4.4 Pathological examples in the case of Gowdy

Before ending this section, let us record the existence of pathological behavior in the class of Gowdy spacetimes. The most obvious example is provided by the flat Kasner solution; letting λ (t,𝜃) = P (t,𝜃) = ln t in Equation (4View Equation) leads to the metric
gFK = − dt2 + d𝜃2 + t2dσ2 + dδ2.

Viewing this metric on (0,∞ ) × ℝ3,

ϕK (t,𝜃, σ,δ) = (tcoshσ, 𝜃,tsinh σ, δ)

yields an isometry to a subset of Minkowski space. Consequently, g FK is flat, and we shall refer to 3 [(0,∞ ) × ℝ ,gF K] as the flat Kasner solution. This solution is past causally geodesically incomplete, but the curvature is clearly bounded. Furthermore, it is extendible. Considering the same solution on the torus, it is also possible to construct an extension; see, e.g., [82].

However, there are more sophisticated examples of pathologies. In [20], the authors demonstrate that given any positive integer n, there is a polarized Gowdy vacuum spacetime with at least n inequivalent maximal extensions.

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