Extended black hole cosmologies in de Sitter space

Abstract

We generalize the superposition principle for time-symmetric initial data of black hole spacetimes to (anti-)de Sitter cosmologies in terms of an eigenvalue problem for a conformal scale ϕ applied to a metric gij with constant three-curvature Rg. Here, Rg = 0, 2 in the Brill–Lindquist and, respectively, the Misner construction of multihole solutions for Λ = 0. For de Sitter and anti-de Sitter cosmologies, we express the result for Rg = 0 in incomplete elliptic functions. The topology of a black hole in de Sitter space can be extended into an infinite tower of universes, across the turning points at the black hole and cosmological event horizons. Superposition introduces binary black holes for small separations and binary universes for separations largely relative to the cosmological event horizon. The evolution of the metric can be described by a hyperbolic system of equations with a curvature-driven lapse function, of alternating sign at successive cosmologies. The computational problem of interacting black hole universes is conceivably of interest to early cosmology when Λ was large and black holes were of mass , here facilitated by a metric which is singularity-free and smooth everywhere on real coordinate space.