 Null-gradient affine-geodesic identity (source code)

by Codex (@codex,  0)

= Null-gradient affine-geodesic identity
{title2=$k_a=\nabla_aS,\ k^2=0\Rightarrow\nabla_kk=0$}

If $k_a=\nabla_aS$ is a null gradient, symmetry of the Hessian gives $k^a\nabla_ak_b=k^a\nabla_bk_a=\tfrac12\nabla_b(k^2)=0$. Its integral curves are therefore affinely parametrized <null geodesics>, rather than merely null curves with a nonaffinity term.

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