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

TL;DR

This paper advances the understanding of causal structures in Lorentzian geometry by exploring stability, new criteria for isocausality, and counterexamples, revealing nuanced properties of causality in various spacetimes.

Contribution

It introduces new developments in causal mapping, analyzes stability of causal structures under perturbations, and provides criteria to distinguish different causal structures, including examples with infinitely many structures on R^2.

Findings

Causal mappings do not preserve two levels of the causality hierarchy.

Causal structures of Minkowski and Einstein static spacetimes are stable, while de Sitter's is unstable.

Infinitely many globally hyperbolic causal structures exist on R^2.

Abstract

Recently ({\em Class. Quant. Grav.} {\bf 20} 625-664) the concept of {\em causal mapping} between spacetimes --essentially equivalent in this context to the {\em chronological map} one in abstract chronological spaces--, and the related notion of {\em 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 time-oriented 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.