Analytic Solutions of the Regge-Wheeler Equation and the Post-Minkowskian Expansion
Analytic Solutions of the Regge-Wheeler Equation and the Post-Minkowskian Expansion
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Regge-Wheeler方程和后Minkowski展开式的解析解
DOI：
10.1143/ptp.96.549
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发表时间：
1996
影响因子：
--
通讯作者：
E. Takasugi
中科院分区：
文献类型：
--
作者：
S. Mano;Hisao Suzuki;E. Takasugi
Regge·Wheeler方程的解析解以一系列具有不同收敛区域的超几何函数和库仑波函数的形式给出。这些解决方案之间的关系已建立。级数解以关于参数 €=2Mw 的后闵可夫斯基展开式给出，M 是黑洞的质量。当将其应用于围绕黑洞的圆形轨道上的粒子的引力辐射时，这种膨胀对应于后牛顿膨胀。这些解决方案对于数值计算也很有用。在之前的工作中，l) 我们以一系列超几何函数的形式提出了 Regge-Wheeler (RW) 方程的解析解。我们证明了本文附录A中给出的超几何函数之间的递推关系，并表明级数系数是在€=2Mw的幂级数中系统确定的，M是黑洞的质量。我们还以一系列库仑波函数的形式提出了解析解，结果与利弗给出的解相同。 2）我们发现这一系列解的特点是重整化角动量是相同的。然后，我们通过匹配这两类解，得到了一个好的解。该方法可以推广到Kerr几何中的Teukolsky方程3)。此时，一系列超几何函数的系数以及一系列库仑波函数的系数满足三项递推关系。关于这些递推关系，Otchik 4) 做出了重要的观察，即两个级数的递推关系是相同的，这使得将这两个级数解联系起来成为可能_*) 在 Otchik 的讨论之后，4) Mano、Suzuki 和 Takasugi 5) 将我们的分析扩展到 Kerr 几何中的 Teukolsky 方程并报告了解析解。我们讨论了这些级数的收敛区域以及不同收敛区域的两个解之间的关系。该级数以 € 展开式表示，该展开式对应于后闵可夫斯基展开式，也对应于后牛顿展开式，当它们应用于围绕黑洞的圆形轨道上的粒子的引力辐射时。在本文中，我们提出了 RW 方程的解析解，并通过重新整理我们之前的工作来讨论这些解的解析性质！）以下*）在 Otchik 的论文中，在两个级数收敛的中间区域研究了超几何函数级数和库仑波函数级数之间的关系，尽管他处理的级数不是 Teukolsky 方程的解。
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 Post Minkowskian 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 gravita tional radiation from a particle in a circular orbit around a black hole. These solutions can also be useful for numerical computations. In 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 character ized 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. Con cerning 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.