Irreducible-Mass Growth and Extractable Energy in a Self-Gravitating Blandford-Znajek Engine

Abstract

Wang recently constructed a self-gravitating split-monopole Blandford-Znajek engine and obtained its secular horizon mass and angular-momentum loss rates. We determine the previously unknown consequences of this evolution: the growth of the irreducible mass, the division of rotational work between outgoing electromagnetic power and horizon dissipation, and the lifetime-integrated energy removed from the horizon. For fixed magnetic flux and impedance matching, one half of the instantaneous rotational work is carried outward, and one half irreversibly increases the horizon area. Integrating Wang's spin-down trajectory, we find in the weak-field limit the exact results A_f/A_0=\sqrt{e} and E_{extr}^H/(M_{H,0}c^2)=1-exp{1/4}/\sqrt{2}\simeq 0.09206. The leading maximum-power spin-flux allocation instead gives E_{\rm extr}^{H}/(M_{H,0}c^2)\simeq 0.04668. These results provide the first irreducible-mass and extractable-energy accounting of Wang's engine. Their extension from quasilocal horizon quantities to energy received at infinity requires a separate flux analysis.