On the space of null geodesics of a spacetime: the compact case, Engel geometry and retrievability

We compute the contact manifold of null geodesics of the family of spacetimes {(S2×S1,g∘-d2c2dt2)}d,c∈N+coprime , with g_∘ the round metric on S² and t the S¹ -coordinate. We find that these are the lens spaces L(2c, 1) together with the pushforward of the canonical contact structure on STS²≅ L(2 , 1) under the natural projection L(2 , 1) → L(2 c, 1) . We extend this computation to Z× S¹ for Z a Zoll manifold. On the other hand, motivated by these examples, we show how Engel geometry can be used to describe the manifold of null geodesics of a certain class of three-dimensional spacetimes, by considering the Cartan deprolongation of their Lorentz prolongation. We characterize the three-dimensional contact manifolds that are contactomorphic to the space of null geodesics of a spacetime. The characterization consists in the existence of an overlying Engel manifold with a certain foliation and, in this case, we also retrieve the spacetime.