9.2 Gowdy to Ernst transformation

In the construction of spikes, the Gowdy to Ernst transformation [71Jump To The Next Citation Point, p. 2963] will play an important role. Let (Q, P ) be a solution to Equations (16View Equation) – (17View Equation) 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
P = τ − P, Q = − e2(P− τ)Q , Q = − e2P Q . (41 ) 1 1τ 𝜃 1𝜃 τ
Note that Equation (17View Equation) ensures that the definitions of Q1 τ and Q1 𝜃 are compatible. Furthermore, (Q1, P1) solve Equations (16View Equation) – (17View Equation). Assuming Q1 (τ0,𝜃0) = q0, we shall denote the corresponding transformation, defined by Equation (41View Equation), 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.

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