(ed.). Gravitational waves (IOP, 2001)(422s).

352 Post-Newtonian computation of binary inspiral waveforms∫ ∫ 1{I L (u) = finite part d 3 4(2l + 1)x dz δ l ˆx L −−1c 2 (l + 1)(2l + 3) δ l+1 ˆx iL ∂ t i}2(2l + 1)+c 4 (l + 1)(l + 2)(2l + 5) δ l+2 ˆx ijL ∂t 2 ij , (17.67)∫ ∫ 1J L (u) = finite part d 3 x dz ε aba b−1}2l + 1−c 2 (l + 2)(2l + 3) δ l+1 ˆx L−1>ac ∂ t bc . (17.68)Here the integrand is evaluated at the instant u + z| x|/c, ε abc is the Levi-Civitasymbol, 〈L〉 is the STF projection, and we employ the notation = τ 00 + τ iic 2 ; i = τ 0ic ; ij = τ ij (17.69)(with τ ii = δ ij τ ij ). The multipole moments I L , J L are valid formally up to anypost-Newtonian order, and constitute a generalization in the nonlinear theory ofthe usual mass and current Newtonian moments (see, [14] for details). It can bechecked that, when considered at the 1PN order, these moments agree with thedifferent expressions obtained in [9] (case of mass moments) and in [10] (currentmoments).17.5.3 Radiative momentsIn linearized theory, where we can neglect the gravitational source term µν in(17.57), as well as the first term in (17.63), the source multipole moments coincidewith the so-called radiative multipole moments, defined as the coefficients of themultipole expansion of the 1/r term in the distance to the source at retarded timest − r/c = constant. However, in full nonlinear theory, the first term in (17.63)will bring another contribution to the 1/r term at future null infinity. Therefore,the source multipole moments are not the ‘measured’ ones at infinity, and sothey must be related to the real observables of the field at infinity which areconstituted by the radiative moments. It has been known for a long time thatthe harmonic coordinates do not belong to the class of Bondi coordinate systemsat infinity, because the expansion of the harmonic metric when r → ∞ witht − r/c = constant involves, in addition to the normal powers of 1/r, somepowers of the logarithm of r. Let us change the coordinates from harmonic tosome Bondi-type or ‘radiative’ coordinates ( X, T ) such that the metric admits apower-like expansion without logarithms when R →∞with T − R/c = constantand R =|X| (it can be shown that the condition to be satisfied by the radiativecoordinate system is that the retarded time T − R/c becomes asymptotically nullat infinity). For the purpose of deriving the formula (17.73) below it is sufficient