Verify Eqs (10.170), (10.173), (10.176) and (10.177) for ¬‚ux compacti¬ca-

tions of M-theory on a Calabi“Yau four-fold.

PROBLEM 10.16

Fill in the details of the Kaluza“Klein compacti¬cation to derive the scalar

potential Eq. (10.168).

PROBLEM 10.17

Derive the formula for Newton™s constant in the context of the strongly

coupled heterotic string Eq. (10.248).

PROBLEM 10.18

Derive the result Eq. (10.221) from the dilatino equation Eq. (10.199).

PROBLEM 10.19

Show that the Einstein ¬eld equations that determine a(t) reduce for the

FRW ansatz to the Friedmann and acceleration equations.

PROBLEM 10.20

When the slow-roll parameters satisfy Eqs (10.289) and (10.290), show that

it is consistent to neglect the corresponding two terms in the FRW equations.

11

Black holes in string theory

Black holes are a fascinating research area for many reasons. On the one

hand, they appear to be a very important constituent of our Universe. There

are super-massive black holes with masses ranging from a million to a billion

solar masses at the centers of most galaxies. The example of M31 is pictured

in Fig. 11.1. Much smaller black holes are formed as remnants of certain

supernovas.

Fig. 11.1. The nuclei of many galaxies, including M31, are quite violent places, and

the existence of supermassive black holes is frequently postulated to explain them.

M15, on the other hand, is one of the most densely packed globular clusters known

in the Milky Way galaxy. The core of this cluster has undergone a core collapse,

and it has a central density cusp with an enormous number of stars surrounding

what may be a central black hole.

From the theoretical point of view, black holes provide an intriguing arena

in which to explore the challenges posed by the reconciliation of general

relativity and quantum mechanics. Since string theory purports to provide

a consistent quantum theory of gravity, it should be able to address these

challenges. In fact, some of the most fascinating developments in string

549

550 Black holes in string theory

theory concern quantum-mechanical aspects of black-hole physics. These

are the subject of this chapter.

The action for general relativity (GR) in D dimensions without any sources

is given by the Einstein“Hilbert action1

√

1

dD x ’gR,

S= (11.1)

16πGD

where GD is the D-dimensional Newton gravitational constant. The classical

equation of motion is the vanishing of the Einstein tensor

1

Gµν = Rµν ’ gµν R = 0, (11.2)

2

or, equivalently (for D > 2), Rµν = 0. Thus, the solutions are Ricci-

¬‚at space-times. Straightforward generalizations are provided by adding

electromagnetic ¬elds, spinor ¬elds or tensor ¬elds of various sorts, such as

those that appear in supergravity theories. Some of the most interesting

solutions describe black holes. They have singularities at which certain

curvature invariants diverge. In most cases these singularities are shielded by

an event horizon, which is a hypersurface separating those space-time points

that are connected to in¬nity by a time-like path from those that are not.

The conjecture that space-time singularities should always be surrounded by

a horizon in physically allowed solutions is known as the cosmic censorship

conjecture.2 Classically, black holes are stable objects, whose mass can only

increase as matter (or radiation) crosses the horizon and becomes trapped

forever. Quantum mechanically, black holes have thermodynamic properties,

and they can decay by the emission of thermal radiation.

Challenges posed by black holes

A long list of challenges is presented by black holes. Some of them have

been addressed by string theory already, while others remain active areas of

research. Here are some of the most important ones:

• Does the existence of black holes and branes imply that quantum me-

chanics must break down and that pure quantum states can evolve into

mixed states? The fact that this super¬cially appears to be the case is

known as the information loss puzzle. String theory is constructed as a

quantum theory, and therefore the answer is expected to be “no.” In fact,

various arguments have been constructed that point quite strongly in that

1 See the Appendix of Chapter 9 for a brief review of Riemannian geometry.

2 This is a modern version of the conjecture. Originally, the conjecture was that, starting from

“good” initial conditions, general relativity never generates naked singularities.

Black holes in string theory 551

direction. However, a complete resolution of the information loss puzzle

undoubtedly requires understanding how string theory makes sense of the

singularity, where quantum gravity e¬ects become very important. So it

is fair to say that this is still an open question.

• Can string theory elucidate the thermodynamic description of black holes?

Does black-hole entropy have a microscopic explanation in terms of a large

degeneracy of quantum states? One of the most important achievements

of string theory in recent times (starting with work of Strominger and

Vafa) is the construction of examples that provide an a¬rmative answer

to this question. This chapter describes explicit string solutions for which

a microscopic derivation of the Bekenstein“Hawking entropy is known.

• Are there black-hole solutions that correspond to single microstates rather

than thermodynamic ensembles? If so, do they have a singularity and a

horizon? Or do these properties arise from thermodynamic averaging?

These questions are currently under discussion. However, since the an-

swers are not yet clear, they will not be addressed further in this chapter.

• What, if anything, renders black-hole singularities harmless in string the-

ory? In some cases, as illustrated by the analysis of the conifold in Chap-

ter 9, the singularity can be “lifted” once nonperturbative states are taken

into account. One natural question is whether string theory can elucidate

the status of the cosmic censorship conjecture?

• Does string theory forbid the appearance of closed time-like curves? Such

causality-violating solutions can be constructed. There needs to be a

good explanation why such solutions should or should not be rejected as

unphysical. It may be that they only occur when sources have unphysical

properties.

• What generalizations of black-hole solutions exist in dimensions D > 4?

The case of ¬ve dimensions is discussed extensively in this chapter, and ex-

plicit supersymmetric black-hole solutions are presented. Black holes fall

into two categories: (1) large black holes that have ¬nite-area horizons in

the supergravity approximation; (2) small black holes that have horizons

of zero area, and hence a naked singularity, in the supergravity approxi-

mation. The small black holes acquire horizons of ¬nite area when stringy

corrections to the supergravity approximation are taken into account. It

seems that large supersymmetric black holes only arise for D ¤ 5. This is

one reason why there has been a lot of interest in the D = 5 case. Another

reason is that nonspherical horizon topologies become possible for D > 4.

The example of D = 5 black rings will be described.

Chapter 12 describes black p-brane solutions. Black branes are higher-

552 Black holes in string theory

dimension generalizations of black-hole solutions. These solutions play an

important role in the context of the AdS/CFT correspondence.

• A recent speculative suggestion is that black holes might be copiously

produced at particle accelerators, such the LHC.3 This prediction hinges

on the possibility of lowering the scale at which gravity becomes strong in

suitably warped backgrounds, such as those discussed in Chapter 10. The

scale might even be as low as the TeV scale. If correct, this would provide

one way of testing string theory at particle accelerators, which would be

quite fantastic.