Structure of the ingoing wave function to 𝒪(𝜖2)

3.5 Structure of the ingoing wave function to $2 𝒪 (𝜖 )$

With the boundary condition discussed in Section 2, we can integrate the ingoing wave Regge–Wheeler function iteratively to higher orders of $𝜖$ in the post-Minkowskian expansion, $𝜖 ≪ 1$ . This was carried out in [91 ] to $𝒪 (𝜖2)$ and in [105

] to $𝒪 (𝜖3)$ (See [71

] for details). Here, we do not recapitulate the details of the calculation since it is already quite involved at $2 𝒪 (𝜖 )$ , with much less space for physical intuition. Instead, we describe the general properties of the ingoing wave function to $𝒪 (𝜖2)$ .

As discussed in Section 2, the ingoing wave Regge–Wheeler function $Xℓ$ can be made real up to $𝒪 (𝜖2ℓ+1 )$ , or to $𝒪 (𝜖5)$ of the post-Minkowski expansion, if we recall $ℓ ≥ 2$ . Choosing the phase of $X ℓ$ in this way, let us explicitly write down the expressions of $(n) Im (ξℓ )$ ( $n = 1,2$ ) in terms of $Re (ξ(mℓ ))$ ( $m ≤ n − 1$ ). We decompose the real and imaginary parts of $ξ(ℓn)$ as

Inserting this expression into Equation (71

), and expanding the result with respect to $𝜖$ (and noting $fℓ(0)= jℓ$ and $g(ℓ0) = 0$ ), we find

[ ( ) ( ) ] X = e−i𝜖ln(z− 𝜖)z j + 𝜖 f(1) + ig(1) + 𝜖2 f(2) + ig(2) + ... ℓ ℓ ℓ ℓ ℓ ℓ [ (1) ( (2) (1) 1 ) ] = z jℓ + 𝜖fℓ + 𝜖2 fℓ + gℓ lnz − --jℓ(lnz )2 + ... [ ( 2 ) ] (1) 2 (2) 1- (1) +iz 𝜖(gℓ − jℓlnz) + 𝜖 gℓ + zjℓ − fℓ ln z + ... . (95 )

Hence, we have

We thus have the post-Minkowski expansion of $X ℓ$ as

( ) ∑∞ n (n) (0) (1) (1) (2) (2) 1 2 X ℓ = 𝜖 X ℓ , with X ℓ = zjℓ, X ℓ = zf ℓ , X ℓ = z fℓ + 2jℓ (ln z) , ... n=0 (97 )

Now, let us consider the asymptotic behavior of $X ℓ$ at $z ≪ 1$ . As we know that $(1) ξℓ$ and $(2) ξℓ$ 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 (84 ), and comparing the terms of the same order on both sides of the equation. We denote the right-hand side of Equation (84 ) by $S (n) ℓ$ .

For $n = 1$ , we have

1 ( 1 4 + z2 ) Re (S(ℓ1)) = -- j′′ℓ + -j′ℓ − ------jℓ z z z2 { 𝒪 (z) for ℓ = 2, = ℓ−3 (98 ) 𝒪 (z ) for ℓ ≥ 3.

Inserting this into Equation (84

) with $n = 1$ , we find

Of course, this behavior is consistent with the full post-Minkowski solution given in Equation (87

For $n = 2$ , we then have

( ) ( ) (2) 1 (1)′′ 1 (1)′ 4 + z2 (1) 1 (1)′ 1 (1) Re (Sℓ ) = -- fℓ + --fℓ − ---2--fℓ − -- 2gℓ + --gℓ z z z { z z 1- ′ 1-- 𝒪 (zℓ− 2) for ℓ = 2, 3, = − z (jℓ lnz ) − z2jℓ ln z + 𝒪 (zℓ− 4) for ℓ ≥ 4. (100 )

This gives

Note that the $ln z$ terms in Equation (100

) arising from $g(ℓ1)$ give the $(ln z)2$ term in $fℓ(2)$ that just cancels the $2 jℓ(ln z)∕2$ term of $(2) Xℓ$ in Equation (97

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

X2 = z3{ 𝒪(1) + 𝜖𝒪 (z) + 𝜖2 [𝒪 (1) + 𝒪 (1)lnz] + ...} , 3 2 X3 = z { 𝒪(z) + 𝜖𝒪 (1) + 𝜖 [𝒪 (z) + 𝒪 (z)lnz] + ...} , (102 ) X = z3{ 𝒪 (z ℓ− 2) + 𝜖𝒪 (zℓ−3) + 𝜖2 [𝒪 (zℓ−4) + 𝒪 (zℓ− 2)ln z] + ...} for ℓ ≥ 4. ℓ

Note that, for $ℓ = 2$ and $3$ , the leading behavior of $X (nℓ)$ at $n = ℓ − 1$ is more regular than the naively expected behavior, $∼ zℓ+1−n$ , which propagates to the consecutive higher order terms in $𝜖$ . This behavior seems to hold for general $ℓ$ , but we do not know a physical explanation for it.

Given a post-Newtonian order to which we want to calculate, by setting $z = 𝒪 (v)$ and $𝜖 = 𝒪 (v3)$ , the above asymptotic behaviors tell us the highest order of $(n) X ℓ$ we need. We also see the presence of $lnz$ terms in $X (ℓ2)$ . The logarithmic terms appear as a consequence of the mathematical structure of the Regge–Wheeler equation at $z ≪ 1$ . The simple power series expansion of $(n) X ℓ$ in terms of $z$ breaks down at $2 𝒪 (𝜖)$ , 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 $𝒪 (v6)$ [99 , 100 ]. 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 $(n) fℓ$ be

Noting Equation (97

) and the equality $∗ e−i𝜖ln(z−𝜖) = e− iz eiz$ , the asymptotic form of $X ℓ$ is expressed as

X → Aince−i(z∗−𝜖ln𝜖) + Arefei(z∗−𝜖ln 𝜖), (104 ) ℓ ℓ { [ℓ ( )] Ainc = 1-iℓ+1e− i𝜖ln𝜖 1 + 𝜖 P (1)+ i Q (1)+ lnz ℓ 2 [ ( } ℓ ( ℓ ) ] } 2 (2) (1) (2) (1) + 𝜖 Pℓ − Qℓ lnz + i Q ℓ + Pℓ lnz + ... . (105 )

Note that

because of our definition of $z ∗$ , $z∗ = z + 𝜖 + ln (z − 𝜖)$ . The phase factor $e−i𝜖ln𝜖$ of $Ainc ℓ$ 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 $Ainℓc$ contains $ln z$ -dependent terms. Since $Ainc ℓ$ should be constant, $P (n) ℓ$ and $Q (n) ℓ$ should contain appropriate $ln z$ -dependent terms which exactly cancel the $ln z$ -dependent terms in Equation (105 ). To be explicit, we must have

(1) (1) P ℓ = pℓ , Q (1)= q(1)− ln z, ℓ ℓ (107 ) P (ℓ2)= p(ℓ2)+ q(ℓ1)lnz − (lnz)2, (2) (2) (1) Q ℓ = qℓ − pℓ lnz,

where $p(n) ℓ$ and $q(n) ℓ$ are constants. These relations can be used to check the consistency of the solution $(n) f$ obtained by integration. In terms of $(n) pℓ$ and $(n) qℓ$ , $inc Aℓ$ is expressed as

Note that the above form of $Ainℓc$ 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.