1.3 Post-Newtonian equations of motion and radiation

By equations of motion we mean the explicit expression of the accelerations of the bodies in terms of the positions and velocities. In Newtonian gravity, writing the equations of motion for a system of particles is trivial; in general relativity, even writing the equations in the case is difficult. The first relativistic term, at the 1PN order, was derived by Lorentz and Droste [156]. Subsequently, Einstein, Infeld and Hoffmann [106] obtained the 1PN corrections by means of their famous “surface-integral” method, in which the equations of motion are deduced from the vacuum field equations, and which are therefore applicable to any compact objects (be they neutron stars, black holes, or, perhaps, naked singularities). The 1PN-accurate equations were also obtained, for the motion of the centers of mass of extended bodies, by Petrova [179] and Fock [112] (see also Ref. [169]).

The 2PN approximation was tackled by Ohta et al. [165167166], 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 [86851048081], building on a non-linear iteration of the metric of two particles initiated in Ref. [11]. The corresponding result for the ADM-Hamiltonian of two particles at the 2PN order was given in Ref. [98] (see also Refs. [195196]). Kopeikin [149] 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 in Ref. [42], following a method based on previous work on wave generation [15].

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 [86851048081], have been used for the study of the radiation damping of the binary pulsar - its orbital  [8182102]. 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 [81]. This is via an “effacing” principle (in the terminology of Damour [81]) 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 Equation (7), do not enter the equations of motion, as has been explicitly verified up to the 2.5PN order by Kopeikin et al. [149127], 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 [134135], who use a variant of the surface-integral approach of Einstein, Infeld and Hoffmann [106], that is valid for compact bodies, independently of the strength of the internal gravity.

The present state of the art is the 3PN approximation. 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 [139140141], and Damour, Jaranowski, and Schäfer [959796], following the line of research of Refs. [16516716698], employ the ADM-Hamiltonian formalism of general relativity; on the other hand, Blanchet and Faye [37383639], and de Andrade, Blanchet, and Faye [103], founding their approach on the post-Newtonian iteration initiated in Ref. [42], compute directly the equations of motion (instead of a Hamiltonian) in harmonic coordinates. The end results have been shown [97103] 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. [103] into a new Lagrangian, whose Legendre transform coincides exactly with the Hamiltonian given in Ref. [95]. 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, both within the ADM-Hamiltonian formalism [96], and the harmonic-coordinates equations of motion [30]. The works [9630] have demonstrated the power of dimensional regularization and its perfect adequateness for the problem of the interaction between point masses in general relativity. Furthermore, an important work by Itoh and Futamase [133132] (using the same surface-integral method as in Refs. [134135]) succeeded in obtaining the complete 3PN equations of motion in harmonic coordinates directly, i.e. without ambiguity and containing the correct value for the parameter .

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 B of this article, we shall present the 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 [21749], 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 [33122], and, independently, by Will and Wiseman [220], using their own formalism (see Refs. [3546] 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. [22150] by application of the formula for tail integrals given in Ref. [29]. 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 [1619]. 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 in Refs. [454044], 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. The latter parameter is itself a linear combination of three unknown parameters, . We shall review the computation of the three parameters , , and by means of dimensional regularization [3132]. In Part B 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 balance 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 [182] at the 1.5PN order (following the pioneering work of Galt’sov et al. [116]), and by Tagoshi and Nakamura [203], using a numerical code, up to the 4PN order. This technique has culminated with the beautiful analytical methods of Sasaki, Tagoshi and Tanaka [194205206] (see also Ref. [160]), who solved the problem up to the extremely high 5.5PN order.