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}^2times mathbb {S}^1, g_circ -frac{d^2}{c^2}dt^2right) right} _{d,cin mathbb {N}^+text { coprime}}, with g_circ the round metric on mathbb {S}^2 and t the mathbb {S}^1-coordinate. We find that these are the lens spaces L(2c, 1) together with the pushforward of the canonical contact structure on STmathbb {S}^2cong L(2,1) under the natural projection L(2,1)rightarrow L(2c,1). We extend this computation to Ztimes mathbb {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.