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 [101,
102,
103,
104,
100
] 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 50s and was continued by Zerilli [212
]. The perturbations of relativistic stars in GR were first
studied in the late 60s 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,
101,
28] 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.
In the next section we attempt to give a mathematical definition of QNMs. The third and fourth section will be devoted to the study of the black hole and stellar QNMs. In the fifth section 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.
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Quasi-Normal Modes of Stars and Black Holes
Kostas D. Kokkotas and Bernd G. Schmidt http://www.livingreviews.org/lrr-1999-2 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |