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 [80, 133
] (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. [20
, 23
].
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. [21
, 22
].