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

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 [79

, 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 $𝜃0$ .