A new recipe for causal completions

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.

The Causal Boundary of Spacetimes Revisited

“…The lack of Hausdorffness in chronological completions cannot be attributed to a defect of our particular approach. On the contrary, it seems a remarkable property intrinsic to the causal boundary approach itself (see [20] for an interesting discussion on this question).…”

“…The procedure proposed in [23], very close to the GKP approach, also fails in simple examples (see [18]). Another approach proposed in [3], and improved later in [25,26] via the Szabados relation (Definition 2.6; see also Section 8), again presents undesirable properties (see [17], [18], [20,Sections 2.2,5]). …”

“…Recently, Marolf and Ross have introduced in [20] an entirely new use of the Szabados relation, including a new topology for the completion, which overcomes important difficulties in previous attempts (see Section 8). In fact, the MR construction satisfies essential requirements to be considered a reasonable completion: (i) the original spacetime becomes densely, chronologically and topologically embedded into the completion and, (ii) any timelike curve in the spacetime has some limit in the completion.…”

“…In fact, the MR construction satisfies essential requirements to be considered a reasonable completion: (i) the original spacetime becomes densely, chronologically and topologically embedded into the completion and, (ii) any timelike curve in the spacetime has some limit in the completion. However, apart from certain "anomalous" limit behaviors in some examples (see [20]), they also admit an annoying failing: not only is their topology not necessarily Hausdorff (which cannot be regarded as unsatisfactory, as we will see later), but it might not even be T 1 , i.e. some point might not be closed (Example 10.5).…”

Recent progress on the notion of global hyperbolicity

“…Here, we point out just the following elements of the solution provided in [18], which takes into account the recent progress in [17,26,33]. The causal boundary ∂ c M is composed by timelike points and non-timelike points.…”