### 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, T^{3}-Gowdy

In the case of T^{3}-topology, there are coordinates such that the metric takes the form

Here , , and are functions of and , is a
constant, and the 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, T^{3}-Gowdy

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 obtain

where we have defined by the relation . An alternate definition of a T^{3}-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.

#### 2.3.3 Coordinate system, S^{3} and S^{2} × S^{1}

In the case of S^{3} and S^{2} × S^{1} topology, the metric can be written

where , , , , where is a constant,
and and are functions of and . 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.