Nonlinear electrodynamics and magnetic black holes
aa r X i v : . [ g r- q c ] A ug Nonlinear electrodynamics and magnetic black holes
S. I. Kruglov Department of Chemical and Physical Sciences, University of Toronto,3359 Mississauga Road North, Mississauga, Ontario L5L 1C6, Canada
Abstract
A model of nonlinear electrodynamics with two parameters, cou-pled with general relativity, is investigated. We study the magnetizedblack hole and obtain solutions. The asymptotic of the metric andmass functions at r → ∞ and r →
0, and corrections to the Reissner-Nordstr¨om solution are found. We investigate thermodynamics ofblack holes and calculate the Hawking temperature and heat capacityof black holes. It is shown that there are phase transitions and atsome parameters of the model black holes are stable.
In last years models of nonlinear electrodynamics (NLED) attract attentionof astrophysicists. Some models of NLED can explain inflation of the uni-verse by NLED coupled to general relativity [1]-[7]. In addition, the initialsingularities in the early universe can be absent in such models [8]. Anotherattractive feature of some NLED is that there is an upper limit on the elec-tric field at the origin of point-like particles and the self-energy of charges isfinite. Thus, in NLED models [9]-[12] there are no problems of singularityand the infinite self-energy of charged particles. In quantum electrodynamicsnonlinear terms appear due to loop corrections [13]-[15]. In this paper weinvestigate a black hole in the framework of new model of NLED with twoparameters. Black holes also were investigated in the framework of NLEDin [16]-[27] and in many other papers. In the model under considerationthe correspondence principle holds so that in the weak field limit NLED isconverted into Maxwell’s electrodynamics.The paper is organised as follows. In section 2 we propose the modelof NLED with two parameters β and γ . In section 3 NLED coupled with E-mail: [email protected] r → r → ∞ are obtained for the pa-rameters γ = 1 / γ = 3 /
4, respectively. We found corrections to theReissner-Nordstr¨om (RN) solution. The thermodynamics of black holes isinvestigated. We calculate the Hawking temperature and heat capacity ofblack holes and demonstrate that black holes undergo phase transitions ofsecond-order. We obtain the range where black holes are stable. Section 6 isdevoted to a conclusion.The units with c = ¯ h = 1, ε = µ = 1 are used and the metric signatureis η = diag( − , , , Let us consider NLED with the Lagrangian density L = − F β F ) γ , (1)where the parameter β has the dimensions of (length) , and γ is the dimen-sionless parameter, F = (1 / F µν F µν = ( B − E ) / F µν = ∂ µ A ν − ∂ ν A µ isthe field strength tensor. At γ = 1 the model was investigated in [12]. At theweak field limit, β F ≪
1, the model (1) becomes Maxwell’l electrodynamics,
L → −F , i.e. the correspondence principle occurs.Making use of Euler-Lagrange equations, from Eq. (1), we obtain fieldequations ∂ µ ( L F F µν ) = 0 , (2)where L F = ∂ L /∂ F . From Eq. (1) we find L F = ( γ − β F ) γ − β F ) γ ) . (3)The electric displacement field D = ∂ L /∂ E , obtained from Eqs. (1) and (3),is given by D = ε E , ε = 1 − ( γ − β F ) γ (1 + ( β F ) γ ) . (4)The magnetic field H = − ∂ L /∂ B becomes H = µ − B , µ − = ε. (5)2he field equations (2), making use of Eqs. (4) and (5), may be written asnonlinear Maxwell’s equations ∇ · D = 0 , ∂ D ∂t − ∇ × H = 0 . (6)The Bianchi identity, ∂ µ ˜ F µν = 0, gives the second pair of nonlinear Maxwell’sequations ∇ · B = 0 , ∂ B ∂t + ∇ × E = 0 . (7)From Eqs. (4) and (5), one obtains D · H = ( ε ) E · B . (8)Because D · H = E · B [28] the dual symmetry is broken. In classical electro-dynamics and in Born-Infeld electrodynamics the dual symmetry holds butin QED with loop corrections the dual symmetry is violated.The causality principle guarantees that the group velocity of excitationsover the background is less than the speed of light and requires L F ≤ γ ≤ ≤ γ ≤ E =0, F = B / T µν = ( γ − β F ) γ − β F ) γ ] F αµ F να − g µν L . (9)The trace of the energy-momentum tensor (9) reads T ≡ T ‘ µµ = 4 γ F ( β F ) γ [1 + ( β F ) γ ] . (10)The trace (10) is not zero because of dimensional parameter β so that thescale invariance is broken. We will study magnetically charged black hole ( E = 0). The action of NLEDcoupled with