Abstract
The study of low regularity (in-)extendibility of Lorentzian manifolds is motivated by the question whether a given solution to the Einstein equations can be extended (or is maximal) as a weak solution. In this paper we show that a timelike complete and globally hyperbolic Lorentzian manifold is C0-inextendible. For the proof we make use of the result, recently established by Sämann (Ann Henri Poincaré 17(6):1429–1455, 2016), that even for continuous Lorentzian manifolds that are globally hyperbolic, there exists a length-maximizing causal curve between any two causally related points.
Highlights
A smooth Lorentzian manifold is said to be Ck-inextendible if it cannot be isometrically embedded as a proper subset into another Lorentzian manifold of the same dimension with a Ck-regular metric
It is a classical result that a smooth timelike geodesically complete Lorentzian manifold is C2-inextendible
Given a complete Riemannian manifold (M, g), it holds that it is C0-inextendible: assuming there exists a C0-extension, one considers a neighborhood of a boundary point and finds a length-minimizer that connects a point in M to this boundary point
Summary
A smooth Lorentzian manifold is said to be Ck-inextendible if it cannot be isometrically embedded as a proper subset into another Lorentzian manifold of the same dimension with a Ck-regular metric. A smooth (at least C2) time-oriented Lorentzian manifold that is timelike geodesically complete and globally hyperbolic is C0-inextendible. Given a complete Riemannian manifold (M, g), it holds that it is C0-inextendible: assuming there exists a C0-extension, one considers a neighborhood of a boundary point and finds a length-minimizer that connects a point in M to this boundary point. The portion of this curve in M has to be an inextendible geodesic, which by the assumption of completeness has to have infinite length. The proofs of Theorems 1 and 2 are given in Sect. 3, where we conclude with some applications
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