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