Black Saturn with a dipole ring

Black Saturn with a dipole ring

Stoytcho S. Yazadjiev*Institut fur Theoretische Physik, Universitat Gottingen, Friedrich-Hund-Platz 1, D-37077 Gottingen, Germany

and Department of Theoretical Physics, Faculty of Physics, Sofia University, 5 James Bourchier Boulevard, Sofia 1164, Bulgaria(Received 6 June 2007; published 12 September 2007)

We present a new stationary, asymptotically flat solution of 5D Einstein-Maxwell gravity describing aSaturn-like black object: a rotating black hole surrounded by a rotating dipole black ring. The solution isgenerated by combining the vacuum black Saturn solution and the vacuum black ring solution withappropriately chosen parameters. Some basic properties of the solution are analyzed and the basicquantities are calculated.

DOI: 10.1103/PhysRevD.76.064011 PACS numbers: 04.50.+h, 04.20.Jb

I. INTRODUCTION

In recent years the higher dimensional gravity is attract-ing much interest. Apart from the fact that the higherdimensional gravity is interesting in its own right, theincreasing amount of works devoted to the study of thehigher dimensional spacetimes is inspired from the stringtheory and the braneworld scenario with large extra dimen-sions. The gravity in higher dimensions exhibits muchricher dynamics and spectrum of solutions than in fourdimensions. One of the most reliable routes for betterunderstanding of higher dimensional gravity and the re-lated topics are the exact solutions. For example, recentlydiscovered exact black rings solutions1 with unusual hori-zon topology [1], demonstrated explicitly that the 5DEinstein gravity exhibits unexpected features completelyabsent in four dimensions. It was shown in [1] that both theblack hole and the black ring can carry the same conservedcharges, the mass and a single angular momentum, andtherefore there is no uniqueness theorem in five dimen-sions. Moreover, the black rings can also carry noncon-served charges which can be varied continuously withoutaltering the conserved charges. This fact leads to continu-ous (classical) nonuniqueness [3].

Among the most fascinating solutions of the 5D vacuumEinstein gravity discovered so far are those describingmany black objects in equilibrium [4,5]. In particular wewill be interested in the so-called black Saturn solutiondescribing a rotating black hole surrounded by a rotatingblack ring. The natural question is whether a similar solu-tion exists in the more general case of nonvacuum 5Dgravity, especially in 5D Einstein-Maxwell gravity. Theaim of this paper is to present a new stationary, asymptoti-cally flat solution of 5D Einstein-Maxwell gravity describ-ing a Saturn-like black object: a rotating black holesurrounded by a rotating dipole black ring. The solution

is generated by combining the vacuum black Saturn solu-tion and the vacuum black ring solution with appropriatelychosen parameters.

II. BLACK SATURN WITH DIPOLE RING

A method for generating exact stationary solutions of 5DEinstein-Maxwell gravity was developed in [6]. Thismethod allows us to construct 5D Einstein-Maxwell solu-tions by combining two stationary solutions of the 5Dvacuum Einstein equations. Through this method, it wasshown that the dipole black ring can be constructed via thescheme

rotating black ring � rotating black ring

! rotating dipole black ring; (1)

where the black ring solutions are with appropriatelychosen parameters. It is natural to expect that the generat-ing scheme

rotating black ring � black saturn

! black Saturn with dipole black ring (2)

will produce a black Saturn solution with a dipole blackring when the parameters of the solutions are appropriatelychosen.

We present below the solution generated via the scheme(2). The spacetime geometry and the electromagnetic fieldare given by

ds2��H2

H1W

�dt�

�!

H2�q

�d

�2

�k2H1PW

Y3�d�2�dz2��H1G

H2Wd 2�W2G�d�

2;

Fz��3pW2

G�

�@�V ; F��

��3pW2

G�

�@zV : (3)

Here k and q are constants and the functions W, Y, V aregiven by

*[email protected];[email protected]

1The black ring solution of [1] is with one rotational parame-ter. Black rings with two independent rotational parameters havebeen constructed recently in [2].

