Constraints in spherically symmetric classical general relativity. II. Identifying the configuration space: A moment of time symmetry

We continue our investigation of the configuration space of general relativity begun in the preceding paper. Here we examine the Hamiltonian constraint when the spatial geometry is momentarily static (MS). We begin with a heuristic description of the presence of apparent horizons and singularities. A peculiarity of MS configurations is that not only do they satisfy the positive quasilocal mass (QLM) theorem, they also satisfy its converse: the QLM is positive everywhere, if and only if the (nontrivial) spatial geometry is nonsingular. 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. As a consequence of the converse, 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 nonsingular 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 some appropriate measure of the material energy content on a given support to a measure of its volume. The sufficiency criteria are cast in the following form: if the material energy exceeds some universal constant times the proper radius ${\mathit{l}}_0$ of the distribution, the geometry will possess an apparent horizon for one constant and a singularity for some other larger constant.

A more appropriate measure of the material energy for casting the necessary criteria is the maximum value of the energy density of matter ${\mathrm{\rho }}_{\max }$: if ${\mathrm{\rho }}_{\max }$${\mathit{l}}_0^2$< some constant the distribution of matter will not possess a singularity for one constant and an apparent horizon for some other smaller constant. These inequalities provide an approximate characterization of the singular (nonsingular) and trapped (nontrapped) partitions on the configuration space. Their strength is gauged by exploiting the exactly solvable piecewise constant density star as a template. Finally, we provide a more transparent derivation of the lower bound on the binding energy conjectured by Arnowitt, Deser, and Misner and proven by Bizon, Malec, and Ó Murchadha and speculate on possible improvements.