Black hole stability in Jordan and Einstein frames

We investigate the classical stability of Schwarzschild black hole in Jordan and Einstein frames that are related by the conformal transformations. For this purpose, we introduce two models of the Brans-Dicke theory and Brans-Dicke-Weyl gravity in the Jordan frame and two corresponding models in the Einstein frame. The former model is suitable for studying the massless spin-2 graviton propagating around the Schwarzschild black hole, while the latter is designed for the massive spin-2 graviton propagating around the black hole. It turns out that the black hole (in) stability is independent of the frame which shows that the two frames are equivalent to each other.