Charging of rotating black holes: Kinetic simulations of black hole magnetospheres

Whether a rotating black hole (BH) immersed in an external magnetic field charges up to the Wald value $Q_{\rm W}=2aMB$, where $M$ and $a$ are the BH mass and spin parameter, and $B$ is the strength of the external field, is a long-standing open question in BH electrodynamics, with consequences for charge separation, particle acceleration and the structure of BH magnetospheres. We address it with axisymmetric general-relativistic particle-in-cell simulations performed with GRZeltron, including self-consistent pair creation. For representative values of the BH spin, we evolve the same asymptotically uniform Wald field from two opposite initial horizon charges, $Q_{0}=0$ and $Q_{0}=Q_{\rm W}$, and track the accumulated charge through the BH horizon. We find that both branches relax within a few tens of gravitational times to the same saturated charge value. Thus, the equilibrium is a dynamical attractor of the kinetic magnetosphere rather than a memory of the initial data. The attractor lies well below $Q_{\rm W}$, at $ξ_{\rm eq}\equiv Q_{\rm eq}/Q_{\rm W}\approx 0.3$ for $a\lesssim0.7$, then falling almost to $ξ_{\rm eq}\approx0$ for large spins $a\approx1$. We derive an analytic expression for $ξ_{\rm eq}$ by requiring equal magnetic fluxes through the horizon associated with positive and negative charges, and find its spin dependence to be in agreement with the simulations. The charge state of an astrophysical BH is therefore driven by kinetic plasma processes, and the Wald charge is an upper bound, not a general equilibrium value. Since $ξ_{\rm eq}<1$, charging does not quench the horizon-infinity potential drop that powers energy-extraction processes such as the Blandford-Znajek mechanism.