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.