Initial Hypersurface Formulation: Hamilton-Jacobi Theory for Strongly Coupled Gravitational Systems

TL;DR

This paper develops a Hamilton-Jacobi framework for strongly coupled gravitational systems, enabling analytical solutions for their evolution and initial conditions, with applications in cosmology and gravitational collapse.

Contribution

It extends previous Hamilton-Jacobi methods to include arbitrary initial hypersurfaces and formulates energy constraints using Lagrange multipliers, providing new tools for analyzing strongly coupled gravity.

Findings

Explicit Green functions for gravity with matter fields and cosmological constant.

Formulation of initial hypersurface conditions via Lagrange multipliers.

Application to cosmology and gravitational collapse problems.

Abstract

Strongly coupled gravitational systems describe Einstein gravity and matter in the limit that Newton's constant G is assumed to be very large. The nonlinear evolution of these systems may be solved analytically in the classical and semiclassical limits by employing a Green function analysis. Using functional methods in a Hamilton-Jacobi setting, one may compute the generating functional (`the phase of the wavefunctional') which satisfies both the energy constraint and the momentum constraint. Previous results are extended to encompass the imposition of an arbitrary initial hypersurface. A Lagrange multiplier in the generating functional restricts the initial fields, and also allows one to formulate the energy constraint on the initial hypersurface. Classical evolution follows as a result of minimizing the generating functional with respect to the initial fields. Examples are given describing Einstein gravity interacting with either a dust field and/or a scalar field. Green functions are explicitly determined for (1) gravity, dust, a scalar field and a cosmological constant and (2) gravity and a scalar field interacting with an exponential potential. This formalism is useful in solving problems of cosmology and of gravitational collapse.