Rotating black holes in Einstein-Maxwell-dilaton gravity

Ahmad Sheykhi

Department of Physics, Shahid Bahonar University, P.O. Box 76175-132, Kerman, IranResearch Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha, Iran

(Received 28 November 2007; published 19 May 2008)

We present a new class of slowly rotating black hole solutions in ðnþ 1Þ-dimensional ðn � 3Þ Einstein-Maxwell-dilaton gravity in the presence of Liouville-type potential for the dilaton field and an arbitrary

value of the dilaton coupling constant. Because of the presence of the dilaton field, the asymptotic

behavior of these solutions are neither flat nor (A)dS. In the absence of a dilaton field, our solution reduces

to the ðnþ 1Þ-dimensional Kerr-Newman modification thereof for small rotation parameter [A. N. Aliev,

Phys. Rev. D 74, 024011 (2006).]. We also compute the angular momentum and the gyromagnetic ratio of

these rotating dilaton black holes.

DOI: 10.1103/PhysRevD.77.104022 PACS numbers: 04.70.Bw, 04.20.Ha, 04.50.�h

I. INTRODUCTION

The motivation for studying higher-dimensional blackholes originates from developments in string/M theory,which is believed to be the most promising approach toquantum theory of gravity in higher dimensions. It wasargued that black holes may play a crucial role in theanalysis of dynamics in higher dimensions as well as inthe compactification mechanisms. In particular to testnovel predictions of string/M theory, microscopic blackholes may serve as good theoretical laboratories [1,2].Another motivation for studying higher-dimensional blackholes comes from the braneworld scenarios, as a newfundamental scale of quantum gravity. For a while it wasthought that the extra spatial dimensions would be of theorder of the Planck scale, making a geometric descriptionunreliable, but it has recently been realized that there is away to make the extra dimensions relatively large and stillbe unobservable. This is if we live on a three-dimensionalsurface (brane) in a higher-dimensional spacetime (bulk)[3,4]. In such a scenario, all gravitational objects such asblack holes are higher dimensional. One of the most inter-esting phenomena predicted within this scenario is thepossibility of the formation in accelerators of higher-dimensional black holes smaller than the size of extradimensions [5].

The pioneering study on higher-dimensional black holeswas done by Tangherlini several decades ago who foundthe analogues of Schwarzschild and Reissner-Nordstromsolutions in higher dimensions [6]. These solutions arestatic with spherical topology. The rotating black holesolutions in higher-dimensional Einstein gravity was foundby Myers and Perry [7]. These solutions [7] have noelectric charge and can be considered as a generalizationof the four-dimensional Kerr solutions. Besides, it wasconfirmed by recent investigations that the gravity inhigher dimensions exhibits much richer dynamics than in

four dimensions. For example, there exists a black ringsolution in five dimensions with the horizon topology ofS2 � S1 [8] which can carry the same mass and angularmomentum as the Myers-Perry solution, and consequentlythe uniqueness theorem fails in five dimensions. While thenonrotating black hole solution to the higher-dimensionalEinstein-Maxwell gravity was found several decades ago[6], the counterpart of the Kerr-Newman solution in higherdimensions, that is the charged generalization of theMyers-Perry solution in ðnþ 1Þ-dimensional Einstein-Maxwell gravity, still remains to be found analytically.Indeed, the case of charged rotating black holes in higherdimensions has been discussed in the framework of super-gravity theories and string theory [9–14]. Recently,charged rotating black hole solutions in ðnþ 1Þ-dimensional Einstein-Maxwell theory with a single rota-tion parameter in the limit of slow rotation has been con-structed in [15] (see also [16,17]). More recently a class ofcharged slowly rotating black hole solutions in Gauss-Bonnet gravity was presented in [18].It is also of great interest to generalize the study to the

