10.2 Relevance of the asymptotic velocity to the issue of curvature blow up

Due to the definition, it is clear that v∞ is a geometric object from the wave-map perspective. Furthermore, it is possible to prove that if v∞ (𝜃0) ⁄= 1, then the curvature blows up along any causal curve ending at 𝜃0; see the proof of [79Jump To The Next Citation Point, Proposition 1.19, p. 989].

Note that the Gowdy metric corresponding to P (τ,𝜃) = τ, Q = 0 and λ(τ,𝜃) = τ is the flat Kasner metric. Consequently, the curvature tensor is in that case identically zero. For this solution, v∞ = 1. In particular, if v ∞(𝜃0) = 1, the Kretschmann scalar need not necessarily be unbounded along a causal curve ending at 𝜃0.

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