Limit structure of Future Null Infinity tangent -- topology of the event horizon and gravitational wave tail

We investigated the relation between the behavior of gravitational wave at late time and the limit structure of future null infinity tangent which will determine the topology of the event horizon far in the future. In the present article, we mainly consider a spacetime with two black holes. Although in most of cases, the black holes coalesce and its event horizon is topologically a single sphere far in the future, there are several possibilities that the black holes never coalesce and such exact solutions as examples. In our formulation, the tangent vector of future null infinity is, under conformal embedding, related to the number of black holes far in the future through the Poincar\'e-Hopf's theorem. Under the conformal embedding, the topology of event horizon far in the future will be affected by the geometrical structure of the future null infinity. In this article, we related the behavior of Weyl curvature to this limit behavior of the generator vector of the future null infinity. We show if Weyl curvature decays sufficiently slowly at late time in the neighborhood of future null infinity, two black holes never coalesce