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188 13 Hypersurface-homogeneous space-timesIt should be noted that hypersurface-homogeneous perfect fluids arealways locally barotropic, provided µ(x) and p(x) have non-zero x derivatives,where x labels the homogeneous hypersurfaces.13.2 Formulations of the field equationsThe Einstein equations for hypersurface-homogeneous space-times reduceto a system of ordinary differential equations. At least for thespatially-homogeneous case, they form a well-posed Cauchy problem(Taub 1951) and, although they have not been completely integrated,their qualitative properties have been discussed in many papers: see Wainwrightand Ellis (1997) for an extensive survey of the results, and e.g. Ryanand Shepley (1975), MacCallum (1973, 1979a), Bogoyavlenskii (1980),and Rosquist and Jantzen (1988) for useful earlier reviews. Methods fromdynamical systems theory which proved fruitful in elucidating these propertieshave also led to ways of restricting the general case to more readilysolvable subcases and thence to new exact solutions (see e.g. Ugglaet al. (1995b) for a summary). Nearly all of these methods were developedinitially for use in the spatially-homogeneous case (‘cosmologies’,for brevity), and we shall therefore describe the methods in this contextalthough they can be adapted to the G 3 on T 3 and H 3 on V 3 cases also(e.g. for an orthonormal tetrad method for G 3 on T 3 see Harness (1982)).The number of degrees of freedom, i.e. the number of essential arbitraryconstants required in a general cosmology for each Bianchi type, has beenstudied by Siklos (1976a) (cp. MacCallum (1979b), Wainwright and Ellis(1997)). Table 13.1 summarizes the results for vacua and perfect fluids.The fluid here may be ‘tilted’, i.e. the velocity u need not coincide withthe normal n to the hypersurfaces of homogeneity. In general such fluidcosmologies can have four more parameters than the corresponding vacuabut Bianchi types I , where u = n, II and VI −1/9 are special cases inwhich the constraints on the Cauchy problem arising from the G 4 α fieldequations are not linearly independent. In the last case, type VI −1/9 , theeffect is an extra degree of freedom in the vacuum solutions, and solutionswhere this is activated are denoted type VI−1/9 ∗ .The residual set of ordinary differential equations to be solved can beformulated in various ways (see e.g. MacCallum (1973), Wainwright andEllis (1997)). One can use a time-independent basis as in (13.20) andparametrize the components g αβ in some suitable way. This is called themetric approach. One may then choose spatial coordinates and a newtime coordinate τ so that the dt of (13.20) is replaced by σ = N(τ)dτ +N α (τ)ω α , N being the lapse and N α the shift. Such a change of basiscould be interpreted as introducing rotation, if σ ∧ dσ ̸= 0, but this