# Charged Rotating Kaluza-Klein Black Holes in Five Dimensions

**ABSTRACT** We construct a new charged rotating Kaluza-Klein black hole solution in the five-dimensional Einstein-Maxwell theory with a Chern-Simon term. The features of the solutions are also investigated. The spacetime is asymptotically locally flat, i.e., it asymptotes to a twisted $\rm S^1$ bundle over the four-dimensional Minkowski spacetime. The solution describe a non-BPS black hole rotating in the direction of the extra dimension. The solutions have the limits to the supersymmetric black hole solutions, a new extreme non-BPS black hole solutions and a new rotating non-BPS black hole solution with a constant twisted $\rm S^1$ fiber.

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**ABSTRACT:**The main focus of this session was the presentation of new higher-dimensional black hole solutions, including black rings, black strings, and multi black holes, and the study of their properties. Besides new asymptotically flat and locally asymptotically flat black objects also new black holes with anti-de Sitter asymptotics were reported. The studies of their properties included the investigation of their stability, their thermodynamics, their analyticity and their existence. Furthermore, the geodesics in such higher-dimensional space-times were investigated.09/2013; - SourceAvailable from: Vladimir Dzhunushaliev[Show abstract] [Hide abstract]

**ABSTRACT:**We consider rotating Lorentzian wormholes with a phantom field in five dimensions. These wormhole solutions possess equal angular momenta and thus represent cohomogeneity-1 configurations. For a given size of the throat, the angular momenta are bounded by the value of the corresponding extremal Myers-Perry black hole, which represents the limiting configuration. With increasing angular momenta the throat becomes increasingly deformed. At the same time, the violation of the null energy condition decreases to zero, as the limiting configuration is approached. Symmetric wormhole solutions satisfy a Smarr-like relation, which is analogous to the Smarr relation of extremal black holes. A stability analysis shows that the unstable mode of the static wormholes solutions vanishes when the angular momentum exceeds some critical value.Physical Review D 09/2013; · 4.69 Impact Factor - SourceAvailable from: export.arxiv.org
##### Article: Thermodynamics of five-dimensional static three-charge STU black holes with squashed horizons

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**ABSTRACT:**We present a new expression for the five-dimensional static Kaluza-Klein black hole solution with squashed $S^3$ horizons and three different charge parameters. This black hole solution belongs to $D = 5$ $N = 2$ supergravity theory, its spacetime is locally asymptotically flat and has a spatial infinity $R \times S^1 \hookrightarrow S^2$. The form of the solution is extraordinary simple and permits us very conveniently to calculate its conserved charges by using the counterterm method. It is further shown that our thermodynamical quantities perfectly obey both the differential and the integral first laws of black hole thermodynamics if the length of the compact extra-dimension can be viewed as a thermodynamical variable.Physics Letters B 11/2013; 726(1). · 4.57 Impact Factor

Page 1

arXiv:0801.0164v1 [hep-th] 30 Dec 2007

OCU-PHYS 286

AP-GR 52

Charged Rotating Kaluza-Klein Black Holes in Five Dimensions

Toshiharu Nakagawa, Hideki Ishihara, Ken Matsuno, and Shinya Tomizawa

Department of Mathematics and Physics,

Graduate School of Science, Osaka City University,

3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, Japan

(Dated: February 2, 2008)

Abstract

We construct a new charged rotating Kaluza-Klein black hole solution in the five-dimensional

Einstein-Maxwell theory with a Chern-Simon term. The features of the solutions are also investi-

gated. The spacetime is asymptotically locally flat, i.e., it asymptotes to a twisted S1bundle over

the four-dimensional Minkowski spacetime. The solution describe a non-BPS black hole rotating

in the direction of the extra dimension. The solutions have the limits to the supersymmetric black

hole solutions, a new extreme non-BPS black hole solutions and a new rotating non-BPS black

hole solution with a constant twisted S1fiber.

