Legendrian Links, Causality, and the Low Conjecture |
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Authors: | Vladimir Chernov Stefan Nemirovski |
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Institution: | 1. Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH, 03755, USA 2. Steklov Mathematical Institute, 119991, Moscow, Russia 3. Mathematisches Institut, Ruhr-Universit?t Bochum, 44780, Bochum, Germany
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Abstract: | Let (X m+1, g) be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of ${\mathbb{R}^{m}}Let (X
m+1, g) be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of
\mathbbRm{\mathbb{R}^{m}} . The Legendrian Low conjecture formulated by Natário and Tod says that two events x, y ? X{x, y \in X} are causally related if and only if the Legendrian link of spheres
\mathfrakSx, \mathfrakSy{{\mathfrak{S}_x,\,\mathfrak{S}_y}} whose points are light geodesics passing through x and y is non-trivial in the contact manifold of all light geodesics in X. The Low conjecture says that for m = 2 the events x, y are causally related if and only if
\mathfrakSx, \mathfrakSy{{\mathfrak{S}_x,\,\mathfrak{S}_y}} is non-trivial as a topological link. We prove the Low and the Legendrian Low conjectures. We also show that similar statements
hold for any globally hyperbolic (X
m+1, g) such that a cover of its Cauchy surface is diffeomorphic to an open domain in
\mathbbRm{\mathbb{R}^{m}} . |
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