Causal structures and causal boundaries

Abstract

We give an up-to-date perspective with a general overview of the theory of causal properties, the derived causal structures, their classification and applications, and the definition and construction of causal boundaries and of causal symmetries, mostly for Lorentzian manifolds but also in more abstract settings.

The causal hierarchy of spacetimes

“…Basic references for this section are [2,27,37,39,40,56], other useful references will be [9,10,19,21,30,31,51].…”

“…Among them causal spaces by Kronheimer and Penrose [30], etiological spaces by Carter [10] and chronological spaces by Harris [25]. Among the applications to spacetimes, a better insight on the meaning of causal boundaries (whose classical construction by Geroch, Kronheimer and Penrose [23] relies on some types of future sets) is obtained, see [16,21] and references therein.…”

“…A fresh viewpoint was introduced by García-Parrado and Senovilla [20,21] by taking into account the following two ideas: (i) the definition of most of the levels of the standard causal hierarchy prevents a bad behavior of some types of causal curves; thus, if the timecones of a metric g on M are included in the timecones of another one g ′ (g ≺ g ′ ), then the causality of g will be at least as good as the causality of g ′ , and (ii) perhaps for some diffeomorphisms Φ, Ψ of M the pull-back metrics satisfy Ψ * g ≺ g ′ ≺ Φ * g; in this case (as the causality of g, Ψ * g, Φ * g must be regarded equivalent), one says that g and g ′ are "isocausal". In this way, one introduces a partial (pre)order in the set of all the spacetimes, which was expected to refine the standard causal ladder.…”

Further properties of causal relationship: causal structure stability, new criteria for isocausality and counterexamples

“…These examples are isocausal in the above sense. Interestingly enough any pair of Lorentzian manifolds V 1 , V 2 are locally causally equivalent, this meaning that one can choose neighbourhoods of the points p 1 ∈ V 1 , p 2 ∈ V 2 which are causally equivalent when regarded as Lorentzian submanifolds (see theorem 4.4 of [15]). …”

Recent progress on the notion of global hyperbolicity

“…Throughout this paper, we will use standard notation in Causality, as in the books and reviews [4,22,29,34,37,40,48]. In particular, a spacetime will be a connected time-oriented n-manifold (n ≥ 2), its chronological and causal relations are denoted ≪, ≤, resp., and these relations determine the chronological and causal futures and pasts I ± (A), J ± (A) of any point or subset A of M .…”