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Quasinormal modes of black holes
Gravitational perturbations of the curvature of Kerr black holes can be
described by two components of the Weyl tensor: and .
Because these are components of the curvature tensor, they have dimensions
of .
Of particular interest is the quantity since it is this term that is
suitable for the study of outgoing waves in the radiative zone. The
formalism for the study of perturbations of rotating black holes was
developed originally by Teukolsky [30] who was able
to separate the differential equation to obtain solutions of the form

(8.1.168) 
where
is a solution to a radial differential
equation, and
is a spinweighted spheroidal wave
function (see [30], equations (4.9) and (4.10)).
The black hole has mass and specific angular momentum
(which has dimensions of length) where is the angular momentum of the
spinning black hole. We shall often refer to the
dimensionless angular momentum parameter,
.
For a Kerr black hole, must be between zero (Schwarzschild limit)
and one (extreme Kerr limit).
The observer of the perturbation is located at radius ,
inclination , and azimuth
(see figure ). The perturbation
itself has the spheroidal eigenvalues and , and has a
(complex) frequency . The constants and are Newton's
gravitational constant and the speed of light.
Figure:
The polar angle, , and the azimuthal angle,
, of the observer relative to the spin axis of a black hole and
the (somewhat arbitrary) axis of perturbation.

The important physical quantities for the study of the gravitational
waves arising from black hole perturbations can be recovered from the
field . In particular, the ``'' and ``'' polarizations
of the strain induced by the gravity waves are found
by [30]

(8.1.169) 
The quantity
is the metric perturbation
that represents the linear polarization state along
and
, while the
quantity
represents the linear
polarization state
along
. The power
radiated towards the observer (per unit solid angle) is

(8.1.170) 
Thus, in order to compute the relevant information about gravitational
waves emitted as perturbations to rotating black hole spacetimes, one
needs to calculate the value of at large radii from the black hole.
The quasinormal modes are resonant modes of the Teukolsky equation
that describe purely outgoing radiation in the wavezone and purely
ingoing radiation at the event horizon. The quasinormal modes are
described by a spectrum of complex eigenvalues (which include the
spectrum of eigenfrequencies ), and
eigenfunctions
and
for each value spheroidal mode
and . These eigenvalues and functions also depend on the mass and
angular momentum of the black hole. We shall only consider
the fundamental () mode since the harmonics of this mode have
shorter lifetimes. For the same reason, we are most interested in the
quadrupole ( and ) mode. The observer is assumed to be at a large
distance; in this case, one can approximate the perturbation as follows:

(8.1.171) 
Here
represents the retarded time,
where is a ``tortoise'' radial parameter. For large radii, the
tortoise radius behaves as
where is the ``radius''
of the black hole event horizon. Thus, we see that the tortoise radius is
nearly equal to the distance of the objects surrounding the black hole,
and we shall view it as the ``distance to the black hole.'' The
parameter represents the amplitude of the perturbation, which has the
dimensions of .
Given the asymptotic form of the perturbation in
equation , we can integrate
equation over the entire sphere and the
interval
to obtain an expression
for the total energy
radiated in terms of the amplitude of the perturbation. Thus, we can
characterize the amplitude by the total amount of energy emitted:
. The gravitational waveform
is found to be

(8.1.172) 
In order to simulate the quasinormal ringing of a black hole, it is necessary
to determine the complex eigenvalues of the desired mode,
and then to compute the spheroidal wave function
. The
routines to perform these computations are discussed in the following sections.
Rather than computing the actual gravitational strain waveforms at the
detector, the routines will calculate the quantity
; the normalization
of these waveforms to the correct source distance is left to the calling
routine. The distance normalization can be computed as follows:

(8.1.173) 
where
is the mass of the sun expressed in
seconds (see equation ).
It will be convenient to write the time dependence of the strain as the
complex function
so that
.
The dimensionless eigenfrequency,
,
depends only on the mode and the dimensionless
angular momentum of the black hole. In terms of this quantity,
the function
is

(8.1.174) 
where is the fractional mass loss due to the radiation in the
excited quasinormal mode.
Next: Function: qn_eigenvalues()
Up: GRASP Routines: Black hole
Previous: GRASP Routines: Black hole
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Bruce Allen
20001119