Black hole electrodynamics is the theory of electrodynamics outside a black hole. This is evident since we cannot gain
information about anything inside the event horizon (i.e. ).
Black hole electrodynamics can be very trivial if you consider just a black hole described by the three usual parameters: mass,
electric charge and angular momentum. Initially simplifying the case by disregarding rotation, we simply get the well known solution
of a point charge. This is of course not physically very interesting, since it seems highly unlikely that any black hole (or any
celestial body) should not be rotating. Adding rotation then we have to use the Kerr metric with the change that charge is present.
This is the Kerr-Newmann geometry where
(1)
A rotating charged black hole creates a magnetic field around the hole because the inertial frame is dragged around the hole.
Far from the black hole at infinity the black hole electric field is that of a point charge, and the magnetic field is a dipole
with magnetic moment .
However, black holes do not even have charges. The ratio of charge to mass, Q/M, cannot exceed
because even charges would be repelled
from the hole, and different charges would neutralize the charge of the hole. So we are forced to consider the elctrodynamic
fields located in the environment outside the black hole.
The domain of a black hole can be separated into three regions (see fig. 1).
1)
The rotating black hole itself and the area near it.
2)
The accretion disk (a region of force-free fields).
3)
An acceleration region outside the plasma.
The magnetic field is frozen into the plasma (flux-freezing) when the conductivity is so high that the electric
field in the reference frame comoving with the plasma vanishes. Then we have that the electric and magnetic fields are perpendicular:
(2)
and that the field is force-free:
(3)
where is the electric charge
density. If the electrodynamic forces dominate over the inertial and gravitational forces, then electric fields and
currents moving with the plasma are parallel to the magnetic field-lines. This condition is violated at the event horizon.
In the acceleration region (zone 3 in fig. 1) both equation 2 and 3 is violated.
In this region the magnetic field of the plasma in region 2 is small enough for inertial forces to dominate.
Conditions 2 and 3 may not
hold near the event horizon, and deviations from these could be important for some physical processes happening there.
Figure 1: Accretion onto a black hole. The dotted line is an example of
an electric current line.
Disk accretion can occur onto supermassive black holes at the center of galaxies and in binary systems where one of
the bodies is a black hole (not necessarily supermassive) and the other (star) has exceeded its critical Roche equipotential
lobe. One example of this is Cyg X-1 with a 33
star losing material to the 16 black hole.
Other good candidates are LMC X-3 and A0620-00.
The accretion disk of a rotating black hole is by the black hole driven into the equatorial plane of the rotation.
This is the Bardeen-Petterson effect (see fig. 2).
It is not until small r < 100M that the disk is equatorial. The force on the disk is gravitational, gravitomagnetic forces (field-line
stretching and recombination) and viscosity. At about r = 2 to 20M the plasma in the disk either falls into the black hole,
or is somehow ejected in either
one or two jets aligned with the rotation axis of the black hole (and the disk).
The processes taking place in the magnetosphere of a black hole is so complex that a full theory has not been developed to describe
it. However, some qualitative features can be deduced.
In the accretion disk around a supermassive black hole gravitational energy is converted into kinetic, thermal and magnetic energy.
The thermal energy ensures that the gas in the disk is highly ionized. The gas is therefore turned into plasma, in which perfect MHD
exists. In other words, flux-freezing is maintained in the plasma of the accretion disk. This is true for the plasma not too close
to the event horizon, where magnetic field lines is separated from the plasma as it is drawn into the black hole. Then the magnetic
field lines might attach themselves to the surface of the black hole, or they may be pushed away from the hole and back into the disk.
Figure 3: The inner region of a black hole. The lines penetrating the accretion disk (which by no means is as thin as
shown here; see fig. 1) are magnetic field-lines. Notice the surface (curled A) and the boundary
(curled C).
Consider figure 3. The closed curve C =
A lies in the accretion disk far
from the black hole, where the flux is frozen into the plasma. We look at it from the point of view of a fiducial observer,
far from the black hole and the disk. Faraday's law applied to the surface and the boundary gives us
(4)
Using the condition that the field is frozen to the plasma at the boundary
(5)
where v is the velocity of the plasma as measured by the observer far from the black hole, equation 4
becomes
(6)
The RHS is the rate with which flux is carried in towards the black hole through the boundary, and the LHS is the flux change per
time interval through the surface. It is now obvious that the magnetic flux must be conserved in time seen from the distant observer.
This is an important conclusion. Now the magnetic field-lines near the black hole can be described: If some irregularity in the
field-lines should happen near the horizon, it will be smeared out with the local velocity of light. Thus the magnetic field near
the horizon will be very clean and uniform, not depending on the state of the magnetic field in the plasma, which is likely to be
very chaotic and disordered. Irregularities in the field near the event horizon posed by the field in the disk will be sorted out
in a time
Flux freezing is an interesting physical phenomenon which, as we have already mentioned, is present in connection
with an accretion disc. This is a known effect in plasma
physics, but perhaps unknown to most physicists. It comes out of well
known electrodynamic laws. Flux freezing is simply that the magnetic flux
through a contour moving with the fluid is constant. Since this holds for all
magnetic field strengths and all kinds of velocity fields, the plasma is able
to amplify or damp the magnetic field by movement. We can deduce this by
following a contour moving with the plasma.
At a time t, the countour is denoted by C at the time t+
dt the contour has moved the distance and been
changed to C' (see figure). Let now be an element of
area on any surface bounded by C,
and an element of area on any surface bounded by C'.
The area bounded by a little element on the countour C,
a little elment on C' and the element is
. Since Maxwell's 2'nd equation on integral form gives us:
we have that the total flux through the closed surface bounded by C and
C' at t+dt is given by:
And then the change in flux from t to t+dt is:
By using Stoke's theorem and
we have:
In the last step we used a vector-relation. So now we se that the flux through a contour
moving with the fluid is constant. Since this holds for all B and all kinds of velocity
fields the plasma is able to amplify or damp a magnetic field. This is called dynamo effect.
From all of this we get that ionized gas falling towards a rotating black hole will start to
rotate around it and carry the poloidal field lines with it. After many rotations the
field has been wound up tightly and the magnetic field has been amplified many times.
This effect is so powerful that the toroidal field in the disc will be many times stronger
than the poloidal field. The disc is off course not rotating as a rigid body. The rotation is differential, and
there is turbulence, and other effects that make it behave chaotically on small scale.
Apart from the differential rotation, the overall picture will remain the same.
In this project we have vizualized a snapshot (no time evolution) of a rotating disc which is chaotic on small scales.
).
(1)
.
because even charges would be repelled
from the hole, and different charges would neutralize the charge of the hole. So we are forced to consider the elctrodynamic
fields located in the environment outside the black hole.
(2) 
(3)
is the electric charge
density. If the electrodynamic forces dominate over the inertial and gravitational forces, then electric fields and
currents moving with the plasma are parallel to the magnetic field-lines. This condition is violated at the event horizon.
In the acceleration region (zone 3 in fig. 1) both equation 2 and 3 is violated.
In this region the magnetic field of the plasma in region 2 is small enough for inertial forces to dominate.
star losing material to the 16 
A lies in the accretion disk far
from the black hole, where the flux is frozen into the plasma. We look at it from the point of view of a fiducial observer,
far from the black hole and the disk. Faraday's law applied to the surface and the boundary gives us
(4)
(5)
(6)
and been
changed to C' (see figure). Let now 
be an element of
area on any surface bounded by C,
and 
an element of area on any surface bounded by C'.
The area bounded by a little element 
on the countour C,
a little elment 
on C' and the element 
. Since Maxwell's 2'nd equation on integral form gives us:
