Hamiltonian formalism in einsteinian gravitation: Geometrical aspect

Abstract

In the Hamiltonian formulation of relativistic theory of gravitation the gravitational field at a given time is defined on a spacelike hypersurface of Riemannian space-time by six gravitational potentials and as many conjugate momenta. The former characterize the metric of the hypersurface and it is shown that the momenta are constructed tensorially only with the two fundamental tensors of the hypersurface since Dirac's velocities are equated with the components of the second fundamental tensor. Invariants of the two fundamental tensors, geometrically interpretable, can then be introduced into the Hamiltonian. It is at once clear from the Gauss and Codazzi equations, relating the two fundamental tensors of a hypersurface that the four secondary constraints, which the conjugate dynamical variables satisfy, are equivalent to four Einstein gravitational equations.