A general homogeneous but anisotropic metric is of the form
with left-invariant 1-forms on space
, which, thanks to homogeneity, can be identified with the
simply transitive symmetry group
as a manifold. The left-invariant 1-forms satisfy the Maurer–Cartan
relations
with the structure constants of the symmetry group. In a matrix parameterization of the symmetry
group, one can derive explicit expressions for
from the Maurer–Cartan form
with generators
of
.
The simplest case of a symmetry group is an Abelian one with , corresponding to the Bianchi
I model. In this case,
is given by
or a torus, and left-invariant 1-forms are simply
in
Cartesian coordinates. Other groups must be restricted to class A models in this context, satisfying
since otherwise there is no standard Hamiltonian formulation [220]. The structure constants can
then be parameterized as
.
A common simplification is to assume the metric to be diagonal at all times, which corresponds to a
reduction technically similar to a symmetry reduction. This amounts to as well
as
for the extrinsic curvature with
. Depending on the structure
constants, there is also non-zero intrinsic curvature quantified by the spin connection components
In the vacuum Bianchi I case the resulting equations are easy to solve by with
[198]. The volume
vanishes for
where the classical singularity
appears. Since one of the exponents
must be negative, however, only two of the
vanish at the
classical singularity, while the third one diverges. This already demonstrates how different the behavior can
be from the isotropic model and that anisotropic models provide a crucial test of any mechanism for
singularity resolution.
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