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

Abstract

the concept of causal mapping between spacetimes -essentially equivalent in this context to the chronological map one in abstract chronological spaces-, and the related notion of causal structure, have been introduced as new tools to study causality in Lorentzian geometry. In the present paper, these tools are further developed in several directions such as: (i) causal mappings -and, thus, abstract chronological ones-do not preserve two levels of the standard hierarchy of causality conditions (however, they preserve the remaining levels as shown in the above reference), (ii) even though global hyperbolicity is a stable property (in the set of all timeoriented Lorentzian metrics on a fixed manifold), the causal structure of a globally hyperbolic spacetime can be unstable against perturbations; in fact, we show that the causal structures of Minkowski and Einstein static spacetimes remain stable, whereas that of de Sitter becomes unstable, (iii) general criteria allow us to discriminate different causal structures in some general spacetimes (e.g. globally hyperbolic, stationary standard); in particular, there are infinitely many different globally hyperbolic causal structures (and thus, different conformal ones) on R 2 , (iv) plane waves with the same number of positive eigenvalues in the frequency matrix share the same causal structure and, thus, they have equal causal extensions and causal boundaries.

Comportamiento de causalidad en el espacio-tiempo de Carter

“…We will not state any of the definitions and properties in Theory of Causality since, if interested, one can find an excellent review on the last advances on Theory of Causality in [3], where all the required definitions, the causal conditions and the interplay between them are carefully studied. It is worth mentioning that a new and interesting ordering of spacetimes has been recently carried out by García-Parrado and Senovilla in [6], [7], and refined by García-Parrado and Sánchez in [8].…”

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].…”

“…Finally, it is worth pointing out the following question regarding stability (recall Section 3.8), also stressed in [19]. In the three first cases of Theorem 4.4, all the spacetimes are isocausal to a fixed one and, thus, the isocausal structure is C 0 -stable in the set of all the metrics on I × S. Nevertheless, in the last case there are different isocausal structures.…”

“…Some examples. In order to study the possible isocausality of two spacetimes, there are two basic naive ideas (see [19]):…”

An Invitation to Lorentzian Geometry

“…Locally, either of these relations characterizes the conformal structure. So, Causality can be identified to conformal geometry in Lorentzian signature (nevertheless, a subtler modification of the notion of Causality has been recently introduced by García-Parrado and Senovilla [57], see also [56]). Remarkably, one can associate a (conformally invariant) causal boundary to every sufficiently well behaved spacetime, this allows to describe the possible asymptotics of timelike curves in a subtle way, ordering them by inclusion of their pasts [50] As a remarkable difference with the Riemannian case, both, completeness and incompleteness are C r unstable, for every r ∈ N, even in the case of Lorentzian metrics on a torus [99], but some results on stability can be still obtained in the Lorentzian case, as C 1 -fine stability in the globally hyperbolic case, see [17,28].…”