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

TL;DR

This paper characterizes the space of null geodesics in certain spacetimes using contact and Engel geometry, computing explicit examples like lens spaces and providing a method to recover spacetimes from their geodesic spaces.

Contribution

It introduces a novel geometric framework linking null geodesics with Engel geometry, extending previous work to new classes of spacetimes and characterizing their geodesic spaces.

Findings

Null geodesics of specific spacetimes form lens spaces with canonical contact structures.

Engel geometry can describe null geodesic manifolds of certain 3D spacetimes.

The paper characterizes contact manifolds that correspond to null geodesic spaces, enabling spacetime retrieval.

Abstract

We compute the contact manifold of null geodesics of the family of spacetimes $\{(\mathbb{S}^2\times\mathbb{S}^1, g_\circ-\frac{d^2}{c^2}dt^2)\}_{d,c\in\mathbb{N}^+\text{ coprime}}$, 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.