# Poisson bracket and Electric and Magnetic Weyl tensor in GR

1. Jul 31, 2013

### 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)

2. Jul 31, 2013

### robphy

3. Jul 31, 2013

### 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.

4. Jul 31, 2013

### 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:

5. Jul 31, 2013

### 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

6. Aug 7, 2013

### Bill_K

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! )
[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
7. Aug 7, 2013

### robphy

8. Aug 13, 2013

### nix00

Thanks, I am working on it.