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Linking, Legendrian linking and causality

Published online by Cambridge University Press:  13 January 2004

José Natário
Affiliation:
Mathematical Institute, 24–29 St Giles', Oxford OX1 3LB Current address: Departamento de Mathematica, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal. E-mail: jnatar@math.ist.utl.pt
Paul Tod
Affiliation:
Mathematical Institute, 24–29 St Giles', Oxford OX1 3LB. tod@math.ox.ac.uk
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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 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.

Type
Research Article
Copyright
2004 London Mathematical Society

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