• Corpus ID: 251594504

Uniqueness and non-uniqueness results for spacetime extensions

@inproceedings{Sbierski2022UniquenessAN,
  title={Uniqueness and non-uniqueness results for spacetime extensions},
  author={Jan Sbierski},
  year={2022}
}

• Jan Sbierski
• Published 16 August 2022
• Mathematics

Given a function f : A → R n of a certain regularity defined on some open subset A ⊆ R m , it is a classical problem of analysis to investigate whether the function can be extended to all of R m in a certain regularity class. If an extension exists and is continuous, then certainly it is uniquely determined on the closure of A . A similar problem arises in general relativity for Lorentzian manifolds instead of functions on R m . It is well-known, however, that even if the extension of a Lorentzian manifold ( M, g ) is analytic, various choices are in general possible at the boundary. This paper establishes a uniqueness condition for extensions of globally hyperbolic Lorentzian manifolds ( M, g ) with a focus on low regularities: any two extensions which are anchored by an inextendible causal curve γ : [ − 1 , 0) → M in the sense that γ has limit points in both extensions, must agree locally around those limit points on the boundary as long as the extensions are at least locally Lipschitz continuous . We also show that this is sharp: anchored extensions which are only H¨older continuous do in general not enjoy this local uniqueness result.