Blanchet and Faye [20, 23], motivated by the previous result, introduced their “improved” Hadamard
regularization, the one we outlined in the previous Section 8.1. This new regularization is mathematically
well-defined and free of ambiguities; in particular it yields unique results for the computation of any of the
integrals occuring in the 3PN equations of motion. Unfortunately, this regularization turned out to be in a
sense incomplete, because it was found in Refs. [21
, 22
] that the 3PN equations of motion involve one and
only one unknown numerical constant, called
, which cannot be determined within the method. The
comparison of this result with the work of Jaranowski and Schäfer [87
, 88
, 89
], on the basis of the
computation of the invariant energy of binaries moving on circular orbits, showed [21
] that
Damour, Jaranowski and Schäfer [60] recovered the value of
given in Eq. (127
) by proving
that this value is the unique one for which the global Poincaré invariance of their formalism is
verified. Since the coordinate conditions associated with the ADM approach do not manifestly
respect the Poincaré symmetry, they had to prove that the Hamiltonian is compatible with the
existence of generators for the Poincaré algebra. By contrast, the harmonic-coordinate conditions
preserve the Poincaré invariance, and therefore the associated equations of motion should be
Lorentz-invariant, as was indeed found to be the case by Blanchet and Faye [21
, 22
], thanks in
particular to their use of a Lorentz-invariant regularization [23
] (hence their determination of
).
The other parameter was computed by Damour, Jaranowski and Schäfer [61
] by means of a
dimensional regularization, instead of a Hadamard-type one, within the ADM-Hamiltonian formalism. Their
result, which in principle fixes
according to Eq. (128
), is
Let us comment that the use of a self-field regularization in this problem, be it dimensional or based on
the Hadamard partie finie, signals a somewhat unsatisfactory situation on the physical point of view,
because, ideally, we would like to perform a complete calculation valid for extended bodies, taking into
account the details of the internal structure of the bodies (energy density, pressure, internal velocities, etc.).
By considering the limit where the radii of the objects tend to zero, one should recover the same result as
obtained by means of the point-mass regularization. This would demonstrate the suitability of
the regularization. This program has been achieved at the 2PN order by Kopeikin [93] and
Grishchuk and Kopeikin [79] who derived the equations of motion of two extended fluid balls,
and proved that for compact bodies the equations depend only on the two masses and
. At the 3PN order we expect that the extended-body approach will give the value of the
regularization parameter
. In the following, we shall prefer to keep
unspecified, until
its value has been confirmed by independent and hopefully more physical methods (like in
Refs. [146, 94, 65]).
Blanchet, Iyer and Joguet [26], in their computation of the 3PN radiation field of two point masses - the
second half of the problem, besides the 3PN equations of motion - used the (standard) Hadamard
regularization and found it necessary to introduce three additional regularization constants
,
and
, which play a role analogous to the equation-of-motion
. Such unknown constants come from the
computation of the 3PN binary’s quadrupole moment
. Some good news is that the total
gravitational-wave flux, in the case of circular orbits, depends in fact only on a single combination of the
three latter constants,
![]() |
http://www.livingreviews.org/lrr-2002-3 |
© Max Planck Society and the author(s)
Problems/comments to |