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

to transform the coordinates according toT − R c = t − r c − 2GMc 3Post-Newtonian wave generation 353( ) rln , (17.70)r 0where M denotes the ADM mass of the source and r 0 is a gauge constant. Inradiative coordinates it is easy to decompose the 1/R term of the metric intomultipoles and to define in that way the radiative multipole moments U L (masstype;where L = i 1 ...i l with l ≥ 2) and V L (current-type; with l ≥ 2).(Actually, it is often simpler to bypass the need for transforming the coordinatesfrom harmonic to radiative by considering directly the TT projection of the spatialcomponents of the harmonic metric at infinity.) The formula for the definition ofthe radiative moments ish TTij= − 4G+∞∑c 2 R È ijab(N)l=2−{1c l N L−2 U abL−2 (T − R/c)l!}2lc(l + 1) N cL−2ε cd(a V b)dL−2 (T − R/c)( ) 1+ OR 2(17.71)where N is the vector N i = N i = X i /R (for instance N L−2 = N i1 ...N il−2 ), andÈ ijab denotes the TT projectorÈ ijab = (δ ia − N i N a )(δ jb − N j N b ) − 1 2 (δ ij − N i N j )(δ ab − N a N b ). (17.72)In the limit of linearized gravity the radiative multipole moments U L , V L agreewith the lth time derivatives of the source moments I L , J L . Let us give, withoutproof, the result for the expression of the radiative mass-quadrupole moment U ijincluding relativistic corrections up to the 3PN or 1/c 6 order inclusively [12, 13].The calculation involves implementing explicitly a post-Minkowskian algorithmdefined in [8] for the computation of the nonlinearities due to the first term of(17.63). We find (U ≡ T − R/c)U ij (U) = M (2)ij(U) + 2 GM+ G c 5 {− 2 7∫ +∞0c 3 ∫ +∞0[ ( )dv M (4)cvij(U − v) ln2r 0+ 11 ]12dv [M (3)aa ](U − v) − 2 7 M(3) aa (U)− 5 7 M(4) aa (U) + 1 7 M(5) aa(U) + 1 3 ε aba J b(U)( GM+ 2[ ( cv× ln 2+ Oc 3 ) 2 ∫ +∞0dv M (5)ij(U − v))+ 57 ( ) ]cv2r 0 70 ln 124 627+2r 0 44 100( ) 1c 7 . (17.73)}