Blandford-Znajek process in vacuo and its holographic dual

Blandford and Znajek discovered a process by which a spinning black hole can transfer rotational energy to a plasma, offering a mechanism for energy and jet emissions from quasars. Here we describe a version of this mechanism that operates with only vacuum electromagnetic fields outside the black hole. The setting, which is not astrophysically realistic, involves either a cylindrical black hole or one that lives in $2+1$ spacetime dimensions, and the field is given in simple, closed form for a wide class of metrics. For asymptotically anti–de Sitter black holes in $2+1$ dimensions, the holographic dual of this mechanism is the transfer of angular momentum and energy, via a resistive coupling, from a rotating thermal state containing an electric field to an external charge density rotating more slowly than the thermal state. In particular, the entropy increase of the thermal state due to Joule heating matches the Bekenstein-Hawking entropy increase of the black hole.

Figure 1

A line current driven opposite to an ambient electric field emits a Poynting flux of energy. This field configuration corresponds to (2). Its generalization to any stationary axisymmetric spacetime (1), allowing for a radial component of the magnetic field, is given by (4). For spinning black cylinder spacetimes, the line current is behind the horizon, and the field is nonsingular on the horizon provided the Znajek condition (9) holds. Energy is then extracted from the black cylinder if there is magnetic flux through the horizon and $0<{\mathrm{\Omega }}_F<{\mathrm{\Omega }}_H$ (10). The nonzero black hole spin is what makes this extraction possible.

Figure 2

Electric field lines for the field (13), with $r_{+}=0.3$, $r_{-}=0.2$, and ${\mathrm{\Omega }}_F={\mathrm{\Omega }}_H/2=1/3$, in units with $\ell =1$. As explained in Appendix pp3, the field lines are intersections of the electric field sheets with a surface of constant ingoing Kerr coordinate $v$, plotted with $r$ and $\overline{\varphi }$ treated as polar coordinates on the Euclidean plane. The field lines for $r<r_{-}$ are not plotted, since $F_3$ is singular at $r_{-}$.