

3.4 More on the inner boundary condition of the
outer solution
In this section, we discuss the inner boundary condition of the
outer solution in more detail. As we have seen in Section 3.3, the boundary condition on
is that it is
regular at
, at least to
, while in the full non-linear level, the horizon
boundary is at
. We therefore investigate to what order in
the condition of
regularity at
can be applied.
Let us consider the general form of the horizon
solution. With
, it is expanded in the form
The lowest order solution
is given by the
polynomial (75). Apart from the common overall factor,
it is schematically expressed as
Thus,
does
not have a term matched with
, but it matches with
. We have
. A term that
matches with
first appears in
. This can be seen from the horizon solution valid
to
,
Equation (79). The second term in the square brackets
of it produces a term
.
This term therefore becomes
higher than the lowest
order term
.
Since
,
this effect first appears at
in the post-Minkowski expansion, while it first
appears at
in the post-Newtonian expansion if we note that
and
. This implies,
in particular, that if we are interested in the gravitational waves
emitted to infinity, we may simply impose the regularity at
as the inner
boundary condition of the outer solution for the calculation up to
6PN order beyond the quadrupole formula.
The above fact that a non-trivial boundary
condition due to the presence of the black hole horizon appears at
in the
post-Minkowski expansion can be more easily seen as follows. Since
as
, we
have
, or
, where
. On
the other hand, from the asymptotic behavior of
at
, the
coefficients
and
must be
of order unity. Then, using the Wronskian argument, we find
Thus, we immediately see that a non-trivial boundary condition
appears at
.
It is also useful to keep in mind the above fact
when we solve for
under the post-Minkowski expansion. It implies that
we may choose a phase such that
and
are complex conjugate to each other,
to
.
With this choice, the imaginary part of
, which reflects the boundary condition at
the horizon, does not appear until
because the Regge-Wheeler equation
is real. Then, recalling the relation of
to
, Equation (71),
for a given
is
completely determined in terms of
for
. That is, we may focus on
solving only the real part of Equation (84).

