3.1. Homogeneous etymological models

We now turn to consider the app.ication of the concepts reviewed in section 2 to general relaiiv;stic systems and specifically to spatially homogeneous solutions of Einstein's gravitational field equations.

Einstein's equations provide ten, coupled, non-linear partial differential equations in four variables for ten unknown metric functions of space and time. Without some simplifying assumptions of symmetry they are intractable. A useful symmetry to impose upon solutions is that >f spatial homo­geneity which simplifies the system to ordinary differential equations whilst leaving a rich range of possible behaviour. Only the simplest homogeneous cosmoiogical models have been solved exactly [12] but the class of admissible three-gecmetries is finite and was completely classified by Bianchi [24] in 1898. The Bianchi classification was first used to classify all possible homogeneous cosmologies by Taub in I95l and was subsequently investigated in more detail by various authors [7,12,27].

The structure constants are anti-symmet ic and satisfy a Jacobi identity,

 

(where [■ • •] denotes the anti-symmetric part).

The three anti-symmetric matrices Cabc have a natural decomposition [29] as

 

In three dimensions spatial homogeneity can be described in terms of three independent sets of curves whose tangent vectors are <f>, (i =1,2, 3). An infinitesimal symmetry transformation takes a point Q with coordinates {*,} into a tother with coordinates + SjcJ where Sx : <f>0 St and is some linear combination of the <f>i at Q; for full details see ref. [28]. If we move aiong two of the in different directions around a closed curve and examine the difference in 4* St at a point when they meet, then the nature of the homogeneous space can be classified by tht structure constants, CX, of the group of motions determined by the non-zero Lie derivatives Zf+^a,

 

 

where nab is symmetric and, like the vector ab, constant; edbc is the anti-symmetric tensor and S;' the Kronecker delta. The Jacobi condition is automatically satisfied when the Ct are expressed m tins fashion if and only if

 

 

We have the freedom to choose new linear combinations of the <f>, as our basis and this can be used to diagonalize the symmetric 'tensor' nab and, by (3.4), set ab = (a, 0,0). The type of homogeneous space is now specified according to the values of a and the eigenvalue s of nah.

Ellis and MacCallum [27] define two broad classes of model, class A where <i„ =') and class B The nine structure constants satisfy the three Jacobi conditions (3.2) and so there remain six in­dependent components of Cbc. The Bianchi classification specifies the homogeneous geometry accord­ing to the values of these components, or equivalent^ those of the n^ and a; table 1.

The relative generality of the different homogeneous geometries can be clearly seen from table 1. We have also indicated the number of independently arbitrary constants that are required to specify the Cauchy data on a hypersarface of constant time in a cosmological model with homogeneous geometry. The Einstein equations in vacuum! aret

 

 

where Rf„ is the Ricci tensor.

In the {fa} basis the components of the Ricci tensor are

 

Table 1

The Bianchi-Behr-Ellis-MacCallum classification of spatially homogeneous cosmo ogical models

The space-time metric} of the spatially homogeneous models may be written in the form

 

 

where e"( = e°(x'). The tensor yab(f) determines the time-dependence of the metric and has six independent components. The three invariant vector fields e° are related to the structure constants of the homogeneous surfaces by

 

 

 

Class  A         B

 

specifying C audi\ data in vacuum

 

f The presence of matter fields is unimportant for our subsequent discussion and they will be assumed t a be absent unless otherwise indicated.

; Wc use -he nutation of ref. [30]. In particular Greek indices run (0,1,2,3) and latins (1,2,3). The metric signature is (H          ).

where ab* 0. Class B is characterized by a»i additional invariant, K defined by

 

$ llie space-time metric tensor should not be confused with the 'metric' (=measure theoretic) propertits of the cosmology.

 

 

where the field equations place two algebraic constraints on the three Kasner indices {/?,}

 

There also exist the following invariance properties

 

The relations (3.15) also allow the {/?,} to be ordered without loss of generality as.

 

Of the Bianchi types listed in table 1 the simplest is type I for which all the structure constants C)r vanish identically. In this case Pah s 0 and the vacuum solution (3.9-3.11) is the well-known Kasner p] space-time

 

 

In their early studies of this space-time in relation to the more general solution of (3.9)—(3.13) Lifshitz and Khalatnikov [31] introduced a convenient re-parametrization of the {/?,} subject to (3.15) by introducing u £ [1, where

 

 

The solution (3.14)-(3.18) illuslrates the typical effect of the 'electric' part of the Weyl cutvafuu with expansion occurring in two orthogonal directions whilst implosion (/?, <0) obtains in the third; the overall volume expands as V-1. This behaviour is not expected to be general amongst homogeneous cosmologies because the Kasner Cauchy data is specified by just one arbitrary constant («). The general case requires four such constants and this occurs for the type VI*,, VIIh, VIII and IX geometries (in the VIh and VIIft models h is one of those four parameters).