dilaton gravity. A dilaton is a kind of scalar field whichappears in the low-energy effective action of string theoryand can be coupled to gravity and gauge fields [19]. Whena dilaton is coupled to Einstein-Maxwell theory, it hasprofound consequences for the black hole solutions. Thisfact may be seen in the case of four-dimensional rotatingEinstein-Maxwell-dilaton (EMd) black holes which do notpossess the gyromagnetic ratio g ¼ 2 of a Kerr-Newmanblack hole [20–23]. Therefore, the study on the rotatingsolutions of Einstein-Maxwell gravity in the presence of adilaton field is well motivated. Of particular interest is toinvestigate the effect of the dilaton field on the physicalproperties of the solutions. The appearance of the dilatonfield changes the asymptotic behavior of the solutions to beneither asymptotically flat nor (A)dS. A motivation toinvestigate nonasymptotically flat, nonasymptotically (A)dS solutions of Einstein gravity is that these might lead topossible extensions of AdS/CFT correspondence. Indeed,*sheykhi@mail.uk.ac.ir

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it has been speculated that the linear dilaton spacetimes,which arise as near-horizon limits of dilatonic black holes,might exhibit holography [24]. Another motivation is thatsuch solutions may be used to extend the range of validityof methods and tools originally developed for, and tested inthe case of, asymptotically flat or asymptotically (A)dSblack holes. It has been shown that the counterterm methodinspired by the AdS/CFT correspondence can be appliedsuccessfully to the computation of the conserved quantitiesof nonasymptotically (A)dS black holes/branes (see e.g.[25–28]). While exact static dilaton black hole/string so-lutions of EMd gravity have been constructed in [29–38],exact rotating black holes solutions with curved horizonshave been obtained only for some limited values of thedilaton coupling constant [21,39–41]. For general dilatoncoupling constant, the properties of charged rotating dila-ton black holes only with infinitesimally small charge [42]or small angular momentum have been investigated[22,23,43–45]. When the horizons are flat, rotating solu-tions of EMd gravity with Liouville-type potential in four[25] and ðnþ 1Þ dimensions have also been constructed[26]. These solutions [25,26] are not black holes anddescribe charged rotating dilaton black strings/branes. Itis worth noting that these rotating solutions [25,26] arebasically obtained by a Lorentz boost from correspondingstatic ones; they are equivalent to static ones locally,although not equivalent globally. So far, charged rotatingdilaton black hole solutions with curved horizons, for anarbitrary value of the dilaton coupling constant in ðnþ1Þ-dimensional EMd theory, have not been constructed. Inthis paper, as a new step to shed some light on this issue forfurther investigation, we present a class of rotating solu-tions in ðnþ 1Þ-dimensional EMd theory. These solutionsdescribe an electrically charged, slowly rotating dilatonblack hole with curved horizons and an arbitrary value ofthe dilaton coupling constant in in ðnþ 1Þ dimensions. Weshall also investigate the effects of the dilaton field as wellas the rotation parameter on the physical quantities such astemperature, entropy, angular momentum, and the gyro-magnetic ratio of these rotating dilaton black holes.

II. BASIC EQUATIONS AND SOLUTIONS

The action of ðnþ 1Þ-dimensional ðn � 3Þ Einstein-Maxwell gravity coupled to a dilaton field can be written

S ¼ 1

16�

Zdnþ1x

ffiffiffiffiffiffiffi�gp �

R� 4

n� 1ðr�Þ2 � Vð�Þ

� e�4��=ðn�1ÞF��F��

�� 1

8�

Z@M

dnxffiffiffiffiffiffiffiffi��

p�ð�Þ;

(1)

where R is the Ricci scalar curvature, � is the dilatonfield, and Vð�Þ is a potential for �. � is a constantdetermining the strength of coupling of the scalar andelectromagnetic field, F�� ¼ @�A� � @�A� is the electro-

magnetic field tensor, and A� is the electromagnetic po-

tential. The last term in Eq. (1) is the Gibbons-Hawkingboundary term which is chosen such that the variationalprinciple is well defined. The manifold M has metric g��

and covariant derivative r�. � is the trace of the extrinsic

curvature �ab of any boundary (boundaries) @M of themanifoldM, with induced metric(s) �ab. In this paper, weconsider the action (1) with a Liouville-type potential,

Vð�Þ ¼ 2�e2��; (2)

where � and � are arbitrary constants. The equations ofmotion can be obtained by varying the action (1) withrespect to the gravitational field g��, the dilaton field �,

and the gauge field A� which yields the following field

equations:

