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4.4 The radiative multipole moments

The leading-order term 1/R of the metric in radiative coordinates, neglecting 2 O(1/R ), yields the operational definition of two sets of STF radiative multipole moments, mass-type UL(U ) and current-type VL(U ). By definition, we have
4G + sum o o 1 { 2l } HTTij (U, N ) = -2--Pijab(N ) -l- NL -2UabL -2(U )- --------NcL -2ecd(aVb)dL-2(U ) c R l=2 c l! c(l + 1) ( 1 ) + O -2- . (59) R
This multipole decomposition represents the generalization, up to any post-Newtonian order (witness the factors of 1/c in front of each of the multipolar pieces) of the quadrupole-moment formalism reviewed in Equation (2View Equation). The corresponding total gravitational flux reads
+ sum oo { } L(U ) = -G--- --(l +-1)(l +-2)-U(1)(U )U(1)(U ) + --------4l(l-+-2)--------V(1)(U)V(1)(U ) . c2l+1 (l- 1)ll!(2l + 1)!! L L c2(l- 1)(l + 1)!(2l + 1)!! L L l=2 (60)
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 Equations (2View Equation, 3View Equation, 4View Equation) reduces to
( ) (2) 1 Uij(U ) = Q ij (U ) + O -2 , (61) c
where Qij is the Newtonian quadrupole given by Equation (3View Equation). 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 UL, VL in terms of the source moments IL,JL, ...,ZL. 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 5.


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