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 Hamiltonian formulation. 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 . 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 one and that anisotropic models provide a crucial test of any mechanism for singularity resolution.
© Max Planck Society and the author(s)