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arXiv:gr-qc/0401100v1 24 Jan 2004

YITP–04–6

A boundary value problem for the five-dimensional stationary

rotating black holes

Yoshiyuki Morisawa∗

Yukawa Institute for Theoretical Physics,

Kyoto University, Kyoto 606-8502, Japan

Daisuke Ida†

Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan

(Dated: January 22, 2004)

Abstract

We study the boundary value problem for the stationary rotating black hole solutions to the

five-dimensional vacuum Einstein equation. Assuming the two commuting rotational symmetry

and the sphericity of the horizon topology, we show that the black hole is uniquely characterized

by the mass, and a pair of the angular momenta.

∗morisawa@yukawa.kyoto-u.ac.jp

†d.ida@th.phys.titech.ac.jp

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I. INTRODUCTION

In recent years there has been renewed interest in higher dimensional black holes in the

context of both string theory and brane world scenario. In particular, the possibility of

black hole production in linear collider is suggested [1, 2, 3, 4]. Such phenomena play

a key role to get insight into the structure of space-time; we might be able to prove the

existence of the extra dimensions and have some information about the quantum gravity.

Since the primary signature of the black hole production in the collider will be Hawking

emission from the stationary black hole, the classical equilibrium problem of black holes is

an important subject. The black holes produced in colliders will be small enough compared

with the size of the extra dimensions and generically have angular momenta, they will be

well approximated by higher dimensional rotating black hole solutions found by Myers and

Perry [5]. The Myers-Perry black hole which has the event horizon with spherical topology

can be regarded as the higher-dimensional generalization of the Kerr black hole. One might

expect that such a black hole solution describes the classical equilibrium state continued

from the black hole production event, if it equips stability and uniqueness like the Kerr

black hole in four-dimensions. The purpose of this paper is to consider the uniqueness and

nonuniqueness of the rotating black holes in higher dimensions.

The uniqueness theorem states that a four-dimensional black hole with regular event hori-

zon is characterized only by mass, angular momentum and electric charge [6, 7]. Recently,

uniqueness and nonuniqueness properties of five or higher-dimensional black holes are also

studied. Emparan and Reall have found a black ring solution of the five-dimensional vacuum

Einstein equation, which describes a stationary rotating black hole with the event horizon

homeomorphic to S2× S1[8]. In a certain parameter region, a black ring and a (Myers-

Perry) black hole can carry the same mass and angular momentum. This might suggest the

nonuniqueness of higher-dimensional stationary black hole solutions. For example, Reall [9]

conjectured the existence of stationary, asymptotically flat higher-dimensional vacuum black

hole admitting exactly two commuting Killing vector fields although all known higher di-

mensional black hole solutions have three or more Killing vector fields. In six or higher

dimensions, Myers-Perry black hole can have an arbitrarily large angular momentum for a

fixed mass. The horizon of such black hole highly spreads out in the plane of rotation and

looks like a black brane in the limit where the angular momentum goes to infinity. Hence,

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Emparan and Myers [10] argued that rapidly rotating black holes are unstable due to the

Gregory-Laflamme instability [11] and decay to the stationary black holes with rippled hori-

zons implying the existence of black holes with less geometric symmetry compared with the

Myers-Perry black holes. For supersymmetric black holes and black rings, string theoretical

interpretation are given by Elvang and Emparan [12]. They showed that the black hole and

the black ring with same asymptotic charges correspond to the different configurations of

branes, giving a partial resolution of the nonuniqueness of supersymmetric black holes in

five dimensions. On the other hand, we have uniqueness theorems for black holes at least in

the static case [13, 14, 15, 16, 17, 18]. Furthermore, the uniqueness of the stationary black

holes is supported by the argument based on linear perturbation of higher dimensional static

black holes [19, 20]. There exist regular stationary perturbations that fall off at asymptotic

region only for vector perturbation, and then the number of the independent modes corre-

sponds to the rank of the rotation group, namely the number of angular momenta carried

by the Myers-Perry black holes [21]. This suggests that the higher-dimensional stationary

black holes have uniqueness property in some sense, but some amendments will be required.

Here we consider the possibility of restricted black hole uniqueness which is consistent with

any argument about uniqueness or nonuniqueness. Though the existence of the black ring

solution explicitly violates the black hole uniqueness, there still be a possibility of black hole

uniqueness for fixed horizon topology [22]. Hence we restrict ourselves to the stationary

black holes with spherical topology.

