Annu. Rev. Astron. Astrophys. 1984. 22:
471-506
Copyright © 1984 by . All rights reserved |

The physics of dense star clusters and of supermassive objects are
complex and poorly understood. In contrast, the final state of such
systems - if gravitational collapse indeed occurs - is comparatively
simple, at least if we accept general relativity. According to the
so-called no-hair theorems, the endpoint of a gravitational collapse,
however messy and asymmetrical it may have been, is a standardized
black hole characterized by just two parameters - mass and spin - and
described exactly by the Kerr metric. If the collapse occurred in a
violent or sudden way, it would take several dynamical time scales for
the hole to settle down; during that period, gravitational waves would
be emitted. But the final state would still be the Kerr solution,
unless the material left behind constituted a strong
perturbation. [The perturbation due to the infalling material in
steady accretion flows is a negligible perturbation by a wide margin
of order (*r*_{g} / *c*) / *t*_{E}.]

The expected spin of the hole - an important influence on its
observable manifestations - depends on the route by which it formed
(see Figure 1). A precursor spinning
fast enough to be significantly
flattened by rotational effects when its radius was
>> *r*_{g} would
probably have more specific angular momentum than the critical value
*GM*/*c*. A massive black hole that forms "in one go" is thus
likely to
have been fed with as much spin as it can accept and to end up near
the top of the range of angular momentum permitted by the Kerr metric;
the same is true for holes that grow by gradual accretion of infalling
galactic gas (11)
(though the expectation is less clear if they grow
by tidal disruption of stars). We should therefore take full
cognizance of the distinctive properties of spinning black holes.

The Kerr metric changes its character, and the event horizon
disappears, if the specific angular momentum
*J* = *J*_{max} = *GM* / *c*. The
so-called cosmic censorship hypothesis would then require that holes
always form with
*J* < *J*_{max}. The Kerr solution then has a
critical radius
called the static limit, within which particles must corotate with the
hole, though they can still escape. This arises because the
frame-dragging is so strong that even light cones necessarily point in
the direction. This critical
surface, with equatorial radius

(17) |

is not the event horizon itself; the latter occurs at a smaller
radius. The region between the event horizon
(*r* = *r*_{EH}) and the
static
limit is called the "ergosphere," because one can in principle extract
energy from it via a process first proposed by Penrose
(96): a
particle entering the ergosphere can split in two in such a way that
one fragment falls into the hole, but the other leaves the ergosphere
with more energy than the original particle. The extra energy comes
from the hole itself. A Kerr hole can be considered to have two kinds
of mass-energy: a fraction associated with its spin, which can be
extracted via the Penrose process, and an "irreducible" mass
(14,
41).
The fraction that can in principle be extracted is

(18) |

which is 29% for a maximally rotating hole. The above limit is an
instance of a general theorem in black hole physics, according to
which the area of the event horizon (a quantity analogous to entropy)
can never decrease: a Kerr hole has smaller surface area than a
Schwarzschild hole of the same mass. There have been various attempts
to incorporate Penrose-style energy extraction into a realistic
astrophysical model (64,
100).
Those mechanisms that involve particle
collisions or scattering operate only for a special subset of
trajectories (14),
and they would be swamped by accompanying
processes. However, a process involving * electromagnetic* effects
- the Blandford-Znajek mechanism
(29)
- seems more promising (and is discussed further below and in
Section 5).

ORBITS The binding energy per unit mass for a circular
orbit of radius around a Schwarzschild hole (with *J* = 0) is

(19) |

For *r* >> *r*_{g} this reduces to *G
M*/2*r*, which is just the Newtonian binding
energy. However, the binding energy has a maximum of 0.057*c* for an
orbit at
*r*_{min} = 6*r*_{g}, with angular momentum
_{min} =
23 *r*_{g}
*c*. Circular
orbits closer in than this have more angular momentum and are less
tightly bound (as for orbits in classical theory when the effective
force law is
*r*^{-n},
with *n* > 3): the orbits have zero
binding energy for
*r* = 4*r*_{g} (with corresponding angular momentum
_{0} =
4*r*_{g} *c*); and for
*r* = 3*r*_{g}, the expression (19.) goes to
infinity, which implies that
photons can move in circular orbits at this radius. In the Kerr
metric, the behavior of orbits depends on their orientation with
respect to the hole and on whether they are corotating or
counterrotating (14).
For corotating equatorial orbits, the innermost
stable orbit moves inward (as compared with the Schwarzschild case);
it becomes more tightly bound, with a smaller
_{min}. For
(*J*/*J*_{max}) > 0.94,
*r*_{min} actually lies within the ergosphere. As
*J* -> *J*_{max} the stable
corotating orbits extend inward toward
*r* = *r*_{g}, and their binding
energy approaches
(1 - 3^{-1/2})*c* = 0.42*c*. These numbers determine the
maximum theoretical efficiency of accretion disks.

**4.2. Three Astrophysically Important Relativistic
Effects**

THE MINIMUM ANGULAR MOMENTUM An important inference from
the above is that there are no stationary
bound orbits whose angular momentum is less than a definite threshold
value: particles whose angular momentum is too low plunge directly
into the hole. This qualitative feature of the orbits means that *no
stationary axisymmetric flow pattern* can extend too close to the
rotation axis of a black hole (even well away from the equatorial
plane) - no such constraint arises for flows around an object with a
"hard" surface. Many authors have suggested that the resultant
"funnels" play a role in the initial bifurcation and collimation of jets.

