I"m trying to understand the exact/technical link between the Eikonal equation and a double-null form of the metric (if such a direct link even exists). R. Wald, in his "General Relativity", doesn't say anything about it.

A function $f$ satisfies (per definition) the Eikonal equation if

$g(\nabla f,\nabla f)=0$

i.e. the gradient field $\nabla f$ is a null vector field.

On the other hand, a metric $g$ has double-null coordinates $(u,v)$ if $g=h+F\dot dudv$ (i.e. no $du^{2}$ and $dv^{2}$ terms appear).

I was wondering if there is a direct link between this function $f$ and the metric double null form?

Thank you for any hints.