Annals of Physics Volume 333 issue 2013 [doi 10.1016_j.aop.2013.03.008] Hendi, S.H. -- Asymptotic...

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Annals of Physics 333 (2013) 282289

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Asymptotic ReissnerNordstrm black holes

S.H. Hendi Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran

Research Institutefor Astrophysics and Astronomy of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran

h i g h l i g h t s

We investigate two types of the BI-like nonlinear electromagnetic fields in the Einsteinian gravity. We analyze the effects of nonlinearity on the electromagnetic field. We examine the influences of the nonlinearity on the geometric properties of the black hole solutions.

a r t i c l e i n f o

Article history:Received 8 December 2012

Accepted 13 March 2013

Available online 27 March 2013

Keywords:

ReissnerNordstrm black hole

Nonlinear electromagnetic field

Geometric properties

a b s t r a c t

We consider two types of BornInfeld like nonlinear electromag-netic fields and obtain their interesting black hole solutions. Theasymptotic behavior of these solutions is the same as that of a

ReissnerNordstrm black hole. We investigate the geometricproperties of the solutions and find that depending on the valueof the nonlinearity parameter, the singularity covered with varioushorizons.

2013 Elsevier Inc. All rights reserved.

1. Introduction

Standard Maxwell theory is confronted with the infinite self-energy of charged point-like particles.The renormalization procedure in quantum electrodynamics theory has been applied to obtain a finitevalue, as deduced from the difference between two infinite forms[1]. TheLagrangian of linearMaxwellfield,LMax= F= F F , has been considered in various theories of gravity[2]. It is well knownthat the ReissnerNordstrm (RN) metric is the spherically symmetric solution of EinsteinMaxwellgravity, which corresponds to the charged static black hole of mass Mand charge Q [3]. Withingravitational field theories, generalization of RN black holes to nonlinear electromagnetic fields playsa crucial role since most of physical systems are intrinsically nonlinear in the nature.

Correspondence to: Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran.

Tel.: +98 7116137006.

E-mail addresses:[email protected],[email protected].

0003-4916/$ see front matter 2013 Elsevier Inc. All rights reserved.

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S.H. Hendi / Annals of Physics 333 (2013) 282289 283

Among nonlinear electromagnetic fields the well-known BornInfeld (BI) theory is quite special aslow energy limiting cases of certain models of string theory [4]. This classical theory has been designedto regulate the self-energy of a point-like charge [5]. Although coupling of the BI theory with Einsteingravity has been studied in 1935 [6], in recent years, black hole solutions of BI theory are of interest [7].

Other kinds of nonlinear electrodynamics, with various motivations, in the context of gravitational

field have been investigated[810]. In this paper, we consider (3 + 1)-dimensional action of Einsteingravity with two kinds of BI-like nonlinear electromagnetic fields. Generalization of Maxwell fieldto nonlinear BI model provides powerful tools for investigation of interesting black hole solutions.In addition, it was found that all order loop corrections to gravity may be regarded as a BI-likeLagrangian [11]. Considering the BI Lagrangian

LBI= 42

1

1 + F22

, (1)

one can expand it for large values of nonlinearity parameter ( ) to obtain

LBI|Large = F+1

2 O(F2

), (2)

which confirms thatLBIreduces to the linear Maxwell Lagrangian for .Moreover, from cosmological point of view, it was shown that the early universe inflation may be

explained with the nonlinear electrodynamics[12]. Also, one can connect the nonlinear electrody-namics with the large scale correlated magnetic fields observed nowadays[13].

Motivated by all the above, we seek four dimensional solutions to Einstein gravity with the BI-likeLagrangian. We consider a recently proposed BI-like model [14], namely Exponential form of nonlinearelectromagnetic field (ENEF) and Logarithmic form of nonlinear electromagnetic field (LNEF), whoseLagrangians are

L(F) =

2 exp

F2 1 , ENEF82 ln

1 + F

82

, LNEF

(3)

in which, for large values of , F2 correction to Maxwell Lagrangian is included. In other words, thetwo forms of the mentioned Lagrangians have the same expansion as presented in Eq.(2).

