In calculating , we need to evaluate the upgoing
solution
, and
the asymptotic amplitude of ingoing and upgoing solutions,
,
, and
in
Equations (20
) and (21
). Evaluation of the incident amplitude
of the
ingoing solution is essential in the calculation. Poisson and
Sasaki [45
] evaluated them, in the case of
a circular orbit around the Schwarzschild black hole, up to
beyond the lowest
order, and obtained the energy flux at the lowest order, using the
method we have described in Section 3. Later, Tagoshi, Mano, and
Takasugi [55] evaluated the
energy absorption rate in the Kerr case to
beyond the lowest
order using the method in Section 4. Since the resulting
formula is very long and complicated, we show it here only to
beyond the
lowest order. The energy absorption rate is given by
We note that the leading terms in are negative
for
,
i.e., the black hole loses energy if
the particle is co-rotating. This is because of the superradiance
for modes with
.