### 4.10 Anisotropies

Anisotropic models provide a first generalization of isotropic ones to more realistic situations. They thus
can be used to study the robustness of effects analyzed in isotropic situations and, at the same time, provide
a large class of interesting applications. In particular, an analysis of the singularity issue is important since
the classical approach to a singularity can be very different from the isotropic one. On the other hand, the
anisotropic approach is deemed to be characteristic even for general inhomogeneous singularities, if the BKL
scenario [38] is correct.
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

This influences the evolution as follows from the Hamiltonian constraint
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.