The plan of the paper is as follows. We give a brief review of the simplest coupling of the nonlin-ear gauge field system(3)to Einstein gravity. Then, we obtain the solutions of the four dimensionalspherical symmetric spacetime in the presence of two classes of nonlinear electromagnetic fields, in-vestigate their properties and compare them with BI solution. The paper ends with some conclusions.

2. (3+ 1)-dimensional black holes with nonlinear electromagnetic field

Let us begin with the (3 + 1)-dimensional nonlinear electromagnetic field in general relativity,with the action

IG= 1

16

M

d4xg[R 2 + L(F)] 1

8

M

d3xK, (4)

whereR is the Ricci scalar, refers to the negative cosmological constant and L(F) is defined inEq. (3). In addition, the secondintegral in Eq. (4) is the boundary term that does not affect the equationsof motion, namely the GibbonsHawking boundary term [15]. The factorsK andare, respectively,the traces of the extrinsic curvature and the induced metric of boundary M of the manifold M. TheEinstein and electromagnetic field equations derived from the above action are

R1

2g(R 2) =

1

2g L(F) 2FFLF

, (5)

gLFF = 0, (6)


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284 S.H. Hendi / Annals of Physics 333 (2013) 282289

whereLF = dL(F)dF . Our main aim here is to obtain charged static black hole solutions of the fieldEqs.(5)and(6)and investigate their properties. We assume a (3 + 1)-dimensional static spacetimeis described by the spherically symmetric line element

ds2

= f(r)dt2

+dr2

f(r) +r2d2 + sin2 d2 . (7)

The electromagnetic tensorF compatible with the metric(7) can involve only a radial electricfieldFtr= Frt. Therefore, we should use the gauge potential ansatz A= h(r)t in the nonlinearelectromagnetic field Eq.(6).Considering the mentioned ansatz, one can show that Eq.(6)reduces to

r

1 +

2h(r)

2h(r) + 2h(r) = 0, ENEF

r

1 +

h(r)

2

2h(r) + 2h(r)

1

h(r)

2

2= 0, LNEF,

(8)

with the following solutions

h(r) =

110

4Q22LWe

LW1/4 5 + LWF

[1] ,

9

4

,

Lw

4

, ENEF

2Q

3r

1

(1 + ) 2F

1

4,

1

2

,

5

4

, 1 2

, LNEF,

(9)

where prime and double primes denote the first and second derivative with respect to r, respectively,

=

1 + Q2r42

and Q is an integration constant which is the electric charge of the black holes. In ad-

dition, LW= LambertW

4Q2

2r4

which satisfies LambertW(x) exp [LambertW(x)] = x and F([a], [b],z)

is a hypergeometric function (for more details, see [16]).

It is notable that the same procedure for the linear Maxwell and nonlinear BI theory leads to

h(r) =

Qr

, Maxwell

Qr

F

1

4,

1

2

,

5

4

, 1 2

, BINEF

(10)

which are completely different with Eq.(9).In order to find the asymptotic behavior of the solutions,we should expand them for large distances (r 1). The radial functionh(r) is expanded as

h(r)|Large r= Q

r+ Q

3

202r5+ O

1

r9, (11)

where = 2, 8 and 1 for BINEF, ENEF and LNEF, respectively. So, we find that for large values ofr, thedominant term of Eq. (11) is the gauge potential of RN black hole. Now, we consider the nonvanishingcomponents ofF . One can show that

Ftr= Frt=Q

r2

eLW

2 , ENEF2

+ 1 , LNEF,(12)

and it has been shown for BI theory[7]

Ftr= Frt=Q

r2.

Differentiating from Eq.(11)or expandingFtrfor large distances, one can obtain

Ftr=Q

r2 Q

3

42r6+ O

1

r10

, (13)

which contain additional radial electric fields apart from the usual Coulomb one.


