Four-Dimensional Black Holes with Scalar Hair in Nonlinear Electrodynamics

Eur. Phys. J. C (2016) 76:677 DOI 10.1140/epjc/s10052-016-4526-6

Regular Article - Theoretical Physics

Four-dimensional black holes with scalar hair in nonlinear electrodynamics

José Barrientos1,2,a, P. A. González3,b, Yerko Vásquez4,c 1 Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile 2 Departamento de Enseñanza de las Ciencias Básicas, Universidad Católica del Norte, Larrondo 1281, Coquimbo, Chile 3 Facultad de Ingeniería, Universidad Diego Portales, Avenida Ejército Libertador 441, Casilla 298-V, Santiago, Chile 4 Departamento de Física y Astronomía, Facultad de Ciencias, Universidad de La Serena, Avenida Cisternas 1200, La Serena, Chile

Received: 14 June 2016 / Accepted: 21 November 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract We consider a gravitating system consisting of 1 Introduction a scalar field minimally coupled to gravity with a self- interacting potential and a U(1) nonlinear electromagnetic Hairy black holes are interesting solutions to Einstein’s the- field. Solving analytically and numerically the coupled sys- ory of gravity and also to certain types of modified gravity tem for both power-law and Born–Infeld type electrodynam- theories. The first attempts to couple a scalar field to gravity ics, we find charged hairy black hole solutions. Then we was done in an asymptotically flat spacetime finding hairy study the thermodynamics of these solutions and we find black hole solutions [1–3], but it was realized that these solu- that at a low temperature the topological charged black hole tions were not physically acceptable as the scalar field was with scalar hair is thermodynamically preferred, whereas the divergent on the horizon and stability analysis showed that topological charged black hole without scalar hair is thermo- they were unstable [4]. Also, asymptotically flat black holes dynamically preferred at a high temperature for power-law with scalar field minimally coupled to gravity were found electrodynamics. Interestingly enough, these phase transi- in [5,6], which evade the no hair theorems by allowing par- tions occur at a fixed critical temperature and do not depend tially negative self-interacting potential, which is in conflict on the exponent p of the nonlinear electrodynamics. with the dominant energy condition. Some of these solutions were found to be stable for some parameter range [6]. On the other hand, by introducing a cosmological constant, hairy black hole solutions with a minimally coupled scalar field Contents and a self-interaction potential in asymptotically dS space were found, but unstable [7,8]. Also, a hairy black hole configuration was reported for a scalar field non-minimally cou- 1 Introduction ...... pled to gravity [9], but a perturbation analysis showed the 2 Four-dimensional black holes with scalar hair instability of the solution [10,11]. In the case of a negative in nonlinear electrodynamics ...... cosmological constant, stable solutions were found numer- 2.1 Power-law electrodynamics ...... ically for spherical geometries [12,13] and an exact solu- 2.2 Born–Infeld type electrodynamics ...... tion in asymptotically AdS space with hyperbolic geome- 3 Thermodynamics ...... try was presented in [14]. A study of general properties of 4 Conclusions ...... black holes with scalar hair with spherical symmetry can be Appendix A: Analytical solutions ...... found in Ref. [15]. Further hairy solutions were reported in References ...... [16–24] with various properties. Furthermore, charged hairy solutions were also found, for instance in [25], a topological black hole dressed with a conformally coupled scalar field and electric charge was studied. An electrically charged black hole solution with a scalar field minimally coupled to grav- a e-mail: [email protected] ity and electromagnetism was presented in [26]. Recently, b e-mail: [email protected] for a gravitating system consisting of a scalar field mini- c e-mail: [email protected] 123 677 Page 2 of 14 Eur. Phys. J. C (2016) 76:677 mally coupled to gravity with a self-interacting potential and associated with each of them. Therefore, it is necessary to U(1) electromagnetic field, exact charged hairy black hole find the conserved charge of the theory. It is worth men- solutions with the scalar field which is regular outside the tioning that the phase-transition phenomena have been ana- event horizon have been found in [27–29]. Also, new hairy lyzed and classified by exploiting Ehrenfest’s