PHYSICAL REVIEW D 76, 064011 (2007)

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http://dx.doi.org/10.1103/PhysRevD.76.064011

W �~�1��5��2 � ~�1�3�

2��2 � ~�1�4�2 � B2

1�3�4� ~�1 ��5�2�4�

�4��5��2 � ~�1�3�2��2 � ~�1�4�

2 � B21 ~�2

1�3�4� ~�1 ��5�2�2�

; (4)

Y ��2 � ~�1�5��5��2 � ~�1�3�

2��2 � ~�1�4�2 � B2

1�3�4� ~�1 ��5�2�4�

�5��2 � ~�1�3��

2 � ~�1�4��2 � ~�2

1��2 ��4�5��

2 ��3�4�; (5)

V � B1�3�4�5� ~�1 ��5��

2 � ~�21��

2 � ~�1�3��2 � ~�1�4�

~�1��5��2 � ~�1�3�

2��2 � ~�1�4�2 � B2

1�3�4� ~�1 ��5�2�4�

: (6)

The functions P, G , G�, ! , H1, and H2 are the metric functions of the vacuum black Saturn solution [4] and they aregiven by the following expressions

P � ��3�4 � �2�2��1�5 � �

2��4�5 � �2�; (7)

G ��3�5

�4; (8)

G� ��2�4

�3�5; (9)

! � 2c1R1

��M0M1

p� c2R2

��M0M2

p� c2

1c2R2

��M1M4

p� c1c2

2R1

��M2M4

p

F��G�

p ; (10)

H1 � F�1�M0 � c21M1 � c

22M2 � c1c2M3 � c

21c

22M4�;

(11)

H2 � F�1 �3

�4

�M0

�1

�2� c2

1M1�2

�1�2� c2

2M2�1�2

�2

� c1c2M3 � c21c

22M4

�2

�1

�; (12)

where

M0 � �2�25��1 ��3�

2��2 ��4�2��2 ��1�2�

2

��2 ��1�4�2��2 ��2�3�

2; (13)

M1 � �21�2�3�4�5�2��1 ��2�

2��2 ��4�2

��1 ��5�2��2 ��2�3�

2; (14)

M2 � �2�3�4�5�2��1 ��2�

2��1 ��3�2��2 ��1�4�

2

��2 ��2�5�2; (15)

M3 � 2�1�2�3�4�5��1 ��3��1 ��5��2 ��4�

��2 ��21��

2 ��22��

2 ��1�4��2 ��2�3�

��2 ��2�5�; (16)

M4 � �21�2�2

3�24��1 ��5�

2��2 ��1�2�2

��2 ��2�5�2; (17)

and

F��1�5��1��3�2��2��4�

2��2��1�3��2��2�3�

��2��1�4��2��2�4��

2��2�5�

��2��3�5�Y5

i�1

��2��2i �: (18)

Finally we have

�i � Ri � �z� ai�; i � 1; 2; 3; 4; 5; (19)

where

Ri ��2 � �z� ai�

2q

(20)

and

~� 1 ��2 � �z� b1�

2q

� �z� b1�: (21)

In all the above expressions ai, b1, c1, c2, and B1 are (real)constants. The presented solution possesses three Killingvectors, namely � � @=@t, � � @=@ , and �� @=@�.

Since �c1; c2� ! ��c1;�c2� and B1 ! �B1 change thedirection of rotation and the sign of the electromagneticfield we can restrict ourselves to the case c1 0 and B1 0. Let us note some limiting cases of the presented solu-tion. For B1 � 0 and b1 � a4 we obtain the black Saturnsolution. Setting further c1 � 0 and a1 � a5 � a4 weobtain the -spinning Myers-Perry black hole. Takinginstead (B1 � 0) c2 � 0 and a2 � a3 we obtain the S1

rotating dipole black ring. These limits will be discussedin more detail in Sec. III J.

STOYTCHO S. YAZADJIEV PHYSICAL REVIEW D 76, 064011 (2007)

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III. ANALYSIS OF THE SOLUTION

In order to analyze the solution we shall use the canoni-cal procedure [7] closely following the analysis of [4].

As we have already mentioned, the generated solutiondescribes the regular black object when the solution pa-rameters are chosen appropriately. Here we will considerthe following ordering of the parameters ai and b1, namely

a1 � a5 � a4 � b1 < a3 � a2: (22)

The other bounds on the dipole parameter b1 different from(22) lead to solutions which are singular and are not ofphysical interest.

