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

64 Mass- and current-quadrupole radiationwave emitted up the axis is twice that emitted in the plane. In this way we cancompletely determine the radiation pattern of a binary system.Notice that, when viewed at an arbitrary angle to the axis, the radiationwill be elliptically polarized, and the degree of ellipticity will directly measurethe inclination of the orbital plane to the line of sight. This is a very specialkind of information, which one cannot normally obtain from electromagneticobservations of binaries. It illustrates the complementarity of the two kinds ofobserving.6.3 Application of the TT gauge to the current-quadrupolefieldNow we turn to the problem of placing next-order terms of the wave field, thecurrent quadrupole and mass octupole, into the TT gauge. Our interest here is tounderstand current-quadrupole radiation in the same physical way as we have justdone for mass-quadrupole radiation. So we shall put the field into the TT gaugeand then see how to separate the current-quadrupole part from the mass-octupole,which we will discard from the present discussion.6.3.1 The field at third order in slow-motionThe next order terms in the non-TT metric bear a simple relationship to the massquadrupoleterms (see equations (6.9)–(6.11)). In each of the metric components,just replace S jk by Ṡ jkl n l to go from one order to the next.This means that we can just skip to the end of the application of the gaugetransformations in equations (6.17) and (6.18) and write the next order of the finalfield, only using S again, not M:¯h TTij = 4 [⊥ ik ⊥ jl Ṡ lkm n m + 1 ]r2 ⊥ij (Ṡ klm n k n l n m − Ṡ k kln l ) , (6.29)or more compactly¯h TTij = 4 r(⊥ ik ⊥ jl ˙˜S klm n m + 1 2 ⊥ij ˙˜S klm n l n k n m ). (6.30)The tilde on S represents a trace-free operation on the first two indices.˙˜S klm = Ṡ lkm − 1 3 δ kl Ṡi im.These are the indices that come from the indices of T jk , so the tensor is symmetricon these. By analogy with the quadrupole calculation, we can also define the TTpart of S ijk by doing the TT projection on the first two indices,S TTijm =⊥k i ⊥ l j S klm − 1 2 ⊥ ij⊥ kl S klm . (6.31)