Physical gradients and boosts

In the first developed sector,

They are orthonormal only at $\alpha =\pi /2$. The Cartesian gradients are

For $0<|g|<1$, define

In an orthonormal spatial basis along and perpendicular to $g$, this is a standard two-dimensional boost plus an unchanged perpendicular coordinate. Consequently,

Thus $\mathcal{B}(g)\in {\mathrm{SO}}^{+}(2,1)$. Here ${\gamma }_h=3$ and ${\gamma }_{\pm }=5/4$.

Put $O_i=\mathrm{diag}(1,\mathrm{Rot}(i\alpha ))$. The affine extensions of the adapted face-coordinate maps are

If a face is $T=q+g\cdot x$, its first boost coordinate is $\gamma (T-g\cdot x)=\gamma q$. Therefore nonresting faces have adapted time $T^{\prime }=0$, whereas resting faces have $T^{\prime }=2+s$.

These are regular ambient affine coordinates away from the singular axis and algebraic extensions to face closures. They do not supply a regular Minkowski neighbourhood of the apex.