Solving the Hamilton-Jacobi Equation for General Relativity

TL;DR

This paper introduces a systematic, gauge-invariant method for solving the Hamilton-Jacobi equation in general relativity with matter, enabling recursive solutions and applications like the Zel'dovich approximation.

Contribution

It presents a new recursive approach to solve the Hamilton-Jacobi equation in general relativity, including matter fields, using a gradient expansion and conformal transformations.

Findings

Derived solutions for irrotational dust and scalar fields at fourth order.

Established a recursive functional method for Hamilton-Jacobi solutions.

Demonstrated evolution of the 3-metric to high order.

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.