short time scales and any correction due to the mass difference between electrons and protons can be neglected. As a result, there is no global charge separation and the plasma is neutral. Note also that the first term in Eq.  accounts for an isotropic scalar law for the current, while the rest represent anisotropies due to the presence of a magnetic field in the comoving frame.

Ideally, it would be desirable to have a well-defined prescription of the conductivity tensor as a function of the fluid properties, $\sigma_{\mu\nu} = \sigma_{\mu\nu} (\rho, \epsilon, b_{\mu})$, which stems from the microphysical properties of the plasma (see Ref. [@Andersson2012] for a recent discussion). In practice, however, we are far from having such a prescription in the extreme physical conditions characterizing merging neutron stars. However, if the collision time scale is much smaller than the electron cyclotron period[^2] or, equivalently, if $\xi \ll |b^{\mu}b_{\mu}|^{-1}$, electrons do not have sufficient time to gyrate perpendicular to the magnetic field lines. Under these conditions, the isotropic part of the conductivity is the dominant one (electrons essentially slide along the magnetic-field lines), and expression can be approximated as $\sigma_{\mu\nu}\approx\sigma g_{\mu\nu}$. As a result, the spatial three-current density in the 