String Theory and M-Theory

556 Black holes in string theory
r=rH
t=-
r=rH
t=
Fig. 11.2. The Schwarzschild black hole in Kruskal–Szekeres coordinates. The solid
lines correspond to the horizon, while the dashed lines correspond to the singularity.
The shaded region describes the part of the diagram in which the Kruskal–Szekeres
coordinates are well defined.
For |u| > |v|,
t = rH log
v
u + v
, (11.19)
u − v
and so the horizon maps to t = ±∞. It takes an infinite amount of
Schwarzschild time to reach the horizon, which reflects the fact that the
horizon is infinitely redshifted for an asymptotic observer. From Fig. 11.2
one can infer that light rays emitted by a source situated inside the black
hole, which means inside the horizon but outside the singularity, never escape
to the region outside the black hole. This is the reason why the surface
r = rH is called the event horizon. In general, such event horizons are null
hypersurfaces, which means that vectors nµ normal to these surfaces satisfy
n 2 = 0. In the case at hand, the horizon is a two-sphere of radius rH times
a null line. In Fig. 11.2, only the null line is shown. It is customary to
say that the horizon is S 2 and leave the null line implicit. 7 In particular, it
follows from Eq. (11.5) that the area of the event horizon is
A = 4πr 2 H = 16π(MG4) 2 . (11.20)
7 There is a theorem to the effect that S 2 is the only possible horizon topology for a black hole in
four dimensions. We will see later that there are other possibilities, besides a sphere, in higher
dimensions.
u