Regularization of the Field of Point Particles

6 Regularization of the Field of Point Particles

Our aim is to compute the metric (and its gradient needed in the equations of motion) at the 3PN order (say) for a system of two point-like particles. A priori one is not allowed to use directly some metric expressions like Eqs. (144) above, which have been derived under the assumption of a continuous (smooth) matter distribution. Applying them to a system of point particles, we find that most of the integrals become divergent at the location of the particles, i.e., when $x → y (t) 1$ or $y (t) 2$ , where $y (t) 1$ and $y (t) 2$ denote the two trajectories. Consequently, we must supplement the calculation by a prescription for how to remove the infinite part of these integrals. At this stage different choices for a “self-field” regularization (which will take care of the infinite self-field of point particles) are possible. In this section we review the:

Hadamard self-field regularization, which has proved to be very convenient for doing practical computations (in particular, by computer), but suffers from the important drawback of yielding some ambiguity parameters, which cannot be determined within this regularization, starting essentially at the 3PN order;
Dimensional self-field regularization, an extremely powerful regularization which is free of any ambiguities (at least up to the 3PN level), and therefore permits to uniquely fix the values of the ambiguity parameters coming from Hadamard’s regularization. However, dimensional regularization has not yet been implemented to the present problem in the general case (i.e., for an arbitrary space dimension $d ∈ ℂ$ ).

The why and how the final results are unique and independent of the employed self-field regularization (in agreement with the physical expectation) stems from the effacing principle of general relativity [142 *] – namely that the internal structure of the compact bodies makes a contribution only at the formal 5PN approximation. However, we shall review several alternative computations, independent of the self-field regularization, which confirm the end results.

6.1 Hadamard self-field regularization

In most practical computations we employ the Hadamard regularization [236 , 381 *] (see Ref. [382 ] for an entry to the mathematical literature). Let us present here an account of this regularization, as well as a theory of generalized functions (or pseudo-functions) associated with it, following the detailed investigations in Refs. [70 *, 72 *].

Consider the class $ℱ$ of functions $F (x)$ which are smooth ( $C ∞$ ) on $ℝ3$ except for the two points $y1$ and $y2$ , around which they admit a power-like singular expansion of the type:³⁹

and similarly for the other point 2. Here $r1 = |x − y1| → 0$ , and the coefficients $1fa$ of the various powers of $r1$ depend on the unit direction $n1 = (x − y1)∕r1$ of approach to the singular point. The powers $a$ of $r1$ are real, range in discrete steps [i.e., $a ∈ (ai)i∈ ℕ$ ], and are bounded from below ( $a0 ≤ a$ ). The coefficients $1fa$ (and $2fa$ ) for which $a < 0$ can be referred to as the singular coefficients of $F$ . If $F$ and $G$ belong to $ℱ$ so does the ordinary product $F G$ , as well as the ordinary gradient $∂iF$ . We define the Hadamard partie finie of $F$ at the location of the point 1 where it is singular as

where $dΩ1 = d Ω(n1 )$ denotes the solid angle element centered on $y1$ and of direction $n1$ . Notice that because of the angular integration in Eq. (160 *), the Hadamard partie finie is “non-distributive” in the sense that

The non-distributivity of Hadamard’s partie finie is the main source of the appearance of ambiguity parameters at the 3PN order, as discussed in Section 6.2.

The second notion of Hadamard partie finie ( $Pf$ ) concerns that of the integral $∫ 3 d xF$ , which is generically divergent at the location of the two singular points $y1$ and $y2$ (we assume that the integral converges at infinity). It is defined by

The first term integrates over a domain $𝒮 (s)$ defined as $3 ℝ$ from which the two spherical balls $r1 ≤ s$ and $r2 ≤ s$ of radius $s$ and centered on the two singularities, denoted $ℬ(y1,s)$ and $ℬ(y2,s)$ , are excised: $𝒮(s) ≡ ℝ3 ∖ ℬ(y1,s) ∪ ℬ(y2,s )$ . The other terms, where the value of a function at point 1 takes the meaning (160 *), are precisely such that they cancel out the divergent part of the first term in the limit where $s → 0$ (the symbol $1 ↔ 2$ means the same terms but corresponding to the other point 2). The Hadamard partie-finie integral depends on two strictly positive constants $s1$ and $s2$ , associated with the logarithms present in Eq. (162 *). We shall look for the fate of these constants in the final equations of motion and radiation field. See Ref. [70 *] for alternative expressions of the partie-finie integral.

