Legendrian links, causality, and the Low conjecture

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 Natário 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 ${\mathbb{R}}^m.$

Version 3 - minor improvements, references added 11 pages, 1 figure

References in corpus (4)

• Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions
• Geometry of contact transformations and domains: orderability versus squeezing
• Globally hyperbolic spacetimes can be defined as "causal" instead of "strongly causal"
• Linking and causality in globally hyperbolic spacetimes

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