

2.1 Teukolsky formalism
In terms of the conventional Boyer-Lindquist coordinates, the
metric of a Kerr black hole is expressed as
where
and
. In the
Teukolsky formalism [61], the gravitational
perturbations of a Kerr black hole are described by a
Newman-Penrose quantity
[38, 39], where
is the Weyl tensor
and
The perturbation equation for
,
, is given by
Here, the operator
is given by
with
. The
source term
is
given by
where
and
,
, and
are the tetrad
components of the energy momentum tensor (
etc.). The bar denotes the
complex conjugation.
If we set
in Equation (6), with appropriate change of the source
term, it becomes the perturbation equation for
. Moreover, it
describes the perturbation for a scalar field (
), a neutrino field
(
), and an
electromagnetic field (
) as well.
We decompose
into the Fourier-harmonic components according
to
The radial function
and the angular function
satisfy the Teukolsky equations with
as
The potential
is given by
where
is the
eigenvalue of
. The angular function
is called the spin-weighted
spheroidal harmonic, which is usually normalized as
In the Schwarzschild limit, it reduces to the spin-weighted
spherical harmonic with
. In the Kerr case, however,
no analytic formula for
is known. The source term
is given by
We mention that for orbits of our interest, which are bounded,
has support only
in a compact range of
.
We solve the radial Teukolsky equation by using
the Green function method. For this purpose, we define two kinds of
homogeneous solutions of the radial Teukolsky equation:
where
, and
is the tortoise coordinate defined by
where
, and where we have fixed the integration constant.
Combining with the Fourier mode
, we see that
has no outcoming
wave from past horizon, while
has no incoming wave at past infinity. Since these
are the properties of waves causally generated by a source, a
solution of the Teukolsky equation which has purely outgoing
property at infinity and has purely ingoing property at the horizon
is given by
where the Wronskian
is given by
Then, the asymptotic behavior at the horizon is
while the asymptotic behavior at infinity is
We note that the homogeneous Teukolsky equation
is invariant under the complex conjugation followed by the
transformation
and
. Thus, we can set
, where the bar denotes the
complex conjugation.
We consider
of a monopole particle of mass
. The energy momentum
tensor takes the form
where
is a geodesic trajectory, and
is the proper
time along the geodesic. The geodesic equations in the Kerr
geometry are given by
where
,
, and
are the energy, the
-component of the
angular momentum, and the Carter constant of a test particle,
respectively, and
Using Equation (28), the tetrad components of the energy
momentum tensor are expressed as
where
and
.
Substituting Equation (10) into Equation (19) and performing integration by part, we
obtain
where
and
denotes
for
simplicity.
For a source bounded in a finite range of
, it is convenient to
rewrite Equation (34) further as
where
Inserting Equation (36) into Equation (26), we obtain
as
where
In this paper, we focus on orbits which are
either circular (with or without inclination) or eccentric but
confined on the equatorial plane. In either case, the frequency
spectrum of
becomes discrete. Accordingly,
in Equation (25) or (26) takes the form,
Then, in particular,
at
is obtained from Equation (14) as
At infinity,
is
related to the two independent modes of gravitational waves
and
as
From Equations (46) and (47), the luminosity averaged over
, where
is the
characteristic time scale of the orbital motion (e.g., a period between the two consecutive
apastrons), is given by
In the same way, the time-averaged angular momentum flux is given
by

