On the space of light rays of a space-time and a reconstruction theorem by Low

Abstract

A reconstruction theorem in terms of the topology and geometrical structures on the spaces of light rays and skies of a given space-time is discussed. This result can be seen as part of Penrose and Low's programme intending to describe the causal structure of a space-time $M$ in terms of the topological and geometrical properties of the space of light rays, i.e., unparametrized time-oriented null geodesics, $\mathcal{N}$. In the analysis of the reconstruction problem it becomes instrumental the structure of the space of skies, i.e., of congruences of light rays. It will be shown that the space of skies $\Sigma$ of a strongly causal skies distinguishing space-time $M$ carries a canonical differentiable structure diffeomorphic to the original manifold $M$. Celestial curves, this is, curves in $\mathcal{N}$ which are everywhere tangent to skies, play a fundamental role in the analysis of the geometry of the space of light rays. It will be shown that a celestial curve is induced by a past causal curve of events iff the legendrian isotopy defined by it is non-negative. This result extends in a nontrivial way some recent results by Chernov \emph{et al} on Low's Legendrian conjecture. Finally, it will be shown that a celestial causal map between the space of light rays of two strongly causal spaces (provided that the target space is null non-conjugate) is necessarily induced from a conformal immersion and conversely. These results make explicit the fundamental role played by the collection of skies, a collection of legendrian spheres with respect to the canonical contact structure on $\mathcal{N}$, in characterizing the causal structure of space-times.