PACS numbers: 04.50.+h 04.70.Bw

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I.INTRODUCTION

In the context of string theory, the five-dimensional Einstein-Maxwell theory with a

Chern-Simon term gathers much attention since it is the bosonic sector of the minimal

supergravity. Supersymmetric (BPS) black hole solutions to the five-dimensional Einstein-

Maxwell equations with a Chern-Simon term have been found by various authors. Based

on the classification of the five-dimensional supersymmetric solutions by Gauntlett.et.al. [1],

they have been constructed on hyper-K¨ ahler base spaces, especially, the Gibbons-Hawking

base space. The first asymptotically flat supersymmetric black hole solution, BMPV

(Breckenridge-Myers-Peet-Vafa) solution, was constructed on the four-dimensional Euclid

space [2]. A supersymmetric black hole solution with a compactified extra dimension on the

Euclidean self-dual Taub-NUT base space was constructed by Gaiotto.et.al [3]. It was ex-

tended to multi-black hole solution with the same asymptotic structure [4]. One of the most

interesting properties is that the possible spatial topology of the horizon of each black hole is

the lens space S3/Zniin addition to S3, where niare the natural numbers . Similarly, black

hole solution on the Eguchi-Hanson space [5], which admits horizon topologies of various

lens spaces L(2n;1) = S3/Z2n(n:natural number), was also constructed. Since the latter

black hole spacetimes are asymptotically locally Euclidean, we cannot locally distinguish

these asymptotic structure.

As solutions in five-dimensional Einstein-Maxwell theory with a positive cosmological

constant, black hole solutions on the Euclid base space [6], the Taub-NUT base space [7]

and the Eguchi-Hanson base space [8] were also constructed. In particular, two-black hole

solution on the Eguchi-Hanson space describes a non-trivial coalescence of black holes. In

refes. [8, 9], the authors compared the two-black hole solution on the Eguchi-Hanson space

with the two-black holes solution on the Euclid space [6], and discussed how the coalescence

of five-dimensional black holes depends on the asymptotic structure of spacetime. Two black

holes with the topology of S3coalesce into a single black hole with the topology of the lens

space L(2;1) = S3/Z2in the case of Eguchi-Hanson space, while two black holes with the

topology of S3coalesce into a single black hole with the topology of S3in the Euclid case.

When the action has the Chern-Simon term, black holes can rotate [10, 11].

The most remarkable feature of the five-dimensional black hole solutions is that they

admit non-spherical topology. Emparan and Reall found the first black ring solution to

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the five-dimensional vacuum Einstein equations [12], which describes a stationary, rotating

black hole with the horizon topology S1× S2. Some supersymmetric black ring solutions

have been also found, based on the construction of the solutions by Gauntlett.et.al. [1].

Elvang et.al. found the first supersymmetric black ring solution with asymptotic flatness

on the four-dimensional Euclidean base space, which is specified by three parameters, mass

and two independent angular momentum components [13]. Gauntlett and Gutowski also

constructed a multi-black ring solution on the same base space [14, 15]. The BPS black rings

with three arbitrary charges and three dipole charges on the flat space were also constructed

in [16, 17]. The BPS black ring solutions on the Taub-NUT base space [16, 18, 19, 20] and

the Eguchi-Hanson space [21]were constructed. The black ring solution on Eguchi-Hanson

space has the same two angular momentum components and the asymptotic structure on

time slices is asymptotically locally Euclidean. The S1-direction of the black ring is along

the equator on a S2-bolt on the Eguchi-Hanson space. It has the limit to the BMPV black

hole with the topology of the lens space L(2;1) = S3/Z2.

In addition to the BPS solutions, the non-BPS black hole solutions have also been studied

by several authors. Cvetic et.al. [22] found a non-extremal, charged and rotating black hole

solution with asymptotic flatness. [30] In the specified limits, the solution reduces to the

known solutions: the same angular momenta case of the Myers-Perry black hole solution [23],

Exact solutions of non-BPS Kaluza-Klein black hole solutions are found in neutral case [24,

25] and charged case [26] . These solutions have a non-trivial asymptotic structure, i.e.,

they asymptotically approach a twisted S1bundle over the four-dimensional Minkowski

spacetime. The horizons are deformed due to this non-trivial asymptotic structure and have

a shape of a squashed S3, where S3is regarded as a twisted bundle over a S2base space.

