6 Conclusion

In this article, we described analytical approaches to calculate gravitational radiation from a particle of mass μ orbiting a black hole of mass M with M ≫ μ, based upon the perturbation formalism developed by Teukolsky. A review of this formalism was given in Section 2. The Teukolsky equation, which governs the gravitational perturbation of a black hole, is too complicated to be solved analytically. Therefore, one has to adopt a certain approximation scheme. The scheme we employed is the post-Minkowski expansion, in which all the quantities are expanded in terms of a parameter 𝜖 = 2M ω where ω is the Fourier frequency of the gravitational waves. For the source term given by a particle in bound orbit, this naturally gives the post-Newtonian expansion.

In Section 3, we considered the case of a Schwarzschild background. For a Schwarzschild black hole, one can transform the Teukolsky equation into the Regge–Wheeler equation. The advantage of the Regge–Wheeler equation is that it reduces to the standard Klein–Gordon equation in the flat-space limit, and hence it is easier to understand the post-Minkowskian or post-Newtonian effects. Therefore, we adopted this method in the case of a Schwarzschild background. However, the post-Minkowski expansion of the Regge–Wheeler equation is not quite systematic, and as one goes to higher orders, the equations to be solved become increasingly complicated. Furthermore, for a Kerr background, although one can perform a transformation similar to the Chandrasekhar transformation, it can be done only at the expense of losing the reality of the equation. Thus, the resulting equation is not quite suited for analytical treatments.

In Section 4, we described a different method, developed by Mano, Suzuki, and Takasugi [69, 68], that directly deals with the Teukolsky equation, and we considered the case of a Kerr background with this method. Although the method is mathematically rather complicated and it is hard to obtain physical insights into relativistic effects, it has great advantage in that it allows a systematic post-Minkowski expansion of the Teukolsky equation, even on the Kerr background. We gave a thorough review on how this method works and how it gives a systematic post-Minkowski expansion.

Finally, in Section 5, we recapitulated the results of calculations of the gravitational waves for various orbits that had been obtained by various authors using the methods described in Sections 3 and 4. These results are useful not only by themselves for the actual case of a compact star orbiting a supermassive black hole, but also because they give us useful insights into higher order post-Newtonian effects even for a system of equal-mass binaries.

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