Dynamics of solutions of the Einstein equations with twisted Gowdy symmetry

Some of the most interesting results on the global dynamics of solutions of the vacuum Einstein equations concern the Gowdy spacetimes whose spatial topology is that of a three-dimensional torus. In this paper certain of these ideas are extended to a wider class of vacuum spacetimes where the spatial topology is that of a non-trivial torus bundle over a circle. Compared to the case of the torus these are topologically twisted. They include inhomogeneous generalizations of the spatially homogeneous vacuum spacetimes of Bianchi types II and VI${}_0$. Using similar procedures it is shown that the vacuum solutions of Bianchi type VII${}_0$ are isometric to a class of Gowdy spacetimes, the circular loop spacetimes, thus establishing links between results in the literature which were not previously known to be related to each other.