Einstein-Euler-Heisenberg Theory and charged black holes
aa r X i v : . [ h e p - t h ] O c t Einstein-Euler-Heisenberg theory and charged black holes
Remo Ruffini a , b , c , Yuan-Bin Wu a , b , c , She-Sheng Xue a , b a Dipartimento di Fisica and ICRA, Sapienza Universit`a di RomaP.le Aldo Moro 5, I-00185 Rome, Italy b ICRANet, P.zza della Repubblica 10, I-65122 Pescara, Italy c ICRANet, University of Nice-Sophia Antipolis, 28 Av. de Valrose,06103 Nice Cedex 2, France
Abstract
Taking into account the Euler-Heisenberg effective Lagrangian of one-loop nonperturbativequantum electrodynamics (QED) contributions, we formulate the Einstein-Euler-Heisenbergtheory and study the solutions of nonrotating black holes with electric and magnetic charges inspherical geometry. In the limit of strong and weak electromagnetic fields of black holes, wecalculate the black hole horizon radius, area, and total energy up to the leading order of QEDcorrections and discuss the black hole irreducible mass, entropy, and maximally extractable en-ergy as well as the Christodoulou-Ruffini mass formula. We find that these black hole quantitiesreceive the QED corrections, in comparison with their counterparts in the Reissner-Nordstr¨omsolution. The QED corrections show the screening effect on black hole electric charges and theparamagnetic effect on black hole magnetic charges. As a result, the black hole horizon area,irreducible mass, and entropy increase; however, the black hole total energy and maximallyextractable energy decrease, compared with the Reissner-Nordstr¨om solution. In addition, weshow that the condition for extremely charged black holes is modified due to the QED correc-tion.PACS: 12.20.-m, 13.40-f, 11.27.+d, 04.70.-sKeywords: Quantum Electrodynamics, Black Holes, Euler-Heisenberg effective Lagrangian ruffi[email protected] Corresponding author. [email protected] [email protected] Introduction
For several decades the nonlinear electromagnetic generalization of the Reissner-Nordstr¨omsolution of the Einstein-Maxwell equations has attracted a great deal of attention. The mostpopular example is the gravitating Born-Infeld (BI) theory [1]. The static charged black holesin gravitating nonlinear electrodynamics were studied in the 1930s [2, 3]. The discovery thatthe string theory, as well as the D-brane physics, leads to Abelian and non-Abelian BI-like La-grangians in its low-energy limit (see, e.g., Refs. [4–6]), has renewed the interest in these kindsof nonlinear actions. Asymptotically flat, static, spherically symmetric black hole solutions forthe Einstein-Born-Infeld theory were obtained in the literature [7, 8].Generalization of the exact solutions of spherically symmetric Born-Infeld black holes witha cosmological constant in arbitrary dimensions has been considered [9–11], as well as in othergravitational backgrounds [12, 13]. Many other models of nonlinear electrodynamics leading tostatic and spherically symmetric structures have been considered in the last decades, such as thetheory with a nonlinear Lagrangian of a general function of the gauge invariants ( F mn F mn and F mn ˜ F mn ) [14–17] or a logarithmic function of the Maxwell invariant ( F mn F mn ) [18], and the the-ory with a generalized nonlinear Lagrangian [19] which can lead to the BI Lagrangian and theweak-field limit of the Euler-Heisenberg effective Lagrangian [20]. The static and sphericallysymmetric black hole, whose gravity coupled to the nonlinear electrodynamics of the weak-field limit of the Euler-Heisenberg effective Lagrangian as a low-energy limit of the Born-Infeldtheory, was studied in Ref. [21]. Some attempts in the obtention of regular (singularity-free)static and spherically symmetric black hole solutions in gravitating nonlinear electrodynamicshave been made [22–26], and the unusual properties of these solutions have been discussed inRefs. [27, 28]. Generalization of spherically symmetric black holes in higher dimension in thetheory with a nonlinear Lagrangian of a function of power of the Maxwell invariant has beenconsidered in the literature [29–32]. Finally, we mention that rotating black branes [33, 34] androtating black strings [35] in the Einstein-Born-Infeld theory have been also considered.The effective Lagrangian of nonlinear electromagnetic fields has been formulated for thefirst time by Heisenberg and Euler using the Dirac electron-positron theory [20]. Schwingerreformulated this nonperturbative one-loop effective Lagrangian within the quantum electro-dynamics (QED) framework [36]. This effective Lagrangian characterizes the phenomenonof vacuum polarization. Its imaginary part describes the probability of the vacuum decayvia the electron-positron pair production. If electric fields are stronger than the critical value E c = m c / e ¯ h , the energy of the vacuum can be lowered by spontaneously creating electron-positron pairs [20, 36, 37]. For many decades, both theorists and experimentalists have beeninterested in the aspects of the electron-positron pair production from the QED vacuum and thevacuum polarization by an external electromagnetic field (see, e.g., Refs. [38, 39]).As a fundamental theory, QED gives an elegant description of the electromagnetic inter-action; moreover, it has been experimentally verified. Therefore, it is important to study theQED effects in black hole physics. As a result of one-loop nonperturbative QED, the Euler-Heisenberg effective Lagrangian deserves to attract more attention in the topic of generalized2lack hole solutions mentioned above. In this article, we adopt the contribution from the Euler-Heisenberg effective Lagrangian to formulate the Einstein-Euler-Heisenberg theory, and studythe solutions of electrically and magnetically charged black holes in spherical geometry. Wecalculate and discuss the QED corrections to the black hole horizon area, entropy, total energy,and the maximally extractable energy.The article is organized as follows. In Sec. 2, we first recall the Euler-Heisenberg effectiveLagrangian. We formulate the Einstein-Euler-Heisenberg theory in Sec. 3. The study of elec-trically charged black holes in the weak electric field case is presented in Sec. 4. The study ofmagnetically charged black holes in both weak and strong magnetic field cases is presented inSec. 5. Then we present the study of black holes with both electric and magnetic charges inthe Einstein-Euler-Heisenberg theory in Sec. 6. A summary is given in Sec. 7. The use of unitswith ¯ h = c = The QED one-loop effective Lagrangian was obtained by Heisenberg and Euler [20] for constantelectromagnetic fields, D L eff = ( p ) Z ¥ dss h e eb s coth ( e e s ) cot ( e b s ) − − e ( e − b ) s i e − is ( m e − i h ) , (1)as a function of two invariants: the scalar S and the pseudoscalar P , S ≡ − F mn F mn = ( E − B ) ≡ e − b , P ≡ − F mn ˜ F mn = E · B ≡ eb , (2)where the field strength is F mn , ˜ F mn ≡ e mnlk F lk /
2, and e = q ( S + P ) / + S , (3) b = q ( S + P ) / − S . (4)The effective Lagrangian reads L eff = L M + D L eff , (5)where L M = S is the Maxwell Lagrangian. Its imaginary part is related to the decay rate of thevacuum per unit volume [20, 36], G V = ae p (cid:229) n = n n pb / e tanh n pb / e exp (cid:18) − n p E c e (cid:19) (6)3or fermionic fields, and G V = ae p (cid:229) n = ( − ) n n n pb / e sinh n pb / e exp (cid:18) − n p E c e (cid:19) (7)for bosonic fields; here, E c = m e c e ¯ h is the critical field. Using the expressions [40] e e s coth ( e e s ) = ¥ (cid:229) n = − ¥ s ( s + t n ) , t n ≡ n p / e e , (8) e b s cot ( e b s ) = ¥ (cid:229) m = − ¥ s ( s − t m ) , t m ≡ m p / e b , (9)one obtains the real part of the Euler-Heisenberg effective Lagrangian (1) (see Refs. [38, 41–45]), ( D L coseff ) P = ( p ) ¥ (cid:229) n , m = − ¥ t m + t n h ¯ d m J ( i t m m e ) − ¯ d n J ( t n m e ) i (10) = − ( p ) " ¥ (cid:229) n = e bt n coth ( e bt n ) J ( t n m e ) − ¥ (cid:229) m = e et m coth ( e et m ) J ( i t m m e ) . (11)The symbol ¯ d i j ≡ − d i j denotes the complimentary Kronecker d , which vanishes for i = j ,and J ( z ) ≡ P Z ¥ ds se − s s − z = − h e − z Ei ( z ) + e z Ei ( − z ) i . (12)Here, P indicates the principle value integral, and Ei ( z ) is the exponential-integral function,Ei ( z ) ≡ P Z z − ¥ dt e t t = log ( − z ) + ¥ (cid:229) k = z k kk ! . (13)Using the series and asymptotic representation of the exponential-integral function Ei ( z ) forlarge z corresponding to weak electromagnetic fields ( e / E c ≪ b / E c ≪ J ( z ) = − z − z − z − z − z + · · · , (14)the weak-field expansion of Eq. (10) is ( D L eff ) P = a m e (cid:8) S + P (cid:9) + pa m e (cid:8) S + SP (cid:9) + · · · , (15)which is expressed in terms of a powers series of weak electromagnetic fields up to O ( a ) , thefirst term was obtained by Heisenberg and Euler in their original article [20].On the other hand, using the series and asymptotic representation of the exponential-integralfunction Ei ( z ) for small z ≪ e / E c ≫ b / E c ≫ J ( z ) = − h e z ln ( z ) + e − z ln ( − z ) i − g h e z + e − z i + O ( z ) , (16)4he leading terms in the strong-field expansion of Eqs. (10) and (11) are given by (see Refs. [38,41, 45, 46]) ( D L coseff ) P = ( p ) ¥ (cid:229) n , m = − ¥ t m + t n h ¯ d n ln ( t n m e ) − ¯ d m ln ( t m m e ) i + · · · (17) = ( p ) (cid:20) ¥ (cid:229) n = e bt n coth ( e bt n ) ln ( t n m e ) − ¥ (cid:229) m = e et m coth ( e et m ) ln ( t m m e ) (cid:21) + · · · . (18)In the case of vanishing magnetic field B = E ≫ E c using lim z → ¥ J ( iz ) = z → z coth ( az ) = / a , Eq. (18) becomes (see Refs. [38, 41, 45]) ( D L coseff ) P = e E p ¥ (cid:229) n = n (cid:20) ln (cid:18) n p E c E (cid:19) + g (cid:21) + · · · (19) = e E p (cid:20) ln (cid:18) p E c E (cid:19) + g (cid:21) − e E p z ′ ( ) + · · · , (20)with the Euler-Mascheroni constant g = . z ( k ) = (cid:229) n / n k ,and z ′ ( ) = p [ g + ln ( p ) −
12 ln A ] ≃ − . , (21)with A = . E = B ≫ E c , Eq. (18) becomes (see Refs. [38, 41, 45]) ( D L coseff ) P = − e B p ¥ (cid:229) m = n (cid:20) ln (cid:18) n p E c B (cid:19) + g (cid:21) + · · · (22) = − e B p (cid:20) ln (cid:18) p E c B (cid:19) + g (cid:21) + e B p z ′ ( ) + · · · . (23)The ( n =
1) term in Eq. (22) is the one obtained by Weisskopf [47].
