Tearing instability in relativistic magnetically dominated plasmas

Many astrophysical sources of high energy emission, such as black hole magnetospheres, superstrongly magnetized neutron stars (magnetars), and probably relativistic jets in Active Galactic Nuclei and Gamma Ray Bursts involve relativistically magnetically dominated plasma. In such plasma the energy density of magnetic field greatly exceeds the thermal and the rest mass energy density of particles. Therefore the magnetic field is the main reservoir of energy and its dissipation may power the bursting emission from these sources, in close analogy to Solar flares. One of the principal dissipative instabilities that may lead to release of magnetic energy is the tearing instability. In this paper we study, both analytically and numerically, the development of tearing instability in relativistically magnetically-dominated plasma using the framework of resistive magnetodynamics. We confirm and elucidate the previously obtained result on the growth rate of the tearing mode: the shortest growth time is the same as in the case of classical non-relativistic MHD, namely $\tau =\sqrt{\tau_a \tau_d}$ where $\tau_a$ is the \Alfven crossing time and $\tau_d$ is the resistive time of a current layer.