On the uniqueness of continuous spacetime extensions in 1+1 dimensions with applications to weak null singularities

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-Nordstr\"om 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.