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

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.

50 pages, Plain Tex, 1 figure available from the authors.

References in corpus (1)

• 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

Cited by in corpus (8)

• Bounds on 2m/R for static spherical objects
• Flat foliations of spherically symmetric geometries
• 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
• Sufficient Conditions for Apparent Horizons in Spherically Symmetric Initial Data
• Necessary Conditions for Apparent Horizons and Singularities in Spherically Symmetric Initial Data
• The inequality between size and charge in spherical symmetry
• Geometric Bounds in Spherically Symmetric General Relativity
• Trapped surfaces in cosmological spacetimes