Physical Foundations of Cosmology

294 Gravitational instability in General Relativity

span the two-dimensional space of the physical perturbations, are

≡φ−

1

a

[

a

(

B−E′

)]′

,≡ψ+

a′

a

(

B−E′

)

. (7.19)

It is easy to see that they do not change under the coordinate transformations

and if and vanish in one particular coordinate system, they will be zero

in any coordinate system. This means we can immediately distinguish physical

inhomogeneities from fictitious perturbations; if both andare equal to zero,

then the metric perturbations (if they are present) are fictitious and can be removed

by a change of coordinates.

Of course one can construct an infinite number of gauge-invariant variables,

since any combination of and will also be gauge-invariant. Our choice of

these variables is justified only by reason of convenience. As with the electric and

magnetic fields in electrodynamics, the potentials andare the simplest possible

combinations and satisfy simple equations of motion (see the following section).

Problem 7.2Using the results in Problem 7.1, verify that

δε=δε−ε′ 0

(

B−E′

)

(7.20)

is the gauge-invariant variable characterizing the energy density perturbations.

Taking into account that the 4-velocity of a fluid in a homogeneous universe is
( 0 )uα=(a, 0 , 0 , 0 ),show that

δu 0 =δu 0 −

[

a

(

B−E′

)]′

, δui=δui−a

(

B−E′

)

,i (7.21)

are the gauge-invariant variables for thecovariantcomponents of the velocity per-

turbationsδuα.

Vector perturbationsFor vector perturbations the metric is

ds^2 =a^2

[

dη^2 + 2 Sidxidη−

(

δij−Fi,j−Fj,i

)

dxidxj

]

, (7.22)

and the variablesSiandFitransform as

Si→S ̃i=Si+ξ⊥′i, Fi→F ̃i=Fi+ξ⊥i. (7.23)

It is obvious that

Vi=Si−Fi′ (7.24)

is gauge-invariant. Only two of the four independent functionsSi,Ficharacterize

physical perturbations; the other two reflect the coordinate freedom. The vari-

ables (7.24) span the two-dimensional space of physical perturbations and describe