scheme [78– black hole solutions, boson stars and numerical rotating hairy 81]. Another point of view to study phase transitions is to black hole solutions were discussed [30–35], as well as time- consider Bragg–Williams’ construction of a free energy func- dependent hairy black holes [36,37]. For a review of hairy tion [82]. Also, it was shown that if the space is flat, then the black holes we refer the reader to [38]. Reissner–Nordström black hole is thermodynamically pre- In this work, we extend our previous work [20,27] and we ferred, whereas if the space is AdS the hairy charged black consider a gravitating system consisting of a scalar field min- hole is thermodynamically preferred at a low temperature imally coupled to gravity with a self-interacting potential and [27]. U(1) nonlinear electromagnetic field. Then we obtain black The work is organized as follows. In Sect. 2 we present the hole solutions for power-law and Born–Infeld type electro- general formalism. Then we derive the field equations and dynamics and we study the thermodynamics and the phase we find hairy black hole solutions. In Sect. 3 we study the transitions between hairy charged black holes dressed with a thermodynamics of our solutions and in Sect. 4 we present scalar field and no hairy charged black holes, focusing on the our conclusions. effects of the nonlinearity of the Maxwell source. The interest in nonlinear electrodynamics arises with the order to eliminate the problem of the infinite energy of the electron by Born 2 Four-dimensional black holes with scalar hair and Infeld [39]. Also, nonlinear electrodynamics emerges in nonlinear electrodynamics in the modern context of the low-energy limit of heterotic string theory [40–42], and it plays an important role in the The four-dimensional Einstein–Hilbert action with a scalar construction of regular black hole solutions [43–48]. Some field minimally coupled to curvature having a self-interacting black holes/branes solutions in a nonlinear electromagnetic potential V (φ) in the presence of a nonlinear electromagnetic field have been investigated for instance in [49–56] and refer- field is ences therein. The thermodynamics of Einstein–Born–Infeld √ 1 1 μν black holes with a negative cosmological constant was stud- I = d4x −g R + L(F2) − g ∇μφ∇νφ − V (φ) , 2κ 2 iedin[57] and for a power-law electrodynamic in [58], where (1) the authors showed that a set of small black holes are locally stable by computing the heat capacity and the electrical per- where κ = 8πG, with G the Newton constant and L(F2) mittivity. The thermodynamics of Gauss–Bonnet black holes an arbitrary function of the electromagnetic invariant F2 = αβ for a power-law electrodynamic was studied in [59]. On Fαβ F . The resulting field equations from the above action the other hand, higher dimensional black hole solutions to are Einstein-dilaton theory coupled to Maxwell field were found 1 (φ) (F) in [60,61] and black hole solutions to Einstein-dilaton the- Rμν − gμν R = κ(Tμν + Tμν ), (2) 2 ory coupled to Born–Infeld and power-law electrodynamics (φ) (F) were found in [62,63]. where the energy-momentum tensors Tμν and Tμν for the The phase transitions have been of great interest since scalar and electromagnetic fields are the discovery of a phase transition by Hawking and Page (φ) 1 ρσ in a four-dimensional Schwarzschild AdS background [64]. Tμν =∇μφ∇νφ − gμν g ∇ρφ∇σ φ + V (φ) , 2 Witten [65] has extended this four-dimensional transition L( 2) to arbitrary dimension and provided a natural explanation (F) 2 d F λ Tμν = gμνL(F ) − 4 Fμ Fνλ, (3) of a confinement/deconfinement transition on the bound- dF2 ary field theory via the AdS/CFT correspondence. However, respectively. Using Eqs. (2) and (3) we obtain the equivalent phase transitions have recently garnered a great deal of atten- equation tion motivated mainly by the relationship between the phase 2 dL(F ) α transitions and holographic superconductivity [66,67]inthe Rμν − κ ∂μφ∂νφ + gμν V (φ) = κ −4 Fμ Fνα dF2 context of the AdS/CFT correspondence. Furthermore, the L( 2) effects of nonlinear electrodynamics on the properties of the 2 d F 2 + 2gμν F − gμνL(F ) . (4) holographic superconductors have recently been investigated dF2 [68–77]. It is well known that these phase transitions can Now, if we consider the following metric ansatz: be obtained by considering black holes as states in a same − grand canonical ensemble and by comparing the free energy ds2 =−f (r)dt2 + f 1(r)dr 2 + a2(r)d 2, (5) 123 Eur. Phys. J. C (2016) 76:677 Page 3 of 14 677

