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 C_1,1 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.