In order to insure the positivity and regularity of thefunctions W, Y, and V we must impose the followingconstraint:

B21 � 2

�a3 � b1��b1 � a4�

�b1 � a5�: (23)

Further, in order to remove the singularity ofH2 we mustimpose the constraint

c21 � 2

�a3 � a1��a4 � a1�

�a5 � a1�(24)

just as for the black Saturn [4].Since the solution is invariant under shifts in z-direction,

its description in terms of ai and b1 is redundant and it isconvenient to introduce new parameters. Following [4] weshall introduce one dimensionful parameter L2 and fourdimensionless parameters �i�i � 1; 2; 3� and � defined asfollows:

L2 � a2 � a1; �i �ai�2 � a1

L2 ; � �b1 � a1

L2 :

(25)

As a consequence of (22) we have

0 � �3 � �2 � �< �1 � 1: (26)

In terms of the new parameters the conditions (23) and(24) take the form

B21 � 2L2 ��1 � �� 2�

�� 3�; (27)

c21 � 2L2 �1�2

�3: (28)

It is also convenient to introduce the dimensionlesscoordinate �z and dimensionless parameter �c2 given by [4]

z � L2 �z� a1; (29)

�c 2 �c2

c1�1� �2�: (30)

A. Asymptotic behavior of the solution

In order to study the asymptotic behavior of the solutionwe shall follow the standard way and we shall introduce theasymptotic coordinates r and � defined by

� � 12r

2 sin2�; z � 12r

2 cos2�: (31)

Then in the asymptotic limit r! 1 we find

W � 1� 2L2

r2 �� 2�; (32)

Y � 1� 2L2

r2 �� 2�sin2�; (33)

H1

H2� 1� 2

L2

r2

f�3�1� �1 � �2� � 2�2�3��1 � �2� �c2 � �2��1 � �2�3�1� �1 � �2�� c22g

�3�1� �2 �c2�2 ; (34)

k2H1P �1

r2 k2�1� �2 �c2�

2; (35)

!

H2� �L

��2�1�2

�3

s�c2

j 1� �2 �c2 j�

4L3

r2

sin2�

�3�1� �2 �c2�3

��2

2�1�3

sf�2

3 � �3��1 � �2��1� �1 � �3�

� �2�1� �3�� c2 � �2�3��1 � �2��1 � �3�

� �1�1� �1 � �2 � �3�� c22

� �1�2��1 � �2�3�2� �1 � �2 � �3�� c32g; (36)

G� � r2cos2�; G � r2sin2�: (37)

The asymptotic behavior of the function V is

V � 2L3��2��1 � �� 2�� 3�

q sin2�

r2 : (38)

Since we are interested in asymptotically flat solutionsfor which gt ! sin2�=r2 and g�� ! 1=r2 we chose theconstants q and k to be

q � L

��2�1�2

�3

s�c2

1� �2 �c2; (39)

k � j1� �2 �c2j�1; (40)

for �2 �c2 � �1. The case �2 �c2 � �1 is singular and willnot be considered here. With this choice for q and k the

BLACK SATURN WITH A DIPOLE RING PHYSICAL REVIEW D 76, 064011 (2007)

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asymptotic metric takes the form

ds2 � �dt2 � dr2 � r2d�2 � r2sin2�d 2 � r2cos2�d�2:

(41)

In what follows we shall show that the periodicities of theangles and � can be chosen to be canonical ones

� � �� 2 (42)

and as a consequence we find that the spacetime is asymp-totically flat.

B. Rod structure and balance conditions

As we have already mentioned the ordering (22) and thecondition (23) insure the positivity and the regularity of thefunctions W and Y. Therefore the rod structure [7] of thesolution presented here is the same as the rod structure ofthe black Saturn. With condition (24) imposed the rodstructure is the following. We have

(i) Semi-infinite rod �z 2 ��1; �3� and finite rod �z 2��2; �1� in direction (0, 1, 0) which are sources of the��-part of the metric

(ii) The source of the -part of the metric is the semi-infinite rod �z 2 �1;1� in direction (0, 0, 1).

(iii) Two finite rods �z 2 ��3; �2� and �z 2 ��1; 1� in direc-tions �1; 0;�DBR

� and �1; 0;�BH �. They correspond

to the location of the dipole black ring horizon andthe location of the black hole, respectively. �DBR

and �BH are the angular velocities of the horizons.

