A geometric description of the intermediate behaviour for spatially homogeneous models

TL;DR

This paper introduces a geometric approach to analyze symmetries in spatially homogeneous models in general relativity, revealing that certain symmetries always exist and relate to self-similarity in asymptotic regimes.

Contribution

It develops a new invariant characterization of symmetries in SH models using the 1+3 formalism and establishes the existence of Kinematic Conformal Symmetries in these models.

Findings

Proper KCS always exists in class A models with LRS or $N_{eta}^{eta}=0$

KCS reduces to self-similarity at asymptotic regimes

Provides a complete set of consistency equations for SH dynamical state space

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 $\gamma -$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 lead 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 the Einstein's Field Equations (EFE), to give a close set of compatibility equations. By specializing to the case of a \emph{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, \emph{at least} for class A models which are Locally Rotationally Symmetric or lying within the invariant subset satisfying $N_{\alpha}^{\alpha}=0 $, a proper KCS \emph{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.