  We performed one-dimensional force-free magnetodynamic numerical simulations
of the propagation of Alfven 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
Alfven waves. We considered axisymmetric and stationary magnetosphere and
perturbed the background magnetosphere to obtain the linear wave equation for
the Alfven wave mode. The numerical results show that the energy of Alfven
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 an additional wave induced by the Alfven wave.
Considering the additional fast magnetosonic wave induced by the Alfven wave,
the energy conservation is recovered. Similar relativistic phenomena, such as
the amplification of Alfven waves and induction of fast magnetosonic waves, are
expected around a spinning black hole.
