Spacetime is locally inertial at points of general relativistic shock wave interaction between shocks from different characteristic families

We prove that spacetime is locally inertial at points of shock wave collision in General Relativity. The result applies for collisions between shock waves from different characteristic families in spherically symmetric spacetimes. We give a constructive proof that there exists 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 was first announced in [16] and the proofs are presented here. This result corrects an error in our earlier publication [15], which led us to the wrong conclusion that such coordinate transformations, which smooth the metric to C1,1, cannot exist. Our result here proves that regularity singularities, (a type of mild singularity introduced in [15]), do not exist at points of two 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 [15], 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.

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M. R. was supported by the Deutsche Forschungsgemeinschaft (DFG), Grant Number RE 3471/2-1, from January 2013 until December 2014. From January 2015 until December 2016, M. R. was supported by CAPES-Brasil, as a Post-Doc of Excellence at IMPA (Instituto Nacional de Matemática Pura e Aplicada) in Rio de Janeiro, Brazil.

Published 14 March 2018