general relativity is I = Z d x √− g (cid:18) κ R + L (cid:19) , (11)3here κ = 8 πG ≡ M − P l , G is Newton’s constant, M P l is the reduced Planckmass, and R is the Ricci scalar. By varying action (11) with respect of themetric and electric potential, we obtain the Einstein and electromagneticfield equations R µν − g µν R = − κ T µν , (12) ∂ µ h √− g L F F µν i = 0 . (13)We consider the line element having the spherical symmetry ds = − f ( r ) dt + 1 f ( r ) dr + r ( dϑ + sin ϑdφ ) . (14)The metric function is defined by the relation [19] f ( r ) = 1 − GM ( r ) r , (15)and the mass function is given by M ( r ) = Z r ρ M ( r ) r dr, (16)where ρ M is the magnetic energy density. The m M = R ∞ ρ M ( r ) r dr is themagnetic mass of the black hole possessing the electromagnetic nature. Themagnetic energy density ( E =0), found from Eq. (9), is ρ M = T = F β F ) γ . (17)We have for the magnetized black hole [19] F = q / (2 r ), where q is a mag-netic charge. For arbitrary value of γ the expression for M ( r ) is complicated.Therefore, we investigate the particular cases γ = 1 / γ = 3 / γ = x = 2 / r/ ( β / √ q ) and making useof Eq. (17) we obtain the magnetic energy density ρ M ( x ) = 1 βx (1 + x ) . (18)4rom Eqs. (16) and (18) one finds the mass function M ( x ) = q / / β / arctan( x ) . (19)From Eq. (19) we obtain the magnetic mass of the black hole m M = M ( ∞ ) = πq / / β / ≃ . q / β / . (20)Making use of Eqs. (15) and (19) we obtain the metric function expressedthrough the dimensionless variable f ( x ) = 1 − √ Gq √ βx arctan x. (21)From Eq. (21) one finds the asymptotic of the metric function at r → r → ∞ f ( r ) = 1 − √ Gq √ β + 2 Gr β − / Gr β / q + O ( r ) r → , (22) f ( r ) = 1 − Gm M r + Gq r − G √ βq √ r + O ( r − ) r → ∞ . (23)It should me mentioned that according to Eq. (22) the black hole does nothave the regular solution because f (0) = 1. Equation (23) gives correctionsto RN solution in the order of O ( r − ). At r → ∞ we have f ( ∞ ) = 1and the spacetime becomes flat. If β = 0 our model becomes Maxwell’selectrodynamics and Eq. (23) is converted into the RN solution. The plots ofthe function f ( x ) for different parameters a = √ β/ ( √ Gq ) are represented inFig. 1. Fig. 1 shows that black holes may have one or no horizons dependingon the parameter a . The horizons of the black hole are defined by equation f ( x + ) = 0 and represented in Table 1.The Ricci scalar found from Eqs. (10) and (12) (for γ = 0 . E = 0) isgiven by R = κ T = κ √ βq r ( √ r + √ βq ) . (24)At r → ∞ the Ricci scalar (24) goes to zero, R →
0, and spacetime becomesflat. 5 f ( x ) Figure 1: The plot of the function f ( x ). Dashed-dotted line correspondsto a = 1, solid line corresponds to a = 0 . a = 0 .
1. Table 1: The horizons of the black hole a x + Let us study the thermal stability of magnetically charged black holes. Weuse the expression for the Hawking temperature T H = κ S π = f ′ ( r + )4 π . (25)Here κ S is the surface gravity and r + is the event horizon. From Eqs. (15)and (16) one obtains the relations as follows: f ′ ( r ) = 2 GM ( r ) r − GM ′ ( r ) r , M ′ ( r ) = r ρ M , M ( r + ) = r + G . (26)6aking use of Eqs. (21), (25) and (26) one finds the Hawking temperature T H = 12 / πβ / √ q x + − x ) arctan( x + ) ! . (27)The plot of the Hawking temperature versus the horizon x + is represented inFig. 2. It follows from Fig. 2 that the Hawking temperature is positive, and + β / q / T H Figure 2: The plot of the function T H √ qβ / vs. x + .therefore, the black hole is stable. The first-order phase transition occurswhen the Hawking temperature changes the sign. The second-order phasetransition takes place if the heat capacity is singular [30]. We use the entropywhich satisfies the Hawking area low S = A/ (4 G ) = πr /G . Than the heatcapacity can be calculated by C q = T H ∂S∂T H ! q = T H ∂S/∂r + ∂T H /∂r + = 2 πr + T H G∂T H /∂r + . (28)According to Eq. (28) the heat capacity diverges if the Hawking temperaturehas extremum. The maximum of the Hawking temperature occurs at x + ≃ . C q = π √ βq [(1 + x ) arctan( x + ) − x + ] x (1 + x ) arctan( x + ) G [ x (1 + 2 x + arctan( x + )) − (1 + x ) arctan ( x + )] . (29)The plot of the function GC q / ( √ βq ) vs. x + is given in Fig. 3. Fig. 3 shows + C q G / ( β / q ) Figure 3: The plot of the function C q G/ ( √ βq ) vs. x + .that indeed at x + ≃ .
