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□h=-16πT

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Everything posted by □h=-16πT

  1. You've got [math](9-x^2)^{\tfrac{1}{2}}[/math] Then you use the chain rule.
  2. No it doesn't. You can get stable orbits about black holes.
  3. Your derivative is wrong by the way. [math]y=-\sqrt{9-x^2} [/math] [math]y'=-\frac{1}{2}(9-x^2)^{-\frac{1}{2}}\frac{d(-x^2)}{dx}=\frac{x}{\sqrt{9-x^2}}[/math] I can't remember how you do coefficient of friction and all the gubbins, but the problem without coefficients of friction just requires that the kid's kinetic energy at the bottom of the pipe be the same as its potential energy at its initial position, height=3. Can you not integrate friction coefficients into conservation of energy.
  4. Well for a weak field we can consider the metric as a small perturbation from flat space-time metric [math]g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}[/math] Where the field h is not necessarilly a tensor, but a function added to each component of the Lorentzian metric. However, if we consider Minkowski space-time as a background, the field h will transform as a (0 2) tensor, so we can consider it as a tensor in such circumstances. If we now give ourselves an expression for the Einstein tensor in terms of our perturbed flat metric we can impose certain simplifying restrictions on h. These are the traceless-transverse gauge and the Lorentz (or de Donder) gauge, the latter being similar to that in classical electrodynamics. If we do this we end up with the weak field version of the field equations of general relativity which serves the basis of linearised field theory (linearised because we only one the terms linear in h) [math]\left(\frac{1}{c^2}\frac{\partial}{\partial t}-\nabla^2\right)h_{\alpha\beta}=-16\pi T_{\alpha\beta}[/math] Where T is the usual symmetric stress energy tensor of the generating field. "A First Course In General Relativity"-B. Schutz has a good discussion on the treatment of gravitational waves in linearised field theory, including its detection and the energy radiated from bodies by their gravitational wave emission. Note that spherically symmetric bodies do not radiate gravitational radiation, as their quadrupole and higher moments vanish due to the symmetry.
  5. Jesus, he polluted myspace with this as well.
  6. Danke, I wish it did work though . I hate functional integrals, as elegant as the whole procedure is for deriving Feynman rules.
  7. Or have you done something that gets rid of that factorial? Cheers, man.
  8. It allows one to take some state defined at some particular time and find the state at a further point in time. So one can remove time dependance, which can then be replaced by evolution operators, and makes things a bit easier in places.
  9. The Hamiltonian is one of the most important elementary quantities in quantum mechanics. It arises because of time, and gives the energy of the system, in a round about way.
  10. Oh, really? Yeah, I guess you're right, after all Hamiltonians and time evolutions opreators play a minor, insignificant role.
  11. I was trying to approach it, as the question suggested, by constructing it out of Wilson lines and using a functional integral. There should be factorials in the exponent that you've missed out, which will muck up the end result.
  12. Angular momentum is conserved for central systems without external forces. That this results in angular momentum conservation arises from the following relationship between turning moment G and angular momentum J in an analogous manner to the conservation of linear momentum from Newton's second law [math]\vec{G}=\frac{d\vec{J}}{dt}[/math]
  13. Hey, I've got this problem from Peskin & Schroeder (chapter 15). I'm not particularly confident with functional integration, as I'm pretty new to it, and working through such a book by myself is pretty tricky in places. Well here goes The Wilson Loop for QED is defined as [math]U_p(z, z)=\exp \left[-ie\oint_pdx^{\mu}A_{\mu}\right][/math] With the Wilson line defined similarly (just change it so that there's not a closed contour integral and with the end points (z,z) changed to (z, y), or whatever you like). Where A is the photon field, the gauge connection asociated with transformations in U(1). Now it says: using functional integration, show that the expectation of the Wilson loop for the electromagnetic field free of fermions is [math]\langle U_p(z, z)\rangle =\exp \left[-e^2\oint_pdx^{\mu}\oint_pdy^{\nu}\frac{g_{\mu\nu}}{8\pi^2(x-y)^2}\right][/math] Where x and y are integrated over the closed loop P. I think the Feynman propogator might be useful here, so to save anyone having to look it up, [math]D_F^{\mu\nu}(x-y)=\int\frac{d^4q}{(2\pi )^4}\frac{-ig^{\mu\nu}e^{-ip\cdot (x-y)}}{q^2+i\epsilon }[/math] (The imaginary term in the denominator of the integrand is the application of the Feynman boundary conditions, ensuring the convergence of the Gaussian integral involved in the derivation of the propogator.) I have a vague idea of how to go about it, but I'm not particularly confident about it, it's finding the relevant starting point that's causing me problems, i.e. putting together and computing the functional integral for the expectation. I'm just going through this to gain some confidence in functional integration etc. so if anyone can give a few pointers as to going about this it'd be much appreciated. This isn't a homework question, if that puts anyone off helping me, I doubt my A level teacher would set something like this . Cheers
  14. The Bohr model only works for hydrogen, as in its derivation it uses a Coloumb potential between two point particles: a proton and an electron. It fails to account for the spin of the electron and proton, as the derivation is non-relativistic, which results in a slight, almost unoticable, discrepency between the predicted emission spectra and the actual emission spectra, known as the lamb shift.
  15. I'm not familiar with the bar notation, so I either didn't see it or neglected it.
  16. That was Galileo's principle of relativity, not Einstein's. Einstein used this to show the invariance of c from Maxwell's equations governing the electromagnetic interaction.
  17. It is non-sensical to use a photon rest frame to conduct any observations. This is simply because such a rest frame is undefined, as the four-velocity is indeterminate (0/0) etc.
  18. Yeah, I think he did. I can't be bothered to get the 1905 paper by Einstein out and check, but I'm pretty sure H. Minkowski introduced the space-time interval (hence the Minkowski interval) and the geometry of SR in 1906 (or '07). The main bulk of "On The Electrodynamics of Moving Bodies" explained the Michelson-Morely experiment concisely and postulating the Lorentz invariance of c supported by a mathematical derivation from the Maxwell equations. Introducing time as an additional property of the universe together with space by itself is nothing special and does not constitute a definition of space-time really. The important result is the invariance of the space-time interval, by themselves spatial and temporal intervals are relative to the observer, whereas an interval in space-time is frame independant. I don't know if you already knew this. Lorenz is the name of a prominant guy in weather models and non-linear dynamics (chaos theory). Lorentz is the guy you're after; he was the dude that, subsequent to the MM experiment and prior to the '05 paper on SR, proposed length contraction and time dilation to explain the results. However his idea was seen merely as a mathematical trick to explain the "negative" results. He conducted other work into the subject as well.
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