Cosmological Application
In cosmology, this classification is used for a homogeneous spacetime of dimension 3+1. The Friedmann-Lemaître-Robertson-Walker metrics are isotropic, which are particular cases of types I, V, and IX. The Bianchi type I models include the Kasner metric as a special case. The Bianchi IX cosmologies include the Taub metric. However, the dynamics near the singularity is approximately governed by a series of successive Kasner (Bianchi I) periods. The complicated dynamics, which essentially amounts to billiard motion in a portion of hyperbolic space, exhibits chaotic behaviour, and is named Mixmaster, and its analysis is referred to as the BKL analysis after Belinskii, Khalatnikov and Lifshitz. More recent work has established a relation of (super-)gravity theories near a spacelike singularity (BKL-limit) with Lorentzian Kac-Moody algebras, Weyl groups and hyperbolic Coxeter groups. Other more recent work is concerned with the discrete nature of the Kasner map and a continuous generalisation.
Read more about this topic: Bianchi Classification
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