Linear instability of hairy black holes in Horndeski theory*

Abstract Horndeski theory constitutes the most general model of scalar-tensor theories. It has attracted much attention in recent years in relation with black holes, celestial dynamics, stability analysis, etc. It is important to note that, for certain subclasses of Horndeski models, one can obtain analytic solutions for the background fields. This facilitates the investigation of the corresponding stability problems in detail. In particular, we aim to determine the constraints to the model or theory under which the stability conditions can be satisfied. In this study, we focused on a subclass of Horndeski theory and a set of analytic background solutions. In addition, the odd-parity gravitational perturbation and 2nd-order Lagrangian were investigated. Through careful analysis, the instability was identified within the neighborhood of the event horizon. This allows exclusion of a specific geometry for the model. Such an instability is implanted in the structure of the corresponding Lagrangian and is not erased by simply adding numerical constraints on the coupling parameters. As a starting point of our research, the current study provides insights for further exploration of the Horndeski theory.