Limit structure of future null infinity tangent - topology of the event horizon and gravitational wave tail

摘要

We investigated the relation between the behaviour of gravitational waves 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 paper, we mainly consider a spacetime with two black holes. Although in most cases, the black holes coalesce and the 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 Poincare-Hopf theorem. Under the conformal embedding, the topology of the event horizon far in the future will be affected by the geometrical structure of the future null infinity. In this paper, we relate the behaviour of Weyl curvature to this limit behaviour of the generator vector of the future null infinity. We show if Weyl curvature decays sufficiently slowly at late time in the neighbourhood of future null infinity, two black holes never coalesce.