1112.1803 (Moritz Reintjes)

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

Moritz Reintjes

We show that the regularity of the gravitational metric tensor in spherically symmetric spacetimes cannot be lifted from $C^{0,1}$ to $C^{1,1}$ by any $C^{1,1}$ coordinate transformation in a neighborhood 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 neighborhood of a point on a smooth {\it single} shock surface. The results thus imply that points of shock wave interaction represent a new kind of spacetime singularity for perfect fluids, singularities that lie in physical spacetime, that can form from the evolution of smooth initial data, but at which the spacetime is not {\it locally Minkowskian} under any coordinate transformation. In particular, at such {\it regularity singularities}, delta function sources in the second derivatives of the gravitational metric tensor exist in all coordinate systems, but due to cancelation, the curvature tensor remains bounded (in $L^\infty$). We announced the main results of this paper in a recent article in the Proceedings of the Royal Society A, [arXiv:1105.0798], where we also sketched the main steps in some of the proofs. In this paper, those results together with their complete proofs and some additional propositions are presented.

View original: http://arxiv.org/abs/1112.1803