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

Post-Newtonian wave generation 351satisfies £h µνpart = Å( µν ). The second term is a retarded solution of thesource-free homogeneous wave equation, i.e. £h µνhom = 0. We denote ∂ L =∂ i1 ...∂ il where L = i 1 ...i l is a multi-index composed of l indices; the lsummations over the indices i 1 ...i l are not indicated in (17.63). The ‘multipolemoments’parametrizing this homogeneous solution are given explicitly by (withu = t − r/c)∫ µνL(u) = finite part∫ 1d 3 x ˆx L dz δ l (z)τ µν (x, u + z|x|/c), (17.64)−1where the integrand contains the post-Newtonian expansion of the pseudostress–energy tensor τ µν , whose structure reads like (17.62). In (17.64), we denote thesymmetric-trace-free (STF) projection of the product of l vectors x i with a hat,so that ˆx L = STF(x L ), with x L = x i 1...x i land L = i 1 ...i l ; for instance,ˆx ij = x i x j − 1 3 δ ij x 2 . The function δ l (z) is given byand satisfies the propertiesδ l (z) =(2l + 1)!!2 l+1 (1 − z 2 ) l , (17.65)l!∫ 1−1dzδ l (z) = 1;lim δ l(z) = δ(z) (17.66)l→+∞(where δ(z) is the Dirac measure). Both terms in (17.63) involve an operationof taking a finite part. This finite part can be defined precisely by means of ananalytic continuation (see [14] for details), but it is in fact basically equivalent totaking the finite part of a divergent integral in the sense of Hadamard [18]. Notice,in particular, that the finite part in the expression of the multipole moments (17.64)deals with the behaviour of the integral at infinity: r →∞(without the finite partthe integral would be divergent because of the factor x L = r l n L in the integrandand the fact that the pseudotensor τ µν is not of compact support).The result (17.63)–(17.64) permits us to define a very convenient notion ofthe source multipole moments (by opposition to the radiative moments definedbelow). Quite naturally, the source moments are constructed from the tencomponents of the tensorial function µνL(u). Among these components fourcan be eliminated using the harmonic gauge condition (17.55), so in the endwe find only six independent source multipole moments. Furthermore, it can beshown that by changing the harmonic gauge in the exterior zone one can furtherreduce the number of independent moments to only two. Here we shall report theresult for the ‘main’ multipole moments of the source, which are the mass-typemoment I L and current-type J L (the other moments play a small role starting onlyat highorder in the post-Newtonian expansion). We have [14]