Charged rotating dilaton black strings

PHYSICAL REVIEW D 71, 044008 (2005)

Charged rotating dilaton black strings

M. H. Dehghani1,2,* and N. Farhangkhah1

1Physics Department and Biruni Observatory, Shiraz University, Shiraz 71454, Iran2Research Institute for Astrophysics and Astronomy of Maragha (RIAAM), Maragha, Iran

(Received 13 December 2004; published 8 February 2005)

*Electronic

1550-7998=20

In this paper we, first, present a class of charged rotating solutions in four-dimensional Einstein-Maxwell-dilaton gravity with zero and Liouville-type potentials. We find that these solutions can present ablack hole/string with two regular horizons, an extreme black hole or a naked singularity provided theparameters of the solutions are chosen suitable. We also compute the conserved and thermodynamicquantities, and show that they satisfy the first law of thermodynamics. Second, we obtain the (n� 1)-dimensional rotating solutions in Einstein-dilaton gravity with Liouville-type potential. We find that thesesolutions can present black branes, naked singularities or spacetimes with cosmological horizon if onechooses the parameters of the solutions correctly. Again, we find that the thermodynamic quantities ofthese solutions satisfy the first law of thermodynamics.

DOI: 10.1103/PhysRevD.71.044008 PACS numbers: 04.20.Jb, 04.40.Nr, 04.50.+h, 04.70.Bw

I. INTRODUCTION

The Einstein equation without a cosmological constanthas asymptotically flat black hole solutions with eventhorizon being a positive constant curvature n-sphere.However, for the Einstein equation with positive or nega-tive cosmological constant [1,2] or Einstein-Gauss-Bonnetequation with or without cosmological constant [3], onecan have also asymptotically anti de Sitter (AdS) or deSitter (dS) black hole solutions with horizons being zero ornegative constant curvature hypersurfaces. These blackhole solutions, whose horizons are not n-sphere, are oftenreferred to as the topological black holes in the literature.Various properties of these topological black holes havebeen investigated in recent years. For example, the ther-modynamics of rotating charged solutions of the Einstein-Maxwell equation with a negative cosmological constantwith zero curvature horizons in various dimensions havebeen studied in Ref. [4], while the thermodynamics ofthese kind of solutions in Gauss-Bonnet gravity havebeen investigated in [5].

All of the above mentioned black holes are the solutionsof field equations in the presence of a long-range gravita-tional tensor field g�� and a long-range electromagneticvector field A�. It is natural then to suppose the existenceof a long-range scalar field too. This leads us to the scalar-tensor theories of gravity, where, there exist one or severallong-range scalar fields. Scalar-tensor theories are not new,and it was pioneered by Brans and Dicke [6], who sought toincorporate Mach’s principle into gravity. Also in the con-text of string theory, the action of gravity is given by theEinstein action along with a scalar dilaton field which isnon minimally coupled to the gravity [7]. The action of(n� 1)-dimensional dilaton Einstein-Maxwell gravitywith one scalar field � and potential V�� can be writtenas [8]

address: [email protected]

05=71(4)=044008(7)$23.00 044008

IG��1

16

ZMdn�1x

��g

p�R�

4

n�1�r��2�V��

�e�4 ��n�1�F��F��

1

8

Z@M

dnx��

p��; (1)

where R is the Ricci scalar, is a constant determining thestrength of coupling of the scalar and electromagnetic field,F�� @�A� � @�A� is the electromagnetic tensor fieldand A� is the vector potential. The last term in Eq. (1) is theGibbons-Hawking boundary term. The manifold M hasmetric g�� and covariant derivative r�. � is the trace ofthe extrinsic curvature �� of any boundary(ies) @M ofthe manifold M, with induced metric(s) �ij. Exact chargeddilaton black hole solutions of the action (1) in the absenceof a dilaton potential [V�� 0] have been constructed bymany authors [9,10]. The dilaton changes the casual struc-ture of the spacetime and leads to curvature singularities atfinite radii. In the presence of Liouville-type potential[V�� 2 exp�2��], static charged black hole solu-tions have also been discovered with a positive constantcurvature event horizons and zero or negative constantcurvature horizons [8,11]. Recently, the properties of theseblack hole solutions which are not asymptotically AdS ordS, have been studied [12].

