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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 [38Jump To The Next Citation Point] is correct.

A general homogeneous but anisotropic metric is of the form

3 2 2 2 ∑ I J ds = − N (t) dt + qIJ(t)ω ⊗ ω I,J=1

with left-invariant 1-forms ωI on space Σ, 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 d ω = − 2C JK ω ∧ ω

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 ω from the Maurer–Cartan form I −1 ω TI = θMC = 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 ωI = 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 standard Hamiltonian formulation [220]. The structure constants can then be parameterized as I I (I) CJK = εJKn.

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 δIJ (I) as well as KIJ = K (I)δIJ for the extrinsic curvature with KI = a˙I. 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 Γ I =-- ---n + ---n − -----n for εIJK = 1 . (36 ) 2 aK aJ aJaK
This influences the evolution as follows from the Hamiltonian constraint
− --1--(a1a˙2a˙3 + a2a˙1a˙3 + a3a˙1a˙2 − (Γ 2Γ 3 − n1Γ 1)a1 − (Γ 1Γ 3 − n2Γ 2)a2 8πG ) − (Γ 1Γ 2 − n3Γ 3)a3 + Hmatter(aI) = 0. (37 )

In the vacuum Bianchi I case the resulting equations are easy to solve by aI ∝ tαI with ∑ αI = ∑ α2 = 1 I I I [198]. The volume a1a2a3 ∝ t vanishes for t = 0 where the classical singularity appears. Since one of the exponents α 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 model and that anisotropic models provide a crucial test of any mechanism for singularity resolution.


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