Gravitational waves emitted by perturbed black holes or relativistic stars are dominated by `quasinormal ringing', damped oscillations at single frequencies which are characteristic of the underlying system. These quasinormal modes have been studied for a long time, often with the intent of describing the time evolution of a perturbation in terms of these modes in a way very similar to a normal-mode analysis. In this review, we summarize how quasinormal modes are defined and computed. We will see why they have been regarded as closely analogous to normal modes, and discover why they are actually quite different. We also discuss how quasinormal modes can be used in the analysis of a gravitational wave signal, such as will hopefully be detected in the near future.

We present an improvement of the continued fraction technique for the numerical calculation of quasinormal modes of Schwarzschild black holes. On this basis it becomes possible to compute quasinormal frequencies with arbitrary imaginary parts. We use this technique for a very accurate numerical determination of the asymptotic behavior of quasinormal frequencies with very large imaginary parts [Im(w)>> 1001.

Quasinormal modes play a prominent role in the literature when dealing with the propagation of linearized perturbations of the Schwarzschild geometry. We show that space-time properties of the solutions of the perturbation equation imply the existence of a unique Green's function of the Laplacetransformed wave equation. This Green's function may be constructed from solutions of the homogeneous time-independent equation, which are uniquely characterized by the boundary conditions they satisfy. These boundary conditions are identified as the boundary conditions usually imposed for quasinormal-mode solutions. It turns out that solutions of the homogeneous equation exist which satisfy these boundary conditions at the horizon and at spatial infinity simultaneously, leading to poles of the Green's function. We therefore propose to define quasinormal-mode frequencies as the poles of the Green's function for the Laplace-transformed equation. On the basis of this definition a new technique for the numerical calculation of quasinormal frequencies is developed. The results agree with computations of Leaver, but not with more recent results obtained by Guinn, Will, Kojima, and Schutz. PACS number(s1: 97.60.Lf, 04.20.Cv, 04.30.+x, 11.1O.Qr

We study small, nonradial oscillations of neutron stars in a general relativistic perturbation treatment, considering different values for the central density of the star. We adapt two techniques used previously for the determination of quasinormal modes of black holes, allowing us for the first time to determine without approximation solutions which satisfy purely outgoing boundary conditions at spatial infinity. We confirm the existence of strongly damped complex normal modes found by Kokkotas and Schutz (w modes). In addition, we identify a new branch of strongly damped modes (WII modes). Our new modes are much more similar to quasinormal modes of black holes than the w modes known before.PACS number(s): 97.60. Jd, 04.30.+x, 95.30.Lz, 95.30.Sf

Computations of the strong field generation of gravitational waves by black hole processes produce waveforms that are dominated by quasinormal (QN) ringing, a damped oscillation characteristic of the black hole. We describe here the mathematical problem of quantifying the QN content of the waveforms generated. This is done in several steps: (i) We develop the mathematics of QN systems that are complete (in a sense to be defined) and show that there is a quantity, the "excitation coefficient," that appears to have the properties needed to quantify QN content. (ii) We show that incomplete systems can (at least sometimes) be converted to physically equivalent complete systems. Most notably, we give a rigorous proof of completeness for a specific modified model problem. (iii) We evaluate the excitation coefficient for the model problem, and demonstrate that the excitation coefficient is of limited utility. We finish by discussing the general question of quantification of QN excitations, and offer a few speculations about unavoidable differences between normal mode and QN systems.