The ratio of the radius S2to that of S1is always larger than one.

Wang proposed that a kind of Kaluza-Klein black hole solutions can be generated by

the ‘squashing transformation’ from black holes with asymptotic flatness [27]. In fact, he

regenerated the five-dimensional Kaluza-Klein black hole solution found by Dobiasch and

Maison [24, 25] from the five-dimensional Myers-Perry black hole solution with two equal

angular momenta.[31]

In this article, using the squashing transformation, we construct a new non-BPS rotating

charged Kaluza-Klein black hole solutions in the five-dimensional Einstein-Maxwell theory

with a Chern-Simon term, which is the generalization of the Kaluza-Klein black hole solution

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in Ref. [24, 25, 26]. Applying the squashing transformation to the Cvetic et.al.’s charged

rotating black hole solution [22] in vanishing cosmological constant case, we obtain the

new Kaluza-Klein black hole solution. We also investigate the features of the solution.

The spacetime is asymptotically locally flat, i.e., it asymptotically approaches a twisted S1

bundle over the four-dimensional Minkowski spacetime. The solution describes a non-BPS

black hole boosted in the direction of the extra dimension. It has various limits, f.g., to the

supersymmetric BMPV black hole solution, to the supersymmetric Kaluza-Klein black hole

solution and to the extreme non-BPS black hole solutions.

The rest of this article is organized as follows. In Sec.II, we present a new Kaluza-Klein

black hole solution in the five-dimensional Einstein-Maxwell theory with a Chern-Simon

term. In Sec.III, we investigate the basic features of the solution. In Sec.IV, we study the

limit of our solution to various known and unknown black hole solutions. In Sec.V, we

summarize the results in this article.

II.SOLUTION

In the metric of squashing Kaluza-Klein black hole in ref. [26], a function of the radial

coordinate k(r), which describes the squashing of the horizons, appeares. Wang pointed

out that the function k(r) would give a transformation from asymptotically flat solutions to

Kaluza-Klein type solutions. He call this squashing transformation.

Applying the squashing transformation to the non-BPS charged rotating black hole so-

lution found by Cvetic et.al. [22], we construct a new charged, rotating Kaluza-Klein black

hole solution to the five-dimensional Einstein-Maxwell theory with a Chern-Simon term.

The metric and the gauge potential of the solution are given by

ds2= −w(r)

h(r)dt2+ k(r)2dr2

w(r)+r2

4

?

k(r)(σ2

1+ σ2

2) + h(r)(f(r)dt + σ3)2

?

, (1)

and

A =

√3q

2r2

?

dt −a

2σ3

?

,(2)

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respectively, where the metric functions w(r),h(r),f(r) and k(r) are defined as

w(r) =(r2+ q)2− 2(m + q)(r2− a2)

r4

,(3)

h(r) = 1 −a2q2

r6+2a2(m + q)

2a

r2h(r)

k(r) =(r2

r4

, (4)

f(r) = −

?2m + q

r2

−q2

r4

?

∞− a2)

, (5)

∞+ q)2− 2(m + q)(r2

(r2

∞− r2)2

, (6)

and the left-invariant 1-forms on S3are given by

σ1= cosψdθ + sinψ sinθdφ, (7)

σ2= −sinψdθ + cosψ sinθdφ,

σ3= dψ + cosθdφ.

(8)

(9)

The coordinates r,θ,φ and ψ run the ranges of 0 < r < r∞, 0 ≤ θ < π, 0 ≤ φ < 2π,

0 ≤ ψ < 4π, respectively. In the case of k(r) = 1, i.e., r∞→ ∞, the metric coincides with

that of the Cvetic et.al.’s solution without a cosmological constant. In this article we assume

that the parameters a,m,q, and r∞appearing in the solutions satisfy the inequalities

m > 0,(10)

q2+ 2(m + q)a2> 0,(11)

(r2

∞+ q)2− 2(m + q)(r2

(m + q)(m − q − 2a2) > 0,

m + q > 0,.

