^{*}

^{†}

^{3}.

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

Prodigious amounts of energy wind up in the rotation of spinning black holes. Although nothing can escape from inside a black hole, that energy can be extracted via a dynamo effect, the “Blandford-Znajek (BZ) process”

Although the principles of the BZ process are fully understood (e.g.,

We begin here with cylindrical black hole (BH) examples, since they are easier to visualize, and the analogy with the astrophysical case is closer. The

In the BZ process

A line current driven opposite to an ambient electric field emits a Poynting flux of energy. This field configuration corresponds to

Consider this model first in flat, empty spacetime. If current flows only along the axis, the Poynting flux emerges from the axis. Of course a line source of Poynting flux does not much resemble the BZ process, so let us introduce a cylindrical black hole and hide the line source behind the event horizon. A solution essentially like the one just described can be placed on this black hole spacetime but, since nothing can escape from a black hole, we expect that energy cannot be extracted from the source behind the horizon. Indeed, if we try to construct such a solution, we find that the fields are singular on the horizon. In effect, the horizon becomes a singular source of energy, which again does not resemble the BZ process. However, if the black hole is spinning with angular velocity

We adopt cylindrical spacetime coordinates

To describe the electromagnetic field we employ the language of differential forms, which greatly simplifies computations, especially in curved spacetime. (For a concise review of differential forms and their application to electromagnetism, see Appendix A of

Let us begin with the simple example in flat spacetime,

The field

The singularity on the axis carries a line current

We are interested in the solution

The Maxwell action is given by

The energy current outward through a surface of constant

The line element

If the regularity condition

The vacuum Maxwell equations are invariant under electric-magnetic duality,

The dual field strength

Dualizing interchanges electric and magnetic quantities, so we change our notation for the parameters accordingly:

All stationary, axisymmetric vacuum solutions to Maxwell’s equations in a

This solution can be placed in particular on the rotating BTZ black hole background

Electric field lines for the field

We now turn to the AdS/CFT dual interpretation of the vacuum BZ process on the spinning BTZ black hole background. A number of previous studies have employed this duality with Einstein-Maxwell theory in three spacetime dimensions, sometimes with other fields included (see e.g.

The conductivity

The response to weak fluxes of electromagnetic energy and angular momentum can be determined using the laws of black hole mechanics (which are derived from Einstein’s equation). The flux formula

The black hole entropy

The rate of entropy generation per unit spacetime volume, due to Ohmic dissipation in the holographic dual field theory process, is given by the rate of energy dissipation divided by the temperature in the corotating frame of the thermal state. The energy dissipation rate is the inner product of the electric field with the charge current in that frame, hence the total rate of entropy generation is

We have thus far treated the electromagnetic field as a test field on the background of a spinning BTZ black hole background and inferred adiabatic evolution from the conservation laws. To more fully probe the duality one should solve the coupled Einstein-Maxwell equations. A similar study was carried out in

We thank S. Hartnoll, G. Horowitz, N. Iqbal, K. Jensen, D. Marolf, A. Speranza, and S. Theisen for helpful discussions and correspondence. This research was initiated at the Peyresq Physics 21 meeting and partially supported there by OLAM. T. J. was also supported in part by NSF Grants No. PHY-1407744 and No. PHY-1708139 at the University of Maryland, and by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. The work of M. J. R. was supported by the Max Planck Gesellschaft through the Gravitation and Black Hole Theory Independent Research Group and by NSF Grant No. PHY-1707571 at Utah State University.

In three spacetime dimensions, the force-free condition

To characterize the regularity condition for the field strength

The spinning BTZ black hole metric is given in Boyer-Lindquist coordinates by

When expressed using these regular coordinates, the dual field