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.

25 pages; v2: minor improvements of the presentation

References in corpus (3)

• Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications
• The global existence, uniqueness and C^1-regularity of geodesics in nonexpanding impulsive gravitational waves
• On holonomy singularities in general relativity and the $C_{\mathrm{loc}}^{0,1}$-inextendibility of spacetimes

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• 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