On a model of magnetically charged black hole with nonlinear electrodynamics
aa r X i v : . [ g r- q c ] M a y On a model of magnetically charged black hole with nonlinearelectrodynamics
S. I. Kruglov Department of Chemical and Physical Sciences, University of Toronto,3359 Mississauga Road North, Mississauga, Ontario L5L 1C6, Canada
Abstract
The Bronnikov model of nonlinear electrodynamics is investigatedin general relativity. The magnetic black hole is considered and we ob-tain a solution giving corrections to the Reissner-Nordstr¨om solution.In this model spacetime at r → ∞ becomes Minkowski’s spacetime.We calculate the magnetic mass of the black hole and the metric func-tion. At some parameters of the model there can be one, two or nohorizons. The Hawking temperature and the heat capacity of blackholes are calculated. We show that a second-order phase transitiontakes place and black holes are thermodynamically stable at somerange of parameters. The black hole (BH) physics is similar to ordinary thermodynamics. Thisanalogy can help of understanding the theory of quantum gravity. The semi-classical analysis of a BH radiation shows that the initial state informationis hidden inside of the event horizon [1]. Some questions were appeared onthe information loss paradox and the unitary of the theory. To answer thesequestions a backreaction and quantum gravity effects should be taken intoaccount. In addition, the semiclassical calculations do not work for lightblack holes. The effects of quantum gravity can correctly describe the shortdistance behavior. At the same time, to avoid the short distance singularity,one can consider regular black holes. A study of regular black holes with-out singularities allows us to treat the minimum size as the Planck length.Not to search for quantum gravity, we concentrate here on the regular BHthermodynamics. Regular black hole solutions may be obtained by couplinggeneral relativity and nonlinear electrodynamics (NLED). E-mail: [email protected] q = 0so that the magnetic field is equal to B = q/r and spacetime possesses aregular center. The entire mass of the BH has the electromagnetic origin.The thermodynamics of black holes and phase transitions are studiedin this paper. In [28], [29] similar issues of the BH thermodynamics werediscussed. Some aspects of black hole physics were studied in [30]-[48].The paper is organized as follows. Field equations and energy-momentumtensor are described in section 2. In section 3 general relativity with NLEDis studied. We obtain the asymptotic of the metric and mass functions at r → r → ∞ . Corrections to the Reissner-Nordstr¨om (RN) solutionare found. In section 4 we calculate the Hawking temperature and the heatcapacity of black holes. It was demonstrated that the second-order phasetransition takes place in black holes. We find the range where black holesare stable. We made a conclusion in section 5.The metric signature is given by η = diag( − , , ,
1) and we explore theunits with c = 1, ε = µ = 1. 2 Field equations of NLED
In this section we consider field equations in Minkowski’s spacetime. Let usstudy NLED, proposed in [27], with the Lagrangian density L = − F cosh q | β F | , (1)where F = (1 / F µν F µν = ( B − E ) /
2, the field tensor is defined as F µν = ∂ µ A ν − ∂ ν A µ and the parameter β has the dimension of (length) . Themodified model was proposed and investigated in [47]. At the weak fieldlimit, β F ≪
1, the Lagrangian density (1) becomes Maxwell’s Lagrangiandensity,
