Analytic solutions of the Teukolsky equation in Kerr geometries are presented in the form of series of hypergeometric functions and Coulomb wave functions. Relations between these solutions are established. The solutions provide a very powerful method not only for examining the general properties of solutions and physical quantities when they are applied to, but also for numerical computations. The solutions are given in the expansion of a small parameter ǫ ≡ 2Mω, M being the mass of black hole, which corresponds to Post-Minkowski expansion by G and to post-Newtonian expansion when they are applied to the gravitational radiation from a particle in circular orbit around a black hole. It is expected that these solutions will become a powerful weapon to construct the theoretical template towards LIGO and VIRGO projects.

549 Analytic solutions of the Regge·Wheeler equation are presented in the form of a series of hypergeometric functions and Coulomb wave functions which have different regions of convergence. Relations among these solutions are established. The series solutions are given as the PostMinkowskian expansion with respect to the parameter €=2Mw, M being the mass of a black hole.This expansion corresponds to the post-Newtonian expansion when they are applied to the gravitational radiation from a particle in a circular orbit around a black hole. These solutions can also be useful for numerical computations. § 1. IntroductionIn a previous work,l) we presented analytic solutions of the Regge-Wheeler (RW) equation in the form of a series of hypergeometric functions. We proved that recurrence relations among hypergeometric functions as given in Appendix A in this text and showed that coefficients of series are systematically determined in a power series of €=2Mw, M being the mass of black hole. We also presented analytic solutions in the form of a series of Coulomb wave functions which turn out to be the same as those given by Leaver. 2 ) We found that the series of solutions is characterized by the renormalized angular momentum which turns out to be identical. Then, we obtained a good solution by matching these two types of solutions.This method can be extended for the Teukolsky equation 3 ) in the Kerr geometry_ In this case, the coefficients of series of hypergeometric functions and also those of a series of Coulomb wave functions satisfy the three term recurrence relations. Concerning these recurrence relations, Otchik 4 ) made the important observation that the recurrence relation for the two series are identical, which made it possible to relate these two series solutions_*) Following the discussion by Otchik,4) Mano, Suzuki and Takasugi 5 ) extended our analysis to the Teukolsky equation in the Kerr geometry and reported analytic solutions. We discussed the convergence regions of these series and the relation between two solutions of different regions of convergence. The series are expressed in the € expansion which corresponds to the Post-Minkowskian expansion and also to the post-Newtonian expansion when they are applied to the gravitational radiation from a particle in circular orbit around a black hole.In this paper, we present analytic solutions of the RW equation and discuss the analytic properties of these solutions by reorganizing our previous work!) following *) In Otchik's paper, the relation between the series of hypergeometric functions and the series of Coulomb wave functions is studied in the intermediate region where both series converge, though the series which he treated are not the solutions of Teukolsky equation.

When a particle moves around a Kerr black hole, it radiates gravitational waves. Some of these waves are absorbed by the black hole. We calculate such absorption of gravitational waves induced by a particle of mass J1 in a circular orbit on an equatorial plane around a Kerr black hole of mass M. We assume that the velocity of the particle v is much smaller than the speed of light c and calculate the energy absorption rate analytically. We adopt an analytic technique for the Teukolsky equation developed by Mano, Suzuki and Takasugi. We obtain the energy absorption rate to O((v/C)8) compared to the lowest order. We find that the black hole absorption occurs at O((V/C)5) beyond the Newtonian-quadrupole luminosity at infinity in the case when the black hole is rotating, which is O((v/c?) lower than the non-rotating case. Using the energy absorption rate, we investigate its effects on the orbital evolution of coalescing compact binaries.

The analytical solutions reported in our previous paper are given as series of hypergeometric or Coulomb wave functions. Using them, we can get the Teukolsky functions analytically in a desired accuracy. For the computation, the deep understanding of their properties is necessary. We sum· marize the main result: The relative normalization between the solutions with a spin weight sand -s is given analytically using the Teukolsky·Starobinsky (T·S) identities. By examining the asymptotic behavior of our solution and combined with the T·S identities and the Wronskian, we found nontrivial identities between the sums of coefficients of the series. These identities will serve to make various expression in simpler forms and also become a powerful tool to test the accuracy of the computation. As an application, we investigated the absorption rate and the evaporation rate of black hole and obtain interesting analytic results.

When a particle moves around a Kerr black hole, it radiates gravitational waves. Some of these waves are absorbed by the black hole. We calculate such absorption of gravitational waves induced by a particle of mass µ in a circular orbit on an equatorial plane around a Kerr black hole of mass M . We assume that the velocity of the particle v is much smaller than the speed of light c and calculate the energy absorption rate analytically. We adopt an analytic technique for the Teukolsky equation developed by Mano, Suzuki and Takasugi. We obtain the energy absorption rate to O((v/c) 8 ) compared to the lowest order. We find that the black hole absorption occurs at O((v/c) 5 ) beyond the Newtonian-quadrapole luminosity at infinity in the case when the black hole is rotating, which is O((v/c) 3 ) lower than the non-rotating case. Using the energy absorption rate, we investigate its effects on the orbital evolution of coalescing compact binaries. typeset using PTPT E X.sty