where d 2 is the metric of the spatial 2 section, which 2 1 2 =− r − ν(6α2 + )r + k − α2ν can have a positive, negative or zero curvature, and Aμ = 3 3 ( ( ), , , ) r At r 0 0 0 is the scalar potential of the electromagnetic − α r(r + ν) 2 2 ln + ν field. We find the following three independent differential r equations: r A (r) + 8Qr˜ (r + ν) t dr. (15) 2( + ν)2 a (r) ∞ r r f (r) + 2 f (r) + 2κV (φ) = 2κL(F2), (6) a(r) To find hairy black hole solutions the differential equations a (r) a (r) f (r) + 2 f (r) + 4 + 2κφ (r)2 f (r) have to be supplemented with the Klein–Gordon equation of a(r) a(r) the scalar field, which in general coordinates reads + κ (φ) = κL( 2), 2 V 2 F (7) dV 2 φ = , a (r) a (r) a (r) k φ f (r) + + f (r) − + κV (φ) d a(r) a(r) a(r) a(r)2 and whose solution for the self-interacting potential V (r) is 2 dL(F ) = κL(F2) + 4κ A (r)2 , (8) t 2 ν2 f (r) dF V (r) = dr. (16) 2 r 2(r + ν)2 where k = 1, 0, −1 parameterizes the curvature of the spatial 2-section. So, if we eliminate the potential V (φ) from the In the following sections we will consider two specific elec- above equations we obtain tromagnetic Lagrangians L(F2): power-law and Born–Infeld type electrodynamics. 1 a (r) + κφ (r)2a(r) = 0, (9) 2 2 2.1 Power-law electrodynamics a (r) a (r) 2k f − 2 + f (r) + ( ) ( ) ( )2 a r a r a r In this section we consider power-law electrodynamics char- dL(F2) acterized by the following Lagrangians [83]: =− κ A (r)2 . 8 t 2 (10) dF L(F2) = η|F2|p, (17) In the following, we will work in units where κ = 1. By considering the scalar field studied in [27], where p is a rational number and the absolute value ensures that any configuration of electric and magnetic fields can be 1 ν φ(r) = √ ln 1 + , (11) described by these Lagrangians. One could also consider the r 2 Lagrangian without the absolute value and the exponent p where ν is a parameter controlling the behavior of the scalar restricted to being an integer or a rational number with an odd field and it has the dimension of length, from Eqs. (9) and denominator [84]. The sign of the coupling constant η will (11) we determine the function a(r), which reads be chosen such that the energy density of the electromagnetic ( ) field is positive; that is, the minus T t F component of the a(r) = r(r + ν). (12) t electromagnetic energy-momentum tensor must be positive, Also, from the Maxwell equations − t (F) = η| 2|p( − )> . T t F 2p 1 0 (18) 2 √ μν dL(F ) ∂μ −gF = 0, (13) This condition is guaranteed in the following cases: p > 1/2 dF2 and η>0orp < 1/2 and η<0. In the following we focus we obtain the following relation: our attention on purely electric field configurations. L( 2) ˜ From the Maxwell equations (14) we obtain ( )d F =− Q , At r (14) dF2 a(r)2 Q A (r) = , (19) t (r(r + ν))1/(2p−1) where Q˜ is an integration constant. We can also determine the metric function f (r) replacing (12) and (14)in(10) where Q is an integration constant related to Q˜ by Q˜ = η2p−1 p(Q2)p/Q.So 2 1 2 f (r) =− r − ν(6α2 + )r + k − α2ν 3 3 (2p − 1)Qr(2−2p)/(1−2p)(1/ν)1/(−1+2p) r At (r) = 2 F1 − 2α r(r + ν)ln 2(p − 1) 2 r + ν 1 2(p − 1) 4p − 3 r 2 × , , , − + . ( + ν) ( )2 dL(F ) C (20) r r At r 2 dr 2p − 1 2p − 1 2p − 1 ν − 8r(r + ν) dF dr r 2(r + ν)2 123 677 Page 4 of 14 Eur. Phys. J. C (2016) 76:677 √ The integration constant C will be chosen in such a way the + (2 + cosh( 2φ))F(φ) electric potential goes to zero asymptotically for 1/2 < p < 3/2, and is given by 2p/(1−2p) p 1 + α1η2 √ , (26) 2(φ/ ) 2−2p 3−2p 4sinh 2 3−2p 1−2p −1+2p C =−Q(1/ν) −1+2p . (21) where we have defined 1 − + ν 1 2p √ 2p/(1−2p) 2φ r F(φ) = ν(1−4p)/(1−2p) −1+e F Thus, for r →∞the electric potential behaves as ( + ν)2 2 1 ∞ r ( − ) − − 3−2p 1 2 p 1 4p 3 r 1 2p − × , , , − dr At (r →∞) ≈ Q r 1 2p +··· , (22) − − − ν 3 − 2p 2p 1 2p 1 2p 1 1√ /( − ) φ 2p 1 2p −1+e 2 r where ... denotes terms that go to zero at large distances. = 2 F1 ∞ (r + 1)2 Equation (22) shows that the field At tends to zero at infinity 1 2(p − 1) 4p − 3 for 1/2 < p < 3/2 and diverges at infinity for p < 1/2 × , , , −r dr. (27) and p > 3/2. However, the above solution is not valid for 2p − 1 2p − 1 2p − 1 = /( − ) > n 1 2p 1 with n 1 being an integer. Fortunately, in The integral above apparently depends on ν; however, it can that case we obtain analytical solutions that we present in the be shown numerically that the integral does not depend on ν; ( ) appendix. We can also determine the metric function f r so, in the last line of the above expression we have set ν = 1 replacing (20)in(15). We find for simplicity. Therefore, the potential V (φ) does not depend on the parameter ν. We have used the reverse type procedure ( ) =− 2 − 1ν( α + ) + − α ν2 f r r 6 2 r k 2 to obtain the potential, chosen in a way that the solution found 3 3 from the remaining equations of motion with a metric ansatz r − 2α r(r + ν)ln and an electric field also solves the Klein–Gordon equation. 2 r + ν Then we can investigate whether our system has a charged 2α η2p p(2p − 1)ν(1−4p)/(1−2p) + 1 r(r + ν) hairy black hole solution. In Fig. 1 we plot the behavior of p − 1 ( ) (φ) the metric function f r and the potential V in Fig. 2, r r 2p/(1−2p) for a choice of parameters ν = 0.5, =−1, α = 1, × F 1 ( + ν)2 2 1 α = 1, p = 2, 3, 5, and k =±1, 0. Also, in Fig. 3 we plot ∞ r 2 1 2(p − 1) 4p − 3 r the potential for other choices of the parameters. The metric × , , , − dr, (23) ( ) 2p − 1 2p − 1 2p − 1 ν function f r changes sign for low values of r, signaling the presence of a horizon, while the potential tends to = V (0) where (V (0)<0) as can be seen in Figs. 2 and 3. We observe a

