However, given its importance here, we will dedicate some space to a review of our fully general-relativistic RMHD framework, which was first presented in Ref. [@Dionysopoulou:2012pp] and represents the extension of the special-relativistic RMHD formalism discussed in Ref. [@Palenzuela:2008sf]. A similar but independent extension has been presented recently in Ref. [@Bucciantini2012], which describes the first 3+1 general-relativistic RMHD implementation in fixed spacetimes.

We start by presenting the augmented Maxwell equations $$\begin{aligned}
  \nabla_{\nu} (F^{\mu \nu} + g^{\mu \nu} \psi) &=& I^{\mu} - \kappa
  n^{\mu} \psi \,,
\label{Maxwell1} \\
 \nabla_{\nu} (^*F^{\mu \nu} + g^{\mu \nu} \phi) &=& - \kappa n^{\mu}
 \phi \,,
\label{Maxwell2} \end{aligned}$$ where $g^{\mu \nu}$ is the 4-metric, $F^{\mu \nu}$ is the Faraday tensor, $^{*\!}F^{\mu \nu}$ is the Maxwell tensor, $I^{\mu}$ is the electric four-current density, and $\phi,\,\psi$ are two auxiliary scalar variables added to the Maxwell equations to control the constraints for the magnetic and electric parts, respectively (see below).

After a standard 3+1 splitting of spacetime, the Maxwell and Faraday tensors can be decomposed in terms of the electric ($E^i$) and magnetic ($B^i$) fields measured by an observer moving along the normal direction $n^{\nu}$ ([i.e., ]{}normal or Eulerian observer) as $$\begin{aligned}
F^{\mu\nu} &=& n^{\mu} E^{\nu} - n^{\nu} E^{\mu} 