We now come to a specific variant of Hadamard’s regularization called the extended Hadamard regularization (EHR) and defined in Refs. [70 *, 72 *]. The basic idea is to associate to any $F ∈ ℱ$ a pseudo-function, called the partie finie pseudo-function $PfF$ , namely a linear form acting on functions $G$ of $ℱ$ , and which is defined by the duality bracket

When restricted to the set $𝒟$ of smooth functions, i.e., $C ∞ (ℝ4)$ , with compact support (obviously we have $𝒟 ⊂ ℱ$ ), the pseudo-function $PfF$ is a distribution in the sense of Schwartz [381 *]. The product of pseudo-functions coincides, by definition, with the ordinary point-wise product, namely $PfF ⋅ PfG = Pf (F G )$ . In practical computations, we use an interesting pseudo-function, constructed on the basis of the Riesz delta function [365 ], which plays a role analogous to the Dirac measure in distribution theory, $δ1(x) ≡ δ(x − y1)$ . This is the delta-pseudo-function $Pf δ1$ defined by

where $(F)1$ is the partie finie of $F$ as given by Eq. (160 *). From the product of $Pfδ1$ with any $PfF$ we obtain the new pseudo-function $Pf (F δ ) 1$ , that is such that

As a general rule, we are not allowed, in consequence of the “non-distributivity” of the Hadamard partie finie, Eq. (161 *), to replace $F$ within the pseudo-function $Pf(F δ1)$ by its regularized value: $Pf (F δ ) ⁄= (F )Pf δ 1 1 1$ in general. It should be noticed that the object $Pf (Fδ ) 1$ has no equivalent in distribution theory.

Next, we treat the spatial derivative of a pseudo-function of the type $PfF$ , namely $∂i(PfF )$ . Essentially, we require [70 *] that the rule of integration by parts holds. By this we mean that we are allowed to freely operate by parts any duality bracket, with the all-integrated (“surface”) terms always being zero, as in the case of non-singular functions. This requirement is motivated by our will that a computation involving singular functions be as much as possible the same as if we were dealing with regular functions. Thus, by definition,

Furthermore, we assume that when all the singular coefficients of $F$ vanish, the derivative of $PfF$ reduces to the ordinary derivative, i.e., $∂i(PfF ) = Pf (∂iF )$ . Then it is trivial to check that the rule (166 *) contains as a particular case the standard definition of the distributional derivative [381 *]. Notably, we see that the integral of a gradient is always zero: $⟨∂i(PfF ),1 ⟩ = 0$ . This should certainly be the case if we want to compute a quantity like a Hamiltonian density which is defined only modulo a total divergence. We pose

where $Pf(∂iF )$ represents the “ordinary” derivative and $Di [F ]$ is the distributional term. The following solution of the basic relation (166 *) was obtained in Ref. [70 *]:

where for simplicity we assume that the powers $a$ in the expansion (159 *) of $F$ are relative integers. The distributional term (168 *) is of the form $Pf (Gδ ) 1$ plus $1 ↔ 2$ ; it is generated solely by the singular coefficients of $F$ .⁴⁰ The formula for the distributional term associated with the $ℓ$ -th distributional derivative, i.e. $DL [F ] = ∂LPfF − Pf ∂LF$ , where $L = i1i2⋅⋅⋅iℓ$ , reads

We refer to Theorem 4 in Ref. [70 *] for the definition of another derivative operator, representing the most general derivative satisfying the same properties as the one defined by Eq. (168 *), and, in addition, the commutation of successive derivatives (or Schwarz lemma).⁴¹

The distributional derivative defined by (167 *) – (168 *) does not satisfy the Leibniz rule for the derivation of a product, in accordance with a general result of Schwartz [380 *]. Rather, the investigation of Ref. [70 *] suggests that, in order to construct a consistent theory (using the ordinary point-wise product for pseudo-functions), the Leibniz rule should be weakened, and replaced by the rule of integration by part, Eq. (166 *), which is in fact nothing but an integrated version of the Leibniz rule. However, the loss of the Leibniz rule stricto sensu constitutes one of the reasons for the appearance of the ambiguity parameters at 3PN order, see Section 6.2.

The Hadamard regularization $(F )1$ is defined by Eq. (160 *) in a preferred spatial hypersurface $t = const$ of a coordinate system, and consequently is not a priori compatible with the Lorentz invariance. Thus we expect that the equations of motion in harmonic coordinates (which manifestly preserve the global Lorentz invariance) should exhibit at some stage a violation of the Lorentz invariance due to the latter regularization. In fact this occurs exactly at the 3PN order. Up to the 2.5PN level, the use of the regularization $(F )1$ is sufficient to get some unambiguous equations of motion which are Lorentz invariant [76 *]. This problem can be dealt with within Hadamard’s regularization, by introducing a Lorentz-invariant variant of this regularization, denoted $[F ]1$ in Ref. [72 *]. It consists of performing the Hadamard regularization within the spatial hypersurface that is geometrically orthogonal (in a Minkowskian sense) to the four-velocity of the particle. The regularization $[F ]1$ differs from the simpler regularization $(F )1$ by relativistic corrections of order $1∕c2$ at least. See [72 *] for the formulas defining this regularization in the form of some infinite power series in $2 1 ∕c$ . The regularization $[F]1$ plays a crucial role in obtaining the equations of motion at the 3PN order in Refs. [69 *, 71 *]. In particular, the use of the Lorentz-invariant regularization $[F ]1$ permits to obtain the value of the ambiguity parameter $ωkinetic$ in Eq. (170a) below.