2−2p 3−2p different behavior of the potential, while it is bounded from − − + α = α − α η p 1 2 p 1 2p , below in Fig. 3, this does not occur in Fig. 2.Itisworthto 2 2 4 1 2 p (24) 1 mention that potentials with similar behavior have been con- − + 1 2p sidered for instance in [17,20,27,29]. However, note that in and the following redefinition of the integration constant Q Fig. 3 the shift of the minimum of the potential depends on has been taken into account: the parameter p. Additionally, in Fig. 4 we plot the metric function f (r) for other choices of the parameters, and we α1 (Q2)p = . (25) observe that for certain values of the parameters the met- (ν2)2p/1−2p ric can describe a black hole with two horizons, r+ and r−. The scalar field potential can be written as Moreover, the metric can describe an extremal black hole = √ √ with degenerate horizons when r+ r−. Note that the full V (φ) = (2 + cosh( 2φ)) + 2α (− 2φ[2 range of the r coordinate covers all the values of a(r) from 3 2 √ √ the center r = rc, where a(rc) = 0, to infinity [15]. So, + cosh( 2φ)]+3sinh( 2φ)) for ν positive, the r coordinate is in the range 0 < r < ∞, √ 2p − 1 φ ( − )/( − ) while for ν negative the range is −ν

f r f r

30 30

20 20

10 10

0 r 0 r 2 4 6 8 10 2 4 6 8 10

f r

0 r 2 4 6 8 10

Fig. 1 The behavior of f (r) for ν = 0.5, =−1, α1 = 1, α2 = 1, η = 1/4, p = 5(dotted line), p = 3(dashed line)andp = 2(dot-dashed line). Left figure for k =−1, right figure for k = 0, and bottom figure for k = 1

V φ V φ 20 200

10 150 0 φ 0.5 1.0 1.5 2.0 100 10

20 50

30 0 φ 40 1 2 3 4 5

50 50

Fig. 2 The behavior of V (φ) for =−1, α1 = 1, α2 = 1, η = 1/4, Fig. 3 The behavior of V (φ) for =−1, α1 = 0.1, α2 =−5, p = 5(dotted line), p = 3(dashed line)andp = 2(dot-dashed line) η = 1/4, p = 3.5(dotted line), p = 3(dashed line)andp = 2 (dot-dashed line) cal values considered (see Fig. 1). Furthermore, there is no curvature singularity outside the horizon; therefore, the met- The electric potential is ric (23) can describe a charged hairy black hole solution for certain values of the parameters. Q 2r + ν 2Q r = / At (r) =− − ln , (29) Equation (23) has an analytical solution for p 3 4given ν2 r(r + ν) ν3 r + ν by

2 1 2 f (r) =− r − ν(6α2 + )r + k − α2ν which goes to zero at inﬁnity, and the self-interacting poten- 3 3 tial of the scalar ﬁeld is r − α r(r + ν) 2 2 ln + ν r √ 2 V (φ) = (2 + cosh( 2φ)) ν(2r + ν) + 2r(r + ν)ln r 3 − / r+ν √ √ √ + 3ηα 2 1 4 . + 2α (− 2φ[2 + cosh( 2φ)]+3 sinh( 2φ)) 1 r(r + ν) 2 √ + η 3/4α 6(φ/ ) − ηα −1/4 (28) 64 2 1 sinh 2 3 12 123 677 Page 6 of 14 Eur. Phys. J. C (2016) 76:677

f r f r 20 20

15 15

10 10

5 5

0 r 0 r 2 4 6 8 10 2 4 6 8 10

5 5 f r 20

0 r 2 4 6 8 10

Fig. 4 The behavior of f (r) for ν = 1, =−1, α1 = 1, η = 1/4, k = 0andα2 =−10 (dot-dashed line), α2 =−23 (dashed line), p = 2. Left figure for k =−1andα2 =−10 (dot-dashed line), α2 =−28 (dotted line). Bottom figure for k = 1andα2 =−10 (dot- α2 =−25 (dashed line), α2 =−32.5(dotted line). Right figure for dashed line), α2 =−20 (dashed line), α2 =−24 (dotted line)

100 100

80 80

60 60

40 40

20 20

0 r 0 r 2 4 6 8 10 2 4 6 8 10

100

0 r 2 4 6 8 10

μνρσ Fig. 5 The behavior of Kretschmann scalar Rμνρσ R (r) as a function of r for ν = 0.5, =−1, α1 = 1, α2 = 1, η = 1/4, p = 5(dotted line), p = 3(dashed line)andp = 2(dot-dashed line). Left figure for k =−1, right figure for k = 0, and bottom figure for k = 1

