On holonomy singularities in general relativity and the $C^{0,1}_{\mathrm{loc}}$-inextendibility of spacetimes

TL;DR

This paper demonstrates that certain cosmological and black hole spacetimes exhibit unbounded holonomy near singularities, leading to their inextendibility in the $C^{0,1}_{\mathrm{loc}}$ class, thus supporting strong cosmic censorship.

Contribution

The paper establishes the $C^{0,1}_{\mathrm{loc}}$-inextendibility of FLRW and spherically symmetric weak null singularities using unbounded holonomy, extending previous results to more general settings.

Findings

Unbounded local holonomy near big-bang singularity in FLRW spacetimes.

Unbounded holonomy implies $C^{0,1}_{\mathrm{loc}}$-inextendibility of certain black hole interior spacetimes.

Extension of strong cosmic censorship to broader classes of spacetimes without mass-inflation assumption.

Abstract

This paper investigates the structure of gravitational singularities at the level of the connection. We show in particular that for FLRW spacetimes with particle horizons a local holonomy, which is related to a gravitational energy, becomes unbounded near the big-bang singularity. This implies the $C^{0,1}_{\mathrm{loc}}$-inextendibility of such FLRW spacetimes. Again using an unbounded local holonomy we also give a general theorem establishing the $C^{0,1}_{\mathrm{loc}}$-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-Nordstr\"om-Vaidya spacetimes as well as to spacetimes which arise from small and generic spherically symmetric perturbations of two-ended subextremal Reissner-Nordstr\"om initial data for the Einstein-Maxwell-scalar field system. In [26], [27] Luk and Oh proved the $C^2$-formulation of strong cosmic censorship for this latter class of spacetimes -- and based on their work we improve this to a $C^{0,1}_{\mathrm{loc}}$-formulation of strong cosmic censorship.