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4.7 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. An analysis in particular 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 [31Jump To The Next Citation Point] is correct.

A general homogeneous but anisotropic metric is of the form

3 2 2 2 sum I J ds = - N (t) dt + qIJ(t)w ox w I,J=1

with left-invariant 1-forms wI on space S, which, thanks to homogeneity, can be identified with the simply transitive symmetry group S as a manifold. The left-invariant 1-forms satisfy the Maurer-Cartan relations

I 1- I J K dw = - 2C JK w /\ w

with the structure constants CI JK of the symmetry group. In a matrix parameterization of the symmetry group, one can derive explicit expressions for I w from the Maurer-Cartan form I -1 w TI = hMC = g dg with generators TI of S.

The simplest case of a symmetry group is an Abelian one with CIJK = 0, corresponding to the Bianchi I model. In this case, S is given by R3 or a torus, and left-invariant 1-forms are simply wI = dxI in Cartesian coordinates. Other groups must be restricted to class A models in this context, satisfying I C JI = 0 since otherwise there is no Hamiltonian formulation. The structure constants can then be parameterized as I I (I) CJK = eJKn.

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 qIJ = a2 dIJ (I) as well as KIJ = K(I)dIJ for the extrinsic curvature with KI = aI. Depending on the structure constants, there is also non-zero intrinsic curvature quantified by the spin connection components

( ) 1 aJ J aK K a2I I GI = -- ---n + ---n - ------n for eIJK = 1. (33) 2 aK aJ aJ aK
This influences the evolution as follows from the Hamiltonian constraint
- --1--(a1a2a3 + a2a1a2 + a3a1a2 - (G2G3 - n1G1)a1 - (G1G3 - n2G2)a2 8pG ) - (G1G2 - n3G3)a3 + Hmatter(aI) = 0. (34)

In the vacuum Bianchi I case the resulting equations are easy to solve by aI oc taI with sum aI = sum a2 = 1 I I I [135]. The volume a1a2a3 oc t vanishes for t = 0 where the classical singularity appears. Since one of the exponents a I must be negative, however, only two of the a I 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.

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