123 Eur. Phys. J. C (2016) 76:677 Page 7 of 14 677

f r At r 0.1 30

0.0 r 20 40 60 80 100 20

0.1

0.2

0 r 2 4 6 8 10 0.3

Fig. 6 The behavior of f (r) for ν = 1, =−1, η = 1/4, α = 1.5, ˜ 1 Fig. 7 The behavior of At (r),forb = 1, ν = 1, and Q = 1 α2 = 0.01, p = 3/4, k =−1(dot-dashed line), k = 0(dashed line) and k = 1(continuous line) ℘ ℘ √ where denotes the -Weierstrass elliptic function, with the 2 2 ×(−28 + 16φ + (31 + 8φ ) cosh( 2φ) Weierstrass invariants g2 and g3 given in (34). In Fig. 7 we √ √ √ √ ˜ − 4cosh(2 2φ) + cosh(3 2φ) − 24 2φ sinh( 2φ)). plot At (r) for b = 1, ν = 1, and Q = 1. ( ) (30) The metric function f r reads

This solution is a special case of more general analytical 2 1 2 f (r) =− r − ν(6α2 + )r + k − α2ν solutions that we show in the appendix. In Fig. 6 we show 3 3 ( ) = / r the behavior of f r for p 3 4 and different values of k. − 2α r(r + ν)ln + 8ν4α2r(r + ν) 2 r + ν 1 r −1 2 2.2 Born–Infeld type electrodynamics ℘ r(r + ν) + ν /12; g2, g3 × dr, (36) ∞ r 2(r + ν)2 In this section we consider a Born–Infeld type Lagrangian given by where we have taken into account the following redeﬁnition: ⎛ ⎞ F2 Q˜ = α ν2, (37) L(F2) = 4b2 ⎝1 − 1 + ⎠ , (31) 1 2b2 and the potential can be written as where b is the Born–Infeld coupling. In the limit b →∞ √ the Maxwell electrodynamics is recovered and in the limit V (φ) = (2 + cosh( 2φ)) b → 0 this Lagrangian vanishes. By inserting (31)in(14), 3 √ √ √ we obtain straightforwardly the electric ﬁeld + 2α2(− 2φ[2 + cosh( 2φ)]+3sinh( 2φ)) ⎛ √ ⎞ ˜ 2φ Q 2 ⎝ be ⎠ A (r) = . (32) + 4b 1 − √ √ t ˜ 2 r 2(r + ν)2 + Q α2(e 2φ − 1)4 + b2e2 2φ b2 1 √ Then, by performing the change of variable u = r(r + ν) + sinh2(φ/ 2) − 64α2 ν2/ 1 √ 12, the scalar potential reads 4(φ/ ) + α2/ 2 csch 2 16 1 b du √ = ˜ , − α2( + ( φ)) (φ) At Q (33) 8 1 2 cosh 2 F 4u3 − g u − g √ √ 2 3 − α2 2(φ/ ) ( φ) (φ), 32 1 sinh 2 sinh 2 G (38) where

1 Q˜ 2 ν2 Q˜ 2 where we have deﬁned g = ν4 − 48 , g =− 144 + ν4 . 2 2 3 2 ν 12 b 216 b √ ℘−1 ( + ν) + ν2/ ; , 2φ − r r 12 g2 g3 F(φ) = ν4 e 1 dr (34) ∞ r 2(r + ν)2 √ 1 −1 Therefore, the solution for the scalar potential is 2φ ℘ (r(r + 1) + 1/12; g , g ) = e −1 2 3 dr, ˜ −1 2 ∞ r 2(r + 1)2 At (r) = Q℘ (r(r + ν) + ν /12; g2, g3), (35) 123 677 Page 8 of 14 Eur. Phys. J. C (2016) 76:677

f r f r

30 30

20 20

10 10

0 r 0 r 2 4 6 8 10 2 4 6 8 10

f r

0 r 2 4 6 8 10

Fig. 8 The behavior of f (r) for =−1, α1 = 0.01, α2 = 1.5, b = 1, ν = 1(dotted line), ν = 2(dashed line)andν = 5(dot-dashed line). Left figure for k =−1, right figure for k = 0, and bottom figure for k = 1

ν2 √ V φ −1 2 G(φ) = ν℘ (csch (φ/ 2) + 1/3); g2, g3 4 40 √ −1 1 2 = ℘ (csch (φ/ 2) + 1/3); g2, g3 . (39) 4 20

Equation (39) apparently depends on the parameter ν;how- 0 φ 0.5 1.0 1.5 2.0 2.5 3.0 ever, it can be shown numerically that these expressions do not depend on ν; for this reason, we have set ν = 1in(39) 20 for simplicity. Therefore, the potential V (φ) does not depend on the parameter ν. 40 Then we can investigate whether our system has a charged hairy black hole solution. In Fig. 8 we plot the behavior of the Fig. 9 The behavior of V (φ) for =−1, b = 1, α = 0.01, α = 1.5 metric function f (r) and the potential V (φ) in Fig. 9,fora 1 2 (dashed line)andα2 =−1.5(dotted line) choice of parameters =−1, α1 = 0.01, α2 = 1.5, b = 1, and ν = 1, 2, 5. The metric function f (r) changes sign for low values of r signaling the presence of a horizon, while the potential tends to = V (0) (V (0)<0), as can be seen in scalar both diverge at the center rc = 1, which is an essential Fig. 9. Additionally, in Fig. 10 we plot the metric function property of these solutions, and the singularity can be located f (r) for other choices of the parameters, and we observe that either in a static region ( f (rc)>0) as the dot-dashed and for certain values of the parameters the metric can describe dashed lines illustrate in Fig. 10 or in the dynamic region a black hole with two horizons, r+ and r−. Moreover, the ( f (rc)<0) as the dotted lines show in Fig. 10. Additionally, metric can describe an extremal black hole with degener- in Fig. 11 we have plotted the behavior of the Kretschmann μνρσ ate horizons when r+ = r−. It is worth mentioning that in scalar Rμνρσ R (r) for ν = 0.5, and it is shown that it Fig. 10 we have set ν =−1. As we mentioned in the previ- is singular at the center rc = 0. The numerical results also ous section, for ν negative the full range of the r coordinate show that there is no curvature singularity outside the hori- is −ν

