In mathematics, the Bianchi classification provides a list of all real 3-dimensional Lie algebras (up to isomorphism). The classification contains 11 classes, 9 of which contain a single Lie algebra and two of which contain a continuum-sized family of Lie algebras. (Sometimes two of the groups are included in the infinite families, giving 9 instead of 11 classes.) The classification is important in geometry and physics, because the associated Lie groups serve as symmetry groups of 3-dimensional Riemannian manifolds. It is named for Luigi Bianchi, who worked it out in 1898.
The term "Bianchi classification" is also used for similar classifications in other dimensions and for classifications of complex Lie algebras.
All the 3-dimensional Lie algebras other than types VIII and IX can be constructed as a semidirect product of R^{2} and R, with R acting on R^{2} by some 2 by 2 matrix M. The different types correspond to different types of matrices M, as described below.
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.
The connected 3-dimensional Lie groups can be classified as follows: they are a quotient of the corresponding simply connected Lie group by a discrete subgroup of the center, so can be read off from the table above.
The groups are related to the 8 geometries of Thurston's geometrization conjecture. 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 Thurston geometry of type S^{2}×R cannot be realized in this way.
The three-dimensional Bianchi spaces each admit a set of three Killing vector fields which obey the following property:
where , the "structure constants" of the group, form a constant order-three tensor antisymmetric in its lower two indices. For any three-dimensional Bianchi space, is given by the relationship
where is the Levi-Civita symbol, is the Kronecker delta, and the vector and diagonal tensor are described by the following table, where gives the ith eigenvalue of ;^{[1]} the parameter a runs over all positive real numbers:
Bianchi type | class | notes | graphical (Fig. 1) | ||||
---|---|---|---|---|---|---|---|
I | 0 | 0 | 0 | 0 | A | describes Euclidean space | at the origin |
II | 0 | 1 | 0 | 0 | A | interval [0,1] along | |
III | 1 | 0 | 1 | -1 | B | the subcase of type VI_{a} with | projects to fourth quadrant of the a = 0 plane |
IV | 1 | 0 | 0 | 1 | B | vertical open face between first and fourth quadrants of the a = 0 plane | |
V | 1 | 0 | 0 | 0 | B | has a hyper-pseudosphere as a special case | the interval (0,1] along the axis a |
VI_{0} | 0 | 1 | -1 | 0 | A | fourth quadrant of the horizontal plane | |
VI_{a} | 0 | 1 | -1 | B | when , equivalent to type III | projects to fourth quadrant of the a = 0 plane | |
VII_{0} | 0 | 1 | 1 | 0 | A | has Euclidean space as a special case | first quadrant of the horizontal plane |
VII_{a} | 0 | 1 | 1 | B | has a hyper-pseudosphere as a special case | projects to first quadrant of the a = 0 plane | |
VIII | 0 | 1 | 1 | -1 | A | sixth octant | |
IX | 0 | 1 | 1 | 1 | A | has a hypersphere as a special case | second octant |
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.
For a given metric:
(where are 1-forms), the Ricci curvature tensor is given by:
where the indices on the structure constants are raised and lowered with which is not a function of .
In cosmology, this classification is used for a homogeneous spacetime of dimension 3+1. The 3-dimensional Lie group is as the symmetry group of the 3-dimensional spacelike slice, and the Lorentz metric satisfying the Einstein equation is generated by varying the metric components as a function of t. 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.^{[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.^{[3]}^{[4]} 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.^{[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. The following pertains to the space part of the metric at a given instant of time t assuming a synchronous frame so that t is the same synchronised time for the whole space.
Homogeneity implies identical metric properties at all points of the space. An exact definition of this concept involves considering sets of coordinate transformations that transform the space into itself, i.e. leave its metric unchanged: if the line element before transformation is
then after transformation the same line element is
with the same functional dependence of γ_{αβ} on the new coordinates. (For a more theoretical and coordinate-independent definition of homogeneous space see homogeneous space). A space is homogeneous if it admits a set of transformations (a group of motions) that brings any given point to the position of any other point. Since space is three-dimensional the different transformations of the group are labelled by three independent parameters.
