Topology of the ambient boundary and the convergence of causal curves

TL;DR

This paper explores how the Zeeman fine topology affects the boundary structure of spacetime and influences the convergence of causal curves, impacting our understanding of singularity formation.

Contribution

It demonstrates that the ambient boundary's topology is constrained by the Zeeman fine topology, affecting causal curve convergence and singularity conditions.

Findings

Zeeman fine topology is necessary for the ambient boundary.

Convergence of causal curves is severely constrained by this topology.

Formulating singularity conditions is impacted by the boundary's topological nature.

Abstract

We discuss the topological nature of the boundary spacetime, the conformal infinity of the ambient cosmological metric. Due to the existence of a homothetic group, the bounding spacetime must be equipped not with the usual Euclidean metric topology but with the Zeeman fine topology. This then places severe constraints to the convergence of a sequence of causal curves and the extraction of a limit curve, and also to our ability to formulate conditions for singularity formation.