the orthonormal frame $e_a = e_a{}^\alpha \partial_\alpha$, $\alpha = t,\varrho, z, \varphi$, where the non-trivial matrix elements are $$\label{frame_matrix}
e_0{}^t = e^{-\gamma \nu}, \quad e_0{}^\varphi = e^{-\gamma \nu} \omega, \quad e_1{}^\varrho = e^{-\mu}, \quad e_2{}^z = e^{-\mu}, \quad e_3{}^\varphi = \frac{e^{\gamma \nu}}{\varrho H}.$$ The corresponding co-frame reads $\alpha^a = e^a{}_\alpha \mathrm dx^\alpha$, where $(e^a{}_\alpha) = (e_a{}^\alpha)^{-1}$ (the inverse matrix), and via the relation $p^\mu \partial_\mu = v^\mu e_\mu$ this frame introduces the new momentum variables $v^0, v^1, v^2, v^3$, given by $$\label{def_frame}
v^0 = e^{\gamma\nu} p^t, \quad v^1 = e^\mu p^\varrho, \quad v^2 = e^\mu p^z, \quad v^3 = \varrho H e^{-\gamma\nu} \left(p^\varphi - \omega p^t\right).$$ In the remainder of this article we work with the coordinates $$ \label{the_coords}
t \in \mathbb R, \quad \varrho \in [0, \infty), \quad \varphi \in [0,2\pi), \quad z \in \mathbb R, \quad (v^0, v^1, v^2, v^3) \in \mathbb R^4$$ on the tangent bundle $T\mathscr M$. In these frame coordinates the mass shell relation (\[m\_s\_r\]) becomes $$-c^2 = -c^2 \left(v^0\right)^2 + \left(v^1\right)^2 + \left(v^2\right)^2 + \left(v^3\right)^2$$ and on $\mathscr P$ we consequently have $$\label{mass_shell_frame}
v^0 = \sqrt{1+\gamma |v|^2}, \quad \mathrm{where}\, |v| = \sqrt{\left(v^1\right)^2 + \left(v^2\right)^2 + \left(v^3\right)^2}.$$

The method of characteristics {#sect_char}
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The Vlasov 