### 4.1 Isotropy

Isotropy reduces the phase space of general relativity to two-dimensions since, up to SU(2)-gauge
freedom, there is only one independent component in an isotropic connection and triad, which is not already
determined by the symmetry. This is analogous to metric variables, where the scale factor is the only
free component in the spatial part of an isotropic metric
The lapse function does not play a dynamical role and correspondingly does not appear in the
Friedmann equation
where is the matter Hamiltonian, is the gravitational constant, and the parameter
takes the discrete values zero or depending on the symmetry group or intrinsic spatial
curvature.
Indeed, can simply be absorbed into the time coordinate by defining proper time through
. This is not possible for the scale factor since it depends on time but multiplies space
differentials in the line element. The scale factor can only be rescaled by an arbitrary constant, which can
be normalized, at least in the closed model where .

One can also understand these different roles of metric components from a Hamiltonian analysis of the
Einstein–Hilbert action

specialized to isotropic metrics (16), whose Ricci scalar is

The action then becomes

(with the spatial coordinate volume ) after integrating by parts, from which one derives the
momenta

illustrating the different roles of and . Since must vanish, is not a degree of freedom but
a Lagrange multiplier. It appears in the canonical action only as a factor
of

such that variation with respect to forces , the Hamiltonian constraint, to be zero. In the presence
of matter, also contains the matter Hamiltonian, and its vanishing is equivalent to the Friedmann
equation.