Excitation of quasinormal ringing of a Schwarzschild black hole

Processes near the event horizon of a black hole excite a ringing of fields (electromagnetic, gravitational perturbation, etc.) at certain complex frequencies, called quasinormal frequencies, characteristic of the hole. Evidence for such oscillations consists almost entirely of their appearance in detailed numerical solutions of specific problems. Despite the importance of quasinormal ringing in the generation of gravitational radiation, little work has been done on clarifying the way in which the ringing is excited, or in estimating the strength of the excitation, without a detailed computer solution. We formulate here the theory of the excitation of ringing of Schwarzschild holes from Cauchy data, in which a coefficient $C_q$ seems to describe the excitation, but is given by a formally divergent integral. The meaning of $C_q$ is shown actually to be an analytic continuation of the integral in the complex frequency plane, and this idea is used as the basis of computational techniques for finding $C_q$. We then demonstrate that $C_q$ does not in general describe the astrophysically interesting quantity, the near-horizon stimulation of the ringing. We introduce two approaches to the correct description. The first uses a modified $C_q$ based on an ad hoc modification of the Cauchy data. The second is based on a series representation of $C_q$; a truncation of this series automatically selects the astrophysically interesting part of $C_q$.