### 8.1 Definitions

A model of structureless point masses is expected to be sufficient to describe the inspiral phase of compact binaries (see the discussion around Eqs. (6, 7, 8)). Thus we want to compute the metric (and its gradient needed in the equations of motion) at the 3PN order for a system of two point-like particles. A priori one is not allowed to use directly the metric expressions (109, 110, 111, 112, 113, 114, 115), as they 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 or , where and denote the two trajectories. Consequently, we must supplement the calculation by a prescription for how to remove the “infinite part” of these integrals. We systematically employ the Hadamard regularization [80133] (see Ref. [134] 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. [2023].

Consider the class of functions which are smooth () on except for the two points and , around which they admit a power-like singular expansion of the type

and similarly for the other point 2. Here , and the coefficients of the various powers of depend on the unit direction of approach to the singular point. The powers of are real, range in discrete steps [i.e. ], and are bounded from below (). The coefficients (and ) for which can be referred to as the singular coefficients of . If and belong to so does the ordinary product , as well as the ordinary gradient . We define the Hadamard “partie finie” of at the location of the singular point 1 as
where denotes the solid angle element centered on and of direction . The Hadamard partie finie is “non-distributive” in the sense that in general. The second notion of Hadamard partie finie () concerns that of the integral , which is generically divergent at the location of the two singular points and (we assume no divergence at infinity). It is defined by
The first term integrates over a domain defined as to which the two spherical balls and of radius and centered on the two singularities are excised: . The other terms, where the value of a function at 1 takes the meaning (118), are such that they cancel out the divergent part of the first term in the limit where (the symbol means the same terms but corresponding to the other point 2). The Hadamard partie-finie integral depends on two strictly positive constants and , associated with the logarithms present in Eq. (119). See Ref. [20] for alternative expressions of the partie-finie integral.

To any we associate the partie finie pseudo-function , namely a linear form acting on , which is defined by the duality bracket

When restricted to the set of smooth functions with compact support (we have ), the pseudo-function is a distribution in the sense of Schwartz [133]. The product of pseudo-functions coincides, by definition, with the ordinary pointwise product, namely . An interesting pseudo-function, constructed in Ref. [20] on the basis of the Riesz delta function [125], is the delta-pseudo-function , which plays a role analogous to the Dirac measure in distribution theory, . It is defined by
where is the partie finie of as given by Eq. (118). From the product of with any we obtain the new pseudo-function , that is such that
As a general rule, we are not allowed, in consequence of the “non-distributivity” of the Hadamard partie finie, to replace within the pseudo-function by its regularized value: . The object has no equivalent in distribution theory.

Next, we treat the spatial derivative of a pseudo-function of the type , namely . Essentially, we require (in Ref. [20]) that the so-called 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 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 a computation valid for regular functions. By definition,

Furthermore, we assume that when all the singular coefficients of vanish, the derivative of reduces to the ordinary derivative, i.e. . Then it is trivial to check that the rule (123) contains as a particular case the standard definition of the distributional derivative [133]. Notably, we see that the integral of a gradient is always zero: . This should certainly be the case if we want to compute a quantity (e.g., a Hamiltonian density) which is defined only modulo a total divergence. We pose
where represents the “ordinary” derivative and the distributional term. The following solution of the basic relation (123) was obtained in Ref. [20]:
where we assume for simplicity that the powers in the expansion (117) of are relative integers. The distributional term (125) is of the form (plus ). It is generated solely by the singular coefficients of (the sum over in Eq. (125) is always finite). The formula for the distributional term associated with the th distributional derivative, i.e. , where , reads
We refer to Theorem 4 in Ref. [20] for the definition of another derivative operator, representing in fact the most general derivative satisfying the same properties as the one defined by Eq. (125) and, in addition, the commutation of successive derivatives (or Schwarz lemma). The distributional derivative (124, 125, 126) does not satisfy the Leibniz rule for the derivation of a product, in accordance with a general theorem of Schwartz [132]. Rather, the investigation in Ref. [20] has suggested that, in order to construct a consistent theory (using the “ordinary” product for pseudo-functions), the Leibniz rule should in a sense be weakened, and replaced by the rule of integration by part (123), which is in fact nothing but an “integrated” version of the Leibniz rule.

The Hadamard regularization is defined by Eq. (118) in a preferred spatial hypersurface of a coordinate system, and consequently is not a priori compatible with the requirement of global Lorentz invariance. Thus we expect that the equations of motion in harmonic coordinates (which, we recall, 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 is sufficient in order to get some Lorentz-invariant equations of motion [25]. To deal with the problem at 3PN a Lorentz-invariant regularization, denoted , was introduced in Ref. [23]. 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 differs from the simpler regularization by relativistic corrections of order at least. See Ref. [23] for the formulas defining this regularization in the form of some infinite power series in the relativistic parameter . The regularization plays a crucial role in obtaining the equations of motion at the 3PN order in Refs. [2122].