Fig. 11.5. Lines with constant „ of the 3-center solution with identical charges.

Multi-center solutions

There are stationary multi-black holes solutions that are known as multi-

center solutions. The reason these exist is that, when each of the black

holes preserves the same supersymmetry charge, this supersymmetry is an

unbroken symmetry of the multi-black hole system. In this case, the BPS

condition result in a no-force condition, which means that the total force

acting on each of the black holes due to the presence of the others exactly

cancels, so that each of them can remain at rest. The various attractive and

repulsive forces due to gravity, scalar ¬elds, and gauge ¬elds are guaranteed

to cancel out due to supersymmetry. This is true even though the ¬eld

con¬gurations are much more complicated than they are for a single black

hole.

The attractor equations can be generalized to the case where are black-

hole horizons, with charges encoded in harmonic three-forms “p , at di¬erent

points xp . In the special case where all of the component black-holes have

the same charges, the ¬‚ow parameter „ is naturally de¬ned to be

1

„= . (11.134)

| x ’ xp |

p

Surfaces with constant „ in the 3-center case are displayed in Fig. 11.5. In

general, the charges are not identical. In order to describe such a solution,

known as a multi-center solution, one has to consider a slightly generalized

metric of the form

ds2 = ’e2U (dt + ωi dxi )2 + e’2U dx · dx. (11.135)

11.5 The attractor mechanism 595

The attractor equations can then be shown to take the form

= 2e’U Im e’i± eK/2 „¦ ,

H

(11.136)

dH § H,

dω = M

where H(x) is a harmonic function of the space-like coordinates (as well as

a di¬erential form in the compact dimensions), is the Hodge star operator

on 3 and ± is the phase of Z( “p ). The ¬rst of these equations is the

¡

generalization of Eq. (11.125), while the second one has no counterpart in

the one-center case. Since each of the horizons is an attractor, the ¬‚ow

equation in this case is called a split attractor ¬‚ow.

If we have N centers in asymptotically ¬‚at space-time, integration gives

N

1

+ 2Im e’i± eK/2 „¦

H=’ “p . (11.137)

| x ’ xp | r=∞

p=1

Acting with the operator d on the second equation in (11.136) gives the

condition

∆H § H = 0. (11.138)

M

Using

1

= ’4πδ (3) (x ’ xp ),

∆ (11.139)

| x ’ xp |

one then obtains

N

1

“p § “q = 2Im e’i± Z(“p ) . (11.140)

r=∞

| xp ’ xq | M

q=1

It can be shown that a multi-center solution exists as long as the this equa-

tion is satis¬ed. It determines the position of the charges. So, for example,

in the two-center case the separation of the horizons is determined by

M “1 § “2

| x1 ’ x2 |= . (11.141)

2Im [e’i± Z(“1 )]r=∞

Black rings

In four dimensions there is a theorem to the e¬ect that the topology of a

black-hole event horizon is necessarily that of a two-sphere. It therefore

came as a surprise when people realized that there are more possibilities in

higher dimensions. In all of the ¬ve-dimensional examples discussed so far,

the horizon has S 3 topology. However, there are also asymptotically ¬‚at

596 Black holes in string theory

supersymmetric solutions in ¬ve dimensions for which the topology of the

horizon is S 1 —S 2 . In fact, there are so many solutions of this type that they

are not uniquely determined by their mass, charges and angular momenta, in

contrast to black holes. These solutions are called black rings. As you might

guess, rotation is required to support this topology. These solutions can be

found by considering N = 2 supergravity coupled to a set of vector multi-

plets in ¬ve dimensions. This can be realized by compactifying M-theory on

a Calabi“Yau three-fold, as was discussed in Chapter 9. This is a concep-

tually beautiful subject, but the formulas tend to get a bit complicated. So

we will just list the essential results without the derivations.20

In order to present the supersymmetric black-ring solutions, let us ¬rst

describe the most general solutions with unbroken supersymmetry. The

scalars in the vector multiplets are real and denoted by Y A . The BPS

equations are then solved by

ds2 = ’f ’2 (dt + ω)2 + f ds2 , (11.142)

5 X

where

4

ds2 hmn dxm dxn .

= (11.143)

X

m,n=1

Here X is a four-dimensional hyperk¨hler space with metric hmn , ω is a

a

one-form on X and f is a scalar function depending on the coordinates

of X. The U (1) ¬eld strength two-forms F A in the vector multiplets are

determined by

F A = d f ’1 Y A (dt + ω) + ˜A , (11.144)

where ˜A are closed self-dual two-forms on X, that is,

˜A = A

4˜ . (11.145)

Moreover, supersymmetry implies that ω and f are determined by

= ’f YA ˜A ,

dω + 4 dω (11.146)

and

2

(f YA ) = 3DABC ˜B ˜C , (11.147)

where DABC are the intersection numbers of two-forms (or dual four-cycles)

describing the geometry of the Y A moduli space. This is the most general

solution preserving supersymmetry in ¬ve dimensions. So, for example, the

¬ve-dimensional three-charge black holes and rotating black holes described

20 For further details see hep-th/0504126.

11.5 The attractor mechanism 597

in Section 11.3 are special cases of this solution, as you are asked to check

in a homework problem.

In order to obtain an example of a black-ring solution, it is su¬cient to

consider the case of one modulus, that is, A = 1 and D111 = 1. The space

X is taken to be Taub“NUT with metric

1