Let $(N, g)$ be an $n$-dimensional Riemannian manifold and $S$ be a submanifold of $N$ of dimension $m$. For $X \in \Gamma(TS)$ define a family of variations in $M$ to be $$\varphi: S \times (-\epsilon, \epsilon) \rightarrow M, \hspace{5mm} \varphi(x, s) = \exp_x(sX)$$
so that the surfaces $\varphi_s$ are generated by geodesics eminating from $S$. Then $\frac{\partial \varphi}{\partial s}|_{s=0} = X$ is the variation field. So the First Variation Formula for any fixed surface $\varphi_s$ is
$${\frac{d}{ds}}\left({Vol}(\varphi_s)\right)_{\{s=0\}} = {\int_{\varphi_s}({div}_{\varphi}X)\Omega_{\varphi_s}}$$
I'm wondering if we let $N$ be Lorentzian with $S$ spacelike if this formula still makes sense? I can't see any mention of this generalisation anywhere but see no reason for it not to be true. The only worry I have is if $X$ is null then the zero inner product could potentially lead to singularity issues. However I only see hand-wavey potential problems, nothing rigorous.
I'm interested in people's thoughts, in particular whether or not $X$'s being null makes a difference.