aa r X i v : . [ h e p - t h ] M a y Asymptotic charged BTZ black hole solutions
S. H. Hendi ∗ Physics Department and Biruni Observatory, Shiraz University, Shiraz 71454, IranResearch Institute for Astrophysics and Astronomy ofMaragha (RIAAM), Maragha, Iran, P.O. Box 55134-441
The well-known (2 + 1)-dimensional Reissner-Nordstr¨om (BTZ) black hole can be general-ized to three dimensional Einstein-nonlinear electromagnetic field, motivated from obtaininga finite value for the self-energy of a pointlike charge. Considering three types of nonlin-ear electromagnetic fields coupled with Einstein gravity, we derive three kinds of black holesolutions which their asymptotic properties are the same as charged BTZ solution. In addi-tion, we calculate conserved and thermodynamic quantities of the solutions and show thatthey satisfy the first law of thermodynamics. Finally, we perform a stability analysis in thecanonical ensemble and show that the black holes are stable in the whole phase space.
I. INTRODUCTION
Nonlinear field theories are of interest to different branches of mathematical physics becausemost physical systems are inherently nonlinear in nature. Nonlinear action of Abelian gaugetheories have been considered in the context of superstring theory. In fact, it has shown that [1]all order loop corrections to gravity may be considered as a Born-Infeld (BI) type Lagrangian [2].Also, the dynamics of D-branes and some soliton solutions of supergravity are governed by theBI action [3]. The first attempt to relate the nonlinear electrodynamics (NLEDs) and gravity wasmade by Hoffmann [4]. Considering NLEDs coupled to the gravitational field (with or withoutscalar field) may lead to black hole solutions with interesting properties [5–9].In addition to the nonlinear BI, other types of NLEDs have been studied in a number of papers[6, 9]. It is known that the NLEDs proposed by Born and Infeld had the aim of obtaining a finitevalue for the self-energy of a point-like charge. So in this paper, we consider three kinds of BI-typefields with the following motivations:First, modifying the linear Maxwell theory to NLED theory may eliminate the problem ofdivergency in electromagnetic field.Second, it is notable that one can find regular black hole solutions of the Einstein field equationscoupled to a suitable NLEDs [9–14]. Also, an interesting property which is common to all the NLED ∗ email address: [email protected] models is the fact that these models satisfy the zeroth and first laws of black hole mechanics [15].Third, it is also remarkable that, BI-type theories are singled out among the classes of NLEDsby their special properties such as the absence of shock waves, birefringence phenomena [16] andenjoying an electric-magnetic duality [17].Fourth, the appropriate world-volume dynamics on a curved D U (1) gauge theory isremarkable [14, 18–20].Fifth, from the point of view of AdS/CFT correspondence in hydrodynamic models, it has beenshown that, unlike gravitational correction, higher-derivative terms for Abelian fields in the form ofNLED do not affect this ratio [21–24]. In addition, in applications of the AdS/CFT correspondenceto superconductivity, NLED theories make crucial effects on the condensation [25] as well as thecritical temperature of the superconductor and its energy gap [26, 27].Sixth, it has been shown that [28, 29] the effects of NLED become important when we investigatesuper-strongly magnetized compact objects, such as pulsars, neutron stars, magnetars and strangequark magnetars. In addition, it has proved that [28] if one consider a NLED to incorporate into thephoton dynamics, Gravitational Redshift depends on the magnetic field pervading the pulsar (whilethe Gravitational Redshift is independent of any background magnetic field in general relativity).Also, since the Gravitational Redshift of magnetized compact objects is connected to the mass–radius relation of the objects, it is important to note that NLED affects on the mass–radius relationof the objects.Motivated by the recent results mentioned above, we study black hole solutions in Einsteingravity with negative cosmological constant coupled to NLED theory. Considering these nonlin-ear