The multipolar-post-Minkowskian method will be implemented systematically, using STF-harmonics to
describe the multipole expansion [210], and looking for a definite algorithm for the approximation scheme [26].
The solution of the system of equations (23, 24) takes the form of a series of retarded multipolar
waves^{7}

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 Equation (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^{8},
, 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 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 Equation (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 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 , but with some moments , that differ from , by non-linear corrections (see Equation (96)). All the multipole moments , , , , , will be computed in Section 5.

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