The 'Regularity Singularity' at Points of General Relativistic Shock Wave Interaction

Abstract

A proof that a new kind of non-removable {\it "regularity singularity"} forms when two shock waves collide within the theory of General Relativity, was first announced in ProcRoySoc A \cite{ReintjesTemple}. In the present paper we give complete proofs of the claims in \cite{ReintjesTemple} and extend the results on the regularity of the Einstein curvature tensor to the full Riemann curvature tensor. The main result is that, in a neighborhood of a point where two shock waves collide in a spherically symmetric spacetime, the gravitational metric tensor cannot be lifted from C0,1 to C1 within the class of C1,1 coordinate transformations. This contrasts Israel's celebrated theorem \cite{Israel}, which states that around each point on a {\it single} shock surface there exist a coordinate system in which the metric is C1,1 regular. Moreover, at points of shock wave interaction, delta function sources exist in the second derivatives of the gravitational metric tensor in all coordinate systems of the C1,1-atlas, but due to cancellation, the Einstein and Riemann curvature tensor remain sup-norm bounded. We conclude that points of shock wave interaction are a new kind of spacetime singularity, (which we name "regularity singularity"), singularities that can form from the evolution of smooth initial data for perfect fluids and that lie in physical spacetime, but at such points {\it locally inertial} coordinates fail to exist.