On a Riemannian manifold $(M,g)$, pregeodesics can be defined either of the following two equivalent ways:
- curves for which the (Levi-Civita) acceleration $\nabla_{\dot\gamma(t)}\dot\gamma(t)$ is proportional to $\dot\gamma(t)$;
- stationary points of the length functional $\ell[\gamma] = \int_a^b\sqrt{|g(\dot\gamma(t),\dot\gamma(t))|}\mathbb d t$.
On a (pseudo-)Riemannian manifold $(M,g)$ pregeodesics can still be defined according to (1), but (2) now leads to some trouble, related to the fact that the integrand $\sqrt{|g(\dot\gamma(t),\dot\gamma(t))|}$ is not differentiable at values of $t\in(a,b)$ for which $g(\dot\gamma(t),\dot\gamma(t))=0$, rendering the function $\epsilon\mapsto S[\gamma+\epsilon \eta]$ nondifferentiable (in general) and hence the variation
$$\frac{\text{d}}{\text{d}\epsilon}\bigg|_{\epsilon=0}S[\gamma+\epsilon \eta]$$
ill defined. It seems to me that, as as long as we restrict ourselves to curves $\gamma$ for which $g(\dot\gamma(t),\dot\gamma(t))$ is nowhere vanishing, the definition (2) is still well-defined and in fact equivalent to (1), but it seems there is no obvious way to extend (2) to curves for which $g(\dot\gamma(t),\dot\gamma(t))$ is not necessarily everywhere nonvanishing.
Question: Is there a precise sense in which we can say that pregeodesics, even those for which $g(\dot\gamma(t),\dot\gamma(t))$ is not nowhere vanishing, are stationary points of the length functional?