L-extensions and L-boundary of conformal spacetimes

TL;DR

This paper explores the L-boundary, a conformally invariant causal boundary concept for spacetimes, establishing conditions for unique L-extensions and their equivalence to canonical extensions in three dimensions.

Contribution

It introduces the notion of L-boundary and L-extensions for conformal spacetimes, proving their existence, uniqueness, and equivalence to canonical extensions in 3D.

Findings

L-boundary provides a natural conformal causal boundary.

L-extensions are characterized by local boundary properties.

In 3D, all L-extensions are equivalent to the canonical extension.

Abstract

The notion of L-boundary, a new causal boundary proposed by R. Low based on constructing a `sky at infinity' for any light ray, is discussed in detail. The analysis of the notion of L-boundary will be done in the 3-dimensional situation for the ease of presentation. The proposed notion of causal boundary is intrinsically conformal and, as it will be proved in the paper, under natural conditions provides a natural extension $\bar{M}$ of the given spacetime $M$ with smooth boundary $\partial M = \bar{M} \backslash M$. The extensions $\bar{M}$ of any conformal manifold $M$ constructed in this way are characterised exclusively in terms of local properties at the boundary points. Such extensions are called L-extensions and it is proved that, if they exist, they are essentially unique. Finally it is shown that in the 3-dimensional case, any L-extension is equivalent to the canonical extension obtained by using the L-boundary of the manifold.