Ja n 19 99 Tetrad Gravity : II ) Dirac ’ s Observables

After a study of the Hamiltonian group of gauge transformations, whose infinitesimal generators are the 14 first class constraints of a new formulation of canonical tetrad gravity on globally hyperbolic, asymptotically flat at spatial infinity, spacetimes with simultaneity spacelike hypersurfaces Στ diffeomorphic to R3, the multitemporal equations associated with the constraints generating space rotations and space diffeomorphisms on the cotriads are given. Their solutions give the dependence of the cotriads on Στ and of their momenta on the six parameters associated with such transformations. The choice of 3-coordinates on Στ , namely the gauge fixing to the space diffeomorphisms constraints, is equivalent to the choice of how to parametrize the dependence of the cotriad on the last three degrees of freedom: namely to the choice of a parametrization of the superspace of 3-geometries. The Shanmugadhasan canonical transformation, corresponding to the choice of 3-orthogonal coordinates on Στ and adapted to 13 of the 14 first class constraints, is found and the superhamiltonian constraint is rewritten in this canonical basis. Some interpretational problems connected with Dirac’s observables are discussed: in particular the gauge interpretation of tetrad gravity based on constraint theory implies that a “Hamiltonian 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 gravitational field) connected by the gauge transformations generated by the constraints, in particular by the superhamiltonian constraint.