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arXiv:1105.3491v2 [hep-th] 2 Jul 2011

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Topology and Uniqueness of Higher Dimensional Black Holes

Daisuke Idaa, Akihiro Ishibashibcand Tetsuya Shiromizud

aDepartment of Physics, Gakushuin University,

Tokyo 171-8588, Japan

bTheory Center, Institute of Particle and Nuclear Studies,

High Energy Accelerator Research Organization (KEK),

Tsukuba, 305-0801, Japan

cDepartment of Physics, Kinki University,

Higashi-Osaka 577-8502, Japan

and

dDepartment of Physics, Kyoto University, Kyoto 606-8502, Japan

We review recent results concerning general properties of higher dimensional black holes.

The topics selected with particular focus are those concerning topology, symmetry, and

uniqueness properties of asymptotically flat vacuum black holes in higher dimensional general

relativity.

Contents

1. Introduction

2. General properties of higher dimensional black holes

2.1. Global structure and area theorem . . . . . . . . . . . . . . . . . . . .

2.2. Apparent horizon. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3. Topological structure of dynamical black hole horizons . . . . . . . . .

3. Stationary black holes in higher dimensions

3.1. Topology theorems in higher dimensions . . . . . . . . . . . . . . . . . 11

3.2. Rigidity theorem in higher dimensions . . . . . . . . . . . . . . . . . . 13

4. Uniqueness of static black holes

4.1. Basic tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2. Outline of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3. Vacuum case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.4. Electro-vacuum case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.5. Other cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5. Uniqueness of stationary, rotating black holes

5.1. Basic strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2. Rotating black holes in 5-dimensions . . . . . . . . . . . . . . . . . . . 32

6. Summary

1

3

3

5

7

10

19

31

38

§1.Introduction

The black hole uniqueness theorem in 4-dimensions is a triumph of classical

general relativity, implying that a tremendous number of black holes existing in our

typeset using PTPTEX.cls ?Ver.0.9?

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2D. Ida, A. Ishibashi and T. Shiromizu

observable universe can be described accurately by the Kerr metric, which possesses

only two parameters. In the course of complete proof of the uniqueness theorem,

there have appeared a number of remarkable results, each of which itself reveals

physically an important property of black holes, such as those concerning topology

and symmetry.1)These results also give us deep insights into thermodynamic aspects

of black holes.2)

It is of great interest to consider generalizations of a number of theorems estab-

lished for 4-dimensional black holes to higher dimensional case. One might expect

that such a generalization could straightforwardly be done by merely replacing “4”

with a general number “D.” However, as is by now well-known, the discovery of the

black ring solution in 5-dimensions3)(as well as a large variety of exact solutions dis-

cussed in Chapter 1 and 4) has drastically changed our view of the issue, highlighting

that the uniqueness theorem no longer holds as it stands in higher dimensions.

In this chapter we shall review general properties of higher dimensional black

holes, attempting to clarify which properties of 4-dimensional black holes can be

straightforwardly generalized to higher dimensions and which properties hold only

in 4-dimensions. There have already been a number of theorems for higher dimen-

sional black holes. This review is, however, not intended to cover the whole relevant

subjects or supply a complete list of existing literature. Rather, we will focus on

some specific topics relevant to topology and uniqueness/non-uniqueness feature of

asymptotically flat vacuum black holes and describe some key ideas and methods for

obtaining the results in higher dimensions.

In the next section we shall first recapitulate how to describe global structure

of black hole spacetimes in general dimensions. Then we discuss topological aspects

of apparent and the event horizon of dynamical black holes. In section 3, we con-

sider stationary black holes in higher dimensional general relativity. After briefly

summarizing a few critical steps in the proof of black hole uniqueness theorems in

4-dimensions, we shall discuss how and to what extent each of the steps can be

generalized to higher dimensional setting. In particular, we review recent results

concerning possible restrictions on the horizon topology and also an enhancement of

Killing symmetries, called the rigidity theorem, in higher dimensions. In section 4,

we review uniqueness theorems for static black holes in higher dimensions. Unique-

ness theorems for asymptotically flat, stationary rotating, vacuum black holes are

discussed in detail in section 5. Section 6 is devoted to summary.

Notations and Conventions

In this chapter, we mainly use the abstract index notation for tensor fields on

a spacetime, where each slot for tangent or cotangent field is denoted by a lower-

case latin index: a,b,c,···. In section 4, a tensor field is written in terms of its

components, where upper-case latin indices M,N,··· run from 0 to D − 1, and

lower-case latin indices i,j,··· run from 1 to D − 1. In section 5, we mainly treat

the 5-dimensional spacetimes. There, a tensor field is mainly written in terms of its

components, where lower-case latin indices i,j,··· run from 1 to 2 and upper-case

latin indices I,J,··· run from 3 to 5.

