Timelike completeness as an obstruction to $C^0$-extensions

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 $C^0$-inextendible. For the proof we make use of the result, recently established by Sämann [17], that even for \emph{continuous} Lorentzian manifolds that are globally hyperbolic, there exists a length-maximizing causal curve between any two causally related points.

v2: pertinent citations added and minor changes, v3: small refinements and improvements; theorem added, v4: added figure, minor changes, version accepted for publication in CMP

References in corpus (2)

• Limit curve theorems in Lorentzian geometry
• Some Remarks on the $C^0$-(in)extendibility of Spacetimes

Cited by in corpus (29)

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• Globally hyperbolic spacetimes can be defined without the 'causal' condition
• The future is not always open
• On the causal hierarchy of Lorentzian length spaces
• Properties of the Null Distance and Spacetime Convergence
• The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature
• The singularity theorems of General Relativity and their low regularity extensions
• On the initial singularity and extendibility of flat quasi-de Sitter spacetimes
• The Hawking-Penrose singularity theorem for $C^1$-Lorentzian metrics
• A Lorentzian analog for Hausdorff dimension and measure
• Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds
• On geodesics in low regularity
• Aspects of $C^0$ causal theory
• Some regularity results for Lorentz-Finsler spaces
• Lorentz meets Lipschitz
• A Note on the Gannon-Lee Theorem
• Polynomial $f(R)$ Palatini cosmology -- dynamical system approach
• Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems
• Remarks on the cosmological constant appearing as an initial condition for Milne-like spacetimes
• Uniqueness and non-uniqueness results for spacetime extensions
• Causal completions as Lorentzian pre-length spaces
• On the asymptotic assumptions for Milne-like spacetimes
• Milne-like spacetimes and their role in Cosmology
• Volume Singularities in General Relativity
• Low regularity extensions beyond Cauchy horizons
• Uniqueness of maximal spacetime boundaries
• A lower semicontinuous time separation function for $C^0$ spacetimes