Physical Foundations of Cosmology

292 Gravitational instability in General Relativity

components and there are four constraints). Thus we have ten functions altogether.
This number coincides with the number of independent components ofδgαβ.
Scalar perturbationsare characterized by the four scalar functionsφ,ψ,B,
E.They are induced by energy density inhomogeneities. These perturbations are
most important because they exhibit gravitational instability and may lead to the
formation of structure in the universe.
Vector perturbationsare described by the two vectorsSiandFiand are related to
the rotational motions of the fluid. As in Newtonian theory, they decay very quickly
and are not very interesting from the point of view of cosmology.
Tensor perturbations hijhave no analog in Newtonian theory. They describe
gravitational waves, which are the degrees of freedom of the gravitational field
itself. In the linear approximation the gravitational waves do not induce any per-
turbations in the perfect fluid.
Scalar, vector and tensor perturbations are decoupled and thus can be studied
separately.

7.1.2 Gauge transformations and gauge-invariant variables

Let us consider the coordinate transformation

xα→x ̃α=xα+ξα, (7.8)

whereξαare infinitesimally small functions of space and time. Ata given point
of the spacetime manifoldthe metric tensor in the coordinate systemx ̃ can be
calculated using the usual transformation law

g ̃αβ(x ̃ρ)=

∂xγ

∂x ̃α

∂xδ

∂x ̃β

gγδ(xρ)≈(^0 )gaβ(xρ)+δgαβ−(^0 )gαδξ,βδ −(^0 )gγβξ,αγ, (7.9)

where we have kept only the terms linear inδgandξ.In the new coordinatesx ̃the
metric can also be split into background and perturbation parts,

g ̃αβ(x ̃ρ)=(^0 )gaβ(x ̃ρ)+δg ̃aβ, (7.10)

where(^0 )gaβis the Friedmann metric (7.3), which now depends onx ̃.Comparing
the expressions in (7.9) and (7.10) and taking into account that

( 0 )gaβ(xρ)≈( 0 )gaβ(x ̃ρ)−( 0 )gaβ,γξγ, (7.11)

we infer the following gauge transformation law:

δgαβ→δg ̃aβ=δgαβ−(^0 )gaβ,γξγ−(^0 )gγβξ,αγ−(^0 )gαδξ,βδ. (7.12)