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

Due to the definition, it is clear that is a geometric object from the wave-map perspective.
Furthermore, it is possible to prove that if , then the curvature blows up along any causal
curve ending at ; see the proof of [79, Proposition 1.19, p. 989].
Note that the Gowdy metric corresponding to , and is the flat
Kasner metric. Consequently, the curvature tensor is in that case identically zero. For this solution,
. In particular, if , the Kretschmann scalar need not necessarily be unbounded along
a causal curve ending at .