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8.2 Regularization ambiguities

The “standard” Hadamard regularization yields some ambiguous results for the computation of certain integrals at the 3PN order, as Jaranowski and Schäfer [87Jump To The Next Citation Point88Jump To The Next Citation Point89Jump To The Next Citation Point] 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, Eq. (118View Equation), and the partie finie of a divergent integral, Eq. (119View Equation) (i.e. without using a theory of pseudo-functions and generalized distributional derivatives as proposed in Refs. [20Jump To The Next Citation Point23Jump To The Next Citation Point]). It was shown in Refs. [87Jump To The Next Citation Point88Jump To The Next Citation Point89Jump 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.

Blanchet and Faye [2023Jump To The Next Citation Point], motivated by the previous result, introduced their “improved” Hadamard regularization, the one we outlined in the previous 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, this regularization turned out to be in a sense incomplete, because it was found in Refs. [21Jump To The Next Citation Point22Jump 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 [87Jump To The Next Citation Point88Jump To The Next Citation Point89Jump To The Next Citation Point], on the basis of the computation of the invariant energy of binaries moving on circular orbits, showed [21Jump To The Next Citation Point] that

41 wkinetic = 24-, (127) 11 1987 wstatic = - ---c- -----. (128) 3 840
Therefore, the ambiguity wkinetic is fixed, while c is equivalent to the other ambiguity wstatic. Note that the harmonic-coordinates 3PN equations of motion as they have been obtained in Refs. [21Jump To The Next Citation Point22Jump To The Next Citation Point] depend also, in addition to c, on two arbitrary constants ' r1 and ' r2 parametrizing some logarithmic terms19; however, these constants are not “physical” in the sense that they can be removed by a coordinate transformation. 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 regularization. Mathematically speaking, the presence of c is (probably) related to the fact that it is impossible to construct a distributional derivative operator, such as (124View Equation, 125View Equation, 126View Equation), satisfying the Leibniz rule for the derivation of the product [132]. The Einstein field equations can be written into 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 they become inequivalent for point particles if the derivative operator violates the Leibniz rule. On the other hand, physically speaking, 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 finally expect c to be independent of the internal structure of the compact bodies.

Damour, Jaranowski and Schäfer [60Jump To The Next Citation Point] recovered the value of wkinetic given in Eq. (127View Equation) by proving that this value is the unique one for which the global Poincaré invariance of their formalism is verified. Since the coordinate conditions associated with the ADM approach do not manifestly respect the Poincaré symmetry, they had to prove that the 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 should be Lorentz-invariant, as was indeed found to be the case by Blanchet and Faye [21Jump To The Next Citation Point22Jump To The Next Citation Point], thanks in particular to their use of a Lorentz-invariant regularization [23Jump To The Next Citation Point] (hence their determination of wkinetic).

The other parameter w static was computed by Damour, Jaranowski and Schäfer [61Jump To The Next Citation Point] by means of a dimensional regularization, instead of a Hadamard-type one, within the ADM-Hamiltonian formalism. Their result, which in principle fixes c according to Eq. (128View Equation), is

1987 wstatic = 0 <====> c = - 3080-. (129)
As Damour et al. [61] argue, clearing up the ambiguity is made possible by the fact that the dimensional regularization, contrary to the Hadamard regularization, respects all the basic properties of the algebraic and differential calculus of ordinary functions. In this respect, the dimensional regularization is certainly better than the Hadamard one, which does not respect the “distributivity” of the product (recall that (F G)1 /= (F )1(G)1) and unavoidably violates at some stage the Leibniz rule for the differentiation of a product.

Let us comment that the use of a self-field regularization in this problem, be it dimensional or based on the Hadamard 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 velocities, 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 has been achieved at the 2PN order by Kopeikin [93] and Grishchuk and Kopeikin [79] who derived the equations of motion of two extended fluid balls, and proved that for compact bodies the equations depend only on the two masses m1 and m 2. At the 3PN order we expect that the extended-body approach will give the value of the regularization parameter c. In the following, we shall prefer to keep c unspecified, until its value has been confirmed by independent and hopefully more physical methods (like in Refs. [1469465]).

Blanchet, Iyer and Joguet [26Jump To The Next Citation Point], in their computation of the 3PN radiation field of two point masses - the second half of the problem, besides the 3PN equations of motion - used the (standard) Hadamard regularization and found it necessary to introduce three additional regularization constants q, k and z, which play a role analogous to the equation-of-motion c. Such unknown constants come from the computation of the 3PN binary’s quadrupole moment Iij. Some good news is that the total gravitational-wave flux, in the case of circular orbits, depends in fact only on a single combination of the three latter constants,

h = q + 2k + z. (130)
To summarize, the final result that we shall derive below for the binary’s orbital phase will involve the two regularization constants: c coming from the equations of motion and h coming from the multipole moments. But, interestingly, we shall find that there is only one unknown coefficient, in the form of a linear combination of c and h.


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