GEOMETRIC BOUNDS IN SPHERICALLY SYMMETRICAL GENERAL-RELATIVITY

We exploit an arbitrary extrinsic time foliation of spacetime to solve the constraints in spherically symmetric general relativity. Among su ch foliations there is a one parameter family, linear and homogeneous in the extrinsic curvature, which permit the momentum constraint to be solved exactly. This family includes, as special cases, the extrinsic time gauges that have been exploited in the past. These foliations ha ve the property that the extrinsic curvature is spacelike with respect to the the spherically symmetric superspace metric. What is remarkabl e is that the linearity can be relaxed at no essential extra cost whic h permits us to isolate a large nonpathological dense subset of all ex trinsic time foliations. We now identify properties of solutions which are independent of the particular foliation within this subset. When the geometry is regular, we can place spatially invariant numerical bo unds on the values of both the spatial and the temporal gradients of t he scalar areal radius R. These bounds are entirely independent of the particular gauge and of the magnitude of the sources. When singularit ies occur, we demonstrate that the geometry behaves in a universal way in the neighborhood of the singularity. These results can be exploite d to develop necessary and sufficient conditions for the existence of both apparent horizons and singularities in the initial data which do not depend sensitively on the foliation. [S0556-2821(97)05224-7].