Linking and causality in globally hyperbolic spacetimes

Abstract

The linking number $lk$ is defined if link components are zero homologous. Our affine linking invariant $alk$ generalizes $lk$ to the case of linked submanifolds with arbitrary homology classes. We apply $alk$ to the study of causality in Lorentz manifolds. Let $M^m$ be a spacelike Cauchy surface in a globally hyperbolic spacetime $(X^{m+1},g)$. The spherical cotangent bundle $S{T}^{\ast }M$ is identified with the space $N$ of all null geodesics in $(X,g).$ Hence the set of null geodesics passing through a point $x\in X$ gives an embedded $(m-1)$-sphere $S_x$ in $N=S{T}^{\ast }M$ called the sky of $x.$ Low observed that if the link $(S_x,S_y)$ is nontrivial, then $x,y\in X$ are causally related. This motivated the problem (communicated by Penrose) on the Arnold's 1998 problem list to apply link theory to the study of causality. The spheres $S_x$ are isotopic to fibers of $(S{T}^{\ast }M{)}^{2m-1}\to M^m.$ They are nonzero homologous and $lk(S_x,S_y)$ is undefined when $M$ is closed, while $alk(S_x,S_y)$ is well defined. Moreover, $alk(S_x,S_y)\in Z$ 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 $\mathbb{Z}$ and $g$ is conformal to $g^{\prime }$ having all the timelike sectional curvatures nonnegative, then $x,y\in X$ are causally related if and only if $alk(S_x,S_y)\ne 0$. We show that $x,y$ in nonrefocussing $(X,g)$ are causally unrelated iff $(S_x,S_y)$ can be deformed to a pair of $S^{m-1}$-fibers of $S{T}^{\ast }M\to M$ by an isotopy through skies. Low showed that if $(ß,g)$ is refocussing, then $M$ is compact. We show that the universal cover of $M$ is also compact.

We added: Theorem 11.5 saying that a Cauchy surface in a refocussing space time has finite pi_1; changed Theorem 7.5 to be in terms of conformal classes of Lorentz metrics and did a few more changes. 45 pages, 3 figures. A part of the paper (several results of sections 4,5,6,9,10) is an extension and development of our work math.GT/0207219 in the context of Lorentzian geometry. The results of sections 7,8,11,12 and Appendix B are new

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