### 9.2 Gowdy to Ernst transformation

In the construction of spikes, the Gowdy to Ernst transformation [71, p. 2963] will play an important
role. Let be a solution to Equations (16) – (17) with . In other words, the solution need
not be periodic in the spatial variable. Then, up to a constant translation in , two smooth functions
and are defined by
Note that Equation (17) ensures that the definitions of and are compatible. Furthermore,
solve Equations (16) – (17). Assuming , we shall denote the corresponding
transformation, defined by Equation (41), by :
We shall refer to this transformation as the Gowdy to Ernst transformation. Note that, even if
is periodic in , the same need not be true of . However, here we are
mainly interested in local (in space) properties of the solutions, and, therefore, this aspect is not
important.