Bianchi universes (s= 3 ) 1333.7.1 Constructing Bianchi universes
The approach of Ellis and MacCallum [45]) uses an orthonormal tetrad based on
the normals to the surfaces of homogeneity (i.e.e 0 =n, the unit normal vector to
these surfaces). The tetrad is chosen to be invariant under the group of isometries,
i.e. the tetrad vectors commute with the KVs. Then we have an orthonormal basis
ea,a= 0 , 1 , 2 ,3, such that equation (3.52) becomes
[ea,eb]=γcab(t)ec (3.85)and all dynamic variables are function of timetonly. The matter variables—
μ(t),p(t),anduα(t)in the case of tilted models—and the commutation functions
γabc(t), which by (3.59) are equivalent to the rotation coefficients, are chosen to
be these variables. The EFE (3.2) are first-order equations for these quantities,
supplemented by the Jacobi identities for theγabc(t), which are also first-order
equations. Thus the equations needed are just the tetrad equations mentioned in
section 3.3, for the case
u ̇α=ωα= 0 =eα(γabc). (3.86)The spatial commutation functionsγαβγ(t)can be decomposed into a time-
dependent matrixnαβ(t)and vectoraα(t), see (3.66), and are equivalent to the
structure constantsCαβγof the symmetry group at each point. In view of (3.86),
the Jacobi identities (3.60) for the spatial vectors now take the simple form
nαβaβ= 0. (3.87)The tetrad basis can be chosen to diagonalizenαβat all times, to attainnαβ =
diag(n 1 ,n 2 ,n 3 ),aα =(a, 0 , 0 ), so that the Jacobi identities are then simply
n 1 a=0. Consequently we define two major classes of structure constants (and
so Lie algebras):
Class A:a=0; and
Class B:a=0.Following Sch ̈ucking’s extension of Bianchi’s work, the classification ofG 3
group types used is as in table 3.2. Given a specific group type at one instant,
this type will be preserved by the evolution equations for the quantitiesnα(t)and
a(t). This is a consequence of a generic property of the EFE: they will always
preserve symmetries in initial data (within the Cauchy development of that data);
see Hawking and Ellis [68].
In some cases, the Bianchi groups allow higher symmetry subcases, i.e. they
are compatible with isotropic (FL) or LRS models, see [45] for details. For us the
interesting point is thatk=0 FL models are compatible with groups of type I
and VII 0 ,k=−1 models with groups of types V and VIIh,andk=+1 models
with groups of type IX.