Linear instability of hairy black holes in Horndeski theory

The Horndeski theory gives the most general model of scalar-tensor theories. It draws a lot of attentions in recent years on its black holes, celestial dynamics, stability analysis, etc. It is important to notice that, for certain subclasses of Horndeski models, one can obtain analytic solutions to the background fields. This provides us with a good opportunity to investigate the corresponding stability problems in details. Specially, we may find out the constraints to the model or theory, under which the stability conditions can be satisfied. In this paper, we focus on a subclass of the Horndeski theory and a set of analytic background solutions are considered. On top of that, the odd-parity gravitational perturbation and the 2nd-order Lagrangian are investigated. With careful analysis, the instability is identified within the neighborhood of the event horizon. We are thus able to exclude a specific geometry for the model. It is interesting to notice that, such an instability is implanted in the structure of the corresponding Lagrangian, and will not be erased by simply adding numerical constraints on the coupling parameters. As a starting point of our research, this current work provides insights into further exploration of Horndeski theories.