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

60 Mass- and current-quadrupole radiationThese can be applied recursively to show, for example, two further very usefulrelationsd 2 M jkdt 2 = 2S jk d 3 M jkl,dt 3 = 6Ṡ ( jkl) (6.8)where the round brackets on indices indicate full symmetrization.Using these relations and notations it is not hard to show thath 00 (t, x i ) = 4 r M + 4 r P j n j + 4 r S jk (t ′ )n j n k + 4 r Ṡ jkl (t ′ )n j n k n l +···(6.9)h 0 j (t, x i ) = 4 r P j + 4 r S jk (t ′ )n k + 4 r Ṡ jkl (t ′ )n k n l +··· (6.10)h jk (t, x i ) = 4 r S jk (t ′ ) + 4 r Ṡ jkl (t ′ )n l +···. (6.11)In these three formulae there are different orders of time-derivatives, but in factthey are evaluated to the same final order in the slow-motion approximation. Onecan see that from the gauge condition h aβ ,β = 0, which relates time-derivativesof some components to space-derivatives of others.In these expressions, one must remember that the moments are evaluatedat the retarded time t ′ = t − r (except for those moments that are constant intime), and they are multiplied by components of the unit vector to the field pointn j = x j /r.6.2 Application of the TT gauge to the mass quadrupole fieldIn the expression for the amplitude that we derived so far, the final terms arethose that represent the current-quadrupole and mass-octupole radiation. Theterms before them represent the static parts of the field and the mass-quadrupoleradiation. In this section we treat just these terms, placing them into the TT gauge.This will be simpler than treating it all at once, and the procedure for the nextterms will be a straightforward generalization.6.2.1 The TT gauge transformationsWe are already in Lorentz gauge, and this can be checked by taking derivativesof the expressions for the field that we have derived above. However, we aremanifestly not in the TT gauge. Making a gauge transformation consists ofchoosing a vector field ξ α and modifying the metric byThe corresponding expression for the potential h αβ ish αβ → h αβ − ξ α,β − ξ β,α . (6.12)h αβ → h αβ + ξ α,β + ξ α,β − η αβ ξ µ ,µ. (6.13)