6.2 Hadamard regularization ambiguities

The standard Hadamard regularization yields some ambiguous results for the computation of certain integrals at the 3PN order, as noticed by Jaranowski & Schäfer [261 *, 262 *, 263 *] 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. (160 *), and the partie finie of a divergent integral, Eq. (162 *), and without using a theory of pseudo-functions and generalized distributional derivatives as in Refs. [70 *, 72 *]. It was shown in Refs. [261 *, 262 *, 263 *] that there are two and only two types of ambiguous terms in the 3PN Hamiltonian, which were then parametrized by two unknown numerical coefficients called $ωstatic$ and $ωkinetic$ .

Progressing concurrently, Blanchet & Faye [70 *, 72 *] introduced the “extended” Hadamard regularization – the one we outlined in Section 6.1 – and obtained [69 *, 71 *] the 3PN equations of motion complete except for one and only one unknown numerical constant, called $λ$ . The new extended Hadamard regularization is mathematically well-defined and yields unique results for the computation of any integral in the problem; however, it turned out to be in a sense “incomplete” as it could not determine the value of this constant. The comparison of the result with the work [261 *, 262 *], on the basis of the computation of the invariant energy of compact binaries moving on circular orbits, revealed [69 *] that

Therefore, the ambiguity $ωkinetic$ is fixed, while $λ$ is equivalent to the other ambiguity $ωstatic$ . Notice that the value (170a) for the kinetic ambiguity parameter $ωkinetic$ , 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 [70 *, 72 *] was defined in such a way that it keeps the Lorentz invariance.

The value of $ω kinetic$ given by Eq. (170a) was recovered in Ref. [162 *] by directly proving that such 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, it was necessary 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 are manifestly Lorentz-invariant, as was found to be the case in Refs. [69 *, 71 *].

The appearance of one and only one physical unknown coefficient $λ$ in the equations of motion constitutes a quite striking fact, that is related specifically with the use of a Hadamard-type regularization.⁴² Technically speaking, the presence of the ambiguity parameter $λ$ is associated with the non-distributivity of Hadamard’s regularization, in the sense of Eq. (161 *). Mathematically speaking, $λ$ is probably related to the fact that it is impossible to construct a distributional derivative operator, such as Eqs. (167 *) – (168 *), satisfying the Leibniz rule for the derivation of the product [380 ]. 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 (167 *) – (168 *) violates the Leibniz rule they become inequivalent for point particles.

Physically speaking, let us also argue that $λ$ 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, one expects 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 $λ$ 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 $λ$ is necessarily the one we compute below, Eq. (172 *), and will be valid for any compact objects, for instance black holes.

The ambiguity parameter $ω static$ , which is in factor of some static, velocity-independent term, and hence cannot be derived by invoking Lorentz invariance, was computed by Damour, Jaranowski & Schäfer [163 *] by means of dimensional regularization, instead of some Hadamard-type one, within the ADM-Hamiltonian formalism. Their result is

As argued in [163 *], 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 superior to Hadamard’s one, which does not respect the distributivity of the product [recall Eq. (161 *)] and unavoidably violates at some stage the Leibniz rule for the differentiation of a product.

The ambiguity parameter $λ$ is fixed from the result (171 *) and the necessary link (170b) provided by the equivalence between the harmonic-coordinates and ADM-Hamiltonian formalisms [69 *, 164 ]. However, $λ$ has also been computed directly by Blanchet, Damour & Esposito-Farèse [61 *] applying dimensional regularization to the 3PN equations of motion in harmonic coordinates (in the line of Refs. [69 *, 71 *]). The end result,

is in full agreement with Eq. (171 *).⁴³ Besides the independent confirmation of the value of $ω static$ or $λ$ , the work [61 *] provides also a confirmation of the consistency of dimensional regularization, since the explicit calculations are entirely different from the ones of Ref. [163 *]: Harmonic coordinates instead of ADM-type ones, work at the level of the equations of motion instead of the Hamiltonian, a different form of Einstein’s field equations which is solved by a different iteration scheme.

Let us comment 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 in Refs. [280 , 234 ] which 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 bodies.⁴⁴ At the 3PN order we expect that the extended-body program should give the value of the regularization parameter $λ$ – probably after a coordinate transformation to remove the terms depending on the internal structure. Ideally, its value should also be confirmed by independent and more physical methods like those of Refs. [407 , 281 , 172 ].

An important work, in several aspects more physical than the formal use of regularizations, is the one of Itoh & Futamase [255 *, 253 *, 254 *], following previous investigations in Refs. [256 , 257 ]. These authors derived the 3PN equations of motion in harmonic coordinates by means of a particular variant of the famous “surface-integral” method à la Einstein, Infeld & Hoffmann [184 ]. The aim is to describe extended relativistic compact binary systems in the so-called strong-field point particle limit which has been defined in Ref. [212 ]. This approach is interesting because it is based on the physical notion of extended compact bodies in general relativity, and is free of the problems of ambiguities. The end result of Refs. [255 *, 253 *] is in agreement with the 3PN harmonic coordinates equations of motion [69 *, 71 *] and is unambiguous, as it does directly determine the ambiguity parameter $λ$ to exactly the value (172 *).

