  X-ray activity of Anomalous X-ray Pulsars and Soft Gamma-Ray Repeaters may
result from the heating of their magnetic corona by direct currents dissipated
by magnetic reconnection. We investigate the possibility that X-ray flares and
bursts observed from AXPs and SGRs result from magnetospheric reconnection
events initiated by development of tearing mode in magnetically-dominated
relativistic plasma. We formulate equations of resistive force-free
electrodynamics, discuss its relation to ideal electrodynamics, and give
examples of both ideal and resistive equilibria. Resistive force-free current
layers are unstable toward the development of small-scale current sheets where
resistive effects become important. Thin current sheets are found to be
unstable due to the development of resistive force-free tearing mode. The
growth rate of tearing mode is intermediate between the short \Alfven time
scale $ \tau_A$ and a long resistive time scale $\tau_R$: $\Gamma \sim
1/(\tau_R \tau_A)^{1/2}$, similar to the case of non-relativistic
non-force-free plasma. We propose that growth of tearing mode is related to the
typical rise time of flares, $\sim 10$ msec. Finally, we discuss how
reconnection may explain other magnetar phenomena and ways to test the model.

Astrophysics