∞− a2) > 0,(12)

(13)

(14)

As will be explained later, the inequalities (10)-(13) are the conditions for the existence of

two horizons, and the condition (14) is the requirement for the absence of closed timelike

curves outside the outer horizon. FIG.1 shows the region of the parameters.

III.FEATURES OF THE SOLUTIONS

A.Asymptotic form

In the coordinate system (t,r,θ,φ,ψ), the metric diverges at r = r∞but we see that this

is an apparent singularity and corresponds to the spatial infinity. To confirm this, introduce

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m

q

mq

= −

mqa

=+2

2

m

qr

−

ra

q

=

+

2

−

∞

∞

()

()

22

2

2

(,)

−a a

22

(,)

−∞

r

∞

r

22

(,)

rar

∞∞

−

222

2

FIG. 1: Region of the parameters in the (q,m)-plane.

a new coordinate defined by

ρ = ρ0

r2

r2

∞− r2,(15)

where the constant ρ0is given by

ρ2

0=(r2

∞+ q)2− 2(m + q)(r2

∞− a2)

4r2

∞

.(16)

In terms of the coordinate ρ, the metric is written as follows

ds2= −V (ρ)dt2+ 2H(ρ)dtσ3+ U(ρ)dρ2+ ρ(ρ + ρ0)(σ2

1+ σ2

2) + W(ρ)σ2

3, (17)

where the functions V (ρ), H(ρ), U(ρ) and W(ρ) are given by

V (ρ) =((r2

∞+ q)ρ + ρ0q)2− 2(m + q)r2

∞ρ(ρ + ρ0)

r4

∞ρ2

,(18)

H(ρ) =a(ρ + ρ0)((q2− (2m + q)r2

∞))ρ + q2ρ0)

2r4

∞ρ2

4r2

,(19)

U(ρ) =

∞ρ2

0ρ(ρ + ρ0)

(q2+ 2a2(m + q))(ρ + ρ0)2− 2mr2

W(ρ) =r6

∞ρ(ρ + ρ0) + r4

∞)ρ + q2ρ0)

∞ρ2,(20)

∞ρ3− a2(ρ + ρ0)2((q2− 2(m + q)r2

4r4

∞ρ2(ρ + ρ0)

. (21)

This new radial coordinate ρ runs from 0 into ∞. For ρ → ∞, which corresponds to the

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limit of r → r∞, the metric behaves as

ds2≃ −(r2

∞+ q)2− 2(m + q)r2

r4

∞

∞

dt2+a(q2− (2m + q)r2

∞)

r4

∞

dtσ3

(22)

+dρ2+ ρ2(σ2

1+ σ2

2) +r6

∞− a2(q2− 2(m + q)r2

4r4

∞)

∞

σ2

3. (23)

Next in order to transform the asymptotic frame into the rest frame, we define the coordi-

nates (˜t,˜ψ) given by

˜ψ = ψ +

2a(q2− (2m + q)r2

∞− a2(q2− 2(m + q)r2

4r4

∞)

r6

∞)t,(24)

˜t =

?

∞ρ2

0

r6

∞− a2(q2− 2(m + q)r2

∞)t.(25)

Then the metric takes the following asymptotic form

ds2≃ −d˜t2+ dρ2+ ρ2(σ2

1+ σ2

2) + L2˜ σ2

3, (26)

where ˜ σ3= d˜ψ + cosθdφ and the size of extra dimension L is given by

L2=r6

∞− a2(q2− 2(m + q)r2

4r4

∞)

∞

.(27)

It should be noted that the coefficient of ˜ σ2

3approaches a constant. Hence the spacetime

is asymptotically locally flat, i.e., the asymptotic form of the metric is a twisted S1bundle

over four-dimensional Minkowski spacetime.

The Komar mass at the infinity, the angular momenta associated with the Killing vector

fields ∂φand ∂ψat the infinity, and the electric charge can be obtained as

MKomar= π2r6

∞(mr2

∞− q2) − 2a4(m + q)q2− a2(q4− 4mq2r2

2r2

∞+ (4m2+ 4mq + 3q2)r4

∞)

∞(r6

∞− a2(q2− 2(m + q)r2

∞))ρ0

L,

Jφ= 0,

Jψ= −πa(a2q3+ 3q2r4

4r4

∞

√3

2πq.

