4 Analytic Solutions of the Homogeneous Teukolsky Equation by Means of the Series Expansion of Special Functions

In this section, we review a method developed by Mano, Suzuki, and Takasugi [68Jump To The Next Citation Point], who found analytic expressions of the solutions of the homogeneous Teukolsky equation. In this method, the exact solutions of the radial Teukolsky equation (14View Equation) are expressed in two kinds of series expansions. One is given by a series of hypergeometric functions and the other by a series of the Coulomb wave functions. The former is convergent at horizon and the latter at infinity. The matching of these two solutions is done exactly in the overlapping region of convergence. They also found that the series expansions are naturally related to the low frequency expansion. Properties of the analytic solutions were studied in detail in [69Jump To The Next Citation Point]. Thus, the formalism is quite powerful when dealing with the post-Newtonian expansion, especially at higher orders.

In many cases, when we study the perturbation of a Kerr black hole, it is more convenient to use the Sasaki–Nakamura equation, since it has the form of a standard wave equation, similar to the Regge–Wheeler equation. However, it is not quite suited for investigating analytic properties of the solution near the horizon. In contrast, the Mano–Suzuki–Takasugi (MST) formalism allows us to investigate analytic properties of the solution near the horizon systematically. Hence, it can be used to compute the higher order post-Newtonian terms of the gravitational waves absorbed into a rotating black hole.

We also note that this method is the only existing method that can be used to calculate the gravitational waves emitted to infinity to an arbitrarily high post-Newtonian order in principle.

 4.1 Angular eigenvalue
 4.2 Horizon solution in series of hypergeometric functions
 4.3 Outer solution as a series of Coulomb wave functions
 4.4 Matching of horizon and outer solutions
 4.5 Low frequency expansion of the hypergeometric expansion
 4.6 Property of ν

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