f r f r 15 15

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5 5

0 r 0 r 1 2 3 4 5 6 7 1 2 3 4 5 6 7

5 5 f r 15

0 r 1 2 3 4 5 6 7

Fig. 10 The behavior of f (r) for =−1, α1 = 1, ν =−1, b = 1, (dotted line), α2 =−10 (dashed line), α2 =−6.5(dot-dashed line). (dotted line), ν = 2(dashed line)andν = 5(dot-dashed line). Left Bottom figure for k = 1andα2 =−20 (dotted line), α2 =−11.5 figure for k =−1andα2 =−15 (dotted line), α2 =−8(dashed line), (dashed line), α2 =−9.5(dot-dashed line) α2 =−2.5(dot-dashed line). Right figure for k = 0andα2 =−15

R Rμνρσ (r) 1 μνρσ ds2 =−N(ρ)2g(ρ)2dt2 + dρ2 + ρ2d 2, (40) 100 g(ρ)2

80 where ρ2 f (ρ) ν2 60 N (ρ)2 = , g (ρ)2 = + ρ2 , (41) ν2 + ρ2 ρ2 4 4 40 and where f (ρ) corresponds to the metric function f (r) eval- 20 =−ν + ν2 + ρ2 uated at r 2 4 . In these coordinates the scalar field is given by 0 r 0.5 1.0 1.5 2.0 ν + ν2 + ρ2 μνρσ 1 4 Fig. 11 The behavior of Kretschmann scalar Rμνρσ R (r) as a φ(ρ) = √ ln . (42) 2 −ν + ν2 + ρ2 function of r for ν = 0.5, =−1, α1 = 1, α2 = 1, b = 1, k =−1 4 (dotted line), k = 0(dashed line)andk = 1(dot-dashed line) Now, we go to the Euclidean time t → iτ, and, for the Euclidean metric associated with (40), the Maxwell equation (13) yields 3 Thermodynamics A (ρ) dL(F2) Q˜ t =− ; (43) In this section we will study the thermodynamics of the N(ρ) dF2 ρ2 hairy black hole solutions found. To compute the conserved charges we will√ apply the Euclidean formalism, we will work here and in the following the prime denotes a derivative in the ρ = r (r + ν) coordinate, in which the metric (5) with respect to ρ. Thus, to apply the Euclidean formalism can be written in the following form: we consider the following action: 123 677 Page 10 of 14 Eur. Phys. J. C (2016) 76:677 β ∞ On the other hand, when the electric potential diverges IE =− N(ρ)H(ρ) + At P dρ + B, (44) 4π ρ+ at infinity (for p > 3/2 in power-law electrodynamics) where the variation of the integral in the metric function (15) diverges too. However, it can be verified numerically that ρ2 (ρ)2 − + ρ( (ρ)2) H(ρ) = g 1 g it cancels out exactly with the contribution coming from 2 2G ρ δBF∞ = β At (∞)δP. So, we can have well defined con- ˜ served charges in that case. 1 4A (ρ) Q + g(ρ)2φ 2 + V (φ) − L(F2) − t The variation of the fields at the horizon yields 2 N(ρ) ρ2 4π (45) δB ρ =− δρ+, (55) G + N(ρ)β is the reduced Hamiltonian which satisfies the constraint δBφρ+ = 0, (56) H(ρ) = 0 and δBFρ+ = β At (ρ+)δP. (57) 4A (ρ)ρ2 dL(F2) P(ρ) =− t , (46) Therefore N(ρ) dF2 β 3 2 δB =− α δν + 2π δρ+ + β δP. (58) which satisfies P(ρ) = 0. By using the Maxwell equation 3 2 ˜ (43) we can show that P(ρ) = 4Q is a constant. Furthermore, Thus, as the Euclidean action is related to the free energy F β = / B is a surface term, 1 T is the period of Euclidean time through IE =−β F, we obtain and finally is the area of the spatial 2 section. We now = − βM + βQ. compute the action when the field equations hold. The con- IE S (59) dition that the permitted geometries should not have conical Then this relation makes it possible to identify the mass (M), ρ singularities at the event horizon + imposes the requirement the entropy (S) and the electric charge (Q)as (ρ ) (ρ )2 N + g + 3 2 ˜ T = . (47) M = α ν , S = ρ+, Q = P = 4 Q. (60) 4π 3 2 4G So, by using the grand canonical ensemble we can fix the Thus, for power-law electrodynamics the electric charge is temperature and the electric potential =−At (ρ+). Then given by the variation of the surface term yields Q = η p−1 α(2p−1)/2pν2, (1) 4 2 p 1 (61) δB = δBG + δBφ + δBF , (48) and for the Born–Infeld type electrodynamics the electric where charge is ∞ 2 2 Q(2) = 4 α1ν . (62) δBG = β N(ρ)ρδg(ρ) , (49) ρ+ Having the temperature, mass, entropy, and electric charge ∞ 2 2 it is possible to study phase transitions between the nonlin- δBφ = β N(ρ)ρ g(ρ) φ δφ , (50) ρ+ early charged black holes with scalar hair and nonlinearly ∞ charged black holes without hair. For this analysis it is con- δBF = β At (ρ)δP . (51) venient to write the temperature of four-dimensional charged ρ+ black holes with scalar hair (with k =−1) as For the variation of the fields at large distances we must 1 1 T = − ( r+ + ( α + )ν) be careful with the integral appearing in the metric function π 2 12 2 4 3 (15). It can be shown numerically that when At tends to zero r+ 2+p 2 p at infinity (for 1/2 < p < 3/2 in power-law electrodynamics − 2α2(2r+ + ν)ln + 2 ηp(Q ) /Q r+ + ν and in Born–Infeld type electrodynamics) the integral does r+ At (r) not contribute to the conserved charges, and the variation of × (2r+ + ν) dr 2( + ν)2 the fields at large distances yields ∞ r r νρ At (r+) 2 1 + r+(r+ + ν) . (63) δB ∞ = β − − α ν + O δν, (52) 2 2 G 2 ρ r+(r+ + ν) 6 νρ 1 On the other hand, in the absence of a scalar field, the field δBφ∞ = β + O δν, (53) 6 ρ equations have as a solution topological nonlinearly charged δBF∞ = β At (∞)δP. (54) black holes, which correspond to the metric presented in 123 Eur. Phys. J. C (2016) 76:677 Page 11 of 14 677

