The Constraints in Spherically Symmetric General Relativity II --- Identifying the Configuration Space: A Moment of Time Symmetry

TL;DR

This paper analyzes the configuration space of general relativity under momentary static conditions, deriving criteria for singularity formation and trapped surfaces, and providing explicit metrics and inequalities related to energy content and geometry.

Contribution

It introduces a supermetric distinguishing singular from non-singular geometries and establishes global criteria for trapped surface and singularity formation in the context of momentary static configurations.

Findings

MS configurations satisfy positive QLM theorem and its converse

Derived an analytical expression for the spatial metric near singularities

Established inequalities relating energy content to volume for singularity criteria

Abstract

We continue our investigation of the configuration space of general relativity begun in I (gr-qc/9411009). Here we examine the Hamiltonian constraint when the spatial geometry is momentarily static (MS). We show that MS configurations satisfy both the positive quasi-local mass (QLM) theorem and its converse. We derive an analytical expression for the spatial metric in the neighborhood of a generic singularity. The corresponding curvature singularity shows up in the traceless component of the Ricci tensor. We show that if the energy density of matter is monotonically decreasing, the geometry cannot be singular. A supermetric on the configuration space which distinguishes between singular geometries and non-singular ones is constructed explicitly. Global necessary and sufficient criteria for the formation of trapped surfaces and singularities are framed in terms of inequalities which relate appropriate measures of the material energy content on a given support to a measure of its volume. The strength of these inequalities is gauged by exploiting the exactly solvable piece-wise constant density star as a template.