### 4.1 Isotropy

Isotropy reduces the phase space of general relativity to be 2-dimensional since, up to SU(2)-gauge
freedom, there is only one independent component in an isotropic connection and triad, respectively, 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
with the matter Hamiltonian and the gravitational constant , and the parameter
taking 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 understand these different roles of metric components also 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.