Spherically symmetric inextendibility of weak null singularities with Christoffel symbols in $L^s_{\text{loc}}$

Abstract

Motivated by the strong cosmic censorship conjecture in general relativity, we prove the inextendibility of spherically symmetric weak null singularities as spherically symmetric Lorentzian manifolds with a continuous metric and Christoffel symbols in $L^{s}_{\text{loc}}$ for $s>1$. The result assumes a suitable blow-up condition on the derivative of the area-radius function transverse to the weak null singularity in combination with a rigidity result on continuous spherically symmetric extensions across null boundaries proven in [1]. In particular we show that these assumptions are satisfied by the Reissner-Nordström-Vaidya spacetime as well as by a class of spacetimes arising from small and generic spherically symmetric perturbations of subextremal Reissner-Nordström under the Einstein-Maxwell-scalar field system.