Linking and causality in (2+1)-dimensional static spacetimes

TL;DR

This paper proves Low's conjecture relating causality of events in certain static spacetimes to the linking of their skies of null geodesics, enhancing understanding of spacetime causality via geometric topology.

Contribution

The paper establishes Low's conjecture for a broad class of static spacetimes, connecting causality with topological linking of null geodesic sets.

Findings

Low's conjecture is proven for static spacetimes.

Causality corresponds to linking of null geodesic sets.

Provides a geometric-topological characterization of causality.

Abstract

Given a (d+1)-dimensional spacetime (M,g), one can consider the set N of all its null geodesics. If (M,g) is globally hyperbolic then this set is naturally a smooth (2d-1)-manifold. The sky of an event x in M is the set X of all null geodesics through x, and is an embedded submanifold of N diffeomorphic to S^{d-1}. Low conjectured that if d=2 then x,y are causally related in M iff X,Y are linked in N. We prove Low's conjecture for a (large) class of static spacetimes.