In order to cure the conical singularities at the locationof a rod the coordinates and � should have periods

� � 2lim�!0

��2g��g

vuut ; �� 2lim�!0

��2g��g��

vuut :

(43)

Let us first consider the rods �z 2 ��1; �3� and �z 2�1;1�. Then the regularity conditions2 (43) fix the periodsof � and to be �� 2 and � � 2. These periodsinsure asymptotic flatness of the metric as it was discussedin (42).

Concerning the regularity on the finite rod �z 2 ��2; �1�we obtain

��2

��1��

3=2

j1��2 �c2 j��1�1��2��1��3��1��2��1��3�

p :

(44)

In order to find the equilibrium (the balancing) conditionfor the black Saturn with dipole black ring we have toimpose that the right-hand side (r.h.s.) of (44) be equal to

2. Solving then for �c2 we find

�c 2�1

�2

�"

��1��3=2��

�1�1��2��1��3��1��2��1��3�p � 1

�;

(45)

where " � 1 for �c2 >��12 and " � �1 for �c2 <��

�12 .

As in the case of black Saturn, we have two separatesectors of the balance condition for the black Saturn witha dipole black ring.

Let us also note that the electromagnetic field is regularon the rods considered here.

C. Black hole horizon

The black hole horizon is located at � � 0 for �z 2��1; 1�. The induced metric on the spacial cross sectionof the horizon is given by

ds2BH �

2L2� �z� �1��z� �3�

��z� �2�w2� �z�d�2 � L2s2

BH

g��z�w� �z�

�1� �z�d 2 �L2��z� �2�d �z2

�1� �z��z� �1��z� �3�g��z�w��z�:

(46)

Here the constant sBH and the function g��z� are formallythe same3 as for the black Saturn solution [4] and are givenby

sBH ��3�1� �1� � �1�2�1� �2��1� �3� �c

22

�3

��1� �1��1� �2��1� �3�

p�1� �2 �c2�

2; (47)

g��z� � 2�1�3�1� �1��1� �2��1� �3��z� �2�

�1� �2 �c2�2��1� �1�

2�3��1��z� �2�

� �3��1 � �2�1� �z�2 � �1�2�2� �z��

� 2�1�2�3�1� �1��1� �2��1� �3��1� �z�

��z� �1� �c2 � �21�2�1� �2�

2

�1� �3�2 �z��z� �1� �c2

2��1: (48)

The function w��z� is defined by

w��z� ��z� �2

�z� �: (49)

The functions g��z� and w��z� are positive for �1 � �z � 1and sBH > 0.

The topology of the horizon is that of S3. Metrically,however, the horizon is distorted S3. This can be seenexplicitly by computing the scalar curvature of the horizonwhich is not constant contrary to the case of the round S3.The distortion is caused by the rotation of the black hole

2Let us note again that we keep imposing the conditions (23)and (24).

3Let us note, however, that the balance condition (45) isdifferent i.e. the parameter �c2 is different.


064011-4

itself and the gravitational attraction of the dipole black ring.The area of the spacial cross section black hole horizon can be found by straightforward calculation and the result is

A BH � 42L3

��2�1� �1�

3

�1� �2��1� �3�

s�3�1� �1� � �1�2�1� �2��1� �3� �c

22

�3�1� �1��1� �2 �c2�2 : (50)

The other horizon quantities which are of interest are the angular velocity and the temperature and they are given by

�BH �

1

L

��2�3

2�1

s�3�1� �1� � �1�1� �2��1� �3� �c2

�3�1� �1� � �1�2�1� �2��1� �3� �c22

�1� �2 �c2�; (51)

TBH �1

2L

��1� �2��1� �3�

2�1� �1�

s�3�1� �1��1� �2 �c2�

2

�3�1� �1� � �1�2�1� �2��1� �3� �c22

: (52)

Let us note that although the horizon area, temperature,and the angular velocity of the black hole in the blackSaturn with dipole ring look at first sight the same as thecorresponding quantities for the black hole in the blackSaturn solution, they are in fact different since the balancecondition (45) (i.e. the parameter �c2) is different.

The investigation of the behavior of the electromagneticfield shows that it is regular on the black hole horizon.