These exact solutions are all static. Recently, one of ushas constructed two classes of magnetic rotating solutionsin four-dimensional Einstein-Maxwell-dilaton gravity withLiouville-type potential [13]. These solutions are not blackholes, and present spacetimes with conic singularities. Tillnow, charged rotating dilaton black hole solutions for anarbitrary coupling constant have not been constructed.Indeed, exact rotating black hole solutions have been ob-tained only for some limited values of the coupling con-stant [14]. For general dilaton coupling, the properties ofcharged dilaton black holes only with infinitesimally smallangular momentum [15] or small charge [16] have beeninvestigated. Our aim here is to construct exact rotating

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M. H. DEHGHANI AND N. FARHANGKHAH PHYSICAL REVIEW D 71, 044008 (2005)

charged dilaton black holes for an arbitrary value of cou-pling constant and investigate their properties.

The outline of our paper is as follows. In Sec. II, weobtain the four-dimensional charged rotating dilaton blackholes/strings, which are not asymptotically flat, AdS or dS,and show that the thermodynamic quantities of these blackstrings satisfy the first law of thermodynamics. Section IIIis devoted to a brief review of the general formalism ofcalculating the conserved quantities, and investigation ofthe first law of thermodynamics for the charged rotatingblack string. In Sec. IV, we construct the (n� 1)-dimensional rotating dilaton black branes, and investigatetheir properties. We finish our paper with some concludingremarks.

II. ROTATING CHARGED DILATON BLACKSTRINGS

The field equation of (n� 1)-dimensional Einstein-Maxwell-dilaton gravity in the presence of one scalar field� with the potential V�� can be written as:

@��g

pe�4 ��n�1�F�� 0; (2)

R�� 4

n� 1

�r��r��

1

4Vg��

�

� 2e�4 ��n�1�

�F��F��

1

2�n� 1�F��F��g��

�;

(3)

r2� �n� 1

8

@V@�

� 2e�4 ��n�1�F��F

��: (4)

In this section we want to obtain the four-dimensionalcharged rotating black hole solutions of the field Eqs. (2)–(4) with cylindrical or toroidal horizons. The metric ofsuch a spacetime with cylindrical symmetry can be writtenas

ds2 � �f�r�� dt� ad’�2 � r2R2�r��a

l2dt� d’

�2

�dr2

f�r��r2

l2R2�r�dz2;

2 � 1�a2

l2; (5)

where the constants a and l have dimensions of length andas we will see later, a is the rotation parameter and l isrelated to the cosmological constant for the case ofLiouville-type potential with constant �. In metric (5),the ranges of the time and radial coordinates are �1<t <1, 0 � r <1. The topology of the two dimensionalspace, t � constant and r � constant, can be (i) S1 � S1

the flat torus T2 model, with 0 � ’< 2, 0 � z < 2l,(ii) R� S1, the standard cylindrical symmetric model with

044008

0 � ’< 2, �1< z<1, and (iii)R2, the infinite planemodel, with �1<’<1, �1< z<1 (this planar so-lution does not rotate).

The Maxwell Eq. (2) for the metric (5) is @��r2R2�r��exp��2 ��F�� 0; which shows that if one choose

R�r� � exp� ��; (6)

then the vector potential can be written as

A� � �qr� 't� � a'’��; (7)

where q is the charge parameter. In order to obtain thefunctions ��r� and f�r�, we write the field Eqs. (3) and (4)for a � 0:

rf00 � 2f0�1� r�0� � 2q2r�3e�2 � � rV�� 0;

(8)

rf00 � 2f0�1� r�0� � 4f� r�00 � 2 �0�

r�1� 2��02 � 2q2r�3e�2 � � rV�� 0; (9)

f0�1� r�0� � f��00 � 2r �02 � 4�0�

� r��1 � q2r�3e�2 � �1

2rV�� 0; (10)

f�00 � f0�0 � 2 f�02 � 2r�1f�0�

q2r�4e�2 � �1

4

@V@�

� 0; (11)

where the ‘‘prime’’ denotes differentiation with respect tor. Subtracting Eq. (8) from Eq. (9) gives:

r�00 � 2 �0 � r�1� 2��02 � 0;

which shows that ��r� can be written as:

��r� �

1� 2 ln�br� c

�; (12)

where b and c are two arbitrary constants. Using theexpression (12) for ��r� in Eqs. (8)–(11), we find thatthese equations are inconsistent for c � 0. Thus, we putc � 0.

