Vladimir Maksimovich Tuchkevich (on his eightieth birthday)

Abstract

We define and study a new kind of relation between two diffeomorphic Lorentzian manifolds called a 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 are sometimes 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 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.