2.3 Coordinate systems

In [16Jump To The Next Citation Point], special coordinate systems are constructed on part of the MGHD. Let us describe the different cases.

2.3.1 Coordinate systems, T3-Gowdy

In the case of T3-topology, there are coordinates such that the metric takes the form

2B 2 2 a a b b gT3 = e (− dt + dπœƒ ) + β„“tnab(dx + g dπœƒ )(dx + g dπœƒ ). (1 )
Here t ∈ (0,∞ ), πœƒ,xa ∈ [0,2π]mod 2π, B and nab are functions of t and πœƒ, β„“ > 0 is a constant, detn = 1 ab and the ga are constants. This is a special case of the form of the metric given in [16Jump To The Next Citation Point, (4.9), p. 116]; see also [16Jump To The Next Citation Point, Theorem 4.2] and [16Jump To The Next Citation Point, (4.12), p. 117].

2.3.2 Working definition, T3-Gowdy

The values of the constants β„“ and a g in Equation (1View Equation) are of no importance in practice. Consequently, β„“ can be taken to equal 1 and a g can be taken to equal 0. In order to arrive at the form of the metric we shall actually be using, let us set β„“ = 1 and ga = 0. Furthermore, we define x1 =: σ, x2 =: δ, n11 =: eP, n12 = ePQ, where we have used the fact that nab are the components of a positive definite matrix. Since nab is also a symmetric matrix with unit determinant, we obtain

−1βˆ•2 λβˆ•2 2 2 P 2 P P 2 −P 2 gT3 = t e (− dt + dπœƒ ) + t[e dσ + 2e Qd σdδ + (e Q + e )d δ ], (2 )
where we have defined λ by the relation t−1βˆ•2eλβˆ•2 = e2B. An alternate definition of a T3-Gowdy spacetime is a manifold of the form I × T 3 with a metric of the form of Equation (2View Equation). Of course, some form of Einstein’s equation should also be enforced.

2.3.3 Coordinate system, S3 and S2 × S1

In the case of S3 and S2 × S1 topology, the metric can be written

gS = e2B (− dT 2 + dψ2 ) + λabdxadxb, (3 )
where a x ∈ [0,2 π]mod 2π, T ∈ (0,π), ψ ∈ [0,π], det λab = β„“sinT sinψ, where β„“ > 0 is a constant, and B and λab are functions of T and ψ. This is the form of the metric given in [16Jump To The Next Citation Point, 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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