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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?

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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

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    $\begingroup$ This is interesting. Is there a good source where I can read more? $\endgroup$ – JohnnyMo1 Jul 28 '15 at 20:23
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    $\begingroup$ Check Hawking/Ellis pp.60--61 and Wald, p.445. Cheers. $\endgroup$ – Ivica Smolić Jul 28 '15 at 21:54

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