A geometric description of the intermediate behaviour for spatially homogeneous models

Abstract

A new approach is suggested for the study of geometric symmetries in general relativity, leading to an invariant characterization of the evolutionary behaviour for a class of spatially homogeneous (SH) vacuum and orthogonal γ-law perfect fluid models. Exploiting the 1 + 3 orthonormal frame formalism, we express the kinematical quantities of a generic symmetry using expansion-normalized variables. In this way, a specific symmetry assumption leads to geometric constraints that are combined with the associated integrability conditions, coming from the existence of the symmetry and the induced expansion-normalized form of Einstein's field equations (EFE), to give a close set of compatibility equations. By specializing to the case of a kinematic conformal symmetry (KCS), which is regarded as the direct generalization of the concept of self-similarity, we give the complete set of consistency equations for the whole SH dynamical state space. An interesting aspect of the analysis of the consistency equations is that, at least for class A models which are locally rotationally symmetric or lying within the invariant subset satisfying Nαα = 0, a proper KCS always exists and reduces to a self-similarity of the first or second kind at the asymptotic regimes, providing a way for the ‘geometrization’ of the intermediate epoch of SH models.