Since the real part of the Euler-Heisenberg effective Lagrangian ( D L coseff ) P of Eq. (10) is ex-pressed in terms of Lorentz invariants ( e , b ) or ( S , P ) , the Euler-Heisenberg effective action inthe curve space-time described by metric g mn can be written as S EH = Z d x √− g L EH , L EH = [ S + ( D L coseff ) P ] . (24)The Einstein and Euler-Heisenberg action is then given by S EEH = − p G Z d x √− gR + S EH , (25)where R is the Ricci scalar. 5he Einstein field equations are G mn ≡ R mn − g mn R = p GT mn , (26)where the energy-momentum tensor is T mn = √− g d S EH d g mn . (27)The electromagnetic field equations and Bianchi identities are given by D m P nm = j n , D m ˜ F mn = , (28)and the displacement fields P nm , D i = P i , and H i = − e i jk P jk are defined as P mn = d L EH d F mn , D i = d L EH d E i , H i = − d L EH d B i . (29)Here, electromagnetic fields are treated as smooth varying fields over all space generated byexternal charge currents j m at infinity.Using functional derivatives, we obtain T mn = − g mn [ S + ( D L coseff ) P ] + (cid:20) d S d g mn d L EH d S + d P d g mn d L EH d P (cid:21) , = − g mn [ S + ( D L coseff ) P ] + (cid:20) ( + A S ) d S d g mn + A P d P d g mn (cid:21) , (30)where two invariants are defined as A S ≡ d ( D L coseff ) P d S ; A P ≡ d ( D L coseff ) P d P . (31)It is straightforward to obtain d S d g mn = F m l F ln , d P d g mn = F m l ˜ F ln = g mn P , (32)and as a result, we rewrite Eq. (30) as T mn = T mn M + g mn [ A P P − ( D L coseff ) P ] + A S F m l F ln , = T mn M ( + A S ) + g mn [ A S S + A P P − ( D L coseff ) P ] , (33)where T mn M = − g mn S + F m l F ln is the energy-momentum tensor of the electromagnetic fields ofthe linear Maxwell theory. Equation (33) is in fact a general result, independent of the explicitform of nonlinear Lagrangian ( D L coseff ) P . Equations (24)-(33) in principle give a completeset of equations for Einstein and Euler-Heisenberg effective theory, together with total charge6 Q ), angular-momentum ( L ), and energy ( M ) conservations. In this article, adopting the Euler-Heisenberg effective Lagrangian (10), we explicitly calculate invariants A S and A P of Eq. (31)as well as the energy-momentum T mn of Eq. (33) in the following cases.It is necessary to point out that in present article, we do not consider the couplings betweenphotons and gravitons that are also induced by QED vacuum polarization effects at the levelof one-fermion loop. Drummond and Hathrell obtained the photon effective action from thelowest term of one-loop vacuum polarization on a general curved background manifold; i.e., agraviton couples to two on-mass-shell photons through a fermionic loop [48], S DH = − a p m e Z d x √− g (cid:0) RF mn F mn − R mn F ms F ns + R mnst F mn F st + D m F mn D s F sn (cid:1) . (34)Further studies of one-loop effective action (34) were made based on the approach of the heat-kernel or “inverse mass” expansion [49, 50], the approach of the so-called “derivative expan-sion” [51, 52], and the consideration of the one-loop one particle irreducible of one gravitoninteracting with any number of photons [53]. This effective action (34) was used to study themodified photon dispersion relation by a generic gravitational background [48] and the possibleconsequences [54–57].At the level of one-loop quantum corrections of the QED theory in the presence of gravita-tional field, the effective Lagrangian (34) should be considered as an addition to the Euler andHeisenberg effective Lagrangian (15) in the weak-field limit. In this article, we try to quantita-tively study the QED corrections in spherically symmetric black holes with mass M and charge Q . In this case, the corrections from the Euler and Heisenberg effective Lagrangian (15) mustbe much larger than the one from the effective Lagrangian (34). Studying the discussion andresult of Ref. [48] for spherical symmetric black holes, we approximately estimate the ratio ofEqs. (15) and (34) around the horizon of black holes with mass M and charge Q . As a result,this ratio is ∼ − (cid:16) QM √ G (cid:17) a Gm e ≫
1. It is not surprising that the electromagnetic coupling e ∼ / √
137 is much larger than the effective gravitational counterpart Gm e ∼ − . Besides,it is expected that calculations involving both the Euler-Heisenberg effective Lagrangian (15)and Eq. (34) are much more complex and tedious. Nevertheless, it is interesting to investigatethe effect of the photon-graviton amplitudes on black hole physics. In this article, for the sakeof simplicity, we first consider only the Einstein-Euler-Heisenberg action (25) as a leading con-tribution in order to gain some physical insight into the QED corrections in black hole physics. = , E = or E = , B = We consider the case of B = E =
0, namely, b = P = e = E = | E | , and S = E / A P = ( D L coseff ) P = − e E p ¥ (cid:229) n = n J ( n p E c / E ) . (35)7sing P Z ¥ ds e − s ( s − z ) = − z h e − z Ei ( z ) − e z Ei ( − z ) i , (36)we calculate dJ ( z ) dz = P Z ¥ ds se − s ( s − z ) = z − P Z ¥ ds e − s ( s − z ) (37)and obtain A S = − e p ¥ (cid:229) n = n J ( n p E c / E ) − e p z ( ) + e p E c E ¥ (cid:229) n = n ˜ J ( n p E c / E ) , (38)where ˜ J ( z ) = e − z Ei ( z ) − e z Ei ( − z ) . (39)Substituting these quantities into Eq. (33), we obtain the expression of the energy-momentumtensor T mn ( e ) . In the case of E = B =
