Poisson bracket and Electric and Magnetic Weyl tensor in GR

[nix00]

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)

 

[robphy]

Possibly useful:
http://en.wikipedia.org/wiki/Ashtekar_variables
http://arxiv.org/abs/gr-qc/9312032 (Giulini's "Ashtekar Variables in Classical General Relativity")
http://projecteuclid.org/euclid.cmp/1103921616 (Ashtekar's "On the symplectic structure of general relativity")
http://arxiv.org/abs/gr-qc/9303032 (Romano's "Geometrodynamics vs. Connection Dynamics")

 

[nix00]

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.

 

[robphy]

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 ]

 

[nix00]

I am currently looking at this so-called quasi-Maxwell equations
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 In your link you mention the book by Hawking and Ellis, I will look at it. Thanks for your comments

 

[Bill_K]

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

Define the electric and magnetic parts of the Riemann tensor: E_ij = R_i0j0, H_ij = ½ ε_ikl R_klj0

He gives the commutators as: (don't ask me to derive 'em - I just work here! )
[E_ij , E_k'l'] = [H_ij , H_k'l'] = (1/4)i (t_ik t_jl + t_il t_jk - t_ij t_kl) ∇⁴ G(x, x')
[E_ij , H_k'l'] = - [H_ij , E_k'l'] = (1/4) ε_kmn (t_im t_jl + t_il t_jm - t_ij t_ml) ∇² G_,0n(x, x')

where t_ij is the tranverse field projection operator: t_ij = δ_ij - (∂/∂x_i) ∇^-2 (∂/∂x_j), 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: E_ij is at point x, while E_k'l' is at point x'.)

 

[robphy]

Great!...
http://repositories.lib.utexas.edu/handle/2152/7896 has a derivation on p. 39.

So now a little(?) more algebra is needed to express this in terms of the analogous parts of Weyl (instead of Riemann).

 

[nix00]

Thanks, I am working on it.