L-extensions and L-boundary of conformal spacetimes

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 M of the given spacetime M with smooth boundary ∂ M = M\ M . The extensions 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.