2.2 Definition of the Gowdy class

Spacetimes admitting two-dimensional isometry groups had been studied prior to [39Jump To The Next Citation Point]. However, the considerations had mainly been limited to the stationary axisymmetric case. To the best of our knowledge, Gowdy was the first one to systematically analyze the consequences of imposing the existence of a two-dimensional isometry group with spacelike orbits [39Jump To The Next Citation Point3837]. As a consequence, a subclass of this family now bears his name. The main objective of the work presented in [39Jump To The Next Citation Point] is to write down a convenient global form for metrics admitting this type of symmetry. However, in the course of the discussion, Gowdy introduces assumptions that exclude a large family of solutions admitting a two-dimensional isometry group. Let us introduce the terminology necessary for describing the discarded class.

2.2.1 Twist constants, two-surface orthogonality

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 Xi, i = 1,2. As pointed out in [16Jump To The Next Citation Point, 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 [16Jump To The Next Citation Point, 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

ca = 𝜖αβγδX α1 X β2∇ γX δa,

a = 1,2. They are constant on the spacetime (this statement is true in the vacuum case [16Jump To The Next Citation Point, 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 [16Jump To The Next Citation Point, Theorem 4.2, p. 117] that the group orbits are orthogonal to the vector field ∂ t, where t 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.

2.2.2 Essential characterizing conditions

To conclude, the essential assumptions characterizing the Gowdy class are that a member of it should

2.2.3 Technical definition

Even though the above list gives the central assumptions, there are some subtleties that have been sorted out in [16Jump To The Next Citation Point]. Thus, the formally inclined are recommended to use the assumptions of [16Jump To The Next Citation Point, Theorem 4.2, p. 117] and of [16Jump To The Next Citation Point, 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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