8.4 Dimensional regularization of the radiation field
We now address the similar problem concerning the binary’s radiation field (3PN beyond the Einstein
quadrupole formalism), for which three ambiguity parameters,
,
,
, have been shown to
appear [45
, 44
] (see Section 8.2).
To apply dimensional regularization, we must use as in Section 8.3 the
-dimensional post-Newtonian
iteration [leading to equations such as (142)]; and, crucially, we have to generalize to
dimensions some
key results of the wave generation formalism of Part A. Essentially we need the
-dimensional analogues
of the multipole moments of an isolated source
and
, Equations (85). The result we find in the
case of the mass-type moments is
where we denote (generalizing Equations (86))
and where for any source densities the underscript
means the infinite series
The latter definition represents the
-dimensional version of the post-Newtonian expansion series (91). At
Newtonian order, Equation (158) reduces to the standard result
with
.
The ambiguity parameters
,
, and
come from the Hadamard regularization of the mass
quadrupole moment
at the 3PN order. The terms corresponding to these ambiguities were found to be
where
,
, and
denote the first particle’s position, velocity, and acceleration. We recall that the
brackets
surrounding indices refer to the symmetric-trace-free (STF) projection. Like in Section 8.3,
we express both the Hadamard and dimensional results in terms of the more basic pHS regularization. The
first step of the calculation [44
] is therefore to relate the Hadamard-regularized quadrupole moment
, 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 coming from the difference between the specific
Hadamard-type regularization scheme used in Ref. [45
] and the pHS one. The pHS part is free of
ambiguities but depends on the gauge constants
and
introduced in the harmonic-coordinates
equations of motion [37
, 38
].
We next use the
-dimensional moment (158) to compute the difference between the dimensional
regularization (DR) result and the pHS one [31
, 32
]. 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
(i.e.
or
) that give rise to poles
, corresponding to
logarithmic ultra-violet (UV) divergences in 3 dimensions. The infra-red (IR) region at infinity
(i.e.
) does not contribute to the difference
. The compact-support terms in the
integrand of Equation (158), proportional to the matter source densities
,
, and
, 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 in (158) near the singularities that
produce poles in
dimensions.
Let
be the non-compact part of the integrand of the quadrupole moment (158) (with
indices
), where
includes the appropriate multipolar factors such as
, so that
We do not indicate that we are considering here only the non-compact part of the moments. Near the
singularities the function
admits a singular expansion of the type (143). In practice, the
various coefficients
are computed by specializing the general expressions of the non-linear
retarded potentials
(valid for general extended sources) to the point particles
case in
dimensions. On the other hand, the analogue of Equation (163) in 3 dimensions is
where
refers to the Hadamard partie finie defined in Equation (124). The difference
between the
DR evaluation of the
-dimensional integral (163), and its corresponding three-dimensional evaluation,
i.e. the partie finie (164), 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
and
. We find that Equation (165)
depends on two constant scales
and
coming from Hadamard’s partie finie (124), and on the
constants belonging to dimensional regularization, which are
and the length scale
defined
by Equation (139). 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 Equation (165), namely
An important fact, hidden in our too-compact notation (166), is that the sum of the two terms in the
right-hand side of Equation (166) does not depend on the Hadamard regularization scales
and
.
Therefore it is possible without changing the sum to re-express these two terms (separately) by means of
the constants
and
instead of
and
, where
,
are the two fiducial scales entering
the Hadamard-regularization result (162). This replacement being made the pHS term in Equation (166) is
exactly the same as the one in Equation (162). At this stage all elements are in place to prove the following
theorem [31
, 32
]:
Theorem 10 The DR quadrupole moment (166) is physically equivalent to the Hadamard-regularized one
(end result of Refs. [45
, 44
]), in the sense that
where
denotes the effect of the same shifts as determined in Theorems 8 and 9, if and only if the HR
ambiguity parameters
,
, and
take the unique values (136). Moreover, the poles
separately
present in the two terms in the brackets of Equation (167) cancel out, so that the physical (“dressed”) DR
quadrupole moment is finite and given by the limit when
as shown in Equation (167).
This theorem finally provides an unambiguous determination of the 3PN radiation field by dimensional
regularization. Furthermore, as reviewed in Section 8.2, several checks of this calculation could be done,
which provide, together with comparisons with alternative methods [96
, 30
, 133
, 132
], independent
confirmations for the four ambiguity parameters
,
,
, and
, and confirm the consistency of
dimensional regularization and its validity for describing the general-relativistic dynamics of compact
bodies.