0, the energy-momentum tensor T mn ( b ) canbe straightforwardly obtained from T mn ( e ) by the discrete duality transformation e → i b , i.e., | E | → i | B | . In principle, using the complete Euler-Heisenberg effective Lagrangian ( D L coseff ) P (10) for arbitrary electromagnetic fields E and B , one can obtain the energy-momentum tensor T mn ( e , b ) of Eq. (33). For the reason of practical calculations, we consider the cases of weakand strong fields. In the weak-field case, using Eq. (15) and calculating Eqs. (30)-(33), we obtain A S = a m e ( S ) + pa m e ( S + P ) + · · · , A P = a m e ( P ) + pa m e ( SP ) + · · · , (40)and T mn = T mn M (cid:20) + (cid:18) a m e (cid:19) S (cid:21) + g mn (cid:18) a m e (cid:19) (cid:2) S + P (cid:3) + · · · , (41)up to the leading order.In strong-field case e / E c ≫ b / E c ≫ A S = ( p ) e + b ¥ (cid:229) n , m = − ¥ ( t m + t n ) n ¯ d n h ( t n − t m ) ln ( t n m e ) − ( t m + t n ) i − ¯ d m h ( t n − t m ) ln ( t m m e ) + ( t m + t n ) io + · · · (42)8nd A P = ( p ) ebe + b ¥ (cid:229) n , m = − ¥ ( t m + t n ) n ¯ d n h (cid:18) t n e + t m b (cid:19) ln ( t n m e ) − ( t m + t n ) e i − ¯ d m h (cid:18) t n e + t m b (cid:19) ln ( t m m e ) − ( t m + t n ) b io + · · · . (43)From Eq. (20) for B = A S = e p (cid:20) (cid:18) p E c E (cid:19) + g − (cid:21) − e p z ′ ( ) + · · · , (44)and the energy-momentum tensor T mn of Eq. (33), T mn = T mn M (cid:26) + e p (cid:20) (cid:18) p E c E (cid:19) + g − (cid:21) − e p z ′ ( ) (cid:27) − g mn e E p + · · · . (45)Analogously, from Eq. (23) for E = A S = e p (cid:20) (cid:18) p E c B (cid:19) + g − (cid:21) − e p z ′ ( ) + · · · , (46)and the energy-momentum tensor T mn = T mn M (cid:26) + e p (cid:20) (cid:18) p E c B (cid:19) + g − (cid:21) − e p z ′ ( ) (cid:27) + g mn e B p + · · · . (47)In the following sections, using the energy-momentum tensors T mn of Eqs. (41), (45), and (47),we try to study the solutions of the Einstein-Euler-Heisenberg theory for nonrotating (spheri-cally symmetric), electrically or magnetically charged black holes. In this section, we study a nonrotating (spherically symmetric) electrically charged black hole.In this spherical symmetry case, the gauge potential is A m ( x ) = [ A ( r ) , , , ] , (48)corresponding to the electric field E ( r ) = − A ′ ( r ) = − ¶ A ( r ) / ¶ r in the radial direction, and themetric field is assumed to be ds = f ( r ) dt − f ( r ) − dr − r d W ; f ( r ) ≡ − Gm ( r ) / r . (49)The metric function f ( r ) and the electric field E ( r ) fulfill the Einstein equations (26) and elec-tromagnetic field equations (28) and their asymptotically flat solutions at r ≫ A ( r ) → − Q p r , E ( r ) → Q p r , Gm ( r ) r → GMr (50)9atisfy the Gauss law, where Q and M are the black hole electric charge and mass seen at infinity.In order to find the solution near to the horizon of the black hole by taking into account theQED effects, we approximately adopt the Euler-Heisenberg effective Lagrangian for constantfields that leads to the energy-momentum tensor (41) or (45) for B =
0. This approximation isbased on the assumption that the macroscopic electric field E ( r ) is approximated as a constantfield E over the microscopic scale of the electron Compton lengths. When the electric fieldof charged black holes are overcritical, electron-positron pair productions take place and theelectric field is screened down to its critical value E c (see Refs. [58–61]). In this article, we studythe QED effects on electrically charged black holes with spherical symmetry, whose electricfield is much smaller than the critical field E c . In this weak electric field case using Eq. (41) weobtain the energy-momentum tensor T mn = T mn M (cid:18) + a E p E c (cid:19) + g mn a E p E c + · · · . (51)As a result, the (0-0) component of Einstein equations is2 m ′ ( r ) r = p h E ( r ) + a p E ( r ) / E c i , (52)which relates to the energy conservation. Analogously, using Eqs. (28) and (29) and the metricof Eq. (49), we obtain the field equation up to the leading order,2 a p E ( r ) / E c + E ( r ) = Q p r , (53)which is the zero component of D m P nm = j n of Eq. (28) in the spherical symmetry case. Thisequation relates to the total charge conservation.A similar case was studied in Ref. [21], in which, however, the effective Lagrangian [the firstterm in Eq. (15)] was considered as a low-energy limit of the Born-Infeld theory; the coefficientsof the S and P terms in Eq. (15) are treated as free parameters, so as to either numerically oranalytically study the properties of spherically symmetric black hole solutions in the Einstein-Euler-Heisenberg system. In the following, in order to analytically study the QED effects on theblack hole solution, we use the Euler-Heisenberg effective Lagrangian (15) and find the blackhole solution by a series expansion in powers of a . Introducing E ( r ) ≡ E ( r ) / E c , up to the firstorder of a , the solution to Eq. (53) is