Isocausal spacetimes may have different causal boundaries

Abstract

We construct an example which shows that two isocausal spacetimes, in the sense introduced recently in [11], may have c-boundaries which are not equal (more precisely, not equivalent, as no bijection between the completions can preserve all the binary relations induced by causality). This example also suggests that isocausality can be useful for the understanding and computation of the c-boundary.

Hausdorff separability of the boundaries for spacetimes and sequential spaces

“…Assume by contradiction that L (σ) ⊂ L * (σ). There exists q ∈ L (σ) such that q ∈ L * (σ) and, necessarily q ∈ D by (10). So, q belongs to both L (κ) and X \ D, which absurd as p ∈ L (κ) ∩ D and L satisfies (9).…”

“…As L τ * is a first order limit operator, Lemma 2.4 ensures that L * (σ) ⊂ L τ * (σ) for all sequence σ ⊂ X. So, taking into account Proposition 3.7, it suffices to prove that L τ * satisfies properties (4), (9) and (10). Properties (4) and (9) follow from properties (A Fin ) and (A Sep ) (recall Lemmas 2.4, 3.6) while (10) is a consequence of Theorem 3.5.…”

“…In fact, the original definition has suffered quite a few modifications. We will consider here the recent redefinition in [9] (which includes an approach to the topology of partial boundaries by Harris [18,19]), and refer also there for extensive bibliography on the topic (see also [20,10,11] Ω · η, where η is the Minkowski metric and Ω is a conformal factor. The conformal factor was defined in such a way that all γ i are geodesics and Ω ≡ 1 outside of the open sets U i as well as on each curve γ i inside U i (see [15] for details).…”

Computability of the causal boundary by using isocausality

“…Figure 2: C-boundary of (V, g). The strain St(P 0 ) is drawn on the left as in [5] and on the right, more conceptually, as an horizontal segment (a "bone").…”

“…For example, most of the steps of the so-called causal ladder of spacetimes are preserved by isocausal spacetimes [10] but not all of them [9,Section 3.2]. And, recently the authors exhibited an example of two isocausal spacetimes with different c-boundaries [5].…”