String Theory Demystified

244 String Theory Demystifi ed
charge Q to a static black hole in string theory, you can arrive at an exotic
black hole that is supersymmetric.
First we defi ne:
∆= 1− +
2 2
mG4
QG
2
r r
4
(14.13)
Notice that we are basically extending the Schwarzschild solution by adding a
Coulomb-type term. The metric for a static, charged black hole is given by
2 2 −1
2 2 2
ds =−∆ dt +∆ dr + r dΩ
(14.14)
This metric has two coordinate singularities which are given by
r MG MG Q G
± = ± −
2 2
( ) (14.15)
4 4
4
The two horizons are denoted by
• r is the outer horizon.
+
• r is the inner horizon.
−
The outer horizon is the event horizon—the point of no return when approaching
the black hole. Now, before stating our next result, we need to talk a little bit
about singularities. Stephen Hawking and Roger Penrose did a great deal of work
on singularities in the context of classical general relativity. They found out some
interesting results about singularities. If a singularity is present in space-time
without a horizon, it is called a naked singularity. This is because the horizon,
like clothing, keeps you from seeing what’s behind the veil. In this case the veil
is provided by the fact that light and hence no information can escape from beyond
the horizon. The singularity is essentially shut off from the rest of the universe.
Hawking and Penrose conjectured that classical physics does not permit the
existence of naked singularities.
Charged black holes are related to this concept in the following way. A charged
black hole with a mass m is limited in the amount of charge Q that it can carry. It
avoids having a naked singularity only if
m G4≥ Q
(14.16)