2.4 The polarized subcase

The polarized Gowdy solutions constitute an interesting subclass. They are characterized by the additional condition that the Killing vectors be mutually orthogonal. In the polarized T3-Gowdy case, the metric can be written
gP = t− 1βˆ•2eλβˆ•2(− dt2 + dπœƒ2) + t(ePdσ2 + e− Pdδ2); (4 )
i.e., it corresponds to setting Q = 0 in Equation (2View Equation). In the case of S3 and S2 × S1, the metric takes the form
2a 2 2 W 2 −W 2 gS,P = e (− dt + dπœƒ ) + sin πœƒsin t(e dx + e dy );

see Equations (5) – (7) of [50Jump To The Next Citation Point]. Clearly, the connection between the coordinates, the metric components and the topology is more complicated in the case of S3 and S2 × S1 than in the case of T3. We shall not go into a detailed discussion of these issues here, but rather refer the interested reader to [21Jump To The Next Citation Point] and [50Jump To The Next Citation Point] for a further discussion. A brief explanation of the origin of the name (in terms of polarizations of gravitational waves) as well as a different characterization is provided at the bottom of [70, p. 65].

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