same regularity as the axially symmetric function on $\mathbb R^n$.

We now introduce a coordinate gauge and the functions in terms of which we will formulate the reduced EVM-system. Consider the four dimensional manifold $\mathscr M$ which is assumed to be homeomorphic to $\mathbb R^4$ and which is equipped with the cylindrical coordinates $t$, $\varrho$, $z$, $\varphi$. A stationary Lorentzian metric is characterised by the four time independent, axially symmetric functions $\nu,\mu,\omega:\mathscr M \to \mathbb R$ and $H: \mathscr M \to \mathbb R_+$. It can be written in the form $$\label{ansatz_metric}
g = -c^2 e^{\frac{2\nu(\varrho, z)}{c^2}}\mathrm dt^2 + e^{2\mu(\varrho, z)}\mathrm d\varrho^2 + e^{2\mu(\varrho, z)}\mathrm dz^2 + \varrho^2H(\varrho, z)^2 e^{-\frac{2\nu(\varrho, z)}{c^2}} \left(\mathrm d\varphi - \omega(\varrho, z) \mathrm dt\right)^2,$$ cf. [@b72] for details.

The electro-magnetic field tensor $F$ is given as the exterior derivative of the electro-magnetic four potential $A\in\Lambda^1(\mathscr M)$, i.e. $F = dA$. With respect to the coordinate co-basis of $t$, $\varrho$, $z$, $\varphi$ the electro-magnetic potential $A$ takes the form $$A = A_t \mathrm dt + A_\varrho \mathrm d\varrho + A_\varphi \mathrm d\varphi + A_z \mathrm dz.$$ We assume that all components are time independent and axially symmetric.

In terms of the electro-magnetic field tensor $F$ the electric field 