corresponding geodesics in two spacetimes?

What conditions are there for two spacetimes to have "corresponding geodesics"?

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Do you mean to ask what you can say about two pseudo-Riemannian metrics on a manifold that have the same geodesics up to reparametrization? If so, this is a classical problem, with some interesting recent progress by A. Bolsinov and V. Matveev. You should consult their recent papers for details. –  Robert Bryant Jan 25 at 21:54