### 3.1 Geometry

General relativity in its canonical formulation [6] describes the geometry of spacetime in terms of
fields on spatial slices. Geometry on such a spatial slice is encoded in the spatial metric
, which presents the configuration variables. Canonical momenta are given in terms of the
extrinsic curvature , which is the derivative of the spatial metric with respect to changes
in the spatial slice. These fields are not arbitrary, since they are obtained from a solution of
Einstein’s equations by choosing a time coordinate defining the spatial slices, and spacetime
geometry is generally covariant. In the canonical formalism this is expressed by the presence of
constraints on the fields: the diffeomorphism constraint and the Hamiltonian constraint. The
diffeomorphism constraint generates deformations of a spatial slice or coordinate changes, and,
when it is satisfied, spatial geometry does not depend on the space coordinates chosen. General
covariance of spacetime geometry for the time coordinate is then completed by imposing the
Hamiltonian constraint. Furthermore, this constraint is important for the dynamics of the theory;
since there is no absolute time, there is no Hamiltonian generating evolution, but only the
Hamiltonian constraint. When it is satisfied, it encodes correlations between the physical fields of
gravity and matter, such that evolution in this framework is relational. The reproduction of a
spacetime metric in a coordinate-dependent way then requires one to choose a gauge and to
compute the transformation in gauge parameters (including the coordinates) generated by the
constraints.
It is often useful to describe spatial geometry not by the spatial metric but by a triad ,
which defines three vector fields orthogonal to each other and normalized at each point. This
specifies all information about the 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 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. Not only can we rotate
the triad vectors, we can 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 one 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 in which the determinant of the metric
vanishes. Points on the boundary are usually singularities at which the classical evolution breaks
down such that both sides will never connect. However, since one expects that quantum gravity
may resolve classical singularities, which is indeed confirmed in loop quantum cosmology, we
will have to keep this issue in mind and not restrict ourselves to only one orientation from the
outset.