The 3PN equations of motion in harmonic coordinates or, more precisely, the associated 3PN Lagrangian, were also derived by Foffa & Sturani [203 ] using another important approach, coined the effective field theory (EFT) [223 *]. Their result is fully compatible with the value (172 *) for the ambiguity parameter $λ$ ; however, in contrast with the surface-integral method of Refs. [255 , 253 ], this does not check the method of regularization because the EFT approach is also based on dimensional self-field regularization.

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 & Joguet [81 *], computing the 3PN compact binary’s mass quadrupole moment $Iij$ , found it necessary to introduce three Hadamard regularization constants $ξ$ , $κ$ , and $ζ$ , which are independent of the equation-of-motion related constant $λ$ . 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, $𝜃 = ξ + 2κ + ζ$ , and the binary’s orbital phase, for circular orbits, involved only the linear combination of $𝜃$ and $λ$ given by $7 ˆ𝜃 = 𝜃 − 3λ$ , as shown in [73 ].

Dimensional regularization (instead of Hadamard’s) has next been applied in Refs. [62 *, 63 *] to the computation of the 3PN radiation field of compact binaries, leading to the following unique determination of the ambiguity parameters:⁴⁵

These values represent the end result of dimensional regularization. However, several alternative calculations provide a check, independent of dimensional regularization, for all the parameters (173). One computes [80 *] the 3PN binary’s mass dipole moment $Ii$ using Hadamard’s regularization, and identifies $Ii$ 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 [174 *]. This yields $ξ + κ = − 9871 ∕9240$ in agreement with Eqs. (173). Next, one considers [65 ] the limiting physical situation where the mass of one of the particles is exactly zero (say, $m2 = 0$ ), 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 $ζ = − 7∕33$ , and represents a direct verification of the global Poincaré invariance of the wave generation formalism (the parameter $ζ$ representing the analogue for the radiation field of the parameter $ωkinetic$ ). Finally, one proves [63 *] that $κ = 0$ by showing that there are no dangerously divergent diagrams corresponding to non-zero $κ$ -values, where a diagram is meant here in the sense of Ref. [151 ].

The determination of the parameters (173) completes the problem of the general relativistic prediction for the templates of inspiralling compact binaries up to 3.5PN order. The numerical values of these parameters indicate, following measurement-accuracy analyses [105 *, 106 *, 159 *, 156 *], that the 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.

6.3 Dimensional regularization of the equations of motion

As reviewed in Section 6.2, work at 3PN order using Hadamard’s self-field regularization showed the appearance of ambiguity parameters, due to an incompleteness of the Hadamard regularization employed for curing the infinite self field of point particles. We give here more details on the determination using dimensional regularization of the ambiguity parameter $λ$ [or equivalently $ωstatic$ , see Eq. (170b)] which appeared in the 3PN equations of motion.

Dimensional regularization was invented as a means to preserve the gauge symmetry of perturbative quantum field theories [391 , 91 , 100 , 131 ]. Our basic problem here is to respect the gauge symmetry associated with the diffeomorphism invariance of the classical general relativistic description of interacting point masses. Hence, we use dimensional regularization not merely as a trick to compute some particular integrals which would otherwise be divergent, but as a powerful tool for solving in a consistent way the Einstein field equations with singular point-mass sources, while preserving its crucial symmetries. In particular, we shall prove that dimensional regularization determines the kinetic ambiguity parameter $ωkinetic$ (and its radiation-field analogue $ζ$ ), and is therefore able to correctly keep track of the global Lorentz–Poincaré invariance of the gravitational field of isolated systems. The dimensional regularization is also an important ingredient of the EFT approach to equations of motion and gravitational radiation [223 ].

The Einstein field equations in $d + 1$ space-time dimensions, relaxed by the condition of harmonic coordinates $∂μh αμ = 0$ , take exactly the same form as given in Eqs. (18 *) – (23 *). In particular the box operator $□$ now denotes the flat space-time d’Alembertian operator in $d + 1$ dimensions with signature $(− 1,1,1,⋅⋅⋅)$ . The gravitational constant $G$ is related to the usual three-dimensional Newton’s constant $GN$ by

where $ℓ0$ denotes an arbitrary length scale. The explicit expression of the gravitational source term $αβ Λ$ involves some $d$ -dependent coefficients, and is given by

When $d = 3$ we recover Eq. (24). In the following we assume, as usual in dimensional regularization, that the dimension of space is a complex number, $d ∈ ℂ$ , and prove many results by invoking complex analytic continuation in $d$ . We shall often pose $𝜀 ≡ d − 3$ .

