Geometric Bounds in Spherically Symmetric General Relativity

TL;DR

This paper develops a framework for solving constraints in spherically symmetric general relativity using a family of extrinsic time foliations, enabling gauge-independent bounds on geometric quantities and revealing universal behavior near singularities.

Contribution

It introduces a large class of extrinsic time foliations that allow exact solutions of the momentum constraint and establishes gauge-independent bounds on geometric gradients.

Findings

Derived spatially invariant bounds on the areal radius gradients.

Identified properties of solutions independent of foliation.

Demonstrated universal behavior of geometry near singularities.

Abstract

We exploit an arbitrary extrinsic time foliation of spacetime to solve the constraints in spherically symmetric general relativity. Among such 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 have the property that the extrinsic curvature is spacelike with respect to the the spherically symmetric superspace metric. What is remarkable is that the linearity can be relaxed at no essential extra cost which permits us to isolate a large non - pathological dense subset of all extrinsic time foliations. We 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 bounds on the values of both the spatial and the temporal gradients of the scalar areal radius, $R$. These bounds are entirely independent of the particular gauge and of the magnitude of the sources. When singularities occur, we demonstrate that the geometry behaves in a universal way in the neighborhood of the singularity.