In this paper, we consider the asymptotically flat, black hole solution to the five-

dimensional vacuum Einstein equation with the regular event horizon homeomorphic to S3,

admitting two commuting spacelike Killing vector fields and stationary (timelike) Killing

vector field. The two spacelike Killing vector fields correspond to the rotations in the (X1-

X2)-plane and (X3-X4)-plane in the asymptotic region ({Xµ} are the asymptotic Cartesian

coordinates), respectively, which are commuting with each other. Along with the argument

by Carter [23], it is possible to construct a timelike Killing vector field tangent to the fixed

points (namely, axis) of the axi-symmetric Killing vector field from the given timelike Killing

vector field. Repeating this procedure for each commuting spacelike Killing vector field, the

obtained timelike Killing vector field is also commuting with both spacelike Killing vector

fields. Hence, it is natural to assume all the three Killing vector fields are commuting with

each other. The five-dimensional vacuum space-time admitting three commuting Killing

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vector fields is described by the nonlinear σ-model [24]. Then the Mazur identity [25] for

this system is derived. We show that the five-dimensional black hole solution with regular

event horizon of spherical topology is determined by three parameters under the appropriate

boundary conditions.

The remainder of the paper is organized as follows. In Section IIA, we give the field

equations for the five-dimensional vacuum space-time admitting three commuting Killing

vector fields. In Section IIB, we introduce the matrix form of field equations to clarify the

hidden symmetry of this system following Maison [24]. Then the Mazur identity which is

useful to show the coincidence of two solutions is derived in Section III. In Section IV,

we determine the boundary conditions. We summarize this paper and make discussions on

related matters in Section V.

II. FIVE-DIMENSIONAL VACUUM SPACE-TIME ADMITTING THREE COM-

MUTING KILLING VECTOR FIELDS

Assuming the symmetry of space-time, the Einstein equations reduce to the equations

for the scalar fields defined on three-dimensional space. Then, we show that the system of

the scalar fields is described by a nonlinear σ-model.

A. Weyl-Papapetrou metrics

We consider the five-dimensional space-time admitting two commuting Killing vector

fields ξI= ∂I, (I = 4,5). The metric can be written in the form

g = f−1γijdxidxj+ fIJ(dxI+ wI

idxi)(dxJ+ wJ

jdxj), (1)

where i,j = 1,2,3, f = det(fIJ). The three-dimensional metric γij, the functions wI

iand

fIJ are independent on the coordinates xI(x4= φ, x5= ψ, and we will later identify ξ4

and ξ5 as Killing vector fields corresponding to two independent rotations in the case of

asymptotically flat space-time). We define the twist potential ωIby

ωI,µ= f fIJ

?

|γ|ǫijµγimγjn∂mwJ

n, (2)

where µ = 1,···,5, γ = det(γij), γijis the inverse metric of γij, and ǫλµν denotes the

totally skew-symmetric symbol such that ǫ123 = 1,ǫIµν = 0. Then the vacuum Einstein

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equation reduces to the field equations for the five scalar fields fIJ and ωI defined on the

three-dimensional space:

D2fIJ = fKLDfIK· DfJL− f−1DωI· DωJ,

D2ωI = f−1Df · DωI+ fJKDfIJ· DωK,

(3)

(4)

and the Einstein equations on the three-dimensional space:

(γ)Rij =

1

4f−2f,if,j+1

4fIJfKLfIK,ifJL,j+1

2f−1fIJωIiωJj, (5)

where D is the covariant derivative with respect to the three-metric γijand the dot denotes

the inner product determined by γij.

Here we assume the existence of another Killing vector field ξ3 = ∂3 which commutes

with the other Killing vectors as [ξ3,ξI] = 0 (we will later identify the ξ3as the stationary

Killing vector field in the case of asymptotically flat space-time). Then the metric can be

written in the Weyl-Papapetrou–type form [26]

g = f−1e2σ(dρ2+ dz2) − f−1ρ2dt2+ fIJ(dxI+ wIdt)(dxJ+ wJdt), (6)

where we denote x3= t, and all the metric functions depend only on ρ and z. Once the

five scalar fields fIJ,ωIare determined, the other metric functions σ and wIare obtained

by solving the following partial derivative equations:

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ρσ,ρ =

1

4f−2[(f,ρ)2− (f,z)2] +1

+1

2f−1fIJ(ωI,ρωJ,ρ− ωI,zωJ,z),

1

4f−2f,ρf,z+1

,ρ= ρf−1fIJωJ,z,

4fIJfMN(fIM,ρfJN,ρ− fIM,zfJN,z)

(7)

1

ρσ,z =

wI

4fIJfMNfIM,ρfJN,z+1

2f−1fIJωI,ρωJ,z, (8)

(9)

wI

,z= −ρf−1fIJωJ,ρ.(10)

The fIJand ωIare given by axi-symmetric solution of the field equations (3) and (4) on the

abstract flat three-space with the metric

γ = dρ2+ dz2+ ρ2dϕ2.(11)

Thus the system is described by the action

S =

?

dρdz ρ

?1

4f−2(∂f)2+1

4fIJfKL∂fIK· ∂fJL+1

2f−1fIJ∂ωI· ∂ωJ

?

.(12)

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