3.5 Structure of the ingoing wave function to 2 π’ͺ (πœ– )

With the boundary condition discussed in Section 2, we can integrate the ingoing wave Regge–Wheeler function iteratively to higher orders of πœ– in the post-Minkowskian expansion, πœ– β‰ͺ 1. This was carried out in [91] to π’ͺ (πœ–2) and in [105Jump To The Next Citation Point] to π’ͺ (πœ–3) (See [71Jump To The Next Citation Point] for details). Here, we do not recapitulate the details of the calculation since it is already quite involved at 2 π’ͺ (πœ– ), with much less space for physical intuition. Instead, we describe the general properties of the ingoing wave function to π’ͺ (πœ–2).

As discussed in Section 2, the ingoing wave Regge–Wheeler function Xβ„“ can be made real up to π’ͺ (πœ–2β„“+1 ), or to π’ͺ (πœ–5) of the post-Minkowski expansion, if we recall β„“ ≥ 2. Choosing the phase of X β„“ in this way, let us explicitly write down the expressions of (n) Im (ξβ„“ ) (n = 1,2) in terms of Re (ξ(mβ„“ )) (m ≤ n − 1). We decompose the real and imaginary parts of ξ(β„“n) as

ξ(β„“n)= f(β„“n)+ ig(β„“n). (94 )
Inserting this expression into Equation (71View Equation), and expanding the result with respect to πœ– (and noting fβ„“(0)= jβ„“ and g(β„“0) = 0), we find
[ ( ) ( ) ] X = e−iπœ–ln(z− πœ–)z j + πœ– f(1) + ig(1) + πœ–2 f(2) + ig(2) + ... β„“ β„“ β„“ β„“ β„“ β„“ [ (1) ( (2) (1) 1 ) ] = z jβ„“ + πœ–fβ„“ + πœ–2 fβ„“ + gβ„“ lnz − --jβ„“(lnz )2 + ... [ ( 2 ) ] (1) 2 (2) 1- (1) +iz πœ–(gβ„“ − jβ„“lnz) + πœ– gβ„“ + zjβ„“ − fβ„“ ln z + ... . (95 )
Hence, we have
(1) (2) 1- (1) gβ„“ = jβ„“ ln z, gβ„“ = − z jβ„“ + fβ„“ lnz, ... (96 )
We thus have the post-Minkowski expansion of X β„“ as
( ) ∑∞ n (n) (0) (1) (1) (2) (2) 1 2 X β„“ = πœ– X β„“ , with X β„“ = zjβ„“, X β„“ = zf β„“ , X β„“ = z fβ„“ + 2jβ„“ (ln z) , ... n=0 (97 )

Now, let us consider the asymptotic behavior of X β„“ at z β‰ͺ 1. As we know that (1) ξβ„“ and (2) ξβ„“ are regular at z = 0, it is readily obtained by simply assuming Taylor expansion forms for them (including possible ln z terms), inserting them into Equation (84View Equation), and comparing the terms of the same order on both sides of the equation. We denote the right-hand side of Equation (84View Equation) by S (n) β„“.

For n = 1, we have

1 ( 1 4 + z2 ) Re (S(β„“1)) = -- j′′β„“ + -j′β„“ − ------jβ„“ z z z2 { π’ͺ (z) for β„“ = 2, = β„“−3 (98 ) π’ͺ (z ) for β„“ ≥ 3.
Inserting this into Equation (84View Equation) with n = 1, we find
{ π’ͺ (z3) for β„“ = 2, Re (ξ(β„“1)) = f(β„“1)= β„“−1 (99 ) π’ͺ (z ) for β„“ ≥ 3.
Of course, this behavior is consistent with the full post-Minkowski solution given in Equation (87View Equation).

For n = 2, we then have

( ) ( ) (2) 1 (1)′′ 1 (1)′ 4 + z2 (1) 1 (1)′ 1 (1) Re (Sβ„“ ) = -- fβ„“ + --fβ„“ − ---2--fβ„“ − -- 2gβ„“ + --gβ„“ z z z { z z 1- ′ 1-- π’ͺ (zβ„“− 2) for β„“ = 2, 3, = − z (jβ„“ lnz ) − z2jβ„“ ln z + π’ͺ (zβ„“− 4) for β„“ ≥ 4. (100 )
This gives
{ (2) (2) π’ͺ(zβ„“) + π’ͺ (zβ„“) ln z − 12jβ„“(ln z)2 for β„“ = 2,3, Re (ξβ„“ ) = fβ„“ = β„“−2 β„“ 1 2 (101 ) π’ͺ(z ) + π’ͺ (z )ln z − 2jβ„“(ln z) for β„“ ≥ 4.
Note that the ln z terms in Equation (100View Equation) arising from g(β„“1) give the (ln z)2 term in fβ„“(2) that just cancels the 2 jβ„“(ln z)βˆ•2 term of (2) Xβ„“ in Equation (97View Equation).