We parametrize the 3PN metric in $d$ dimensions by means of some retarded potentials $V$ , $Vi$ , $Wˆij$ , $...$ , which are straightforward $d$ -dimensional generalizations of the potentials used in three dimensions and which were defined in Section 5.3. Those are obtained by post-Newtonian iteration of the $d$ -dimensional field equations, starting from appropriate definitions of matter source densities generalizing Eqs. (145):

As a result all the expressions of Section 5.3 acquire some explicit $d$ -dependent coefficients. For instance we find [61 *]

Here $−1 □ ret$ means the retarded integral in $d + 1$ space-time dimensions, which admits, though, no simple expression generalizing Eq. (31 *) in physical $(t,x)$ space.⁴⁶

As reviewed in Section 6.1, the generic functions $F (x)$ we have to deal with in 3 dimensions, are smooth on $ℝ3$ except at $y1$ and $y2$ , around which they admit singular Laurent-type expansions in powers and inverse powers of $r ≡ |x − y | 1 1$ and $r ≡ |x − y | 2 2$ , given by Eq. (178 *). In $d$ spatial dimensions, there is an analogue of the function $F$ , which results from the post-Newtonian iteration process performed in $d$ dimensions as we just outlined. Let us call this function $(d) F (x)$ , where $x ∈ ℝd$ . When $r1 → 0$ the function $F (d)$ admits a singular expansion which is more involved than in 3 dimensions, as it reads

The coefficients $(𝜀) f1p,q(n1)$ depend on $𝜀 = d − 3$ , and the powers of $r1$ involve the relative integers $p$ and $q$ whose values are limited by some $p0$ , $q0$ and $q1$ as indicated. Here we will be interested in functions $F (d)(x)$ which have no poles as $𝜀 → 0$ (this will always be the case at 3PN order). Therefore, we can deduce from the fact that $F(d)(x )$ is continuous at $d = 3$ the constraint

For the problem at hand, we essentially have to deal with the regularization of Poisson integrals, or iterated Poisson integrals (and their gradients needed in the equations of motion), of the generic function $F (d)$ . The Poisson integral of $F (d)$ , in $d$ dimensions, is given by the Green’s function for the Laplace operator,

where $&tidle; k$ is a constant related to the usual Eulerian $Γ$ -function by⁴⁷

We need to evaluate the Poisson integral at the point $x′ = y1$ where it is singular; this is quite easy in dimensional regularization, because the nice properties of analytic continuation allow simply to get $[P (d)(x′)]x′=y 1$ by replacing $x ′$ by $y1$ inside the explicit integral (180 *). So we simply have

It is not possible at present to compute the equations of motion in the general $d$ -dimensional case, but only in the limit where $𝜀 → 0$ [163 *, 61 *]. The main technical step of our strategy consists of computing, in the limit $𝜀 → 0$ , the difference between the $d$ -dimensional Poisson potential (182 *), and its Hadamard 3-dimensional counterpart given by $(P )1$ , where the Hadamard partie finie is defined by Eq. (160 *). But we must be precise when defining the Hadamard partie finie of a Poisson integral. Indeed, the definition (160 *) stricto sensu is applicable when the expansion of the function $F$ , for $r1 → 0$ , does not involve logarithms of $r1$ ; see Eq. (160 *). However, the Poisson integral $P (x ′)$ of $F (x)$ will typically involve such logarithms at the 3PN order, namely some $ln r′1$ where $r′ ≡ |x′ − y1| 1$ formally tends to zero (hence $ln r′ 1$ is formally infinite). The proper way to define the Hadamard partie finie in this case is to include the $′ lnr1$ into its definition; we arrive at [70 *]

The first term follows from Hadamard’s partie finie integral (162 *); the second one is given by Eq. (160 *). Notice that in this result the constant $s1$ entering the partie finie integral (162 *) has been “replaced” by $r′1$ , which plays the role of a new regularization constant (together with $r′2$ for the other particle), and which ultimately parametrizes the final Hadamard regularized 3PN equations of motion. It was shown that $′ r1$ and $′ r2$ are unphysical, in the sense that they can be removed by a coordinate transformation [69 *, 71 *]. On the other hand, the constant $s2$ remaining in the result (183 *) is the source for the appearance of the physical ambiguity parameter $λ$ . Denoting the difference between the dimensional and Hadamard regularizations by means of the script letter $𝒟$ , we pose (for what concerns the point 1)

That is, $𝒟P1$ is what we shall have to add to the Hadamard-regularization result in order to get the $d$ -dimensional result. However, we shall only compute the first two terms of the Laurent expansion of $𝒟P 1$ when $𝜀 → 0$ , say $𝒟P = a 𝜀−1 + a + 𝒪 (𝜀) 1 −1 0$ . This is the information we need to clear up the ambiguity parameter. We insist that the difference $𝒟P1$ comes exclusively from the contribution of terms developing some poles $∝ 1∕𝜀$ in the $d$ -dimensional calculation.