(28)

∞− 2(2m + q)r6

∞− a2(q2− 2(m + q)r2

∞)

∞)L,

?r6

Q = −

(29)

B.Near-Horizon geometry and regularity

As is seen later, the solution within the region of the parameters in FIG.1 has two horizons,

an event horizon at r = r+> 0 and an inner horizon at r = r−> 0, which are determined

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by the equation w(r) = 0, where r±are given by

r2

±= m ±

?

(m + q)(m − q − 2a2). (30)

In the coordinate system (t,r,θ,φ,ψ), the metric diverges at r = r+and r = r−but these

are apparent. In order to remove this apparent divergence, we introduce the coordinates

(t′,ψ′) such that

dt = dt′+

?h(r)k(r)

w(r)

?h(r)f(r)k(r)

w(r)

dr,(31)

dψ = dψ′−

dr. (32)

Then the metric takes the form of

ds2= −w(r)

h(r)dt′2−2k(r)

?h(r)dt′dr +r2

4

?

k(r)(σ2

1+ σ2

2) + h(r)(f(r)dt′+ σ′

3)2?

.(33)

This metric well behaves at r = r±, i.e., it is smooth at this place.

Moreover, define the coordinates (v,ψ′′) given by

v = t,(34)

ψ′′= ψ′+ f(r+)t,(35)

and then the metric can be rewritten as follows

ds2= −

?w(r)

+r2

2h(r)(f(r) − f(r+))σ′′

h(r)+r2

4h(r)(f(r) − f(r+))2

?

dv2−

2k(r)

?h(r)dvdr

1+ σ2

3dt +r2

4

?

k(r)(σ2

2) + h(r)σ

′′2

3

?

.(36)

The Killing vector field V = ∂vbecomes null at r = r+and furthermore, V is hypersurface

orthogonal from Vµ∝ dr there.

These mean that the hypersurfaces r = r+is a Killing horizon. Similarly, r = r−is also

a Killing horizon. We should also note that in the coordinate system (v,φ,ψ′′,r,θ), each

component of the metric form (36) is analytic on and outside the black hole horizon. Hence

the spacetime has no curvature singularity on and outside the black hole horizon.

The surface gravity κ of the black hole is obtained as

κ =

r2

+− r2

∞− r2

−

r∞(r2

−)

?

(r2

(r2

∞− r2

∞− r2

+)

−)

?

r6

r6

∞− a2(q2− 2(m + q)r2

+− a2(q2− 2(m + q)r2

∞)

+).(37)

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Next, we investigate the shape of the horizon, especially, the aspect ratio of S2base space to

the S1fiber, which characterize the squashing of S3and is denoted by k(r+)/h(r+). In the

case of k(r+)/h(r+) > 1, the event horizon is called oblate, where the radius of S2larger than

that of S1. In the case of k(r+)/h(r+) < 1, the event horizon is called prolate, where the

radius of S2smaller than that of S1. FIG.2 shows the oblate region and the prolate region

in the (m,q)-plane. The shaded region and unshaded region are the oblate region and the

prolate region, respectively. At the boundary of two regions, the ratio is k(r+)/h(r+) = 1,

where the horizon becomes a round S3. Thus unlike the static solution [26] obtained by two

of the present authors, the horizon admits a prolate shape in addition to a round S3.

(,)

−a a

22

oblate

prolate

m

q

(,)

−∞

r

∞

r

22

(,)

rar

∞∞

−

222

2

(()/,/)213323

22

aa

−

@

FIG. 2: The aspect ratio of the outer horizon.

C.Parameters region and absence of CTCs

We consider the condition that the spacetime has non-degenerate horizons and no CTCs

outside the horizons.

