Charged Black Holes

There are a variety of ways to generalize the conventional Schwarzschild black hole. By going to higher dimensions we can consider not only black holes, but black strings, black membranes, and so forth. Typically black strings and branes are studied as systems of infinite extent, and therefore have infinite entropy. For this reason they can store infinite amounts of information. Higher dimensional Schwarzschild black holes are quite similar to their four-dimensional counterparts.

Another way to generalize the ordinary black hole is to allow it to carry gauge charge and/or angular momentum. In this lecture we will describe the main facts about charged black holes. The most important fact about them is that they cannot evaporate away completely. They have ground states with very special and simplifying features.

Thus, let us consider electrically charged black holes. The metric for a Reissner-Nordstrom black hole is

 (1.1)

The electric field is given by the familiar Coulomb law

 (1.2)

If the electric field is too strong at the horizon, it will cause pair production of electrons, which will discharge the black hole in the same manner as a nucleus with   Z >> 137 is discharged. Generally the horizon occurs at r ∼ MG, and the threshold field for unsupressed pair production is E ∼ m²_e, where m_e is the electron mass. Pair production is exponentially suppressed if

 (1.3)

or alternativley M²/Q >> 1/m²_e G² .

For Q² >M² G the metric in equation 6.0.1 has a time-like singularity with no horizon to cloak it. Such “naked singularities” indicate a breakdown of classical relativity visible to a distant observer. The question is not whether objects with Q² > M²G can exist. Clearly they can. The electron is such an object. The question is whether they can be described by classical general relativity. Clearly they cannot. Accordingly we restrict our attention to the case M² > Q²/G or M² /Q > Q/G.A Reissner–Nordstrom black hole that saturates this relationship M² = Q²/G is called an extremal black hole. Thus equation 6.0.3 is satisfied if

Black holes with charge >> 10⁴⁴ can only discharge by exponentially suppressed tunneling processes. For practical purposes we regard them as stable.

The Reissner–Nordstrom solution has two horizons, an outer one and an inner one. They are defined by

 (1.4)

where r₊ (r₋) refers to the outer (inner) horizon:

 (1.5)

The metric can be rewritten in the form

 (1.6)

Note that in the extremal limit M² = Q²/G the inner and outer horizons merge at          r_± = MG.

To examine the geometry near the outer horizon, let us begin by computing the distance from r₊ to an arbitrary point r > r₊. Using equation 6.0.6 we compute the distance ρ to be

 (1.7)

We define the following

 (1.8)

We find

 (1.9)

The radial-time metric is given by

 (1.10)

Expanding equation 1.9 near the horizon r₊ one finds

 (1.11)

Note that the proper distance becomes infinite for extremal black holes. For non-extremal black holes, equation 1.10 becomes

 (1.12)

where

 (1.13)

Evidently the horizon geometry is again well approximated by Rindler space. The charge density on the horizon is Q/4πr²₊.Since r+ ∼ MG the charge density is ∼ Q/4πM²G² .Th us for near extremal black holes, the charge density is of the form

 (1.14)

For very massive black holes the charge density becomes vanishingly small. Therefore the local properties of the horizon cannot be distinguished from those of a Schwarzschild black hole. In particular, the temperature at a small distance ρ_o from the horizon is 1/2πρ_o. From equation 1.13 we can compute the temperature as seen at infinity.

(1.15)

Using

 (1.16)

We find

 (1.17)

As the black hole tends to extremality, the horizon becomes progressively more removed from any fiducial observer. From equation 1.9 we see that as Δ → 0

 (1.18)

Thus for a fiducial observer at a fixed value of r the horizon recedes to infinite proper distance as Δ → 0.

In the limit Δ → 0 the geometry near the horizon simplifies to the form

 (1.19)

which, although infinitely far from any fiducial observer with r _-= r₊, is approximately Rindler.

We note from equation 1.17 that in the extremal limit the temperature at infinity tends to zero. The entropy, however, does not tend to zero. This can be seen in two ways, by focusing either on the region very near the horizon or the region at infinity. As we have seen, the local properties of the horizon even in the extreme limit are identical to the Schwarzschild case from which we deduce an entropy density 1/4G.Accordingly ,

 (1.20)

We can deduce this result by using the first law together with equation 1.17

to obtain S = Area/4G as a general rule.

The fact that the temperature goes to zero in the extreme limit indicates that the evaporation process slows down and does not proceed past the point Q = M√G. In other words, the extreme limit can be viewed as the ground state of the charged black hole. However it is unusual in that the entropy does not also tend to zero. This indicates that the ground state is highly degenerate with a degeneracy ∼ e^S. Whether this degeneracy is exact or only approximate can-not presently be answered in the general case. However in certain supersymmetric cases the supersymmetry requires exact degeneracy.

The metric in equation 1.19 for extremal black holes can be written in a form analogous by introducing a radial variable

 (1.21)

The metric then takes the form

 (1.22)

Obviously the physics near R/r₊→ 0 is identical to Rindler space, from which it follows that the horizon will have the usual properties of temperature, entropy, and a thermal atmosphere including particles of high angular momenta trapped near the horizon by a centrifugal barrier.

Although the external geometry of an extreme or near extreme Reissner– Nordstrom black hole is very smooth with no large curvature, one can nevertheless expect important quantum effects in its structure. To understand why, consider the fact that as Δ → 0 the horizon recedes to infinity. Classically, if we drop the smallest amount of energy into the extreme black hole, the location of the horizon, as measured by its proper distance, jumps an infinite amount. In other words, the location of the horizon of an extremal black hole is very unstable. Under these circumstances, quantum fluctuations can be expected to make the location very uncertain. Whether this effect leads to a lifting of the enormous degeneracy of ground states or any other physical phenomena is not known at present except in the supersymmetric case.