R �� ¼ 4

n� 1

�@��@��þ 1

4g��Vð�Þ

�

þ 2e�4��=ðn�1Þ�F��F

��

� 1

2ðn� 1Þ g��F��F��

�; (3)

r2� ¼ n� 1

8

@V

@�� �

2e�4��=ðn�1ÞF��F

��; (4)

r�ðe�4��=ðn�1ÞF��Þ ¼ 0: (5)

We would like to find ðnþ 1Þ-dimensional rotating solu-tions of the above field equations. For infinitesimal rota-tion, we can solve Eqs. (3)–(5) to first order in the angularmomentum parameter a. Inspection of the ðnþ1Þ-dimensional Kerr solutions shows that the only term inthe metric changes to OðaÞ is gt. Similarly, the dilaton

field does not change to OðaÞ and A is the only compo-

nent of the vector potential that changes. Therefore, forinfinitesimal angular momentum up to OðaÞ, we can takethe following form of the metric

ds2 ¼ �UðrÞdt2 þ dr2

UðrÞ � 2afðrÞsin2dtdþ r2R2ðrÞðd2 þ sin2d2 þ cos2d�2

n�3Þ; (6)

where a is a parameter associated with its angular momen-tum, and d�2

n�3 denotes the metric of an unit ðn� 3Þsphere. The functions UðrÞ, RðrÞ, and fðrÞ should bedetermined. In the particular case a ¼ 0, this metric re-duces to the static and spherically symmetric cases. Forsmall a, we can expect to have solutions with UðrÞ still afunction of r alone. The t component of the Maxwellequations can be integrated immediately to give

Ftr ¼ qe4��=ðn�1Þ

ðrRÞn�1; (7)

where q is an integration constant related to the electric

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charge of the solutions. Defining the electric charge viaQ ¼ 1

4�

Rexp½�4��=ðn� 1Þ��Fd�, we get

Q ¼ q!n�1

4�; (8)

where !n�1 represents the area of the unit ðn� 1Þ-sphere.In general, in the presence of rotation, there is also a vectorpotential in the form

A ¼ aqhðrÞsin2: (9)

Notice that for an infinitesimal rotation parameter, theelectric field (7) does not change from the static case. Inorder to solve the system of Eqs. (3)–(5) for four unknownfunctions fðrÞ, RðrÞ, �ðrÞ, and hðrÞ, we take the followingansatz:

RðrÞ ¼ e2��=ðn�1Þ: (10)

Inserting (10), the Maxwell fields (7) and (9), and themetric (6), into the field Eqs. (3)–(5), one can show thatthese equations have the following solutions:

UðrÞ ¼ � ðn� 2Þð�2 þ 1Þ2b�2�r2�

ð�2 � 1Þð�2 þ n� 2Þ � m

rðn�1Þð1��Þ�1

þ 2q2ð�2 þ 1Þ2b�2ðn�2Þ�

ðn� 1Þð�2 þ n� 2Þ r2ðn�2Þð��1Þ; (11)

fðrÞ ¼ mð�2 þ n� 2Þbðn�3Þ�

�2 þ 1rðn�1Þð��1Þþ1

� 2q2ð�2 þ 1Þbð1�nÞ�

n� 1r2ðn�2Þð��1Þ; (12)

�ðrÞ ¼ ðn� 1Þ�2ð�2 þ 1Þ ln

�b

r

�; hðrÞ ¼ rðn�3Þð��1Þ�1:

(13)

Here b is an arbitrary constant and � ¼ �2=ð�2 þ 1Þ. Inthe above expressions,m appears as an integration constantand is related to the ADM (Arnowitt-Deser-Misner) massof the black hole. According to the definition of mass dueto Abbott and Deser [46], the mass of the solution is [38]

M ¼ bðn�1Þ�ðn� 1Þ!n�1

16�ð�2 þ 1Þ m: (14)

For ð� ¼ 0 ¼ �Þ the mass of the black hole reduces to

M ¼ ðn� 1Þ!n�1

16�m: (15)

In order to fully satisfy the system of equations, we musthave

� ¼ 2

�ðn� 1Þ ; � ¼ ðn� 1Þðn� 2Þ�2

2b2ð�2 � 1Þ : (16)

