### 3.1 Basic assumptions

We consider the case of a test particle with mass in a nearly circular orbit around a black hole with
mass . For a nearly circular orbit, say at , what we need to know is the behavior of
at . In addition, the contribution of to comes mainly from , where
is the orbital angular frequency.
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

where is a function related to as
The ingoing wave boundary condition of is derived from Equations (65) and (71) as
The above form of the Regge–Wheeler equation is used in Sections 3.2, 3.3, 3.4, and 3.5.
It should be noted that if we reinstate 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 [7, 12] 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.