Quasinormal modes of Reissner–Nordström–anti-de Sitter black holes: Scalar, electromagnetic, and gravitational perturbations

We study scalar, electromagnetic and gravitational perturbations of a Reissner--Nordstr\"om--anti-de Sitter (RN-AdS) spacetime, and compute its quasinormal modes (QNM's). We confirm and extend results previously found for Schwarzschild--anti-de Sitter (S-AdS) black holes. For ``large'' black holes, whose horizon is much larger than the AdS radius, different classes of perturbations are almost exactly isospectral; this isospectrality is broken when the black hole's horizon radius is comparable to the AdS radius. We provide very accurate fitting formulas for the QNM's, which are valid for large black holes having charge Q<Q₄ₗₓ/3 (Q₄ₗₓ being the extremal value of the charge). Electromagnetic and axial perturbations of large black holes are characterized by the existence of pure-imaginary (purely damped) modes. The damping of these modes tends to infinity as the black hole charge approaches the extremal value; if the corresponding mode amplitude does not tend to zero in the same limit, this implies that extremally charged RN-AdS black holes are marginally unstable. This result is relevant in view of the AdS--conformal field theory conjecture, since, according to it, the AdS QNM's give the time scales for the approach to equilibrium in the corresponding conformal field theory.