L → −F , and as result, the correspondence principle holds. Fieldequations found from Eq. (1), by the variation of the action correspondingto Lagrangian density (1) with respect to the 4-potential A µ , are given by ∂ µ ( L F F µν ) = 0 , (2)where L F = ∂ L /∂ F = | β F | / tanh q | β F | q | β F | − q | β F | . (3)With the help of Eq. (1) one obtains the electric displacement field D = ∂ L /∂ E D = ε E , ε = −L F . (4)We find the magnetic field from the relation H = − ∂ L /∂ B , H = µ − B , µ − = −L F = ε. (5)Making use of Eqs. (4) and (5) one can represent field equations (2) as theMaxwell equations ∇ · D = 0 , ∂ D ∂t − ∇ × H = 0 . (6)From the identity ∂ µ ˜ F µν = 0, where ˜ F µν is the dual tensor, we obtain thesecond pair of nonlinear Maxwell’s equations ∇ · B = 0 , ∂ B ∂t + ∇ × E = 0 . (7)3he relation, followed from Eqs. (4) and (5), is D · H = ( ε ) E · B . (8)Because D · H = E · B , according to the criterion of [49], the dual symmetryis broken. In classical electrodynamics and in Born-Infeld electrodynamicsthe dual symmetry occurs but in QED, due to quantum corrections, the dualsymmetry is violated.The symmetrical energy-momentum tensor can be found from the relation T µν = L F F αµ F να − g µν L . (9)From Eqs. (1), (3) and (9) one obtains the energy-momentum tensor trace T ≡ T µµ = 2 F q | β F | sinh q | β F | cosh q | β F | . (10)As T 6 = 0, the scale invariance is broken. At β = 0 NLED (1) becomesMaxwell’s electrodynamics, T = 0, and the scale invariance is recovered. The action of general relativity with NLED is given by I = Z d x √− g (cid:18) κ R + L (cid:19) , (11)where κ = 8 πG ≡ M − P l , G is the Newton constant and M P l is the reducedPlanck mass. The stability of a black hole with action (11) was studiedin [36]. We investigate the magnetically charged black hole, and therefore, E = 0, B = 0. The Einstein and the electromagnetic field equations followfrom action (11), R µν − g µν R = − κ T µν , (12) ∂ µ (cid:16) √− g L F F µν (cid:17) = 0 . (13)We use the line element with the spherical symmetry ds = − f ( r ) dt + 1 f ( r ) dr + r ( dϑ + sin ϑdφ ) , (14)4nd the metric function is given by [27] f ( r ) = 1 − GM ( r ) r . (15)The mass function is defined as follows: M ( r ) = Z r ρ M ( r ) r dr = Z ∞ ρ M ( r ) r dr − Z ∞ r ρ M ( r ) r dr, (16)with ρ M being the magnetic energy density and m M = R ∞ ρ M ( r ) r dr is themagnetic mass of the black hole . From Eq. (9) at E =0 we find the magneticenergy density ρ M = T = −L = F cosh q | β F | , (17)where F = B / q / (2 r ), and q is a magnetic charge. It is convenient tointroduce the dimensionless parameter x = 2 / r/ ( β / √ q ). Then using Eqs.(16) and (17) one obtains the mass function M ( x ) = m M − q / / β / tanh (cid:18) x (cid:19) . (18)The mass function presented by Eq. (18) has the same form as the solutiongiven in [27]. We calculate the magnetic mass of the black hole m M = Z ∞ ρ M ( r ) r dr = q / / β / . (19)Making use of Eqs. (15) and (18) one finds the metric function f ( x ) = 1 − − tanh (1 /x ) bx , (20)with b = √ β/ ( √ Gq ). With the help of Eq. (20) we obtain the asymptoticof the metric function at r → ∞ f ( r ) = 1 − Gm M r + Gq r − G √ βq √ r + Gβq r + O ( r − ) . (21)Equation (21) gives the corrections to the RN solution which are in the orderof O ( r − ). At r → ∞ , f ( ∞ ) = 1, and the spacetime becomes flat. It is easyto verify that lim x → + f ( x ) = 1 . (22)5 f ( x ) Figure 1: The plot of the function f ( x ). The dashed-dotted line correspondsto b = 0 .