In Euclidean space the homogeneity of space is expressed by the invariance of the metric under parallel displacements (translations) of the Cartesian coordinate system. 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. These forms are written as where the Latin index (a) labels three independent vectors (coordinate functions); these vectors are called a frame field or triad. The Greek letters label the three space-like curvilinear coordinates. A spatial metric invariant is constructed under the given group of motions with the use of the above forms:
(eq. 6a) |
i.e. the metric tensor is
(eq. 6b) |
where the coefficients η_{ab}, which are symmetric in the indices a and b, are functions of time. The choice of basis vectors is dictated by the symmetry properties of the space and, in general, these basis vectors are not orthogonal (so that the matrix η_{ab} is not diagonal).
The reciprocal triple of vectors is introduced with the help of Kronecker delta
(eq. 6c) |
In the three-dimensional case, the relation between the two vector triples can be written explicitly
(eq. 6d) |
where the volume v is
with e_{(a)} and e^{(a)} regarded as Cartesian vectors with components and , respectively. The determinant of the metric tensor eq. 6b is γ = ηv^{2} where η is the determinant of the matrix η_{ab}.
The required conditions for the homogeneity of the space are
(eq. 6e) |
The constants are called the structure constants of the group.
Proof of eq. 6e |
---|
The invariance of the differential forms means that where the on the two sides of the equation are the same functions of the old and new coordinates, respectively. Multiplying this equation by , setting and comparing coefficients of the same differentials dx^{α}, one finds These equations are a system of differential equations that determine the functions for a given frame. In order to be integrable, these equations must satisfy identically the conditions Calculating the derivatives, one finds Multiplying both sides of the equations by and shifting the differentiation from one factor to the other by using eq. 6c, one gets for the left side: and for the right, the same expression in the variable x. Since x and x' are arbitrary, these expression must reduce to constants to obtain eq. 6e. |
Multiplying by , eq. 6e can be rewritten in the form
(eq. 6f) |
Equation 6e can be written in a vector form as
where again the vector operations are done as if the coordinates x^{α} were Cartesian. Using eq. 6d, one obtains
(eq. 6g) |
and six more equations obtained by a cyclic permutation of indices 1, 2, 3.
The structure constants are antisymmetric in their lower indices as seen from their definition eq. 6e: . Another condition on the structure constants can be obtained by noting that eq. 6f can be written in the form of commutation relations
(eq. 6h) |
for the linear differential operators
(eq. 6i) |
In the mathematical theory of continuous groups (Lie groups) the operators X_{a} satisfying conditions eq. 6h are called the generators of the group. The theory of Lie groups uses operators defined using the Killing vectors instead of triads . Since in the synchronous metric none of the γ_{αβ} components depends on time, the Killing vectors (triads) are time-like.
The conditions eq. 6h follow from the Jacobi identity
and have the form
(eq. 6j) |
It is a definite advantage to use, in place of the three-index constants , a set of two-index quantities, obtained by the dual transformation
(eq. 6k) |
where e_{abc} = e^{abc} is the unit antisymmetric symbol (with e_{123} = +1). With these constants the commutation relations eq. 6h are written as
(eq. 6l) |
The antisymmetry property is already taken into account in the definition eq. 6k, while property eq. 6j takes the form
(eq. 6m) |
The choice of the three frame vectors in the differential forms (and with them the operators X_{a}) is not unique. They can be subjected to any linear transformation with constant coefficients:
(eq. 6n) |
The quantities η_{ab} and C^{ab} behave like tensors (are invariant) with respect to such transformations.
The conditions eq. 6m are the only ones that the structure constants must satisfy. 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. 6n. 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 C^{ab}, by the following simple method (C. G. Behr, 1962).
