  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
$\alpha$-H\"older continuous Lorentzian metric admit a
$\mathcal{C}^{1,\frac{\alpha}{4}}$-parametrization.
