Solving the Hamilton-Jacobi Equation for General Relativity

Abstract

We demonstrate a systematic method for solving the Hamilton-Jacobi equation for general relativity with the inclusion of matter fields. The generating functional is expanded in a series of spatial gradients. Each term is manifestly invariant under reparameterizations of the spatial coordinates (``gauge-invariant''). At each order we solve the Hamiltonian constraint using a conformal transformation of the 3-metric as well as a line integral in superspace. This gives a recursion relation for the generating functional which then may be solved to arbitrary order simply by functionally differentiating previous orders. At fourth order in spatial gradients, we demonstrate solutions for irrotational dust as well as for a scalar field. We explicitly evolve the 3-metric to the same order. This method can be used to derive the Zel'dovich approximation for general relativity.