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It is easy to find that all the mentioned electric fields vanish at large values ofr, as they should be.Furthermore, one may expect to obtain a finite value for the nonlinear electric fields at r= 0. It isinteresting to mention that, despite ENEF, the electric field of LNEF is finite at the origin. One can findthatFENEFtr has a finite valuenearthe origin, which is the same as the electric fields of BINEF and LNEF,

butFENEF

tr

diverges atr=

0, as it occurs for Maxwell field.Now, we should find a suitable function f(r) to satisfy all components of Eq. (5). Considering

Eq.(9),one may show that therrandttcomponents of Eq.(5)can be simplified as

Eqtt= Eqrr= f(r) +f(r) + r2 1

r r2g(r) = 0, (14)

where

g(r) =

1 Qr2

LW

(1 LW) , ENEF

41 ln 2

+1 , LNEF.

(15)

Other nonzero components of Eq.(5)can be written as

Eq = Eq=

d

dr 1

r

Eqtt= 0, (16)

and therefore it is sufficient to solveEqtt= 0. After some cumbersome calculations, the solutions ofEq.(14)can be written as

f(r)= 1 2Mr

r2

3

+

Q

34Q

26eLW

r4L5W

1/8

1 + LW+4

5

L2WF[1] , 94 , LW

4 , ENEF

8Q2F

12

, 14

,

54

, 1 2

3r2

42r2

ln

(21)

2e7/3

3

42

r2 ln ( 1) drr

, LNEF,

(17)

where Mis the integration constant which is the total mass of spacetime (the integral term of Eq. (17)will be solved in theAppendix). In order to compare our solutions with the BI and RN black holes, weshould present the BI and RN solutions. Considering Eq. (10),it has been shown that the field Eq.(5)reduces to the Eqs.(14)and(16)with

g(r) =Q

2

r3 , RN

2 (1 ) , BINEF,(18)

and the following solutions

f(r) = 1 2Mr

r2

3+

Q2

r2 , RN

4Q2

3r2F

1

4,

1

2

,

5

4

, 1 2

, BINEF.

(19)

We should note that although Eq.(17)is different with RN and BI solutions, as previously mentioned,

they display the same asymptotic behavior. The metric functions of nonlinear electromagnetic fieldsare expanded as

f(r) = 1 2Mr

r2

3+ Q

2

r2 Q

4

40r62+ O

1

r10

. (20)


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Considering the first correction term in Eq. (20), we should stress that the this equation contains boththe usual RN terms as well as an analytic function ofO( 1

r6), in which one can ignore it to obtain RN

solution for large values ofr.In what follows, we are going to investigate the geometric nature of the obtained solutions. At first,

we look for the existence of curvature singularities and their horizons. Given the metric in Eq.(7),we

can compute the Ricci and the Kretschmann scalars

R = f(r) 4f(r)r

2f(r)r2

, (21)

RR = f2(r) +

2f(r)

r

2+

2f(r)

r2

2(22)

wheref(r)is the metric function. Also one can show that other curvature invariants (such as Riccisquare) are functions of f, f/r and f/r2, and therefore it is sufficient to study the Ricci and theKretschmann scalars for the investigation of the spacetime curvature. After a number of manipula-tions, we find these scalars diverge in the vicinity of the origin

limr0+

R = , (23)

limr0+

RR = , (24)

which confirms that the spacetime given by Eqs.(7)and(17)has an essential singularity at r= 0.In order to investigate the asymptotic behavior of the solution, we should simplify Eqs.(21)and

(22)for large distances. Expanding these equations and keep the first order nonlinear correction, wecan obtain

limr

R = 4 + Q4

22r8+ O

1

r12 , (25)lim

rRR

= 83

2 + 48M2

r6 96MQ

2

r7 + 56Q

4

r8 2 Q

4

2l2r8+ O

1

r11

, (26)

where the last term is the leading nonlinear correction to the RN black hole solutions. Eqs. (25)and(26)show that the asymptotic behavior of the obtained solutions is adS for < 0.

