No Regularity Singularities Exist at Points of General Relativistic Shock Wave Interaction between Shocks from Different Characteristic Families

TL;DR

This paper proves that in spherically symmetric spacetimes, coordinate transformations can smooth the gravitational metric at shock wave collision points from Lipschitz continuous to twice differentiable, showing regularity singularities do not exist in this context.

Contribution

It provides a constructive proof that regularity singularities are absent at shock interactions from different characteristic families, extending Israel's work with a new proof strategy.

Findings

Coordinate transformations raise metric regularity from C^{0,1} to C^{1,1} near shock interactions.

Regularity singularities do not exist at shock wave interactions in spherically symmetric spacetimes.

The proof corrects previous misconceptions about the non-existence of such smooth transformations.

Abstract

We give a constructive proof that coordinate transformations exist which raise the regularity of the gravitational metric tensor from $C^{0,1}$ to $C^{1,1}$ in a neighborhood of points of shock wave collision in General Relativity. The proof applies to collisions between shock waves coming from different characteristic families, in spherically symmetric spacetimes. Our result here implies that spacetime is locally inertial and corrects an error in our earlier RSPA-publication, which led us to the false conclusion that such coordinate transformations, which smooth the metric to $C^{1,1}$, cannot exist. Thus, our result implies that regularity singularities, (a type of mild singularity introduced in our RSPA-paper), do not exist at points of interacting shock waves from different families in spherically symmetric spacetimes. Our result generalizes Israel's celebrated 1966 paper to the case of such shock wave interactions but our proof strategy differs fundamentally from that used by Israel and is an extension of the strategy outlined in our original RSPA-publication. Whether regularity singularities exist in more complicated shock wave solutions of the Einstein Euler equations remains open.