approximately given by E ( r ) = E Q (cid:18) − a p E Q + · · · (cid:19) , (54)where E Q ≡ E Q ( r ) ≡ Q / ( p r E c ) . We find that the electric field E ( r ) is smaller than Q / p r ,due to the charge screening effect of the vacuum polarization. Substituting this solution (54)into the Einstein equation (52), we obtain the integration m ( r ) = M − Z ¥ r p r dr h E ( r ) + a p E ( r ) / E c i . (55)10his equation clearly shows that the energy-mass function m ( r ) of Eq. (49) is the total grav-itational mass M (attractive) “screened down” by the electromagnetic energy (repulsive). Inthe Maxwell theory ( D L coseff ) P = E ( r ) = Q / ( p r ) , we obtain the Reissner-Nordstr¨omsolution m ( r ) = M − Q / p r . In the Euler-Heisenberg system, it is not proper to make the inte-gration in Eq. (55), since the integrand comes from the Euler-Heisenberg effective Lagrangian,which is valid only for constant fields. In order to gain some physical insight into the energy-mass function (55), we integrate Eq. (55) to the leading order of a , m ( r ) ≈ M − Q p r (cid:20) − a p ( p ) Q r E c (cid:21) = M − Q p r h − a p E Q i , (56)which shows the QED correction to the Reissner-Nordstr¨om solution. Due to the QED vacuumpolarization effect, the black hole charge Q is screened Q → Q h − a p E Q i / . (57)As a consequence, the electrostatic energy of Eq. (56) is smaller than Q / ( p r ) in the Reissner-Nordstr¨om solution.Moreover, we study the QED correction to the black hole horizon. For this purpose, wedefine the horizon radius r H at which the function f ( r ) of Eq. (49) vanishes, i.e., f ( r H ) = Gm ( r H ) r H = . (58)Using the energy-mass function m ( r ) of Eq. (56), we obtain GMr H − GQ p r H h − a p E Qh i = , (59)where E Qh ≡ E Q ( r H ) . Up to the leading order of a , we obtain r H + = GM + r G M − GQ p h − a p E Q + i , (60) r H − = GM − r G M − GQ p h − a p E Q − i , (61)where E Q + ≡ E Q ( r H + ) and E Q − ≡ E Q ( r H − ) . Equation (60) shows that the black hole horizonradius r H + becomes larger than the Reissner-Nordstr¨om one r + given by Eq. (60) for setting a =
0. The black hole horizon area 4 p r H + becomes larger than the Reissner-Nordstr¨om one4 p r + given by Eq. (60) for setting a =
0. This is again due to the black hole charge Q screenedby the QED vacuum polarization (57).In the Reissner-Nordstr¨om solution, the extreme black hole solution is given by r + = r − or4 p GM = Q . In our case, this is given by r H + = r H − = r H yielding G M − GQ p h − a p E Qh i = . (62)11rom Eqs. (60) and (61), we obtain4 p r H = p G M = GQ h − a p E Qh i = GQ (cid:20) − a p G Q E c (cid:21) , (63) r H ± ≈ Q h − a p E Qh i / = Q (cid:20) − a p ( E c Q ) (cid:21) / . (64)In Eq. (64) we adopt G / p =
1. Due to the QED correction, the condition of extremely electri-cally charged black holes with spherical symmetry changes from M = Q / p to M = Q p (cid:20) − a p ( E c Q ) (cid:21) / . (65)This implies that for a given M , the black holes are allowed to carry more charge Q than theReissner-Nordstr¨om case. These results show that when the black hole mass M is fixed, thehorizon area and radius of the extremely electrically charged black hole are the same as theextreme Reissner-Nordstr¨om one. However, when the black hole charge Q is fixed, the blackhole horizon area and radius are smaller than those of the extreme Reissner-Nordstr¨om blackhole. The reason is that the charge screening effect decreases the electrostatic energy; hence,this leads to a smaller mass M for the extreme black hole.Now we turn to the maximal energy extractable from a black hole. As pointed out inRef. [62], the surface area S a of the black hole horizon is related to the irreducible mass M ir of the black hole S a = p G M ir = p r H + , (66)where r H + is given by Eq. (60). The surface area of the black hole horizon cannot be decreasedby classical processes [62–64]. Any transformation of the black hole which leaves fixed the irre-ducible mass is called reversible [62, 63]. Any transformation of the black hole which increasesits irreducible mass, for instance, the capture of a particle with nonzero radial momentum atthe horizon, is called irreversible. In irreversible transformations there is always some kineticenergy that is irretrievably lost behind the horizon. Note that transformations which arbitrar-ily close to reversible ones are the most efficient transformations for extracting energy froma black hole [62, 63]. Following the same argument presented in Ref. [62], and including theleading-order QED correction (56), we obtain the Christodoulou-Ruffini mass formula M = M ir + Q p GM ir h − a p E Q + i , (67)where the electrostatic energy of the black hole is reduced for the reason that the black holecharge is screened down by the QED vacuum polarization effect (57).The properties of the surface area S a of the black hole horizon and irreducible mass M ir canalso been understood from the concepts of information theory [65]. The black hole entropy S en