FIG.1 shows the region of the parameters in a (q,m)-plane. The horizons appear at r > 0

satisfying w(r) = 0, i.e., the quadratic equation with respect to r2

(r2+ q)2− 2(m + q)(r2− a2) = 0.(38)

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Page 10

The necessary and sufficient condition that this quadratic equation has two different roots

within the range of 0 < r2< r2

∞is

m > 0,(39)

q2+ 2(m + q)a2> 0,(40)

(r2

∞+ q)2− 2(m + q)(r2

(m + q)(m − q − 2a2) > 0.

∞− a2) > 0,(41)

(42)

As shown in FIG.3, the region satisfying the inequalities (39)-(42) consist of two regions:

Σ = {(q,m)| m − q > 2a2, m + q > 0, m < −q + (q + r2

{(q,m)| m−q < 2a2, m+q < 0, m > −(q2+2a2q)/(2a2)}. However, as will be seen below,

one of the regions Σ′is excluded from the requirement for the absence of CTCs outside the

∞)2/(2(r2

∞− a2))} and Σ′=

black hole horizon.

The positivity of the components gφφand gψψoutside the horizons assures the absence

of CTCs there. Since gφφis always positive definite outside the horizons, it is sufficient to

consider the parameter region such that

gψψ= h(r) =r6+ 2a2(m + q)r2− a2q2

r6

> 0 (43)

outside the horizons. It is noted that the function h(r) is a monotony increasing function

of r. Hence, as a result, the condition for the absence of CTC outside the outer horizons

is h(r+) > 0. The curve in FIG.3 which enter only the region Σ′and has endpoints at the

points (0,0) and (−a2,a2) denotes h(r+) = 0. Since the function h(r+) takes a maximum

value of a zero in the region Σ′, it becomes non positive throughout the region Σ′. In

contrast, it takes positive values in the region Σ. This is why in addition to the inequalities

(39)-(42), we impose the inequality m + q > 0 on the parameters.

D.Ergo region

In our solutions, an ergo surface is located at r = re(r+< re< r∞) satisfying the cubic

equation with respect to r2

− A2r6+ 4r4− 2(aA − 2)((aA − 2)(m + q) + 2q)r2+ (aA − 2)2q2= 0,

where the constant A is defined by

2a(q2− (2m + q)r2

r6

(44)

A = −

∞)

∞− a2(q2− 2(m + q)r2

∞).(45)

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Page 11

⁄

S'

m

q

mq

= −

mqa

=+2

2

m

qr

−

ra

q

=

+

2

−

∞

∞

()

()

22

2

2

m

qa q

a

= −

+

2

22

2

2

h r ( )

+= 0

(,)

−a a

22

(,)

−∞

r

∞

r

22

(,)

rar

∞∞

−

222

2

FIG. 3: In the region Σ, since h(r+) > 0, the solution has no CTC everywhere outside the horizon,

but in the region Σ′, since h(r+) ≤ 0, it has CTCs outside the horizon. The curve which enter only

the region Σ′and has endpoints at the points (0,0) and (−a2,a2) denotes h(r+) = 0.

In fact, for r2

+< r2< r2

∞, this equation has a single positive root within the region of

parameters Σ, the boundaries ∂Σ−= {(q,m) | m + q = 0,− r2

{(q,m) | m−q−2a2= 0, −a2< q ≤ r2

the region of parameters (10)-(14) and ∂Σ±.

∞≤ q < −a2} and ∂Σ+=

∞−2a2}. Thus, there is always an ergo region within

IV.VARIOUS LIMITS

1.

r∞→ ∞

In the limit of r∞ → ∞ with the other parameters fixed, where the size of an extra

dimension becomes infinite, the function k(r) takes the limit of k(r) → 1. Then the metric

coincides with that of the asymptotically flat solutions obtained by Cvetic. et.al. [22]. In

the specified case of q → 0, the solution reduces to the five-dimensional Myers-Perry black

hole solution [23] with two equal angular momenta.

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Page 12

2.

q → 0

In the limit of q → 0, the solution coincides with the one obtained by Gibbons et.al. [24,

25]. This solutions has only a angular momentum in the direction of an extra dimension.

