Geodesic equation for a conformally flat metric
= Geodesic equation for a conformally flat metric
For $g_{\mu\nu}=\varphi^2\eta_{\mu\nu}$,
$$
\Gamma^\mu_{\alpha\beta}
=\delta^\mu_\alpha\partial_\beta\log\varphi
+\delta^\mu_\beta\partial_\alpha\log\varphi
-\eta_{\alpha\beta}\partial^\mu\log\varphi.
$$
For a null curve, the last term drops from the <geodesic equation>; the remaining acceleration is parallel to the tangent and can be removed by reparametrization.