Kaluza-Klein rotating multi-black-hole configurations with electromagneticfield in Einstein-Maxwell-dilaton gravity

Stoytcho S. Yazadjiev*

Department of Theoretical Physics, Faculty of Physics, Sofia University, 5 James Bourchier Boulevard, Sofia 1164, Bulgaria(Received 27 September 2012; published 27 November 2012)

We present a new solution in 5D Einstein-Maxwell-dilaton gravity describing an equilibrium configu-

ration of extremal rotating black holes with lens space horizon topologies. The basic properties of the

solution are investigated and the basic physical quantities are calculated. It is shown that the black-hole

horizons are superconducting in the sense that they expel the magnetic flux lines.

DOI: 10.1103/PhysRevD.86.107504 PACS numbers: 04.50.Gh, 04.50.Cd

In the last decade the higher dimensional gravity hasattracted a lot of interest and it is now a well-establishedarea of the modern theoretical and mathematical physics.The spacetimes with compact extra dimensions (Kaluza-Klein spacetimes) take a special place since they are muchmore realistic than the asymptotically flat spacetimes.During the last years Kaluza-Klein black holes were exten-sively studied and many exact solutions were found but alot remains to be done in this direction since the spectrumof black-hole solutions in the Kaluza-Klein case is muchricher than in the asymptotically flat case. For recentreview on the subject we refer the reader to Ref. [1].

In the present paper we deal with multi-black-holeKaluza-Klein spacetimes. Such multi-black-hole configu-rations are very interesting since they provide us withvaluable insight into the black-hole theory in spacetimeswith compact extra dimensions and especially into theblack-hole interactions in the strong field regime. Veryrecently a Kaluza-Klein rotating vacuum multi-black-hole solution in five dimensions was studied in Ref. [2].The authors show that the previously found solution ofRef. [3] can be interpreted as an equilibrium configurationof extremal rotating black holes held apart by the repulsivespin-spin interaction. The solution was generalized to thecase of 5D Einstein-Maxwell gravity and 5D minimalsupergravity in Ref. [4].

The purpose of the present paper is to present a newsolution describing an equilibrium configuration of extremalrotating black holes with a self-gravitating electromagneticfield in the 5D Einstein-Maxwell-dilaton (EMD) gravity andto study some of its basic properties. Our solution differsfrom those in Ref. [4] in that it is not charged and themagnetic fluxes through the horizons are zero.

We consider the five dimensional EMD gravity given bythe action

S¼ 1

16�

Zd5x

ffiffiffiffiffiffiffi�gp ½R�2g��@�’@�’�e�2�’F��F���;

(1)

where R is the Ricci scalar curvature with respect tothe spacetime metric g��, F�� is the Maxwell tensor, ’

is the dilaton field and � is the dilaton coupling constant.The field EMD equations are obtained by varying theaction:

R �� ¼ 2r�’r�’þ 2e�2�’

�F��F

�� � g��

6F��F��

�;

(2)

r�r�’ ¼ ��

2e�2�’F��F��; (3)

r�½e�2�’F��� ¼ 0: (4)

In the present paper we will consider the case � ¼ffiffiffiffiffiffiffiffi8=3

p. For this value of the dilaton coupling parameter

we have found the following solution:

ds2 ¼V�1=3f�H�2dt2þ½ ffiffiffi2

p ðH�1�1ÞdtþLdc

þ ffiffiffi2

pcosh�W�2gþV2=3H2½dx2þdy2þdz2�; (5)

A�dx� ¼ � sinh�

2ðH�1 � 1Þ2dt� sinh�ffiffiffi

2p ðH�1 � 1ÞLdc

� cosh� sinh�H�1W; (6)

effiffiffiffiffiffi2=3

p’ ¼ V1=3; (7)

where the metric function V and the 1-form W are given by

V ¼ cosh2��H�2sinh2�; (8)

W ¼ Xi

mi

z� zijR� Rij

ðx� xiÞdy� ðy� yiÞdxðx� xiÞ2 þ ðy� yiÞ2

; (9)

and H is a harmonic function on the three dimensional flatspace explicitly given by

H ¼ 1þXi

mi

jR� Rij (10)

with point sources located at Ri ¼ ðxi; yi; ziÞ. The 1-form

W and the harmonic function H satisfy the followingequation on the three dimensional flat space:*yazad@phys.uni-sofia.bg

PHYSICAL REVIEW D 86, 107504 (2012)

1550-7998=2012=86(10)=107504(4) 107504-1 � 2012 American Physical Society

r� W ¼ rH: (11)

The parameters �, L mi run in the ranges�1< �<1,L > 0 and mi � 0. In the case � ¼ 0 our solution reducesto the vacuum solution of Ref. [2]. In the general case theabove metric possesses only two Killing fields @

@t and@@c .

