It is often useful to describe spatial geometry not by the spatial metric but by a triad which
defines three vector fields which are orthogonal to each other and normalized in each point. This
yields all information about spatial geometry, and indeed the inverse metric is obtained from
the triad by
where we sum over the index
counting the triad vector fields.
There are differences, however, between metric and triad formulations. First, the set of triad
vectors can be rotated without changing the metric, which implies an additional gauge freedom
with group SO(3) acting on the index
. Invariance of the theory under those rotations is
then guaranteed by a Gauss constraint in addition to the diffeomorphism and Hamiltonian
constraints.
The second difference will turn out to be more important later on: We can not only rotate the triad
vectors but also reflect them, i.e., change the orientation of the triad given by . This does not
change the metric either, and so could be included in the gauge group as O(3). However, reflections are not
connected to the unit element of O(3) and thus are not generated by a constraint. It then has to be seen
whether or not the theory allows to impose invariance under reflections, i.e., if its solutions are reflection
symmetric. This is not usually an issue in the classical theory since positive and negative orientations on the
space of triads are separated by degenerate configurations where the determinant of the metric vanishes.
Points on the boundary are usually singularities where the classical evolution breaks down
such that we will never connect between both sides. However, since there are expectations that
quantum gravity may resolve classical singularities, which indeed are confirmed in loop quantum
cosmology, we will have to keep this issue in mind and not restrict to only one orientation from the
outset.
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