Dynamical extensions for shell-crossing singularities

TL;DR

This paper develops global weak solutions for spherically symmetric dust-filled spacetimes with shell-crossing singularities, using conservation laws and functions of bounded variation, replacing singularities with shock waves.

Contribution

It introduces a novel approach to solving Einstein's equations with shell-crossing singularities by employing weak solutions and shock wave techniques.

Findings

Solutions are weak and globally defined for shell-crossing singularities.

In the marginally bound case, solutions satisfy conservation laws.

Solutions are not unique after the shell-crossing, with metric discontinuities.

Abstract

We derive global weak solutions of Einstein's equations for spherically symmetric dust-filled space-times which admit shell-crossing singularities. In the marginally bound case, the solutions are weak solutions of a conservation law. In the non-marginally bound case, the equations are solved in a generalized sense involving metric functions of bounded variation. The solutions are not unique to the future of the shell-crossing singularity, which is replaced by a shock wave in the present treatment; the metric is bounded but not continuous.