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‘Topology and Uniqueness in Higher Dimensional Black Holes’3

We use the natural unit where the speed of light c and the reduced Planck

constant ? are set to unity.

§2. General properties of higher dimensional black holes

We begin with noting that the uniqueness feature of a stationary black hole is

related to its thermodynamic aspects in the sense that a thermodynamically equi-

librium system can completely be characterized by a small number of state param-

eters.2)The idea of black hole thermodynamics has originated from Bekenstein’s

interpretation4)of Hawking’s area theorem5)as the 2nd law of thermodynamics,

as well as from the black hole mechanics due to Bardeen, Carter and Hawking.6)

While the latter concerns a stationary equilibrium configuration, the former involves

a dynamical process concerning the total area of all black holes in the universe.

Therefore, before going into discussion of stationary black holes, in this section we

shall discuss general circumstances that can include some dynamical processes such

as a formation and evolution of black holes, to which the area theorem becomes

relevant.

2.1. Global structure and area theorem

First of all, in order to define an isolated black hole in general context, one needs

to introduce a suitable notion of “infinity” and associated asymptotic structure.

In 4-dimensions this is usually, and elegantly, done in the conformal framework,

in which an unphysical spacetime (˜M,˜ gab), conformally isometric to our physical

spacetime (M,gab) in M∩˜ M, plays a role. A desired notion of infinity and asymptotic

flatness can be defined by specifying the behavior of the conformal metric ˜ gabnear

a conformal null boundary I = ∂˜

M. If one further imposes an additional condition

that every maximally extended null geodesic in M has past and future endpoints

on null boundary I in ˜M, then I is divided into disjoint sets of the future and

past null infinity, I+and I−. Such a spacetime is called asymptotically simple.

A spacetime (M,gab) is said to be weakly asymptotically simple at null infinity if

(˜M,˜ gab) has a neighborhood of I which is isometric to a neighborhood of I for some

asymptotically simple spacetime. The notion of strong asymptotic predictability

is then defined such that the closure of M ∩ J−(I+) is contained in a globally

hyperbolic open subset of˜M. A black hole region B is defined as the complement

of J−(I+) and the future event horizon H as the boundary of B in M. As such,

H is a null hypersurface ruled by null geodesics. Since B is a future set, every null

geodesic generator of H is future inextendible, but in general admits a past end

point. With the set of these definitions, general properties of 4-dimensional black

holes are studied by using the global method, which consists of a number of general

results concerning causal structure, behavior of causal geodesic congruence, etc, as

in Refs. 7),8).

A key equation for the global method is the Raychaudhuri equation for causal

geodesics, which together with certain energy condition, governs the occurrence of (a

pair of) conjugate points. The structure of the Raychaudhuri equation is unchanged

in higher dimensions as far as (higher dimensional version of) general relativity is

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4D. Ida, A. Ishibashi and T. Shiromizu

considered; it takes the form for, e.g., a surface orthogonal null geodesic congruence

in D-dimensions,

d

dλθ = −

with ka= (d/dλ)a, θ, ˆ σabbeing the tangent of a null geodesic with affine parameter

λ of the congruence, its expansion and the shear. Therefore, once well-defined no-

tions of asymptotic flatness at null infinity and strong asymptotic predictability are

formulated in higher dimensions, one can apply general results in 4-dimensions–more

specifically, Propositions and Theorems in Section 12.2 of Wald8)–to higher dimen-

sions. Note that the predictability is needed, in particular to show the area theorem

without demanding that null geodesic generators of the event horizon be complete.

Note also that in order for a weakly asymptotically simple spacetime in D-dimensions

to be consistent with the asymptotic simplicity under the additional condition on

maximally extended null geodesics mentioned above, each component of I has to be

topologically R × SD−2. This, combined with the topological censorship,9)ensures

that the domain of outer communication is simply connected.

For D = even-spacetime dimensions, there exists a stable notion of conformal

null infinity and weak asymptotic simplicity [see Ref. 10) for definition], and therefore

there is no obstruction to apply general results [i.e., those in Section 12.2 of Ref. 8)]

to higher-even-dimensional spacetimes.

