On weak solutions in Einstein theory and beyond

TL;DR

Shell-crossing singularities (SCS) in spherical dust collapse pose a challenge to general relativity because they push the problem into a regime where weak solutions of the conservation laws in Painlevé-Gullstrand (PG) coordinates are invoked to extend spacetime. The authors show that weak solutions are non-unique and, when one imposes metric continuity across the shock within a single PG chart, the resulting dynamics imply shocks that can move with superluminal speeds or cause signature changes, raising physical concerns. By contrast, applying Israel junction conditions with separate interior and exterior coordinates yields a physically consistent, signature-preserving evolution (e.g., $\\dot{R}^2 = \frac{R_S}{2R}\left(\frac{R_S}{8R}+1\right)$ for a classical thin shell) but requires a discontinuity in PG time across the shell. The work argues that weak solutions are not adequate for GR dust collapse and that a relativistic Rankine-Hugoniot formulation or an Israel-based treatment (possibly extended to effective quantum-corrected models) provides a more physical description of shocks beyond SCS, with a qualitative post-bounce scenario involving a time-like shock that eventually exits into a second asymptotic region. This has implications for modeling gravitational collapse and suggests directions for numerical implementations and extensions to more general distributions of matter.

Abstract

In spherical symmetry, gravitational collapse of dust may give rise to the so-called shell-crossing singularities, beyond which spacetime can be extended using weak solutions to the integrated version of the equations of motion. We argue that the paradigm of weak solutions is ill-suited for dynamical extension beyond shell crossings through shock waves because it is based implicitly on an assumption that turns out to be unphysical for the shock and leads to the unwanted prospect of shock waves of dust particles moving faster than light.