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 ${x, y \in X}$ 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 ${\mathbb{R}^{m}}$ .