However, when spacetime dimension is odd, one needs to be more careful; The

conformal method for defining null infinity would not in general work since the

unphysical metric fails to be smooth at conformal null infinity I when radiation

is present around I.11)This is essentially due to the fact that the leading fall-off

behavior of gravitational radiation near null infinity is in proportion to a half-integer

power of the conformal factor, Ω, when D is odd. Therefore, for the case of odd

spacetime dimensions, we need to formulate a sensible definition of null infinity that

can be used to define asymptotic flatness and some equivalent notion of the strong

asymptotic predictability [see Ref. 12) for such an attempt to define asymptotic

flatness in 5-dimensions without using the conformal method]∗).

In the following when we discuss a black hole in a dynamical, non-stationary

spacetime, we simply assume that a sensible definition of infinity and asymptotic

flatness, equivalent to I and weak asymptotic simplicity above, are formulated, and

we just use the same symbol I to denote thus defined infinity, even if the spacetime

dimension is odd and gravitational radiation is present near infinity. This should be

kept in mind when, e.g., a topology changing process of cross-sections of the event

horizon is considered, since such a phenomenon can only occur in non-stationary,

dynamical spacetime where gravitational radiation is likely to generate.

With the above caveat concerning definitions of null infinity and the predictabil-

1

D − 2θ2− ˆ σabˆ σab− Rabkakb, (2.1)

∗)One may also want to consider non-asymptotically flat spacetimes, such as asymptotically

Kaluza-Klein spacetimes. For that case, we would not be able to use the standard conformal

approach to defining null infinity since the compactified dimensions shrink to a single point by the

conformal transformation. We again need to formulate a suitable definition of infinity, presumably

by dealing with the physical spacetime metric and its asymptotic expansion. Note however the

stationary case discussed below.

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‘Topology and Uniqueness in Higher Dimensional Black Holes’5

ity, the standard proof of the theorem [Prop. 9.2.8 in Ref. 7), Prop. 12.2.2 in Ref. 8)]

that under the null convergence condition, an apparent horizon (see below) is con-

tained in the black hole region is straightforwardly generalized to arbitrary dimen-

sions. In particular, the standard proof of the Hawking’s black hole area theorem7)

[Theorem 12.2.6 in Ref. 8)] is generalized to arbitrary higher dimensions.

2.2. Apparent horizon

For some purposes, instead of dealing with the event horizon, one is more inter-

ested in an apparent horizon which, in a sense, defines a black hole region in a local

manner and plays a role in particular in numerical studies [See Chapter 9]. This

notion can be straightforwardly generalized to higher dimensional case as discussed

below.

Let us consider a partial Cauchy surface, S, in D-dimensional spacetime M,

which is assumed to be an (D − 1)-dimensional connected hypersurface smoothly

embedded in M. A closed connected (D − 2)-surface, T, smoothly embedded in

S is called a trapped surface if the expansion of the congruence of outgoing light

rays orthogonal to T is non-positive at each point of T.

trapped surface is well-defined only when the notion of “outgoing” has a definite

meaning. The asymptotic flatness and the orientability of S and T do not give the

unique definition of ‘out direction’ on T. In many cases, it is assumed that S is

also simply connected. In this case, the out direction of T can be defined in terms

of the Z2-intersection numbers of curves from T to the spatial infinity. However,

since the simple connectedness of S might be too restrictive, it is worth giving

another example of a condition that determines the out direction of T without an

ambiguity. Possible such conditions are that S is orientable and that T separates S

into two disconnected parts. In other words, it is required that S\T = Sin∪ Sout

and Sin∩ Sout= ∅ hold, where Soutis determined by the property that it contains

a neighborhood of the spatial infinity. This clearly ensures that T is orientable and

defines the out direction of T in an obvious way. This condition also applies even

when S is not simply connected.

Under the above conditions, the inner region Sinwith respect to a trapped surface

T is called the inside region Sin(T) of T. The inside region of T will be a closed

subregion of S, whose topological boundary in S consists of T. Then, the trapped

region of S is defined to be the topological sum of Sin(T) over all possible T. The

trapped region might not be a closed region or a smooth region of S. For simplicity,

we however only consider the case where the trapped region is a smooth closed region

of S. Then, the apparent horizon on S is defined to be the topological boundary

of the trapped region. It turns out that the apparent horizon on S is a marginally

outer trapped surface, or in other words, that the expansion of the congruence of the

outgoing light rays orthogonal to the apparent horizon is zero everywhere on it.7)

Now let us consider topological aspects of apparent horizons. Hawking13)has

shown under the dominant energy condition that the apparent horizon in 4-dimensional

spacetime must be diffeomorphic to a 2-sphere or possibly to a 2-torus. Hawking’s

proof of the horizon topology theorem takes two steps: First (i) it is derived under

This definition of the