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 $\{({\mathbb{S}}^2\times {\mathbb{S}}^1,g_{\circ }-\frac{d^2}{c^2}d{t}^2){\}}_{d,c\in {\mathbb{N}}^{+}\text{ coprime}}$, 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 $ST{\mathbb{S}}^2\cong L(2,1)$ under the natural projection $L(2,1)\to L(2c,1)$. We extend this computation to $Z\times {\mathbb{S}}^1$ 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.

22 pages, 2 figures, to appear in Mathematische Zeitschrift