On the Canonical Contact Structure of the Space of Null Geodesics of a Spacetime

TL;DR

This paper explores the canonical contact structure of null geodesic spaces in specific spacetimes, providing explicit calculations for certain manifolds and examining the role of Engel geometry in understanding spacetime structures.

Contribution

It computes null geodesic spaces and their contact structures for  imes \u0012 with specific metrics, and investigates how Engel geometry can recover spacetime from null geodesic data.

Findings

Null geodesic spaces are lens spaces L(2c,1) for the given metrics.

Contact structures are pushforwards of canonical structures on tangent sphere bundles.

Engel geometry helps recover spacetime from null geodesic and sky data.

Abstract

The space of null geodesics of a spacetime carries a canonical contact structure which has proved to be key in the discussion of causality in spacetimes. However, not much progress has been made on its nature and not many explicit calculations for specific spacetimes can be found over the literature. We compute the spaces of null geodesics and their canonical contact structures for the manifold $\mathbb{S}^2\times\mathbb{S}^1$ equipped with the family of metrics $\{g_c = g_\circ-\frac{1}{c^2}dt^2 \}_{c\in\mathbb{N}^+}$. We obtain that these are the lens spaces $L(2c,1)$ and that the contact structures are the pushforward of the canonical contact structure on $ST\mathbb{S}^2\cong L(2,1)$ under the projection map. We also study the applicability of Engel geometry on the discussion of three-dimensional spacetimes. We show that, for a particular type of three-dimensional spacetimes, one can obtain the space of null geodesics and its contact structure solely from the information of the Lorentz prolongation of the spacetime. We present an approach that makes use of this result to recover the spacetime from its space of null geodesics and skies.