Linking, Legendrian linking and causality

TL;DR

This paper explores the relationship between causality and Legendrian linking of skies in spacetime, proving a conjecture in specific cases and proposing a refined linking concept for higher dimensions.

Contribution

It proves Low's conjecture for certain cases and introduces Legendrian linking as a more suitable concept in higher-dimensional spacetimes.

Findings

Proved causality-linking conjecture for specific cases in d=2.

Suggested Legendrian linking as a better model for d=3.

Applied contact structure and knot polynomial techniques.

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.