12s introduced as the measure of information about a black hole interior which is inaccessible toan exterior observer and is proportional to the surface area S a of the black hole horizon [65] S en = S a / = p r H + . (68)The physical content of the concept of the black hole entropy derives from the generalizedsecond law of thermodynamics: when common entropy in the black hole exterior plus the blackhole entropy never decreases [65]. In the Einstein-Euler-Heisenberg theory, the black holeirreducible mass of Eq. (66) and entropy of Eq. (68) with the QED correction are determined bythe horizon radius r H + of Eq. (60) for charged black holes and Eq. (63) for extreme black holes.Now we consider the physical interpretation of the electromagnetic term in Eq. (67). Thisterm represents the maximal energy extractable from a black hole, which can be obtained byevaluating the conserved Killing integral [38, 66] Z S + t x m + T mn d S n = p Z ¥ r H + r T dr , (69)where S + t is the spacelike hypersurface in the space-time region that is outside the horizon r > r H + described by the equation t = constant, with d S n as its surface element vector. x m + is the static Killing vector field. This electromagnetic term in Eq. (67) is the total energy ofthe electromagnetic field and includes its own gravitational binding energy. Using the energy-momentum tensor of Eq. (51) and weak-field solution (54), we obtain the maximal energyextractable from an electrically charged black hole e ex = Q p r H + h − a p E Q + i . (70)This shows that the black hole maximal extractable energy decreases in comparison with theReissner-Nordstr¨om case ( Q / p r + ). This can be explained by the following: (i) the chargescreening effect decreases the electrostatic energy; (ii) the black hole horizon radius r H + ofEq. (60) increases, leading to the decrease of the maximally extractable energy, because themost efficient transformations that extract energy from a black hole occur near the horizon. Forthe extremely electrically charged black hole, the maximally extractable energy is the same asthat in the Reissner-Nordstr¨om case, when the black hole mass M is fixed; however, it becomessmaller than the Reissner-Nordstr¨om one when the black hole electric charge Q is fixed. Now we turn to study the Einstein-Euler-Heisenberg theory (41) and (47) in the presence of themagnetic field B . As shown by Eq. (6), the magnetic field B does not contribute to the pair-production rate so that the process of the electron-positron pair production does not occur for astrong magnetic field B . For this reason, we consider black holes with strong magnetic fields.The conventional black hole with electric and magnetic fields is the rotating charged black hole13f the Kerr-Newman black hole [67]. However, the solution to a rotating charged black holein the Einstein-Euler-Heisenberg theory is rather complicated, and we do not consider it in thiswork. For the sake of simplicity, we study the nonrotating magnetically charged black hole withspherical symmetry in order to investigate the QED corrections in the presence of the magneticfield B in the Einstein-Euler-Heisenberg theory.For a nonrotating magnetically charged black hole with magnetic charge Q m , the tensor F mn compatible with spherical symmetry can involve only a radial magnetic field F = − F . In theEinstein-Maxwell theory, the field equations (28) give (see, e.g., Refs. [68, 69]) F = Q m sin q p , (71)and the gauge potential will be (see, e.g., Refs. [68]) A m ( x ) = [ , , , Q m ( − cos q ) / p ] . (72)The metric is similar to the one of nonrotating electrically charged black holes, ds = f ( r ) dt − f ( r ) − dr − r d W , f ( r ) ≡ − Gm ( r ) / r , (73)where m ( r ) is the mass-energy function. In the Einstein-Maxwell theory, the metric func-tion f ( r ) of magnetically charged black holes with spherical symmetry is given by (see, e.g.,Refs. [68]) f ( r ) = − GMr + GQ m p r , (74)where M is the black hole mass seen at infinity. Using Eq. (41), we obtain the energy-momentum tensor for the weak magnetic field B case, T mn = T mn M (cid:18) − a B p E c (cid:19) + g mn a B p E c + · · · . (75)Similar to the analysis of electrically charged black holes with spherical symmetry, we obtainthe (0-0) component of Einstein equations,2 m ′ ( r ) r = p h B ( r ) − a p B ( r ) / E c i . (76)For the magnetically charged black hole with spherical symmetry, only a radial magnetic field ispresent. The field equations (28) give B ( r ) = Q m / ( p r ) (see, e.g., Refs. [21, 28]). Substituting B ( r ) into the Einstein equation (76), we obtain the mass-energy function m ( r ) = M − Z ¥ r p r dr h B ( r ) − a p B ( r ) / E c i . (77)14eglecting the QED correction of the Euler-Heisenberg effective Lagrangian, Eq. (77) gives m ( r ) = M − Q m / p r , which is the solution of the magnetically charged Reissner-Nordstr¨omblack hole in the Einstein-Maxwell theory. Making the integration in Eq. (77), one obtains [21] m ( r ) = M − Q m p r (cid:20) − a p ( p ) Q m r E c (cid:21) = M − Q m p r h − a p B Q i , (78)where B Q ≡ B Q ( r ) ≡ Q m / ( p r E c ) . As shown in Eq. (78), taking into account the QED vacuumpolarization effect, the total magnetostatic energy is smaller than Q m / p r in the magneticallycharged Reissner-Nordstr¨om case. This can be understood as follows. In the