A. Solutions with V�� 0

We begin by looking for the solutions in the absence of apotential (V�� 0). In this case, it is easy to solve thefield Eqs. (8)–(11). One obtains

f�r� � r��1��C��1� 2�q2

V0r�;

where C is an arbitrary constant, � � 2 2=�1� 2� andV0 � b�. In the absence of a nontrivial dilaton ( � 0 ��), this solution does not exhibit a well-defined spacetimeat infinity. Indeed, should be greater than or equal to one.In order to study the general structure of these solutions,

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CHARGED ROTATING DILATON BLACK STRINGS PHYSICAL REVIEW D 71, 044008 (2005)

we first look for the curvature singularities. It is easy toshow that the Kretschmann scalar R��+R��+ diverges atr � 0, it is finite for r � 0 and goes to zero as r! 1.Thus, there is an essential singularity located at r � 0.Also, it is notable to mention that the Ricci scaler is finiteevery where except at r � 0, and goes to zero as r! 1.The spacetime is asymptotically flat for � 1, while it isneither asymptotically flat nor (A)dS for > 1. Thisspacetime presents a naked singularity with a regular cos-mological horizon at

rc ��1� 2�q2

CV0;

provided C> 0, and no cosmological horizon for C< 0.

B. Solutions with Liouville-type potentials

Now we consider the solutions of Eqs. (8)-(11) with aLiouville-type potential V�� 2 exp�2��. One mayrefer to as the cosmological constant, since in theabsence of the dilaton field � the action (1) reduces tothe action of Einstein-Maxwell gravity with cosmologicalconstant. The only case that we find exact solutions for anarbitrary values of with R�r� and ��r� given in Eqs. (6)and (12) is when � � . It is easy then to obtain thefunction f�r� as

f�r� � r��V0�1� 2�2

� 2 � 3�r2�1��

mr�

�1� 2�q2

V0r2

�:

(13)

In the absence of a nontrivial dilaton ( � 0 � �), thesolution reduces to the asymptotically AdS and dS chargedrotating black string for � �3=l2 and �3=l2,respectively [2,4]. As one can see from Eq. (13), thereis no solution for �

��3

pwith a Liouville potential ( �

0). In order to investigate the casual structure of the space-time, we consider it for different ranges of separately.

For >��3

p, as r goes to infinity the dominant term is

the second term, and therefore the spacetime has a cosmo-logical horizon for positive values of the mass parameter,despite the sign of the cosmological constant .

For <��3

pand large values of r, the dominant term is

the first term, and therefore there exists a cosmologicalhorizon for > 0, while there is no cosmological horizonsif < 0 . Indeed, in the latter case ( <

��3

pand < 0) the

spacetimes associated with the solution (13) exhibit avariety of possible casual structures depending on thevalues of the metric parameters , m, q, and . One canobtain the casual structure by finding the roots of f�r� � 0.Unfortunately, because of the nature of the exponents in(13), it is not possible to find explicitly the location ofhorizons for an arbitrary value of . But, we can obtainsome information by considering the temperature of thehorizons.

044008

One can obtain the temperature and angular velocity ofthe horizon by analytic continuation of the metric. Theanalytical continuation of the Lorentzian metric by t! i.and a! ia yields the Euclidean section, whose regularityat r � rh requires that we should identify .� .� �h and’� ’� i�h�h, where �h and �h are the inverseHawking temperature and the angular velocity of the hori-zon. It is a matter of calculation to show that

Th �f0�rh�4

�r��3h

4 V0�1� 2��3� 2�V0mrh � 4�1� 2�q2

� ��1� 2�r��3

h

4 V0�V2

0r4�2�h � q2; (14)

�h �a

l2: (15)

Eq. (14) shows that the temperature is negative for the twocases of (i) >

��3

pdespite the sign of , and (ii) positive

despite the value of . As we argued above in these twocases we encounter with cosmological horizons, and there-fore the cosmological horizons have negative temperature.Numerical calculations shows that the temperature of theevent horizon goes to zero as the black hole approaches theextreme case. Thus, one can see from Eq. (14) that thereexist extreme black holes only for negative and <

��3

p,

if rh � 4�1� 2�q2=�mcrit�3� 2�V0, where mcrit is themass of extreme black hole. If one substitutes this rh intothe equation f�r� � 0, then one obtains the condition forextreme black string as:

mcrit �4�1� 2�q2

V0�3� 2�

��V2

0

q2

��1� 2�=4

: (16)

Indeed, the metric of Eqs. (5) and (13) has two inner andouter horizons located at r� and r�, provided the massparameter m is greater than mcrit. We will have an extremeblack string in the case of m � mcrit, and a naked singu-larity if m<mcrit. Note that Eq. (16) reduces to the criticalmass obtained in Ref. [4] in the absence of dilaton field.

