Cosmological Perturbation Theory, 26.4.2011 version

15 OTHER GAUGES 31Figure 3: Slicing and threading the perturbed spacetime.In the gauge transformation,˜x α = x α + ξ α , (15.2)or˜η = η + ξ 0x i = x i + ξ i , (15.3)ξ 0 changes the slicing, and ξ i changes the threading. From the 4-scalar transformation law,δ˜s = δs − ¯s ′ ξ 0 , we see that perturbations in 4-scalars, e.g., δρ and δp depend only on the slicing.15.2 Comoving GaugeWe say that the slicing is comoving, if the time slices are orthogonal to the fluid 4-velocity.For scalar perturbations such a slicing always exists. This condition turns out to be equivalentto the condition that the fluid velocity perturbation v i equals the shift vector B i . For scalarperturbations,Comoving slicing ⇔ v = B . (15.4)From the gauge transformation equations (7.7) and (9.29),ṽ = v + ξ ′˜B = B + ξ ′ + ξ 0 (15.5)we see that we get to comoving slicing by ξ 0 = v − B.We say that the threading is comoving, if the threads are world lines of comoving (with thefluid) observers, i.e., the velocity perturbation vanishes, v i = 0. For scalar perturbations,Comoving threading ⇔ v = 0. (15.6)