fields in (2 + 1)-dimensional spacetime, with asymptotic BTZ (Banados–Teitelboim–Zanelli)behavior, help us to find a simple mechanism for analyzing the properties of the solutions.Physicists believe that one of the great achievements in gravity is discovery of the (2 + 1)-dimensional BTZ black hole solutions [30–32]. In fact, BTZ black holes provide a simplified modelto investigate and find some conceptual issues such as black hole thermodynamics [33–35], quantumgravity, string and gauge theory and specially, in the context of the AdS/CFT conjecture [36, 37].Furthermore, BTZ solutions perform a central role to improve our perception of gravitationalinteraction in low dimensional spacetime [38]. Generalization of BTZ black hole and its propertiesto higher dimensions and also its near-horizon solutions have been studied in [39–47].Since in this paper, we consider three classes of the Einstein-NLED field and expect to obtainasymptotic BTZ black hole, we want to discuss about two significant properties of charged BTZsolutions. First, it is notable that for (2 + 1)-dimensional charged BTZ solutions, A t and theelectromagnetic field are proportional to ln r and r − , respectively. Second, the charge term of thelaps function in horizon flat (2 + 1)-dimensional BTZ solutions, is a logarithmic function of r .Organization of the paper is as follows: at first, we give a brief review of the field equationsof Einstein gravity sourced by the NLED field. Then we consider three dimensional spacetimeand find relative solutions. After that we investigate their properties, especially singularity andasymptotic behavior of them. Then, we obtain conserved and thermodynamic quantities of theblack holes, in which satisfy the first law of thermodynamics. Also, we analyze the local stability incanonical ensemble and at last we confirm that obtained solutions are asymptotic BTZ. We finishour paper with some conclusions. II. (2 + 1) -DIMENSIONAL BLACK HOLES WITH NLED FIELD
The (2 + 1)-dimensional action of Einstein gravity with NLED field in the presence of cosmo-logical constant is given by I G = − π Z M d x √− g [ R −
2Λ + L ( F )] − π Z ∂ M d x √− γK, (1)where R is the Ricci scalar, Λ refers to the negative cosmological constant which in general isequal to − /l for asymptotically anti-deSitter solutions, in which l is a scale length factor. InEq. (1), L ( F ) is the Lagrangian of NLED field. Here, we consider three classes of Born-Infeld-like NLED fields, namely Born-Infeld nonlinear electromagnetic (BINEF), Exponential form ofnonlinear electromagnetic field (ENEF) and Logarithmic form of nonlinear electromagnetic field(LNEF) in which their Lagrangians are L ( F ) = β (cid:16) − q F β (cid:17) , BINEF β (cid:16) exp( − F β ) − (cid:17) , ENEF − β ln (cid:16) F β (cid:17) , LNEF . (2)In this equation, β is called the nonlinearity parameter, the Maxwell invariant F = F µν F µν inwhich F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic field tensor and A µ is the gauge potential.It is natural to expect that the nonlinear electromagnetic field appears as an effective theoryof string/M—theory. For instance, one of the subgroup of the E × E or SO (32) gauge group is U (1). Ignoring all other gauge fields leaves us with the effective quartic order of U (1) Lagrangian,( F µν F µν ) [48–51]. In addition, Natsuume [52] considered the next order correction terms inthe heterotic string effective action of a magnetically-charged black hole [53] and obtained the( F µν F µν ) term as a subset of all possible α corrections to the bosonic sector of supergravity,which is the same order as the Gauss-Bonnet term [48] L cor = α h a (cid:0) R µνρσ R µνρσ − R µν R µν + R (cid:1) + b ( F µν F µν ) i . (3)Furthermore, calculating one-loop approximation of QED, it has shown that [54] the effectiveLagrangian is given by L eff = c ( F µν F µν ) + d ( F µν F µν ) . (4)Euler and Heisenberg have shown that the correction contain logarithmic form in the electro-magnetic field strength appear in the calculation of exact 1-loop corrections for electrons in auniform electromagnetic field background [55], which is description of vacuum polarization effects.So, these corrections are a