D. Dipole black ring horizon

The dipole black ring horizon is located at � � 0 for �z 2��3; �2�. The induced metric on the spacial cross section ofthe horizon is

ds2DBR �

2L2��2 � �z��z� �3�

��1 � �z�w2��z�d�2 � L2s2

DBR

f��z�w��z�

��1 � �z�d 2 �L2y3

DBRd �z2

��2 � �z��z� �3�f��z�w��z�:

(53)

Here the constant sDBR and the function f� �z� are formallythe same as for the black Saturn solution [4] and are givenby

sDBR �

��2��2 � �3�

�1��1 � �3��1� �3�

s

��3 � �3��1 � �2� �c2 � �1�2�1� �3� �c

22�

�3�1� �2 �c2�2 ; (54)

f��z� � 2�1�3��1 � �3��1� �3��1� �z��1� �2 �c2�2

��2 � �3��1��3��2��1 � �z�

� �3��2�1� �1�2� �z�� 1�1� �z�2��

� 2�1�2�3�1� �3��1� �z��2 � �z� �c2

� �1�22�1� �3�

2 �z��2 � �z� �c22��1: (55)

The function w� �z� and the constant yDBR are defined by

w��z� �� z��1 � �z�

��1 � �z��2 � �z� � ��1��2��3�

��z� �3�; (56)

yDBR �� 3

�2 � �3: (57)

The functions f��z� and w��z� are positive for �3 � �z ��2 and sDBR > 0, yDBR > 0. The topology of the horizon isS2 S1 where S1 is parametrized by the angular coordi-nate and has radius depending on �z. The two-sphere isparametrized by the coordinates ��z; �� and is metricallydistorted.

The horizon area is found by straightforward integrationand the result is

ADBR � 42L3

��2�2�� 3�

3

�1��1 � �3��1� �3�

s

�3 � �3��1 � �2� �c2 � �1�2�1� �3� �c2

2

�3�1� �2 �c2�2 : (58)

The angular velocity and the temperature of the horizonare given by

�DBR �

1

L

��1�3

2�2

s�3 � �2�1� �3� �c2

�3 � �3��1 � �2� �c2 � �1�2�1� �3� �c22

�1� �2 �c2�; (59)

TDBR �1

2L

��1�1� �3��1 � �3��2 � �3�

2

2�2�� 3�3

s

�3�1� �2 �c2�

2

�3 � �3��1 � �2� �c2 � �1�2�1� �3� �c22

: (60)

The analysis shows that the electromagnetic field isregular on the black ring horizon.

E. ADM mass and angular momentum of the blackSaturn with dipole black ring

The ADM mass and angular momentum can be foundfrom the asymptotic form of the metric (32)–(37) and theresult is


064011-5

MADM �34L2�� 2� �

34L2 f�3�1� �1 � �2� � 2�2�3��1 � �2� �c2 � �2��1 � �2�3�1� �1 � �2�� c2

2g

�3�1� �2 �c2�2 ; (61)

JADM �L3

�3�1� �2 �c2�3

��2

2�1�3

sf�2

3 � �3��1 � �2��1� �1 � �3� � �2�1� �3�� c2 � �2�3��1 � �2��1 � �3�

� �1�1� �1 � �2 � �3�� c22 � �1�2��1 � �2�3�2� �1 � �2 � �3�� c

32g: (62)

Let us note that the ADM mass is positive as a conse-quence of the ordering (26).

F. Komar masses and angular momenta

The definition of the Komar mass and angular momen-tum is well known, namely

MKomar �3

32

Z@�?d~�; JKomar �

1

16

Z@�?d~� ;

(63)

where ~� and ~� are one-forms dual to the timelike Killingvector � and the spacelike Killing vector � . Here @� is aboundary of any spacelike hypersurface �. From a physi-cal point of view the Komar integrals measure the mass and

angular momentum contained in @�. When @� is a three-sphere at infinity the Komar integrals coincide with theADM mass and angular momentum of an asymptoticallyflat spacetime. When dealing with multiblack objects con-figurations the Komar integrals evaluated on the horizoncross sections are of special interest since they give theintrinsic mass and angular momenta of the black objects.These intrinsic quantities for the black objects in oursolution are the following:

MBHKomar �

3L2

4

�3�1� �1� � �1�2�1� �2��1� �3� �c22

�3�1� �2 �c2�;

(64)

