Bianchi classification
In mathematics, the Bianchi classification provides a list of all real 3-dimensional Lie algebras (up to isomorphism).
The classification of 3-dimensional complex Lie algebras is similar except that types VIII and IX become isomorphic, and types VI and VII both become part of a single family of Lie algebras.
More precisely, seven of the 8 geometries can be realized as a left-invariant metric on the simply connected group (sometimes in more than one way).
The three-dimensional Bianchi spaces each admit a set of three Killing vector fields
;[1] the parameter a runs over all positive real numbers: The standard Bianchi classification can be derived from the structural constants in the following six steps: The Bianchi spaces have the property that their Ricci tensors can be separated into a product of the basis vectors associated with the space and a coordinate-independent tensor.
The Bianchi type I models include the Kasner metric as a special case.
[2] 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; its analysis is referred to as the BKL analysis after Belinskii, Khalatnikov and Lifshitz.
[5][6][7] Other more recent work is concerned with the discrete nature of the Kasner map and a continuous generalisation.
[8][9][10] In a space that is both homogeneous and isotropic the metric is determined completely, leaving free only the sign of the curvature.
Assuming only space homogeneity with no additional symmetry such as isotropy leaves considerably more freedom in choosing the metric.
Homogeneity implies identical metric properties at all points of the space.
Since space is three-dimensional the different transformations of the group are labelled by three independent parameters.
Each translation is determined by three parameters — the components of the displacement vector of the coordinate origin.
All these transformations leave invariant the three independent differentials (dx, dy, dz) from which the line element is constructed.
In the general case of a non-Euclidean homogeneous space, the transformations of its group of motions again leave invariant three independent linear differential forms, which do not, however, reduce to total differentials of any coordinate functions.
6f can be written in the form of commutation relations for the linear differential operators In the mathematical theory of continuous groups (Lie groups) the operators Xa satisfying conditions eq.
The theory of Lie groups uses operators defined using the Killing vectors
Since in the synchronous metric none of the γαβ components depends on time, the Killing vectors (triads) are time-like.
6h follow from the Jacobi identity and have the form It is a definite advantage to use, in place of the three-index constants
But among the constants admissible by these conditions, there are equivalent sets, in the sense that their difference is related to a transformation of the type eq.
The question of the classification of homogeneous spaces reduces to determining all nonequivalent sets of structure constants.
This can be done, using the "tensor" properties of the quantities Cab, by the following simple method (C. G. Behr, 1962).
The asymmetric tensor Cab can be resolved into a symmetric and an antisymmetric part.
Equation 6p shows that the vector ab (if it exists) lies along one of the principal directions of the tensor nab, the one corresponding to the eigenvalue zero.
The Jacobi identities take the form: The only remaining freedoms are sign changes of the operators Xa and their multiplication by arbitrary constants.
This permits to simultaneously change the sign of all the na and also to make the quantity a positive (if it is different from zero).
Thus one arrives at the Bianchi classification listing the possible types of homogeneous spaces classified by the values of a, n1, n2, n3 which is graphically presented in Fig.
To do this one must resolve the spatial components of four-vectors and four-tensors along the triad of basis vectors of the space:
by two-index symbols Cab and the transformations: one gets the "homogeneous" Ricci tensor expressed in structure constants: Here, all indices are raised and lowered with the local metric tensor ηab The Bianchi identities for the three-dimensional tensor Pαβ in the homogeneous space take the form Taking into account the transformations of covariant derivatives for arbitrary four-vectors Ai and four-tensors Aik the final expressions for the triad components of the Ricci four-tensor are: In setting up the Einstein equations there is thus no need to use explicit expressions for the basis vectors as functions of the coordinates.
