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 [83Jump To The Next Citation Point] 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 [99Jump To The Next Citation Point]. They obtained the formulae for the luminosity to 4PN order numerically by using the least-square fitting method. They found the logv 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 [91Jump To The Next Citation Point] found an analytic method and obtained formulae that were needed to calculate the gravitational waves to 4PN order. Then, Tagoshi and Sasaki [100Jump To The Next Citation Point] 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 [105Jump To The Next Citation Point]. Fujita and Iyer [39Jump To The Next Citation Point] extended this work and derived 5.5PN waveforms. UpdateJump To The Next Update Information

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 [94Jump To The Next Citation Point] 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 [101Jump To The Next Citation Point], 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 [95Jump To The Next Citation Point, 96Jump To The Next Citation Point] 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 [85Jump To The Next Citation Point]. The black hole absorption in the case of a rotating black hole appears at 2.5PN order [46Jump To The Next Citation Point]. Using a new analytic method to solve the homogeneous Teukolsky equation found by Mano, Suzuki, and Takasugi [68Jump To The Next Citation Point], the black hole absorption in the Kerr case was calculated by Tagoshi, Mano, and Takasugi [98Jump To The Next Citation Point] 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. UpdateJump To The Next Update Information

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 [70Jump To The Next Citation Point] 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 [30Jump To The Next Citation Point]. Sago et al. [90Jump To The Next Citation Point] 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 O (v5)  [89Jump To The Next Citation Point]. In Ref. [48Jump To The Next Citation Point], 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 5 O (v ) were derived.

In the rest of the paper, we use the units c = G = 1.


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