Legendrian links, causality, and the Low conjecture

TL;DR

This paper proves the Legendrian and classical Low conjectures, establishing a topological and contact-geometric criterion for causality between events in certain globally hyperbolic spacetimes.

Contribution

It confirms the Low and Legendrian Low conjectures, linking causality with Legendrian and topological linking of light geodesics in globally hyperbolic spacetimes.

Findings

Proves the Legendrian Low conjecture for general globally hyperbolic spacetimes.

Establishes causality corresponds to non-trivial Legendrian links of light geodesics.

Extends results to covers of Cauchy surfaces diffeomorphic to open subsets of Euclidean space.

Abstract

Let $(X^{m+1}, g)$ be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of $\mathbb R^m$. The Legendrian Low conjecture formulated by Nat\'ario and Tod says that two events $x,y\in\ss$ are causally related if and only if the Legendrian link of spheres $\mathfrak S_x, \mathfrak S_y$ whose points are light geodesics passing through $x$ and $y$ is non-trivial in the contact manifold of all light geodesics in $X$. The Low conjecture says that for $m=2$ the events $x,y$ are causally related if and only if $\mathfrak S_x, \mathfrak S_y$ is non-trivial as a topological link. We prove the Low and the Legendrian Low conjectures. We also show that similar statements hold for any globally hyperbolic $(X^{m+1}, g)$ such that a cover of its Cauchy surface is diffeomorphic to an open domain in $\mathbb R^m.$