with the perturbed metric having the form
which leads to a variation of the Einstein equations i.e.
By assuming a decomposition into tensor spherical harmonics
for each
of the form
the perturbation problem is reduced to a single wave equation,
for the function
(which is a combination of the various components of
). It should be pointed out that equation (20
) is an expansion for scalar quantities only. From the 10
independent components of the
only
,
, and
transform as scalars under rotations. The
,
,
, and
transform as components of two-vectors under rotations and can
be expanded in a series of vector spherical harmonics while the
components
,
, and
transform as components of a
tensor and can be expanded in a series of tensor spherical
harmonics (see [202
,
212
,
152
] for details). There are two classes of vector spherical
harmonics (polar
and
axial) which are build out of combinations of the Levi-Civita volume
form and the gradient operator acting on the scalar spherical
harmonics. The difference between the two families is their
parity. Under the parity operator
a spherical harmonic with index
transforms as
, the polar class of perturbations transform under parity in the
same way, as
, and the axial perturbations as
. Finally, since we are dealing with spherically symmetric
spacetimes the solution will be independent of
m, thus this subscript can be omitted.
The radial component of a perturbation outside the event horizon satisfies the following wave equation,
where
is the ``tortoise'' radial coordinate defined by
and M is the mass of the black hole.
For ``axial'' perturbations
is the effective potential or (as it is known in the
literature) Regge-Wheeler potential [173], which is a single potential barrier with a peak around
r
=3
M, which is the location of the unstable photon orbit. The
form (23) is true even if we consider scalar or electromagnetic test
fields as perturbations. The parameter
takes the values 1 for scalar perturbations, 0 for
electromagnetic perturbations, and -3 for gravitational
perturbations and can be expressed as
, where
s
=0, 1, 2 is the spin of the perturbing field.
For ``polar'' perturbations the effective potential was derived by Zerilli [212] and has the form
where
Chandrasekhar [54] has shown that one can transform the equation (21) for ``axial'' modes to the corresponding one for ``polar''
modes via a transformation involving differential operations. It
can also be shown that both forms are connected to the
Bardeen-Press [38] perturbation equation derived via the Newman-Penrose formalism.
The potential
decays exponentially near the horizon,
, and as
for
.
From the form of equation (21) it is evident that the study of black hole perturbations will
follow the footsteps of the theory outlined in section
2
.
Kay and Wald [117] have shown that solutions with data of compact support are
bounded. Hence we know that the time independent Green function
is analytic for
Re
(s)>0. The essential difficulty is now to obtain the solutions
(cf. equation (10
)) of the equation
(prime denotes differentiation with respect to
) which satisfy for real, positive
s
:
To determine the quasi-normal modes we need the analytic continuations of these functions.
As the horizon () is a regular singular point of (26
), a representation of
as a converging series exists. For
it reads:
The series converges for all complex
s
and |
r
-1|<1 [162]. (The analytic extension of
is investigated in [115
].) The result is that
has an extension to the complex
s
plane with poles only at negative real integers. The
representation of
is more complicated: Because infinity is a singular point no
power series expansion like (28
) exists. A representation coming from the iteration of the
defining integral equation is given by Jensen and Candelas [115
], see also [159
]. It turns out that the continuation of
has a branch cut
due to the decay
for large
r
[115].
The most extensive mathematical investigation of quasi-normal
modes of the Schwarzschild solution is contained in the paper by
Bachelot and Motet-Bachelot [35]. Here the existence of an infinite number of quasi-normal modes
is demonstrated. Truncating the potential (23
) to make it of compact support leads to the estimate (16
).
The decay of solutions in time is not exponential because of
the weak decay of the potential for large
r
. At late times, the quasi-normal oscillations are swamped by the
radiative tail [166,
167
]. This tail radiation is of interest in its own right since it
originates on the background spacetime. The first authoritative
study of nearly spherical collapse, exhibiting radiative tails,
was performed by Price [166,
167].
Studying the behavior of a massless scalar field propagating on a fixed Schwarzschild background, he showed that the field dies off with the power-law tail,
at late times, where
P
=1 if the field is initially static, and
P
=2 otherwise. This behavior has been seen in various
calculations, for example the gravitational collapse simulations
by Cunningham, Price and Moncrief [66,
67,
68]. Today it is apparent in any simulation involving evolutions of
various fields on a black hole background including
Schwarzschild, Reissner-Nordström [106], and Kerr [132
,
133
]. It has also been observed in simulations of axial oscillations
of neutron stars [18
], and should also be present for polar oscillations.
