Suppose $g$ is a Lorentzian metric on $\mathbb{R}^4$, then consider the variational problem of finding extrema of

$$F(\gamma) = \int_a^b \| \dot{\gamma}(s) \|_g ds$$

The so-called "null geodesics" are defined as curves for which the integrand $\| \dot{\gamma}(s) \|_g$ is zero, i.e. they have zero "speed" measured by $g$.

Are such curves necessarily extrema of $F$? The fact that they are called "geodesics" suggests that this is true, but I have not found a proof anywhere...

Suppose $g$ is a Lorentzian metric on $\mathbb{R}^4$, then consider the variational problem of finding extrema of

$$F(\gamma) = \int_a^b \| \dot{\gamma}(s) \|_g ds$$

The so-called "null curves" are defined as curves for which the integrand $\| \dot{\gamma}(s) \|_g$ is zero, i.e. they have zero "speed" measured by $g$.

Are such curves necessarily extrema of $F$?

NOTE: I changed the wording of the question: I replaced "null geodesic" with "null curve"