Points of general relativistic shock wave interaction are ‘regularity singularities’ where space–time is not locally flat

Abstract

We show that the regularity of the gravitational metric tensor in spherically symmetric space-times cannot be lifted from C 0,1 to C 1,1 within the class of C 1,1 coordinate transformations in a neighbourhood 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 neighbourhood of a point on a smooth single shock surface. The results thus imply that points of shock wave interaction represent a new kind of regularity singularity for perfect fluids evolving in space-time, singularities that make perfectly good sense physically, that can form from the evolution of smooth initial data, but at which the space-time is not locally Minkowskian under any coordinate transformation. In particular, at regularity singularities, delta function sources in the second derivatives of the metric exist in all coordinate systems of the C 1,1 -atlas, but due to cancellation, the full Riemann curvature tensor remains supnorm bounded.

Points of general relativistic shock wave interaction are ‘regularity singularities’ where space–time is not locally flat

“…However, the larger question as to whether regularity singularities exist in more complicated shock wave solutions of the Einstein-Euler equations remains an open problem. Thus, the larger issues laid out in Reintjes & Temple [1] remain valid.…”

“…We now identify the mistake in Reintjes & Temple [1]. To start, all of the mathematics in section 2-6 of Reintjes & Temple [1] remains correct without change.…”

“…With apologies, we must inform the reader that the result announced in our RSPA publication [1] is false. Although the issues and motivations regarding the problem as to whether regularity singularities exist at points of shock wave interaction in GR set out in Reintjes & Temple [1] are valid, there is a flaw in the final chapter of our proof that cannot be fixed.…”

“…The mistake is in the last section, and this propagates into a mistake in the proof of the main theorem. Specifically, our proof in Reintjes & Temple [1] is based on the violation of the identity (7.9), which according to the argument there, is required for the smoothing coordinates to exist. But, this violation was due to the error of neglected terms that should have been present in our ansatz for the Jacobians in To construct Jacobians that satisfy the smoothing condition, we correct the expression for the Jacobians J μ α in (7.1) of Reintjes [2] as follows…”

“…To start, all of the mathematics in section 2-6 of Reintjes & Temple [1] remains correct without change. The mistake is in the last section, and this propagates into a mistake in the proof of the main theorem.…”