10.1 The binary’s multipole moments
The general expressions of the source multipole moments given by Theorem 6 (Eqs. (80)) are first to be
worked out explicitly for general fluid systems at the 3PN order. For this computation one uses the
formula (88), and we insert the 3PN metric coefficients (in harmonic coordinates) expressed in Eq. (109,
110, 111) by means of the retarded-type elementary potentials (113, 114, 115). Then we specialize each of
the (quite numerous) terms to the case of point-particle binaries by inserting, for the matter stress-energy
tensor
, the standard expression made out of Dirac delta-functions. The infinite self-field of
point-particles is removed by means of the Hadamard regularization (see Section 8). This computation has
been performed by Blanchet and Schäfer [30] at the 1PN order, and by Blanchet, Damour and Iyer [18] at
the 2PN order; we report below the most accurate 3PN results obtained by Blanchet, Iyer and
Joguet [26
].
The difficult part of the analysis is to find the closed-form expressions, fully explicit in terms of the
particle’s positions and velocities, of many non-linear integrals. We refer to [26
] for full details; nevertheless,
let us give a few examples of the type of technical formulas that are employed in this calculation. Typically
we have to compute some integrals like
where
and
. When
and
, this integral is perfectly
well-defined (recall that the finite part
deals with the bound at infinity). When
or
, our basic ansatz is that we apply the definition of the Hadamard partie finie provided by
Eq. (119). Two examples of closed-form formulas that we get, which do not necessitate the Hadamard
partie finie, are (quadrupole case
)
We denote for example
; the constant
is the one pertaining to the finite-part process
(see Eq. (36)). One example where the integral diverges at the location of the particle 1 is
where
is the Hadamard-regularization constant introduced in
Eq. (119).
The crucial input of the computation of the flux at the 3PN order is the mass quadrupole moment
,
since this moment necessitates the full 3PN precision. The result of Ref. [26
] for this moment (in the case of
circular orbits) is
where we pose
and
. The third term is the 2.5PN radiation-reaction term,
which does not contribute to the energy flux for circular orbits. The two important coefficients are
and
, whose expressions through 3PN order are
These expressions are valid in harmonic coordinates via the post-Newtonian parameter
given by
Eq. (145). As we see, there are two types of logarithms in the moment: One type involves the length
scale
related by Eq. (148) to the two gauge constants
and
present in the 3PN
equations of motion; the other type contains the different length scale
coming from the general
formalism of Part aaa - indeed, recall that there is a
operator in front of the source
multipole moments in Theorem 6. As we know, that
is pure gauge; it will disappear from our
physical results at the end. On the other hand, we have remarked that the multipole expansion
outside a general post-Newtonian source is actually free of
, since the
’s present in
the two terms of Eq. (67) cancel out. We shall indeed find that the constants
present in
Eqs. (158) are compensated by similar constants coming from the non-linear wave “tails of
tails”. More seriously, in addition to the harmless constants
and
, there are three
unknown dimensionless parameters in Eqs. (158), called
,
and
. These parameters
reflect some incompleteness of the Hadamard self-field regularization (see the discussion in
Section 8.2).
Besides the 3PN mass quadrupole (157, 158), we need also the mass octupole moment
and current quadrupole moment
, both of them at the 2PN order; these are given by [26]
Also needed are the 1PN mass
-pole, 1PN current
-pole (octupole), Newtonian mass
-pole and
Newtonian current
-pole:
These results permit one to control what can be called the “instantaneous” part, say
, of the total
energy flux, by which we mean that part of the flux that is generated solely by the source multipole
moments, i.e. not counting the “non-instantaneous” tail integrals. The instantaneous flux is defined by the
replacement into the general expression of
given by Eq. (60) of all the radiative moments
and
by the corresponding (
th time derivatives of the) source moments
and
. Actually, we
prefer to define
by means of the intermediate moments
and
. Up to the 3.5PN order we
have
The time derivatives of the source moments (157, 158, 159, 160) are computed by means of the
circular-orbit equations of motion given by Eq. (146, 147); then we substitute them into Eq. (161) (for
circular orbits there is no difference at this order between
,
and
,
). The net result is
The Newtonian approximation,
, is the prediction of the Einstein quadrupole
formula (4), as computed by Landau and Lifchitz [97]. The self-field regularization ambiguities arising at
the 3PN order are the equation-of-motion-related constant
and the multipole-moment-related constant
(see Section 8.2).