Next we outline the way we obtain, starting from the computation of the “difference”, the 3PN equations of motion in dimensional regularization, and show how the ambiguity parameter $λ$ is determined. By contrast to $′ r1$ and $′ r2$ which are pure gauge, $λ$ is a genuine physical ambiguity, introduced in Refs. [70 *, 71 *] as the single unknown numerical constant parametrizing the ratio between $s2$ and $r′2$ [where $s2$ is the constant left in Eq. (183 *)] as

where $m1$ and $m2$ are the two masses. The terms corresponding to the $λ$ -ambiguity in the acceleration $a1 = dv1∕dt$ of particle 1 read simply

where the relative distance between particles is denoted $y1 − y2 ≡ r12n12$ (with $n12$ being the unit vector pointing from particle 2 to particle 1). We start from the end result of Ref. [71 *] for the 3PN harmonic coordinates acceleration $a1$ in Hadamard’s regularization, abbreviated as HR. Since the result was obtained by means of the specific extended variant of Hadamard’s regularization (in short EHR, see Section 6.1) we write it as

where $a(EHR) 1$ is a fully determined functional of the masses $m1$ and $m2$ , the relative distance $r12n12$ , the coordinate velocities $v1$ and $v2$ , and also the gauge constants $′ r1$ and $′ r2$ . The only ambiguous term is the second one and is given by Eq. (186 *).

Our strategy is to extract from both the dimensional and Hadamard regularizations their common core part, obtained by applying the so-called “pure-Hadamard–Schwartz” (pHS) regularization. Following the definition in Ref. [61 *], the pHS regularization is a specific, minimal Hadamard-type regularization of integrals, based on the partie finie integral (162 *), together with a minimal treatment of “contact” terms, in which the definition (162 *) is applied separately to each of the elementary potentials $V$ , $Vi$ , etc. (and gradients) that enter the post-Newtonian metric. Furthermore, the regularization of a product of these potentials is assumed to be distributive, i.e., $(F G )1 = (F )1(G)1$ in the case where $F$ and $G$ are given by such elementary potentials; this is thus in contrast with Eq. (161 *). The pHS regularization also assumes the use of standard Schwartz distributional derivatives [381 ]. The interest of the pHS regularization is that the dimensional regularization is equal to it plus the “difference”; see Eq. (190 *).

To obtain the pHS-regularized acceleration we need to substract from the EHR result a series of contributions, which are specific consequences of the use of EHR [70 , 72 ]. For instance, one of these contributions corresponds to the fact that in the EHR the distributional derivative is given by Eqs. (167 *) – (168 *) which differs from the Schwartz distributional derivative in the pHS regularization. Hence we define

where the $δa1$ ’s denote the extra terms following from the EHR prescriptions. The pHS-regularized acceleration (188 *) constitutes essentially the result of the first stage of the calculation of $a1$ , as reported in Ref. [193 ].

The next step consists of evaluating the Laurent expansion, in powers of $𝜀 = d − 3$ , of the difference between the dimensional regularization and the pHS (3-dimensional) computation. As we reviewed above, this difference makes a contribution only when a term generates a pole $∼ 1∕𝜀$ , in which case the dimensional regularization adds an extra contribution, made of the pole and the finite part associated with the pole [we consistently neglect all terms $𝒪 (𝜀)$ ]. One must then be especially wary of combinations of terms whose pole parts finally cancel but whose dimensionally regularized finite parts generally do not, and must be evaluated with care. We denote the above defined difference by

It is made of the sum of all the individual differences of Poisson or Poisson-like integrals as computed in Eq. (184 *). The total difference (189 *) depends on the Hadamard regularization scales $r′ 1$ and $s2$ (or equivalently on $λ$ and $r′ 1$ , $r′ 2$ ), and on the parameters associated with dimensional regularization, namely $𝜀$ and the characteristic length scale $ℓ0$ introduced in Eq. (174 *). Finally, the result is the explicit computation of the $𝜀$ -expansion of the dimensional regularization (DR) acceleration as

With this result we can prove two theorems [61 ].

Theorem 8. The pole part $∝ 1∕𝜀$ of the DR acceleration (190 *) can be re-absorbed (i.e. renormalized) into some shifts of the two “bare” world-lines: $y1 → y1 + ξ1$ and $y2 → y2 + ξ2$ , with $ξ1,2 ∝ 1∕ 𝜀$ say, so that the result, expressed in terms of the “dressed” quantities, is finite when $𝜀 → 0$ .

The situation in harmonic coordinates is to be contrasted with the calculation in ADM-type coordinates within the Hamiltonian formalism, where it was shown that all pole parts directly cancel out in the total 3PN Hamiltonian: No renormalization of the world-lines is needed [163 ]. The central result is then:

Theorem 9. The renormalized (finite) DR acceleration is physically equivalent to the Hadamard-regularized (HR) acceleration (end result of Ref. [71 *]), in the sense that

where $δξa1$ denotes the effect of the shifts on the acceleration, if and only if the HR ambiguity parameter $λ$ entering the harmonic-coordinates equations of motion takes the unique value (172 *).

