Electromagnetic Energy Extraction in Kerr Black Holes through Frame-Dragging Magnetospheres

It is argued that the zero-angular-momentum-observers (ZAMOs) circulating with the frame-dragging-angular-velocity $\omega$ plays a leading part in energy extraction. When the condition $\Omega_{\rm F}<\Omega_{\rm H}$ is satisfied, where $\Omega_{\rm H}$ and $\Omega_{\rm F}$ are the horizon and field-line (FL) angular-velocities (AVs), they will see that the null surface S$_{\rm N}$ with $\omega_{\rm N}=\Omega_{\rm F}$ always exists in the force-free magnetosphere. The pivotal ZAMO-measured FLAV $\Omega_{\rm ZF} \equiv \Omega_{\rm F} - \omega$ changes sign on this surface S$_{\rm N}$ where the force-free and freezing-in conditions break down. The force-free magnetosphere is divided on this surface, with particle-current sources on it. The outer domain ${D}_{\rm (out)}$ outside S$_{\rm N}$ spins forward ($\Omega_{\rm ZF}>0$), whereas the inner domain ${D}_{\rm (in)}$ inside spins backward ($\Omega_{\rm ZF}<0$). Because the electric field ${\bf E}_{\rm p}$ reverses direction there, the Poynting flux reverses direction as well from outward to inward, though the positive angular momentum always flows outwardly. Electromagnetic self-extraction of energy will be possible only through the frame-dragged magnetosphere, with the inner domain ${D}_{\rm (in)}$ nested between the horizon and the surface S$_{\rm N}$, when $\Omega_{\rm F}<\Omega_{\rm H}$ is ensured by the 1st and 2nd laws of thermodynamics.