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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. (6View Equation, 7View Equation, 8View Equation)). 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 (109View Equation, 110View Equation, 111View Equation, 112View Equation, 113View Equation, 114View Equation, 115View Equation), 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 x --> y1(t) or y2(t), where y1(t) and y2(t) 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 [80133Jump To The Next Citation Point] (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. [20Jump To The Next Citation Point23Jump To The Next Citation Point].

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

s um A n (- N, F (x) = ra 1fa(n1) + o(rn), (117) a <a<n 1 1 0
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 r 1 are real, range in discrete steps [i.e. a (- (a ) i 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 singular point 1 as
integral d_O_1- (F )1 = 4p 1f0(n1), (118)
where d_O_1 = d_O_(n1) denotes the solid angle element centered on y1 and of direction n1. The Hadamard partie finie is “non-distributive” in the sense that (F G)1 /= (F )1(G)1 in general. The second notion of Hadamard partie finie (Pf) concerns that of the integral integral d3xF, which is generically divergent at the location of the two singular points y1 and y2 (we assume no divergence at infinity). It is defined by
integral { integral a+3 ( ) ( ) } Pf d3x F = lim d3x F + 4p sum s----- F-- + 4p ln -s (r3F ) + 1 <--> 2 .(119) s1s2 s-->0 S(s) a + 3 ra1 1 s1 1 1 a+3<0
The first term integrates over a domain S(s) defined as R3 to which the two spherical balls r1 < s and r2 < s of radius s and centered on the two singularities are excised: 3 S(s) =_ R \B(y1, s) U B(y2, s). The other terms, where the value of a function at 1 takes the meaning (118View Equation), are 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. (119View Equation). See Ref. [20Jump To The Next Citation Point] for alternative expressions of the partie-finie integral.

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

integral A G (- F , <Pf F,G > = Pf d3x F G. (120)
When restricted to the set D of smooth functions with compact support (we have D < F), the pseudo-function Pf F is a distribution in the sense of Schwartz [133Jump To The Next Citation Point]. The product of pseudo-functions coincides, by definition, with the ordinary pointwise product, namely Pf F.Pf G = Pf(F G). An interesting pseudo-function, constructed in Ref. [20Jump To The Next Citation Point] on the basis of the Riesz delta function [125], is the delta-pseudo-function Pf d1, which plays a role analogous to the Dirac measure in distribution theory, d1(x) =_ d(x - y1). It is defined by
integral 3 A F (- F , <Pf d1,F > = Pf d x d1F = (F )1, (121)
where (F)1 is the partie finie of F as given by Eq. (118View Equation). From the product of Pf d1 with any Pf F we obtain the new pseudo-function Pf(F d1), that is such that
A G (- F , <Pf(F d1),G > = (FG)1. (122)
As a general rule, we are not allowed, in consequence of the “non-distributivity” of the Hadamard partie finie, to replace F within the pseudo-function Pf(F d1) by its regularized value: Pf(F d1) /= (F )1 Pf d1. The object Pf(F d1) has no equivalent in distribution theory.

Next, we treat the spatial derivative of a pseudo-function of the type Pf F, namely @i(Pf F ). Essentially, we require (in Ref. [20Jump To The Next Citation Point]) 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,

A F, G (- F , <@ (Pf F ),G > = - <@ (Pf G), F >. (123) i i
Furthermore, we assume that when all the singular coefficients of F vanish, the derivative of Pf F reduces to the ordinary derivative, i.e. @i(Pf F ) = Pf(@iF). Then it is trivial to check that the rule (123View Equation) 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: <@i(Pf F ),1> = 0. 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
@i(Pf F) = Pf(@iF ) + Di[F], (124)
where Pf(@iF ) represents the “ordinary” derivative and Di[F ] the distributional term. The following solution of the basic relation (123View Equation) was obtained in Ref. [20Jump To The Next Citation Point]:
( [1 sum 1 ] ) Di[F ] = 4p Pf ni1 -r1 1f-1 + --k1f-2-k d1 + 1 <--> 2, (125) 2 k>0r1
where we assume for simplicity that the powers a in the expansion (117View Equation) of F are relative integers. The distributional term (125View Equation) is of the form Pf(Gd1) (plus 1 <--> 2). It is generated solely by the singular coefficients of F (the sum over k in Eq. (125View Equation) is always finite). The formula for the distributional term associated with the lth distributional derivative, i.e. DL[F ] = @L Pf F - Pf @LF, where L = i1i2...il, reads
sum l DL[F ] = @i...i Di [@i ...iF ]. (126) k=1 1 k-1 k k+1 l
We refer to Theorem 4 in Ref. [20Jump To The Next Citation Point] for the definition of another derivative operator, representing in fact the most general derivative satisfying the same properties as the one defined by Eq. (125View Equation) and, in addition, the commutation of successive derivatives (or Schwarz lemma)18. The distributional derivative (124View Equation, 125View Equation, 126View Equation) does not satisfy the Leibniz rule for the derivation of a product, in accordance with a general theorem of Schwartz [132Jump To The Next Citation Point]. Rather, the investigation in Ref. [20Jump To The Next Citation Point] 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 (123View Equation), which is in fact nothing but an “integrated” version of the Leibniz rule.

The Hadamard regularization (F )1 is defined by Eq. (118View Equation) in a preferred spatial hypersurface t = const. 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 (F )1 is sufficient in order to get some Lorentz-invariant equations of motion [25Jump To The Next Citation Point]. To deal with the problem at 3PN a Lorentz-invariant regularization, denoted [F ]1, was introduced in Ref. [23Jump To The Next Citation Point]. 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 2 1/c at least. See Ref. [23Jump To The Next Citation Point] for the formulas defining this regularization in the form of some infinite power series in the relativistic parameter 1/c2. The regularization [F ]1 plays a crucial role in obtaining the equations of motion at the 3PN order in Refs. [21Jump To The Next Citation Point22Jump To The Next Citation Point].


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