiar with basic attenuation, so if I assume that the gas captures or reflects some light, which equates to the heat per unit area (the units for radiative forcing), then I would write an equation like the following, with $F$ being the forcing in $W/m^2$ and $C$ being the ppm concentration. _The $1-e^x$ form is appropriate because we need the reflected quantity, versus the $e^x$ transmitted quantity_ $$F \propto \left( 1- e^{-\lambda C} \right)$$ What is fundamentally different about the greenhouse effect? I hesitate to give a specific reference because it seems to occur so frequently, but I will note that IPCC 2007 report is said to make such a statement. Wikipedia gives the following equation. $$\Delta F = 5.35 \times \ln {C \over C_0}~\mathrm{W}~\mathrm{m}^{-2} $$ I would love to see just a simple and clean takedown of why $ln$ and not $exp$.


How feasible is it to use the dielectric constant of a liquid to determine specific gravity?

I homebrew my own beer and I'm looking for way to cheaply monitor the specific gravity of my brew during the fermentation process. I came across this thread that seems to indicate capacitance may be used to determine specific gravity. Is there a direct relationship between the dielectric constant and/or capacitance of a liquid and its specific gravity?


Charging a Capacitor?

How is the charge +q given to the one plate to the capacitor with the help of a battery, does the battery remove electrons from the plate and +ve charge gets accumulated on one plate?Does it? And one plate gets charge induction of other plate occurs which creates potential difference between the two plates and then the +ve charge of the 2nd plate is grounded and passes through to the ground. Is this process correct?


Why is there a factor of 1/2 in the interaction energy of an induced dipole with the field that induces it?

In this paper, there's the following sentence: > ...and the factor 1/2 takes into account that the dipole moment is an > induced, not a permanent one. Without any further explanation. I looked through Griffiths' electrodynamics to see if this was a standard sort of thing, but couldn't find anything. I was thinking it might be because the field of the dipole itself opposes the inducing field, but that doesn't quite seem right for some reason.