α = Eq. (15) of Ref. [63] by setting the parameter 0 and 0.04 n = 3. The non-hairy black hole we consider here is also a generalization of the metric considered in [58], where a 0.02 thermodynamic study of spherically symmetric black holes charged with power-law electrodynamics but with no cos- F 0.00 mological constant was performed. The solution is 1 0.02 ds2 =−f˜(ρ)dt2 + dρ2 + ρ2d 2, f˜(ρ) 0.04 p( − )2( 2)p ˜(ρ) = − ρ2 − m + η 2 2p 1 q , f k 2 (64) 0.10 0.12 0.14 0.16 0.18 0.20 3 ρ − (3 − 2p)ρ 2p 1 T − 3−2p ˜ 1 2p − At (ρ) = qρ 1 2p . = − 3 − 2p Fig. 12 The behavior of F F1 F0 as a function of the temperature T with k =−1, = 1, =−3, α1 = 0.05, α2 =−1, η = 0.5 with The temperature, entropy, mass, and electric charge are given, p = 0.9(continuous line)andp = 1.2(dashed line) respectively, by 0.02 1+2p 1 1 p 2 p 1−2p TBH = − − ρ+ + η2 (1 − 2p)(q ) ρ+ , 4π ρ+ 0.01 2 S = 2π ρ+, BH ⎛ ⎞ 2 2 p 2 1−2p 2 p ⎜ ρ+ 2 (2p − 1) ρ+ (q ) ⎟ F 0.00 M =− ρ+ ⎝1 + − η ⎠ , BH 3 3 − 2p

