Black Holes in Multi-Metric Gravity II: Hairy Solutions and Linear Stability of the Non- and Partially Proportional Branches

TL;DR

This paper analyzes the linear stability of non-proportional and partially proportional black hole solutions in ghost-free multi-metric gravity, introduces a method to construct black holes with massive graviton hair, and explores their asymptotic properties.

Contribution

It provides the first stability analysis of non-proportional and partially proportional black hole branches and offers a general method to construct multi-metric black holes with graviton hair.

Findings

Proportional branch exhibits Gregory-Laflamme instability for small horizons.

Non-proportional and partially proportional branches' stability analyzed.

Constructed explicit hairy black hole solutions with massive graviton modes.

Abstract

Owing to our work in part I of this series of papers, it is understood that the analytically known black hole solutions in the theory of ghost free multi-metric gravity can be split into three distinct classes, and that one of these classes - the proportional branch - exhibits the Gregory-Laflamme instability at linear level in the metric perturbations, whenever the black hole horizon size is smaller than (roughly) the Compton wavelength of the theory's lightest massive graviton. In this first of two sequels, we determine the linear stability of the two remaining classes of black hole solutions - the non-proportional and partially proportional branches - and discuss how our results likely differ at nonlinear level. We also give a general prescription to construct multi-metric solutions describing black holes endowed with massive graviton hair, which may constitute the end state of the instability in the proportional branch. We utilise a tractable example model involving 3 metrics to see how this works in practice, and determine the asymptotic form of its corresponding hairy solutions at infinity, where one can clearly see the individual contributions from each of the graviton mass modes.