Linking and Causality in Globally Hyperbolic Space-times

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 $$\mathcal N$$ 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 $$\mathfrak S_x$$ in $$\mathcal N=ST^*M$$ called the sky of x. Low observed that if the link $$(\mathfrak S_x, \mathfrak S_y)$$ is nontrivial, then $$x,y\in X$$ 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 $$\mathfrak S_x$$ are isotopic to the fibers of $$(ST^*M)^{2m-1}\to M^m.$$ They are nonzero homologous and the classical linking number lk $$(\mathfrak S_x, \mathfrak S_y)$$ is undefined when M is closed, while alk $$(\mathfrak S_x, \mathfrak S_y)$$ is well defined. Moreover, alk $$(\mathfrak S_x, \mathfrak S_y)\in {\mathbb{Z}}$$ 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 $$\widehat{g}$$ that has all the timelike sectional curvatures nonnegative, then $$x, y\in X$$ are causally related if and only if alk $$(\mathfrak S_x, \mathfrak S_y)\neq 0$$ . We prove that if alk takes values in $${\mathbb{Z}}$$ and y is in the causal future of x, then alk $$(\mathfrak S_x, \mathfrak S_y)$$ 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 $$(\mathfrak S_x, \mathfrak S_y)$$ can be deformed to a pair of S m-1-fibers of $$ST^*M\to M$$ 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.