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 is the subset of N defined by all null geodesics through that event, and is an embedded Legendrian submanifold of N diffeomorphic to a (d-1)-dimensional sphere. It was conjectured by Low that for d=2 two events are causally related iff their skies 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.