measured in the rest frame comoving with the fluid, and should be contrasted with ${q}:=-I^{\mu} n_{\mu}$, which is instead the charge density measured by the Eulerian observer. Similarly, $e_\nu$ and $\tilde{\sigma}_{\mu\nu}$ are, respectively, the electric field and the electrical conductivity of the medium (which is a rank-2 symmetric tensor) as measured in the same frame. In collisional plasmas the current in the comoving frame can be considered to be carried by the mobile electrons, with charge $e$, and the conductivity tensor becomes $$\tilde{\sigma}^{\mu\nu}=\sigma (g^{\mu\nu}+\xi^2 b^\mu b^\nu + \xi
\epsilon^{\mu\nu\alpha\beta} u_\alpha u_\beta)\,,
\label{eq:conductivity}$$ with $b^{\mu}$ being the magnetic field in the comoving frame, $\xi:=e\tau_e/m_e$, $\sigma:=n_ee\xi/(1+\xi^2+b^2)$, and $\tau_e$ the electron collision time scale.

Expressing the four-current density of Eq.  in terms of the fields measured by the Eulerian observer we arrive at $$\begin{aligned}
I^\mu =& q n^\mu+q v^\mu+W\sigma[E^\mu+\epsilon^{\mu\nu\alpha} v_\nu
  B_\alpha - (v_\nu E^\nu)v^{\mu}] + \nonumber \\
\phantom{=} & W\sigma\xi^2(E^\alpha
B_\alpha)[B^\mu-\epsilon^{\mu\nu\alpha}v_\nu E_\alpha-(v_\nu
  B^\nu)v^\mu)]\,
\label{eq:ohm}\end{aligned}$$ In deriving Eq.  we have made the implicit assumption that the collision frequency between particles is much larger than the typical oscillation frequency of the plasma, which, we recall, is defined as $\omega_{_{\rm P}} := (4\pi n_e
e^2/m_e)^{1/2}$. This implies that electrons and ions can reach equilibrium on very 