The chart based approach to studying the global structure of a spacetime induces a coordinate invariant boundary

Abstract

I demonstrate that the chart based approach to the study of the global structure of Lorentzian manifolds induces a homeomorphism of the manifold into a topological space as an open dense set. The topological boundary of this homeomorphism is a chart independent boundary of ideal points equipped with a topological structure and a physically motivated classification. I show that this new boundary contains all other boundaries that can be presented as the topological boundary of an envelopment. Hence, in particular, it is a generalisation of Penrose's conformal boundary. I provide three detailed examples: the conformal compactification of Minkowski spacetime, Scott and Szekeres' analysis of the Curzon singularity and Beyer and Hennig's analysis of smooth Gowdy symmetric generalised Taub-NUT spacetimes.

Generalizations of the abstract boundary singularity theorem

“…Hence, the theorem extends the 'standard' singularity theorems, e.g., the Penrose and Hawking singularity theorems [2, section 8.2], by showing that they actually produce genuine singularities, at least according to the abstract boundary classification of boundary points [1, theorem 4.13]. For further details about the abstract boundary please refer to one of [1,[3][4][5].…”

What actually happens when you approach a gravitational singularity?

“…Our very recent paper [3] completes a body of work [4,5,6,7,8,9,10,11] that presents a solution to all these issues-the Abstract Boundary. The Abstract Boundary is an algorithm that takes a spacetime and produces a compact, non-Hausdorff, locally Euclidean topological space into which the original spacetime is homeomorphically embedded.…”

“…In the Kerr solution, the origin in Boyer-Lindquist coordinates is equivalent to the disc in Kerr-Schild coordinates. See [8] for the creation of the necessary embedding from a chart.…”

“…In particular, it contains, and thus generalises, both the classical coordinate dependent methods of studying singularities and Penrose's conformal boundary [2] (see Figure (3)). Concrete examples of this can be found in [8]. One can therefore perform computations in the charts of N using the pushforward of g by φ.…”

“…Our recent result, the Endpoint Theorem [3], shows that for any sequence (x i ) i∈N ⊂ M with no accumulation points, there exists an element [x] ∈ B(M) and an embedding φ so that φ(x i ) → x. This theorem establishes that M ∪ B(M) is compact in the topology given in [5,6] or [8] and thereby shows that B(M) is large enough to capture all possible singular behaviour. Most importantly, it provides locations (the endpoints) for the gravitational singularities associated with the incomplete, inextendible, causal geodesics predicted by the singularity theorems.…”