### 4.4 The radiative multipole moments

The leading-order term of the metric in radiative coordinates, neglecting , yields the
operational definition of two sets of STF radiative multipole moments, mass-type and current-type
. By definition, we have
This multipole decomposition represents the generalization, up to any post-Newtonian order (witness the
factors of in front of each of the multipolar pieces) of the quadrupole-moment formalism reviewed in
Eq. (2). The corresponding total gravitational flux reads
Notice that the meaning of such formulas is rather empty, because we do not know yet how the radiative
moments are given in terms of the actual source parameters. Only at the Newtonian level do we know this
relation, which from the comparison with the quadrupole formalism of Eqs. (2, 3, 4) reduces to
where is the Newtonian quadrupole given by Eq. (3). Fortunately, we are not in such bad shape
because we have learned from Theorem 4 the general method that permits us to compute the radiative
multipole moments , in terms of the source moments , , , . Therefore,
what is missing is the explicit dependence of the source multipole moments as functions of the
actual parameters of some isolated source. We come to grips with this question in the next
section.