298 Gravitational instability in General Relativity
The components ofδT
α
βcan also be decomposed into scalar, vector and tensor
pieces; each piece contributes only to the evolution of the corresponding perturba-
tion.
Scalar perturbationsThe left hand side in (7.37) is gauge-invariant and depends
only on the metric perturbations. Therefore it can be expressed entirely in terms of
the potentials and.The direct calculation ofδG
α
βfor the metric (7.17) gives
the equations:
− 3 H(
′+H
)
= 4 πGa^2 δT
0
0 , (7.38)(
′+H)
,i=^4 πGa(^2) δT^0
i, (7.39)
[
′′+H( 2 + )′+
(
2 H′+H^2
)
+^12 ( −)
]
δij
−^12 ( −),ij=− 4 πGa^2 δT
i
j.(7.40)
We have to stress that these equations can be derived without imposing any gauge
conditions and they are valid in an arbitrary coordinate system. To obtain the explicit
form of the equations for the metric perturbations in a particular coordinate system,
we simply have to express andin (7.38)–(7.40) through these perturbations.
For example, in the synchronous coordinate system, we would use the expressions
in (7.28).
Problem 7.5Write down the equations forψsandEsin the synchronous coordinate
system. (HintDo not forget thatEsenters the definition ofδT
α
β.)
The equations for the metric perturbations in the conformal-Newtonian coordi-
nate system obviously coincide with (7.38)–(7.40). Therefore calculations in these
coordinates are identical to calculations in terms of the gauge-invariant potentials,
with the advantage that we need not carryBandEthrough the intermediate for-
mulae.
Problem 7.6Derive (7.38)–(7.40). (Hint: The direct calculation ofδG
α
βin terms
of the gauge-invariant potentials is rather tedious. However, it can be significantly
simplified if we take into account that these potentials coincide with the metric
perturbations in the conformal-Newtonian coordinate system. Therefore, calculate
the Einstein tensor for the metric (7.26) and then replaceφl,ψlwith and
respectively. It is convenient to calculateδGαβin two steps: (a) seta=1 in (7.26)
and find the Einstein tensor in this case, (b) make a conformal transformation to an
arbitrarya(t)and calculateδGαβusing formulae (5.110), (5.111), whereF=a^2 .)