Gowdy to Ernst transformation

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 $(Q, P )$ be a solution to Equations (16

) – (17

) with $𝜃 ∈ ℝ$ . In other words, the solution need not be periodic in the spatial variable. Then, up to a constant translation in $Q1$ , two smooth functions $Q 1$ and $P 1$ are defined by

Note that Equation (17

) ensures that the definitions of $Q1 τ$ and $Q1 𝜃$ are compatible. Furthermore, $(Q1, P1)$ solve Equations (16

) – (17

). Assuming $Q1 (τ0,𝜃0) = q0$ , we shall denote the corresponding transformation, defined by Equation (41

), by $GE q0,τ0,𝜃0$ :

(Q1, P1) = GEq ,τ ,𝜃 (Q, P). 0 0 0

We shall refer to this transformation as the Gowdy to Ernst transformation. Note that, even if $(Q, P )$ is periodic in $𝜃$ , the same need not be true of $(Q1, P1)$ . However, here we are mainly interested in local (in space) properties of the solutions, and, therefore, this aspect is not important.