3.5 Structure of the ingoing wave function to

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

As discussed in Section 2, the ingoing wave Regge–Wheeler function can be made real up to , or to of the post-Minkowski expansion, if we recall . Choosing the phase of in this way, let us explicitly write down the expressions of () in terms of (). We decompose the real and imaginary parts of as

Inserting this expression into Equation (71), and expanding the result with respect to (and noting and ), we find
Hence, we have
We thus have the post-Minkowski expansion of as

Now, let us consider the asymptotic behavior of at . As we know that and are regular at , it is readily obtained by simply assuming Taylor expansion forms for them (including possible 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 .

For , we have

Inserting this into Equation (84) with , we find
Of course, this behavior is consistent with the full post-Minkowski solution given in Equation (87).

For , we then have

This gives
Note that the terms in Equation (100) arising from give the term in that just cancels the term of in Equation (97).

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

Note that, for and , the leading behavior of at is more regular than the naively expected behavior, , which propagates to the consecutive higher order terms in . This behavior seems to hold for general , but we do not know a physical explanation for it.

Given a post-Newtonian order to which we want to calculate, by setting and , the above asymptotic behaviors tell us the highest order of we need. We also see the presence of terms in . The logarithmic terms appear as a consequence of the mathematical structure of the Regge–Wheeler equation at . The simple power series expansion of in terms of breaks down at , and we have to add logarithmic terms to obtain the solution. These logarithmic terms will give rise to terms in the wave-form and luminosity formulae at infinity, beginning at  [99, 100]. It is not easy to explain physically how these terms appear. But the above analysis suggests that the terms in the luminosity originate from some spatially local curvature effects in the near-zone.

Now we turn to the asymptotic behavior at . For this purpose, let the asymptotic form of be

Noting Equation (97) and the equality , the asymptotic form of is expressed as
Note that
because of our definition of , . The phase factor of 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 contains -dependent terms. Since should be constant, and should contain appropriate -dependent terms which exactly cancel the -dependent terms in Equation (105). To be explicit, we must have

where and are constants. These relations can be used to check the consistency of the solution obtained by integration. In terms of and , is expressed as

Note that the above form of implies that the so-called tail of radiation, which is due to the curvature scattering of waves, will contain 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 terms in the luminosity. That is, it is not due to the wave propagation effect but due to some near-zone curvature effect.