The asymmetric tensor C^{ab} can be resolved into a symmetric and an antisymmetric part. The first is denoted by n^{ab}, and the second is expressed in terms of its dual vector a_{c}:
(eq. 6o) |
Substitution of this expression in eq. 6m leads to the condition
(eq. 6p) |
By means of the transformations eq. 6n the symmetric tensor n^{ab} can be brought to diagonal form with eigenvalues n_{1}, n_{2}, n_{3}. Equation 6p shows that the vector a_{b} (if it exists) lies along one of the principal directions of the tensor n^{ab}, the one corresponding to the eigenvalue zero. Without loss of generality one can therefore set a_{b} = (a, 0, 0). Then eq. 6p reduces to an_{1} = 0, i.e. one of the quantities a or n_{1} must be zero. The Jacobi identities take the form:
(eq. 6q) |
The only remaining freedoms are sign changes of the operators X_{a} and their multiplication by arbitrary constants. This permits to simultaneously change the sign of all the n_{a} and also to make the quantity a positive (if it is different from zero). Also all structure constants can be made equal to ±1, if at least one of the quantities a, n_{2}, n_{3} vanishes. But if all three of these quantities differ from zero, the scale transformations leave invariant the ratio h = a^{2}(n_{2}n_{3})^{−1}.
Thus one arrives at the Bianchi classification listing the possible types of homogeneous spaces classified by the values of a, n_{1}, n_{2}, n_{3} which is graphically presented in Fig. 3. In the class A case (a = 0), type IX (n^{(1)}=1, n^{(2)}=1, n^{(3)}=1) is represented by octant 2, type VIII (n^{(1)}=1, n^{(2)}=1, n^{(3)}=–1) is represented by octant 6, while type VII_{0} (n^{(1)}=1, n^{(2)}=1, n^{(3)}=0) is represented by the first quadrant of the horizontal plane and type VI_{0} (n^{(1)}=1, n^{(2)}=–1, n^{(3)}=0) is represented by the fourth quadrant of this plane; type II ((n^{(1)}=1, n^{(2)}=0, n^{(3)}=0) is represented by the interval [0,1] along n^{(1)} and type I (n^{(1)}=0, n^{(2)}=0, n^{(3)}=0) is at the origin. Similarly in the class B case (with n^{(3)} = 0), Bianchi type VI_{h} (a=h, n^{(1)}=1, n^{(2)}=–1) projects to the fourth quadrant of the horizontal plane and type VII_{h} (a=h, n^{(1)}=1, n^{(2)}=1) projects to the first quadrant of the horizontal plane; these last two types are a single isomorphism class corresponding to a constant value surface of the function h = a^{2}(n^{(1)}n^{(2)})^{−1}. A typical such surface is illustrated in one octant, the angle θ given by tan θ = |h/2|^{1/2}; those in the remaining octants are obtained by rotation through multiples of π/2, h alternating in sign for a given magnitude |h|. Type III is a subtype of VI_{h} with a=1. Type V (a=1, n^{(1)}=0, n^{(2)}=0) is the interval (0,1] along the axis a and type IV (a=1, n^{(1)}=1, n^{(2)}=0) is the vertical open face between the first and fourth quadrants of the a = 0 plane with the latter giving the class A limit of each type.
The Einstein equations for a universe with a homogeneous space can reduce to a system of ordinary differential equations containing only functions of time with the help of a frame field. To do this one must resolve the spatial components of four-vectors and four-tensors along the triad of basis vectors of the space:
where all these quantities are now functions of t alone; the scalar quantities, the energy density ε and the pressure of the matter p, are also functions of the time.
The Einstein equations in vacuum in synchronous reference frame are^{[11]}^{[12]}^{[note 1]}
(eq. 11) |
(eq. 12) |
(eq. 13) |
where is the 3-dimensional tensor , and P_{αβ} is the 3-dimensional Ricci tensor, which is expressed by the 3-dimensional metric tensor γ_{αβ} in the same way as R_{ik} is expressed by g_{ik}; P_{αβ} contains only the space (but not the time) derivatives of γ_{αβ}. Using triads, for eq. 11 one has simply
The components of P_{(a)(b)} can be expressed in terms of the quantities η_{ab} and the structure constants of the group by using the tetrad representation of the Ricci tensor in terms of quantities ^{[13]}
After replacing the three-index symbols by two-index symbols C^{ab} 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 η_{ab}
The Bianchi identities for the three-dimensional tensor P_{αβ} in the homogeneous space take the form
Taking into account the transformations of covariant derivatives for arbitrary four-vectors A_{i} and four-tensors A_{ik}
the final expressions for the triad components of the Ricci four-tensor are:
(eq. 11a) |
(eq. 12a) |
(eq. 13a) |
In setting up the Einstein equations there is thus no need to use explicit expressions for the basis vectors as functions of the coordinates.