Geometrodynamics regained: A Lagrangian approach

Abstract

The closing relation between two super-Hamiltonians is cast into a condition on the super-Lagrangian by a functional Legendre transformation. It is shown that the ADM super-Lagrangian provides the unique representation of the ``group'' of deformations of a spacelike hypersurface embedded in a Riemannian space-time when the intrinsic geometry gij of the hypersurface is allowed as the sole configuration variable. No such uniqueness exists for the super-Lagrangians of source fields. As an illustration, the most general super-Lagrangian for a scalar field with nonderivative gravitational coupling is recovered from the closing relation.