Points in general relativistic shock wave interaction are “regularity singularities” where spacetime is not locally flat

Abstract

We show that the regularity of the gravitational metric tensor in spher-
ically symmetric spacetimes cannot be lifted from C0,1 to C1,1 within the class of
C1,1 coordinate transformations in a neighborhood of a point of shock wave inter-
action 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 [5]
which states that such coordinate transformations always exist in a neighborhood
of a point on a smooth single shock surface. The results thus imply that points of
shock wave interaction represent a new kind of singularity for perfect fluids evolving
in spacetime, singularities that make perfectly good sense physically, that can form
from the evolution of smooth initial data, but at which the spacetime is not locally
Minkowskian under any coordinate transformation. In particular, at such singu-
larities, delta function sources in the second derivatives of the gravitational metric
tensor exist in all coordinate systems of the C1,1 atlas, but due to cancelation, the
curvature tensor remains uniformly bounded.