A new recipe for causal completions

TL;DR

This paper introduces a novel method for constructing ideal points at infinity in spacetime, directly pairing future and past sets, resulting in improved causal properties and topologies for spacetime completions.

Contribution

It presents a new approach to spacetime completion by directly identifying ideal points as pairs of future and past sets, differing from previous quotient-based methods.

Findings

Provides a construction with better causal properties

Introduces two topologies on the completed space

Successfully applies the method to plane wave solutions

Abstract

We discuss the asymptotic structure of spacetimes, presenting a new construction of ideal points at infinity and introducing useful topologies on the completed space. Our construction is based on structures introduced by Geroch, Kronheimer, and Penrose and has much in common with the modifications introduced by Budic and Sachs as well as those introduced by Szabados. However, these earlier constructions defined ideal points as equivalence classes of certain past and future sets, effectively defining the completed space as a quotient. Our approach is fundamentally different as it identifies ideal points directly as appropriate pairs consisting of a (perhaps empty) future set and a (perhaps empty) past set. These future and past sets are just the future and past of the ideal point within the original spacetime. This provides our construction with useful causal properties and leads to more satisfactory results in a number of examples. We are also able to endow the completion with a topology. In fact, we introduce two topologies, which illustrate different features of the causal approach. In both topologies, the completion is the original spacetime together with endpoints for all timelike curves. We explore this procedure in several examples, and in particular for plane wave solutions, with satisfactory results.