p^\nu = -c^2 m_p^2, \qquad x \in \mathscr M, p \in T_x\mathscr M$$ is interpreted as the particles’ rest mass. It can be shown that it stays constant along the characteristic curves of the Vlasov equation (\[eq\_vlasov\]). Consequently the particle distribution function $f$ describing an ensemble of particles where all particles have the same rest mass $m_p$ can be assumed to be supported on the mass shell $\mathscr P_{m_p}$, a seven dimensional submanifold of $T\mathscr M$ which is defined to be $$\mathscr P_{m_p} = \{(x,p)\in T\mathscr M \,:\, g_{\mu\nu}(x)p^\mu p^\nu = -c^2 m_p^2, \; p\,\mathrm{is\,future\,pointing}\}.$$ In the remainder of this article we assume $m_p=1$ for all particles, and we denote the corresponding mass shell simply by $\mathscr P$. The volume form $\mathrm{dvol}_{\mathscr P_x}$ on the mass shell fibre $\mathscr P_x$ over $x\in \mathscr M$ is given by $$\mathrm{dvol}_{\mathscr P_x} = \frac{\sqrt{|\det(g_{\mu\nu}(x))|}}{-p_0}\, \mathrm d p^1 \wedge \mathrm dp^2 \wedge \mathrm dp^3,$$ and the transport operator $\mathfrak T$ is given by $$\label{def_transport_op}
\mathfrak T = p^\mu \partial_\mu + \left(q F^\gamma{}_{\mu} \, p^\mu - \Gamma^\gamma_{\alpha\beta} p^\alpha p^\beta\right) \partial_{p^\gamma}.$$ It is tangent to any mass shell $\mathscr P$ [@sz14].

Assume that we have a solution $(g,f,F)$ of the EVM system and that on $\mathscr 