In order to be able to describe the class excluded by Gowdy’s assumptions, let us note some of the consequences of the essential symmetry assumption – that there is a compact and connected two-dimensional Lie group acting effectively on the initial hypersurface. Since the initial hypersurface is a three-dimensional manifold, [59, Theorem 6, p. 453] implies that the Lie group has to be T2; i.e., U(1) × U(1). In particular, there are initially two Killing fields that then extend to two spacelike Killing fields on the development. We shall denote them by , . As pointed out in [16, p. 101], the symmetry assumptions imply that the topology of the initial hypersurface has to be T3, S2 × S1, S3 or one of the Lens spaces. Since the Lens spaces have S3 as a universal covering space, we shall consider them to be subsumed under that case. It is also possible to describe how the group acts on the manifold in some detail [16, p. 102]. In particular, there must be “axes” at which one of the Killing fields vanish for all topologies except T3. Returning to the discarded class of solutions, let us define the functions
. They are constant on the spacetime (this statement is true in the vacuum case [16, p. 103] but not necessarily in the presence of matter) and are referred to as the twist constants. Gowdy assumes them to vanish [39, p. 211]. Note, however, that he calls this specialization two-surface orthogonality, since it implies [16, Theorem 4.2, p. 117] that the group orbits are orthogonal to the vector field , where is the areal time coordinate (i.e. the function that to each point of the spacetime associates the area of the group orbit containing the point). In the case of S2 × S1, S3 and Lens space topology, the twist constants have to vanish due to the existence of the axes. However, in the case of T3 topology, there is a class of solutions with nonvanishing twist constants called T2-symmetric spacetimes. The behavior of solutions belonging to this class is much more complicated than that of those belonging to the Gowdy class of spacetimes.
To conclude, the essential assumptions characterizing the Gowdy class are that a member of it should
Even though the above list gives the central assumptions, there are some subtleties that have been sorted out in . Thus, the formally inclined are recommended to use the assumptions of [16, Theorem 4.2, p. 117] and of [16, Theorem 6.1, p. 128–129] as a definition of Gowdy initial data. The Gowdy class of solutions is then defined as the MGHDs of Gowdy initial data.
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