The precise shifts $ξ1$ and $ξ2$ needed in Theorem 9 involve not only a pole contribution $∝ 1∕𝜀$ , but also a finite contribution when $𝜀 → 0$ . Their explicit expressions read:⁴⁸

where $GN$ is Newton’s constant, $ℓ0$ is the characteristic length scale of dimensional regularization, cf. Eq. (174 *), where $aN 1$ is the Newtonian acceleration of the particle 1 in $d$ dimensions, and $q-≡ 4πe γE$ depends on Euler’s constant $γ ≃ 0.577 E$ .

6.4 Dimensional regularization of the radiation field

We now address the similar problem concerning the binary’s radiation field – to 3PN order beyond Einstein’s quadrupole formalism (2 *) – (3 *). As reviewed in Section 6.2, three ambiguity parameters: $ξ$ , $κ$ and $ζ$ , have been shown to appear in the 3PN expression of the quadrupole moment [81 *, 80 *].

To apply dimensional regularization, we must use as in Section 6.3 the $d$ -dimensional post-Newtonian iteration leading to potentials such as those in Eqs. (177); and we have to generalize to $d$ dimensions some key results of the wave generation formalism of Part A. Essentially, we need the $d$ -dimensional analogues of the multipole moments of an isolated source $IL$ and $JL$ in Eqs. (123). Here we report the result we find in the case of the mass-type moment:

in which we denote, generalizing Eqs. (124),

and where for any source densities the underscript $[ℓ]$ means the infinite series

The latter definition represents the $d$ -dimensional version of the post-Newtonian expansion series (126 *). At Newtonian order, the expression (193) reduces to the standard result $(d) ∫ d 2 IL = d xρxˆL + 𝒪 (1 ∕c )$ with $ρ = T 00∕c2$ denoting the usual Newtonian density.

The ambiguity parameters $ξ$ , $κ$ and $ζ$ come from the Hadamard regularization of the mass quadrupole moment $I ij$ at the 3PN order. The terms corresponding to these ambiguities were found to be

where $y1$ , $v1$ and $a1$ denote the first particle’s position, velocity and acceleration (and the brackets $⟨⟩$ surrounding indices refer to the STF projection). Like in Section 6.3, we express both the Hadamard and dimensional results in terms of the more basic pHS regularization. The first step of the calculation [80 *] is therefore to relate the Hadamard-regularized quadrupole moment $(HR) Iij$ , for general orbits, to its pHS part:

In the right-hand side we find both the pHS part, and the effect of adding the ambiguities, with some numerical shifts of the ambiguity parameters ( $ξ → ξ + 1∕22$ , $ζ → ζ + 9∕110$ ) due to the difference between the specific Hadamard-type regularization scheme used in Ref. [81 *] and the pHS one. The pHS part is free of ambiguities but depends on the gauge constants $′ r1$ and $′ r2$ introduced in the harmonic-coordinates equations of motion [69 *, 71 *].

We next use the $d$ -dimensional moment (193) to compute the difference between the dimensional regularization (DR) result and the pHS one [62 *, 63 *]. As in the work on equations of motion, we find that the ambiguities arise solely from the terms in the integration regions near the particles, that give rise to poles $∝ 1∕ 𝜀$ , corresponding to logarithmic ultra-violet (UV) divergences in 3 dimensions. The infra-red (IR) region at infinity, i.e., $|x| → + ∞$ , does not contribute to the difference between $DR$ and $pHS$ . The compact-support terms in the integrand of Eq. (193), proportional to the matter source densities $σ$ , $σ a$ , and $σ ab$ , are also found not to contribute to the difference. We are therefore left with evaluating the difference linked with the computation of the non-compact terms in the expansion of the integrand of (193) near the singularities that produce poles in $d$ dimensions.

Let $F(d)(x )$ be the non-compact part of the integrand of the quadrupole moment (193) (with indices $L = ij$ ), where $(d) F$ includes the appropriate multipolar factors such as $ˆxij$ , so that

We do not indicate that we are considering here only the non-compact part of the moments. Near the singularities the function $F (d)(x)$ admits a singular expansion of the type (178 *). In practice, the various coefficients $f(𝜀) 1 p,q$ are computed by specializing the general expressions of the non-linear retarded potentials $V$ , $Va$ , $ˆ Wab$ , etc. (valid for general extended sources) to point particles in $d$ dimensions. On the other hand, the analogue of Eq. (198 *) in 3 dimensions is

where $Pf$ refers to the Hadamard partie finie defined in Eq. (162 *). The difference $𝒟Iij$ between the DR evaluation of the $d$ -dimensional integral (198 *) and its corresponding three-dimensional evaluation (199 *), reads then

