1.1 Potential difference

The potential difference between the (north) pole of the black hole and its equator, as measured by an stationary observer v^μ ∼ ξ^μ_{(t )}+ Ω^Fξ^μ_(φ), is

 (1.1)

 (1.2)

One has

 (1.3)

 (1.4)

Thus

 (1.5)

For a stationary observer co-rotating with a black hole ΔU vanishes. For Ω^F ≠ Ω^H there is a non-vanishing electric potential.

One can easily ‘predict’ this effect by using the analogy of a black hole horizon with a conducting surface with the effective surface resistance R^H = 4π/c = 377 Ω. It is well known that a rotation of a metallic conducting sphere in an external homogeneous magnetic field generates a difference in the potentials between the pole and the equator of the sphere. Such a device is known as a unipolar inductor.

1.2 Black hole magnetosphere and efficiency of the power generating process

Astrophysical black holes are surrounded by plasma. In the most important case for astrophysics, the conductivity of the plasma is so high that the electric field in the reference frame, comoving with the plasma, vanishes, and the magnetic lines of force are ‘frozen’ into the plasma. In this case, the electric and magnetic fields in an arbitrary reference frame are perpendicular to each other (degenerate fields):

  E · B = 0.    (1.6)

Note that this condition is only an approximation and generally a small longitudinal electric field is present. To solve problems concerning the configuration of fields, currents, and charge distributions, it is only necessary that the inequality

|E · B| << |E² − B²|       (1.7)

is satisfied. Small deviations from the exact equation in the neighborhood of a black hole may prove to be important for a number of astrophysical processes. To simplify the problem, it is usually assumed that the system (a black hole, surrounding plasma, and the electromagnetic field) is stationary and axisymmetric.

Denote by ṽ^μ a vector of an observer comoving with the plasma. Then one has

 (1.8)

In the reference frame of this observer, the electric field vanishes. This property is also valid for any frame which is moving with respect to ṽ^μ with the velocity along the magnetic field. Let us choose a special solution of equation (1.8) which meets the symmetry property

 (1.9)

where Ω^F is a function of r and θ. Let us stress that the vector ṽ^μ, which has evident physical meaning, must always be timelike, while the vector v^μ can be spacelike. This happens near the horizon if Ω^F ≈ Ω^H.

In the force-free approximation, the rotational energy of the black hole is extracted

 (1.10)

This energy is transferred along magnetic lines of force into a region located far away from the black hole where the force-free condition is violated; energy is pumped into accelerated particles, and so forth.

The power is maximal when Ω^F = Ω^H /2. Macdonald and Thorne  demonstrated that this condition is likely to be implemented in the described model.

In order of magnitude, the power of the ‘electric engine’ outlined here is

 (1.11)

Here B is the magnetic field strength in the neighborhood of the black hole. Sometimes this electric engine is described in terms of concepts taken from electrical.

1.3 Black hole as a unipolar inductor

The lines of constant θ are equipotential curves at the horizon since the field E^H is meridional. Hence, the potential difference between two equipotential lines (marked by l and 2) is

 (1.12)

where d l is the distance element along a meridian of the black hole surface and ΔU^H is the difference between the values of UH on the equipotentials 1 and 2. The approximate equality is valid if Ω^F ≈ Ω^H /2, maximal Ω^H, and the equipotentials 2 and 1 corresponding to the equatorial and polar regions, respectively.

However, ΔU^H can be written in terms of the surface current i^H and resistance:

 (1.13)

where Δl is the distance along the meridian between the equipotentials 2 and 1. Substituting the expression for i^H, we obtain

 (1.14)

where

 (1.15)

is the total resistance between the equipotential lines 2 and 1. (If the equipotentials 2 and 1 correspond to the equator and to θ ≈ π/4, the integration of (1.15) yields ΔZ ^H ≈ 30 Ω).

These formulas permit the conclusion that in this model the rotating black hole acts as a battery with an e.m.f. of order

 (1.16)

and an internal resistance of about 30 Ω.

This mechanism (and a number of its variants) has been employed in numerous papers for the explanation of the activity of the nuclei of galaxies and quasars.