In the case of T3-topology, there are coordinates such that the metric takes the form[16, (4.9), p. 116]; see also [16, Theorem 4.2] and [16, (4.12), p. 117].
The values of the constants and in Equation (1) are of no importance in practice. Consequently, can be taken to equal 1 and can be taken to equal 0. In order to arrive at the form of the metric we shall actually be using, let us set and . Furthermore, we define , , , , where we have used the fact that are the components of a positive definite matrix. Since is also a symmetric matrix with unit determinant, we obtainT3-Gowdy spacetime is a manifold of the form with a metric of the form of Equation (2). Of course, some form of Einstein’s equation should also be enforced.
In the case of S3 and S2 × S1 topology, the metric can be written[16, Theorem 6.3, p. 133], though it should again be pointed out that it is not claimed that the coordinates with respect to which the metric takes this form cover the entire MGHD.
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