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Poisson bracket and Electric and Magnetic Weyl tensor in GR

  1. Jul 31, 2013 #1
    In order to understand how related are the theories of General Relativity and Electromagnetism, I am looking at the electric and magnetic parts of the Weyl tensor (in the ADM formalism) and compare them with the ones from Maxwell's theory.

    I have tried to look at the Poisson bracket, but the computations are long and there are many possibilities for me to have made a mistake.

    Do you know if it has been done somewhere ?

    (a research on internet gives a lot of results for $E_{ab}$ and $B_{ab}$ but no one talks about the Poisson brackets)
     
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  3. Jul 31, 2013 #2

    robphy

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  4. Jul 31, 2013 #3
    Thanks Robphy for your answer, but you may have misunderstood the meaning of E_{ab} which is not related here to the densitized triad used in the Ashtekar variables. It's true I can reexpress the components in terms of these variables, but I am still looking at an expression for the Poisson brackets.
     
  5. Jul 31, 2013 #4

    robphy

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    I understood E_{ab} as the electric-part of Weyl.
    I haven't seen the Poisson bracket expressions involving E_{ab} and B_{ab}.
    However, it may be that the methods in the links I posted show analogues of the calculation you seek.

    [Personally, I have been curious about the so-called quasi-Maxwell equations,
    where the gravitational field equations can be cast into a form which resembles those from electromagnetism:
    https://www.physicsforums.com/showthread.php?p=691492#post691492 ]
     
  6. Jul 31, 2013 #5
    I am currently looking at this so-called quasi-Maxwell equations :smile:
    They are simply derived from a geometric point of view, without referring to the Einstein's equations, expressed either in terms of the spatial metric and its conjugate momentum (q_{ab}, P^{cd}) or in terms of the Ashtekar variables (A^i_a, E^b_j). In both cases, I know how to deal with the Poisson brackets for these variables but because the electric and magnetic part of the Weyl tensor involve them in a really complicated way, I don't know if my results are correct or not :smile: In your link you mention the book by Hawking and Ellis, I will look at it. Thanks for your comments
     
  7. Aug 7, 2013 #6

    Bill_K

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    From "Dynamical Theory of Groups and Fields" by Bryce DeWitt:

    Define the electric and magnetic parts of the Riemann tensor: Eij = Ri0j0, Hij = ½ εikl Rklj0

    He gives the commutators as: (don't ask me to derive 'em - I just work here! :smile:)
    [Eij , Ek'l'] = [Hij , Hk'l'] = (1/4)i (tik tjl + til tjk - tij tkl) ∇4 G(x, x')
    [Eij , Hk'l'] = - [Hij , Ek'l'] = (1/4) εkmn (tim tjl + til tjm - tij tml) ∇2 G,0n(x, x')

    where tij is the tranverse field projection operator: tij = δij - (∂/∂xi) ∇-2 (∂/∂xj), and ∇-2 is the Green's function for the Laplacian operator. G(x, x') is the "commutator Green's function" (often written G~) of the wave equation for m = 0.

    (The primes on the indices indicate location: Eij is at point x, while Ek'l' is at point x'.)
     
    Last edited: Aug 7, 2013
  8. Aug 7, 2013 #7

    robphy

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  9. Aug 13, 2013 #8
    Thanks, I am working on it.
     
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