Topologies on the future causal completion

TL;DR

This paper thoroughly analyzes the topology $ au_+$ on the Geroch-Kronheimer-Penrose future completion of spacetimes, providing new insights into convergence, extending to low-regularity spaces, and explicit calculations for specific models.

Contribution

It offers a complete characterization of convergence differences between topologies, extends the framework to low-regularity spaces, and computes examples for multiply warped spaces.

Findings

Characterization of convergence differences between topologies

Extension to chr. spaces and low-regularity spacetimes

Explicit calculations for multiply warped spaces

Abstract

On the Geroch-Kronheimer-Penrose future completion $IP(X)$ of a spacetime $X$, there are two frequently used topologies. We systematically examine $\tau_+$, the stronger (metrizable) of them, which is the coarsest causally continuous topology, obtaining a variety of novel results, among them a complete characterization of the difference in convergence between both topologies. In our framework, we can allow for $X$ being a chr. space and consequently for the interpretation of $IP$ as an idempotent functor on a category that includes spacetimes of very low regularity. Furthermore, we explicitly calculate $(IP(X), \tau_+)$ for multiply warped chronological spaces.