r

Note that, from Eqs (11.14) and (11.15), it follows that

r

u2 ’ v 2 = ’ 1 er/rH . (11.17)

rH

Di¬erent regions of space-time determined by this metric are represented in

the Kruskal diagram shown in Fig. 11.2. Equation (11.17) shows that the

event horizon r = rH corresponds to u = ±v, which is represented by a pair

of solid lines in Fig. 11.2. Equation (11.17) also shows that v 2 < u2 when

r > rH . The metric in the u, v coordinates can be analytically extended to

the region in between the horizon and the singularity. In these coordinates

the curvature singularity at r = 0 corresponds to the hyperbola v 2 ’ u2 = 1.

This is a pair of space-like curves represented by dashed lines in Fig. 11.2.

Thus the space-time is well de¬ned for

v 2 < u2 + 1.

’∞ < u < +∞ and (11.18)

As can be seen from Eq. (11.16), the singularity at the horizon is no longer

present in these coordinates.

The Schwarzschild geometry in Kruskal“Szekeres coordinates displays more

space-time regions than those represented by the original Schwarzschild co-

ordinates, which are only good for r > rH . The additional regions are

unphysical in the sense that a physical black hole that forms by collapse

would only have the future singularity (with u > 0) and not the past one

(with u < 0). The latter behaves like a time-reversed black hole and is

sometimes called a white hole.

The Kruskal“Szekeres coordinates have the additional advantage that

geodesics take a very simple form. The equation ds = 0 is satis¬ed by

lines with the property du = ±dv (and ¬xed position on the two-sphere).

This means that null geodesics are 45o lines in Fig. 11.2.

556 Black holes in string theory

v

r=rH

r=rH

t=

t=-

u

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 de¬ned.

For |u| > |v|,

u+v

t = rH log , (11.19)

u’v

and so the horizon maps to t = ±∞. It takes an in¬nite amount of

Schwarzschild time to reach the horizon, which re¬‚ects the fact that the

horizon is in¬nitely 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 es-

cape 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

n2 = 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 = 4πrH = 16π(M G4 )2 .

2

(11.20)

7 There is a theorem to the e¬ect 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.

11.1 Black holes in general relativity 557

Reissner“Nordstr¨m black hole

o

Reissner“Nordstr¨m metric in spherical coordinates

o

The generalization of the Schwarzschild black hole to one with electric charge

Q, but no angular momentum, is called the Reissner“Nordstr¨m black hole.

o

Charged black holes play a very special role in string theory, because in some

cases they are supersymmetric. Thus, by the usual BPS-type reasoning, they

can provide information about string theory at strong coupling. In four

dimensions the metric of a Reissner“Nordstr¨m black hole can be written in

o

the form

ds2 = ’∆ dt2 + ∆’1 dr2 + r2 d„¦2 , (11.21)

2

where

2M G4 Q2 G4

∆=1’ + . (11.22)

r2

r

This metric is a solution to Einstein™s equations in the presence of an electric

¬eld

1

Gµν = Rµν ’ Rgµν = 8πG4 Tµν , (11.23)

2

where Tµν is in general the energy“momentum tensor for this ¬eld

1

Tµν = Fµρ Fν ρ ’ gµν Fρσ F ρσ . (11.24)

4

Since the problem has spherical symmetry, the only nonvanishing component

of the U (1) electric ¬eld strength is given by the radial component of the

electric ¬eld Er

Q

Ftr = Er = 2 , (11.25)

r

as is veri¬ed in Exercise 11.1. The Reissner“Nordstr¨m metric can be gener-

o

alized to include magnetic charges as well as electric charges, which results

in a nonvanishing component Fθφ . This generalization is described in Exer-

cise 11.2.

Singularities

The metric components in Eq. (11.21) are singular for three values of r.

The dependence of the function ∆(r) which illustrates these singularities is

shown in Fig. 11.3. There is a physical curvature singularity at r = 0, which

can be veri¬ed by computing again the scalar Rµνρσ Rµνρσ . In addition, the

factor gtt in the metric vanishes for

(M G4 )2 ’ Q2 G4 ,

r = r± = M G 4 ± (11.26)

558 Black holes in string theory

∆(r) Q >G4M 2

Q =G4M 2

(0,0)

Q <G4M 2

r

__

G4 M

Q 2=0

r’ r+

1

__ __

G4 M G4 M

Fig. 11.3. Plots of the function ∆(r) for the Reissner“Nordstr¨m black hole

o

which are referred to as the inner horizon and the outer horizon. The outer

horizon, r = r+ , is the event horizon in this case. Note that it is only present