Matter perturbations such as and
obey the following transformation rule
One can construct a number of gauge-invariant quantities unchanged under the transformation (6.20):
We note that the tensor perturbation is invariant under the gauge transformation [412].
We can choose specific gauge conditions to fix the gauge degree of freedom. After fixing a gauge, two
scalar variables and
are determined accordingly. The Longitudinal gauge corresponds to the gauge
choice
and
, under which
and
. In this gauge one has
and
, so that the line element (without vector and tensor perturbations) is given by
The uniform-field gauge corresponds to which fixes
. The spatial threading
is
fixed by choosing either
or
(up to an integration constant in the former case). For this
gauge choice one has
. Since the spatial curvature
on the constant-time hypersurface is
related to
via the relation
, the quantity
is often called the curvature
perturbation on the uniform-field hypersurface. We can also choose the gauge condition
or
.
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