As mentioned previously, Wang regenerated it [27] via the squashing transformation for the

five-dimensional Myers-Perry black hole solutions.

3.

m → −q

Taking the limit of m → −q with introducing new coordinates (˜t, ˜ r) and the parameters

(˜Q,R∞,˜J) defined as

r2=4(R2

∞˜ r + R∞˜Q)

˜ r + R∞

R2

∞

R2

∞= 4R2

,(46)

t =

∞−˜Q

∞,

˜t,(47)

r2

(48)

q = −4˜Q,

a = −2˜J

(49)

˜Q,

(50)

we obtain the following metric

ds2= −H−2?

d˜t +

˜J

R2

∞

Hkσ3

?2

+ Hds2

T−NUT,(51)

where ds2

T−NUTis the metric on the Euclidean self-dual Taub-NUT space written in the

Gibbons-Hawking coordinate and is given by

ds2

T−NUT= Hk(d˜ r2+ ˜ r2dΩ2

S2) + R2

∞H−1

kσ2

3

(52)

H and Hkare harmonic functions on the three-dimensional Euclid space and are expressed

in the coordinate system as follows

H = 1 +

˜Q

R∞˜ r,

(53)

Hk= 1 +R∞

˜ r

.(54)

This coincides with the metric of the supersymmetric black hole solutions with a compactified

extra dimension on the Euclidean self-dual Taub-NUT space in Ref [3]. This spacetime has

12

Page 13

an ergo region. Moreover, in the case of r∞→ ∞, the metric can be written as follows

q

R2

ds2= −

?

1 +

?−2?

dt −

aq

2R2σ3

?2

+

?

1 +

q

R2

??dR2+ R2dΩ2

S3?,(55)

where R2= r2+ q and dΩ2

S3 is the metric on a unit three-sphere. This is the metric of the

BMPV black hole solutions written in terms of the Gibbons-Hawking coordinate.

4.

a → 0

The case of a → 0 corresponds to the metric of the static non-BPS Kaluza-Klein black

hole solution with a squashed horizon obtained by two of authors [26].

M, Q, J,

M, Q,

M, J,

M=Q, J, r1

r1

r1

r1

Gaiotto-Strominger-Yin

Gibbons-Wiltshire

Dobiasch-Maison

Ishihara-Matsuno

M, Q, J

M, Q

Cvetic-Lu-Pope

Cvetic-Youm

M=Q, J

Breckenridge-Myers-Peet-Vafa

M, J

Myers-Perry *

(5-dim. Reissner-Nordstrom)

Tangherlini

Squashing Transformation

Asymptotic Flat Black Holes

Kaluza-Klein Black Holes

@

FIG. 4: Various limits: This figure shows the relation between black hole solutions with asymptot-

ically flatness and Kaluza-Klein black holes. * We consider, here, the Myers-Perry solution in the

special case of two equal angular momenta.

5.

m → q + 2a2

In the case of m → q+2a2, two horizons degenerate, although this is not a BPS solution.

In this case, the metric can be written in the form

ds2= −Γ

r4

?

dt +ω

2Γσ3

?2

+r4

∆k2dr2+r2

4

?

k(σ2

1+ σ2

2) +∆

Γσ2

3

?

,(56)

13

Page 14

where the functions ∆, Γ, ω and k(r) in the metric are given by

∆ = (r2− (q + 2a2))2,

Γ = r4− 2(q + 2a2)r2+ q2,

ω = a((3q + 4a2)r2− q2),

k(r) =(r2

(r2

(57)

(58)

(59)

∞− (q + 2a2))2

∞− r2)2

. (60)

This spacetime has an ergo region. It is noted that from Eq.(37), the surface gravity of the

black hole vanishes.

6.

