Perturbations of a Schwarzschild black hole in torsion bigravity

In this paper we pursue the study of linear perturbations around a Schwarzschild black hole in a generalized Einstein-Cartan theory of gravity, called torsion bigravity. This theory contains both massless and massive spin-2 excitations. Here we consider non spherically-symmetric perturbations with generic multipolarity $L \geq 1$. We extend the conclusion of linear stability, previously obtained for $L=0$ [Phys. Rev. D 104, 024032], to the generic $L \geq 1$ case. We prove that the mass $\kappa$ of the massive spin-2 excitation must be large enough, namely $\kappa r_h > \sqrt{1+\eta}$, to avoid the presence of singularities in the perturbation equations. The perturbation equations are shown to have a triangular structure, where massive spin-2 excitations satisfy decoupled equations, while the Einstein-like massless spin-2 ones satisfy inhomogeneous equations sourced by the massive spin-2 sector. We study quasi-bound states, and exhibit some explicit complex quasi-bound frequencies. We briefly discuss the issue of superradiance instabilities.