1, the solid line corresponds to b = 0 . b = 1.Equation (22) shows that the black hole is regular without conical singularity.The RN solution is recovered at β = 0. The plot of the function f ( x ) isrepresented in Fig. 1 for different parameters b = √ β/ ( √ Gq ). Accordingto Fig. 1 at b > . b ≃ . b < . x h are defined by the equation f ( x h ) = 0. Thenfrom Eq. (20) we come to the equation b = 1 − tanh(1 /x h ) x h . (23)The plot of the function b ( x h ) is represented in Fig. 2. From Eq. (23) weobtain the inner x − and the outer x + horizons of the black hole which aregiven in Table 1. It follows from Eqs. (10) at F = q / (2 r ) that at r → ∞ the energy-momentum trace becomes zero. As a result, in accordance withEq. (12) the Ricci scalar R = κ T vanishes and spacetime becomes flat.Let us discuss the equation of state of the BH inside the Cauchy horizon.From the expression for the pressure p = L +( E − B ) L F / h b Figure 2: The plot of the function b ( x h ).Table 1: The inner and outer horizons of the black hole b x − x + E = 0) p = − ρ M − F L F /
3. This equationholds for any radius r . At r →
0, inside the Cauchy horizon, p → ρ M → F L F →
0. As a result, the equation of state for a de Sitter spacetime p = − ρ at r → r → We will study the possible phase transitions and thermal stability of magne-tized black holes. Let us calculate the Hawking temperature which is given7y T H = κ S π = f ′ ( r h )4 π , (24)where κ S is the surface gravity and r h is the horizon. Making use of Eqs.(15) and (16) we obtain the relations as follows: f ′ ( r ) = 2 GM ( r ) r − GM ′ ( r ) r , M ′ ( r ) = r ρ M , M ( r h ) = r h G . (25)It follows from Eq. (20) that f ′ ( x ) = 1 bx − bx coth (1 /x ) − tanh(1 /x ) bx . Then one can verify that indeed this equation leads to (25) taking into ac-count b = √ β/ ( √ Gq ), x = 2 / r/ ( β / √ q ), F = q / (2 r ) and Eqs. (17)-(19). With the help of Eqs. (17) and (23) - (25), we find the Hawkingtemperature T H = 12 / πβ / √ q x h − x h [1 − tanh(1 /x h )] cosh (1 /x h ) ! = 1 + b / πβ / √ qx h − b ) x h ! . (26)The plot of the function T H ( x h ) is given in Fig. 3. The Hawking temperaturebecomes zero at x h ≃ .
56 ( b = 0 . x h > .
56 the Hawking temperature is positive and the black hole isstable. When x h < .
56 the Hawking temperature becomes negative and theblack hole is unstable. The maximum of the Hawking temperature holds at x + ≃ .
95 and the heat capacity is singular. In this point the second-orderphase transition takes place. We calculate the heat capacity from the relation C q = T H ∂S∂T H ! q = T H ∂S/∂r h ∂T H /∂r h = 2 πr h T H G∂T H /∂r h . (27)The entropy obeys the Hawking area low S = A/ (4 G ) = πr h /G . In Figs.4 and 5 one can find the plots of the function GC q / ( √ βq ) vs. the horizon x h for different values of x h . According to Figs. 4 and 5 the heat capacitypossesses a discontinuity at x + ≃ .
95, and therefore, the second-order phasetransition of the black hole occurs. When 1 . ≤ x h < .
95 the black hole isstable and at x + > .
95 the heat capacity becomes negative and the blackhole is unstable. 8 h β / q / T H Figure 3: The plot of the function T H √ qβ / vs. horizons x h . We have investigated the Bronnikov model of NLED which at the weak fieldlimit becomes Maxwell’s electrodynamics and the correspondence principleholds. NLED coupled with the gravitational field was considered. The mag-netized black holes were studied and we obtained the metric and the massfunctions. At r → ∞ we obtained corrections to the RN solution that arein the order of O ( r − ). Physical values of the theory depend on the pa-rameter of NLED β . The Hawking temperature and the heat capacity ofblack holes were calculated and we demonstrated that second-order phasetransitions take place in black holes for definite parameters β . The thermo-dynamic stabilities of black holes were studied and it was shown that in therange 1 . ≤ x + < .
95 the black holes are stable. In the framework of thenon-commutative model with two horizons [50] similar phase transitions canhappen. Phase transitions may appear also in alternative theories of gravity[51]. It should be mentioned that in models of black holes with two horizonsthe Hawking temperature is connected with each horizons. This leads insome models to the existence of a minimal and maximal temperature in theblack body radiation [52]. Author of [53] found conditions in Massive Gravity9 h G C q / ( q ( β ) . ) Figure 4: The plot of the function C q G/ ( √ βq ) vs. x h .for an observer to hold in the order to agree with the black-hole tempera-ture. Within the Generalized Uncertainty Principle authors of [54] shownthat there can exist a black hole remnant with a mass M P l corresponding toa maximal temperature M P l . There exists also a minimum length l P l . References [1] Hawking, S. W.
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