The “standard” Hadamard regularization yields some ambiguous results for the computation of certain integrals at the 3PN order, as Jaranowski and Schäfer [139140141] first noticed in their computation of the equations of motion within the ADM-Hamiltonian formulation of general relativity. By standard Hadamard regularization we mean the regularization based solely on the definitions of the partie finie of a singular function, Equation (122), and the partie finie of a divergent integral, Equation (124) (i.e. without using a theory of pseudo-functions and generalized distributional derivatives as proposed in Refs. [3639]). It was shown in Refs. [139140141] that there are two and only two types of ambiguous terms in the 3PN Hamiltonian, which were then parametrized by two unknown numerical coefficients and .

Motivated by the previous result, Blanchet and Faye [3639] introduced their “extended” Hadamard regularization, the one we outlined in 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, the extended Hadamard regularization turned out to be in a sense incomplete, because it was found [3738] 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 [139140], on the basis of the computation of the invariant energy of compact binaries moving on circular orbits, showed [37] that

Therefore, the ambiguity is fixed, while is equivalent to the other ambiguity . Notice that the value (132) for the kinetic ambiguity parameter , which is in factor of some velocity dependent terms, is the only one for which the 3PN equations of motion are Lorentz invariant. Fixing up this value was possible because the extended Hadamard regularization [3639] was defined in such a way that it keeps the Lorentz invariance.

Damour, Jaranowski, and Schäfer [95] recovered the value of given in Equation (132) by directly proving that this value is the unique one for which the global Poincaré invariance of the ADM-Hamiltonian formalism is verified. Since the coordinate conditions associated with the ADM formalism do not manifestly respect the Poincaré symmetry, they had to prove that the 3PN 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 at 3PN order should be manifestly Lorentz-invariant, as was indeed found to be the case in Refs. [3738].

The appearance of one and only one physical unknown coefficient in the equations of motion constitutes a quite striking fact, that is related specifically with the use of a Hadamard-type regularization. Technically speaking, the presence of the ambiguity parameter is associated with the non-distributivity of Hadamard’s regularization, in the sense of Equation (123). Mathematically speaking, is probably related to the fact that it is impossible to construct a distributional derivative operator, such as Equations (129, 130, 131), satisfying the Leibniz rule for the derivation of the product [198]. The Einstein field equations can be written in many different forms, by shifting the derivatives and operating some terms by parts with the help of the Leibniz rule. All these forms are equivalent in the case of regular sources, but since the derivative operator (129, 130, 131) violates the Leibniz rule they become inequivalent for point particles. Finally, physically speaking, let us argue that has its root in the fact that in a complete computation of the equations of motion valid for two regular extended weakly self-gravitating bodies, many non-linear integrals, when taken individually, start depending, from the 3PN order, on the internal structure of the bodies, even in the “compact-body” limit where the radii tend to zero. However, when considering the full equations of motion, we expect that all the terms depending on the internal structure can be removed, in the compact-body limit, by a coordinate transformation (or by some appropriate shifts of the central world lines of the bodies), and that finally is given by a pure number, for instance a rational fraction, independent of the details of the internal structure of the compact bodies. From this argument (which could be justified by the effacing principle in general relativity) the value of is necessarily the one we compute below, Equation (135), and will be valid for any compact objects, for instance black holes.

The ambiguity parameter , which is in factor of some static, velocity-independent term, and hence cannot be derived by invoking Lorentz invariance, was computed by Damour, Jaranowski, and Schäfer [96] by means of dimensional regularization, instead of some Hadamard-type one, within the ADM-Hamiltonian formalism. Their result is

As Damour et al. [96] argue, clearing up the static ambiguity is made possible by the fact that dimensional regularization, contrary to Hadamard’s regularization, respects all the basic properties of the algebraic and differential calculus of ordinary functions: associativity, commutativity and distributivity of point-wise addition and multiplication, Leibniz’s rule, and the Schwarz lemma. In this respect, dimensional regularization is certainly better than Hadamard’s one, which does not respect the distributivity of the product (recall Equation (123)) and unavoidably violates at some stage the Leibniz rule for the differentiation of a product.

