Analytic Solutions of the Teukolsky Equation in Kerr-de Sitter and Kerr-Newman-de Sitter Geometries

Abstract

An analytic solution of the Teukolsky equation in Kerr-de Sitter and Kerr-Newman-de Sitter geometries is presented, and the properties of this solution are examined. In particular, we show that our solution satisfies the Teukolsky-Starobinsky identities explicitly and fix the relative normalization between solutions with spin weights s and −s. §1. IntroductionIn a series of papers, Mano, Suzuki and Takasugi 1), 2) have constructed an analytic solution of the perturbation equation of massless fields in Kerr geometries which is commonly called the Teukolsky equation. 3) This solution enabled us to investigate various properties of black holes analytically, 4) the scattering problem of a particle emitted in a black hole, an analytical expression for the absorption rate by a Kerr black hole, which was one of the main interests of Chandrasekhar, 5) and the Teukolsky-Starobinsky identities.Our solution is expressed in the form of a series of hypergeometric functions and Coulomb wave functions. The coefficients of this series are determined by solving three-term recurrence relations. Two series have different convergence regions, and they are matched in the region where both series converge. This method turns out to be quite powerful in practical calculations and is successfully applied to examine gravitational waves from a binary in a post-Newtonian expansion; to construct the template of the gravitational wave emitted from a particle moving around a Kerr black hole 6) and the rate of absorption of a gravitational wave by the black hole. 7)In a previous paper, 8) Suzuki, Takasugi and Umetsu extended this method to solve the perturbation equations of massless fields (the Teukolsky equation) in Kerrde Sitter and Kerr-Newman-de Sitter geometries. We found transformations such that both the angular and the radial equations are reduced to Heun's equation. 9) The solution of Heun's equation is expressed in the form of a series of hypergeometric functions, and its coefficients are determined by three-term recurrence relations, similarly to the Kerr geometry case. It should be noted that electromagnetic fields and gravitational fields couple to each other in Kerr-Newman-de Sitter geometries and do not obey the equation we considered there, although the fields obey the * )

Quasinormal modes of black holes: From astrophysics to string theory

“…The (3 + 1) Schwarzschild-de Sitter and ReissnerNordström de Sitter black holes were proven to be stable as well 96 , and the stability of the Schwarzschild-anti-de Sitter solution was shown in 153 . The stability of Kerrde Sitter black holes was shown recently in 287 , and of Kerr-AdS black holes in 163 . Finally, the D = 4 string theory inspired black holes, such as dilaton black holes, Born-Infeld black holes, and Gauss-Bonnet coupled to dilaton black holes, were shown to be gravitationally stable as well 92,[288][289][290][291][292] .…”

Spin-2 spectrum of defect theories

“…These series converge in a region around a given singular point, so to determine m 2 in the case at hand one must match the expansions around the points s = 0 and s = 1 (for an example of this procedure see [67,68]). For the purposes of this work, simpler numerical algorithms will be sufficient.…”

Superradiant instability of Kerr-de Sitter black holes in scalar-tensor theory

“…For more general BHs, there is no explicit analytical expression of λ. For Kerr spacetime, ∆ θ = 1 and the above equation becomes a spheroidal equation which has been studied in detail in [25][26][27][28][29][30][31][32]. And λ can be expanded as a series of aω for small aω, whose explicit form can be found in [26,27,32].…”