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
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