Stability analysis of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>)-AdS black holes

We study the stability of the f(R)-AdS (Schwarzschild–AdS) black hole obtained from f(R) gravity. In order to resolve the difficulty of solving fourth-order linearized equations, we transform f(R) gravity into scalar–tensor theory by introducing two auxiliary scalars. In this case, the linearized curvature scalar becomes a dynamical scalaron, showing that all linearized equations are second order. Using the positivity of gravitational potentials and S-deformed technique allows us to guarantee the stability of f(R)-AdS black hole if the scalaron mass squared satisfies the Breitenlohner–Freedman bound. This is confirmed by computing quasinormal frequencies of the scalaron for the f(R)-AdS black hole.