p2p(q2)p 0.01 Q = 2η , (65) BH q 0.02 where 0.15 0.16 0.17 0.18 0.19 0.20 ( − ) T = 2p 3 . q 3−2p (66) 1−2p Fig. 13 The behavior of F = F −F as a function of the temperature (1 − 2p)ρ+ 1 0 T with k =−1, = 1, =−3, α1 = 0.05, α2 =−1, η = 0.5 with So, the horizon radius ρ+ can be written as a function of p = 2(continuous line)andp = 3(dashed line) the temperature and of the electric potential. Now, in order to find phase transitions, we must consider both black holes in the same grand canonical ensemble, i.e., at the same T function of T or only. This is similar to what happens in and . Making T and equal for both black holes and by [85] for a charged hairy black hole in linear electrodynamics. considering the free energy F, F = F(T,)= M − TS− Q, (67) 4 Conclusions we plot the difference of the free energies for the nonlinearly charged black hole (F1) and the nonlinearly charged We have considered a gravitating system consisting of black hole with scalar hair (F0), F = F1 − F0 as a func- a scalar field minimally coupled to gravity with a self- tion of the temperature in Figs. 12 and 13.InFig.12 we interacting potential and a U(1) nonlinear electromagnetic show that there is a second-order phase transition at the fixed field. We solved the coupled field equations with a profile critical temperature Tc ≈ 0.16 for low values of p, and the of the scalar field which falls sufficiently fast outside the topological nonlinearly charged hairy black hole dominates black hole horizon. For a range of values of the scalar field for temperatures lower than critical, whereas for tempera- parameter, which characterizes its behavior, we found numer- tures higher than critical the topological nonlinearly charged ically charged hairy black hole solutions in power-law and black hole is thermodynamically preferred. In Fig. 13 we Born–Infeld type electrodynamics, with the scalar field reg- show that there are no phase transitions for high values of p ular everywhere outside and on the event horizon. Also, in and the charged black hole without scalar hair is thermody- the case of power-law electrodynamics, we found analytical namically preferred. It is worth mentioning that since ν and hairy black hole solutions for some special values of the expo- Q˜ are related, the temperature and the electric potential are nent p in the range 1/2 < p < 1. Then we studied the ther- not independent; thus, we can express the free energy as a modynamics for both electrodynamics and phase transitions 123 677 Page 12 of 14 Eur. Phys. J. C (2016) 76:677 of our black hole solutions in power-law electrodynamics, × 1 (− ν2 ) j+1( ( + ν))n− j−1 and we showed that there is a second-order phase transition 2 r r for low values of p. Moreover, at a low temperature, the topo- − n1 − − + 1 2n 2i 1 r . logical nonlinearly charged hairy black hole is thermody- − ln ν2 n 1 n − i r + ν namically preferred, whereas the topological charged black ν − i=1 2 hole without scalar hair is thermodynamically preferred at (A2) a high temperature for power-law electrodynamics. Interest- ingly enough, these phase transitions occur at a fixed critical So, using this expression, we find that the metric function is temperature and do not depend on the exponent p of the given by nonlinearity electrodynamics. On the other hand, we showed that there is no phase transition for high values of p.This 2 1 2 f (r) =− r − ν(6α2 + )r + k − α2ν picture is consistent with the findings of the application of 3 3 r the AdS/CFT correspondence to condensed matter systems. − 2α r(r + ν)ln 2 r + ν In these systems there is a critical temperature below which the system undergoes a phase transition to a hairy black hole 3 + 1 (n + 1) (n+1) 1 + 2 2 2n η Q n −r(r + ν) configuration at a low temperature. This corresponds in the n ν2 ( − ) 2 n 1 boundary field theory to the formation of a condensation of ν × I 1(r; n + 1) + I 2(r; n + 1) the scalar field. 2 ⎛ ⎞ n−2 j n − i − Acknowledgements We would like to thank the anonymous referee for + ( + ν) ⎝ 2 2 1 ⎠ 1 r r + n − i − 1 ν2 j 1 valuable comments which help us to improve the quality of our paper. j=1 i=1 (n − 1) − This work was partially funded by Comisión Nacional de Ciencias y 2 ν Tecnología through FONDECYT Grants 11140674 (PAG), by Direc- × I 1(r; n − j + 1) + I 2(r; n − j + 1) ción de Investigación y Desarrollo de la Universidad de La Serena (Y.V.) 2 ⎛ ⎞ and partially supported by grants from Comisión Nacional de Investi- n−1 1 2n − 2i − 1 gación Científica y Tecnológica CONICYT, Doctorado Nacional 2016 − ⎝ ⎠ n−1 − (21160784) (J.B.O.). P.A.G. and J.B.O. acknowledge the hospitality of − ν2 ν4 = n i 2 i 1 the Universidad de La Serena, where part of this work was carried out. P.A.G. acknowledges the hospitality of the National Technical Univer- r r × ν + (r + ν)ln ν + r ln , sity of Athens. r + ν r + ν Open Access This article is distributed under the terms of the Creative (A3) Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, where we have defined and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative rdr r I 2(r; n) ≡ = Commons license, and indicate if changes were made. (r(r + ν))n ν(n − 1)(r(r + ν))n−1 Funded by SCOAP3. 2n − 3 + I 1(r; n − 1). (A4) ν(n − 1)

Appendix A: Analytical solutions = 1 + 1 > On the other hand, for p 2 2m−1 , with m 1 being 1 < < an integer number, so 2 p 1, the electric field is given In this appendix we present some analytical hairy black hole by solutions with power-law electrodynamics for some particu- = 1 + 1 > ( ) = ( ; ), lar values of the exponent p.Forp 2 2n , with n 1 At r QI3 r m (A5) 1 < < being an integer number, so 2 p 1, the electric field is given by where ν + dr 2 r At (r) = QI1(r; n), (A1) I 3(r; m) ≡ = 2m−1 ν2 2m−3 2m−3 (r(r + ν)) 2 − (r(r + ν)) 2 ⎛ 2 2 ⎞ where we have defined ν − j + r m1 m − i − ν + 2 ⎝ 2 2 2 ⎠ dr + r − − ( ; ) ≡ = 2 2m 3 2m 1 − i − 1 I 1 r n ν2 2 j=1 i=1 2 (r(r + ν))n − (n − 1)(r(r + ν))n−1 ⎛ 2 ⎞ 1 ν n−2 j × , (A6) + r − − ν2 + 2m−2 j−3 2 ⎝ 2n 2i 1⎠ (− ) j 1( ( + ν)) 2 + 2 r r (n − 1) n − i − 1 j=1 i=1 123 Eur. Phys. J. C (2016) 76:677 Page 13 of 14 677 and the metric function is 13. E. 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