with
M
is the mass and
the rotational parameter of the Kerr metric. The zeros of
are
and determine the horizons. For
the spacetime admits locally a timelike Killing vector. In the
ergosphere region
the Killing vector
which is timelike at infinity, becomes spacelike. The scalar
wave equation for the Kerr metric is
where
for scalar, electromagnetic or gravitational perturbations,
respectively. As the Kerr metric outside the horizon
is globally hyperbolic, the Cauchy problem for the scalar wave
equation (37
) is well posed for data on any Cauchy surface. However, the
coefficient of
becomes negative in the ergosphere. This implies that the time
independent equation we obtain after the Fourier or Laplace
transformation is not elliptic!
For linear hyperbolic equations with time independent
coefficients, we know that solutions determined by data with
compact support are bounded by
, where
is independent of the data. It is not known whether all such
solutions are bounded in time, i.e. whether they are stable.
Assuming harmonic time behavior
, a separation in angular and radial variables was found by
Teukolsky [196]:
Note that in contrast to the case of spherical harmonics, the
separation is
-dependent. To be a solution of the wave equation (37
), the functions
R
and
S
must satisfy
where
,
, and
E
is the separation constant.
For each complex
and positive integer
m, equation (39
) together with the boundary conditions of regularity at the
axis, determines a singular Sturm-Liouville eigenvalue problem.
It has solutions for eigenvalues
,
. The eigenfunctions are the spheroidal (oblate) harmonics
. They exist for all complex
. For real
the spheroidal harmonics are complete in the sense that any
function of
, absolutely integrable over the interval [-1, 1], can be
expanded into spheroidal harmonics of fixed
m
[181]. Furthermore, functions
absolutely integrable over the sphere can be expanded into
For general complex
such an expansion is not possible. There is a countable number
of ``exceptional values''
where no such expansion exists [148
].
Let us pick one such solution
and consider some solutions
of (40
) with the corresponding
. Then
is a solution of (37
). Is it possible to obtain ``all'' solutions by summing over
and integrating over
? For a solution in spacetime for which a Fourier transform in
time exists at any space point (square integrable in time), we
can expand the Fourier transform in spheroidal harmonics because
is real. The coefficient
will solve equation (40
). Unfortunately, we only know that a solution determined by data
of compact support is exponentially bounded. Hence we can only
perform a Laplace transformation.
We proceed therefore as in section
2
. Let
be the Laplace transform of a solution determined by data
and
, while
is analytic in
s
for real
, and has an analytic continuation onto the half-plane
. For real
s
we can expand
into a converging sum of spheroidal harmonics [148]
satisfies the radial equation (9
) with
. This representation of
does, however, not hold for all complex values in the half-plane
on which it is defined. Nevertheless is it true that for all
values of
s
which are not exceptional an expansion of the form (42
) exists
.
To define quasi-normal modes we first have to define the
correct Green function of (40) which determines
from the data
. As usual, this is done by prescribing decay for real
s
for two linearly independent solutions
and analytic continuation. Out of the Green function for
and
for real
s
we can build the Green function
by a series representation like (42
). Analytic continuation defines
G
on the half plane
. For non exceptional
s
we have a series representation. (We must define
G
by this complicated procedure because the partial differential
operator corresponding to (39
), (40
) is not elliptic in the ergosphere.)
Normal and quasi-normal modes appear as poles of the analytic
continuation of
G
. Normal modes are determined by poles with
Re
(s)>0 and quasi-normal modes by
Re
(s)<0. Suppose all such values are different from the
exceptional values. Then we have always the series expansion of
the Green function near the poles and we see that they appear as
poles of the radial Green function of
.
To relate the modes to the asymptotic behavior in time we study the inverse Laplace transform and deform the integration path to include the contributions of the poles. The decay in time is dominated either by the normal mode with the largest (real) eigenfrequency or the quasi-normal mode with the largest negative real part.
It is apparent that the calculation of the QNM frequencies of
the Kerr black hole is more involved than the Schwarzschild and
Reissner-Nordström cases. This is the reason that there have only
been a few attempts [135,
185
,
123
,
160] in this direction.
The quasi-normal mode frequencies of the Kerr-Newman black
hole have not yet been calculated, although they are more general
than all other types of perturbation. The reason is the
complexity of the perturbation equations and, in particular,
their non-separability. This can be understood through the
following analysis of the perturbation procedure. The equations
governing a perturbing massless field of spin
can be written as a set of
wavelike equations in which the various different helicity
components of the perturbing field are coupled not only with each
other but also with the curvature of the background space, all
with four independent variables as coordinates over the manifold.
The standard problem is to decouple the
equations or at least some physically important subset of them
and then to separate the decoupled equations so as to obtain
ordinary differential equations which can be handled by one of
the previously stated methods. For a discussion and estimation of
the QNM frequencies in a restrictive case refer to [124
].
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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 |