### 5.8 Adiabatic evolution of Carter constant for orbit with small eccentricity and small inclination angle
around a Kerr black hole

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In the Schwarzschild case, the particle’s trajectory is characterized by the energy and the
z-component angular momentum . In the adiabatic approximation, the rates of change of and
are equated with those radiated to infinity as gravitational waves or with those absorbed into the
black hole horizon, in accordance with the conservation of and . On the other hand, in
the Kerr case, the Carter constant, , is also necessary to specify the particle’s trajectory.
In this case, the rate of change of is not directly related to the asymptotic gravitational
waves. Mino [70] proposed a new method for evaluating the average rate of change of the Carter
constant by using the radiative field in the adiabatic approximation. He showed that the average
rate of change of the Carter constant as well as the energy and angular momentum can be
obtained by the radiative field of the metric perturbation. The radiative field is defined as half the
retarded field minus half the advanced field. Mino’s work gave a proof of an earlier work by
Gal’tsov [46] in which the radiative field is used to evaluate the average rate of change of the
energy and the angular momentum without proof. Inspired by this new development, it was
demonstrated in [53] and [31] that the time-averaged rates of change of the energy and the angular
momentum can be computed numerically for generic orbits. A first step toward explicit calculation of
the rate of change of the Carter constant was done in the case of a scalar charged particle
in [30]. After that a simpler formula for the average rate of change of the Carter constant was
found in [90, 89]. This new formula relates the rate of change of the Carter constant to the
flux evaluated at infinity and on the horizon. Based on the new formula, in [89], the rate of
change of the Carter constant for orbits with small eccentricities and inclinations is derived
analytically up to by using the MK method discussed in Section 4. In Ref. [48], the
method was extended to the case of the orbits with small eccentricity but arbitrary inclination
angle.
First we show the results for the small inclination case [89]. We define the radius and the
eccentricity as

In Ref. [89], the parameter which expresses a small inclination from the equatorial plane is defined as
Note that this definition of is different from in Equation (197). By solving Equation (209) with
respect to and , we obtain
By using Equations (4.9) and (4.12) in [89], we find the azimuthal angular frequency as
where and . From the definition of , the parameter is
expressed with as
By using the above formula, we can express defined in Equation (197) in terms of above as
The average rate of change of , and become

The rate of change of becomes
We can use Equation (214) to express these equations in terms of . We obtain

We note that if we set , and agree, respectively, with the rate of emission of
the energy and the azimuthal angular momentum radiated to infinity, Equations (195) and (196) in
Section 5.5. We also note that by using the transformation from to , Equation (215), we can
directly show that Equations (202) and (203) in Section 5.6 agree respectively with and
in Equations (220) and (221) with .