Breakdown of force-free electrodynamics in electric zones

It is shown that force-free electrodynamics (FFE) breaks down in regions where $B^2 -E^2 <0$ (electric zones) even if $\pmb{E}\cdot\pmb{B} =0$. Spontaneous creation of such regions will inevitably lead to plasma oscillations that will subsequently decay over a few periods via anomalous heating and, under certain conditions, emission of high energy quanta, until the system relaxes to a state in which $B^2-E^2 \sim 0$. For M87, assuming pair plasma with order unity multiplicity, the inverse Compton cooling time is estimated to be shorter than the dynamical time when $E^2/B^2-1 > (10^4/\sigma)^2$ roughly, where $\sigma$ is the magnetization. If the electric zone is weak, the global system will maintain a nearly force-free state, however, the force-free condition, $F^{\mu\nu}J_\nu=0$, will be broken at the order of the access electric field and cannot describe wave dynamics. Our analysis does not support recent claims, that creation of electric zones can trigger a transition to force-free turbulence which, when generated in the ergosphere of a Kerr black hole, can lead to extraction of the black hole rotational energy. Whether some secondary electromagnetic modes produced in the decaying electric zone can extract the BH energy is yet an open question.