A Bubble-Time Canonical Formalism for Geometrodynamics

A functional differential version of the ADM canonical formalism is proposed. The existence of a canonical transformation separating canonical variables into internal coordinates, energy-momentum densities, and two pairs of true dynamical variables is assumed. The evolution of dynamical variables is governed by functional differential Hamilton's equations. They satisfy certain integrability conditions ensuring the internal path independence of dynamical evolution. The change of dynamical variables along any spacelike hypersurface is given by their Lie derivatives. This allows an elimination of 3∞3 components of the Hamilton equation, leading to a functional differential Hamilton equation based on a single bubble time. The Hamilton-Jacobi theory is built along the same lines. The formalism is illustrated in the mini-phase-space of the cylindrical Einstein-Rosen wave.