José Natário and Paul Tod

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.

2000 Mathematical Subject Classification: 83C60 (primary), 53D10, 57M25 (secondary).

Keywords: manifold of light rays, contact structure, sky, wavefront, Legendrian invariants, linking, Legendrian linking.

E-mail:
jnatar@math.ist.utl.pt
tod@maths.ox.ac.uk