Spacetime is Locally Inertial at Points of General Relativistic Shock Wave Interaction between Shocks from Different Characteristic Families

Abstract

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