One may also note that in the absence of a nontrivialdilaton (� ¼ � ¼ 0), the above solutions reduce to

UðrÞ ¼ 1� m

rn�2þ 2q2

ðn� 1Þðn� 2Þr2ðn�2Þ ; (17)

fðrÞ ¼ mðn� 2Þr2�n

� 2q2

ðn� 1Þr2ðn�2Þ ; (18)

hðrÞ ¼ r2�n; (19)

which describe an ðnþ 1Þ-dimensional Kerr-Newmanblack hole in the limit of slow rotation [15]. The metriccorresponding to (11)–(13) is neither asymptotically flatnor (anti)–de Sitter. In order to study the physical proper-ties of these solutions, we first look for the curvaturesingularities. In the presence of a dilaton field, theKretschmann scalar R����R

���� diverges at r ¼ 0, it is

finite for r � 0 and goes to zero as r ! 1. Thus, there is anessential singularity located at r ¼ 0. As one can see from(11), the solution is ill-defined for the string case where� ¼ 1. The cases with �> 1 and �< 1 should be consid-ered separately. For �> 1, we have cosmological hori-zons, while there are no cosmological horizons if �< 1(see Fig. 1). In fact, in the latter case (�< 1) the space-times exhibit a variety of possible casual structures de-pending on the values of the metric parameters (see Figs. 2and 3). One can obtain the casual structure by finding theroots of fðrÞ ¼ 0. Unfortunately, because of the nature ofthe exponent in (11), it is not possible to find analyticallythe location of the horizons. To get better understanding onthe nature of the horizons, we plot in Figs. 4 and 5, the massparameter m as a function of the horizon radius for differ-ent values of dilaton coupling constant �. It is easy to showthat the mass parameter m of the dilaton black hole can beexpressed in terms of the horizon radius rh as

–3

–2

–1

0

1

2

3

f(r)

0.2 0.4 0.6 0.8 1 1.2 1.4r

FIG. 1. The function fðrÞ versus r for q ¼ 1, b ¼ 1, m ¼ 2,and n ¼ 4. � ¼ 0:5 (bold line) and � ¼ 1:3 (continuous line).

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mðrhÞ ¼ � ðn� 2Þð�2 þ 1Þ2b�2�

ð�2 � 1Þðnþ �2 � 2Þ rn�2þ�ð3�nÞh

þ 2q2ð�2 þ 1Þ2b�2�ðn�2Þ

ðn� 1Þðnþ �2 � 2Þ rðn�3Þð��1Þ�1h : (20)

These figures show that for a given value of � or q, thenumber of horizons depend on the choice of the value ofthe mass parameter m. We see that, up to a certain value ofthe mass parameter m, there are two horizons, and as wedecrease the m further, the two horizons meet. In this casewe get an extremal black hole with mass mext. The entropyof the black hole typically satisfies the so-called area law ofthe entropy which states that the entropy of the black holeis a quarter of the event horizon area [47]. This nearuniversal law applies to almost all kinds of black holes,including dilaton black holes, in Einstein gravity [48].Since the surface gravity and the area of the event horizon

do not change up to the linear order of the rotating pa-rameter a, we can easily show that the Hawking tempera-ture and the entropy of dilaton black hole on the outer eventhorizon rþ can be written as

Tþ ¼ �

2�¼ f0ðrþÞ

4�; S ¼ A

4; (21)

where � is the surface gravity and A is the horizon area.Then, one can get

Tþ ¼ � ð�2 þ n� 2Þm4�ð�2 þ 1Þ rðn�1Þð��1Þ

þ

þ b�2�ðn� 2Þð�2 þ 1Þ2�ð1� �2Þ r2��1

þ ; (22)

–3

–2

–1

0

1

2

3

f(r)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2r

FIG. 2. The function fðrÞ versus r for q ¼ 1, b ¼ 1, m ¼ 2,and n ¼ 4. � ¼ 0 (bold line), � ¼ 0:5 (continuous line),and� ¼ 0:7 (dashed line).

0

1

2

3

4

5

m

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2rh

FIG. 5. The function mðrhÞ versus rh for � ¼ 0:5, b ¼ 1,and n ¼ 4. q ¼ 0:5 (bold line), q ¼ 1 (continuous line), andq ¼ 1:5 (dashed line).

