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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 N particles is trivial; in general relativity, even writing the equations in the case N = 2 is difficult. The first relativistic term, at the 1PN order, was derived by Lorentz and Droste [156]. Subsequently, Einstein, Infeld and Hoffmann [106Jump To The Next Citation Point] 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. [165Jump To The Next Citation Point167Jump To The Next Citation Point166Jump To The Next Citation Point], who considered the post-Newtonian iteration of the Hamiltonian of N 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 [86Jump To The Next Citation Point85Jump To The Next Citation Point104Jump To The Next Citation Point80Jump To The Next Citation Point81Jump To The Next Citation Point], 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. [98Jump To The Next Citation Point] (see also Refs. [195196]). Kopeikin [149Jump To The Next Citation Point] 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 g ab) generated by two point masses were computed in Ref. [42Jump To The Next Citation Point], following a method based on previous work on wave generation [15Jump To The Next Citation Point].

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 [8685Jump To The Next Citation Point1048081Jump To The Next Citation Point], have been used for the study of the radiation damping of the binary pulsar - its orbital P [81Jump To The Next Citation Point82102]. 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 [81Jump To The Next Citation Point]. This is via an “effacing” principle (in the terminology of Damour [81Jump To The Next Citation Point]) for the internal structure of the bodies. As a result, the equations depend only on the “Schwarzschild” masses, m1 and m2, of the compact objects. Notably their compactness parameters K1 and K2, defined by Equation (7View Equation), do not enter the equations of motion, as has been explicitly verified up to the 2.5PN order by Kopeikin et al. [149Jump To The Next Citation Point127Jump To The Next Citation Point], 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 [134Jump To The Next Citation Point135Jump To The Next Citation Point], who use a variant of the surface-integral approach of Einstein, Infeld and Hoffmann [106Jump To The Next Citation Point], that is valid for compact bodies, independently of the strength of the internal gravity.

The present state of the art is the 3PN approximation4. 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 [139Jump To The Next Citation Point140Jump To The Next Citation Point141Jump To The Next Citation Point], and Damour, Jaranowski, and Schäfer [95Jump To The Next Citation Point97Jump To The Next Citation Point96Jump To The Next Citation Point], following the line of research of Refs. [16516716698Jump To The Next Citation Point], employ the ADM-Hamiltonian formalism of general relativity; on the other hand, Blanchet and Faye [37Jump To The Next Citation Point38Jump To The Next Citation Point36Jump To The Next Citation Point39Jump To The Next Citation Point], and de Andrade, Blanchet, and Faye [103Jump To The Next Citation Point], founding their approach on the post-Newtonian iteration initiated in Ref. [42Jump To The Next Citation Point], compute directly the equations of motion (instead of a Hamiltonian) in harmonic coordinates. The end results have been shown [97Jump To The Next Citation Point103Jump To The Next Citation Point] 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. [103Jump To The Next Citation Point] into a new Lagrangian, whose Legendre transform coincides exactly with the Hamiltonian given in Ref. [95Jump To The Next Citation Point]. The 3PN equations of motion, however, depend on one unspecified numerical coefficient, wstatic in the ADM-Hamiltonian formalism and c 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 [96Jump To The Next Citation Point], and the harmonic-coordinates equations of motion [30Jump To The Next Citation Point]. The works [96Jump To The Next Citation Point30Jump To The Next Citation Point] 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 [133Jump To The Next Citation Point132Jump To The Next Citation Point] (using the same surface-integral method as in Refs. [134Jump To The Next Citation Point135Jump To The Next Citation Point]) 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 c.

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 E).

The second sub-problem, that of the computation of the energy flux L, has been carried out by application of the wave-generation formalism described previously. Following earliest computations at the 1PN level [217Jump To The Next Citation Point49Jump To The Next Citation Point], at a time when the post-Newtonian corrections in L had a purely academic interest, the energy flux of inspiralling compact binaries was completed to the 2PN order by Blanchet, Damour and Iyer [33Jump To The Next Citation Point122Jump To The Next Citation Point], and, independently, by Will and Wiseman [220Jump To The Next Citation Point], using their own formalism (see Refs. [35Jump To The Next Citation Point46Jump To The Next Citation Point] 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. [22150Jump To The Next Citation Point] by application of the formula for tail integrals given in Ref. [29Jump To The Next Citation Point]. 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 [16Jump To The Next Citation Point19Jump To The Next Citation Point]. 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. [45Jump To The Next Citation Point40Jump To The Next Citation Point44Jump To The Next Citation Point], 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 c in the equations of motion, and a new parameter h coming from the computation of the 3PN quadrupole moment. The latter parameter is itself a linear combination of three unknown parameters, h = q + 2k + z. We shall review the computation of the three parameters q, k, and z by means of dimensional regularization [31Jump To The Next Citation Point32Jump To The Next Citation Point]. 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 (5View Equation), supposed to be valid at this order.

The post-Newtonian flux L, 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 n --> 0, where n = m/m). 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 [203Jump To The Next Citation Point], using a numerical code, up to the 4PN order. This technique has culminated with the beautiful analytical methods of Sasaki, Tagoshi and Tanaka [194205Jump To The Next Citation Point206] (see also Ref. [160]), who solved the problem up to the extremely high 5.5PN order.


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