kind of effective actions in quantum electrodynamics.Furthermore, logarithmic form of the electrodynamic Lagrangians, like BI electrodynamics,removes divergences in the electric field. Although the exponential form of BI-like NLED does notcancel the divergency of electric field at r = 0, but its singularity is too much weaker than in, e.g.,Einstein-Maxwell gravity.From the cosmological point of view, these BI-like NLEDs have also been used to explain anequation of state of radiation for inflation [56]. As an example of a BI-like Lagrangian with alogarithmic term, four dimensional asymptotically flat solutions of Einstein gravity was discussedin [9]. Expanding L ( F )’s for large values of β , one can write L ( F ) | Large β = −F + F β − F β + F β + O (cid:16) β (cid:17) , BINEF −F + F β − F β + F β + O (cid:16) β (cid:17) , ENEF −F + F β − F β + F β + O (cid:16) β (cid:17) , LNEF (5)which confirm that L ( F )’s reduce to the standard Maxwell form L ( F ) = −F , for β −→ ∞ andalso the leading first correction of Maxwell theory has ( F µν F µν ) form.Furthermore, we should note that the second integral in Eq. (1) is the Gibbons-Hawking surfaceterm [57] which is chosen such that the variational principle is well defined. The factor K is thetrace of the extrinsic curvature K ab of any boundary ∂ M of the manifold M , with induced metric γ ab . Although three dimensional solution of Einstein-BI gravity has been investigated before [58],but we present it again with two motivations: firstly, one can confirm that our BI solution is morecompact and we discuss about the geometry of BI black hole and its horizon precisely, and secondly,in order to compare the solutions of the exponential and logarithmic Lagrangian with BI solution,we need to present BI solution here.Varying the action (1) with respect to the gravitational field g µν and the gauge field A µ , thefield equations are obtained as R µν − g µν ( R − αT µν , (6) ∂ µ (cid:0) √− gL F F µν (cid:1) = 0 , (7)where T µν = 12 g µν L ( F ) − F µλ F λν L F , (8)and L F = dL ( F ) d F . Our main aim here is to obtain charged static black hole solutions of the fieldequations (6) - (8) and investigate their properties. We assume (2 + 1)-dimensional metric has thefollowing form ds = − f ( r ) dt + dr f ( r ) + r dθ (9)Using the gauge potential ansatz A µ = h ( r ) δ µ in the NLED fields equation (7) leads to thefollowing differential equations rh ′′ ( r ) + h ′ ( r ) (cid:20) − (cid:16) h ′ ( r ) β (cid:17) (cid:21) = 0 , BINEF r h β h ′ ( r ) i h ′′ ( r ) + h ′ ( r ) = 0 , ENEF r (cid:20) (cid:16) h ′ ( r )2 β (cid:17) (cid:21) h ′′ ( r ) + h ′ ( r ) (cid:20) − (cid:16) h ′ ( r )2 β (cid:17) (cid:21) = 0 , LNEF , (10)with the following solutions h ( r ) = q × (cid:2) r l (1 + Γ) (cid:3) , BINEF βr √ L W q + Ei (cid:16) , L W (cid:17) + ln (cid:16) q l β (cid:17) + γ − , ENEF2 ln (cid:2) r l (1 + Γ) (cid:3) − r β q (1 − Γ) − , LNEF , (11)where Γ = q q r β , q is an integration constant which is related to the electric charge of the blackhole. In addition, L W = LambertW ( q r β ) which satisfies LambertW ( x ) exp [ LambertW ( x )] = x , γ = γ (0) ≃ . Ei (1 , x ) = ∞ R e − xz z dz (for more details, see [59]). FIG. 1: F tr versus r for l = 1, q = 1 and β = 5 (Bold line), β = 10 (solid line), β = 15 (dashed line) and β −→ ∞ (BTZ solution) (dotted line). ”BINEF (left), ENEF (middle) and LNEF (right)” It is easy to show that the non-vanishing components of the electromagnetic field tensor can bewritten in the following form F tr = qr × Γ − , BINEF rβ √ L W q , ENEF β r q (Γ − , LNEF . (12)Considering Fig. 1, it is interesting to note that all three types of the mentioned electromagneticfields have finite values near the origin and they vanish at large values of r , as it should be.In addition, this figure shows that the effect of nonlinearity parameter, β , on the strength ofelectromagnetic fields is more considerable for small values of distances. Furthermore, Fig. 2shows that the mentioned NLED fields have different values for the fixed parameters and one maythink they have finite values at the origin ( r → F ENEFtr divergesat r = 0 (see table A for more details). r = 10 − − − − − − r −→ F BINEFtr .
