4.1 One-Killing vector reductions

Let us start by considering the simplest example of symmetry reduction corresponding to one-dimensional spatial isometry groups. In practice one considers R or U(1) and, hence, the spacetime metrics are required to have a single Killing vector field. These models are interesting because they retain important features of full general relativity, in particular diffeomorphism invariance, an infinite number of physical degrees of freedom and a non-linear character.

The local aspects of the one-Killing vector reductions were first considered by Geroch [96]. In that paper he developed a method to dimensionally reduce gravity by defining a way to “project” 3+1 dimensional geometric objects to the 2+1 dimensional space of orbits of the Killing field (required to have a non-vanishing norm). Here the world “local” refers to the fact that some topological aspects are sidestepped in a first look; however, if the quotient itself is well behaved (for example, it is Haussdorff) the projection is globally defined and has a clear geometrical meaning. The most important result of this work was to show that the reduced system can be interpreted as 2+1 GR coupled to certain matter fields with a concrete geometrical meaning: the norm and twist of the four dimensional Killing vector field (a scalar and a one-form field respectively). This link between one-Killing vector reductions and 2+1 dimensional gravity theories opened the door to quantum treatments relying on techniques specially tailored for lower dimensional models. The Geroch method can be adapted to treat symmetry reductions. For example it allows one to write the four-dimensional scalar curvature as a curvature on the 2+1 dimensional orbit manifold plus some extra terms involving the norm of the Killing. This is very useful to write the 3+1 dimensional action as a 2+1 dimensional one.

The global aspects and the Hamiltonian formalism (for vacuum GR) have been discussed by Moncrief in the case when the symmetry group is U(1) with compact Cauchy surfaces [175Jump To The Next Citation Point]. The spatial slices in this case can be taken to be U(1) bundles (or rather S1) over the sphere (though the analysis can be extended to arbitrary surfaces). The discussion presented in [175] is relevant to studying some of the compact Gowdy models, in particular those with the S2 × S1 and S3 spatial topologies, though it is possible to employ other approaches that rely on the Geroch reduction as discussed in [22Jump To The Next Citation Point]. The non-compact case with asymptotically-flat two-geometries (in the sense relevant in 2+1 gravity developed in [15Jump To The Next Citation Point]) has been studied by Varadarajan [212].

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