Geodesic scaling identity

On $a^{-1}{I}_{p,v}$, the curve $\eta (t)={\gamma }_{p,v}(at)$ is geodesic by [F2], with $\eta (0)=p$ and ${\eta }^{\mathrm{\prime }}(0)=av$. The unique maximal solution in [F1] extends every solution with those initial data, so $a^{-1}{I}_{p,v}\subseteq I_{p,av}$ and $\eta (t)={\gamma }_{p,av}(t)$ throughout that domain.

If $a\mathrm{\ne }0$ and $I_{p,av}$ were larger than $a^{-1}{I}_{p,v}$, then $s\mapsto {\gamma }_{p,av}(s\mathrm{/}a)$ would extend ${\gamma }_{p,v}$ beyond $I_{p,v}$, contradicting maximality; applying the same argument in the other direction proves $I_{p,av}=a^{-1}{I}_{p,v}$. If $a=0$, both initial velocity and the right-hand curve are zero/constant, and [F1] gives the global domain $\mathbb{R}$. This also treats dimensions zero and one and all signs of $a$. The domains are open, so no finite endpoint is included. The only choice principle is the explicitly inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ for the geodesic-flow construction; no new selection is made.