Now, we investigate the effects of nonlinear electromagnetic fields and the nonlinearity param-eter, .Figs. 1and2 show that the nonlinearity parameter has effect on the value of the inner andouter horizons in addition to the minimum value of the metric function (whenf(r) has a minimum).Furthermore, considering Eq. (19), we find that due to the fact that the metric function of the RN blackhole is positive near the origin as well as large values ofr, depending on the value of the metric param-

eters, one can obtain a black hole with two inner and outer horizons, an extreme black hole with zerotemperature and a naked singularity. It is so interesting to mention that for the nonlinear solutions,a new situation may appear. In other words, one can show that near the origin (r 0), the metricfunction, Eq.(17),may be positive, zero or negative for > c,= cor < c, respectively (seeFigs. 1and2for more details). We should note that considering limr0+f(r)= 0, one may obtaincas a function of, Qand M, numerically. The new situation appears for < c, in which the blackholes have one non-extreme horizon with positive temperature as it happens for (adS) Schwarzschildsolutions (uncharged solutions). In addition,Figs. 1and2 show that for > c, we may obtain anextreme value for the nonlinearity parameter (ext) to achieve a black hole with two horizons, anextreme black hole and a naked singularity for c< < ext, = extand > ext, respectively.

3. Conclusions

In this paper, we have considered two classes of the BI-like nonlinear electromagnetic fields inthe Einsteinian gravity. Expansion of both the mentioned nonlinear Lagrangians, for large values ofnonlinearity parameter, is the same as that of BI theory and so we called them BI-like fields.


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Fig. 1. f(r) (ENEF branch) versusrfor = 1,q = 2,M= 2.5, and = 0.5 < c(solid line: black hole with a non-extremehorizon),c < = 1 < ext(bold line: black hole with two horizons), = ext= 3 (dotted line: black hole with an extremehorizon) and = 10 > ext(dashed line: naked singularity).

Fig. 2. f(r) (LNEF branch) versusrfor = 1,q = 2,M= 2.5, and = 0.1 < c(solid line: black hole with a non-extremehorizon),c < = 0.5 < ext (bold line: black hole with two horizons), = ext= 1.1 (dotted line: black hole with anextreme horizon) and = 5> ext(dashed line: naked singularity).

Regarding four dimensional spherically symmetric spacetime, we have obtained the compatibleelectromagnetic fields and found that, in contrast with Maxwell field, the mentioned nonlinear


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electromagnetic fields have finite values in the vicinity of the origin. In addition, we have found thatFENEFtr diverges atr= 0, but its divergency is much slower than the Maxwell one. In other words, thebehavior of theFLNEFtr is very close to BI field, but in the neighborhood of the origin,F

ENEFtr has a special

behavior between FBINEFtr and FRNtr . Expansion of the nonlinear electromagnetic fields for large rshowed

that the asymptotic behavior of all of them is exactly the same as the linear Maxwell field.

Then, we have solved the gravitational field equations and obtained asymptotic AdS solutions witha curvature singularity atr= 0. We have found that depending on the values of the nonlinearity pa-rameter, , the singularity is covered with various horizons. To state the matter differently, we shouldnote that one can obtain two nonlinearity values, c(critical value) and ext (extreme value), in whichthe singularity covered with a non-extreme horizon for < c, two horizons when c< < ext, anextreme horizon for = ext, andthere is a naked singularity otherwise. We have plotted some figuresfor more clarifications. Finally, we should remark that the (3+1)-dimensional nonlinear charged blackholes described here have similar asymptotic properties to RN black holes which are well studied.

It would be interesting to carry out a study of the thermodynamic and dynamic properties ofthe mentioned solutions. Also, it is worthwhile to generalize these four dimensional solutions to therotating and higher dimensional cases. We leave these problems for future works.

Acknowledgments

We would like to thank an anonymous referee for his/her constructive comments. We are indebtedto H. Mohammadpour for reading the manuscript. We also wish to thank Shiraz University ResearchCouncil. This work has been supported financially by Research Institute for Astronomy & Astrophysicsof Maragha (RIAAM), Iran.

Appendix

In order to complete our discussion, we may solve the following integral r2 ln ( 1) dr= q

3/2( 1)1/423/43/2

14

3F

1

4,

1

4,

11

4

,

5

4,

5

4

,

1 2

1425

( 1)F

5

4,

5

4,

11

4

,

9

4,

9

4

,

1 2

[4 + 3 ln( 1)]9( 1) F

34

,7

4

,

1

4

,

1 2

+ [4 + ln( 1)] F14 , 74 , 54 , 1 2 .

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