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3.3 Outer solution; e « 1

We now solve Equation (70View Equation) in the limit e« 1, i.e., by applying the post-Minkowski expansion to it. In this section, we consider the solution to O(e). Then we match the solution to the horizon solution given by Equation (80View Equation) at e « z « 1.

By setting

sum oo ql(z) = enq(ln)(z), (83) n=0
each (n) ql (z) is found to satisfy
[d2 2 d ( l(l + 1) )] d [1 d ( )] ----+ -----+ 1- -------- q(nl) = e- iz--- ----- eizz2q(ln-1)(z) . (84) dz2 z dz z2 dz z3dz
Equation (84View Equation) is an inhomogeneous spherical Bessel equation. It is the simplicity of this equation that motivated the introduction of the auxiliary function ql [49Jump To The Next Citation Point].

The zeroth-order solution (0) ql satisfies the homogeneous spherical Bessel equation, and must be a linear combination of the spherical Bessel functions of the first and second kinds, jl(z) and nl(z). Here, we demand the compatibility with the horizon solution (80View Equation). Since jl(z) ~ zl and nl(z) ~ z-l-1, nl(z) does not match with the horizon solution at the leading order of e. Therefore, we have

(0) (0) ql (z) = a l jl(z). (85)
The constant a(l0) represents the overall normalization of the solution. Since it can be chosen arbitrarily, we set (0) al = 1 below.

The procedure to obtain (1) ql (z) was described in detail in [49Jump To The Next Citation Point]. Using the Green function G(z, z') = jl(z<)nl(z>), Equation (84View Equation) may be put into an indefinite integral form,

(n) integral z [1 (n-1) ]' integral z [1 (n-1) ]' ql = nl dzz2e-izjl -3(eizz2ql (z))' - jl dzz2e-iznl -3(eizz2ql (z))' . (86) z z
The calculation is tedious but straightforward. All the necessary formulae to obtain q(nl) for n < 2 are given in the Appendix of [49Jump To The Next Citation Point] or Appendix D of [34Jump To The Next Citation Point]. Using those formulae, for n = 1 we have
q(1) = a(1)j + b(1)n l l l l l (l - 1)(l + 3) ( l2 - 4 2l- 1 ) + ----------------jl+1 - ---------- + -------- jl-1 2(l + 1)(2l + 1) 2l(2l + 1) l(l- 1) sum l- 2( 1 1 ) +Rl,0j0 + -- + ------ Rl,mjm - 2Dnjl + ijlln z. (87) m=1 m m + 1
Here, nj D l and Rl,m are functions defined as follows. The function nj D l is given by
1 Dnjl = --[jlSi(2z) - nl (Ci(2z) - g- ln 2z)], (88) 2
where integral Ci(x) = - x oo dtcos t/t and integral Si(x) = x0 dtsint/t. The function Rm,k is defined by Rm,k = z2(nmjk - jmnk). It is a polynomial in inverse powers of z given by
12(m -k-1) ( ) ( )m - k- 1-2r { sum r-G(m----k---r)G--m--+-12---r--- 2- R = - (- 1) r!G(m - k - 2r)G(k + 3 + r) z for m &gt; k, (89) m,k r=0 2 - Rk,m for m &lt; k.

Here, we again perform the matching with the horizon solution (80View Equation). It should be noted that (1) ql, given by Equation (87View Equation), is regular in the limit z --> 0 except for the term (1) b l nl. By examining the asymptotic behavior of Equation (87View Equation) at z« 1, we find b(1)= 0 l, i.e., the solution is regular at z = 0. As for (1) a l, it only contributes to the renormalization of (0) a l. Hence, we set a(1l) = 0 and the transmission amplitude Atlrans is determined to O(e) as

trans (l---2)!(l-+-2)! l+1 2 Al = (2l)!(2l + 1)! e [1 - ieal + O(e )]. (90)
It may be noted that this explicit expression for trans A l is unnecessary for the evaluation of gravitational waves at infinity. It is relevant only for the evaluation of the black hole absorption.

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