993 5 .
000 5 .
000 5 .
000 5 .
000 5 .
000 5 . F ENEFtr .
941 15 .
91 23 .
16 37 .
33 53 .
18 169 . ∞ F LNEFtr .
512 9 .
999 10 .
00 10 .
00 10 .
00 10 .
00 10 . F BT Ztr ∞ Table A: F tr for β = 5, q = 1, l = 1 and small values of r .To find the metric function f ( r ), one may use any components of Eq. (6). Considering the FIG. 2: F tr versus r in ENEF (Bold line), BINEF (solid line), LNEF (dashed line) and BTZ solution (dottedline) for l = 1, q = 1 and β = 5. function h ( r ), the nontrivial independent components of the field equation, (6), are f ′ ( r ) − rl − rβ (1 − Γ) = 0 ,f ′′ ( r ) − l − β (cid:0) − Γ − (cid:1) = 0 . , BINEF f ′ ( r ) − rl + rβ h − Γ √ L W (1 − L W ) i = 0 ,f ′′ ( r ) − l + β (cid:16) − Γ √ L W (cid:17) = 0 . , ENEF f ′ ( r ) − rl + 8 rβ ln (cid:20) − (cid:16) βr (1 − Γ) q (cid:17) (cid:21) + β r (1 − Γ) q (cid:20) − (cid:16) βr (1 − Γ) q (cid:17) (cid:21) = 0 ,f ′′ ( r ) − l + 8 β ln (cid:18) − (cid:16) βr (1 − Γ) q (cid:17) (cid:19) = 0 . , LNEF . (13)The solutions of Eq. (13) can be written as f ( r ) = r l − M + r β (1 − Γ) + q h − (cid:16) r (1+Γ)2 l (cid:17)i , BINEF βrq (1 − L W ) √ L W − β r + q h ln (cid:16) β l q (cid:17) − Ei (cid:16) , L W (cid:17) − γ + 3 i , ENEF4 β r (cid:2) ln (cid:0) Γ+12 (cid:1) + 3 (cid:3) − q h ln (cid:16) β r (Γ − q l (cid:17) + − − i , LNEF , (14)where M is the integration constant which is related to mass parameter. A. Properties of the solutions
It is easy to show that for the metric (9), the Ricci and the Kretschmann scalars are R = − f ′′ ( r ) − f ′ ( r ) r (15) R µνρσ R µνρσ = f ′′ ( r ) + 2 (cid:18) f ′ ( r ) r (cid:19) , (16)where prime and double primes denote first and second derivative with respect to r , respectively.Also one can show that other curvature invariants (such as Ricci square) are functions of f ′′ and f ′ /r and so it is sufficient to study the Ricci and the Kretschmann scalars for the investigation ofthe spacetime curvature. Considering Eq. (14), one can expand the Ricci and the Kretschmannscalars near the origin R = ζ r + O ( r ) , BTZ ζ r + O ( r ) , BINEF ζ √ L W r + ζ r √ L W + O ( r ) , ENEF ζ r + ζ ln r + O ( r ) , LNEF (17) R µνρσ R µνρσ = ξ r + ξ r + O ( r ) , BTZ ξ r + ξ r + O ( r ) , BINEF ξ L W r + ξ r + ξ r L W + ξ √ L W r + ξ r √ L W + O ( r ) , ENEF ξ r + ξ ln rr + ξ r + ξ (ln r ) + ξ ln r + O ( r ) , LNEF (18)where ζ i ’s and ξ i ’s are functions of β , l and q . We should note that the functions L W and √ L W go to infinity for r −→
0, but much weaker than r − and r − . So it is easy to find thatlim r −→ + R = ∞ , (19)lim r −→ + R µνρσ R µνρσ = ∞ , (20)which confirm that, like BTZ black hole, the metric given by Eqs. (9) and (14) has an essentialtimelike singularity at r = 0. We should note that the singularity strength is different for BTZ,BINEF, ENEF and LNEF solutions and also the nonlinearity of electromagnetic field reduces thestrength of singularity.In order to consider the asymptotic behavior of the solution, we calculate the Ricci and theKretschmann scalars for large values of r lim r −→∞ R = − l + 2 q r + χq β r + O (cid:18) r (cid:19) , (21)lim r −→∞ R µνρσ R µνρσ = 12 l − q r l + q (cid:0) β l − χ (cid:1) β l r + O (cid:18) r (cid:19) , (22) FIG. 3: f ( r ) versus r for l = 1, q = 1, M = 1 .
