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

Abstract

We compute the contact manifold of null geodesics of the family of spacetimes $$\left\{ \left( \mathbb {S}^2\times \mathbb {S}^1, g_\circ -\frac{d^2}{c^2}dt^2\right) \right\} _{d,c\in \mathbb {N}^+\text { coprime}}$$ S 2 × S 1 , g ∘ - d 2 c 2 d t 2 d , c ∈ N + coprime , with $$g_\circ $$ g ∘ the round metric on $$\mathbb {S}^2$$ S 2 and t the $$\mathbb {S}^1$$ S 1 -coordinate. We find that these are the lens spaces L(2c, 1) together with the pushforward of the canonical contact structure on $$ST\mathbb {S}^2\cong L(2,1)$$ S T S 2 ≅ L ( 2 , 1 ) under the natural projection $$L(2,1)\rightarrow L(2c,1)$$ L ( 2 , 1 ) → L ( 2 c , 1 ) . We extend this computation to $$Z\times \mathbb {S}^1$$ Z × S 1 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.