Lorentz Meets Lipschitz

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.

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C.L. and A.L. were supported by the DFG-grant SFB/TRR 191 “Symplectic structures in Geometry, Algebra and Dynamics”, C.S. was supported by research grant J4305 of the Austrian Science Fund FWF.

Published 14 September 2022