Linking and Causality in Globally Hyperbolic Space-times |
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Authors: | Vladimir?V?Chernov Yuli?B?Rudyak |
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Institution: | (1) Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755, USA;(2) Department of Mathematics, University of Florida, 358 Little Hall, Gainesville, FL 32611-8105, USA |
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Abstract: | The classical linking number lk is defined when link components are zero homologous. In 15] we constructed the affine linking
invariant alk generalizing lk to the case of linked submanifolds with arbitrary homology classes. Here we apply alk to the
study of causality in Lorentzian manifolds.
Let M
m
be a spacelike Cauchy surface in a globally hyperbolic space-time (X
m+1, g). The spherical cotangent bundle ST
*
M is identified with the space of all null geodesics in (X,g). Hence the set of null geodesics passing through a point gives an embedded (m−1)-sphere in called the sky of x. Low observed that if the link is nontrivial, then are causally related. This observation yielded a problem (communicated by R. Penrose) on the V. I. Arnold problem list 3,4]
which is basically to study the relation between causality and linking. Our paper is motivated by this question.
The spheres are isotopic to the fibers of They are nonzero homologous and the classical linking number lk is undefined when M is closed, while alk is well defined. Moreover, alk if M is not an odd-dimensional rational homology sphere. We give a formula for the increment of alk under passages through Arnold
dangerous tangencies. If (X,g) is such that alk takes values in and g is conformal to that has all the timelike sectional curvatures nonnegative, then are causally related if and only if alk . We prove that if alk takes values in and y is in the causal future of x, then alk is the intersection number of any future directed past inextendible timelike curve to y and of the future null cone of x.
We show that x,y in a nonrefocussing (X, g) are causally unrelated if and only if can be deformed to a pair of S
m-1-fibers of by an isotopy through skies. Low showed that if (X, g) is refocussing, then M is compact. We show that the universal cover of M is also compact. |
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