Fθφ = p(r, t) sin θ.

Taking into account the Bianchi identity, ‚r Fθφ = ‚t Fθφ = 0, one obtains

Fθφ = p sin θ,

where p is a constant. This ¬eld can then be inserted in the Einstein equation

to determine the functions A and B. 2

EXERCISE 11.2

Show that the parameters q and p in the previous exercise are electric and

magnetic charges.

SOLUTION

As discussed in Chapter 8, magnetic and electric charge are given by

π 2π

1 1

Qmag = F= dθ dφ Fθφ

4π 4π 0 0

and

π 2π

1 1

Qel = F= dθ dφ ( F )θφ .

4π 4π 0 0

Inserting Fθφ = p sin θ in the ¬rst integral gives Qmag = p. To evaluate the

electric charge it is necessary to compute the dual of the electric ¬eld:

√

( F )θφ = ’gF rt = eA+B r2 sin θe’2(A+B) Ftr = q sin θ.

Thus Qel = q.

EXERCISE 11.3

Show that the near-horizon geometry of a D = 4 extremal Reissner“Nord-

str¨m black hole is AdS2 — S 2 .

o

SOLUTION

Near the horizon r ≈ 0. In this limit Eq. (11.30) becomes

’2

r0 r0 2

ds2 = ’ dt2 + dr2 + r0 d„¦2 .

2

2

r r

2

Setting r = r0 /r, and dropping the tilde,

˜

r0 2

ds2 = ’dt2 + dr2 + r0 d„¦2 .

2

2

r

562 Black holes in string theory

This gives a constant negative curvature in the r and t directions, which is

AdS2 . Similarly, in the angular directions one has a sphere, with constant

positive curvature. In each case the radius of curvature is r0 . As a result,

the geometry in the near-horizon limit is AdS2 — S 2 . This is also known as

the Bertotti“Robinson metric. 2

11.2 Black-hole thermodynamics

Entropy and temperature

Classical black holes behave like thermodynamical objects characterized by

a temperature and an entropy. The microscopic quantum origin of these

features is addressed in Section 11.4. For now, let us consider the thermo-

dynamic description, that is, the macroscopic description of black holes.

Given a static metric, such as the D = 4 Schwarzschild metric Eq. (11.4),

there is an elementary method of computing the temperature. The key

point to recall is that a system that has a temperature T = β ’1 is periodic

in Euclideanized time „ = it with period β. A simple way to understand

this fact is to recall that a thermodynamic partition function is given by

Z = Tr e’βH ,

where H is the Hamiltonian of the system. Since quantum mechanical evolu-

tion by a time interval t is given by e’iHt , the trace corresponds to imposing

a periodicity β in Euclidean time.

The way to determine the temperature of a black hole is to consider its

analytic continuation to Euclidean time and then to examine the period-

icity of this coordinate. This period is determined by requiring that the

Euclideanized metric is regular at the horizon. This may sound like a cook-

book recipe, but it is by far the easiest way to carry out the computation. It

can be con¬rmed in a variety of ways, for example by showing that a black

hole emits blackbody radiation at the computed temperature.

In order to examine the vicinity of the horizon, let us de¬ne ρ by

r = rH (1 + ρ2 ), (11.35)

and expand the Euclideanized version of the Schwarzschild metric Eq. (11.4)

about ρ = 0. This gives

d„ 1

2

ds2 ≈ 4rH dρ2 + ρ2

2

+ d„¦2 . (11.36)

42

2rH

The ¬rst two terms describe a ¬‚at plane in polar coordinates provided that

11.2 Black-hole thermodynamics 563

the period of „ is

β = 4πrH = 8πM G4 . (11.37)

Thus the temperature of the Schwarzschild black hole is T = 1/(8πM G4 ).

Since the temperature decreases as M increases, the speci¬c heat is negative.

Very massive black holes are accurately described by classical solutions of

Einstein™s theory of general relativity. Classically, black holes are stable and

black, which means that nothing can ever escape from inside the horizon.

Thus the mass can only increase as matter falls through the horizon. If

one takes the thermodynamic interpretation of black holes into account, the

analogy suggests that

dM = T dS, (11.38)

where M is the mass of the black hole, T is its temperature and S is the

black hole™s entropy. The black-hole entropy should be taken into account

in the second law of thermodynamics,

dS/dt ≥ 0. (11.39)

The entropy of black holes added to the entropy of their surroundings always

has to increase with time.

For a Schwarzschild black hole, β = 1/T = 8πM G4 . Requiring that

S ’ 0 as M ’ 0, to ¬x an integration constant, one obtains

S = 4πM 2 G4 . (11.40)

Bekenstein“Hawking entropy formula

From Eq. (11.4) it follows that the area A of the event horizon of a Schwarz-

schild black hole is given by

A = 4πrH = 16π(M G4 )2 ,

2

(11.41)

so the entropy can be written in the form

A

S= . (11.42)

4G4

This is one-quarter of the area of the horizon measured in units of the

Planck length. This relation, known as the Bekenstein“Hawking (BH) en-

tropy formula, appears to be universally valid (for any black hole in any

dimension), at least when A is su¬ciently large. For an arbitrary (not nec-

essarily Schwarzschild) black hole in D dimensions, the formula becomes

A

S= , (11.43)

4GD

564 Black holes in string theory

where A is the volume (usually called the area) of the (D ’ 2)-dimensional

horizon. According to this formula, the entropy of a D = 4 Reissner“

Nordstr¨m black hole is

o

2

S = πr+ /G4 . (11.44)

For an extremal Reissner“Nordstr¨m black hole this is S = πM 2 G4 .

o

Hawking radiation

When an object has a ¬nite temperature, it emits thermal radiation, which