Without loss of generality we will consider the case oftwo black holes with R1 ¼ ð0; 0; 0Þ and R2 ¼ ð0; 0; aÞwhere a > 0. In this case the flat three metric, the harmonic

function H and the 1-form W are given by

ds23 ¼ dx2 þ dy2 þ dz2 ¼ dR2 þ R2ðd�2 þ sin2�d2Þ;(12)

H ¼ 1þm1

Rþ m2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R2 þ a2 � 2Ra cos�p ; (13)

W ¼�m1 cos�þm2

R cos�� affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ a2 � 2aR cos�

p�d: (14)

In addition to the Killing fields @@t and

@@c , in the case

under consideration there is one more Killing field @@ . The

coordinates satisfy�1< t <1, 0<R<1, 0 � � � �,0 � < 2� and 0 � c < 2�.

Using a standard approach one can show that the pointsources R ¼ R1 and R ¼ R2 correspond to smooth hori-zons for the metric. The coordinates we use to writesolutions are actually singular on the horizons and there-fore we should introduce new coordinates that cover thehorizons too. We will do so only for the horizon at R ¼ 0since the other horizon can be treated analogously. Near

the horizon we introduce the new coordinates and ~cgiven by

d ¼ dt� cosh�m1d

�m1

R

�; (15)

d ~c ¼dc �ffiffiffi2

pL

dt�ffiffiffi2

pcosh�m1

L

dR

R�

ffiffiffi2

pcosh�m2

Ld:

(16)

In the new coordinates the metric takes the form

ds2¼ cosh�2=3�

�R2

m21

d2þ2cosh�ddR

þm21cosh

2�

�d�2þ2

�Lffiffiffi

2p

m1 cosh�d ~c þcos�d

�2�

þ4cosh�Rd

�Lffiffiffi

2p

m1 cosh�d ~c þcos�d

��: (17)

The spatial cross section of R ¼ 0 has the followinginduced metric:

dl2¼m21cosh

4=3�

�d�2þ2

�Lffiffiffi

2p

m1 cosh�d ~cþcos�d

�2�:

(18)

In order for the above metric to be smooth, just as in thevacuum case [2], the following quantization conditionmust be imposed:

m1 ¼ L

2ffiffiffi2

pcosh�

n1; (19)

where n1 is an integer. With this condition imposed themetric (18) describes the standard smooth metric on thelens space Lðn1; 1Þ. In this way we obtained the analyticalextension of our metric across the surface R ¼ 0. In acompletely analogous way one can build the analyticalextension of the metric across the surface R2 with thefollowing quantization condition:

m2 ¼ L

2ffiffiffi2

pcosh�

n2; (20)

where n2 is an integer. Therefore the topology of thesecond horizon is Lðn2; 1Þ.The asymptotic behavior of the solution is the following:

ds2 ’�1þ 2msinh2�

R

��1=3���1� 2m

R

�dt2

þ�1þ 2mcosh2�

R

�ðdR2 þ R2d�2Þ

þ n2L2

4

�� 2

R cosh�dtþ 2dc

nþ cos�d

�2�; (21)

A�dx� ’ � sinh�m2

2R2dtþ sinh�mffiffiffi

2p

RLdc

� cosh� sinh�m cos�d; (22)

with m ¼ Pimi and n ¼ P

ini. From the explicit form ofthe asymptotic metric it is clear that the topology of spatialinfinity is Lðn; 1Þ.It is worth noting that in the case of two black holes there

is an additional Killing vector @@ and the techniques based

on the notion of interval structure [5] can be applied andthey give the same results as the above analysis.We proceed further with calculating the masses and

angular momenta. The mass of each black hole is givenby the Komar integral

Mi ¼ �3

32�

ZHi

?d�; (23)

where ? is the Hodge duality operator and � is the 1-formcorresponding to the timelike Killing vector. The explicitcalculation gives the following result:

Mi ¼ 3

2�Lmi: (24)

The total mass of the multi-black-hole configuration is

M ¼ �3

32�

Z1?d� ¼ 3

2�L

Xi

mi

�1þ 1

3sinh2�

�: (25)

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The deference M�PiMi � 0 reflects the contribution

of the electromagnetic and dilaton fields to the total mass.The angular momentum of each black hole is defined by

the Komar integral

Jci ¼ 1

16�

ZHi

?d�; (26)

where � is 1-form corresponding to the Killing field @@c .

