4.2 Horizon solution in series of hypergeometric functions

As in Section 3, we focus on the ingoing wave function of the radial Teukolsky equation (14View Equation). Since the analysis below is applicable to any spin, |s| = 0, 1∕2, 1, 3∕2, and 2, we do not specify it except when it is needed. Also, the analysis is not restricted to the case aω ≪ 1 unless so stated explicitly. For general spin weight s, the homogeneous Teukolsky equation is given by
( ) ( 2 ) Δ −s-d- Δs+1 dR-ℓmω- + K--−-2is(r-−-M--)K--+ 4isωr − λ R ℓm ω = 0. (113 ) dr dr Δ
As before, taking account of the symmetry ¯ R ℓm ω = R ℓ− m−ω, we may assume 𝜖 = 2M ω > 0 if necessary.

The Teukolsky equation has two regular singularities at r = r±, and one irregular singularity at r = ∞. 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 x in place of z (= ωr) as

x = z+-−--z, (114 ) 𝜖κ
where
∘ ------ a z± = ωr ±, κ = 1 − q2, q = ---. (115 ) M
For later convenience, we also introduce τ = (𝜖 − mq )∕κ and 𝜖± = (𝜖 ± τ )∕2. Taking into account the structure of the singularities at r = r±, we put the ingoing wave Teukolsky function Riℓnmω as
in i𝜖κx −s−i(𝜖+τ)∕2 i(𝜖− τ)∕2 R ℓm ω = e (− x) (1 − x ) pin(x ). (116 )
Then the radial Teukolsky equation becomes
′′ ′ x(1 − x)pin + [1 − s − i𝜖 − iτ − (2 − 2iτ)x]pin + [iτ(1 − iτ) + λ + s(s + 1 )]pin = 2i𝜖κ[− x(1 − x)pin′ + (1 − s + i𝜖 − iτ)xpin] + [𝜖2 − i𝜖κ(1 − 2s)]pin, (117 )
where a prime denotes d∕dx. The left-hand side of Equation (117View Equation) is in the form of a hypergeometric equation. In the limit 𝜖 → 0, noting Equation (110View Equation), we find that a solution that is finite at x = 0 is given by
pin(𝜖 → 0) = F (− ℓ − iτ,ℓ + 1 − iτ, 1 − s − iτ,x). (118 )
For a general value of 𝜖, Equation (117View Equation) suggests that a solution may be expanded in a series of hypergeometric functions with 𝜖 being a kind of expansion parameter. This idea was extensively developed by Leaver [64Jump To The Next Citation Point]. 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 [64Jump To The Next Citation Point].

The essential point is to introduce the so-called renormalized angular momentum ν, which is a generalization of ℓ, 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 [ν(ν + 1) − λ − s(s + 1)]pin to both sides of Equation (117View Equation) to rewrite it as

x(1 − x)pin′′ + [1 − s − i𝜖 − iτ − (2 − 2iτ)x]pin′ + [iτ(1 − iτ) + ν (ν + 1)]pin = ′ 2i𝜖κ [− x(1 − x)pin + (1 − s + i𝜖 − iτ)xpin] +[ν(ν + 1) − λ − s(s + 1) + 𝜖2 − i𝜖κ(1 − 2s)]pin. (119 )
Of course, no modification is done to the original equation, and ν is just an irrelevant parameter at this stage. A trick is to consider the right-hand side of the above equation as a perturbation, and look for a formal solution specified by the index ν in a series expansion form. Then, only after we obtain the formal solution, we require that the series should converge, and this requirement determines the value of ν. Note that, if we take the limit 𝜖 → 0, we must have ν → ℓ (or ν → − ℓ − 1) to assure [ν (ν + 1) − λ − s(s + 1)] → 0 and to recover the solution (118View Equation).

