New approach to the quasinormal modes of a black hole

We describe a new analytic approach to the problem of black-hole oscillations, which has been investigated numerically thus far. Our treatment is based on a connection between the quasinormal modes and the bound states of the inverted black-hole effective potentials. Approximate analytic formulas for the quasinormal frequencies of Schwarzschild, Reissner-Nordstr\"om, and slowly rotating Kerr black holes are provided. We find that a real quasinormal frequency for an extreme Kerr black hole has vanishing amplitude in the ordinary (i.e., nonsuperradiant) regime; therefore, extreme Kerr black holes are not marginally unstable in this case. These results are significant for the question of the stability of a black hole as well as for the late-time behavior of radiation from gravitationally collapsing configurations.