Horizon solution in series of hypergeometric functions

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 (14

). 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

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

where

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

Then the radial Teukolsky equation becomes

where a prime denotes $d∕dx$ . The left-hand side of Equation (117

) is in the form of a hypergeometric equation. In the limit $𝜖 → 0$ , noting Equation (110

), we find that a solution that is finite at $x = 0$ is given by

For a general value of $𝜖$ , Equation (117

) 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 [64

]. 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 [64

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 (117 ) 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 (118

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 [68

(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 (120

) into Equation (119

) and using the above recurrence relations, we obtain a three-term recurrence relation among the expansion coefficients $an$ . It is given by

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 (120

) is determined by the asymptotic behaviors of the coefficients $aνn$ at $n → ± ∞$ . We thus discuss properties of the three-term recurrence relation (123

) and the role of the parameter $ν$ in detail.

The general solution of the recurrence relation (123 ) 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 [49 ], Page 31) when there exists a pair of solutions that satisfy

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

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 [49

], Page 31) that the continued fraction (127

) converges if and only if the recurrence relation (123

) possesses a minimal solution as $n → ∞$ , and the same for the continued fraction (128

) as $n → − ∞$ .

Analysis of the asymptotic behavior of (123 ) 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)

and another set of two independent solutions that behave as

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

Since the recurrence relation (123 ) possesses minimal solutions as $n → ±∞$ , the continued fractions on the right-hand sides of Equations (127 ) and (128 ) 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

Thus, if we choose $ν$ such that it satisfies the implicit equation for $ν$ , Equation (133

), for a certain $n$ , we obtain a unique minimal solution ${fνn}$ that is valid over the entire range of $n$ , $− ∞ < n < ∞$ , that is

Note that if Equation (133

) 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 (120 ). For the minimal solution $ν {fn }$ , together with the properties of the hypergeometric functions $pn+ν$ for large $|n |$ , we find

Thus, the series of hypergeometric functions (120

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

where $Rn+1$ and $Ln+1$ are those given by the continued fractions (131

) and (132

), 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

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 (123

) 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 (133 ) or (136 ), $ν + k$ with an arbitrary integer $k$ is also a solution, since $ν$ appears only in the combination of $n + ν$ . 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 $𝜖 → 0$ , Equation (118 ), 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 $𝜖 → 0$ .