Twisting Gowdy

I have previously written about the Gowdy solutions as the simplest model case of spatially inhomogeneous solutions of the Einstein vacuum equations and about the generalization of this to the Einstein-Maxwell equations. I have also written about spatially homogeneous solutions (Bianchi models). I have just posted a paper where I explore connections between these two classes of solutions (Gowdy and Bianchi) in the hope of learning more about both. The basic idea is simple. A Bianchi model has Killing vectors which form a three-dimensional Lie algebra. Most of these (all except types VIII and IX) have a two-dimensional Abelian subalgebra. When this is the case we can simply forget about one Killing vector and end up with a spacetime with two commuting Killing vectors. This looks very like a Gowdy spacetime. There are, however, two potential problems. The first is that Gowdy spacetimes have a certain discrete symmetry in addition to their two Killing vectors. Almost all Bianchi types do satisfy this condition, the exception being type {\rm VI}{}_{\frac19}. In what follows I ignore this exceptional type. The other problem is that a Gowdy spacetime is supposed to have a compact Cauchy surface and the Killing vectors are assumed to exist globally. If these conditions are weakened then more alternatives are allowed. Let me call this situation a locally Gowdy spacetime. (This is not standard terminology.) Assume that the Killing vector fields have no zeroes. Then in the Gowdy case the Cauchy surface has the topology of a three-torus T^3. In the locally Gowdy case the existence of a compact Cauchy surface is closely linked to those Bianchi types belonging to Bianchi Class A. I now restrict to that class. Then the additional Bianchi types which are possible are II, {\rm VI}{}_0 and {\rm VII}{}_0.

In Gowdy spacetimes there are two of the Einstein equations for functions P(t,\theta) and Q(t,\theta) which are of central importance. I call them the Gowdy equations. In order to describe Gowdy spacetimes the functions P and Q should be periodic in \theta. To accomodate Bianchi models of types II and {\rm VI}{}_0 more complicated boundary conditions are required. The twisted Gowdy solutions are solutions of the Gowdy equations which satisfy these boundary conditions but need not be homogeneous. They represent finite inhomogeneous perturbations of the corresponding Bianchi models. Periodic boundary conditions can accommodate type {\rm VII}{}_0, which then corresponds to what are called circular loop spacetimes. In the twisted cases the topology of a compact Cauchy surface is more complicated than that of T^3. For type II and type {\rm VI}{}_0 it corresponds to a manifold admitting geometric structures of type Nil and Sol in the sense of Thurston, respectively. These are twisted topologies which are torus bundles over a circle.

A lot is understood about the global dynamics of Gowdy solutions, mainly due to the work of Hans Ringström. By uncovering certain connections between different classes of solutions I have been able to transfer some of these results to the twisted Gowdy case. Unfortunately I was not able to obtain a general analysis of the late-time behaviour of twisted Gowdy solutions. The analytical techniques which were successful in the untwisted case do not seem to adapt well. This leaves a challenge for the future.

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