Dirac Brackets in General Relativity

Question

I want to calculate Dirac brackets of different phase space variables in gravity.

In case of electrodynamics, one does the same using the following steps:

1. Looking at the momenta to find that $\Pi^A_0 = 0$.

2. Taking Poisson bracket of the above constraint with the Hamiltonian and setting it to zero to get the Gauss constraint.

3. Since both are first class constraints, fixing gauges to make them second class.

4. Using the Dirac bracket formalism for these second class constraints, and calculating Dirac brackets instead of Poisson brackets for the same.

For the case of doing the same with gravitation, I am unsure of a few things. The formalism more or less goes like as follows, and I will ask my doubts after this.

1. One does the ADM $3+1$ decomposition of the metric and splits the Einstein-Hilbert Lagrangian using that.

2. Specifically, since the canonical momenta corresponding to the shift and the lapse function vanishes, one identifies their coefficients in the massaged form of Einstein-Hilbert as Hamiltonian and momenta constraints.

So if we start with pure gravity in $(3+1)D$, the decomposition tells us that we had 12 d.o.f. We need to reduce it to 4 d.o.f. This is where my doubts start.

The d.o.f. counting argument indicates that the four constraints should be either:

a. First class, and we would need to choose 4 gauges for the same to make them second class. 3 of them can fix the diffeomorphism on the slices, I am not sure of the other one. (In case this option is correct, what is the meaning of fixing the fourth gauge here?)

b. Second class, but then we would need more constraints. (I don't think this option is correct).

So my question is this: How do I get an exact expression for the Dirac brackets of phase space variables (as I indicated previously in points 3 & 4 for electromagnetism) here for the gravitation case? Is there any reference where this problem is treated in depth?

Answer 1

I think this is the answer to my question. Basically we have a family of first class constraints. The family comes because of the fact that the algebra doesn't close, but one can slightly modify the constraints, and then use a gauge choice to close the algebra. The necessity of gauge fixing indicates that the constraints are first class.

One can now use the $d$ diffeos to fix up the gauge, and eliminate $d$ degrees of freedom using the constraints. So as asked in the question for 4 dimensions using $ADM$ decomposition, we have $12−4−4=4$ degrees of freedom for the graviton in the phase space, which amounts to the 2 polarizations.