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

TL;DR

This paper explores how the late-time behavior of gravitational waves and the structure of future null infinity influence the topology of the event horizon, especially in black hole spacetimes with potential non-coalescence.

Contribution

It establishes a relation between the Weyl curvature decay rate at late times and the event horizon topology, extending understanding of black hole coalescence scenarios.

Findings

Slow decay of Weyl curvature prevents black hole coalescence.

Future null infinity's tangent structure determines event horizon topology.

Topological differences depend on the geometrical structure of null infinity.

Abstract

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.