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5.9 Inhomogeneous models

Homogeneous models provide a rich generalization of isotropic ones, but inhomogeneities lead to stronger qualitative differences. To start with, at least at the kinematical level one has infinitely many degrees of freedom and is thus always dealing with field theories. Studying field theoretical implications does not require going immediately to the full theory since there are many inhomogeneous models of physical interest.

We will describe some 1-dimensional models with one inhomogeneous coordinate x and two others parameterizing symmetry orbits. A general connection is then of the form (with coordinate differentials wy and wz depending on the symmetry)

A = Ax(x)/\x(x) dx + Ay(x)/\y(x)wy + Az(x)/\z(x)wz + field independent terms (53)
with three real functions AI (x) and three internal directions /\I(x) normalized to 1 tr(/\2I) = - 2, which in general are independent of each other. The situation in a given point x is thus similar to general homogeneous models with nine free parameters. Correspondingly, there are not many simplifications from this general form, and one needs analogs of the diagonalization employed for homogeneous models. What is required mathematically for simplifications to occur is a connection with internally perpendicular components, i.e., tr(/\I /\J) = - 12dIJ in each point. This arises in different physical situations.
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