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

TL;DR

This paper investigates the uniqueness of continuous spacetime extensions across null boundaries in 1+1 dimensions, revealing non-uniqueness at the $C^1$ level and establishing conditions for rigidity in spherically symmetric cases, with implications for higher dimensions.

Contribution

It demonstrates that continuous extensions are not uniquely determined by the metric's continuity, identifies conditions for rigidity in symmetric cases, and constructs examples showing non-uniqueness at higher regularity levels.

Findings

$C^0$-structure is not uniquely determined by continuity.

Rigidity holds for strongly spherically symmetric extensions.

Constructed examples show non-uniqueness at $C^1$ level.

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-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.