Since the area law of entropy is universal, and applies toall kinds of black holes and black strings in Einsteingravity, the entropy per unit length of the string is

S � V0r

2�1� 2��1

h

2l; (17)

where rh is the horizon radius. One may note that theentropy of the extreme black hole is q=�2l

��

p�,

which is independent of the coupling constant .Finally, it is worthwhile to mention about the asymptotic

behavior of these spacetimes. The asymptotic form of themetric given by Eqs. (5) and (13) for the nonrotating casewith no cosmological horizon ( <

��3

pand < 0) is:

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ds2��V0�1� 2�2

2� 2�3�r2=�1�

2�dt2

�� 2�3�

V0�1� 2�2r�2=�1� 2�dr2

�r2=�1� 2�

�d’2�

dz2

l2

�:

Note that gtt, for example, goes to infinity as r! 1, butwith a rate slower than that of AdS spacetimes. Indeed, theform of the Ricci and Kretschmann (K) scalars for largevalues of r are:

R �6� 2 � 2�V0

� 2 � 3�r�2 2=�1� 2�;

K �12� 4 � 2 2 � 2�V2

0

� 2 � 3�22r�4 2=�1� 2�:

As one may note, these quantities go to zero as r! 1, butwith a slower rate than those of asymptotically flat space-times and do not approach nonzero constants as in the caseof asymptotically AdS spacetimes.

III. THE CONSERVED QUANTITIES AND FIRSTLAW OF THERMODYNAMICS

The conserved charges of the string can be calculatedthrough the use of the substraction method of Brown andYork [17]. Such a procedure causes the resulting physicalquantities to depend on the choice of reference back-ground. For asymptotically (A)dS solutions, the way thatone deals with these divergences is through the use ofcounterterm method inspired by (A)dS/CFT correspon-dence [18]. However, in the presence of a nontrivial dilatonfield, the spacetime may not behave as either dS (> 0) orAdS (< 0). In fact, it has been shown that with theexception of a pure cosmological constant potential, where� � 0, no AdS or dS static spherically symmetric solutionexist for Liouville-type potential [19]. But, as in the case ofasymptotically AdS spacetimes, according to the domain-wall/QFT (quantum field theory) correspondence [20],there may be a suitable counterterm for the stress-energytensor which removes the divergences. In this paper, wedeal with the spacetimes with zero curvature boundary[Rabcd�� 0], and therefore the counterterm for thestress-energy tensor should be proportional to �ab. Thus,the finite stress-energy tensor in �n� 1�-dimensionalEinstein-dilaton gravity with Liouville-type potentialmay be written as

Tab �1

8

��ab ��ab �

n� 1

leff�ab

�; (18)

where leff is given by

l2eff ��n� 1�3�2 � 4n�n� 1�

8e�2��: (19)

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As � goes to zero, the effective leff of Eq. (19) reduces tol � n�n� 1�=2 of the (A)dS spacetimes. The first twoterms in Eq. (18) is the variation of the action (1) withrespect to �ab, and the last term is the counterterm whichremoves the divergences. One may note that the counter-term has the same form as in the case of asymptoticallyAdS solutions with zero curvature boundary, where l isreplaced by leff . To compute the conserved charges of thespacetime, one should choose a spacelike surface B in @Mwith metric �ij, and write the boundary metric in ADMform:

�abdxadxa � �N2dt2 � �ij�d’i � Vidt��d’j � Vjdt�;

where the coordinates ’i are the angular variables parame-trizing the hypersurface of constant r around the origin,andN and Vi are the lapse and shift functions, respectively.When there is a Killing vector field 1 on the boundary, thenthe quasilocal conserved quantities associated with thestress tensors of Eq. (18) can be written as

Q �1� �ZBdn�1’

��

pTabn

a1b; (20)

where � is the determinant of the metric �ij, 1 and na arethe Killing vector field and the unit normal vector on theboundary B . For boundaries with timelike (1 � @=@t) androtational (& � @=@’) Killing vector fields, one obtains thequasilocal mass and angular momentum

M �ZBdn�1’

��

pTabna1b; (21)

J �ZBdn�1’

��

pTabn

a&b; (22)

provided the surface B contains the orbits of &. Thesequantities are, respectively, the conserved mass, angularand linear momenta of the system enclosed by the bound-ary B. Note that they will both be dependent on thelocation of the boundary B in the spacetime, althougheach is independent of the particular choice of foliationB within the surface @M.

