Null-gradient affine-geodesic identity ($k_a={\nabla }_{a}S,k^2=0\Rightarrow {\nabla }_{k}k=0$)

If $k_a={\nabla }_{a}S$ is a null gradient, symmetry of the Hessian gives $k^a{\nabla }_a{k}_b=k^a{\nabla }_b{k}_a=\frac{1}{2}{\nabla }_b(k^2)=0$. Its integral curves are therefore affinely parametrized null geodesics, rather than merely null curves with a nonaffinity term.