Gravity and Strings

8.3 The electric Reissner–Nordström solution 231
r = 0
r h
r h
r h
Fig. 8.2. Part of the Penrose diagram of an extreme Reissner–Nordström black hole. The
complete diagram has an infinite number of “universes.”
We see that the RN BH provides a very interesting playground on which to test
cosmic censorship. We will see that the relation between cosmic censorship and
positivity of the energy can be translated into supersymmetry (BPS) bounds.
8. The limiting case M = 2|q| between the naked singularity and the regular BH is
very special. When M = 2|q| the two horizons coincide, r+ = r− = G (4)
N M, and
there is no change of signature across the resulting horizon (which is a degenerate
Killing horizon), which still has a non-vanishing area
-
-
+
+
+
I 0
I 0
I 0
Aextreme = 4πr 2 + = 4π G (4)
N M 2 . (8.78)
This object is an extreme RN (ERN) BH and it will play a central role in much of
what follows. Some of the properties of ERN BHs are the following.
(a) The proper distance to the horizon along radial directions at constant time,
lim
r2→r+
r2
r1
ds = lim
r2→r+
r2
r1
dr 1 − r+
r
−1
=∞, (8.79)
diverges. This does not happen along timelike or null directions, though an
observer can cross it in a finite proper time.
(b) The Penrose diagram is drawn in Figure 8.2. As we see, the causal structure
is completely different from that of any regular RN BH no matter how close to
the extreme limit it is. Thus, we can expect physical properties of the family of
RN BHs to be discontinuous at the extreme limit.