Linking, Legendrian linking and causality

Abstract

The set N of all null geodesics of a globally hyperbolic $(d + 1)$-dimensional spacetime $(M, g)$ is naturally a smooth $(2d - 1)$-dimensional contact manifold. The sky of an event x in M is the subset X of N consisting of all null geodesics through x, and is an embedded Legendrian submanifold of N diffeomorphic to $S^{(d - 1)}$. It was conjectured by Low that for $d = 2$ two events x and y are causally related if and only if X and Y are linked (in an appropriate sense). We use the contact structure and knot polynomial calculations to prove this conjecture in certain particular cases, and suggest that for $d = 3$ smooth linking should be replaced with Legendrian linking.