Periodic analogues of the Kerr solutions: a numerical study

TL;DR

This paper numerically investigates periodic coaxial configurations of rotating 3+1 vacuum black holes, revealing existence conditions, asymptotic behaviors, and singularity formation related to angular momentum and separation distance.

Contribution

It provides the first numerical analysis of rotating coaxial black hole arrays, extending static solutions and identifying critical parameters for their existence and stability.

Findings

Solutions exist only above a critical separation depending on area and angular momentum.

Configurations have Lewis's cylindrical asymptotic similar to Stockum's cylinders.

Below the critical separation, solutions develop singularities due to excessive rotational energy.

Abstract

In recent years black hole configurations with non standard topology or with non-standard asymptotic have gained considerable attention. In this article we carry out numerical investigations aimed to find periodic coaxial configurations of co-rotating 3+1 vacuum black holes, for which existence and uniqueness has not yet been theoretically proven. The aimed configurations would extend Myers/Korotkin-Nicolai's family of non-rotating (static) coaxial arrays of black holes. We find that numerical solutions with a given value for the area A and for the angular momentum J of the horizons appear to exist only when the separation between consecutive horizons is larger than a certain critical value that depends only on A and |J|. We also establish that the solutions have the same Lewis's cylindrical asymptotic as Stockum's infinite rotating cylinders. Below the mentioned critical value the rotational energy appears to be too big to sustain a global equilibrium and a singularity shows up at a finite distance from the bulk. This phenomenon is a relative of Stockum's asymptotic's collapse, manifesting when the angular momentum (per unit of axial length) reaches a critical value compared to the mass (per unit of axial length), and that results from a transition in the Lewis's class of the cylindrical exterior solution. This remarkable phenomenon seems to be unexplored in the context of coaxial arrays of black holes. Ergospheres and other global properties are also presented in detail.