The mass and angular momentum per unit length of thestring when the boundary B goes to infinity can be calcu-lated through the use of Eqs. (21) and (22),

M �V0

8l

��3� 2� 2 � 2 � 1

�1� 2�

�m;

J ��3� 2�V0

8l�1� 2� ma:

(23)

For a � 0 ( � 1), the angular momentum per unit lengthvanishes, and therefore a is the rotational parameter of thespacetime. Note that Eq. (23) is valid only for <

��3

p,

which the spacetime has no cosmological horizons. Ofcourse, one may note that these conserved charges reduce

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to the conserved charges of the rotating black string ob-tained in Ref. [4] as ! 0.

Next, we calculate the electric charge of the solutions.To determine the electric field we should consider theprojections of the electromagnetic field tensor on specialhypersurfaces. The normal to such hypersurfaces for thespacetimes with a longitudinal magnetic field is

u0 �1

N; ur � 0; ui � �

Vi

N;

and the electric field is E� � g�� exp��2 ��F��u�.

Then the electric charge per unit length Q can be foundby calculating the flux of the electric field at infinity,yielding

Q � q2l: (24)

The electric potential U, measured at infinity with respectto the horizon, is defined by [21]

U � A�8�jr!1 � A�8

�jr�r� ;

where 8 � @t ��@’ is the null generators of the eventhorizon. One obtains

U �q

r�: (25)

Finally, we consider the first law of thermodynamics forthe black string. Although it is difficult to obtain the massM as a function of the extensive quantities S, J and Q foran arbitrary values of , but one can show numerically thatthe intensive thermodynamic quantities, T, � and U cal-culated above satisfy the first law of thermodynamics,

dM � TdS ��dJ �UdQ: (26)

IV. THE ROTATING BLACK BRANES IN VARIOUSDIMENSIONS

In this section we look for the rotating uncharged solu-tions of field Eqs. (2)–(4) in �n� 1� dimensions withLiouville-type potential. we first obtain the static solutionand then generalize it to the case of rotating solution withall the rotation parameters.

A. Static Solutions

The metric of a static (n� 1)-dimensional spacetimewith an (n� 1)-dimensional flat submanifold dX2 can bewritten as

ds2 � �f�r�dt2 �dr2

f�r��r2

l2e2��dX2: (27)

The field equations (2)–(4) for the above metric become

rn� 1

f00 � f0�1� �r�0� �4r

�n� 1�2e2�� 0; (28)

044008

rn� 1

f00 � f0�1� �r�0� �4r

�n� 1�2e2��

2�n� 2�rf��r�00 � 2��0 � r

��2 �

4

�n� 1�2

��02

�� 0;

(29)

f0�1� �r�0� � �f�r�00 � �n� 1��2�0 � �r�02��

�n� 2��r��1 �2rn� 1

e2�� 0;

(30)

f�00 �f0�0 ��n�1��f�02�r�1f�0 �

1

2�e2��

��0:

(31)

Subtracting Eq. (28) from Eq. (29) gives

�r�00 � 2��0 � r��2 �

4

�n� 1�2

��02 � 0;

which indicates that ��r� can be written as:

��r� ��n� 1�2�

4� �n� 1�2�2 ln�cr� d

�; (32)

where c and d are two arbitrary constants. Substituting��r� of Eq. (32) into the field equations (28)–(31), onefinds that they are consistent only for d � 0. Putting d � 0,then f�r� can be written as

f�r� �8V0

�n� 1�3�2 � 4n�n� 1�r2� �mr1��n�1��;

V0 � ��2c2�1��;� � 4f�n� 1�2�2 � 4g�1:(33)

One may note that there is no solution for �n� 1�2�2 �4n � 0. It is worthwhile to note that the Ricci scalar goesto zero as r! 1.

The Kretschmann scalar diverges at r � 0, it is finite forr � 0, and goes to zero as r! 1. Thus, there is anessential singularity located at r � 0. Again, the spacetimeis neither asymptotically flat nor (A)dS, but has a regularevent horizon for negative at:

rh �

(��n� 1�2�2 � 4nm

V0

)1=��n�1��1

; (34)

provided �n� 1�2�2 � 4n < 0. If �n� 1�2�2 � 4n > 0,then the spacetime has a cosmological horizon with radiusgiven in Eq. (34) for positive values of . The Hawkingtemperature of the event or cosmological horizon is:

Th � ��V0

2 r2��1h ;

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which is positive for event horizon (< 0) and negativefor cosmological horizon (> 0).

