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.
© Max Planck Society and the author(s)