4.5 Low frequency expansion of the hypergeometric expansion

So far, we have considered exact solutions of the Teukolsky equation. Now, let us consider their low frequency approximations and determine the value of ν. We solve Equation (136View Equation) with n = 0,
βν0 + α ν0R1 + γν0L −1 = 0, (171 )
with a requirement that ν → ℓ as 𝜖 → 0.

To solve Equation (174View Equation), we first note the following. Unless the value of ν is such that the denominator in the expression of ανn or γνn happens to vanish, or βνn happens to vanish in the limit 𝜖 → 0, we have αν = 𝒪 (𝜖) n, γ ν= 𝒪 (𝜖) n, and β ν= 𝒪(1) n. Also, from the asymptotic behavior of the minimal solution ν fn as n → ±∞ given by Equation (134View Equation), we have Rn (ν) = 𝒪 (𝜖) and L −n(ν) = 𝒪 (𝜖) for sufficiently large n. Thus, except for exceptional cases mentioned above, the order of aνn in 𝜖 increases as |n| increases. That is, the series solution naturally gives the post-Minkowski expansion.

First, let us consider the case of R (ν) n for n > 0. It is easily seen that αν = 𝒪(𝜖) n, γν = 𝒪 (𝜖) n, and ν βn = 𝒪 (1) for all n > 0. Therefore, we have Rn(ν) = 𝒪 (𝜖) for all n > 0.

On the other hand, for n < 0, the order of L−n (ν ) behaves irregularly for certain values of n. For the moment, let us assume that L −1(ν) = 𝒪 (𝜖). We see from Equations (124View Equation) that α ν0 = 𝒪 (𝜖), γ0ν= 𝒪(𝜖), since ν = ℓ + 𝒪 (𝜖). Then, Equation (174View Equation) implies βν0 = 𝒪 (𝜖2). Using the expansion of λ given by Equation (110View Equation), we then find ν = ℓ + 𝒪 (𝜖2) (i.e., there is no term of 𝒪 (𝜖) in ν). With this estimate of ν, we see from Equation (128View Equation) that L −1(ν) = 𝒪 (𝜖) is justified if L −2(ν) is of order unity or smaller.

The general behavior of the order of L− n(ν ) in 𝜖 for general values of s is rather complicated. However, if we assume s to be a non-integer and ℓ ≥ |s|, and τ = (𝜖 − mq )∕κ = 𝒪 (1 ), it is relatively easily studied. With the assumption that ν = ℓ + 𝒪 (𝜖2), we find there are three exceptional cases:

These imply that L−2ℓ−1(ν) = 𝒪 (1∕𝜖), L −ℓ−1(ν) = 𝒪 (1), and 2 L−ℓ(ν) = 𝒪 (𝜖 ), respectively. To summarize, we have

ν R (ν) = -fn--= 𝒪 (𝜖) for all n > 0, n fνn−1 ν L (ν) = -f−ℓ--= 𝒪 (𝜖2), −ℓ fν−ℓ+1 ν L (ν) = f−ℓ−1-= 𝒪 (1), (172 ) −ℓ−1 fν−ℓ ν L (ν) = f−2ℓ−1 = 𝒪 (1∕𝜖), −2ℓ−1 fν−2ℓ ν Ln (ν) = -fn--= 𝒪 (𝜖) for all the other n < 0. fνn+1
With these results, we can calculate the value of ν to 𝒪 (𝜖), which is given by
1 ( s2 [(ℓ + 1)2 − s2]2 (ℓ2 − s2)2 ) ν = ℓ + ------ − 2 − --------+ ------------------------− ------------------ 𝜖2 + 𝒪 (𝜖3).(173 ) 2ℓ + 1 ℓ(ℓ + 1) (2ℓ + 1)(2ℓ + 2)(2 ℓ + 3) (2ℓ − 1)2ℓ(2ℓ + 1)
Now one can take the limit of an integer value of s. In particular, the above holds also for s = 0. Interestingly, ν is found to be independent of the azimuthal eigenvalue m to 2 𝒪 (𝜖 ).

The post-Minkowski expansion of homogeneous Teukolsky functions can be obtained with arbitrary accuracy by solving Equation (123View Equation) to a desired order, and by summing up the terms to a sufficiently large |n |. The first few terms of the coefficients fν n are explicitly given in [68Jump To The Next Citation Point]. A calculation up to a much higher order in 𝒪(𝜖) was performed in [98Jump To The Next Citation Point], in which the black hole absorption of gravitational waves was calculated to 8 𝒪 (v ) beyond the lowest order.

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