Physical Foundations of Cosmology

290 Gravitational instability in General Relativity

x

t

t= const, ε =const

t∼=const, ε∼=const

Fig. 7.1.

the hypersurfaces of constant energy: in this caseδε=0 in spite of the presence
of therealinhomogeneities.
To resolve real and fictitious perturbation modes in General Relativity, it is
necessary to have a full set of variables. To be precise we need both the matter field
perturbations and the metric perturbations.
In this chapter we introduce gauge-invariant variables, which do not depend on
the particular choice of coordinates and have a clear physical interpretation. We
apply the formalism developed to study the behavior of relativistic perturbations in
a few interesting cases. To simplify the formulae we consider only a spatially flat
universe. The generalization of the results obtained to nonflat universes is largely
straightforward.

7.1 Perturbations and gauge-invariant variables

Inhomogeneities in the matter distribution induce metric perturbations which can
be decomposed into irreducible pieces. In the linear approximation different types
of perturbations evolve independently and therefore can be analyzed separately. In
this section we first classify metric perturbations, then determine how they trans-
form under general coordinate (gauge) transformations and finally construct the