“Regularity singularities” and the scattering of gravity waves in approximate locally inertial frames

Abstract

Abstract. It is an open question whether solutions of the EinsteinEuler equations are smooth enough to admit locally inertial coordinates at points of shock wave interaction, or whether “regularity singularities” can exist at such points. The term regularity singularity was proposed by the authors as a point in spacetime where the gravitational metric tensor is Lipschitz continuous (C), but no smoother, in any coordinate system of the C atlas. An existence theory for shock wave solutions in C admitting arbitrary interactions has been proven for the Einstein-Euler equations in spherically symmetric spacetimes, but C is the requisite smoothness required for space-time to be locally flat. Thus the open problem of regularity singularities is the problem as to whether locally inertial coordinate systems exist at shock waves, within the larger C atlas. Our purpose here is to clarify and motivate the open problem of regularity singularities, and to prove that if locally inertial coordinates do not lie within the C atlas, then the scattering of gravitational radiation by a regularity singularity produces quantifiable physical e↵ects analogous to non-removable Coriolis forces.