Let us denote the formal solution specified by a value of ν by pνin. We express it in the series form,

∑∞ p νin = an pn+ν(x ), n=−∞ (120 ) pn+ν(x) = F (n + ν + 1 − iτ,− n − ν − iτ;1 − s − i𝜖 − iτ ;x).
Here, the hypergeometric functions pn+ ν(x) satisfy the recurrence relations [68Jump To The Next Citation Point],
(n + ν + 1 − s − i𝜖)(n + ν + 1 − iτ ) xpn+ ν = − ----2-(n-+--ν +-1)(2n-+--2ν +-1)-----pn+ν+1 [ ] 1- ----iτ(s +-i𝜖)------ + 2 1 + (n + ν)(n + ν + 1) pn+ ν − (n-+--ν +-s +-i𝜖)(n-+-ν-+-iτ)pn+ν−1, (121 ) 2 (n + ν)(2n + 2ν + 1) ′ (n + ν + iτ)(n + ν + 1 − iτ)(n + ν + 1 − s − i𝜖) x(1 − x)pn+ ν = -----------------------------------------------pn+ν+1 [2(n + ν + 1)(2n + 2ν + 1]) 1- ----iτ(1 −-iτ)----- + 2(s + i𝜖) 1 + (n + ν)(n + ν + 1) pn+ν − (n-+--ν +-1 −-iτ)(n-+-ν-+-iτ)(n-+-ν-+-s-+-i𝜖)pn+ν−1, (122 ) 2(n + ν )(2n + 2ν + 1)
Inserting the series (120View Equation) into Equation (119View Equation) and using the above recurrence relations, we obtain a three-term recurrence relation among the expansion coefficients an. It is given by
ν ν ν αnan+1 + β nan + γnan−1 = 0, (123 )
where
ν i𝜖κ(n + ν + 1 + s + i𝜖)(n + ν + 1 + s − i𝜖)(n + ν + 1 + iτ ) αn = ----------------(n-+-ν-+-1)(2n-+-2ν-+-3)-----------------, 2 2 βν = − λ − s(s + 1) + (n + ν)(n + ν + 1) + 𝜖2 + 𝜖(𝜖 − mq ) +-𝜖(𝜖-−-mq-)(s-+-𝜖)-, (124 ) n (n + ν )(n + ν + 1 ) ν i𝜖κ(n + ν − s + i𝜖)(n + ν − s − i𝜖)(n + ν − iτ) γn = − -------------(n-+-ν)(2n-+-2ν-−-1)-------------.
The convergence of the series (120View Equation) is determined by the asymptotic behaviors of the coefficients aνn at n → ± ∞. We thus discuss properties of the three-term recurrence relation (123View Equation) and the role of the parameter ν in detail.

The general solution of the recurrence relation (123View Equation) is expressed in terms of two linearly independent solutions (1) {f n } and (2) {fn } (n = ±1, ±2, ...). According to the theory of three-term recurrence relations (see [49Jump To The Next Citation Point], Page 31) when there exists a pair of solutions that satisfy

( ) f(n1) fn(1) lni→m∞ -(2)-= 0 nl→im−∞ --(2) = 0 , (125 ) fn fn
then the solution {f(n1)} is called minimal as n → ∞ (n → − ∞). Any non-minimal solution is called dominant. The minimal solution (either as n → ∞ or as n → − ∞) is determined uniquely up to an overall normalization factor.

The three-term recurrence relation is closely related to continued fractions. We introduce

R ≡ -an-, L ≡ -an--. (126 ) n an−1 n an+1
We can express Rn and Ln in terms of continued fractions as
-----γνn------ -γ-νn- αnγνn+1- αn+1γνn+2- Rn = − βν + ανRn+1 = − β ν− ⋅ βν − ⋅ βν − ⋅ ..., (127 ) n nν nν n+1 ν n+2ν Ln = − ----α-n------= − -αn--⋅ αn-−1γn ⋅ αn-−2γn−1 ⋅ .... (128 ) βνn + γνnLn −1 βνn− βνn−1− βνn−2−
These expressions for Rn and Ln are valid if the respective continued fractions converge. It is proved (see [49Jump To The Next Citation Point], Page 31) that the continued fraction (127View Equation) converges if and only if the recurrence relation (123View Equation) possesses a minimal solution as n → ∞, and the same for the continued fraction (128View Equation) as n → − ∞.

Analysis of the asymptotic behavior of (123View Equation) shows that, as long as ν is finite, there exists a set of two independent solutions that behave as (see, e.g., [49], Page 35)

a(1n) i𝜖κ a(2n) 2i lim n-(1)-= ---, lim ---(2)- = ---, (129 ) n→ ∞ an−1 2 n→∞ na n−1 𝜖κ
and another set of two independent solutions that behave as
(1) (2) bn--- i𝜖κ -bn--- 2i- n→lim−∞ n (1) = − 2 , nl→im− ∞ (2) = − 𝜖κ. (130 ) bn+1 nbn+1

Thus, {a(1)} n is minimal as n → ∞ and {b(1)} n is minimal as n → − ∞.

