Net Charge Accretion in Magnetized Kerr Black Holes

We investigate the charging process of a rotating Kerr black hole of mass $M$ and angular momentum $J$ immersed in a stationary, axisymmetric, asymptotically uniform magnetic field of strength $B_{0}$. In Wald's classic analysis (Wald 1974), which was based on the assumption of vanishing injection energy, the black hole was predicted to acquire a universal "saturation charge" $Q_{\mathrm{w}}=2B_{0}J$. However, the physical mechanism that sets the saturation charge must ultimately be governed by the competition between the absorption rates of positively and negatively charged particles. Motivated by this observation, we revisit the problem in the framework of a simple accretion model, where two dilute, equivalent fluxes of charged particles of opposite signs are injected from infinity along the magnetic field lines. The problem then reduces to that of individual particle motion in the electromagnetic field of the magnetized Kerr black hole. Using a combination of numerical and analytical tools, we determine the domains of absorption and establish both lower and upper bounds on the corresponding absorption cross sections. At $Q=Q_\mathrm{w}$ these bounds reveal a systematic difference between the two charge signs. In particular, for sufficiently strong magnetic fields, the lower bound on the absorption cross section for the "attracted" charge exceeds the upper bound for the "repelled" one. This charge accretion imbalance (which we find to become extreme at the limit of large $B_{0}$) indicates a persistent net charge accretion at $Q=Q_{\mathrm{w}}$, implying that the actual saturation charge must differ from Wald's charge $Q_{\mathrm{w}}$.