Lemma: Adapted charts

These are proper time-oriented Lorentz-affine charts on regular ambient neighbourhoods. The nonresting face has adapted time coordinate zero. The boost factors for $V$ and $U$ are respectively $3$ and $5/4$.

Proof. In a spatial orthonormal basis along $g$ and perpendicular to it, $B(g)$ is a $1+1$ boost and an unchanged perpendicular coordinate. Direct multiplication gives $B(g{)}^{\mathsf{T}}{\eta }_{3}B(g)={\eta }_3$, determinant one, and positive $00$-entry. Its time coordinate is $\gamma (\tau -g\cdot x)$, proving adaptation. The gradient norms in Lemma lem:graphs give the two boost factors.