On holonomy singularities in general relativity and the Cloc0,1-inextendibility of space-times

Abstract

This paper investigates the structure of gravitational singularities at the level of the connection. We show in particular that for FLRW space-times with particle horizons a local holonomy, which is related to a gravitational energy, becomes unbounded near the big-bang singularity. This implies the $C_{\mathrm{loc}}^{0,1}$-inextendibility of such FLRW space-times. Again using an unbounded local holonomy, we also give a general theorem establishing the $C_{\mathrm{loc}}^{0,1}$-inextendibility of spherically symmetric weak null singularities which arise at the Cauchy horizon in the interior of black holes. Our theorem does not presuppose the mass-inflation scenario and in particular applies to the Reissner–Nordström-Vaidya space-times, as well as to space-times which arise from small and generic spherically symmetric perturbations of two-ended subextremal Reissner–Nordström initial data for the Einstein–Maxwell scalar field system. In previous work, Luk and Oh proved the $C^2$-formulation of strong cosmic censorship for this latter class of space-times—and based on their work we improve this to a $C_{\mathrm{loc}}^{0,1}$-formulation of strong cosmic censorship.