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.

中文翻译：

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洛伦兹遇见利普希茨

我们表明，Lipschitz 连续洛伦兹度量的最大因果曲线承认 $\mathcal{C}^{1,1}$-参数化，并且他们在此参数化中解决了 Filippov 意义上的测地线方程。我们的证明表明，最大因果曲线要么无处不在，要么无处不在。此外，证明证明了 $\alpha$-Hölder 连续洛伦兹度量的最大因果曲线允许 $\mathcal{C}^{1,\frac{\alpha}{4}}$-参数化。