1.4 Brief historical notes
Let us briefly review some of the past work on post-Newtonian calculations in black-hole perturbation
theory. Although the literature on numerical calculations of gravitational waves emitted by a particle
orbiting a black hole is abundant, there are not so many papers that deal with the post-Newtonian
expansion of gravitational waves, mainly because such an analysis was not necessary until recently, when
the construction of accurate theoretical templates for the interferometric gravitational wave detectors
became an urgent issue.
In the case of orbits in the Schwarzschild background, one of the earliest papers was by
Gal’tsov, Matiukhin and Petukhov [47], who considered the case when a particle is in a slightly
eccentric orbit around a Schwarzschild black hole, and calculated the gravitational waves up to
1PN order. Poisson [83
] considered a circular orbit around a Schwarzschild black hole and
calculated the waveforms and luminosity to 1.5PN order at which the tail effect appears. Cutler,
Finn, Poisson, and Sussman [24] worked on the same problem numerically by applying the
least-square fitting technique to the numerically evaluated data for the luminosity, and obtained a
post-Newtonian formula for the luminosity to 2.5PN order. Subsequently, a highly accurate numerical
calculation was carried out by Tagoshi and Nakamura [99]. They obtained the formulae for the
luminosity to 4PN order numerically by using the least-square fitting method. They found
the 
terms in the luminosity formula at 3PN and 4PN orders. They concluded that,
although the convergence of the post-Newtonian expansion is slow, the luminosity formula
accurate to 3.5PN order will be good enough to represent the orbital phase evolution of coalescing
compact binaries in theoretical templates for ground-based interferometers. After that, Sasaki [91]
found an analytic method and obtained formulae that were needed to calculate the gravitational
waves to 4PN order. Then, Tagoshi and Sasaki [100] obtained the gravitational waveforms and
luminosity to 4PN order analytically, and confirmed the results of Tagoshi and Nakamura. These
calculations were extended to 5.5PN order by Tanaka, Tagoshi, and Sasaki [105]. Fujita and Iyer [39]
extended this work and derived 5.5PN waveforms. Update
In the case of orbits around a Kerr black hole, Poisson calculated the 1.5PN order corrections to the
waveforms and luminosity due to the rotation of the black hole, and showed that the result agrees with the
standard post-Newtonian effect due to spin-orbit coupling [84]. Then, Shibata, Sasaki, Tagoshi,
and Tanaka [94] calculated the luminosity to 2.5PN order. They calculated the luminosity
from a particle in circular orbit with small inclination from the equatorial plane. They used
the Sasaki–Nakamura equation as well as the Teukolsky equation. This analysis was extended
to 4PN order by Tagoshi, Shibata, Tanaka, and Sasaki [101], in which the orbits of the test
particles were restricted to circular ones on the equatorial plane. The analysis in the case of
slightly eccentric orbit on the equatorial plane was also done by Tagoshi [95, 96] to 2.5PN
order.
Tanaka, Mino, Sasaki, and Shibata [104] considered the case when a spinning particle is in a circular
orbit near the equatorial plane of a Kerr black hole, based on the Papapetrou equations of motion for a
spinning particle [79] and the energy momentum tensor of a spinning particle by Dixon [29]. They derived
the luminosity formula to 2.5PN order which includes the linear order effect of the particle’s
spin.
The absorption of gravitational waves into the black hole horizon, appearing at 4PN order in the
Schwarzschild case, was calculated by Poisson and Sasaki for a particle in a circular orbit [85]. The black
hole absorption in the case of a rotating black hole appears at 2.5PN order [46]. Using a new analytic
method to solve the homogeneous Teukolsky equation found by Mano, Suzuki, and Takasugi [68], the black
hole absorption in the Kerr case was calculated by Tagoshi, Mano, and Takasugi [98] to 6.5PN order
beyond the quadrupole formula.
If gravity is not described by the Einstein theory but by the Brans–Dicke theory, there will appear
scalar-type gravitational waves as well as transverse-traceless gravitational waves. Such scalar-type
gravitational waves were calculated to 2.5PN order by Ohashi, Tagoshi, and Sasaki [77] in the
case when a compact star is in a circular orbit on the equatorial plane around a Kerr black
hole.
Update
In the above works the energy and angular momentum flux at infinity or the absorption rate at
the horizon were evaluated. In the Kerr case, in order to specify the evolution of particle’s
trajectory under the influence of radiation reaction, we need to determine the rate of change of the
Carter constant which is not directly related to the asymptotic gravitational waves. Mino [70]
proved that the average rate of change of the Carter constant can be evaluated by using the
radiative field (i.e., retarded minus advanced field) in the adiabatic approximation. An explicit
calculation of the rate of change of the Carter constant was done in the case of a scalar charged
particle in [30]. Sago et al. [90] extended Mino’s work and found a simpler formula for the
average rate of change of the Carter constant. They derived analytically the rate of change of
the Carter constant as well as the energy and the angular momentum of a particle for orbits
with small eccentricities and inclinations up to 
[89]. In Ref. [48], the method was
extended to the case of the orbits with small eccentricity but arbitrary inclination angle, and the
rate of change of the energy, angular momentum and the Carter constant up to 
were
derived.
In the rest of the paper, we use the units 
.
