Linear stability of nonbidiagonal black holes in massive gravity

TL;DR

This paper proves that nonbidiagonal static black holes in massive gravity are mode stable with a quasinormal spectrum identical to Schwarzschild black holes, contrasting with bidiagonal solutions which are radially unstable.

Contribution

It demonstrates the mode stability of nonbidiagonal black holes in massive gravity and analyzes their perturbation spectrum, highlighting differences from bidiagonal solutions and implications for spinning cases.

Findings

Nonbidiagonal black holes are mode stable with Schwarzschild-like quasinormal modes.

Bidiagonal black holes exhibit radial instability.

Spinning nonbidiagonal black holes are not affected by superradiant instability.

Abstract

We consider generic linear perturbations of a nonbidiagonal class of static black-hole solutions in massive (bi)gravity. We show that the quasinormal spectrum of these solutions coincides with that of a Schwarzschild black hole in general relativity, thus proving that these solutions are mode stable. This is in contrast to the case of bidiagonal black-hole solutions which are affected by a radial instability. On the other hand, the full set of perturbation equations is generically richer than that of a Schwarzschild black hole in general relativity, and this affects the linear response of the black hole to external perturbations. Finally, we argue that the generalization of these solutions to the spinning case does not suffer from the superradiant instability, despite the fact that the theory describes a massive graviton.