general relativity - Can you recover a spacetime from its null geodesics?

So, I know that you can learn a lot about a spacetime from its causal structure, but can one completely recover the metric of a spacetime, just knowing the equations for the null geodesics in it?

If so, how would one go about doing this?

Answer

You can only recover conformally related spacetimes from its null geodesics, that is, the class of spacetimes related by the transformation

$g_{\mu\nu} \rightarrow \Omega^2(x) g_{\mu\nu}$

which possess a different matter content