The 2PN approximation was tackled by Otha et al. [105, 107
, 106
], who considered the post-Newtonian
iteration of the Hamiltonian of
point-particles. We refer here to the Hamiltonian as the
Fokker-type Hamiltonian, which is obtained from the matter-plus-field Arnowitt-Deser-Misner
(ADM) Hamiltonian by eliminating the field degrees of freedom. The result for the 2PN and
even 2.5PN equations of binary motion in harmonic coordinates was obtained by Damour and
Deruelle [56
, 55
, 67
, 51
, 52
], building on a non-linear iteration of the metric of two particles initiated in
Ref. [2]. The corresponding result for the ADM-Hamiltonian of two particles at the 2PN order was given in
Ref. [63
] (see also Refs. [130, 131]). Kopeikin [93
] derived the 2.5PN equations of motion for two
extended compact objects. The 2.5PN-accurate harmonic-coordinate equations as well as the
complete gravitational field (namely the metric
) generated by two point masses were
computed by Blanchet, Faye and Ponsot [25
], following a method based on previous work on wave
generation [6
].
Up to the 2PN level the equations of motion are conservative. Only at the 2.5PN order appears the first
non-conservative effect, associated with the gravitational radiation reaction. The (harmonic-coordinate)
equations of motion up to that level, as derived by Damour and Deruelle [56, 55, 67, 51, 52
], have been
used for the study of the radiation damping of the binary pulsar - its orbital
[52
]. It is important to
realize that the 2.5PN equations of motion have been proved to hold in the case of binary systems of
strongly self-gravitating bodies [52
]. This is via an “effacing” principle (in the terminology of Damour [52])
for the internal structure of the bodies. As a result, the equations depend only on the “Schwarzschild”
masses,
and
, of the compact objects. Notably their compactness parameters
and
, defined by Eq. (7
), do not enter the equations of motion, as has been explicitly
verified up to the 2.5PN order by Kopeikin [93
] and Grishchuk and Kopeikin [79
], who made a
“physical” computation, à la Fock, taking into account the internal structure of two self-gravitating
extended bodies. The 2.5PN equations of motion have also been established by Itoh, Futamase
and Asada [83, 84], who use a variant of the surface-integral approach of Einstein, Infeld and
Hoffmann [69
], that is valid for compact bodies, independently of the strength of the internal
gravity.
The present state of the art is the 3PN
approximation3.
To this order the equations have been worked out independently by two groups, by means of different
methods, and with equivalent results. On the one hand, Jaranowski and Schäfer [87, 88
, 89
], and
Damour, Jaranowski and Schäfer [60
, 62
, 61
], following the line of research of Refs. [105, 107, 106, 63
],
employ the ADM-Hamiltonian formalism of general relativity; on the other hand, Blanchet and
Faye [21
, 22
, 20
, 23
], and de Andrade, Blanchet and Faye [66
], founding their approach on the
post-Newtonian iteration initiated in Ref. [25
], compute directly the equations of motion (instead of a
Hamiltonian) in harmonic coordinates. The end results have been shown [62, 66
] to be physically
equivalent in the sense that there exists a unique “contact” transformation of the dynamical variables that
changes the harmonic-coordinates Lagrangian obtained in Ref. [66
] into a new Lagrangian, whose Legendre
transform coincides exactly with the Hamiltonian given in Ref. [60
]. The 3PN equations of motion,
however, depend on one unspecified numerical coefficient,
in the ADM-Hamiltonian formalism and
in the harmonic-coordinates approach, which is due to some incompleteness of the Hadamard self-field
regularization method. This coefficient has been fixed by means of a dimensional regularization in
Ref. [61
].
So far the status of the post-Newtonian equations of motion is quite satisfying. There is mutual
agreement between all the results obtained by means of different approaches and techniques, whenever it is
possible to compare them: point particles described by Dirac delta-functions, extended post-Newtonian
fluids, surface-integrals methods, mixed post-Minkowskian and post-Newtonian expansions, direct
post-Newtonian iteration and matching, harmonic coordinates versus ADM-type coordinates, and different
processes or variants of the regularization of the self field of point particles. In Part bbb of this article, we
shall present the most complete results for the 3PN equations of motion, and for the associated
Lagrangian and Hamiltonian formulations (from which we deduce the center-of-mass energy
).
The second sub-problem, that of the computation of the energy flux , has been carried out by
application of the wave-generation formalism described previously. Following earliest computations at the
1PN level [149, 30
], at a time when the post-Newtonian corrections in
had a purely academic interest,
the energy flux of inspiralling compact binaries was completed to the 2PN order by Blanchet,
Damour and Iyer [18
, 77], and, independently, by Will and Wiseman [152
], using their own
formalism (see Refs. [19, 27
] for joint reports of these calculations). The preceding approximation,
1.5PN, which represents in fact the dominant contribution of tails in the wave zone, had been
obtained in Refs. [153, 31
] by application of the formula for tail integrals given in Ref. [17
].
Higher-order tail effects at the 2.5PN and 3.5PN orders, as well as a crucial contribution of tails
generated by the tails themselves (the so-called “tails of tails”) at the 3PN order, were obtained by
Blanchet [7
, 10
]. However, unlike the 1.5PN, 2.5PN and 3.5PN orders that are entirely composed
of tail terms, the 3PN approximation also involves, besides the tails of tails, many non-tail
contributions coming from the relativistic corrections in the (source) multipole moments of the
binary. These have been “almost” completed by Blanchet, Iyer and Joguet [26
, 24], in the
sense that the result still involves one unknown numerical coefficient, due to the use of the
Hadamard regularization, which is a combination of the parameter
in the equations of
motion, and a new parameter
coming from the computation of the 3PN quadrupole moment.
In Part bbb of this article, we shall present the most up-to-date results for the 3.5PN energy
flux and orbital phase, deduced from the energy equation (5
), supposed to be valid at this
order.
The post-Newtonian flux , which comes from a “standard” post-Newtonian calculation,
is in complete agreement (up to the 3.5PN order) with the result given by the very different
technique of linear black-hole perturbations, valid in the “test-mass” limit where the mass of one of
the bodies tends to zero (limit
, where
). Linear black-hole perturbations,
triggered by the geodesic motion of a small mass around the black hole, have been applied to this
problem by Poisson [120] at the 1.5PN order (following the pioneering work of Galt’sov et
al. [75]), and by Tagoshi and Nakamura [135
], using a numerical code, up to the 4PN order.
This technique has culminated with the beautiful analytical methods of Sasaki, Tagoshi and
Tanaka [129, 137
, 138] (see also Ref. [102]), who solved the problem up to the extremely high 5.5PN
order.
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