Leaver [136
] has studied in detail these tails and associated this power low
tail with the branch-cut integral along the negative imaginary
axis in the complex
plane. His suggestion that there will be radiative tails
observable at
and
has been verified by Gundlach, Price, and Pullin [106]. Similar results were arrived at recently by Ching et
al. [62] in a more extensive study of the late time behavior. In a
nonlinear study Gundlach, Price, and Pullin [107] have shown that tails develop even when the collapsing field
fails to produce a black hole. Finally, for a study of tails in
the presence of a cosmological constant refer to [49], while for a recent study, using analytic methods, of the
late-time tails of linear scalar fields outside Schwarzschild and
Kerr black holes refer to [36,
37].
Using the properties of the waves at the horizon and infinity
given in equation (27) one can search for the quasi-normal mode frequencies since
practically the whole problem has been reduced to a boundary
value problem with
being the complex eigenvalue. The procedure and techniques used
to solve the problem will be discussed later in section
6, but it is worth mentioning here a simple approach to calculate
the QNM frequencies proposed by Schutz and Will [180
]. The approach is based on the standard WKB treatment of wave
scattering on the peak of the potential barrier, and it can be
easily shown that the complex frequency can be estimated from the
relation
where
is the peak of the potential barrier. For
and
n
=0 (the fundamental mode) the complex frequency is
, which for a
black hole corresponds to a frequency of 1.2 kHz and
damping time of 0.55 ms. A few more QNM frequencies for
and 4 are listed in table
1
.
Figure
2
shows some of the modes of the Schwarzschild black hole. The
number of modes for each harmonic index
is infinite, as was mathematically proven by Bachelot and
Motet-Bachelot [35
]. This was also implied in an earlier work by Ferrari and
Mashhoon [85
], and it has been seen in the numerical calculations in [25
,
157
]. It can be also seen that the imaginary part of the frequency
grows very quickly. This means that the higher modes do not
contribute significantly in the emitted gravitational wave
signal, and this is also true for the higher
modes (octapole etc.).
As is apparent in figure
2
that there is a special purely imaginary QNM frequency. The
existence of ``algebraically special'' solutions for
perturbations of Schwarzschild, Reissner-Nordström and Kerr black
holes were first pointed out by Chandrasekhar [57]. It is still questionable whether these frequencies should be
considered as QNMs [137] and there is a suggestion that the potential might become
transparent for these frequencies [11]. For a more detailed discussion refer to [144].
As a final comment we should mention that as the order of the
modes increases the real part of the frequency remains constant,
while the imaginary part increases proportionally to the order of
the mode. Nollert [157] derived the following approximate formula for the asymptotic
behavior of QNMs of a Schwarzschild black hole,
where
, 0.7545 and 2.81 for
2, 3 and 6, correspondingly, and
. The above relation was later verified in [10] and [143].
For large values of
the distribution of QNMs is given by [164,
86
,
85
,
113
]
For a mathematical proof refer to [39].
The perturbations of Reissner-Nordström black holes, due to
the spherical symmetry of the solution, follow the footsteps of
the analysis that we have presented in this section. Most of the
work was done during the seventies by Zerilli [213], Moncrief [153,
154] and later by Chandrasekhar and Xanthopoulos [55,
209]. For an extensive discussion refer to [56]. We have again wave equations of the form (21
), one for each parity with potentials which are like (23
) and (24
) plus extra terms which relate to the charge of the black hole.
An interesting feature of the charged black holes is that any
perturbation of the gravitational (electromagnetic) field will
also induce electromagnetic (gravitational) perturbations. In
other words, any perturbation of the Reissner-Nordström spacetime
will produce both electromagnetic and gravitational radiation.
Again it has been shown that the solutions for the odd parity
oscillations can be deduced from the solutions for even parity
oscillations and vice versa [55]. The QNM frequencies of the Reissner-Nordström black hole have
been calculated by Gunter [108
], Kokkotas, and Schutz [129
], Leaver [137
], Andersson [9], and lately for the nearly extreme case by Andersson and
Onozawa [26].
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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 |