3.3 Outer solution; πœ– β‰ͺ 1

We now solve Equation (70View Equation) in the limit πœ– β‰ͺ 1, i.e., by applying the post-Minkowski expansion to it. In this section, we consider the solution to π’ͺ (πœ–). Then we match the solution to the horizon solution given by Equation (80View Equation) at πœ– β‰ͺ z β‰ͺ 1.

By setting

∑∞ ξβ„“(z) = πœ–nξ(β„“n)(z), (83 ) n=0
each (n) ξβ„“ (z) is found to satisfy
[ d2 2 d ( β„“(β„“ + 1) )] d [ 1 d ( ) ] ----+ -----+ 1 − -------- ξ(nβ„“) = e− iz--- ----- eizz2ξ(β„“n−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 ξβ„“ [91Jump To The Next Citation Point].

The zeroth-order solution (0) ξβ„“ satisfies the homogeneous spherical Bessel equation, and must be a linear combination of the spherical Bessel functions of the first and second kinds, jβ„“(z ) and nβ„“(z). Here, we demand the compatibility with the horizon solution (80View Equation). Since jβ„“(z) ∼ zβ„“ and nβ„“(z) ∼ z−β„“−1, n β„“(z) does not match with the horizon solution at the leading order of πœ–. Therefore, we have

(0) (0) ξβ„“ (z) = α β„“ jβ„“(z). (85 )
The constant α(β„“0) represents the overall normalization of the solution. Since it can be chosen arbitrarily, we set (0) αβ„“ = 1 below.

The procedure to obtain (1) ξβ„“ (z) was described in detail in [91Jump To The Next Citation Point]. Using the Green function G (z,z′) = jβ„“(z< )n β„“(z> ), Equation (84View Equation) may be put into an indefinite integral form,

(n) ∫ z [ 1 (n−1) ]′ ∫ z [ 1 (n−1) ]′ ξβ„“ = n β„“ dzz2e−izjβ„“ -3(eizz2ξβ„“ (z))′ − jβ„“ dzz2e−iznβ„“ -3(eizz2ξβ„“ (z))′ . (86 ) z z
The calculation is tedious but straightforward. All the necessary formulae to obtain ξ(nβ„“) for n ≤ 2 are given in the Appendix of [91Jump To The Next Citation Point] or Appendix D of [71Jump To The Next Citation Point]. Using those formulae, for n = 1 we have
ξ(1) = α (1)j + β (1)n β„“ β„“ β„“ β„“ β„“ (β„“ − 1)(β„“ + 3 ) ( β„“2 − 4 2β„“ − 1 ) + ----------------jβ„“+1 − ---------- + -------- jβ„“−1 2(β„“ + 1)(2β„“ + 1) 2β„“(2β„“ + 1) β„“(β„“ − 1) ∑β„“− 2( 1 1 ) +R β„“,0j0 + -- + ------ R β„“,mjm − 2Dnjβ„“ + ijβ„“ln z. (87 ) m=1 m m + 1
Here, nj D β„“ and R β„“,m are functions defined as follows. The function nj D β„“ is given by
1 Dnjβ„“ = --[jβ„“Si(2z) − nβ„“ (Ci (2z) − γ − ln 2z)], (88 ) 2
where ∫ Ci(x) = − x∞ dtcos tβˆ•t and ∫ Si(x ) = x0 dt sin tβˆ•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 ||{ ∑ r-Γ-(m-−-k-−-r-)Γ--m--+-12-−-r--- 2- R = − (− 1) r!Γ (m − k − 2r)Γ (k + 3 + r) z for m > k, (89 ) m,k || r=0 2 ( − Rk,m for m < k.

Here, we again perform the matching with the horizon solution (80View Equation). It should be noted that (1) ξβ„“, given by Equation (87View Equation), is regular in the limit z → 0 except for the term (1) ββ„“ nβ„“. By examining the asymptotic behavior of Equation (87View Equation) at z β‰ͺ 1, we find β (1)= 0 β„“, i.e., the solution is regular at z = 0. As for (1) αβ„“, it only contributes to the renormalization of (0) α β„“. Hence, we set α (β„“1) = 0 and the transmission amplitude Atrβ„“ans is determined to π’ͺ (πœ–) as

trans (β„“-−-2)!(β„“-+-2)! β„“+1 2 Aβ„“ = (2β„“)!(2β„“ + 1)! πœ– [1 − iπœ–aβ„“ + π’ͺ (πœ–)]. (90 )
It may be noted that this explicit expression for trans A β„“ 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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