The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature

Abstract

In this work, we prove a synthetic splitting theorem for globally hyperbolic Lorentzian length spaces with global non-negative timelike curvature containing a complete timelike line. Just like in the case of smooth spacetimes, we construct complete, timelike asymptotes which, via triangle comparison, can be shown to fit together to give timelike lines. To get a control on their behaviour, we introduce the notion of parallelity of timelike lines in the spirit of the splitting theorem for Alexandrov spaces and show that asymptotic lines are all parallel. This helps to establish a splitting of a neighbourhood of the given line. We then show that this neighbourhood has the timelike completeness property and is hence inextendible, which globalises the local result.

43 pages, 6 figures. Updates in V2: local and global versions of the main theorem are incorporated in a single statement. Minor fixes throughout the document

References in corpus (6)

• Null distance and convergence of Lorentzian length spaces
• The singularity theorems of General Relativity and their low regularity extensions
• Comparison theorems for Lorentzian length spaces with lower timelike curvature bounds
• Gluing constructions for Lorentzian length spaces
• Gluing of Lorentzian length spaces and the causal ladder
• Synthetic versus distributional lower Ricci curvature bounds

Cited by in corpus (7)

• The singularity theorems of General Relativity and their low regularity extensions
• Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds
• Gluing of Lorentzian length spaces and the causal ladder
• A lower semicontinuous time separation function for $C^0$ spacetimes
• The representation of spacetime through time functions
• A Toponogov globalisation result for Lorentzian length spaces
• Concavity of spacetimes