  We introduce a new formalism to study perturbations of Hassan-Rosen bigravity
theory, around general backgrounds for the two dynamical metrics. In
particular, we derive the general expression for the mass term of the
perturbations and we explicitly compute it for cosmological settings. We study
tensor perturbations in a specific branch of bigravity using this formalism. We
show that the tensor sector is affected by a late-time instability, which sets
in when the mass matrix is no longer positive definite.
