The space of light rays: Causality and $L$-boundary

TL;DR

This paper explores the space of light rays in conformal Lorentz manifolds, introduces the $L$-boundary as a new causal boundary concept based solely on the geometry of this space, and discusses its properties and potential generalizations.

Contribution

It introduces the $L$-boundary, a new causal boundary for spacetimes derived from the geometry of the space of light rays, independent of the spacetime's metric.

Findings

The $L$-boundary is constructed for 3-dimensional manifolds.

The properties of the $L$-boundary characterize the spacetime extension.

The approach is proposed to be generalizable to higher dimensions.

Abstract

The space of light rays $\mathcal{N}$ of a conformal Lorentz manifold $(M,\mathcal{C})$ is, under some topological conditions, a manifold whose basic elements are unparametrized null geodesics. This manifold $\mathcal{N}$, strongly inspired on R. Penrose's twistor theory, keeps all information of $M$ and it could be used as a space complementing the spacetime model. In the present review, the geometry and related structures of $\mathcal{N}$, such as the space of skies $\Sigma$ and the contact structure $\mathcal{H}$, are introduced. The causal structure of $M$ is characterized as part of the geometry of $\mathcal{N}$. A new causal boundary for spacetimes $M$ prompted by R. Low, the $L$-boundary, is constructed in the case of $3$-dimensional manifolds $M$ and proposed as a model of its construction for general dimension. Its definition only depends on the geometry of $\mathcal{N}$ and not on the geometry of the spacetime $M$. The properties satisfied by the $L$-boundary $\partial M$ permit to characterize the obtained extension $\overline{M}=M\cup \partial M$ and this characterization is also proposed for general dimension.