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 T^{2}; 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 T^{3}, S^{2} × S^{1}, S^{3} or
one of the Lens spaces. Since the Lens spaces have S^{3} 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 T^{3}. 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 S^{2} × S^{1}, S^{3} and Lens space topology, the twist constants have to vanish due to the
existence of the axes. However, in the case of T^{3} topology, there is a class of solutions with
nonvanishing twist constants called T^{2}-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

- be a globally-hyperbolic cosmological spacetime,
- admit an effective action by isometries by a two-dimensional compact Lie group with spacelike orbits,
- be such that the twist constants vanish.

Even though the above list gives the central assumptions, there are some subtleties that have been sorted out in [16]. 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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