Cosmological Perturbation Theory, 26.4.2011 version

5 SEPARATION INTO SCALAR, VECTOR, AND TENSOR PERTURBATIONS 9Applying the gauge transformation law (4.24) now separately to the different metric perturbationcomponents, we get first˜δg 00 ≡ −2a 2 Ã= δg 00 + a 2 (−ξ ρ ,0 η ρ0 − ξ σ ,0η 0σ − 2 a′a η 00ξ 0 )from which we obtain the gauge transformation law= −2a 2 A + a 2 (+ξ 0 ,0 + ξ 0 ,0 + 2 a′a ξ0 ), (4.26)Similarly, from δg 0i we obtainà = A − ξ 0 ,0 − a′a ξ0 . (4.27)˜B i = B i + ξ i ,0 − ξ 0 ,i , (4.28)and from δg ij ,− ˜Dδ ij + Ẽij = −Dδ ij + E ij − 1 2 (ξi ,j + ξ j ,i ) − a′a ξ0 δ ij . (4.29)The trace of 1 2 (ξi ,j + ξj ,i ) is ξk ,k, so we can write12 (ξi ,j + ξ j ,i ) = 1 3 δ ijξ k ,k + 1 2 (ξi ,j + ξ j ,i ) − 1 3 δ ijξ k ,k , (4.30)where the last two terms are the traceless part, and we can separate Eq. (4.29) into˜D = D + 1 3 ξk ,k + a′a ξ0Ẽ ij = E ij − 1 2 (ξi ,j + ξ j ,i ) + 1 3 δ ijξ k ,k . (4.31)5 Separation into Scalar, Vector, and Tensor PerturbationsIn GR perturbation theory there are two kinds of coordinate transformations of interest. Oneis the gauge transformation just discussed, where the coordinates of the background are keptfixed, but the coordinates in the perturbed spacetime are changed, changing the correspondencebetween the points in the background and the perturbed spacetime.The other kind is one where we keep the gauge, i.e, the correspondence between the backgroundand perturbed spacetime points, fixed, but do a coordinate transformation in the backgroundspacetime. This then induces a corresponding coordinate transformation in the perturbedspacetime. Our background coordinate system was chosen to respect the symmetries ofthe background, and we do not want to lose this property. In cosmological perturbation theorywe have chosen the background coordinates to respect its homogeneity property, which gives usa unique slicing of the spacetime into homogeneous t = const. spacelike slices. Thus we do notwant to change this slicing. This leaves us:1. homogeneous transformations of the time coordinate, i.e., reparameterizations of time, ofwhich we already had an example, when we switched from cosmic time t to conformal timeη,