Linear perturbations in force-free black hole magnetospheres -- I. General theory

Abstract

A Lagrangian perturbation theory for the treatment of small disturbances of forcefree electromagnetic fields is developed in a covariant way. The geometrical nature of degenerate electromagnetic fields permits us to develop a Lagrangian treatment. A Lagrangian displacement similar to the one used in the perturbation theory of the ideal fluid is introduced. In the Lagrangian displacement, the basic equation is given as a set of two-component second-order partial differential equations. The action that yields our basic equation is also obtained. Using this action, conserved quantities and the energy-momentum tensor of disturbances are studied. Further, it is shown that there exists intrinsic arbitrariness in our theory. It corresponds to the so-called trivial displacement and arises from the gauge freedom of the electromagnetic fields. We apply the theory to the stationary and axisymmetric force-free black hole magnetosphere. The boundary condition at the event horizon is considered. It is shown that the boundary condition for pure-mode solutions takes the same form as that of the case of electromagnetic waves and gravitational waves scattered by the black hole in vacuum. Lastly, in a special limiting example, we discuss the consistency of our theory with the result from the black hole physics.