135 the black hole undergoes the second-order phasetransition. At 0 ≤ x + < .
135 the black hole is stable and at x + > .
135 theheat capacity is negative and the black hole becomes unstable. γ = x = 2 / r/ ( β / √ q ),one finds the magnetic energy density ρ M ( x ) = 1 βx (1 + x ) . (30)8rom Eqs. (16) and (30) we obtain the mass function M ( x ) = q / / β / " ln x − x + 1( x + 1) + 2 √ x − √ ! . (31)From Eq. (31) one finds the magnetic mass of the black hole m M = M ( ∞ ) = πq / / √ β / ≃ . q / β / . (32)By virtue of Eqs. (15) and (31) we obtain the metric function f ( x ) = 1 − Gq √ βx " ln x − x + 1( x + 1) + 2 √ x − √ ! . (33)Making use of Eq. (33) one finds the asymptotic of the metric function at r → r → ∞ f ( r ) = 1 + πGq / √ β / r − G √ qr / β / + 2 / Gr β / q + O ( r ) r → , (34) f ( r ) = 1 − Gm M r + Gq r − Gβ / q / / r + O ( r − ) r → ∞ . (35)It follows from Eq. (34) that f (0) = ∞ , and therefore, the black hole solutionis not the regular solution. In accordance with Eq. (35) corrections to RNsolution are in the order of O ( r − ). When r → ∞ the spacetime becomesMinkowski’s spacetime. At β = 0 the model is transformed into Maxwell’selectrodynamics and Eq. (35) becomes the RN solution. The plots of thefunction f ( x ) for different parameters a = √ β/ ( √ Gq ) are represented inFig. 4. Fig. 4 shows different behaviour of the function f ( x ) dependingon the parameter a . At a > .
21 there are no horizons and we have nakedsingularity. When a ≃ .
21 the extreme singularity occurs. At a < .
21 onehas two horizons. The inner x − and outer x + horizons of the black hole aregiven by the equation f ( x ) = 0 and are in Table 2.The Ricci scalar due to Eqs. (10) and (12), for γ = 3 / E = 0) at r → ∞ , approaches to zero, R →
0, and spacetime becomes Minkowski’sspacetime. 9 f ( x ) Figure 4: The plot of the function f ( x ). Dashed-dotted line corresponds to a = 1, solid line corresponds to a = 0 .
21 and dashed line corresponds to a = 0 . By virtue of Eqs. (25), (26) and (30) we obtain the Hawking temperature T H = 12 / πβ / √ q x + − x + ( x + 1) (cid:20) ln x − x + +1( x + +1) + 2 √ (cid:16) x + − √ (cid:17)(cid:21) . (36)The plot of the function T H ( x + ) is given in Fig. 5. According to Fig. 5 theHawking temperature is positive at 0 . > x + >
0, and the black hole isunstable for x + > . x + ≃ . ∂T H ∂x + = 12 / πβ / √ q (cid:20) − x a x − x + + β / q / T H Figure 5: The plot of the function T H √ qβ / vs. x + . + 6 (cid:18) (2 x −
1) ln (cid:18) x − x + +1( x + +1) (cid:19) + 2 √ x −
1) arctan (cid:16) x + − √ (cid:17) + 6 x (cid:19) ( x + 1) (cid:20) ln (cid:18) x − x + +1( x + +1) (cid:19) + 2 √ (cid:16) x + − √ (cid:17)(cid:21) (cid:21) . (37)From Eqs. (28), (36) and (37) one can find the heat capacity. The plotof the function GC q / ( √ βq ) vs. x + is given in Fig. 6. Fig. 6 shows thatat x + ≃ . < x + < . . > x + > . + G C q / ( q ( β ) . ) Figure 6: The plot of the function C q G/ ( √ βq ) vs. x + . We have proposed new NLED model with two independent