Tetrad Gravity: II) Dirac's Observables

Abstract

After a study of the Hamiltonian group of gauge transformations of canonical tetrad gravity on globally hyperbolic, asymptotically flat at spatial infinity, spacetimes with Cauchy hypersurfaces $Σ_τ$ diffeomorphic to $R^3, we find the dependence of the cotriads on $Σ_τ$ and of their momenta on the six parameters associated with rotations and space diffeomorphisms. The choice of 3-coordinates on $Σ_τ$ is equivalent to the parametrization of the cotriads with the last three degrees of freedom individuating the 3-geometries. The Shanmugadhasan canonical transformation, corresponding to the choice of 3-orthogonal coordinates on $Σ_τ$ and adapted to 13 of the 14 first class constraints, and the interpretation of the gauge transformations are given. The gauge interpretation of tetrad gravity based on constraint theory implies that a "Hamiltonian kinematical gravitational field" is an equivalence class of pseudo-Riemannian spacetimes modulo the Hamiltonian group of gauge transformations: it includes a conformal 3-geometry and all the different 4-geometries (standard definition of a kinematical gravitational field, $Riem M^4/Diff M^4$) connected to it by the gauge transformations. A "Hamiltonian Einstein or dynamical gravitational field" is a kinematical one which satisfies the Hamilton-Dirac equations generated by the ADM energy: it coincides with the standard Einstein or dynamical gravitational field, namely a 4-geometry solution of Einstein's equations, since the Hilbert and ADM actions both generate Einstein's equations so that the kinematical Hamiltonian gauge transformations are dynamically restricted to the spacetime diffeomorphisms of the solutions of Einstein's equations.