Causal relationship: a new tool for the causal characterization of Lorentzian manifolds

Abstract

We define and study a new kind of relation between two diffeomorphic Lorentzian manifolds called causal relation, which is any diffeomorphism characterized by mapping every causal vector of the first manifold onto a causal vector of the second. We perform a thorough study of the mathematical properties of causal relations and prove in particular that two given Lorentzian manifolds (say V and W ) may be causally related only in one direction (say from V to W , but not from W to V ). This leads us to the concept of causally equivalent (or isocausal in short) Lorentzian manifolds as those mutually causally related and to a definition of causal structure over a differentiable manifold as the equivalence class formed by isocausal Lorentzian metrics upon it.Isocausality is a more general concept than the conformal relationship, because we prove the remarkable result that a conformal relation ϕ is characterized by the fact of being a causal relation of the particular kind in which both ϕ and ϕ −1 are causal relations. Isocausal Lorentzian manifolds are mutually causally compatible, they share some important causal properties, and there are one-to-one correspondences, which sometimes are non-trivial, between several classes of their respective future (and past) objects. A more important feature is that they satisfy the same standard causality constraints. We also introduce a partial order for the equivalence classes of isocausal Lorentzian manifolds providing a classification of all the causal structures that a given fixed manifold can have.By introducing the concept of causal extension we put forward a new definition of causal boundary for Lorentzian manifolds based on the concept of isocausality, and thereby we generalize the traditional Penrose's constructions of conformal infinity, diagrams and embeddings. In particular, the concept of causal diagram is given.Many explicit clarifying examples are presented throughout the paper.

The causal hierarchy of spacetimes

“…A timelike version of the previous theorem (in fact easier to prove), where < and causal are replaced with ≪ and timelike, also holds (for smooth versions see [20]). …”

“…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

“…Clearly the relation "≺" is a preorder in the set of all the diffeomorphic Lorentzian manifolds. Other basic properties easy to prove are the following [14]. Proposition 2.1 (Basic properties of causal mappings).…”

Causal cones, cone preserving transformations and causal structure in special and general relativity

“…An ordering which is reflexive and transitive is called quasi -ordering. This ordering was developed in a generalized sense by Sorkin and Woolgar [7] and these concepts were further developed by Garcia Parrado and Senovilla [30,31] and S. Janardhan and Saraykar [8] to prove many interesting results in causal structure theory in GR.…”