  Force-Free Flectrodynamics (FFE) is a non-linear system of equations modeling
the evolution of the electromagnetic field, in the presence of a magnetically
dominated relativistic plasma. This configuration arises on several
astrophysical scenarios, which represent exciting laboratories to understand
physics in extreme regimes. We show that this system, when restricted to the
correct constraint submanifold, is symmetric hyperbolic. In numerical
applications is not feasible to keep the system in that submanifold, and so, it
is necessary to analyze its structure first in the tangent space of that
submanifold and then in a whole neighborhood of it. As already shown by
Pfeiffer, a direct (or naive) formulation of this system (in the whole tangent
space) results in a weakly hyperbolic system of evolution equations for which
well-possednes for the initial value formulation does not follows.
  Using the generalized symmetric hyperbolic formalism due to Geroch, we
introduce here a covariant hyperbolization for the FFE system. In fact, in
analogy to the usual Maxwell case, a complete family of hyperbolizers is found,
both for the restricted system on the constraint submanifold as well as for a
suitably extended system defined in a whole neighborhood of it. A particular
symmetrizer among the family is then used to write down the pertaining
evolution equations, in a generic ($3+1$)-decomposition on a background
spacetime. Interestingly, it turns out that for a particular choice of the
lapse and shift functions of the foliation, our symmetrized system reduces to
the one found by Pfeiffer. Finally we analyze the characteristic structure of
the resulting evolution system.