ρ0→ 0 with ρ±finite

Here we consider the limit of ρ0→ 0, where the function k(r) → 0. We introduce the

new parameters ρ±defined as

ρ±=

r2

±

r2

∞− r2

±

ρ0. (61)

In terms of r±, the constant ρ0defined in Eq.(16) is written in the form

ρ2

0=(r2

∞− r2

+)(r2

4r2

∞

∞− r2

−)

.(62)

Then, the metric can rewritten and as

ds2= −¯V dT2+ Udρ2+ ρ(ρ + ρ0)dΩ2

where T = 2ρ0t/r∞, and the metric functions¯V ,U,W, H in Eqs.(18)-(21) are rewritten in

S2 + Wσ2

3+ 2HdTσ3,(63)

the form

¯V =

?

1 −ρ+

ρ

??

1 −ρ−

?

1 −ρ−

ρ

?

−m + q

2r2

∞

?a

ρ0

?2?

1 +ρ0

ρ

?2

,(64)

U =

?

1 +ρ0

ρ

?

1 −ρ+

∞ρ3− a2(ρ + ρ0)2(q2(ρ + ρ0) − 2(m + q)r2

4r4

H = −(ρ + ρ0)(r2

4r3

From Eq.(61), the following equations holds

ρ

??

ρ

?,(65)

W =r6

∞ρ)

∞ρ2(ρ + ρ0)

,(66)

∞(2m + q)ρ − q2(ρ + ρ0)

∞ρ2

?a

ρ0

?

. (67)

r2

∞− r2

+=r2

∞

ρ+ρ0,

r2

∞− r2

−=r2

∞

ρ−ρ0.(68)

14

Page 15

In order that ρ±are finite in the limit of ρ0→ 0, it is necessary that r±→ r∞. Therefore,

from Eq.(30), a pair of parameters (q,m) must take either limit of

(q,m) → (−r2

∞,r2

∞)(69)

or

(q,m) → (r2

∞− 2a2,r2

∞).(70)

Furthermore, from Eqs.(30) and (68), the parameters (q,m) must behave as

(q,m) ≃ (−r2

∞+ β1ρ0+ β−

2ρ2

0, r2

∞− β1ρ0+ β+

2ρ2

0),(71)

or

(q,m) ≃ (r2

∞− 2a2− β1ρ0− β−

2ρ2

0, r2

∞− β1ρ0+ β+

2ρ2

0),(72)

respectively, where the constants β±

2satisfy β−

2+ β+

2= β2and the constants (β1,β2) are

given by

(β1,β2) =

?

2(ρ++ ρ−),2(ρ+− ρ−)2

4ρ+ρ−− a2

∞− β1ρ0+ β+

dT2+

?

?4ρ+ρ−− a2

2ρ

?

2ρ2

dρ2

??

dTσ3,

.(73)

(i) In the case of (q,m) ≃ (−r2

∞+ β1ρ0+ β−

2ρ2

0, r2

0), the metric becomes

ds2= −4(ρ − ρ+)(ρ − ρ−) − a2

4ρ2

1 −ρ+

ρ

1 −ρ−

ρ

?

+ρ2dΩ2

S2 +4ρ+ρ−− a2

4

σ2

3+ a(74)

where the coordinates ψ and T are transformed as

ψ → ψ −

a(ρ+− ρ−)

√ρ+ρ−(4ρ+ρ−− a2)T,

4ρ+ρ−

4ρ+ρ−− a2T.

(75)

T →

?

(76)

A coefficient of σ2

3, i.e., the size of the S1fiber, takes the constant value.

(ii) In the case of (q,m) ≃ (r2

∞−2a2−β1ρ0−β−

2ρ2

0, r2

∞−β1ρ0+β+

2ρ2

0), the metric reduces

to

ds2= −16ρ2

+ρ2

−(ρ − ρ+)(ρ − ρ−) − a2(a2− 6ρ+ρ−)2

16ρ2

+ρ2

S2 +(4ρ+ρ−− a2)(2ρ+ρ−+ a2)2

16ρ2

−ρ2

dT2

+

dρ2

??

?

1 −ρ+

+aa2− 6ρ+ρ−

2ρ+ρ−ρ

ρ

1 −ρ−

ρ

? + ρ2dΩ2

?

+ρ2

−

σ2

3

(4ρ+ρ−− a2)(2ρ+ρ−+ a2)2

16ρ2

+ρ2

−

dTσ3

(77)

15

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