B. Rotating solutions with all the rotation parameters

The rotation group in �n� 1�-dimensions is SO�n� andtherefore the number of independent rotation parametersfor a localized object is equal to the number of Casimiroperators, which is �n=2 � k, where �n=2 is the integerpart of n=2. The generalization of the metric (27) with allrotation parameters is

ds2��f�r�� dt�

Xki�1

aid<i

�2

�r2

l4e2��

Xki�1

�aidt� l2d<i�2

�r2

l2e2��

Xki�1

�aid<j�ajd<i�2�

dr2

f�r��r2

l2e2��dX2;

2�1�Xki�1

a2il2; (35)

where ai’s are k rotation parameters, f�r� is given inEq. (33), and dX2 is now the Euclidean metric on the �n�k� 1�-dimensional submanifold with volume Vn�k�1.

The conserved mass and angular momentum per unitvolume Vn�k�1 of the solution calculated on the boundaryB at infinity can be calculated through the use of Eqs. (21)and (22),

M � �2�k�nV�n�1�=2

0

16ln�k�1

�

�n� 1�

�n�

�n� 1�2�2

4

�� 2 � 1�

�m; (36)

J i � �2�k�nV�n�1�=2

0

16ln�k�1

�n�

�n� 1�2�2

4

� mai: (37)

The entropy per unit volume Vn�k�1 of the black braneis

S � �2�k ��2V0�

�n�1�=2r�n�1��h

4ln�k�1; (38)

where rh is the horizon radius. Again, it is a matter ofcalculation to show that the thermodynamic quantitiescalculated in this section satisfy the first law of thermody-namics,

dM � TdS �Xki�1

�idJ i; (40)

where �i � � l2��1ai is the ith component of angularvelocity of the horizon.

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V. CLOSING REMARKS

Till now, no explicit rotating charged black hole solu-tions have been found except for some dilaton couplingsuch as �

��3

p[14] or �1 when the string threeform

Habc is included [22]. For general dilaton coupling, theproperties of charged dilaton black holes have been inves-tigated only for rotating solutions with infinitesimallysmall angular momentum [15] or small charge [16]. Inthis paper we obtained a class of charged rotating blackhole solutions with zero and Liouville-type potentials. Wefound that these solutions are neither asymptotically flatnor (A)dS. In the case of V�� 0, the solution (whichincludes the string theoretical case, � 1) presents ablack string with a regular event horizon, provided thecharge parameter does not vanish. This solution has notinner horizons, and is acceptable only for � 1. Thus, ithas not a counterpart for Einstein gravity without dilaton( � 0).

In the presence of Liouville-type potential, we obtainedexact solutions provided � � �

��3

p. These solutions

reduce to the charged rotating black string of [2,4]. Wefound that these solutions have a cosmological horizon for(i) >

��3

pdespite the sign of , and (ii) positive values of

, despite the magnitude of . For <��3

p, the solutions

present black strings with outer and inner horizons if m>mcrit, an extreme black hole if m � mcrit, and a nakedsingularity if m<mcrit. The Hawking temperature of allthe above horizons were computed. We found that theHawking temperature is negative for inner and cosmologi-cal horizons, and it is positive for outer horizons. We alsocomputed the conserved and thermodynamics quantities ofthe four-dimensional rotating charged black string, andfound that they satisfy the first law of thermodynamics.

Next, we constructed the rotating uncharged solutionsof (n� 1)-dimensional dilaton gravity with all rota-tion parameters. If �n� 1�2�2 � 4n < 0, then these solu-tions present a black branes for < 0, and a naked singu-larity for > 0. If �n� 1�2�2 � 4n > 0, the solutionshave a cosmological horizon for positive , whilethey are not acceptable for negative values of . Againwe found that the thermodynamic quantities of theblack brane solutions satisfy the first law of thermo-dynamics.

Note that the �n� 1�-dimensional rotating solutionsobtained here are uncharged. Thus, it would be interestingif one can construct rotating solutions in �n� 1� dimen-sions in the presence of electromagnetic field. One mayalso attempt to generalize these kinds of solutions obtainedhere to the case of two-term Liouville potential.

ACKNOWLEDGMENTS

This work has been supported by Research Institute forAstronomy and Astrophysics of Maragha, Iran

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