Cosmological Perturbation Theory, 26.4.2011 version

3 THE PERTURBED UNIVERSE 43 The Perturbed UniverseWe write the metric of the perturbed (around FRW(0)) universe asg µν = ḡ µν + δg µν = a 2 (η µν + h µν ), (3.1)where h µν , as well as h µν,ρ and h µν,ρσ are assumed small. Since we are doing first-order perturbationtheory, we shall drop from all equations all terms with are of order O(h 2 ) or higher, andjust write “=” to signify equality to first order in h µν . Here the perturbation h µν is not a tensorin the perturbed universe, neither is η µν , but we defineh µ ν ≡ η µρ h ρν , h µν ≡ η µρ η νσ h ρσ . (3.2)One easily finds (exercise), that the inverse metric of the perturbed spacetime isg µν = a −2 (η µν − h µν ) (3.3)(to first order).We shall now give different names for the time and space components of the perturbedmetric, defining[ ]−2A −Bi[h µν ] ≡(3.4)−B i −2Dδ ij + 2E ijwhereD = − 1 6 hi i (3.5)carries the trace of the spatial metric perturbation h ij , and E ij is traceless,δ ij E ij = 0. (3.6)Since indices on h µν are raised and lowered with η µν , we immediately have[ ][h µν −2A +Bi] ≡+B i −2Dδ ij + 2E ij(3.7)On B i and E ij we do not raise/lower indices (or if we do, it is just the same thing, B i = δ ij B j =B i ).The line element is thusds 2 = a 2 (η) { −(1 + 2A)dη 2 − 2B i dηdx i + [(1 − 2D)δ ij + 2E ij ] dx i dx j} . (3.8)The function A(η,x i ) is called the lapse function, and B i (η,x i ) the shift vector.4 Gauge TransformationsThe association between points in the background spacetime and the perturbed spacetime isvia the coordinate system {x α }. As we noted earlier, for a given coordinate system in thebackground, there are many possible coordinate systems in the perturbed spacetime, all closeto each other, that we could use. In GR perturbation theory, a gauge transformation means acoordinate transformation between such coordinate systems in the perturbed spacetime. (It maybe helpful to temporarily forget at this point what you have learned about gauge transformationsin other field theories, e.g. electrodynamics, so that you can learn the properties of this concepthere with a fresh mind, without preconceptions.)