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

350 Post-Newtonian computation of binary inspiral waveformsIn addition, r∂ λ h µν should be bounded in the same limit. Actually we often adopt,for technical reasons, the more restrictive condition that the field is stationarybefore some finite instant −Ì in the past (refer to [8] for details). With theno-incoming radiation condition (17.58) or (17.59) we transform the differentialEinstein equation (17.56) into the equivalent integro-differential systemh µν = 16πG £ −1c 4 R τ µν , (17.60)where £ −1Rdenotes the standard retarded inverse d’Alembertian given by(£ −11Rτ)(x, t) =−4π17.5.2 Source moments∫d 3 x ′|x − x ′ | τ(x′ , t −|x − x ′ |/c). (17.61)In this section we shall solve the field equations (17.55) and (17.56) in the exteriorof the isolated source by means of a multipole expansion, parametrized by someappropriate source multipole moments. The particularity of the moments we shallobtain, is that they are defined from the formal post-Newtonian expansion of thepseudotensor τ µν , supposing that the latter expansion can be iterated to any order.Therefore, these source multipole moments are physically valid only in the caseof a slowly-moving source (slow internal velocities; weak stresses). The generalstructure of the post-Newtonian expansion involves besides the usual powers of1/c some arbitrary powers of the logarithm of c, sayτ µν (t, x, c) = ∑ p,q(ln c) qc p τ pq µν (t, x), (17.62)where the overbar denotes the formal post-Newtonian expansion, and whereτ pq µν are the functional coefficients of the expansion (p, q are integers, includingzero). Now, the general multipole expansion of the metric field h µν , denotedby Å(h µν ), is found by requiring that when re-developed into the near-zone,i.e. in the limit where r/c → 0 (this is equivalent with the formal re-expansionwhen c →∞), it matches with the multipole expansion of the post-Newtonianexpansion h µν (whose structure is similar to (17.62)) in the sense of themathematical technics of matched asymptotic expansions. We find [11, 14] thatthe multipole expansion Å(h µν ) satisfying the matching is uniquely determined,and is composed of the sum of two terms,(−) ll!l=0(17.63)The first term, in which £ −1Ris the flat retarded operator (17.61), is a particularsolution of the Einstein field equations in vacuum (outside the source), i.e. itÅ(h µν ) = finite part£ −1R [Å(µν )] − 4Gc 4 +∞ ∑∂ L{ 1r µνL (t − r/c) }.