Conjectures on the Relations of Linking and Causality in Causally Simple Spacetimes

TL;DR

This paper explores the relationship between linking and causality in causally simple spacetimes, proposing a generalized conjecture and demonstrating a weakened version of it, with implications for understanding spacetime causal structures.

Contribution

It generalizes the Legendrian Low conjecture to causally simple spacetimes and proves a weakened version, suggesting a new approach to linking causality in these spacetimes.

Findings

Proved a weakened version of the generalized conjecture.

In known examples, causally simple spacetimes embed into globally hyperbolic spacetimes.

The space of light rays in such spacetimes relates to that in globally hyperbolic spacetimes.

Abstract

We formulate the generalization of the Legendrian Low conjecture of Natario and Tod (proved by Nemirovski and myself before) to the case of causally simple spacetimes. We prove a weakened version of the corresponding statement. In all known examples, a causally simple spacetime $(X, g)$ can be conformally embedded as an open set into some globally hyperbolic $(\widetilde X, \widetilde g)$ and the space of light rays in $(X, g)$ is an open submanifold of the space of light rays in $(\widetilde X, \widetilde g)$. If this is always the case, this provides an approach to solving the conjectures relating causality and linking in causally simples spacetimes.