Causality and skies: is non-refocussing necessary?

Abstract

The causal structure of a strongly causal, null pseudo-convex, space-time M is completely characterized in terms of a partial order on its space of skies defined by means of a class of non-negative Legendrian isotopies called sky isotopies. It is also shown that such partial order is determined by the class of future causal celestial curves, that is, curves in the space of light rays which are tangent to skies and such that they determine non-negative sky isotopies. It will also be proved that the space of skies Σ equipped with Low’s (or reconstructive) topology is homeomorphic and diffeomorphic to M under the only additional assumption that M separates skies, that is, that different events determine different skies. The sky-separating property of M is sharp and the previous result provides an answer to the question about the class of space-times whose causal structure, topological and differentiable structure can be reconstructed from their spaces of light rays and skies. These results can be understood as a Malament–Hawking-like theorem stated in terms of the partial order defined on the space of skies.