Physical Foundations of Cosmology

7 Gravitational instability in General Relativity

The Newtonian analysis of gravitational instability has limitations. It clearly fails
for perturbations on scales larger than the Hubble radius. In the case of a relativistic
fluid we have to use General Relativity for both short-wavelength and long-wave-
length perturbations. This theory gives us a unified description for any matter on
all scales. Unfortunately, the physical interpretation of the results obtained is less
transparent in General Relativity than in Newtonian theory. The main problem
is the freedom in the choice of coordinates used to describe the perturbations. In
contrast to the homogeneous and isotropic universe, where the preferable coordinate
system is fixed by the symmetry properties of the background, there are no obvious
preferable coordinates for analyzing perturbations. The freedom in the coordinate
choice, or gauge freedom, leads to the appearance of fictitious perturbation modes.
These fictitious modes do not describe any real inhomogeneities, but reflect only
the properties of the coordinate system used.
To demonstrate this point let us consider anundisturbedhomogeneous isotropic
universe, whereε(x,t)=ε(t). In General Relativity any coordinate system is al-
lowed, and we can in principle decide to use a “new” time coordinate ̃t,related to
the “old” timetviat ̃=t+δt(x,t). Then the energy density ̃ε

(

t ̃,x

)

≡ε

(

t

(

̃t,x

))

on the hypersurfacet ̃=const depends, in general, on the spatial coordinatesx
(Figure 7.1). Assuming thatδtt,we have

ε(t)=ε

(

̃t−δt(x,t)

)

ε

(

t ̃

)

−

∂ε

∂t

δt≡ε

(

t ̃

)

+δε

(

x, ̃t

)

. (7.1)

The first term on the right hand side must be interpreted as the background en-
ergy density in the new coordinate system, while the second describes a linear
perturbation. This perturbation is nonphysical and entirely due to the choice of the
new “disturbed” time. Thus we can “produce”fictitiousperturbations simply by
perturbing the coordinates. Moreover, we can “remove” arealperturbation in the
energy density by choosing the hypersurfaces of constant time to be the same as

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