Energy self-extraction of a Kerr black hole through its frame-dragged force-free magnetosphere

It is shown that when only the condition} $0<\Omega_{\rm F}<\Omega_{\rm H}$ is satisfied, the Kerr black hole frame-drags its surrounding force-free magnetosphere with the field-line-angular-velocity (FLAV) $\Omega_{\rm F}$, where $\Omega_{\rm H}$ is the horizon angular-velocity. Then, the zero-angular-momentum-observers (ZAMOs) circulating with the frame-dragging-angular-velocity $\omega$ will see that the `null surface' S$_{\rm N}$ where $\omega_{\rm N}=\Omega_{\rm F}$ always exists. They will see that the outer domain $D_{\rm (out)}$ outside S$_{\rm N}$ is prograde-rotating with $\Omega_{{\rm F} \omega}>0$, whereas the inner domain $D_{\rm (in)}$ inside is retrograde-rotating with $\Omega_{{\rm F} \omega}<0$, where $\Omega_{{\rm F} \omega}=\Omega_{{\rm F}} - \omega$ denotes the ZAMO-FLAV. `This surface' S$_{\rm N}$ must be the magneto-centrifugal divider of the force-free magnetosphere, with a kind of plasma-shed on it. Subsequently, the force-free and freezing-in conditions break down on S$_{\rm N}$, thereby allowing the particle-current sources to be set up on S$_{\rm N}$. `This surface' also is the ZAM-surface S$_{\rm ZAMD}$, on which no flow of angular momentum nor electric current can cross. Because the electric field ${\bf E}_{\rm p}$ reverses sign on S$_{\rm N}$, the Poynting flux reverses direction from outward to inward on S$_{\rm N}$. An 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 `this surface' S$_{\rm N}$, in order to comply with the first and second laws of thermodynamics.