Inserting Equations (99View Equation) and (101View Equation) into the relevant expressions in Equation (97View Equation), we find

X2 = z3{ π’ͺ(1) + πœ–π’ͺ (z) + πœ–2 [π’ͺ (1) + π’ͺ (1)lnz] + ...} , 3 2 X3 = z { π’ͺ(z) + πœ–π’ͺ (1) + πœ– [π’ͺ (z) + π’ͺ (z)lnz] + ...} , (102 ) X = z3{ π’ͺ (z β„“− 2) + πœ–π’ͺ (zβ„“−3) + πœ–2 [π’ͺ (zβ„“−4) + π’ͺ (zβ„“− 2)ln z] + ...} for β„“ ≥ 4. β„“
Note that, for β„“ = 2 and 3, the leading behavior of X (nβ„“) at n = β„“ − 1 is more regular than the naively expected behavior, ∼ zβ„“+1−n, which propagates to the consecutive higher order terms in πœ–. This behavior seems to hold for general β„“, but we do not know a physical explanation for it.

Given a post-Newtonian order to which we want to calculate, by setting z = π’ͺ (v) and πœ– = π’ͺ (v3), the above asymptotic behaviors tell us the highest order of (n) X β„“ we need. We also see the presence of lnz terms in X (β„“2). The logarithmic terms appear as a consequence of the mathematical structure of the Regge–Wheeler equation at z β‰ͺ 1. The simple power series expansion of (n) X β„“ in terms of z breaks down at 2 π’ͺ (πœ–), and we have to add logarithmic terms to obtain the solution. These logarithmic terms will give rise to lnv terms in the wave-form and luminosity formulae at infinity, beginning at π’ͺ (v6) [99Jump To The Next Citation Point, 100Jump To The Next Citation Point]. It is not easy to explain physically how these lnv terms appear. But the above analysis suggests that the lnv terms in the luminosity originate from some spatially local curvature effects in the near-zone.

Now we turn to the asymptotic behavior at z = ∞. For this purpose, let the asymptotic form of (n) fβ„“ be

f(n)→ P (n)j + Q (n)n as z → ∞. (103 ) β„“ β„“ β„“ β„“ β„“
Noting Equation (97View Equation) and the equality ∗ e−iπœ–ln(z−πœ–) = e− iz eiz, the asymptotic form of X β„“ is expressed as
X → Aince−i(z∗−πœ–lnπœ–) + Arefei(z∗−πœ–ln πœ–), (104 ) β„“ β„“ { [β„“ ( )] Ainc = 1-iβ„“+1e− iπœ–lnπœ– 1 + πœ– P (1)+ i Q (1)+ lnz β„“ 2 [ ( } β„“ ( β„“ ) ] } 2 (2) (1) (2) (1) + πœ– Pβ„“ − Qβ„“ lnz + i Q β„“ + Pβ„“ lnz + ... . (105 )
Note that
( r − 2M ) ωr∗ = ω r + 2M ln -------- = z∗ − πœ–ln πœ–, (106 ) 2M
because of our definition of z ∗, z∗ = z + πœ– + ln (z − πœ–). The phase factor e−iπœ–lnπœ– of Ainc β„“ originates from this definition, but it represents a physical phase shift due to wave propagation on the curved background.

As one may immediately notice, the above expression for Ainβ„“c contains ln z-dependent terms. Since Ainc β„“ should be constant, P (n) β„“ and Q (n) β„“ should contain appropriate ln z-dependent terms which exactly cancel the ln z-dependent terms in Equation (105View Equation). To be explicit, we must have

(1) (1) P β„“ = pβ„“ , Q (1)= q(1)− ln z, β„“ β„“ (107 ) P (β„“2)= p(β„“2)+ q(β„“1)lnz − (lnz)2, (2) (2) (1) Q β„“ = qβ„“ − pβ„“ lnz,
where p(n) β„“ and q(n) β„“ are constants. These relations can be used to check the consistency of the solution (n) f obtained by integration. In terms of (n) pβ„“ and (n) qβ„“, inc Aβ„“ is expressed as
inc 1 β„“+1 −iπœ–lnπœ–[ ( (1) (1)) 2( (2) (2)) ] Aβ„“ = -i e 1 + πœ– pβ„“ + iqβ„“ + πœ– pβ„“ + iqβ„“ + ... . (108 ) 2

Note that the above form of Ainβ„“c implies that the so-called tail of radiation, which is due to the curvature scattering of waves, will contain lnv terms as phase shifts in the waveform, but will not give rise to such terms in the luminosity formula. This supports our previous argument on the origin of the lnv terms in the luminosity. That is, it is not due to the wave propagation effect but due to some near-zone curvature effect.


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