Lorentz meets Lipschitz

Abstract

We show that maximal causal curves for a Lipschitz continuous Lorentzian metric admit a ${\mathcal{C}}^{1,1}$-parametrization and that they solve the geodesic equation in the sense of Filippov in this parametrization. Our proof shows that maximal causal curves are either everywhere lightlike or everywhere timelike. Furthermore, the proof demonstrates that maximal causal curves for an $\alpha$-Hölder continuous Lorentzian metric admit a ${\mathcal{C}}^{1,\frac{\alpha }{4}}$-parametrization.

arXiv comment25 pages; v2: minor improvements of the presentation

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Cited by in corpus (9)

• The singularity theorems of General Relativity and their low regularity extensions
• The Hawking-Penrose singularity theorem for $C^1$-Lorentzian metrics
• Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds
• A Note on the Gannon-Lee Theorem
• Remarks on the cosmological constant appearing as an initial condition for Milne-like spacetimes
• On the asymptotic assumptions for Milne-like spacetimes
• Spacetime singularities and curvature blow-ups
• Cut-and-paste for impulsive gravitational waves with $\Lambda$: The mathematical analysis
• A lower semicontinuous time separation function for $C^0$ spacetimes