In the Schwarzschild case, Chandrasekhar found
that the Teukolsky equation can be transformed to the Regge-Wheeler
equation, which has the standard form of a wave equation with
solutions having regular asymptotic behaviors at horizon and
infinity [11]. The Regge-Wheeler equation was
originally derived as an equation governing the odd parity metric
perturbation [47
]. The existence of this
transformation implies that the Regge-Wheeler equation can describe
the even parity metric perturbation simultaneously, though the
explicit relation of the Regge-Wheeler function obtained by the
Chandrasekhar transformation with the actual metric perturbation
variables has not been given in the literature yet.
Later, Sasaki and Nakamura succeeded in
generalizing the Chandrasekhar transformation to the Kerr
case [50, 51
]. The
Chandrasekhar-Sasaki-Nakamura transformation was originally
introduced to make the potential in the radial equation
short-ranged, and to make the source term well-behaved at the
horizon and at infinity. Since we are interested only in bound
orbits, it is not necessary to perform this transformation.
Nevertheless, because its flat-space limit reduces to the standard
radial wave equation in the Minkowski spacetime, it is convenient
to apply the transformation when dealing with the post-Minkowski or
post-Newtonian expansion, at least at low orders of expansion.
We transform the homogeneous Teukolsky equation to the Sasaki-Nakamura equation [50, 51], which is given by
The functionThe relation between and
is
If we set , this transformation reduces to the Chandrasekhar
transformation for the Schwarzschild black hole [11].
The explicit form of the transformation is
The asymptotic behavior of the ingoing wave
solution which
corresponds to Equation (20
) is
In the following sections, we present a method of
post-Newtonian expansion based on the above formalism in the case
of the Schwarzschild background. In the Kerr case, although a
post-Newtonian expansion method developed in previous
work [52, 58
] was based on the
Sasaki-Nakamura equation, we will not present it in this paper.
Instead, we present a different formalism, namely the one developed
by Mano, Suzuki, and Takasugi which allows us to solve the
Teukolsky equation in a more systematic manner, albeit very
mathematical [33
]. The reason is that the
equations in the Kerr case are already complicated enough even if
one uses the Sasaki-Nakamura equation, so that there is not much
advantage in using it. In contrast, in the Schwarzschild case, it
is much easier to obtain physical insight into the role of
relativistic corrections if we deal with the Regge-Wheeler
equation.