1 Introduction

Helioseismology and asteroseismology are well known terms in classical astrophysics. From the beginning of the century the variability of Cepheids has been used for the accurate measurement of cosmic distances, while the variability of a number of stellar objects (RR Lyrae, Mira) has been associated with stellar oscillations. Observations of solar oscillations (with thousands of nonradial modes) have also revealed a wealth of information about the internal structure of the Sun [204]. Practically every stellar object oscillates radially or nonradially, and although there is great difficulty in observing such oscillations there are already results for various types of stars (O, B, …). All these types of pulsations of normal main sequence stars can be studied via Newtonian theory and they are of no importance for the forthcoming era of gravitational wave astronomy. The gravitational waves emitted by these stars are extremely weak and have very low frequencies (cf. for a discussion of the sun  [70], and an important new measurement of the sun’s quadrupole moment and its application in the measurement of the anomalous precession of Mercury’s perihelion  [163]). This is not the case when we consider very compact stellar objects i.e. neutron stars and black holes. Their oscillations, produced mainly during the formation phase, can be strong enough to be detected by the gravitational wave detectors (LIGO, VIRGO, GEO600, SPHERE) which are under construction.

In the framework of general relativity (GR) quasi-normal modes (QNM) arise, as perturbations (electromagnetic or gravitational) of stellar or black hole spacetimes. Due to the emission of gravitational waves there are no normal mode oscillations but instead the frequencies become “quasi-normal” (complex), with the real part representing the actual frequency of the oscillation and the imaginary part representing the damping.

In this review we shall discuss the oscillations of neutron stars and black holes. The natural way to study these oscillations is by considering the linearized Einstein equations. Nevertheless, there has been recent work on nonlinear black hole perturbations [100, 101, 102, 103, 104] while, as yet nothing is known for nonlinear stellar oscillations in general relativity.

The study of black hole perturbations was initiated by the pioneering work of Regge and Wheeler [173] in the late 1950s and was continued by Zerilli [212]. The perturbations of relativistic stars in GR were first studied in the late 1960s by Kip Thorne and his collaborators [202, 198, 199, 200]. The initial aim of Regge and Wheeler was to study the stability of a black hole to small perturbations and they did not try to connect these perturbations to astrophysics. In contrast, for the case of relativistic stars, Thorne’s aim was to extend the known properties of Newtonian oscillation theory to general relativity, and to estimate the frequencies and the energy radiated as gravitational waves.

QNMs were first pointed out by Vishveshwara [207] in calculations of the scattering of gravitational waves by a Schwarzschild black hole, while Press [164] coined the term quasi-normal frequencies. QNM oscillations have been found in perturbation calculations of particles falling into Schwarzschild [73] and Kerr black holes [76, 80] and in the collapse of a star to form a black hole [66, 67, 68]. Numerical investigations of the fully nonlinear equations of general relativity have provided results which agree with the results of perturbation calculations; in particular numerical studies of the head-on collision of two black holes [30, 29] (cf. Figure 1) and gravitational collapse to a Kerr hole [191]. Recently, Price, Pullin and collaborators [170, 31, 100, 27] have pushed forward the agreement between full nonlinear numerical results and results from perturbation theory for the collision of two black holes. This proves the power of the perturbation approach even in highly nonlinear problems while at the same time indicating its limits.

In the concluding remarks of their pioneering paper on nonradial oscillations of neutron stars Thorne and Campollataro [202] described it as “just a modest introduction to a story which promises to be long, complicated and fascinating”. The story has undoubtedly proved to be intriguing, and many authors have contributed to our present understanding of the pulsations of both black holes and neutron stars. Thirty years after these prophetic words by Thorne and Campollataro hundreds of papers have been written in an attempt to understand the stability, the characteristic frequencies and the mechanisms of excitation of these oscillations. Their relevance to the emission of gravitational waves was always the basic underlying reason of each study. An account of all this work will be attempted in the next sections hoping that the interested reader will find this review useful both as a guide to the literature and as an inspiration for future work on the open problems of the field.

Figure 1: QNM ringing after the head-on collision of two unequal mass black holes [29]. The continuous line corresponds to the full nonlinear numerical calculation while the dotted line is a fit to the fundamental and first overtone QNM.

In the next Section 2 we attempt to give a mathematical definition of QNMs. Sections 3 and 4 will be devoted to the study of the black hole and stellar QNMs. In Section 5 we discuss the excitation and observation of QNMs and finally in the sixth section we will mention the more significant numerical techniques used in the study of QNMs.