Extended black hole cosmologies in de Sitter space

TL;DR

This paper develops a generalized superposition principle for black hole cosmologies in de Sitter space, expressing solutions via eigenvalue problems and elliptic functions, revealing an infinite tower of universes connected through horizons.

Contribution

It introduces a new eigenvalue-based method to construct black hole solutions in de Sitter space, extending previous models to include multiple universes and complex topologies.

Findings

Black hole topologies extend into infinite universe towers.

Superposition yields binary black holes and universes depending on separation.

The metric evolution follows a hyperbolic system with curvature-driven lapse.

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 $\Delta_g\phi={1/8}(R_g-2\Lambda)\phi$ for a conformal scale $\phi$ applied to a metric $g_{ij}$ with constant three-curvature $R_g$. Here, $R_g=0,2$ in the Brill-Lindquist and, respectively, Misner construction of multihole solutions for $\Lambda=0$. For de Sitter and anti-de Sitter cosmologies, we express the result for $R_g=0$ 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 large relative to the cosmological event horizon. The evolution of the metric can be described by a hyperbolic system of equations with 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 $\Lambda$ was large and black holes were of mass $<{1/3}\Lambda^{-1/2}$, here facilitated by a metric which is singularity-free and smooth everywhere on real coordinate space.