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 (Xm+1, g). The spherical cotangent bundle ST*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 is an element of X gives an embedded (m-1)-sphere G(x) in N=ST*M called the sky of x. Low observed that if the link (G(x), G(y)) is nontrivial, then x, y is an element of X 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 G(x) are isotopic to the fibers of (ST*M)(2m-1) -> M-m. They are nonzero homologous and the classical linking number lk(G(x,) G(y)) is undefined when M is closed, while alk(G(x,) G(y)) is well defined. Moreover, alk(G(x,) G(y))is an element of 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 Z and g is conformal to (g) over cap that has all the timelike sectional curvatures nonnegative, then x, y is an element of X are causally related if and only if alk(G(x,) G(y)) not equal 0. We prove that if alk takes values in Z and y is in the causal future of x, then alk(G(x,) G(y)) 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 (G(x,) G(y)) can be deformed to a pair of Sm-1-fibers of ST*M -> M 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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