Since the recurrence relation (123View Equation) possesses minimal solutions as n → ±∞, the continued fractions on the right-hand sides of Equations (127View Equation) and (128View Equation) converge for an = a(n1) and an = b(n1). In general, however, a (1n) and b(n1) do not coincide. Here, we use the freedom of ν to obtain a consistent solution. Let ν {fn } be a sequence that is minimal for both n → ± ∞. We then have expressions for ν ν fn∕fn−1 and ν ν fn ∕fn+1 in terms of continued fractions as

fn γ ν αnγν αn+1γν &tidle;Rn ≡ -----= − --νn-⋅ -ν-n+1-⋅ --ν--n+2-⋅ ..., (131 ) fn−1 β n− βn+1− βn+2− &tidle; -fn-- -α-νn- αn−1γnν αn−2γnν−-1 Ln ≡ f = − β ν− ⋅ βν − ⋅ βν − ⋅ .... (132 ) n+1 n n− 1 n− 2
This implies
&tidle;R &tidle;L = 1. (133 ) n n−1
Thus, if we choose ν such that it satisfies the implicit equation for ν, Equation (133View Equation), for a certain n, we obtain a unique minimal solution {fνn} that is valid over the entire range of n, − ∞ < n < ∞, that is
-fνn-- i𝜖κ -fνn-- i𝜖κ nli→m∞ n fν = 2 , nl→im−∞ n fν = − 2 . (134 ) n−1 n+1
Note that if Equation (133View Equation) for a certain value of n is satisfied, it is automatically satisfied for any other value of n.

The minimal solution is also important for the convergence of the series (120View Equation). For the minimal solution ν {fn }, together with the properties of the hypergeometric functions pn+ν for large |n |, we find

ν ν lim n fn+1pn+ν+1(x)-= − lim nfn−-1pn+-ν−1(x) = i𝜖κ [1 − 2x + ((1 − 2x)2 − 1)1∕2]. (135 ) n→ ∞ fνnpn+ν(x) n→− ∞ fnνpn+ ν(x) 2
Thus, the series of hypergeometric functions (120View Equation) converges for all x in the range 0 ≥ x > − ∞ (in fact, for all complex values of x except at |x| = ∞), provided that the coefficients are given by the minimal solution.

Instead of Equation (133View Equation), we may consider an equivalent but practically more convenient form of an equation that determines the value of ν. Dividing Equation (123View Equation) by an, we find

ν ν ν β n + α nRn+1 + γnLn− 1 = 0, (136 )
where Rn+1 and Ln+1 are those given by the continued fractions (131View Equation) and (132View Equation), respectively. Although the value of n in this equation is arbitrary, it is convenient to set n = 0 to solve for ν.

For later use, we need a series expression for Rin with better convergence properties at large |x|. Using analytic properties of hypergeometric functions, we have

Rin = R ν+ R −ν−1, (137 ) 0 0
where
R ν0 = ei𝜖κx(− x)−s−(i∕2)(𝜖+τ)(1 − x)(i∕2)(𝜖+ τ)+ν ∑∞ × fν---Γ (1-−-s-−-i𝜖 −-iτ-)Γ (2n-+-2ν-+-1-)- n=− ∞ nΓ (n + ν + 1 − iτ) Γ (n + ν + 1 − s − i𝜖) × (1 − x )nF(− n − ν − iτ,− n − ν − s − i𝜖;− 2n − 2ν;-1---). (138 ) 1 − x
This expression explicitly exhibits the symmetry of R in under the interchange of ν and − ν − 1. This is a result of the fact that ν(ν + 1) is invariant under the interchange ν ↔ − ν − 1. Accordingly, the recurrence relation (123View Equation) has the structure that − ν− 1 f−n satisfies the same recurrence relation as fnν.

Finally, we note that if ν is a solution of Equation (133View Equation) or (136View Equation), ν + k with an arbitrary integer k is also a solution, since ν appears only in the combination of n + ν. Thus, Equation (133View Equation) or (136View Equation) 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 𝜖 → 0, Equation (118View Equation), we must have ν → ℓ (or ν → − ℓ − 1 by symmetry). Thus, we impose a constraint on ν such that it must continuously approach ℓ as 𝜖 → 0.


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