Physical Foundations of Cosmology

296 Gravitational instability in General Relativity

class of synchronous coordinate systems. From (7.18), it follows that if the condi-
tionsφs=0 andBs=0 are satisfied in some coordinate systemxα≡(η,x),then
they will also be satisfied in another coordinate systemx ̃α,related toxαby

η ̃=η+

C 1

a

, x ̃i=xi+C 1 ,i

∫

dη

a

+C 2 ,i, (7.27)

where C 1 ≡C 1

(

xj

)

and C 2 ≡C 2

(

xj

)

are arbitrary functions of the spatial
coordinates.This residual coordinate freedom leads to the appearance of unphysi-
cal gauge modes, which render the interpretation of the results difficult, especially
on scales larger than the Hubble radius.
If we know a solution for perturbations in terms of the gauge-invariant variables
or, equivalently, in the conformal-Newtonian coordinate system, then the behavior
of perturbations in the synchronous coordinate system can easily be determined
without needing to solve the Einstein equations again. Using the definitions in
(7.19) we have

=

1

a

[

aEs′

]′

,=ψs−

a′

a

E′s. (7.28)

These two equations can easily be resolved to expressψsandEsin terms of the
gauge-invariant potentials:

Es=

∫

1

a

(∫η

a dη ̃

)

dη, ψs=+

a′

a^2

∫

a dη. (7.29)

Similarly, from (7.20) it follows that the energy density perturbations are

δεs=δε−

ε 0 ′

a

∫

a dη. (7.30)

The constants of integration arising in these formulae correspond to unphysical,
fictitious modes.

Problem 7.3Impose thecomovinggauge conditions

φ= 0 ,δui
=−

1

a^2

δu
i+

1

a

B,i= 0 , (7.31)

whereδui
are thecontravariantspatial components of the potential 4-velocity, and
find the metric perturbations in the comoving coordinate system in terms of the
gauge-invariant variables. Do these conditions fix the coordinates uniquely?