0

1

2

3

4

5

m

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2rh

FIG. 4. The function mðrhÞ versus rh for q ¼ 1, b ¼ 1, andn ¼ 4. � ¼ 0 (bold line), � ¼ 0:5 (continuous line), and � ¼0:6 (dashed line).

–4

–2

0

2

4

f(r)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2r

FIG. 3. The function fðrÞ versus r for � ¼ 0:5, b ¼ 1, m ¼2, and n ¼ 4. q ¼ 0:5 (bold line), q ¼ 1 (continuous line), andq ¼ 1:5 (dashed line).

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S ¼ bðn�1Þ�rðn�1Þð1��Þþ !n�1

4: (23)

Equation (22) shows that for �> 1 the temperature isnegative. As we argued above, in this case we encountercosmological horizons, and therefore the cosmologicalhorizons have negative temperature. Numerical calcula-tions show that the temperature of the event horizon goesto zero as the black hole approaches the extreme case. It isa matter of calculation to show that the mass parameter ofthe extremal black hole can be written (�< 1)

mext ¼ 2ðn� 2Þð�2 þ 1Þ2b�2�

ð1� �2Þð�2 þ n� 2Þ rð2�nÞð��1Þþ�þ : (24)

In summary, the metric of Eqs. (6) and (11)–(13) canrepresent a black hole with inner and outer event horizonslocated at r� and rþ, provided m>mext, an extreme blackhole in the case of m ¼ mext, and a naked singularity ifm<mext. It is worth noting that in the absence of a non-trivial dilaton field (� ¼ � ¼ 0), expressions (22) and (24)reduce to that of an ðnþ 1Þ-dimensional asymptoticallyflat black hole.

Finally, we calculate the angular momentum and thegyromagnetic ratio of these rotating dilaton black holeswhich appear in the limit of slow rotation parameter. Theangular momentum of the dilaton black hole can be calcu-lated through the use of the quasilocal formalism of Brownand York [49]. According to the quasilocal formalism, thequantities can be constructed from the information thatexists on the boundary of a gravitating system alone.Such quasilocal quantities will represent information aboutthe spacetime contained within the system boundary, justlike Gauss’s law. In our case the finite stress-energy tensorcan be written as

Tab ¼ 1

8�ð�ab ���abÞ; (25)

which is obtained by variation of the action (1) with respectto the boundary metric �ab. To compute the angular mo-mentum of the spacetime, one should choose a spacelikesurface B in @M with metric �ij, and write the boundary

metric in ADM form

�abdxadxa ¼ �N2dt2 þ �ijðd’i þ VidtÞðd’j þ VjdtÞ;

where the coordinates ’i are the angular variables parame-terizing 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 on the boundary, thenthe quasilocal conserved quantities associated with thestress tensors of Eq. (25) can be written as

Qð Þ ¼ZBdn�1’

ffiffiffiffi�

pTabn

a b; (26)

where � is the determinant of the metric �ij, and na are

the Killing vector field and the unit normal vector on the

boundary B. For boundaries with a rotational (& ¼ @=@’)Killing vector field, one obtains the quasilocal angularmomentum

J ¼ZBdn�1’

ffiffiffiffi�

pTabn

a&b; (27)

provided the surfaceB contains the orbits of &. Finally, theangular momentum of the black holes can be calculatedthrough the use of Eq. (27). We find

J ¼ ðn� �2Þð�2 þ n� 2Þb2ðn�2Þ�!n�1

8�nðn� 2Þð�2 þ 1Þ2 ma: (28)

For a ¼ 0, the angular momentum vanishes, and thereforea is the rotational parameter of the dilaton black hole. Forð� ¼ � ¼ 0Þ, the angular momentum reduces to the angu-lar momentum of the ðnþ 1Þ-dimensional Kerr black holes

J ¼ ma!n�1

8�: (29)

Combining Eq. (15) with Eq. (29) we get

J ¼ 2Ma

n� 1: (30)

Next we calculate the gyromagnetic ratio of these rotatingdilaton black holes. The magnetic dipole moment for thisslowly rotating dilaton black hole is

� ¼ Qa: (31)

Therefore, the gyromagnetic ratio is given by

g ¼ 2�M

QJ¼ nðn� 1Þðn� 2Þð�2 þ 1Þ

ðn� �2Þð�2 þ n� 2Þbðn�3Þ� : (32)

1

2

3

4

5

6

g

0 0.2 0.4 0.6 0.8 1alpha

FIG. 6. The behavior of the gyromagnetic ratio g versus � forb ¼ 1. n ¼ 3 (bold line), n ¼ 4 (continuous line), and n ¼ 5(dashed line).

