Causality, linking and refocusing spacetimes
V.Chernov (based on a joint work with Yuli Rudyak)
We define the invariant $alk$ that generalizes the linking number to the
case of nonzero homologous submanifolds and apply it to the study of
causality in globally hyperbolic spacetimes $(X, g)$. The space $N$ of null
geodesics in $(X, g)$ is identified with the spherical cotangent bundle
$ST^*M$ of a Cauchy surface $M$. All the null geodesics passing through
$x\in X$ form a sky $S_x\subset N=ST^*M$ of $x.$
As it was observed by Low, if the link $(S_x, S_y)$ is nontrivial, then
$x,y\in X$ are causally related. We give many examples of spacetimes for
which $alk(S_x, S_y)$\neq 0$ if and only if $x, y\in X$ are causally
related. For nonrefocussing $(X, g)$ we show that two events are causally
unrelated iff the link of their skies can be deformed to a pair of
$S^{m-1}$-fibers of $ST^*M\to M$ by an isotopy through skies.
Low proved that if $(X, g)$ is refocusing, then $M$ is a closed manifolds.
We prove that the universal cover of $M$ also is a closed manifold and hence
$\pi_1(M)$ is finite. We discuss relations between refocusing spacetimes and
$Y^l_m$ Riemannian manifolds studied by Berard Bergeri et al.