MDBRKomar �

3L2

4

�2�1� �1� �2� �c2��3 � �3��1 � �2� �c2 � �1�2�1� �3� �c22�

�3�1� �2 �c2�2 ; (65)

JBHKomar � �L

3 �c2

��1�2

2�3

s�3�1� �1� � �1�2�1� �2��1� �3� �c

22

�3�1� �2 �c2�2 ; (66)

JDBRKomar � L3

��2

2�1�3

s��3 � �2��1 � �3� �c2 � �1�2�1� �2� �c

22��3 � �3��1 � �2� �c2 � �1�2�1� �3� �c

22�

�3�1� �2 �c2�3 : (67)

Usually one thinks that the Komar mass of every blackobject in a self-gravitating configuration should be posi-tive. However, when the objects are under strong gravita-tional interaction the Komar masses can become negativefor such tightly bound configurations. Numerical solutionsof black holes with negative Komar mass but with positiveADM mass were presented in [8,9] for the 5D Einstein-Maxwell gravity with a Chern-Simons term. For our solu-tion the situation is the same as for the vacuum blackSaturn solution [4]. The balance condition (45) with " �1 leads to the inequality ��1

2 < �c2 < �1� �2��1 which

guarantees that MBHKomar > 0 and MDBR

Komar > 0. The balancecondition (45) with " � �1 leads to the inequality �c2 <��1

2 and as a consequence we have MBHKomar < 0 and

MDBRKomar > 0.

G. Dipole charge and potential

The dipole charge is defined as

Q �1

4

IS2F; (68)

where S2 is a sphere on the black ring horizon. For oursolution we find

Q � �L

��3�� 2�� 3�

2��1 � ��

s: (69)

Finding explicitly the potential B (H � dB) of the dualform H � �F seems to be a formidable task at least in thecanonical coordinates ��; z�. That is why the direct com-putation of the dipole potential � is not possible. In orderto find � we shall proceed in the following way. First it isclear that � is proportional to

��3pB1 i.e.

� � ��3pB1; (70)

where the dimensionless constant � is a function of thesolution parameters. What is important is that this constantdoes not depend on the parameters a2 and c2. Therefore �is the same for the dipole black ring solution which isobtained in the limit c2 � 0 and a2 � a3. Hence we find� � �

2 and therefore we have


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� � �

��3p

2B1 � �L

��3��1 � �� 2�

2�� 3�

s: (71)

H. Smarr relations

Straightforward calculations show that the followingSmarr-like relations are satisfied:

MBHKomar �

3

2

�TBH ABH

4� JBH

Komar�BH

�; (72)

MDBRKomar �

3

2

�TDBR ADBR

4� JDBR

Komar�DBR

�; (73)

MADM � MBHKomar �M

DBRKomar �

12Q�: (74)

The term 12Q� can be interpreted as the energy of the

electromagnetic ‘‘hair’’ of the dipole black ring.

I. Ergosurfaces and closed timelike curves

As a consequence of the way the solution was generatedand the positiveness of W the existence/nonexistence ofergosurfaces and closed timelike curves depends only onthe seed vacuum black Saturn solution i.e. the dipolesolution inherits the existence/nonexistence of ergosurfa-ces and closed timelike curves from the vacuum blackSaturn solution. Our preliminary investigations show thatthere are ergosurfaces for the black hole and the black ring.However, the solution is too involved in the canonicalcoordinates and the explicit description is not possible.Concerning the possible existence of closed timelikecurves we should say that no sign for their existence isseen. The same conclusion is also reached in [4]. Theproblem, however, remains open.

J. Limits of the solution

The limits of the balance black Saturn with dipole blackring are more or less clear from the way the solution wasgenerated. By removing the black hole from the configu-ration we obtain the dipole black ring [3,6]. The formalprocedure is as follows. First we set the angular momentumof the black hole to zero by taking �c2 � 0. Then the blackhole is removed by setting �1 � 1.

Removing the dipole black ring from the configurationwe obtain a Myers-Perry black hole with one angularmomentum. This is achieved as follows. First we set � ��2 which eliminates the electromagnetic field from theblack ring. Then we take the limit �3 ! �2 and finallyset �2 � 0.

It is tempting to consider the limit in which the blackring is removed but the electromagnetic field is preservedi.e. to repeat the above procedure without setting � � �2

expecting that the result is a dipole black hole.Unfortunately the described limit is singular as the analysis

shows. One can show that there is no regular limit of amerger of the black hole and the dipole black ring.