5, and β = 1 (two horizons: left figure) and β = 0 . f ( r ) versus r for l = 1, q = 1, β = 2 and M > M ext (Bold line), M = M ext (solid line) and M < M ext (dashed line), where M ext = 0 .
94 for BINEF, M ext = 0 . M ext = 0 .
96 for LNEF. ”BINEF(left), ENEF (middle) and LNEF (right)” where χ = 2, 8 and 1 for BINEF, ENEF and LNEF, respectively. Equations (21) and (22) confirmthat the asymptotic behavior of the obtained solutions is AdS. In addition, it is easy to find thatthe laps function of charged BTZ black hole is positive for both r −→ r −→ ∞ and thereforedepend on the values of the metric parameters, one can obtain a black hole with two horizons, anextreme black hole and a naked singularity. But for the nonlinear charged solutions, (Eq. (14)),we should not that depends on the value of the nonlinearity parameter, β , the function f ( r ) iszero, positive or negative near the origin (see Figs. 3 and 4 for more details). In other word, onecan find that for r −→ ∞ , the lapse function (Eq. (14)) is positive but near the origin ( r −→ β > β c , β = β c and β < β c ,respectively. Considering lim r −→ + f ( r ) = 0, we may find β c as a function of l , q and M . It is sointeresting to note that, unlike charged BTZ black hole, the metric function of nonlinear chargedblack hole solutions (Eq. (14)) behaves like as uncharged (Schwarzschild) solution (Fig. 3, right),for β < β c . In other word, for β < β c , the function f ( r ) has one real positive root at r = r + , where f ′ ( r = r + ) = 0. B. Conserved and thermodynamics quantities
The Hawking temperature of the black hole on the outer horizon r + , may be obtained throughthe use of the definition of surface gravity, T + = β − = 12 π r −
12 ( ∇ µ χ ν ) ( ∇ µ χ ν )where χ = ∂/∂t is the Killing vector. One obtains T + = r + l Γ + + q ( Γ − − β l ) r + β l (1+Γ + ) , BINEF h r + (cid:0) l − β (cid:1) + 2 qβ p L W + (cid:16) L W + − (cid:17)i , ENEF r + l + q ln (cid:18) exp(1) ( ) (cid:19) r + Γ + (Γ + − − q +4 β r ln (cid:18) exp(2) ( ) (cid:19) r + Γ + , LNEF (23)where Γ + = r q r β and L W + = LambertW ( q β r ).The electric potential U , measured at infinity with respect to the horizon, is defined by [60] U = A µ χ µ | reference − A µ χ µ | r = r + = (24) − q × (cid:2) r + l (1 + Γ + ) (cid:3) , BINEF √ L W + βr + q + Ei (cid:16) , L W + (cid:17) + ln (cid:16) q l β (cid:17) + γ − , ENEF2 ln (cid:2) r + l (1 + Γ + ) (cid:3) − r β q (1 − Γ + ) − , LNEF . (25)where in the reference, U should vanish.More than thirty years ago, Bekenstein argued that the entropy of a black hole is a linearfunction of the area of its event horizon, which so called area law [61]. Since the area law of theentropy is universal, and applies to all kinds of black holes in Einstein gravity [61, 62], thereforethe entropy of the obtained charged black hole solutions is equal to