Such difference depends only on the UV behaviour of the integrands, and can therefore be computed “locally”, i.e., in the vicinity of the particles, when $r1 → 0$ and $r2 → 0$ . We find that Eq. (200 *) depends on two constant scales $s1$ and $s2$ coming from Hadamard’s partie finie (162 *), and on the constants belonging to dimensional regularization, i.e., $𝜀 = d − 3$ and $ℓ0$ defined by Eq. (174 *). The dimensional regularization of the 3PN quadrupole moment is then obtained as the sum of the pHS part, and of the difference computed according to Eq. (200 *), namely

An important fact, hidden in our too-compact notation (201 *), is that the sum of the two terms in the right-hand side of Eq. (201 *) does not depend on the Hadamard regularization scales $s 1$ and $s 2$ . Therefore it is possible without changing the sum to re-express these two terms (separately) by means of the constants $r′1$ and $r′2$ instead of $s1$ and $s2$ , where $r1′$ , $r′2$ are the two fiducial scales entering the Hadamard-regularization result (197 *). This replacement being made the pHS term in Eq. (201 *) is exactly the same as the one in Eq. (197 *). At this stage all elements are in place to prove the following theorem [62 *, 63 *].

Theorem 10. The DR quadrupole moment (201 *) is physically equivalent to the Hadamard-regularized one (end result of Refs. [81 *, 80 *]), in the sense that

where $δξIij$ denotes the effect of the same shifts as determined in Theorems 8 and 9, if and only if the HR ambiguity parameters $ξ$ , $κ$ and $ζ$ take the unique values reported in Eqs. (173). Moreover, the poles $1∕𝜀$ separately present in the two terms in the brackets of Eq. (202 *) cancel out, so that the physical (“dressed”) DR quadrupole moment is finite and given by the limit when $𝜀 → 0$ as shown in Eq. (202 *).

This theorem finally provides an unambiguous determination of the 3PN radiation field by dimensional regularization. Furthermore, as reviewed in Section 6.2, several checks of this calculation could be done, which provide independent confirmations for the ambiguity parameters. Such checks also show the powerfulness of dimensional regularization and its validity for describing the classical general-relativistic dynamics of compact bodies.

	Abstract
1	Introduction
	1.1	Analytic approximations and wave generation formalism
	1.2	The quadrupole moment formalism
	1.3	Problem posed by compact binary systems
	1.4	Post-Newtonian equations of motion
	1.5	Post-Newtonian gravitational radiation
A	Post-Newtonian Sources
2	Non-linear Iteration of the Vacuum Field Equations
	2.1	Einstein’s field equations
	2.2	Linearized vacuum equations
	2.3	The multipolar post-Minkowskian solution
	2.4	Generality of the MPM solution
	2.5	Near-zone and far-zone structures
3	Asymptotic Gravitational Waveform
	3.1	The radiative multipole moments
	3.2	Gravitational-wave tails and tails-of-tails
	3.3	Radiative versus source moments
4	Matching to a Post-Newtonian Source
	4.1	The matching equation
	4.2	General expression of the multipole expansion
	4.3	Equivalence with the Will–Wiseman formalism
	4.4	The source multipole moments
5	Interior Field of a Post-Newtonian Source
	5.1	Post-Newtonian iteration in the near zone
	5.2	Post-Newtonian metric and radiation reaction effects
	5.3	The 3.5PN metric for general matter systems
	5.4	Radiation reaction potentials to 4PN order
B	Compact Binary Systems
6	Regularization of the Field of Point Particles
	6.1	Hadamard self-field regularization
	6.2	Hadamard regularization ambiguities
	6.3	Dimensional regularization of the equations of motion
	6.4	Dimensional regularization of the radiation field
7	Newtonian-like Equations of Motion
	7.1	The 3PN acceleration and energy for particles
	7.2	Lagrangian and Hamiltonian formulations
	7.3	Equations of motion in the center-of-mass frame
	7.4	Equations of motion and energy for quasi-circular orbits
	7.5	The 2.5PN metric in the near zone
8	Conservative Dynamics of Compact Binaries
	8.1	Concept of innermost circular orbit
	8.2	Dynamical stability of circular orbits
	8.3	The first law of binary point-particle mechanics
	8.4	Post-Newtonian approximation versus gravitational self-force
9	Gravitational Waves from Compact Binaries
	9.1	The binary’s multipole moments
	9.2	Gravitational wave energy flux
	9.3	Orbital phase evolution
	9.4	Polarization waveforms for data analysis
	9.5	Spherical harmonic modes for numerical relativity
10	Eccentric Compact Binaries
	10.1	Doubly periodic structure of the motion of eccentric binaries
	10.2	Quasi-Keplerian representation of the motion
	10.3	Averaged energy and angular momentum fluxes
11	Spinning Compact Binaries
	11.1	Lagrangian formalism for spinning point particles
	11.2	Equations of motion and precession for spin-orbit effects
	11.3	Spin-orbit effects in the gravitational wave flux and orbital phase
	Acknowledgments
	References
	Footnotes
	Updates
	Figures
	Tables