Irreducible-Mass Balance for Magnetized Null Horizons with an Internal Current Sheet

RESUMO

We begin with a general question: how does the irreducible mass of a magnetized black hole change when the horizon is not stationary? The null Raychaudhuri equation gives an exact area balance for any smooth null horizon, without assuming stationarity or axisymmetry. When the horizon is axisymmetric and admits an integrable quasilocal Hamiltonian, this geometrical identity becomes an energy balance. The equation keeps area growth, rotational and electromagnetic work, nonstationary focusing, canonical-flux corrections, and internal-boundary terms separate. We then turn to Wang's self-gravitating split-monopole Blandford-Znajek engine. In the weak-field and slow-rotation regime, P/M_H\ll1 and a/M_H\ll1, our general balance reproduces Wang's mass- and angular-momentum loss rates. The irreducible mass, M_{irr}=\sqrt{A_H/(16\pi)}, reveals what those rates do not show by themselves. It measures the part of the rotational work absorbed irreversibly by the horizon. At fixed magnetic flux and under impedance matching, half of the instantaneous rotational work leaves as electromagnetic power. The other half increases the horizon area and M_{irr}. The assumptions behind this reduction are derived explicitly. A minimal nondissipative world-volume action makes the current-sheet contribution vanish through O(p^2\epsilon^2). Direct power counting of Wang's perturbative fields, together with a compatible canonical-flux prescription, gives a relative focusing and canonical correction of O(p^2)+O(\epsilon^2). These estimates mark the range in which the reduced Blandford-Znajek trajectory remains controlled. Extending that trajectory to high spin requires an additional extrapolation. Relating the quasilocal horizon loss to energy measured at infinity requires a separate asymptotic flux calculation.