Abstract:Motivated by weak null singularities in black hole interiors, we study 1+1 dimensional Lorentzian manifolds $(M,g)$ which admit a continuous spacetime extension across a null boundary $v=0$, where $v<0$ is a null coordinate. We study the degree to which such extensions are unique up to the boundary. Firstly, we find that in general not even the $C^0$-structure of the extension is uniquely determined by the assumption that the metric extends continuously. However, we exhibit an interesting local-global relation regarding the $C^0$-structure which in particular entails its rigidity for ''strongly spherically symmetric'' continuous extensions across the Cauchy horizon of the Reissner-Nordström spacetime. Secondly, we construct continuous extensions which have the same $C^0$-structure, but do not have equivalent $C^1$-structures. This construction also carries over to weak null singularities in 3+1 dimensions. Understanding the uniqueness properties of continuous spacetime extensions to the boundary is of importance for the study of low-regularity inextendibility problems.
From: Peter Cameron [view email]
[v1]
Mon, 17 Nov 2025 14:34:31 UTC (857 KB)
[v2]
Tue, 23 Jun 2026 07:14:10 UTC (858 KB)
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