Solution

Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 52 / 1 / c / Solution

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.

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 Incoming links: Geodesic equation for a conformally flat metric