Dimensional regularization was invented as a means to preserve the gauge symmetry of perturbative
quantum field theories [202, 51, 57, 73]. 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 (and its radiation-field analogue
), and is therefore able to
correctly keep track of the global Lorentz-Poincaré invariance of the gravitational field of isolated
systems.
The Einstein field equations in space-time dimensions, relaxed by the condition of
harmonic coordinates
, take exactly the same form as given in Equations (9
, 14
). In
particular
denotes the flat space-time d’Alembertian operator in
dimensions. The
gravitational constant
is related to the usual three-dimensional Newton’s constant
by
We parametrize the 3PN metric in dimensions by means of straightforward
-dimensional
generalizations of the retarded potentials
,
,
,
, and
of Section 7. Those are obtained
by post-Newtonian iteration of the
-dimensional field equations, starting from the following definitions of
matter source densities
As reviewed in Section 8.1, the generic functions we have to deal with in 3 dimensions,
say , are smooth on
except at
and
, around which they admit singular
Laurent-type expansions in powers and inverse powers of
and
, given by
Equation (121
). In
spatial dimensions, there is an analogue of the function
, which results from
the post-Newtonian iteration process performed in
dimensions as we just outlined. Let
us call this function
, where
. When
the function
admits
a singular expansion which is a little bit more complicated than in 3 dimensions, as it reads
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
. The Poisson integral of
, in
dimensions, is given by the Green’s function for the Laplace
operator,
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 is
determined. By contrast to
and
which are pure gauge,
is a genuine physical ambiguity,
introduced in Refs. [36
, 38
] as the single unknown numerical constant parametrizing the ratio between
and
(where
is the constant left in Equation (148
)) as
Our strategy is to express both the dimensional and Hadamard regularizations in terms of their common
“core” part, obtained by applying the so-called “pure-Hadamard-Schwartz” (pHS) regularization. Following
the definition of Ref. [30], the pHS regularization is a specific, minimal Hadamard-type regularization
of integrals, based on the partie finie integral (124
), together with a minimal treatment of
“contact” terms, in which the definition (124
) is applied separately to each of the elementary
potentials
(and gradients) that enter the post-Newtonian metric in the form given
in Section 7. Furthermore, the regularization of a product of these potentials is assumed to
be distributive, i.e.
in the case where
and
are given by such
elementary potentials (this is in contrast with Equation (123
)). The pHS regularization also
assumes the use of standard Schwartz distributional derivatives [199]. The interest of the pHS
regularization is that the dimensional regularization is equal to it plus the “difference”; see
Equation (155
).
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 [36, 39]. For instance, one of these
contributions corresponds to the fact that in the EHR the distributional derivative is given by
Equations (129
, 130
) which differs from the Schwartz distributional derivative in the pHS regularization.
Hence we define
The next step consists of evaluating the Laurent expansion, in powers of , 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
, 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
). One must then be especially wary of combinations of
terms whose pole parts finally cancel (“cancelled poles”) but whose dimensionally regularized finite parts
generally do not, and must be evaluated with care. We denote the above defined difference by
Theorem 8 The pole part of the DR acceleration (155
) can be re-absorbed (i.e. renormalized)
into some shifts of the two “bare” world-lines:
and
, with, say,
, so that the result, expressed in terms of the “dressed” quantities, is finite when
.
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 [96]. A central result is then as
follows:
Theorem 9 The renormalized (finite) DR acceleration is physically equivalent to the Hadamard-regularized
(HR) acceleration (end result of Ref. [38]), in the sense that
The precise shifts and
needed in Theorem 9 involve not only a pole contribution
(which would define a
renormalization by minimal subtraction (MS)), but also a finite contribution when
. Their explicit expressions
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