Points of General Relativisitic Shock Wave Interaction are "Regularity Singularities" where Spacetime is Not Locally Flat

TL;DR

This paper demonstrates that points of shock wave interaction in General Relativity create a new type of singularity where spacetime cannot be locally flattened by coordinate transformations, despite the metric remaining physically meaningful.

Contribution

It establishes that shock wave interaction points are 'regularity singularities' where the metric's regularity cannot be improved, revealing a novel class of physically relevant spacetime singularities.

Findings

Shock interactions prevent lifting metric regularity to $C^{1,1}$

Curvature remains bounded despite metric singularities

Shock interaction points form a new class of singularities

Abstract

We show that the regularity of the gravitational metric tensor in spherically symmetric spacetimes cannot be lifted from $C^{0,1}$ to $C^{1,1}$ within the class of $C^{1,1}$ coordinate transformations in a neighborhood of a point of shock wave interaction in General Relativity, without forcing the determinant of the metric tensor to vanish at the point of interaction. This is in contrast to Israel's Theorem which states that such coordinate transformations always exist in a neighborhood of a point on a smooth single shock surface. The results thus imply that points of shock wave interaction represent a new kind of singularity for perfect fluids evolving in spacetime, singularities that make perfectly good sense physically, that can form from the evolution of smooth initial data, but at which the spacetime is not locally Minkowskian under any coordinate transformation. In particular, at such singularities, delta function sources in the second derivatives of the gravitational metric tensor exist in all coordinate systems of the $C^{1,1}$ atlas, but due to cancelation, the curvature tensor remains uniformly bounded.