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This paper investigates the structure of gravitational singularities at thelevel of the connection. We show in particular that for FLRW spacetimes withparticle 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 usingan unbounded local holonomy we also give a general theorem establishing the$C^{0,1}_{\mathrm{loc}}$-inextendibility of spherically symmetric weak nullsingularities which arise at the Cauchy horizon in the interior of black holes.Our theorem does not presuppose the mass-inflation scenario and in particularapplies to the Reissner-Nordstr\"om-Vaidya spacetimes as well as to spacetimeswhich arise from small and generic spherically symmetric perturbations oftwo-ended subextremal Reissner-Nordstr\"om initial data for theEinstein-Maxwell-scalar field system. In [26], [27] Luk and Oh proved the$C^2$-formulation of strong cosmic censorship for this latter class ofspacetimes -- and based on their work we improve this to a$C^{0,1}_{\mathrm{loc}}$-formulation of strong cosmic censorship.

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