one-quarter of the area of thehorizon, i.e., S = πr + . (26)1The electric charge of the black holes, Q , can be found by calculating the flux of the electromagneticfield at infinity, yielding Q = πq , (27)for all three types of the mentioned NLED fields.The present spacetime (9), have boundary with timelike ( ξ = ∂/∂t ) Killing vector field. It isstraightforward to show that for the quasi-local mass, we can write M = Z B dϕ √ σT ab n a ξ b = π M. (28)Here, we check the first law of thermodynamics for our solutions. We obtain the mass as afunction of the extensive quantities S and Q . One may then regard the parameters S , and Q as acomplete set of extensive parameters for the mass M ( S, Q ) M ( S, Q ) = S πl + β S (1 − Υ) π + Q h − (cid:16) S (Υ+1) πl (cid:17)i π , BINEF βSQ π √ Π − β S π + Q (cid:20) − γ − EI − βS √ Π Q +ln (cid:18) π β l Q (cid:19)(cid:21) π , ENEF β S [ ( Υ+12 )] π + Q (cid:18) − − − ln (cid:20) βS √ Υ2 − πlQ (Υ+1) − (cid:21)(cid:19) π , LNEF , (29)where Υ = r (cid:16) QSβ (cid:17) , EI = Ei (cid:0) , Π2 (cid:1) and Π = LambertW ( Q β S ). Now, we should differentiate M ( S, Q ) to obtain d M ( S, Q ) = (cid:18) ∂ M ∂S (cid:19) Q dS + (cid:18) ∂ M ∂Q (cid:19) S dQ, (30)where (cid:18) ∂ M ∂S (cid:19) Q = (Υ+1)+ Q β S ( − l β Υ ) πl S Υ(1+Υ) , BINEF − βQl (Π − S √ Π [ l β − ] πl √ Π , ENEF Sβ [ Υ − Υ ] h ln ( Υ2 ) + l β +2 i π Υ(Υ − , LNEF , (31)and (cid:18) ∂ M ∂Q (cid:19) S = − Q ( Υ +Υ ) ln [ Sπl (1+Υ) ] π Υ(1+Υ) , BINEF Q (cid:20) − γ − EI +ln (cid:18) π β l Q (cid:19)(cid:21) − βS √ Π π , ENEF Q (cid:20) S β Q − ln (cid:18) βS √ Υ2 − πlQ (cid:19) − − (cid:21) πS β (Υ − , LNEF , (32)2At this point, we should replace S and Q from Eqs. (26) and (27), and rewrite (cid:0) ∂ M ∂S (cid:1) Q and (cid:16) ∂ M ∂Q (cid:17) S which are the same as Eqs. (23) and (25), respectively. In other word, we could define the intensiveparameters conjugate to extensive quantities S and Q . These quantities are the temperature andthe electric potential T = (cid:18) ∂ M ∂S (cid:19) Q , U = (cid:18) ∂ M ∂Q (cid:19) S , (33)where the intensive quantities calculated by Eq. (33) coincide with Eqs. (23) and (25). Thus,these quantities satisfy the first law of thermodynamics d M = T dS + U dQ. (34)
C. Thermodynamic stability in the canonical ensemble
Now, we investigate the thermodynamic stability of nonlinear charged black hole solutions in thecanonical ensemble. The stability of a thermodynamic system with respect to the small variationsof the thermodynamic coordinates, can be studied in the canonical ensemble which the chargeis fixed parameter. In other word, the positivity of the heat capacity C Q = T + / ( ∂ M /∂S ) Q issufficient to ensure the local stability. It is straightforward to show that ( ∂ M /∂S ) Q is (cid:18) ∂ M ∂S (cid:19) Q = 1 πl + q πr (1+Γ + )Γ + , BINEF β π (cid:18) e LW +2 − (cid:19) , ENEF β π ln (cid:16) + (cid:17) , LNEF . (35)It is clear to find that ( ∂ M /∂S ) Q is positive for BINEF and LNEF branches. Considering ENEFbranch, one may find that e LW +2 ≥ ∂ M /∂S ) Q is positive (see Fig. 5 for moredetails). Since the temperature of the black hole solutions is positive, it is clear that the blackholes are stable in the canonical ensemble. III. ASYMPTOTIC BTZ SOLUTIONS