magnetic field B of the black holes, the vacuum polarization effect results in a positive magnetic polarization M . Then the magnetic H field defined B = H + M is smaller than the magnetic field B . Themagnetostatic energy density e EM (cid:181) B · H decreases. This shows that in weak magnetic fields,the vacuum polarization effect exhibits the paramagnetic property.Compared to the result of the electrically charged black hole in the first order of a , Eqs. (56)and (78) have the same expression. One can obtain Eq. (78) by simply replacing E Q in Eq. (56)by B Q , namely, replacing Q by Q m because of the duality symmetry (see, e.g., Ref. [68]).Similar to the analysis of electric charged black holes, we obtain the horizon radii r H + and r H − of the magnetically charged black hole, up to the leading order of a , r H + = GM + r G M − GQ m p h − a p B Q + i , (79) r H − = GM − r G M − GQ m p h − a p B Q − i , (80)where B Q + ≡ B Q ( r H + ) and B Q − ≡ B Q ( r H − ) . The result (79) shows that the black hole hori-zon radius r H + increases in comparison with the magnetically charged Reissner-Nordstr¨om one r + . This is again due to the paramagnetic effect of the vacuum polarization that decreases themagnetostatic energy of the black hole.Now we turn to the extreme black hole ( r H + = r H − = r H ). Similarly, we have G M − GQ m p h − a p B Qh i = , (81)where B Qh ≡ B Q ( r H ) , and we obtain the black hole horizon area and radius4 p r H = p G M = GQ m h − a p B Qh i = GQ m (cid:20) − a p G Q m E c (cid:21) , (82) r H ≈ Q m h − a p B Qh i / = Q m (cid:20) − a p ( E c Q m ) (cid:21) / . (83)In the second line, we adopt G / p =
1. The QED correction changes the condition of extremelymagnetically charged black holes with spherical symmetry from M = Q m / p to M = Q m p (cid:20) − a p ( E c Q m ) (cid:21) / . (84)15he properties of the horizon area and radius of the extremely magnetically charged black holeare the same as their counterparts in the extremely electrically charged black hole, given by theduality transformation Q ↔ Q m .Following the same argument presented in Ref. [62], we obtain the Christodoulou-Ruffinimass formula M = M ir + Q m p GM ir h − a p B Q + i (85)for magnetically charged black holes with spherical symmetry in the Einstein-Euler-Heisenbergtheory. One is able to obtain the irreducible mass M ir by substituting Eq. (79) into Eq. (66), andthe black hole entropy S en by substituting Eq. (79) into Eq. (68). The irreducible mass M ir and the black hole entropy S en in terms of black hole horizon radius r H + Eq. (79) have thesame paramagnetic property in the presence of the QED vacuum polarization effect, as alreadydiscussed.As shown in Eq. (85), the maximal energy extractable from a magnetically charged blackhole is e ex = Q m p r H + h − a p B Q + i , (86)where r H + is given by Eq. (79). The result shows that the maximal energy extractable froma magnetically charged black hole is smaller than Q m p r + of the magnetically charged Reissner-Nordstr¨om black hole. The reasons are the following: (i) the vacuum polarization effect de-creases the magnetostatic energy; (ii) the black hole horizon radius r H + of Eq. (79) increases,therefore the maximally extractable energy decreases. The maximal energy extractable from anextremely magnetically charged black hole is the same as that from an extremely magneticallycharged Reissner-Nordstr¨om black hole when the black hole mass M is fixed, while it decreaseswhen the black hole magnetic charge Q m is fixed, as we have already discussed at the end ofSec. 4 for the case of the extremely electrically charged black hole. In this section, we study the magnetically charged black holes with a strong magnetic field B ( r ) .From Eq. (47), we obtain the energy-momentum tensor of the magnetically charged black holewith spherical symmetry in the strong magnetic field case. Analogous to the weak magneticfield case of magnetically charged black holes with spherical symmetry, we obtain the (0-0)component of Einstein equations2 m ′ ( r ) p r = p (cid:26) B ( r ) + e B p (cid:20) ln (cid:18) p E c B (cid:19) + g − p z ′ ( ) (cid:21)(cid:27) , (87)16nd the field equations (28) give B ( r ) = Q m / ( p r ) . Substituting this magnetic field B ( r ) intothe Einstein equation (87), we obtain m ( r ) ≈ M − Z ¥ r p r dr (cid:26) B + e B p (cid:20) ln (cid:18) p E c B (cid:19) + g − p z ′ ( ) (cid:21)(cid:27) (88) ≈ M − Q m p r (cid:26) + a p (cid:20) ln (cid:18) p B Q (cid:19) + g + − p z ′ ( ) (cid:21)(cid:27) . (89)This result is valid for B ≫ E c , for which the value of ln ( p / B Q ) + g + − p z ′ ( ) is negative.As a result, Eq. (89) shows that the total magnetostatic energy in the presence of the vacuumpolarization is smaller than Q m / p r of the magnetically charged Reissner-Nordstr¨om blackhole. Similar to the weak-field case, this is again due to the paramagnetic effect of the vacuumpolarization that decreases the magnetostatic energy of black holes. In the strong magnetic fieldcase, the QED vacuum polarization effect is much larger than the result (78) in the weak-fieldcase, where the QED correction term in Eq. (78) is small for the smallness of a / ( p ) and B Q . This result (89) shows a significant QED effect of the vacuum polarization on the energyof magnetically charged black holes in the strong magnetic field case.Now we turn to the study of the black hole horizon radius and area in the strong magneticfield case. Using the condition f ( r H ) =