parameters β and γ . In this model the correspondence principle takes place so that forweak fields the model is converted into Maxwell’s electrodynamics. NLEDcoupled with the gravitational field was investigated. We have studied themagnetized black holes and found the asymptotic of the metric and massfunctions at r → r → ∞ for the parameters γ = 1 / γ = 3 / eferences [1] R. Garc´ıa-Salcedo and N. Breton, Int. J. Mod. Phys. A , 4341 (2000)[arXiv:gr-qc/0004017].[2] C. S. Camara, M. R. de Garcia Maia, J. C. Carvalho and J. A. S. Lima,Phys. Rev. D , 123504 (2004) [arXiv:astro-ph/0402311].[3] E. Elizalde, J. E. Lidsey, S. Nojiri and S. D. Odintsov, Phys. Lett. B , 1 (2003) [arXiv:hep-th/0307177].[4] M. Novello, S. E. Perez Bergliaffa and J. M. Salim, Phys. Rev. D ,127301 (2004) [arXiv:astro-ph/0312093].[5] M. Novello, E. Goulart, J. M. Salim and S. E. Perez Bergliaffa, Class.Quant. Grav. , 3021 (2007) [arXiv:gr-qc/0610043].[6] D. N. Vollick, Phys. Rev. D , 063524 (2008) [arXiv:0807.0448].[7] S. I. Kruglov, Phys. Rev. D , 123523 (2015) [arXiv:1601.06309]; Int. J.Mod. Phys. A , 1650058 (2016) [arXiv:1607.03923]; Int. J.Mod. Phys.D , 1640002 (2016) [arXiv:1603.07326].[8] M. Novello, E. Goulart, J. M. Salim and S. E. Perez Bergliaffa, Class.Quant. Grav. , 3021 (2007) [arXiv:gr-qc/0610043].[9] M. Born and L. Infeld, Proc. Royal Soc. (London) A , 425 (1934).[10] D. M. Gitman and A. E. Shabad, Eur. Phys. J. C , 3186 (2014)[arXiv:1410.2097].[11] C. V. Costa, D. M. Gitman and A. E. Shabad, Phys. Scripta , 074012(2015) [arXiv:1312.0447].[12] S. I. Kruglov, Ann. Phys. , 299 (2015) [arXiv:1410.0351]; Ann. Phys.(Berlin) , 397 (2015) [arXiv:1410.7633]; Commun. Theor. Phys. ,59 (2016) [arXiv:1511.03303].[13] W. Heisenberg and H. Euler, Z. Physik, , 714 (1936)[arXiv:physics/0605038].[14] J. Schwinger, Phys. Rev. , 664 (1951).1315] S. L. Adler, Ann. Phys. (N.Y.) , 599 (1971).[16] J. M. Bardeen, in Proc. Int. Conf. GR5, Tbilisi, p. 174, 1968.[17] I. Dymnikova, Gen. Rev. Grav. , 235 (1992).[18] E. Ay´on-Beato, A. Gar´cia, Phys. Rev. Lett. , 5056 (1998)[arXiv:gr-qc/9911046].[19] K. A. Bronnikov, Phys. Rev. D , 044005 (2001).[20] N. Breton, Phys. Rev. D , 124004 (2003) [arXiv:hep-th/0301254].[21] S. A. Hayward, Phys. Rev. Lett. , 31103 (2006) [arXiv:gr-qc/0506126].[22] J. P. S. Lemos and V. T. Zanchin, Phys. Rev. D , 124005 (2011)[arXiv:1104.4790].[23] A. Flachi and J. P.S. Lemos, Phys. Rev. D , 024034 (2013)[arXiv:1211.6212].[24] S. H. Hendi, Ann. Phys. , 282 (2013) [arXiv:1405.5359].[25] L. Balart and E. C. Vagenas, Phys. Rev. D , 124045 (2014)[arXiv:1408.0306].[26] S. I. Kruglov, Phys. Rev. D , 044026 (2016) [arXiv:1608.04275]; Eu-rophys. Lett. , 60006 (2016) [arXiv:1611.02963]; Ann. Phys. (Berlin) , 588 (2016) [arXiv:1607.07726].[27] V. P. Frolov, Phys. Rev. D , 104056 (2016) [arXiv:1609.01758].[28] G. W. Gibbons and D. Rasheed, Nucl. Phys. B (1995) 185[arXiv:hep-th/9506035].[29] A. E. Shabad, V. V. Usov, Phys. Rev. D , 105006 (2011)[arXiv:1101.2343].[30] P. C. W. Davies, Rep. Prog. Phys.41