Perturbations of Cosmological and Black Hole Solutions in Massive gravity and Bi-gravity

We investigate perturbations of a class of spherically symmetric solutions in massive gravity and bi-gravity. The background equations of motion for the particular class of solutions we are interested in reduce to a set of the Einstein equations with a cosmological constant. Thus, the solutions in this class include all the spherically symmetric solutions in general relativity, such as the Friedmann-Lema\^{i}tre-Robertson-Walker solution and the Schwarzschild (-de Sitter) solution, though the one-parameter family of two parameters of the theory admits such a class of solutions. We find that the equations of motion for the perturbations of this class of solutions also reduce to the perturbed Einstein equations at first and second order. Therefore, the stability of the solutions coincides with that of the corresponding solutions in general relativity. In particular, these solutions do not suffer from non-linear instabilities which often appear in the other cosmological solutions in massive gravity and bi-gravity.