One-dimensional Force-free Numerical Simulations of Alfvén Waves around a Spinning Black String

We performed one-dimensional force-free magnetodynamic numerical simulations of the propagation of Alfvén waves along magnetic field lines around a spinning black-hole-like object, the Banados–Teitelboim–Zanelli black string, to investigate the dynamic process of wave propagation and energy transport with Alfvén waves. We considered an axisymmetric and stationary magnetosphere and perturbed the background magnetosphere to obtain the linear wave equation for the Alfvén wave mode. The numerical results show that the energy of Alfvén waves monotonically increases as the waves propagate outwardly along the rotating curved magnetic field line around the ergosphere, where energy seems not to be conserved, in the case of energy extraction from the black string by the Blandford–Znajek mechanism. The apparent breakdown of energy conservation suggests the existence of a wave induced by the Alfvén wave. Considering the additional fast magnetosonic wave induced by the Alfvén wave, the energy conservation is confirmed. Similar relativistic phenomena, such as the amplification of Alfvén waves and induction of fast magnetosonic waves, are expected around a spinning black hole.