i read in a "a first course on loop quantum gravity" written by Gambini that: "General relativity is invariant under spatial coordinate transformations. The constraint associated with the lapse (usually called Hamiltonian or superHamiltonian constraint) represents the invariance of the theory under the choice of any possible deformation of the spatial surface. It is akin to the reparametrization invariance of time we discussed in the mechanical example, except that because one can reparametrize time at each point in space, the end result is a deformation of the spatial three surface when viewed within space-time. The theory has therefore six configuration degrees of freedom ($g_{ij}$ is a symmetric $3 \times 3$ matrix) and four constraints, which leaves us with two degrees of freedom, just like Maxwell theory."

My first question is about hese 2 dof. what are they in GR? and how to find them in the EM case?

In this analogy we also have in electromagnetism case 6 variables $E_x$, $E_y$, $E_z$, $B_x$, $B_y$ and $B_z$. and at the end we only have two degrees of freedom for the light. we should have four constraints on a phase space (12 dimensions or more). What are these four constraints? And what are the orbits?

  • $\begingroup$ This question (v3) is also addressed in e.g this and this Phys.SE answers. $\endgroup$ – Qmechanic Jun 25 '20 at 15:37