Hamiltonian Time Evolution for General Relativity

TL;DR

This paper derives a Hamiltonian formulation for general relativity that allows explicit parameter time evolution even when constraints are not satisfied, providing a well-posed system for the constraints based on the ADM formalism.

Contribution

It introduces a Hamiltonian time evolution framework for general relativity with freely specified slicing density, extending the ADM formalism to non-constraint-satisfying scenarios.

Findings

Constraint algebra becomes a well-posed evolution system.

The system reduces to the twice-contracted Bianchi identity when R_{ij}=0.

The Hamiltonian constraint determines the metric volume element and lapse function.

Abstract

Hamiltonian time evolution in terms of an explicit parameter time is derived for general relativity, even when the constraints are not satisfied, from the Arnowitt-Deser-Misner-Teitelboim-Ashtekar action in which the slicing density $\alpha(x,t)$ is freely specified while the lapse $N=\alpha g^{1/2}$ is not. The constraint ``algebra'' becomes a well-posed evolution system for the constraints; this system is the twice-contracted Bianchi identity when $R_{ij}=0$. The Hamiltonian constraint is an initial value constraint which determines $g^{1/2}$ and hence $N$, given $\alpha$.