## 3 Linearized Vacuum Equations

In what follows we solve the field equations (12, 13), in the vacuum region outside the compact-support
source, in the form of a formal non-linearity or post-Minkowskian expansion, considering the field variable
as a non-linear metric perturbation of Minkowski space-time. At the linearized level (or
first-post-Minkowskian approximation), we write:
where the subscript ”ext” reminds us that the solution is valid only in the exterior of the source, and where
we have introduced Newton’s constant as a book-keeping parameter, enabling one to label very
conveniently the successive post-Minkowskian approximations. Since is a dimensionless variable, with
our convention the linear coefficient in Eq. (22) has the dimension of the inverse of - a mass
squared in a system of units where . In vacuum, the harmonic-coordinate metric coefficient
satisfies
We want to solve those equations by means of an infinite multipolar series valid outside a time-like world
tube containing the source. Indeed the multipole expansion is the correct method for describing the physics
of the source as seen from its exterior (). On the other hand, the post-Minkowskian series is
physically valid in the weak-field region, which surely includes the exterior of any source, starting at a
sufficiently large distance. For post-Newtonian sources the exterior weak-field region, where both
multipole and post-Minkowskian expansions are valid, simply coincides with the exterior .
It is therefore quite natural, and even, one would say inescapable when considering general
sources, to combine the post-Minkowskian approximation with the multipole decomposition.
This is the original idea of the “double-expansion” series of Bonnor [33], which combines the
-expansion (or -expansion in his notation) with the -expansion (equivalent to the
multipole expansion, since the th order multipole moment scales like with the source
radius).
The multipolar-post-Minkowskian method will be implemented systematically, using STF-harmonics to
describe the multipole expansion [142], and looking for a definite algorithm for the approximation scheme [14].
The solution of the system of equations (23, 24) takes the form of a series of retarded multipolar
waves

where , and where the functions are smooth functions of the retarded time
[], which become constant in the past, when . It is evident,
since a monopolar wave satisfies and the d’Alembertian commutes with the
multi-derivative , that Eq. (25) represents the most general solution of the wave equation (23) (see
Section 2 in Ref. [14] for a proof based on the Euler-Poisson-Darboux equation). The gauge
condition (24), however, is not fulfilled in general, and to satisfy it we must algebraically decompose
the set of functions , , into ten tensors which are STF with respect to all
their indices, including the spatial indices , . Imposing the condition (24) reduces the
number of independent tensors to six, and we find that the solution takes an especially simple
“canonical” form, parametrized by only two moments, plus some arbitrary linearized gauge
transformation [142, 14].
Theorem 1 The most general solution of the linearized field equations (23, 24), outside some
time-like world tube enclosing the source (), and stationary in the past (see Eq. (19)), reads
The first term depends on two STF-tensorial multipole moments, and , which are arbitrary
functions of time except for the laws of conservation of the monopole: , and dipoles:
, . It is given by
The other terms represent a linearized gauge transformation, with gauge vector of the type (25), and
parametrized for four other multipole moments, say , , and .

The conservation of the lowest-order moments gives the constancy of the total mass of the source, ,
center-of-mass position,
, total linear momentum , and total angular momentum,
. It is always possible to achieve by translating the origin of our coordinates to
the center of mass. The total mass is the Arnowitt-Deser-Misner (ADM) mass of the Hamiltonian
formulation of general relativity. Note that the quantities , , and include the
contributions due to the waves emitted by the source. They describe the “initial” state of the source, before
the emission of gravitational radiation.
The multipole functions and , which thoroughly encode the physical properties of the
source at the linearized level (because the other moments , , parametrize a gauge
transformation), will be referred to as the mass-type and current-type source multipole moments. Beware,
however, that at this stage the moments are not specified in terms of the stress-energy tensor of the
source: the above theorem follows merely from the algebraic and differential properties of the vacuum
equations outside the source.
For completeness, let us give the components of the gauge-vector entering Eq. (26):

Because the theory is covariant with respect to non-linear diffeomorphisms and not merely with
respect to linear gauge transformations, the moments , , do play a physical role
starting at the non-linear level, in the following sense. If one takes these moments equal to zero
and continues the calculations one ends up with a metric depending on and only,
but that metric will not describe the same physical source as the one constructed from the six
moments , , . In other words, the two non-linear metrics associated with the sets
of multipole moments {, , 0, , 0} and {, , , , } are not
isometric. We point out in Section 4.2 below that the full set of moments {, , ,
, } is in fact physically equivalent to some reduced set {, , 0, , 0},
but with some moments , that differ from , by non-linear corrections (see
Eq. (90)). All the multipole moments , , , , , will be computed in
Section 5.