Toward a new definition of quasinormal modes in a Schwarzschild black hole

Abstract: Quasinormal modes of Schwarzschild black holes are defined by their properties at infinite distances. It is believed that the quasinormal mode spectrum is discrete, but the derivation makes assumptions about finite distances. We prove that the spectrum is continuous if the usual definition is used, so a new definition is required if a discrete spectrum is desired. Within the usual definition, any frequency corresponds to a valid mode, and each frequency is two-fold degenerate. Even within the continuous spectrum, the fundamental least-damped discrete quasinormal mode should dominate the dynamics, due to its lack of an incoming component at finite distances. On the other hand, the discrete overtones are very similar to the continuum modes, since all highly-damped modes have very small incoming components. Furthermore, the continuous spectrum may be complete.