Here we would like to find that for large distance ( r >> r which3 FIG. 5: e LambertW ( x )2 versus x , which shows that e LambertW ( x )2 ≥ x ≥ leads to h ( r ) | Large r = q ln (cid:16) rl (cid:17) + q r β − q r β + O (cid:0) r (cid:1) , BINEF q r β − q r β + O (cid:0) r (cid:1) , ENEF q r β − q r β + O (cid:0) r (cid:1) , LNEF . (36)It is easy to find that for large r , the dominant (first) term of h ( r ) for all the mentioned NLEDfields is the same as one in BTZ solution [30, 31]. Differentiating from Eq. (36) or expanding Eq.(12) for large distance, one can obtain F tr = qr + − q β r + q β r + O (cid:0) r (cid:1) , BINEF − q β r + q β r + O (cid:0) r (cid:1) , ENEF − q β r + q β r + O (cid:0) r (cid:1) , LNEF , (37)which its dominant (first) term is similar to the electrical field of (2 + 1)-dimensional Reissner-Nordstr¨om black hole.Now, we focus on the metric function. Straightforward calculations show that expansion of f ( r )4in Eq. (14) for r >>
1, leads to the following equation f ( r ) = r l − M − q ln (cid:16) rl (cid:17) + − q r β + q r β + O (cid:0) r (cid:1) , BINEF − q r β + q r β + O (cid:0) r (cid:1) , ENEF − q r β + q r β + O (cid:0) r (cid:1) , LNEF . (38)One may ignore small charge terms to find laps function of (2 + 1)-dimensional BTZ black hole. IV. CONCLUSIONS
Considering the nonlinear electromagnetic fields in various sciences is one of the interests butwith cumbersome calculations. In this paper, we introduced three kinds of NLED theories whichin the weak field approximation (large values of nonlinearity parameter: β −→ ∞ ) become theusual linear Maxwell theory. In addition, as we have shown, presented solutions have the followingproperties:First, obtained electromagnetic fields have regular behavior for large values of distance andthey are finite near the origin. In addition, we showed that the effect of nonlinearity parameter,on the strength of electromagnetic fields is more considerable for small values of distances. It is sointeresting that F ENEFtr diverges at r = 0, but its divergency is very slower than the electromagneticfield of BTZ solution.Second, all obtained solutions have a timelike curvature singularity at r = 0, and they areasymptotic AdS. In other word, the NLED fields have no effect on the existence of singularity andasymptotic behavior, but we should note that the nonlinearity of electromagnetic field reduces thestrength of singularity. Furthermore, for small values of the nonlinearity parameter, ( β < β c ), thesingularity covered with a non-extreme horizon. In other word, in this case the horizon geometryof nonlinear charged black holes is close to the horizon of uncharged (Schwarzschild) black holesolution.Third, obtained solutions have different temperature and electric potential, but the same entropyand electric charge. 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