LENSE-THIRRING PRECESSION An orbit around a spinning (Kerr) hole that does not lie in the equatorial plane precesses around the hole's spin axis with an angular velocity [discussed by Bardeen & Petterson (13)] of

(20) |

This precession has a time scale longer than the orbital
period by a
factor of ~ (*r* / *r*_{g})^{3/2}(*J* /
*J*_{max})^{-1}. However,
if material spirals slowly
inward (at a rate controlled by viscosity) in a time much exceeding
the orbital time, then the effects of this precession can mount
up. The important consequence follows that the flow pattern near a
black hole, within the radius where
2 /
_{BP}, is less
than the inflow
time, can be axisymmetric with respect to the hole irrespective of the
infalling material's original angular momentum vector. The
Lense-Thirring precession, an inherently relativistic effect, thus
guarantees that a wide class of flow patterns near black holes will be
axisymmetric - an important simplification of the problem.

ELECTROMAGNETIC PROPERTIES OF BLACK HOLES
Interactions of black holes with magnetic fields imposed on their
surroundings can have important astrophysical effects. When a hole
forms from collapsing magnetized material, the magnetic field outside
the horizon decays ("redshifts away") on the collapse time scale
*r*_{g} / *c*. But if, for instance, an external
electric field were applied to
a Schwarzschild hole, then after transients had decayed, a modified
field distribution would be established where the electric field
appeared to cross the horizon normally. The event horizon (or
"surface") of the hole thus behaves in some respects like a conductor
(47,
76,
134).
It does not have *perfect* conductivity, however: if it
did, electromagnetic flux would never be able to penetrate the
horizon. Comparing the decay time scale for transients around a black
hole (*r*_{g} / *c*) with the time scale
(*r*_{g}^{2} /
4
) appropriate to a sphere of
radius *r*_{g} and conductivity
, we can associate a surface
resistivity
of 377 with the
horizon. This analogy can be put on a more rigorous
basis (134),
and the "resistance" of a black hole is found to be
*Z*_{H} 100
ohms. More generally, a Kerr black hole behaves like a
* spinning* conductor. A simple discussion
(98) valid for
*J* << *J*_{max} shows
that a
hole embedded in a uniform magnetic field *B*_{0} would
acquire a quadrupole distribution of electric charge

(21) |

where
_{H} = (*J* /
*J*_{max}) *c* / 2*r*_{EH} is
the effective angular velocity of the hole.
The corresponding poloidal electric field in a nonrotating frame is

(22) |

Just as in a classical "unipolar inductor," power can be extracted
by allowing a current flow between a spinning hole's equator and
poles. The maximum electric potential drop is
~ *B*_{0} *r*_{g}(*J* /
*J*_{max}), where *B*_{0}
is the imposed field. This can be very large, as it is when a similar
argument is applied to spinning magnetized neutron stars in
conventional models for pulsars.

For the fiducial field strength *B*_{E} (Equation 6), this
emf is

(23) |

A single test charge introduced into this electromagnetic field will extract from the hole an energy of this order. However, the magnetosphere is unlikely just to contain a few "test charges"; indeed, the bare minimum charge density needed to modify the imposed field is

(24) |

(cf. Equation 12), and pair production generates far more
charges than
this (see Section 5). Just as in pulsars
(8), a realistic
magnetospheric current system and plasma distribution, though very
hard to calculate, is likely to "short-out" the electric field. A
relevant parameter is then
_{H}, the angular
velocity of the field lines
at large distance from the hole. This is related to the ratio of the
effective resistance
*Z*_{} to the
resistance of the hole *Z*_{H}:

(25) |

In the charge-starved limit, corresponding to infinite
resistance at infinity,
^{F} = 0. The
"matched" case when
*Z*_{} =
*Z*_{F}(^{F} =
½ ^{H})
corresponds to the maximum power extraction for a given
*B*_{0}. This power is of order

(26) |

The efficiency in this case is lower than when
*Z*_{} ->
(zero power), in
the sense that half of the power is dissipated in the hole, and raises
its irreducible mass; nevertheless, 9.2% of the rest energy could be
extracted while slowing down a hole that started off with
*J* = *J*_{max}.

Electromagnetic extraction of energy from black holes seems a realistic and important possibility. Its astrophysical context is discussed in Section 5.

SUMMARY The results of this section can be summarized by saying that three distinctively relativistic features of black holes are important in models for galactic nuclei:

- There is a definite lower limit to the angular momentum of any
stably orbiting material.
- The Lense-Thirring precession enforces axisymmetry on any
inward-spiraling flow pattern near the hole; consequently, any
directed outflow initiated in the relativistic domain will be aligned
with the hole's spin axis and will squirt in a constant direction
(irrespective of the provenance of the infalling gas), except insofar
as precession or accretion processes can reorient the hole's spin
(105).
- A rotating hole's latent spin energy can be tapped by externally applied magnetic fields; this can provide a power source far exceeding that from the accretion process itself.