Magnetospheres of many astrophysical objects can be accurately described by the low-inertia (or "force-free") limit of MHD. We present a new numerical method for the solution of equations of force-free relativistic MHD based on the finite-difference time-domain approach, with a prescription for handling the spontaneous formation of current sheets. We use this method to study the time-dependent evolution of pulsar magnetospheres in both aligned and oblique magnetic geometries. For the aligned rotator, we confirm the general properties of the time-independent solution of Contopoulos et al. For the oblique rotator, we present the three-dimensiional structure of the magnetosphere and compute, for the first time, the spin-down power of pulsars as a function of the inclination of the magnetic axis. We find that the pulsar spin-down luminosity is L ≈ (μ 2 Ω / c 3 )(1 + α), where μ is the stellar dipole moment, Ω * is the rotation frequency, and α is the magnetic inclination angle. We also discuss the effects of current sheet resistivity and reconnection on the structure and evolution of the magnetosphere.
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A 2006 study studied this question.