Perturbations of Kerr-de Sitter Black Holes and Heun's Equations

Abstract

It is well known that the perturbation equations of massless fields for the Kerr-de Sitter geometry can be written in the form of separable equations. The equations have five definite singularities, so it has been thought that their analysis would be difficult. We show that these equations can be transformed into Heun's equations, for which we are able to use a known technique to analyze solutions. We reproduce known results for the Kerr geometry and de Sitter geometry in the confluent limits of Heun's functions. Our analysis can be extended to Kerr-Newman-de Sitter geometry for massless fields with spin 0 and ~. §1. IntroductionOne of the most non-trivial aspects of the perturbation equations for Kerr geometry is the separability of the radial and the angular parts. Carter 1) first found that the scalar wave function is separable in the Kerr-Newman-de Sitter geometries. Later, this observation was extended for spin 1/2, electromagnetic fields, gravitational perturbations and gravitinos for the Kerr geometries and even for the Kerr-de Sitter class of geometries. These perturbation equations are called Teukolsky equations. 2 ) Except for electromagnetic and gravitational perturbations, the separability persists even for the Kerr-Newman-de Sitter solutions. An important application of this fact is the proof of the stability of the Kerr black hole. 3) Though the Teukolsky equations for Kerr geometries are separable, both spheroidal and radial equations have two regular singularities and one irregular singularity, so that the solutions cannot be written in a single form of any special functions, but they can be expressed as a series of special functions whose coefficients satisfy three term recurrence relations. 4) -7) The solution of the angular equation is expressed in the form of a series of Jacobi functions. 4 ) The solution of the radial equation is rather complicated, because we need a solution which is valid in the entire region extending from the outer horizon to infinity. The solutions are written in the form of a series of confluent hypergeometric functions 5) which are convergent around infinity and hypergeometric functions 6), 7) which are convergent around the outer horizon. By matching these solutions in the region where both solutions are convergent, we can obtain a solution which is valid from the outer horizon to infinity. 6)-8) The great benefit of this kind of solution is that the coefficients of the series are obtained through the post Minkowskian expansion. 6), 7) This technique has been *)

Isomonodromy, Painlevé transcendents and scattering off of black holes

“…This is done in the next section. A similar result has been reported by [19] for massless perturbations of spin s = 0, 1 2 , 1, 3 2 , 2 for Kerr-(A)dS (the so-called Teukolsky master equation) and for s = 0, 1 2 for Kerr-Newman-(A)dS. One can show that Teukolsky master equation reduces to conformally coupled Klein-Gordon equation for scalar perturbations, being those perturbations of the Weyl tensor.…”

“…The fact that we had, for generic values for the curvature coupling ξ, mass M , angular momentum a = J/M , NUT charge b and cosmological constant Λ, a Fuchsian equation is a remarkable fact in itself. We found further that, while the generic coupling has 5 regular singular points for the radial equation, 3 the conformally coupled case ξ = 1/6 has one apparent singularity, and the ensuing equation is of Heun type [19,20].…”

“…We note at this point that, for the de Sitter case, t 0 is a real number, which can be taken to be between 1 and ∞, whereas for the anti-de Sitter case, it is a pure phase |t 0 | = 1. Now, we define 19) and…”

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

“…For Kerr-de Sitter BH, eq. (3.2) is a generalized spheroidal equation and the explicit expansion of separation constant λ in small aω is [33,34] …”

The analytic solution for the power series expansion of Heun function

“…The Heun function also appears in Kerr-Newman-de Sitter Geometry for massless fields with spin 0 and 1 2 . The angular and radial Teukolsky equations eventually transform into the Heun function [3,4] In this paper I will construct the power series expansion of Heun function in closed forms analytically with the three-term recurrence formula [5]. Heun's equation is a second-order linear ordinary differential equation of the form [6,9] With the condition ǫ = α + β − γ − δ + 1.…”