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

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

With the boundary condition discussed in Section 2, we can integrate the ingoing wave Regge-Wheeler function iteratively to higher orders of $e$ in the post-Minkowskian expansion, $e « 1$ . This was carried out in [49 ] to $O(e2)$ and in [60

] to $O(e3)$ (See [34

] for details). Here, we do not recapitulate the details of the calculation since it is already quite involved at $2 O(e )$ , with much less space for physical intuition. Instead, we describe the general properties of the ingoing wave function to $O(e2)$ .

As discussed in Section 2, the ingoing wave Regge-Wheeler function $X l$ can be made real up to $2l+1 O(e )$ , or to $5 O(e )$ of the post-Minkowski expansion, if we recall $l > 2$ . Choosing the phase of $Xl$ in this way, let us explicitly write down the expressions of $(n) Im(ql )$ ( $n = 1,2$ ) in terms of $Re(q(m)) l$ ( $m < n - 1$ ). We decompose the real and imaginary parts of $q(n) l$ as

Inserting this expression into Equation (71

), and expanding the result with respect to $e$ (and noting $(0) fl = jl$ and $(0) gl = 0$ ), we find

[ ( ) ( ) ] X = e-ieln(z- e)z j + e f(1) + ig(1) + e2 f(2) + ig(2) + ... l l l l l l [ (1) ( (2) (1) 1 ) ] = z jl + efl + e2 fl + gl lnz - --jl(lnz)2 + ... [ ( 2 ) ] (1) 2 (2) 1- (1) +iz e(gl - jllnz) + e gl + zjl - fl ln z + ... . (95)

Hence, we have

(1) (2) 1- (1) gl = jl ln z, gl = - z jl + fl lnz, ... (96)

We thus have the post-Minkowski expansion of $Xl$ as

( ) sum oo n (n) (0) (1) (1) (2) (2) 1 2 Xl = e 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 $Xl$ at $z« 1$ . As we know that $(1) ql$ and $(2) ql$ are regular at $z = 0$ , it is readily obtained by simply assuming Taylor expansion forms for them (including possible $lnz$ 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- '' 1-' 4 +-z2 Re (Sl ) = z jl + zjl- z2 jl { O(z) for l = 2, = O(zl -3) for l > 3. (98)

Inserting this into Equation (84

) with $n = 1$ , we find

{ 3 Re (q(1)) = f(1)= O(z ) for l = 2, (99) l l O(zl-1) for l > 3.

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(S l ) = -- 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

{ (2) (2) O(zl) + O(zl) ln z - 12jl(ln z)2 for l = 2,3, Re(q l ) = fl = l-2 l 1 2 (101) O(z ) + O(z )ln z - 2jl(ln z) for l > 4.

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

) arising from $g(1l)$ give the $(ln z)2$ term in $f(l2)$ that just cancels the $2 jl(ln z)/2$ term of $(2) X l$ in Equation (97

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

X2 = z3{O(1) + eO(z) + e2 [O(1) + O(1) lnz] + ...}, 3 2 X3 = z {O(z) + eO(1) + e [O(z) + O(z) lnz] + ...}, (102) X = z3{ O(zl- 2) + eO(zl -3) + e2 [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 $e$ . 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 $e = O(v3)$ , the above asymptotic behaviors tell us the highest order of $(n) Xl$ we need. We also see the presence of $ln z$ terms in $X(2l)$ . 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 $2 O(e )$ , and we have to add logarithmic terms to obtain the solution. These logarithmic terms will give rise to $ln v$ terms in the wave-form and luminosity formulae at infinity, beginning at $O(v6)$ [56 , 57 ]. It is not easy to explain physically how these $ln v$ 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 $z = oo$ . For this purpose, let the asymptotic form of $(n) f l$ be

f(n)--> P (n)j + Q(n) n as z --> oo . (103) l l l l l

Noting Equation (97

) and the equality $* e-ieln(z-e) = e-izeiz$ , the asymptotic form of $Xl$ is expressed as

X -- > Aince-i(z*-elne) + Arefei(z*-eln e), (104) l l { [l ( )] Ainc = 1-il+1e- ielne 1 + e P (1)+ i Q(1) + lnz l 2 [( } l ( l )] } 2 (2) (1) (2) (1) + e Pl - Ql lnz + i Q l + Pl lnz + ... . (105)

Note that

( r- 2M ) wr* = w r + 2M ln -------- = z* - eln e, (106) 2M

because of our definition of $z*$ , $z* = z + e + ln(z - e)$ . The phase factor $e- ieln e$ 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 $lnz$ -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 $(n) ql$ , $inc A l$ is expressed as

inc 1 l+1 -ielne[ ( (1) (1)) 2( (2) (2)) ] Al = -i e 1 + e pl + iql + e pl + iql + ... . (108) 2

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.