Hamilton-Jacobi theory for general relativity with matter fields

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Abstract
A nonlinear perturbative calculational scheme for solving the Hamilton-Jacobi equation for general relativity with matter fields is described. By taking advantage of the invariance of the Hamilton-Jacobi equation under spatial reparametrizations, the generating functional may be written in terms of a spatial gradient expansion. The solution in superspace, which describes an ensemble of inhomogeneous universes, may be reduced to a solution of a finite dimensional field space at each order of the expansion. Exact solutions of the separated Hamilton-Jacobi equations of order zero and two are given for gravity interacting with dust and scalar fields. Dust fields allow for a quantum formulation with a positive-conserved probability density. Contrary to what is usually required, the new Hamiltonian density does not vanish strongly, but may be written in a form which is quadratic in weakly vanishing terms. The authors' approach is covariant, and they show how to introduce an arbitrary time parameter into the Hamilton-Jacobi formalism. This formalism may be applied with profit to the nonlinear evolution of dust fields in matter-dominated as well as vacuum-dominated universes (inflationary cosmologies).