

1.2 Post-Newtonian expansion of gravitational
waves
The post-Newtonian expansion of general relativity assumes that the
internal gravity of a source is small so that the deviation from
the Minkowski metric is small, and that velocities associated with
the source are small compared to the speed of light,
. When we consider the
orbital motion of a compact binary system, these two conditions
become essentially equivalent to each other. Although both
conditions may be violated inside each of the compact objects, this
is not regarded as a serious problem of the post-Newtonian
expansion, as long as we are concerned with gravitational waves
generated from the orbital motion, and, indeed, the two bodies are
usually assumed to be point-like objects in the calculation. In
fact, recently Itoh, Futamase, and Asada [29, 30] developed a new
post-Newtonian method that can deal with a binary system in which
the constituent bodies may have strong internal gravity, based on
earlier work by Futamase and Schutz [22, 23]. They
derived the equations of motion to 2.5PN order and obtained a
complete agreement with the Damour-Deruelle equations of
motion [16, 15], which assumes
the validity of the point-particle approximation.
There are two existing approaches of the
post-Newtonian expansion to calculate gravitational waves: one
developed by Blanchet, Damour, and Iyer (BDI) [7
, 6
] and another by Will and Wiseman
(WW) [66] based on previous
work by Epstein, Wagoner, and Will [18, 64]. In both
approaches, the gravitational waveforms and luminosity are expanded
in time derivatives of radiative multipoles, which are then related
to some source multipoles (the relation between them contains the
“tails”). The source multipoles are expressed as integrals over the
matter source and the gravitational field. The source multipoles
are combined with the equations of motion to obtain explicit
expressions in terms of the source masses, positions, and
velocities.
One issue of the post-Newtonian calculation
arises from the fact that the post-Newtonian expansion can be
applied only to the near-zone field of the source. In the
conventional post-Newtonian formalism, the harmonic coordinates are
used to write down the Einstein equations. If we define the
deviation from the Minkowski metric as
the Einstein equations are schematically written in the form
together with the harmonic gauge condition,
, where
is the
D’Alambertian operator in flat-space time,
,
and
represents the non-linear terms in the Einstein equations. The
Einstein equations (2) are integrated using the flat-space
retarded integrals. In order to perform the post-Newtonian
expansion, if we naively expand the retarded integrals in powers of
, there appear
divergent integrals. This is a technical problem that arises due to
the near-zone nature of the post-Newtonian approximation. In the
BDI approach, in order to integrate the retarded integrals, and to
evaluate the radiative multipole moments at infinity, two kinds of
approximation methods are introduced. One is the multipolar
post-Minkowski expansion, which can be applied to a region outside
the source including infinity, and the other is the near-zone,
post-Newtonian expansion. These two expansions are matched in the
intermediate region where both expansions are valid, and the
radiative multipole moments are evaluated at infinity. In the WW
approach, the retarded integrals are evaluated directly, without
expanding in terms of
, in the region outside the source in a novel way.
The lowest order of the gravitational waves is
given by the Newtonian quadrupole formula. It is standard to refer
to the post-Newtonian formulae (for the waveforms and luminosity)
that contain terms up to
beyond the Newtonian quadrupole
formula as the (
)PN formulae. Evaluation of gravitational waves
emitted to infinity from a compact binary system has been
successfully carried out to the 3.5 post-Newtonian (PN) order
beyond the lowest Newtonian quadrupole formula (apart from an
undetermined coefficient that appears at 3PN order) in the BDI
approach [7
, 6
]. The computation of the 3.5PN
flux requires the 3.5PN equations of motion. See a review by
Blanchet [5] for
details on post-Newtonian approaches. Up to now, both approaches
give the same answer for the gravitational waveforms and luminosity
to 2PN order.

