In the case of orbits in the Schwarzschild
background, one of the earliest papers was by Gal’tsov, Matiukhin
and Petukhov [25], 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 [43] 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 [14] 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 [56
]. 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 [49
] found an analytic method and
obtained formulae that were needed to calculate the gravitational
waves to 4PN order. Then, Tagoshi and Sasaki [57
] 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 [60
].
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 [44]. Then,
Shibata, Sasaki, Tagoshi, and Tanaka [52] 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 [58
], 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 [53
] to 2.5PN order.
Tanaka, Mino, Sasaki, and Shibata [59] 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 [41] and the energy momentum tensor of a spinning particle by Dixon [17]. 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 [45]. The black hole absorption in
the case of a rotating black hole appears at 2.5PN order [24]. Using a
new analytic method to solve the homogeneous Teukolsky equation
found by Mano, Suzuki, and Takasugi [33
], the black hole absorption in
the Kerr case was calculated by Tagoshi, Mano, and
Takasugi [55
] 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 [40] in the case when a compact star is in a circular orbit on the equatorial plane around a Kerr black hole.
In the rest of the paper, we use the units .