contradicts the known classical behavior, namely that the mass can only

increase, discussed earlier. This paradox led Hawking to consider quantum

corrections to the classical description. He argued that the gravitational

¬elds at the horizon are strong enough for quantum mechanical pair pro-

duction in the vicinity of the horizon to lead to the emission of thermal

radiation. Roughly speaking, one particle in the virtual pair falls into the

hole, and the other one is emitted as a physical on-shell particle. For large

black holes, this can be demonstrated reliably using quantum ¬eld theory in

a classical curved space-time background geometry. Since gravity is treated

classically, the black hole has to be big for this analysis to be reliable. In this

way, Hawking argued that a black hole emits radiation, and as a consequence

it loses mass. The outgoing radiation is thermal, when back-reaction can

be neglected. Thus, the black hole behaves as if it were a black body with

the temperature computed earlier. The classical statement that nothing can

escape from a black hole is undermined by quantum e¬ects. The fact that

the entropy of the black hole decreases when Hawking radiation is emitted

is consistent with the second law of thermodynamics when the entropy of

the emitted radiation is taken into account.

Pure states and mixed states

Hawking has argued that quantum mechanics breaks down when gravity is

taken into account. First, by a semi-classical analysis, he argued that black

holes emit thermal radiation at a temperature determined by the parameters

of the black hole (mass, charge, and angular momentum). Such radiation

has no correlations, and therefore is in a mixed state, characterized by a

density matrix. On the other hand, a collapsing shell of matter that forms a

black hole can be in a pure quantum state. Thus, he argued, pure states can

evolve into mixed states in a quantum theory of gravity. This contradicts

the basic tenet of unitary evolution in quantum mechanics, and it is referred

to as information loss or loss of quantum coherence.

11.2 Black-hole thermodynamics 565

The AdS/CFT conjecture, described in Chapter 12, certainly would ap-

pear to contradict this reasoning, since the AdS space in which black holes

can form is dual to a unitary conformal ¬eld theory. Thus string solutions,

at least ones that are asymptotically AdS, probably provide counterexam-

ples to Hawking™s claim. That said, it should be admitted that it is an

extremely subtle matter to explain in detail where Hawking™s argument for

information loss breaks down. This question has been discussed extensively

in the literature, but it is not yet completely settled.

EXERCISE 11.4

Show that the temperature of a D = 4 Reissner“Nordstr¨m black hole is

o

(M G4 )2 ’ Q2 G4

T= .

2

2πr+

What happens to this temperature in the extremal limit?

SOLUTION

Using the same reasoning as in Section 11.2, we set r = r+ (1+ρ2 ) and expand

the Euclideanized metric of the Reissner“Nordstr¨m black hole about ρ = 0

o

to get

2

3

4r+ (r+ ’ r’ )d„ r+ ’ r’ 2

2

dρ2 + ρ2

ds = + d„¦ .

2

r+ ’ r’ 4r+

2r+

The value of β that follows from this expression is

2

4πr+

β= ,

r+ ’ r’

which leads to a temperature

(M G4 )2 ’ Q2 G4

r+ ’ r’

T= = .

2 2

4πr+ 2πr+

√

In the extremal limit, M G4 = |Q|, this gives a vanishing temperature

T = 0. 2

EXERCISE 11.5

Estimate the Schwarzschild radius, temperature, and entropy of a one solar

mass Schwarzschild black hole. Estimate its lifetime due to the emission of

Hawking radiation. The sun has a mass of M = 2.0 — 1033 g.

566 Black holes in string theory

SOLUTION

Reinstating h, c and kB by dimensional analysis, in order to express these

¯

quantities in ordinary units, gives

hc3

2G4 M ¯

∼ 6.0 — 10’8 K,

∼ 3.0 — 103 m,

rH = T=

2

c 8πM G4 kB

πR2 c3 G2 M 3

A

∼ 1.0 — 1077 , ∆t ∼ 4 4 ∼ 1066 years.

S= =

4G4 G4 h

¯ ±¯ c

h

The value of the coe¬cient ± is about 10’3 . 2

11.3 Black holes in string theory

This section considers supersymmetric (and hence extremal) black holes

that have ¬nite entropy in the supergravity approximation. These include

three-charge black holes in ¬ve dimensions and four-charge black holes in

four dimensions, which can be interpreted as approximations to solutions

of toroidally compacti¬ed string theory. For this class of compacti¬cations,

¬nite-horizon-area black-hole solutions that are asymptotically ¬‚at only exist

in the supergravity approximation in four and ¬ve dimensions. The reason

for this can be explained by referring to the extremal Reissner“Nordstr¨m o

solutions given in Eqs (11.30) and (11.32). In each case the coe¬cient of

dr2 takes the form

2

D’3

grr = 1 + (r0 /r) . (11.45)

D’3

This behaves near the horizon (r = 0) like grr ∼ (r0 /r)2 , which is necessary

to obtain a ¬nite horizon radius and area. It appears that constructions

obtained by string-theory or M-theory compacti¬cation always give an outer

exponent that is a positive integer. This can only correspond to 2/(D ’ 3)

if D = 4 or D = 5. In the multi-charge examples that are discussed in this

section, the expression 1 + (r0 /r)D’3 is replaced by a product of factors that

can have di¬erent radii, but the same conclusion still applies.

For all other supersymmetric black holes, including any supersymmetric

solution for D > 5, the horizon has zero radius in the supergravity approxi-

mation. To obtain a nonzero radius in these cases, it is necessary to include

stringy corrections, that is, corrections to the Einstein“Hilbert action that

are higher order in the curvature tensor. This is discussed in Section 11.6.

11.3 Black holes in string theory 567

Extremal three-charge black holes for D = 5

The simplest nontrivial example for which the entropy can be calculated