The explicit calculation gives

Jci ¼ L2ffiffiffi2

p �mi: (27)

The angular velocity of each black hole is �H ¼ffiffi2

pL

which, combined with the expressions for the mass andangular momentum, results in the following relation:

Mi ¼ 3

2�HJ

ci : (28)

For the total angular momentum of the configuration wefind

Jc ¼ 1

16�

Z1?d� ¼ L2ffiffiffi

2p �

Xi

mi ¼Xi

Jci : (29)

Therefore the electromagnetic field gives no contributionto the total angular momentum.

The total electric charge of the configuration is defined by

Q ¼ 1

4�

Z�e�2�’ ? F; (30)

and the calculations give Q ¼ 0. This result can be easilyseen from the asymptotic behavior At � 1

R2 . The electric

charge of each black hole in the configuration is also zero,Qi ¼ 0. The nontrivial quantity that characterizes the elec-tromagnetic field is the magnetic flux � through the basespace S21 of the S1 fibration Lðn; 1Þ at infinity, namely,

� ¼ 1

4�

ZS21

F ¼ cosh� sinh�m: (31)

It is also interesting to find the magnetic flux through aportion of the horizon cross sections. It is known however,that in the general case, the magnetic flux lines are expelledfrom the extremal black holes, more precisely, the compo-nents of the field strength normal to the horizon vanish[6,7]. In our case we have

FjHi¼ 0; (32)

and therefore, within the framework of the present solu-tion, the black-hole horizons are superconducting in thesense that they exhibit a ‘‘Meissner effect’’ typical for thesuperconductors. Even more, the electric field also van-ishes on the horizons.We now are at a position to derive a Smarr-like formula

giving a relation between the Komar mass and the mag-netic flux. For this purpose we consider the 1-form ¼i�i�e

�2�’ ? F which is closed dði�i�e�2�’ ? FÞ ¼ 0 as a

consequence of the Killing symmetries and the Maxwellequations. The 1-form is invariant under the Killingsymmetries and therefore it can be viewed as defined onthe orbit (factor) space spacetime/isometry group. Sincethe factor space is simply connected [5], there exists apotential � such that d� ¼ i�i�e

�2�’ ? F. The explicit

expression for � in our case is

�¼�cosh�sinh�

2

1�H�2

cosh2��H�2sinh2�þ tanh�

2; (33)

where we have fixed the arbitrary constant in the definitionof � so that � vanishes on the horizons. Using then thesame approach as in Ref. [8] it can be shown that thefollowing relation is satisfied:

M ¼ Xi

Mi þ �L�ð1Þ�: (34)

This is in fact Eq. (25), as can be checked.The ergosurface, defined by gð�; �Þ ¼ 0, exists always

since gð�; �ÞjHi> 0 and gð�; �Þj1 < 0 and its topology

depends on the point sources’ configuration just as in thevacuum case.As a final remark it is worth noting that our preliminary

investigations show that there exist solutions describingrotating extremal multi-black-hole configurations with aself-gravitating electromagnetic field, more general thanthe solutions presented in Ref. [4] and in the present paper.The results will be presented elsewhere.

The partial financial supports from the BulgarianNational Science Fund under Grant No. DMU-03/6, and bySofia University Research Fund under Grant No. 148/2012are gratefully acknowledged.

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[2] K. Matsuno, H. Ishihara, M. Kimura, and T. Tatsuoka,Phys. Rev. D 86, 044036 (2012).

[3] G. Clement, Gen. Relativ. Gravit. 18, 861(1986).

[4] K. Matsuno, H. Ishihara, M. Kimura, and T. Tatsuoka,arXiv:1208.5536.

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