A conformal boundary for space-times based on light-like geodesics: the 3-dimensional case

TL;DR

This paper introduces and analyzes a new conformal boundary, called the l-boundary, based on light-like geodesics for 3-dimensional space-times, showing it forms a smooth manifold with boundary and relating it to existing causal boundaries.

Contribution

It extends the l-boundary concept to 3-dimensional space-times, demonstrating its smooth manifold structure and exploring its relation to the Geroch-Kronheimer-Penrose boundary.

Findings

The completed space-time with the l-boundary is a smooth manifold with boundary.

The l-boundary is invariant under conformal diffeomorphisms.

Examples illustrate the properties and advantages of the l-boundary.

Abstract

A new causal boundary, which we will term the $l$-boundary, inspired by the geometry of the space of light rays and invariant by conformal diffeomorphisms for space-times of any dimension $m\geq 3$, proposed by one of the authors (R.J. Low, The space of null geodesics (and a new causal boundary), Lecture Notes in Physics, 692, Springer, 2006, 35--50) is analyzed in detail for space-times of dimension 3. Under some natural assumptions it is shown that the completed space-time becomes a smooth manifold with boundary and its relation with Geroch-Kronheimer-Penrose causal boundary is discussed. A number of examples illustrating the properties of this new causal boundary as well as a discussion on the obtained results will be provided.