Interactions of magnetohydrodynamic waves with gravitomagnetic fields, and their possible roles in black-hole magnetospheres.

Abstract

The magnetospheres of rotating, magnetized black holes are thought to generate some of the jets observed in quasars and active galactic nuclei. Previous research on such magnetospheres has focused on stationary configurations. This paper is an initial, exploratory study of dynamical magnetospheres. Because a dynamical study in a rotating hole's Kerr spacetime would be exceedingly difficult, this paper introduces a class of simpler, plane-symmetric or cylindrically symmetric model spacetimes in which to explore the dynamics. These model spacetimes preserve the key physical features of the Kerr geometry: they have a Kerr-type gravitomagnetic potential (shift function), a Kerr-type horizon, and a Kerr-type asymptotically flat region far from the horizon. This first exploratory study is restricted to the simplest of these spacetimes, one with the planar metric $d{s}^{2}=\ensuremath{-}d{t}^{2}+{(dx+\ensuremath{\beta}dt)}^{2}+d{y}^{2}+d{z}^{2}$, gravitomagnetic potential $\ensuremath{\beta}={V}_{F}(tanhz\ensuremath{-}1)$, and horizon lateral velocity $\frac{\mathrm{dx}}{\mathrm{dt}}=2{V}_{F}$. In this spacetime the asymptotically flat region is at $z\ensuremath{\gg}1$, and the horizon has been pushed off to $z=\ensuremath{-}\ensuremath{\infty}$. We treat the dynamical magnetospheres using the 3+1 formalism of general-relativistic magnetohydrodynamics (GRMHD) developed in an earlier paper. Kerr-type models of stationary magnetospheres are built in this spacetime as exact solutions to the fully nonlinear equations of GRMHD. In these solutions the magnetic field, under the influence of the horizon's lateral motion, is driven to move laterally with velocity $\frac{\mathrm{dx}}{\mathrm{dt}}={V}_{F}$; and plasma particles (${e}^{+}{e}^{\ensuremath{-}}$ pairs) are created at $z=0$, and are then driven up to relativistic velocities by magnetic-gravitomagnetic coupling, as they flow off to "infinity" ($z=+\ensuremath{\infty}$) and down toward the "horizon" ($z=\ensuremath{-}\ensuremath{\infty}$). Weak perturbations of these analytic magnetospheres are studied using a linearization of the GRMHD perturbation equations. The linearized perturbation equations are Fourier analyzed in $t$ and $x[\mathrm{exp}(\ensuremath{-}i\ensuremath{\omega}t+i{k}_{x}x)]$ and are solved numerically to obtain the $z$ dependence of the perturbations. The numerical solutions describe the response of the magnetosphere to oscillatory driving forces in the plasma-injection plane, $z=0$. This models the response of a Kerr hole's magnetosphere to oscillatory driving forces near the plasma-production region---forces that might arise when lumpy magnetic fields, anchored in an accretion disk, orbit the hole, pressing inward on the magnetosphere. In our model spacetime the magnetosphere responds resonantly at frequencies (as measured at infinity) $\ensuremath{\omega}={k}_{x}{V}_{F}$, i.e., at frequencies for which the perturbations are stationary as seen in the field lines' rest frame. The analog for a Kerr magnetosphere would be resonant responses at $\ensuremath{\omega}=m{\ensuremath{\Omega}}_{F}$, where $m$ is the azimuthal quantum number of the perturbation and ${\ensuremath{\Omega}}_{F}$ is the field-line angular velocity (roughly equal to half the horizon angular velocity). Such resonances, if they occur in real black-hole magnetospheres, would show up as a modulation of the jet's outflowing energy flux.