An analytic effective model for hairy black holes

TL;DR

This paper introduces a universal analytical effective model for hairy black holes, enabling the calculation of thermodynamic properties based on a fiducial bald black hole, thus simplifying the analysis of complex hairy black hole solutions.

Contribution

The paper proposes a new effective model that analytically describes hairy black holes using a fiducial bald black hole, capturing key properties with broad applicability.

Findings

Model accurately predicts thermodynamics of asymptotically flat BHs with hair in 4D and 5D.

Effective model's validity is limited to near the fiducial BH limit.

Model's applicability extends to certain AdS and static colored black holes, with noted limitations.

Abstract

Hairy black holes (BHs) have macroscopic degrees of freedom which are not associated with a Gauss law. As such, these degrees of freedom are not manifest as quasi-local quantities computed at the horizon. This suggests conceiving hairy BHs as an interacting system with two components: a "bald" horizon coupled to a "hairy" environment. Based on this idea we suggest an effective model for hairy BHs -- typically described by numerical solutions -- that allows computing analytically thermodynamic and other quantities of the hairy BH in terms of a fiducial bald BH. The effective model is universal in the sense that it is only sensitive to the fiducial BH, but not to the details of the hairy BH. Consequently, it is only valid in the vicinity of the fiducial BH limit. We discuss, quantitatively, the accuracy of the effective model for asymptotically flat BHs with synchronised hair, both in $D=4$ (including self-interactions) and $D=5$ spacetime dimensions. We also discuss the applicability of the model to synchronised BHs in $D=5$ asymptotically $AdS$ and static $D=4$ coloured BHs, exhibiting its limitations.