Affine Linking Numbers and Causality Relations for Wave Fronts

Abstract

Two wave fronts $W_1$ and $W_2$ that originated at some points of the manifold $M^n$ are said to be causally related if one of them passed through the origin of the other before the other appeared. We define the causality relation invariant $CR (W_1, W_2)$ to be the algebraic number of times the earlier born front passed through the origin of the other front before the other front appeared. Clearly, if $CR(W_1, W_2)\neq 0$, then $W_1$ and $W_2$ are causally related. If $CR(W_1, W_2)= 0$, then we generally can not make any conclusion about fronts being causally related. However we show that for front propagation given by a complete Riemannian metric of non-positive sectional curvature, $CR(W_1, W_2)\neq 0$ if and only if the two fronts are causally related. The models where the law of propagation is given by a metric of constant sectional curvature are the famous Friedmann Cosmology models. The classical linking number $lk$ is a $Z$-valued invariant of two zero homologous submanifolds. We construct the affine linking number generalization $AL$ of the $lk$ invariant to the case of linked $(n-1) $-spheres in the total space of the unit sphere tangent bundle $(STM)^{2n-1}\to M^n$. For all $M$, except of odd-dimensional rational homology spheres, $AL$ allows one to calculate the value of $CR(W_1, W_2)$ from the picture of the two wave fronts at a certain moment. This calculation is done without the knowledge of the front propagation law and of their points and times of birth. Moreover, in fact we even do not need to know the topology of $M$ outside of a part $\bar M$ of $M$ such that $W_1$ and $W_2$ are null-homotopic in $\bar M$.