CONSTRAINTS IN SPHERICALLY SYMMETRICAL CLASSICAL GENERAL-RELATIVITY .2. IDENTIFYING THE CONFIGURATION-SPACE - A MOMENT OF TIME SYMMETRY

We continue our investigation of the configuration space of general re lativity begun in the preceding paper. Here we examine the Hamiltonian constraint when the spatial geometry is momentarily static (MS). We b egin with a heuristic description of the presence of apparent horizons and singularities. A peculiarity of MS configurations is that not onl y do they satisfy the positive quasilocal mass (QLM) theorem, they als o satisfy its converse: the QLM is positive everywhere, if and only if the (nontrivial) spatial geometry is nonsingular. We derive an analyt ical expression for the spatial metric in the neighborhood of a generi c singularity. The corresponding curvature singularity shows up in the traceless component of the Ricci tenser. As a consequence of the conv erse, if the energy density of matter is monotonically decreasing, the geometry cannot be singular. A. supermetric on the configuration spac e which distinguishes between singular geometries and nonsingular ones is constructed explicitly. Global necessary and sufficient criteria f or the formation of trapped surfaces and singularities are framed in t erms of inequalities which relate some appropriate measure of the mate rial 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 l(0) of the distribution, the geometry will possess an apparent horizon for o ne constant and a singularity for some other larger constant. A more a ppropriate measure of the material energy for casting the necessary cr iteria is the maximum value of the energy density of matter rho(max): if rho(max)l(0)(2) < some constant the distribution of matter will not possess a singularity for one constant and an apparent horizon for so me other smaller constant. These inequalities provide an approximate c haracterization of the singular (nonsingular) and trapped (nontrapped) partitions on the configuration space. Their strength is gauged by ex ploiting the exactly solvable piecewise constant density star as a tem plate. 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 O Murchadha and speculate on possible improvements.