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It was argued in [22,23] that the dilaton field can modifythe gyromagnetic ratio of the asymtotically flat and asym-totically (A)dS four-dimensional black holes. Our resulthere confirms their arguments. However, in contrast to thegyromagnetic ratio of the asymtotically flat or (A)dS four-dimensional dilatonic black holes which turned out to beg � 2, in our case in which the solutions are neitherasymtotically flat nor (A)dS, we get g � 2 in four dimen-sions. We have shown the behavior of the gyromagneticratio g of the dilatonic black hole ð�< 1Þ versus � inFigs. 6 and 7. From these figures we find out that for smallb the gyromagnetic ratio increases with increasing�, whilefor large b and n � 4 the gyromagnetic ratio decreaseswith increasing �. In the absence of a nontrivial dilatonð� ¼ � ¼ 0Þ, the gyromagnetic ratio reduces to

g ¼ n� 1; (33)

which is the gyromagnetic ratio of the ðnþ 1Þ-dimensionalKerr-Newman black holes (see e.g. [15]).

III. CONCLUSION

Though the nonrotating black hole solution to thehigher-dimensional Einstein-Maxwell gravity was found

several decades ago [6], the counterpart of the Kerr-Newman solution in higher dimensions, that is the chargedgeneralization of the Myers-Perry solution [7] in ðnþ1Þ-dimensional Einstein-Maxwell gravity, still remains tobe found analytically. Recently, charged rotating blackhole solutions in ðnþ 1Þ-dimensional Einstein-Maxwelltheory have been constructed with a single rotation pa-rameter in the limit of slow rotation parameter [15]. In thispaper, as a next step to shed some light on this issue forfurther investigation, we further generalized these slowlyrotating black hole solutions [15] by including a dilatonfield and a Liouville-type potential for the dilaton field inthe action. These solutions describe charged rotating dila-ton black holes with an arbitrary dilaton coupling constantin the limit of slow rotation parameter. In contrast to therotating black holes in the Einstein-Maxwell theory, whichare asymptotically flat, the charged rotating dilaton blackholes we found here are neither asymptotically flat nor (A)dS. Our strategy for constructing these solutions was theperturbative technique suggested in [22]. We first studiedcharged black hole solutions in ðnþ 1Þ-dimensionalEinstein-Maxwell-dilaton gravity. Then, we consideredthe effect of adding a small amount of rotation parametera to the black hole. We discarded any terms involving a2 orhigher power in a. Inspection of the Kerr-Newman solu-tions shows that the only term in the metric changes toOðaÞ is gt. Similarly, the dilaton does not change to OðaÞand A is the only component of the vector potential that

changes to OðaÞ. We showed that in the absence of adilaton field ð� ¼ 0 ¼ �Þ, our solutions reduce to the ðnþ1Þ-dimensional Kerr-Newman modification thereof forsmall rotation parameter [15]. We computed temperatureand entropy of black holes, which did not change to OðaÞfrom the static case. We also obtained the angular momen-tum and the gyromagnetic ratio of these rotating dilatonblack holes. Interestingly enough, we found that the dilatonfield can modify the gyromagnetic ratio of the ðnþ1Þ-dimensional rotating dilaton black holes. This is inagreement with the arguments in [22,23].

ACKNOWLEDGMENTS

This work has been supported financially by theResearch Institute for Astronomy and Astrophysics ofMaragha, Iran.

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2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

g

0 0.2 0.4 0.6 0.8 1alpha

FIG. 7. The behavior of the gyromagnetic ratio g versus � forn ¼ 4. b ¼ 2 (bold line), b ¼ 3 (continuous line), and b ¼ 4(dashed line).

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