Given two points in Lorentzian spacetime $p,q\in M$, is it true that there is only a unique null geodesic (up to affine reparametrization) that connects that the two points?
On the one hand, it seems that the answer is "yes", since null geodesics are obtained from the geodesic equation which have unique answers given initial data and all I need to fix is the endpoint $q$ if initial data starts at point $p$. On the other hand, I can imagine a case of, say, gravitational lensing, where null rays from a source (point $p$) behind a gravitational lens are bent along two different null directions and reach the Earth (point $q$), which seems to imply that the answer is "no".
I am trying to understand the physics so I am avoiding full dive into uniqueness proof right now, unless that's the only way to go.
Edit: I should have been more careful in excluding the standard counterexamples, such as those with spherical spatial topology. If I have to choose a sufficiently good restriction, it would be to let $M$ be Schwarzschild spacetime --- in particular, the exterior of the Schwarzschild black holes or spherical stars. Naively, I expect this to be non-unique but I am not sure if the solution is best dealt with using caustics and stuff used for singularity theorems.