  The role of the Taub time gauge in cosmology is linked to the use of the
densitized lapse function instead of the lapse function in the variational
principle approach to the Einstein equations. The spatial metric variational
equations then become the Ricci evolution equations, which are then
supplemented by the Einstein constraints which result from the variation with
respect to the densitized lapse and the usual shift vector field. In those
spatially homogeneous cases where the least disconnect occurs between the
general theory and the restricted symmetry scenario, the recent adjustment of
the conformal approach to solving the initial value problem resulting from
densitized lapse considerations is seen to be inherent in the use of
symmetry-adapted metric variables. The minimal distortion shift vector field is
a natural vector potential for the new York thin sandwich initial data approach
to the constraints, which in this case corresponds to the diagonal spatial
metric gauge. For generic spacetimes, the new approach suggests defining a new
minimal distortion shift gauge which agrees with the old gauge in the Taub time
gauge, but which also makes its defining differential equation agree with the
vector potential equation for solving the supermomentum constraint in any time
gauge.
