Lemma: Chart validity

The matrices are proper future-oriented Lorentz matrices. The affine maps send their respective face planes to the displayed horizontal planes. They are regular charts only away from $\mathcal{W}$; their formulas extend to cell closures as affine formulas.

Proof. For ${\eta }_{2+1}=\mathrm{diag}(-1,1,1)$, multiply to check $B^{\mathsf{T}}{\eta }_{2+1}B={\eta }_{2+1}$, $\det B=1$, and positive time component. For the $h$-plane, $3t-2\xi -2\eta =3s$. For the $U_i^{-}$-plane, $(5t-3\eta )/4=5s/2$, and similarly for $U_i^{+}$. The axial extension changes none of these identities.