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8.2 Hadamard regularization ambiguities

The “standard” Hadamard regularization yields some ambiguous results for the computation of certain integrals at the 3PN order, as Jaranowski and Schäfer [139Jump To The Next Citation Point140Jump To The Next Citation Point141Jump To The Next Citation Point] 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 (122View Equation), and the partie finie of a divergent integral, Equation (124View Equation) (i.e. without using a theory of pseudo-functions and generalized distributional derivatives as proposed in Refs. [36Jump To The Next Citation Point39Jump To The Next Citation Point]). It was shown in Refs. [139Jump To The Next Citation Point140Jump To The Next Citation Point141Jump To The Next Citation Point] that there are two and only two types of ambiguous terms in the 3PN Hamiltonian, which were then parametrized by two unknown numerical coefficients wstatic and wkinetic.

Motivated by the previous result, Blanchet and Faye [36Jump To The Next Citation Point39Jump To The Next Citation Point] 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 [37Jump To The Next Citation Point38Jump To The Next Citation Point] that the 3PN equations of motion involve one and only one unknown numerical constant, called c, which cannot be determined within the method. The comparison of this result with the work of Jaranowski and Schäfer [139Jump To The Next Citation Point140Jump To The Next Citation Point], on the basis of the computation of the invariant energy of compact binaries moving on circular orbits, showed [37Jump To The Next Citation Point] that

41 wkinetic = 24-, (132) 11 1987 wstatic = - ---c- -----. (133) 3 840
Therefore, the ambiguity wkinetic is fixed, while c is equivalent to the other ambiguity wstatic. Notice that the value (132View Equation) for the kinetic ambiguity parameter wkinetic, 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 [36Jump To The Next Citation Point39Jump To The Next Citation Point] was defined in such a way that it keeps the Lorentz invariance.

Damour, Jaranowski, and Schäfer [95Jump To The Next Citation Point] recovered the value of wkinetic given in Equation (132View Equation) 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. [37Jump To The Next Citation Point38Jump To The Next Citation Point].

The appearance of one and only one physical unknown coefficient c in the equations of motion constitutes a quite striking fact, that is related specifically with the use of a Hadamard-type regularization24. Technically speaking, the presence of the ambiguity parameter c is associated with the non-distributivity of Hadamard’s regularization, in the sense of Equation (123View Equation). Mathematically speaking, c is probably related to the fact that it is impossible to construct a distributional derivative operator, such as Equations (129View Equation, 130View Equation, 131View Equation), 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 (129View Equation, 130View Equation, 131View Equation) violates the Leibniz rule they become inequivalent for point particles. Finally, physically speaking, let us argue that c 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 c 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 c is necessarily the one we compute below, Equation (135View Equation), and will be valid for any compact objects, for instance black holes.

The ambiguity parameter wstatic, 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 [96Jump To The Next Citation Point] by means of dimensional regularization, instead of some Hadamard-type one, within the ADM-Hamiltonian formalism. Their result is

wstatic = 0. (134)
As Damour et al. [96Jump To The Next Citation Point] 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 (123View Equation)) and unavoidably violates at some stage the Leibniz rule for the differentiation of a product.

The ambiguity parameter c is fixed from the result (134View Equation) and the necessary link (133View Equation) provided by the equivalence between the harmonic-coordinates and ADM-Hamiltonian formalisms [37Jump To The Next Citation Point97Jump To The Next Citation Point]. However, c was also been computed directly by Blanchet, Damour, and Esposito-Farèse [30Jump To The Next Citation Point] applying dimensional regularization to the 3PN equations of motion in harmonic coordinates (in the line of Refs. [37Jump To The Next Citation Point38Jump To The Next Citation Point]). The end result,

1987 c = - -----, (135) 3080
is in full agreement with Equation (134View Equation)25. Besides the independent confirmation of the value of wstatic or c, the work [30Jump To The Next Citation Point] provides also a confirmation of the consistency of dimensional regularization, because the explicit calculations are entirely different from the ones of Ref. [96Jump To The Next Citation Point]: 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 m1 and m2 of the compact bodies26. At the 3PN order we expect that the extended-body program should give the value of the regularization parameter c (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 [133Jump To The Next Citation Point132Jump To The Next Citation Point], 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. [133Jump To The Next Citation Point132Jump To The Next Citation Point] is in agreement with the 3PN harmonic coordinates equations of motion [37Jump To The Next Citation Point38Jump To The Next Citation Point] and, moreover, is unambiguous, as it does determine the ambiguity parameter c to exactly the value (135View Equation).

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 [45Jump To The Next Citation Point], in their computation of the 3PN compact binary’s mass quadrupole moment Iij, found it necessary to introduce three Hadamard regularization constants q, k, and z, which are additional to and independent of the equation-of-motion related constant c. 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, h = q + 2k + z, and the binary’s orbital phase, for circular orbits, involves only the linear combination of h and c given by ^ h = h - 7c/3, as shown in [40Jump To The Next Citation Point].

Dimensional regularization (instead of Hadamard’s) has next been applied by Blanchet, Damour, Esposito-Farèse, and Iyer [31Jump To The Next Citation Point32Jump To The Next Citation Point] to the computation of the 3PN radiation field of compact binaries, leading to the following unique values for the ambiguity parameters27:

9871 q = - ----, 9240 k = 0, (136) 7 z = - --. 33
These values represent the end result of dimensional regularization. However, several alternative calculations provide a check, independent of dimensional regularization, for all the parameters (136View Equation). Blanchet and Iyer [44Jump To The Next Citation Point] compute the 3PN binary’s mass dipole moment Ii using Hadamard’s regularization, and identify Ii with the 3PN center of mass vector position Gi, already known as a conserved integral associated with the Poincaré invariance of the 3PN equations of motion in harmonic coordinates [103Jump To The Next Citation Point]. This yields q + k = - 9871/9240 in agreement with Equation (136View Equation). Next, we consider [34Jump To The Next Citation Point] the limiting physical situation where the mass of one of the particles is exactly zero (say, m = 0 2), 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 Iij in the limit m2 = 0. The result is z = - 7/33, and represents a direct verification of the global Poincaré invariance of the wave generation formalism (the parameter z represents the analogue for the radiation field of the equation-of-motion related parameter wkinetic)28. Finally, k = 0 is proven [32Jump To The Next Citation Point] by showing that there are no dangerously divergent “diagrams” corresponding to non-zero k-values, where a diagram is meant here in the sense of Ref. [87].

The determination of the parameters (136View Equation) 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 [19Jump To The Next Citation Point]). The relevant combination of the parameters (136View Equation) entering the 3PN energy flux in the case of circular orbits is now fixed to be

11831 h =_ q + 2k + z = -------. (137) 9240
Numerically, h -~ - 1.28041. 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
^ 7- 1039- h =_ h - 3c = 4620 . (138)
The fact that the numerical value of this parameter is quite small, ^h - ~ 0.22489, indicates, following measurement-accuracy analyses [59Jump To The Next Citation Point58Jump To The Next Citation Point91], 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.
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