Legendrian links, causality, and the Low conjecture

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 $x,y\in \text{ss}$ are causally related if and only if the Legendrian link of spheres $\mathfrak S_x, \mathfrak S_y$whosepointsarelightgeodesicspassingthrough$x$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 ${\mathbb{R}}^m\mathrm{.}$