The future is not always open

Abstract

We demonstrate the breakdown of several fundamentals of Lorentzian causality theory in low regularity. Most notably, chronological futures (defined naturally using locally Lipschitz curves) may be non-open, and may differ from the corresponding sets defined via piecewise $C^1$-curves. By refining the notion of a causal bubble from [CG:12],we characterize spacetimes for which such phenomena can occur, and also relate these to the possibility of deforming causal curves of positive length into timelike curves (push-up). The phenomena described here are, in particular, relevant for recent synthetic approaches to low regularity Lorentzian geometry where, in the absence of a differentiable structure, causality has to be based on locally Lipschitz curves.

Minor amendments. Final version. 17 pages, 4 figures

References in corpus (6)

• Causality theory for closed cone structures with applications
• Limit curve theorems in Lorentzian geometry
• Inextendibility of spacetimes and Lorentzian length spaces
• Maximizers in Lipschitz spacetimes are either timelike or null
• On the proof of the $C^0$-inextendibility of the Schwarzschild spacetime
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Cited by in corpus (21)

• Singularity theorems for $C^1$-Lorentzian metrics
• The singularity theorems of General Relativity and their low regularity extensions
• 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
• Aspects of $C^0$ causal theory
• A Note on the Gannon-Lee Theorem
• Time functions on Lorentzian length spaces
• Causality theory of spacetimes with continuous Lorentzian metrics revisited
• Remarks on the cosmological constant appearing as an initial condition for Milne-like spacetimes
• Uniqueness and non-uniqueness results for spacetime extensions
• On the asymptotic assumptions for Milne-like spacetimes
• Synthetic versus distributional lower Ricci curvature bounds
• Further observations on the definition of global hyperbolicity under low regularity
• Cut-and-paste for impulsive gravitational waves with $\Lambda$: The mathematical analysis
• Low regularity extensions beyond Cauchy horizons
• A lower semicontinuous time separation function for $C^0$ spacetimes
• A Toponogov globalisation result for Lorentzian length spaces
• Results on Lorentzian metric spaces
• Causal bubbles in globally hyperbolic spacetimes
• Ordered Locales