On the space of light rays of a spacetime 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 spacetime is discussed. This result can be seen as a part of Penrose and Low’s programme intending to describe the causal structure of a spacetime M in terms of the topological and geometrical properties of the space of light rays, i.e., unparametrized time-oriented null geodesics, . In the analysis of the reconstruction problem, the structure of the space of skies, i.e., of congruences of light rays, becomes instrumental. It will be shown that the space of skies Σ of a strongly causal non-refocusing spacetime M carries a canonical differentiable structure diffeomorphic to the original manifold M. Celestial curves, this is, curves in 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 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 , in characterizing the causal structure of spacetime.