The Bianchi models are arguably the most important among the minisuperspaces. They describe spatially homogeneous (but generally non-isotropic) cosmologies. These spacetimes are obtained (see [193, 224] for a pedagogical presentation) by requiring that the spacetime admits a foliation by smooth three-dimensional hypersurfaces that are the orbits of a group of isometries . When the action is required to be simply transitive (i.e., for each pair of points there exist a unique element of such that ) its dimension must be 3. In addition to the Bianchi models there are other spatially-homogeneous spacetimes for which the group action is not simply transitive (or does not have a subgroup with a simply transitive action). These are the Kantowski–Sachs models with and such that the spatial homogeneous hypersurfaces are . Metrics for the Bianchi models are parameterized by functions of the “time” variable that labels the sheets of the spacetime foliation and can be conveniently written by using a basis of invariant one forms. The Killing vector fields of the metric induced on each are in one to one correspondence with the right invariant vector fields in the group and satisfy the commutation rules of the Lie algebra of .
The Einstein field equations reduce in these cases to a system of ordinary differential equations. Bianchi models are classified as type A and type B depending on some invariant properties encoded in the structure constants of the isometry group. If they satisfy the condition the resulting model is type A, otherwise it is called type B. Only the type A ones satisfy the principle of symmetric criticality and can be quantized in a straightforward way .
Two main approaches are possible to studying the classical dynamics of minisuperspace models and, in particular, the Bianchi models: The covariant spacetime textbook approach (see, for example, ) that directly looks for the spatially-homogeneous solutions to the Einstein field equations, and the Hamiltonian one that can be applied when the principle of symmetric criticality holds. Of course, they are ultimately equivalent, but the descriptions that they provide for the classical dynamics of these systems are surprisingly different. A very good account of these issues can be found in . Among the points that are worthwhile singling out, maybe the most striking one refers to the identification and counting of the number of degrees of freedom. As can be seen, these numbers generically disagree in the case of open spatial slices. This can be easily shown  for the Bianchi I model for R3 spatial slices. From the covariant point of view the family of solutions of Bianchi type I is fully described by a single parameter; on the other hand the Hamiltonian analysis (this is a constrained Hamiltonian system) tells us that the number of phase-space degrees of freedom is ten corresponding to five physical degrees of freedom. The resolution of this problem  requires a careful understanding of several issues:
The bottom line can be summarized by saying that the extra structure present in the Hamiltonian framework provides us with sharper tools to separate gauge and symmetries than the purely geometric point of view of the standard covariant approach . If one is interested in the quantization of these minisuperspace reductions, the Hamiltonian framework is the natural (and essentially unavoidable) starting point.
It is obvious that essentially all the points discussed here will also be relevant in the case of midisuperspaces, though to our knowledge the current analyses of this issue are far from complete – and definitely much harder – because one must deal with infinite dimensional spaces. In this case, as we will see, the gauge symmetry remaining after the symmetry reduction will include a non-trivial class of restricted diffeomorphisms. This is, in fact, one of the main reasons to study these symmetry reductions as they may shed some light on the difficult issue of dealing with diffeomorphism invariance in full quantum gravity. A final interesting point that we want to mention is the problem of understanding how minisuperspace models sit inside the full superspace. This has been discussed by Jantzen in .
Living Rev. Relativity 13, (2010), 6
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