K. Nonuniqueness

The balanced solution depends on four dimensionlessparameters ��1; �2; �3; �� and one dimensionful parameterL. Two of the parameters can be fixed by fixing the massand angular momentum of the configuration. Therefore thesolution exhibits threefold continuous nonuniqueness.

IV. DISCUSSIONS

In this paper we presented a new asymptotically flatsolution of 5D Einstein-Maxwell gravity describing aSaturn-like black configuration: a rotating black hole sur-rounded by a rotating dipole black ring. The solution wasgenerated by combining the vacuum black Saturn solutionand the vacuum black ring solution with appropriatelychosen parameters along the lines of the method developedin [6]. Some basic properties of the solution were analyzedand the basic quantities were calculated. It is interesting tosee in detail how the presence of the dipole charge affectsthe physics of the black Saturn. This however is verydifficult and requires numerical methods since the dipoleblack Saturn depends on many parameters in a nontrivialmanner. The physical properties of the dipole black Saturnare currently under numerical investigation and the resultswill be presented elsewhere. Preliminary results show that,more or less, many of the physical properties of the dipolesolution are similar to those of the vacuum black Saturnsolution which are thoroughly enough discussed in [4]. Inparticular, the dipole black Saturn exhibits effects likerotational frame-dragging and countering frame-dragging.Respectively, the differences between the dipole blackSaturn and the vacuum black Saturn are inherited fromthe differences of the dipole black ring and the vacuumblack ring [3]. In particular, the dipole charge increases theself-interaction of the ring and larger angular momentum isneeded to balance the ring in comparison with the vacuumblack Saturn solution. Another important point is that thedipole black Saturn is expected to be stable near theextremal non-BPS limit [10].

Lets us finish with prospects for future investigations.The question whether there is a more general dipole Saturnsolution than the one presented here remains open. If sucha solution exists, most probably it has to be generated viathe scheme

black Saturn � black Saturn

! new dipole black Saturn? (75)

provided that the potential singularities can be removed byan appropriate choice of the solution parameters. Since theformal operation denoted by

Lis not commutative it is

also interesting to consider the generating scheme


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black Saturn � rotating black ring

! new dipole black rings? (76)

which will generate a new dipole black ring solutionprovided the potential singularities can be removed.

Our solution can be extended to a solution of the 5DEinstein-Maxwell-dilaton gravity with an arbitrary dilatoncoupling parameter via the solution generating methodsdeveloped in [11,12]. Moreover these dipole solutions canbe uplifted to supergravity solutions.

Finally the methods of [6,11,12] can be applied to thevacuum solution of [5] describing two (or more) rotating

back rings at equilibrium in order to generate configura-tions with two (or more) dipole black rings.

ACKNOWLEDGMENTS

The author would like to thank the Alexander vonHumboldt Foundation for the support and the Institut furTheoretische Physik Gottingen for its kind hospitality.The partial support by the Bulgarian National ScienceFund under Grant No. MUF04/05 (MU 408) is alsoacknowledged.

[1] R. Emparan and H. Reall, Phys. Rev. Lett. 88, 101101(2002).

[2] A. Pomeransky and A. Sen’kov, arXiv:hep-th/0612005.[3] R. Emparan, J. High Energy Phys. 03 (2004) 064.[4] H. Elvang and P. Figueras, J. High Energy Phys. 05 (2007)

050.[5] H. Iguchi and T. Mishima, Phys. Rev. D 75, 064018

(2007).[6] S. Yazadjiev, Phys. Rev. D 73, 104007 (2006).

[7] T. Harmark, Phys. Rev. D 70, 124002 (2004).[8] J. Kunz and F. Navarro-Lerida, Phys. Lett. B 643, 55

(2006).[9] J. Kunz and F. Navarro-Lerida, Mod. Phys. Lett. A 21,

2621 (2006).[10] H. Elvang, R. Emparan, and P. Figueras, J. High Energy

Phys. 05 (2007) 056.[11] S. Yazadjiev, J. High Energy Phys. 07 (2006) 036.[12] S. Yazadjiev, Gen. Relativ. Gravit. 39, 601 (2007).


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Black Saturn with a dipole ring