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Linking, Legendrian Linking and Causality
Author(s) -
Natário José,
Tod Paul
Publication year - 2004
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611503014424
Subject(s) - mathematics , submanifold , knot (papermaking) , geodesic , pure mathematics , diffeomorphism , conjecture , manifold (fluid mechanics)
The set N of all null geodesics of a globally hyperbolic ( d + 1)‐dimensional spacetime ( M, g ) is naturally a smooth (2 d − 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.