The ambiguity parameter is fixed from the result (134) and the necessary link (133) provided by the equivalence between the harmonic-coordinates and ADM-Hamiltonian formalisms [3797]. However, was also been computed directly by Blanchet, Damour, and Esposito-Farèse [30] applying dimensional regularization to the 3PN equations of motion in harmonic coordinates (in the line of Refs. [3738]). The end result,

is in full agreement with Equation (134). Besides the independent confirmation of the value of or , the work [30] provides also a confirmation of the consistency of dimensional regularization, because the explicit calculations are entirely different from the ones of Ref. [96]: Harmonic coordinates are used instead of ADM-type ones, the work is at the level of the equations of motion instead of the Hamiltonian, and a different form of Einstein’s field equations is solved by a different iteration scheme.

Let us comment here that the use of a self-field regularization, be it dimensional or based on Hadamard’s 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 velocity field, 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 was undertaken at the 2PN order by Kopeikin et al. [149127] who derived the equations of motion of two extended fluid balls, and obtained equations of motion depending only on the two masses and of the compact bodies. At the 3PN order we expect that the extended-body program should give the value of the regularization parameter (maybe after some gauge transformation to remove the terms depending on the internal structure). Ideally, its value should be confirmed by independent and more physical methods (like those of Refs. [214150101]).

An important work, in several respects more physical than the formal use of regularizations, is the one of Itoh and Futamase [133132], following previous investigations in Refs. [134135]. These authors derived the 3PN equations of motion in harmonic coordinates by means of a particular variant of the famous “surface-integral” method introduced long ago by Einstein, Infeld, and Hoffmann [106]. The aim is to describe extended relativistic compact binary systems in the strong-field point particle limit defined in Ref. [115]. This approach is very interesting because it is based on the physical notion of extended compact bodies in general relativity, and is free of the problems of ambiguities due to the Hadamard self-field regularization. The end result of Refs. [133132] is in agreement with the 3PN harmonic coordinates equations of motion [3738] and, moreover, is unambiguous, as it does determine the ambiguity parameter to exactly the value (135).

We next consider the problem of the binary’s radiation field, where the same phenomenon occurs, with the appearance of some Hadamard regularization ambiguity parameters at 3PN order. More precisely, Blanchet, Iyer, and Joguet [45], in their computation of the 3PN compact binary’s mass quadrupole moment , found it necessary to introduce three Hadamard regularization constants , , and , which are additional to and independent of the equation-of-motion related constant . The total gravitational-wave flux at 3PN order, in the case of circular orbits, was found to depend on a single combination of the latter constants, , and the binary’s orbital phase, for circular orbits, involves only the linear combination of and given by , as shown in [40].

Dimensional regularization (instead of Hadamard’s) has next been applied by Blanchet, Damour, Esposito-Farèse, and Iyer [3132] to the computation of the 3PN radiation field of compact binaries, leading to the following unique values for the ambiguity parameters:

These values represent the end result of dimensional regularization. However, several alternative calculations provide a check, independent of dimensional regularization, for all the parameters (136). Blanchet and Iyer [44] compute the 3PN binary’s mass dipole moment using Hadamard’s regularization, and identify with the 3PN center of mass vector position , already known as a conserved integral associated with the Poincaré invariance of the 3PN equations of motion in harmonic coordinates [103]. This yields in agreement with Equation (136). Next, we consider [34] the limiting physical situation where the mass of one of the particles is exactly zero (say, ), and the other particle moves with uniform velocity. Technically, the 3PN quadrupole moment of a boosted Schwarzschild black hole is computed and compared with the result for in the limit . The result is , and represents a direct verification of the global Poincaré invariance of the wave generation formalism (the parameter represents the analogue for the radiation field of the equation-of-motion related parameter ). Finally, is proven [32] by showing that there are no dangerously divergent “diagrams” corresponding to non-zero -values, where a diagram is meant here in the sense of Ref. [87].

The determination of the parameters (136) completes the problem of the general relativistic prediction for the templates of inspiralling compact binaries up to 3PN order (and actually up to 3.5PN order as the corresponding tail terms have already been determined [19]). The relevant combination of the parameters (136) entering the 3PN energy flux in the case of circular orbits is now fixed to be

Numerically, . The orbital phase of compact binaries, in the adiabatic inspiral regime (i.e. evolving by radiation reaction), involves at 3PN order a combination of parameters which is determined as
The fact that the numerical value of this parameter is quite small, , indicates, following measurement-accuracy analyses [595891], that the 3PN (or, even better, 3.5PN) order should provide an excellent approximation for both the on-line search and the subsequent off-line analysis of gravitational wave signals from inspiralling compact binaries in the LIGO and VIRGO detectors.