Thus, if we express the Regge-Wheeler
equation (63) in terms of a non-dimensional variable
, with a
non-dimensional parameter
, we are interested in the behavior
of
at
with
, where
is the characteristic
orbital velocity. The post-Newtonian expansion assumes that
is much smaller than
the velocity of light:
. Consequently, we have
in the post-Newtonian expansion.
To obtain (which we denote below by
for simplicity)
under these assumptions, we find it convenient to rewrite the
Regge-Wheeler equation in an alternative form. It is
It should be noted that if we recover the
gravitational constant , we have
. Thus, the expansion in terms of
corresponds to the
post-Minkowski expansion, and expanding the Regge-Wheeler equation
with the assumption
gives a set of iterative wave equations on the flat
spacetime background. One of the most significant differences
between the black hole perturbation theory and any theory based on
the flat spacetime background is the presence of the black hole
horizon in the former case. Thus, if we naively expand the
Regge-Wheeler equation with respect to
, the horizon boundary condition becomes
unclear, since there is no horizon on the flat spacetime. To
establish the boundary condition at the horizon, we need to treat
the Regge-Wheeler equation near the horizon separately. We thus
have to find a solution near the horizon, and the solution obtained
by the post-Minkowski expansion must be matched with it in the
region where both solutions are valid.
It may be of interest to note the difference between the matching used in the BDI approach for the post-Newtonian expansion [6, 7] and the matching used here. In the BDI approach, the matching is done between the post-Minkowskian metric and the near-zone post-Newtonian metric. In our case, the matching is done between the post-Minkowskian gravitational field and the gravitational field near the black hole horizon.