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

Application of the TT gauge to the mass quadrupole field 61For the different components this implies changesδh 00 = ξ 0,0 + ξ j , j (6.14)δh 0 j = ξ 0, j + ξ j,0 (6.15)δh jk = ξ j,k + ξ k, j − δ jk ξ µ ,µ (6.16)where δ jk is the Kronecker delta (unit matrix). In practice, when takingderivatives, the algebra is vastly simplified by the fact that we are keeping onlyr −1 terms in the potentials. This means that spatial derivatives do not act on1/r but only on t ′ = t − r. It follows that ∂t ′ /∂x j =−n j , and ∂h(t ′ )/∂x j =−ḣ(t ′ )n j .It is not hard to show that the following vector field puts the metric into theTT gauge to the order we are working:ξ 0 = 1 r Pk k + 1 r P jk n j n k + 1 r Sl lkn k + 1 r Sijk n i n j n k , (6.17)ξ i = 4 r Mi + 4 r Pij n j − 1 r Pk kn i − 1 r P jk n j n k n i + 4 r Sijk n j n k− 1 r Sl lkn k n i − 1 r S jlk n j n l n k n i . (6.18)6.2.2 Quadrupole field in the TT gaugeThe result of applying this gauge transformation to the original amplitudes ish TT00 = 4M r , (6.19)h TT0i = 0, (6.20)h TTij = 4 [⊥ ik ⊥ jl S lk + 1 ]r2 ⊥ij (S kl n k n l − S k k) . (6.21)Remember that here we are not including Ṡ jkl , because it is a third-order effect.The notation ⊥ ik represents the projection operator perpendicular to thedirection n i to the field point.⊥ jk = δ jk − n j n k . (6.22)It can be verified that this tensor is transverse to the direction n iprojection, in the sense that it projects to itselfand is a⊥ jk n k = 0, ⊥ jk ⊥ k l = ⊥ jl . (6.23)The spherical component of the field is not totally eliminated in this gaugetransformation: the time–time component of the metric must contain the constantNewtonian field of the source. (In fact we have succeeded in eliminating the