i=1

The radial size of the horizon is

rH = r0 [»(r0 )]1/6 , (11.65)

and thus the area of the horizon is

A = 2π 2 rH = 2π 2 r0 cosh ±1 cosh ±2 cosh ±3 .

3 3

(11.66)

The entropy is then given by

3

A 2πr0 V6

S= = cosh ±1 cosh ±2 cosh ±3 . (11.67)

9

4G5 p

To convert to string units, one would replace 9 by gs 9 .

2

p s

These formulas have a suggestive interpretation. Let us imagine that,

in addition to there being Q1 D1-branes wrapping the y 1 circle, there are

also Q1 anti-D1-branes wrapping the same circle. Similarly, anti-D5-branes

and right-moving Kaluza“Klein excitations can be introduced. Then the net

charge in each case is

Qi = Q i ’ Q i i = 1, 2, 3. (11.68)

The three types of electric charge have Qi ∼ sinh 2±i . By identifying Qi ∼

exp(2±i ) and Qi ∼ exp(’2±i ) one interprets the net charge as a di¬erence of

brane and antibrane contributions and the expression for Mi ∼ cosh 2±i as

the sum of brane and antibrane contributions. This also allows the entropy

in Eq. (11.67) to be rewritten in form

3

A

S= = 2π ( Qi + Qi ), (11.69)

4G5

i=1

which is a nice generalization of Eq. (11.55).

Rotating supersymmetric black holes for D = 5

In ¬ve dimensions it is possible for a three-charge black hole to rotate and

still be supersymmetric.13 This is not possible in four dimensions, where all

13 A rotating time-independent black-hole solution in four dimensions is known as a Kerr black

hole. The solution under consideration here is quite di¬erent from that one.

574 Black holes in string theory

rotating black holes, even extremal ones, are not supersymmetric. The key is

to note that the rotation group in ¬ve dimensions is SO(4) ∼ SU (2)—SU (2).

Supersymmetry requires restricting the rotation to one of the two SU (2) fac-

tors, which corresponds to simultaneous rotation, with equal angular mo-

mentum, in two orthogonal planes. There are more general ways in which a

¬ve-dimensional black hole can rotate, of course, but this is the only one that

is supersymmetric. It preserves 1/8 of the original 32 supersymmetries, just

like the previous examples. To describe this case, let us introduce angular

coordinates as follows:

x1 = r cos θ cos ψ, x2 = r cos θ sin ψ, (11.70)

x3 = r sin θ cos φ, x4 = r sin θ sin φ. (11.71)

Then

dxi dxi = dr2 + r2 d„¦2 (11.72)

3

describes Euclidean space for

d„¦2 = dθ2 + sin2 θdφ2 + cos2 θdψ 2 , 0 ¤ θ ¤ π/2, 0 ¤ φ, ψ ¤ 2π. (11.73)

3

The metric of the desired supersymmetric rotating black hole is a relatively

simple generalization of Eq. (11.46)

a a 2

ds2 = ’»’2/3 dt ’ sin2 θdφ + 2 cos2 θdψ + »1/3 dr2 + r2 d„¦2 ,

3

r2 r

(11.74)

where » is again given by Eq. (11.47). This metric describes simultaneous

rotation in the 12 and 34 planes. The parameter a is related to J12 = J34 = J

by

πa

J= . (11.75)

4G5

The area of the horizon at r = 0, and hence the entropy, is computed in

Exercise 11.7 and shown to be

A

= 2π Q1 Q5 n ’ J 2 .

S= (11.76)

4G5

Extremal four-charge black holes for D = 4

The metric and entropy

The construction of supersymmetric black holes in four dimensions is quite

similar to the ¬ve-dimensional case. Before proposing a speci¬c brane real-

ization, let us write down the metric and explore its properties. The analog

11.3 Black holes in string theory 575

of Eq. (11.46) is

ds2 = ’»’1/2 dt2 + »1/2 dr2 + r2 d„¦2 , (11.77)

2

where

4

ri

»= 1+ . (11.78)

r

i=1

This reduces to Eq. (11.30) when all four ri are equal. We can read o¬ the

mass of the black hole from the large distance behavior of gtt using Eqs (11.7)

and (11.8). The result is

4

ri

M= Mi with Mi = . (11.79)

4G4

i=1

The area of the horizon, which is located at r = 0, is

√

A = 4π r1 r2 r3 r4 . (11.80)

Putting these facts together, the resulting entropy is

A

S= = 16πG4 M1 M2 M3 M4 . (11.81)

4G4

Type IIA brane construction

It still remains to relate the four masses to four electric (or magnetic)

charges. This requires some sort of brane construction involving four types

of branes or excitations. To be speci¬c, let us consider the type IIA theory

compacti¬ed on a six torus that is a product of six circles with coordinates

y 1 , . . . , y 6 and radii R1 , . . . , R6 . A brane con¬guration that preserves 1/8 of

the N = 8 supersymmetry, and therefore is suitable, is the following: Q1

D2-branes wrapped on the y 1 and y 6 circles, Q2 D6-branes wrapped on all

six circles, Q3 NS5-branes wrapped on the ¬rst ¬ve circles, and Q4 units of

Kaluza“Klein momentum on the ¬rst circle. The masses that correspond to

these types of excitations are

1

M1 = (2πR1 )(2πR6 )TD2 Q1 = (R1 R6 )Q1 ,

gs 3

s

1

M2 = (2πR1 ) · · · (2πR6 )TD6 Q2 = (R1 · · · R6 )Q2 ,

gs 7

s

(11.82)

1

M3 = (2πR1 ) · · · (2πR5 )TNS5 Q3 = (R1 · · · R5 )Q3 ,

2 6

gs s