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
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