Coordinate systems

2.3 Coordinate systems

In [16

], 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

Here $t ∈ (0,∞ )$ , $𝜃,xa ∈ [0,2π]mod 2π$ , $B$ and $nab$ are functions of $t$ and $𝜃$ , $ℓ > 0$ is a constant, $detn = 1 ab$ and the $ga$ are constants. This is a special case of the form of the metric given in [16

, (4.9), p. 116]; see also [16

, Theorem 4.2] and [16

, (4.12), p. 117].

2.3.2 Working definition, T³-Gowdy

The values of the constants $ℓ$ and $a g$ in Equation (1 ) are of no importance in practice. Consequently, $ℓ$ 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 $ℓ = 1$ 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

where we have defined $λ$ by the relation $t−1∕2eλ∕2 = e2B$ . 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 (2

). 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

where $a x ∈ [0,2 π]mod 2π$ , $T ∈ (0,π)$ , $ψ ∈ [0,π]$ , $det λab = ℓsinT sinψ$ , where $ℓ > 0$ is a constant, and $B$ and $λab$ are functions of $T$ and $ψ$ . This is the form of the metric given in [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.

2.3 Coordinate systems

2.3.1 Coordinate systems, T3-Gowdy

2.3.2 Working definition, T3-Gowdy

2.3.3 Coordinate system, S3 and S2 × S1

2.3.1 Coordinate systems, T³-Gowdy

2.3.2 Working definition, T³-Gowdy

2.3.3 Coordinate system, S³ and S² × S¹