Linear perturbations in force-free black hole magnetospheres .2. Wave propagation

Short wavelength small perturbations in the force-free black hole magnetosphere are investigated. A geometric optical approximation for the linear perturbation theory of the force-free electromagnetic field is developed for this purpose. Two wave modes arise from the force-free electromagnetic held: one corresponds to the fast magnetosonic mode and the other corresponds to the Alfven mode in the magnetohydrodynamics. The general properties of these two modes are studied, along with wave propagation in the stationary and axisymmetric black hole magnetosphere and the interaction of the waves with the black hole. The fast magnetosonic mode tends to the linearly polarized light in the force-free limit. Thus the super-radiant scattering occurs in this mode. We interpret this effect by analogy with the negative energy wave appearing in the non-relativistic hydrodynamics. Specific properties of the Alfven mode result from the algebraic peculiarity of the dispersion relation. It is also shown that the gauge arbitrariness in the force-free electromagnetic field appears in this mode. We give a general solution in the stationary and axisymmetric magnetosphere. The light surface behaves as a one-way membrane to the Alfven wave. The Alfven wave also enters a negative energy state in which the super-radiance cannot occur. The underlying physical difference between the two modes is clarified.