The Teukolsky equation has two regular singularities at , and one irregular singularity at . This implies that it cannot be represented in the form of a single hypergeometric equation. However, if we focus on the solution near the horizon, it may be approximated by a hypergeometric equation. This motivates us to consider the solution expressed in terms of a series of hypergeometric functions.
We define the independent variable in place of () as. Leaver obtained solutions of the Teukolsky equation expressed in a series of the Coulomb wave functions. The MST formalism is an elegant reformulation of the one by Leaver .
The essential point is to introduce the so-called renormalized angular momentum , which is a generalization of , to a non-integer value such that the Teukolsky equation admits a solution in a convergent series of hypergeometric functions. Namely, we add the term to both sides of Equation (117) to rewrite it as
Let us denote the formal solution specified by a value of by . We express it in the series form,,
The general solution of the recurrence relation (123) is expressed in terms of two linearly independent solutions and (, ). According to the theory of three-term recurrence relations (see , Page 31) when there exists a pair of solutions that satisfyminimal as (). Any non-minimal solution is called dominant. The minimal solution (either as or as ) is determined uniquely up to an overall normalization factor.
The three-term recurrence relation is closely related to continued fractions. We introduce, Page 31) that the continued fraction (127) converges if and only if the recurrence relation (123) possesses a minimal solution as , and the same for the continued fraction (128) as .
Analysis of the asymptotic behavior of (123) shows that, as long as is finite, there exists a set of two independent solutions that behave as (see, e.g., , Page 35)
Thus, is minimal as and is minimal as .
Since the recurrence relation (123) possesses minimal solutions as , the continued fractions on the right-hand sides of Equations (127) and (128) converge for and . In general, however, and do not coincide. Here, we use the freedom of to obtain a consistent solution. Let be a sequence that is minimal for both . We then have expressions for and in terms of continued fractions as
The minimal solution is also important for the convergence of the series (120). For the minimal solution , together with the properties of the hypergeometric functions for large , we find
Instead of Equation (133), we may consider an equivalent but practically more convenient form of an equation that determines the value of . Dividing Equation (123) by , we find
For later use, we need a series expression for with better convergence properties at large . Using analytic properties of hypergeometric functions, we have
Finally, we note that if is a solution of Equation (133) or (136), with an arbitrary integer is also a solution, since appears only in the combination of . Thus, Equation (133) or (136) contains an infinite number of roots. However, not all of these can be used to express a solution we want. As noted in the earlier part of this section, in order to reproduce the solution in the limit , Equation (118), we must have (or by symmetry). Thus, we impose a constraint on such that it must continuously approach as .
This work is licensed under a Creative Commons License.