### 5.7 Absorption of gravitational waves by a black hole

In this section, we evaluate the energy absorption rate by a black hole. The energy flux formula is given
by [107]
where , and
In calculating , we need to evaluate the upgoing solution , and the asymptotic
amplitude of ingoing and upgoing solutions, , , and in Equations (19)
and (20). Evaluation of the incident amplitude of the ingoing solution is essential in the
calculation. Poisson and Sasaki [85] 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 [98] 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
where
and is the polygamma function. We see that the absorption effect begins at beyond
the quadrupole formula in the case . If we set in the above formula, we have
which was obtained by Poisson and Sasaki [85].
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
.