The black hole perturbation theory was originally
developed as a metric perturbation theory. For non-rotating
(Schwarzschild) black holes, a single master equation for the
metric perturbation was derived by Regge and Wheeler [47] for the so-called odd parity
part, and later by Zerilli [67] for the
even parity part. These equations have the nice property that they
reduce to the standard Klein-Gordon wave equation in the flat-space
limit. However, no such equation has been found in the case of a
Kerr black hole so far.
Then, based on the Newman-Penrose null-tetrad
formalism, in which the tetrad components of the curvature tensor
are the fundamental variables, a master equation for the curvature
perturbation was first developed by Bardeen and Press [4] for a
Schwarzschild black hole without source (), and by Teukolsky [61
] for a Kerr black hole with
source (
).
The master equation is called the Teukolsky equation, and it is a
wave equation for a null-tetrad component of the Weyl tensor
or
. In the source-free
case, Chrzanowski [13]
and Wald [65] developed a
method to construct the metric perturbation from the curvature
perturbation.
The Teukolsky equation has, however, a rather
complicated structure as a wave equation. Even in the flat-space
limit, it does not reduce to the standard Klein-Gordon form. Later,
Chandrasekhar showed that the Teukolsky equation can be transformed
to the form of the Regge-Wheeler or Zerilli equation for the
source-free Schwarzschild case [11]. A generalization of this to
the Kerr case with source was done by Sasaki and
Nakamura [50
, 51
]. They gave a transformation
that brings the Teukolsky equation to a Regge-Wheeler type equation
that reduces to the Regge-Wheeler equation in the Schwarzschild
limit. It may be noted that the Sasaki-Nakamura equation contains
an imaginary part, suggesting that either it is unrelated to a
(yet-to-be-found) master equation for the metric perturbation for
the Kerr geometry or implying the non-existence of such a master
equation.
As mentioned above, an important difference between the black-hole perturbation approach and the conventional post-Newtonian approach appears in the structure of the Green function used to integrate the wave equations. In the black-hole perturbation approach, the Green function takes account of the curved spacetime effect on the wave propagation, which implies complexity of its structure in contrast to the flat-space Green function. Thus, since the system is linear in the black-hole perturbation approach, the most non-trivial task is the construction of the Green function.
There are many papers that deal with a numerical evaluation of the Green function and calculations of gravitational waves induced by a particle. See Breuer [9], Chandrasekhar [12], and Nakamura, Oohara, and Kojima [35] for reviews and for references on earlier papers.
Here, we are interested in an analytical
evaluation of the Green function. One way is to adopt the
post-Minkowski expansion assuming . Note that, for bound orbits, the
condition
is equivalent to the condition for the
post-Newtonian expansion,
. If we can calculate the Green function to a
sufficiently high order in this expansion, we may be able to obtain
a rather accurate approximation of it that can be applicable to a
relativistic orbit fairly close to the horizon, possibly to a
radius as small as the inner-most stable circular orbit (ISCO),
which is given by
in the case of a Schwarzschild black hole.
It turns out that this is indeed possible. Though there arise some complications as one goes to higher PN orders, they are relatively easy to handle as compared to situations one encounters in the conventional post-Newtonian approaches. Thus, very interesting relativistic effects such as tails of gravitational waves can be investigated easily. Further, we can also easily investigate convergence properties of the post-Newtonian expansion by comparing a numerically calculated exact result with the corresponding analytic but approximate result. In this sense, the analytic black-hole perturbation approach can provide an important test of the post-Newtonian expansion.