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
[56, 57
]. 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.