Net Charge Accretion in Magnetized Kerr Black Holes

Abstract

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 $Q_{\mathrm{w}}$.

Figures (8)

• Figure 1: The critical impact parameter $b_{1}^{\mathrm{univ}}$ in the universal limit $\varepsilon\to-\infty$, obtained numerically as a function of the spin parameter $\alpha$.
• Figure 2: Evolution of the critical impact parameter for an attracted particle, $b_{1}^{-}$, in the large-$\varepsilon$ limit for three initial energies (different shapes and colors) at two values of the spin parameter $\alpha$ (different panels). The values of $b_{1}^{-}$ for all initial energies converge toward the universal limit for the same $\alpha$.
• Figure 3: Evolution with $b$ of the allowed region for particle presence (green) for $E=\sqrt{2}$, $\alpha=0.3$, and $\varepsilon=10$. The (non-equatorial) critical point occurs at $b_{0}=1.87777$. The gray region represents the region $r<r_{+}$. Panel a shows the allowed region for $b<b_{0}$ which permits the absorption of the charged particle; panel b shows the disconnection of the horizon from infinity, precisely at the critical point $b_{0}$; and panel c shows the disconnected allowed regions for $b>b_{0}$.
• Figure 4: Maps of allowed (green) and forbidden (white) regions for particle presence around a critical point. The gray region around the origin represents the zone $r<r_{+}$. In panel a the equatorial critical point has $n_{\mathrm{u},rr}<0$ and can therefore correspond to a (dis)connection from infinity to the horizon, while in panel b the critical point has $n_{\mathrm{u},rr}>0$ and cannot correspond to a (dis)connection from infinity to the horizon. The parameters for both panels are $E=3,\,\alpha=0.5$, while $\varepsilon=-100$ for panel a and $\varepsilon=15$ for panel b.
• Figure 5: The upper bound on the accretion cross section for repelled charges $\sigma_{\mathrm{max}}^{+}$, the lower bound on the accretion cross section for attracted charges $\sigma_{\mathrm{min}}^{-}$, and their difference $\Delta\sigma=\sigma_{\mathrm{min}}^{-}-\sigma_{\mathrm{max}}^{+}$ as a function of $\mathopen{}\mathclose{\left|\varepsilon\right|$, for the case $E=\sqrt{2}}$, $\alpha=0.75$. Presented as the purple dashed line is the lower bound for attracted charges obtained through the universal EOM.