The Constraints in Spherically Symmetric General Relativity I --- Optical Scalars, Foliations, Bounds on the Configuration Space Variables and the Positivity of the Quasi-Local Mass

TL;DR

This paper analyzes the constraints in spherically symmetric general relativity, introducing new variables, foliations, and bounds to establish positivity of quasi-local mass and characterize the configuration space.

Contribution

It introduces a new family of spacetime foliations, establishes positivity of quasi-local mass, and characterizes the configuration space using optical scalar variables.

Findings

Quasi-local mass is positive when constraints are satisfied.

A family of valid spacetime foliations is identified.

Gravitational binding energy is always negative.

Abstract

We examine the constraints of spherically symmetric general relativity with one asymptotically flat region, exploiting both the traditional metric variables and variables constructed from the optical scalars. With respect to the latter variables, there exist two linear combinations of the Hamiltonian and momentum constraints which are related by time reversal. We introduce a one-parameter family of linear extrinsic time foliations of spacetime. The values of the parameter yielding globally valid gauges correspond to the vanishing of a timelike vector in the superspace of spherically symmetric geometries. We define a quasi-local mass on spheres of fixed proper radius which we prove is positive when the constraints are satisfied. Underpinning the proof are various local bounds on the configuration variables. We prove that a reasonable definition of the gravitational binding energy is always negative. Finally, we provide a tentative characterization of the configuration space of the theory in terms of closed bounded trajectories on the parameter space of the optical scalars.