Causality and skies: is non-refocussing necessary?

TL;DR

This paper explores how the causal structure and topology of certain space-times can be fully reconstructed from the space of light rays and skies, using Legendrian isotopies and causal curves, without requiring non-refocussing conditions.

Contribution

It introduces a new characterization of causal structures via Legendrian isotopies and establishes conditions under which the space of skies reconstructs the space-time.

Findings

Causal structure is characterized by a partial order on the space of skies.

The space of skies is homeomorphic and diffeomorphic to the space-time under sky-separating conditions.

The results provide a formulation of the Malament-Hawking theorem based on skies.

Abstract

It is shown that if $M$ is a strongly causal free of naked singularities space-time, then its causal structure is completely characterized by a partial order in the space of skies defined by means of a class non-negative Legendrian isotopies. It is also proved 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-Legendrian isotopies. It will also be proved that the space of skies $\Sigma$ 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 points determine different skies. The sky-separating property of $M$ being weaker than the "non-refocussing" property encountered in the previous literature is sharp and the previous result provides the answer to the question of what is the class of space-times whose causal structure, topology and differentiable structure can be reconstructed from their spaces of light rays and skies. Finally, the previous results allow a formulation of Malament-Hawking theorem in terms of the partial order defined on the space of skies.