Conformally Embedded Spacetimes and the Space of Null Geodesics

Hedicke, Jakob; Suhr, Stefan

doi:10.1007/s00220-019-03499-0

Cited by 11 publications

(15 citation statements)

References 14 publications

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“…Though less known causal simplicity has also attracted considerable interest, particularly in the study of the space of lightlike geodesics [22,5,17]. It plays an important role in questions of geodesic connectedness (e.g.…”

Section: Introductionmentioning

confidence: 99%

Globally hyperbolic spacetimes can be defined without the ‘causal’ condition

Hounnonkpe

Minguzzi

2019

Class. Quantum Grav.

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Reasonable spacetimes are non-compact and of dimension larger than two. We show that these spacetimes are globally hyperbolic if and only if the causal diamonds are compact. That is, there is no need to impose the causality condition, as it can be deduced. We also improve the definition of global hyperbolicity for the non-regular theory (non C 1,1 metric) and for general cone structures by proving the following convenient characterization for upper semi-continuous cone distributions: causality and the causally convex hull of compact sets is compact. In this case the causality condition cannot be dropped, independently of the spacetime dimension. Similar results are obtained for causal simplicity. Let us start by recalling a result of causality theory [32, Thm. 4.12]. Theorem 2.1. The following properties are equivalent: (i) J is closed in the topology of M × M , (ii) J + (p) and J − (p) are closed for every p ∈ M , (iii) J + (K) and J − (K) are closed for every compact subset K ⊂ M , (iv) the causal diamonds are closed.(v) the causally convex hulls of compact sets are closed.This result is a bit surprising because we are not demanding causality, a fact that will prove important, as we shall see. It will follow from the more general Prop. 2.3.Lemma 2.2. Let (M, g) be a spacetime, and let K = {p, q} with p, q ∈ M , thenbut if J + (q)∩J − (p) = ∅ then q ≤ p, thus J ± (q) = J ± (p), hence J + (q)∩J − (p) = J + (p) ∩ J − (q). Thus in any case we have J + (K) ∩ J − (K) = J + (p) ∩ J − (q).The other cases are analogous or trivial.

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Section: Introductionmentioning

confidence: 99%

Globally hyperbolic spacetimes can be defined without the ‘causal’ condition

Hounnonkpe

Minguzzi

2019

Class. Quantum Grav.

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“…Our counterexample is a static spacetime based on a corresponding Riemannian counterexample, that enjoys a certain limiting property for minimizing geodesics, but fails to be convex. This counterexample adds to the list of recently found counterexamples involving the notion of causal simplicity [14,10].…”

Section: Introductionmentioning

confidence: 72%

Causal simplicity and (maximal) null pseudoconvexity

Hedicke,

Minguzzi,

Schinnerl

et al. 2021

Preprint

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“…In Ref. [9], Hedicke and Suhr showed that the space of null geodesics of a n-dimensional causally simple spacetime M for n > 2 is Hausdorff if it admits an open conformal embedding into a globally hyperbolic spacetime. And by Theorem 2.7, M is null pseudoconvex.…”

Section: Resultsmentioning

confidence: 99%

“…Remark 2.6. We note that Theorem 2.5 is not true for three-dimensional spacetimes because the null pseudoconvexity and strong causality lift to Lorentzian covers but, there are three-dimensional spacetimes which show that causal simplicity doesn't lift [8,9,19]. So, we immediately conclude that these causally simple spacetimes, [8, Example 2.3.]…”

Section: -2)mentioning

confidence: 89%

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Causally simple spacetimes and naked singularities

Vatandoost¹,

Pourkhandani²,

Ebrahimi³

2021

Preprint

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In this paper, We prove a conjecture which states that if M is a nakedly singular future boundary or nakedly singular past boundary spacetime, then the space of null geodesics, N, is non-Hausdorff. Also, we show that every two-dimensional strongly causal spacetime M is causally simple if and only if it is null pseudoconvex. As a result, it implies the converse of the conjecture for two-dimension but there are examples that refute it for more dimensions.

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Conformally Embedded Spacetimes and the Space of Null Geodesics

Cited by 11 publications

References 14 publications

Globally hyperbolic spacetimes can be defined without the ‘causal’ condition

Globally hyperbolic spacetimes can be defined without the ‘causal’ condition

Causal simplicity and (maximal) null pseudoconvexity

Causally simple spacetimes and naked singularities

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