on a four dimensional manifold $\mathscr M$, a particle distribution function $f \in C^1(T\mathscr M; \mathbb R_+)$, defined on the tangent bundle of $\mathscr M$, and an electro-magnetic field tensor $F\in\Lambda^2(T \mathscr M)$ such that the EVM-system, $$\begin{aligned}
G_{\mu\nu} &= \frac{8\pi}{c^4} \left(T_{\mu\nu} + \tau_{\mu\nu}\right), \label{eq_einstein} \\
T_{\mu\nu} &= g_{\mu\alpha} g_{\nu\beta} \frac{c}{m_p} \int_{\mathscr P_x} f(x,p) p^\alpha p^\beta \, \mathrm{dvol}_{\mathscr P_x}, \label{eq_em_tensor} \\
\tau_{\mu\nu} &= \frac{1}{4\pi} \left(-\frac 1 4 g_{\mu\nu} F_{\alpha\beta} F^{\alpha\beta} + F_{\nu\alpha} F_{\mu}^{\;\;\alpha} \right), \label{el_em_tensor} \\
\mathfrak T (f) &=0, \label{eq_vlasov} \\
\mathrm dF &= 0, \label{maxwell_eq_1} \\
\nabla_\alpha F^{\alpha\beta} &= - 4\pi qJ^\beta,\quad J^\beta =\frac 1 c \int_{\mathscr P_x} f(x,p)p^\beta \mathrm{dvol}_{\mathscr P_x}, \label{maxwell_eq_2} \end{aligned}$$ is satisfied. Here $G_{\mu\nu}$ is the Einstein tensor and we choose units such that $G=1$ ($G$ is the gravitational constant) but we leave $c$ as parameter in the system.

We give a brief explanation of the involved quantities, consult however e.g. [@sz14] for a more detailed introduction to the EVM-system. The particle distribution function $f=f(x,p)$ describes the particle number density at a certain point in $x \in \mathscr M$ with a certain four-momentum $p \in T_x\mathscr M$. The particle number can be obtained via integration. The quantity $m_p$, defined by the relation $$\label{m_s_r}
g_{\mu\nu}(x) p^\mu 