0, we obtain the horizon radii r H + and r H − up to theleading order of a , r H + = GM + r G M − GQ m p h + a p K NR + i , (90) r H − = GM − r G M − GQ m p h + a p K NR − i , (91)where K NR + = ln (cid:18) p B Q + (cid:19) + g + − p z ′ ( ) , (92) K NR − = ln (cid:18) p B Q − (cid:19) + g + − p z ′ ( ) . (93)Equation (90) shows that the horizon radius r H + increases in comparison with the magneticallycharged Reissner-Nordstr¨om one r + . This is again due to the paramagnetic effect of the vacuumpolarization that decreases the magnetostatic contribution to the total energy of black holes.For the case of the extreme black hole ( r H + = r H − = r H ), we have G M − GQ m p h + a p K NR i = , (94)where K NR = ln (cid:18) p B Qh (cid:19) + g + − p z ′ ( ) . (95)17s a result, we obtain 4 p r H = p G M = GQ m h + a p K NR i , (96) r H ≈ Q m h + a p K NR i / . (97)Similar to the weak magnetic field case, the QED correction changes the condition of extremelymagnetically charged black holes with spherical symmetry from M = Q m / p to M = Q m p h + a p K NR i / . (98)These results show that the horizon area and radius of the extreme black hole are the same astheir counterparts of the extremely magnetically charged Reissner-Nordstr¨om black hole, whenthe black hole mass M is fixed. Whereas, the black hole magnetic charge Q m is fixed, Eqs. (96)and (97) show that the black hole horizon area and radius become smaller than their counterpartsof extremely magnetically charged Reissner-Nordstr¨om black holes. We have discussed thisbehavior in Eqs. (62)-(65) for the case of extremely electrically charged black holes.Analogously, we obtain the Christodoulou-Ruffini mass formula in the strong-field case ofmagnetically charged black holes, M = M ir + Q p GM ir h + a p K NR + i . (99)It is straightforward to obtain irreducible mass M ir by substituting Eq. (90) into Eq. (66), andthe black hole entropy S en by substituting Eq. (90) into Eq. (68). Analogous to the case of theelectrically charged black hole, the black hole irreducible mass M ir and entropy S en in the strongmagnetic field case depend on the black hole horizon radius r H + of Eqs. (90) and (96). Equation(99) indicates that the maximal energy extractable from a magnetically charged black hole is e ex = Q m p r H + h + a p K NR + i . (100)The properties of the maximally extractable energy in the strong magnetic field case are similarto those of the magnetically charged black hole in the weak magnetic field case. However, theQED correction of the vacuum polarization effect to the energy of the magnetically chargedblack hole in the strong magnetic field case is much more significant in comparison with that inthe weak magnetic field case. If the spherically symmetric (nonrotating) black hole is both electrically and magneticallycharged, electric and magnetic fields do not vanish. As shown in Eq. (11), both invariants S and P contribute to the Euler-Heisenberg effective Lagrangian. The metric takes the same18orm as the metric of Eq. (49) for electrically charged black holes with spherical symmetry. Inthis case, the tensor F mn compatible with spherical symmetry can involve only a radial electricfield F = − F and a radial magnetic field F = − F , and the gauge potential is (see, e.g.,Ref. [68]) A m ( x ) = [ A ( r ) , , , Q m ( − cos q ) / p ] . (101)In the Einstein-Maxwell theory, A ( r ) = − Q / ( p r ) , and the metric function f ( r ) of Eq. (49) isgiven by (see, e.g., Ref. [68]) f ( r ) = − GMr + GQ p r + GQ m p r . (102)In the Einstein-Euler-Heisenberg theory, we study the spherically symmetric black hole withelectric and magnetic charges in the weak-field case. Using Eq. (41), we derive the energy-momentum tensor with a radial electric field E and a radial magnetic field B , T mn = T mn M (cid:20) + a p E c ( E − B ) (cid:21) + g mn a p E c (cid:2) ( E − B ) + ( E · B ) (cid:3) + · · · . (103)Analogous to the analysis of electrically/magnetically charged black holes with spherical sym-metry, we obtain the (0-0) component of Einstein equations,2 m ′ ( r ) r = p (cid:20) E ( r ) + B ( r ) + a p E ( r ) / E c − a p B ( r ) / E c + a p E c E ( r ) B ( r ) (cid:21) . (104)In addition, we obtain the field equations from Eq. (28) (see also Ref. [21]), E ( r ) + a p E ( r ) / E c + a B p E c E ( r ) = Q p r , (105) B ( r ) = Q m p r . (106)Note that the mixing terms of the electric and magnetic fields in Eqs. (104) and (105) comefrom the contribution of the invariant P . Introducing E ( r ) ≡ E ( r ) / E c , we have E ( r ) = E Q − a p E Q − a p B Q E Q + · · · , (107)up to the first order of a . We substitute the solutions of (106) and (107) into the Einsteinequation (104) and obtain the mass-energy function m ( r ) = M − Z ¥ r p r dr E c h E Q + B Q − a p E Q − a p B Q − a p B Q E Q i . (108)Disregarding the QED correction of the Euler-Heisenberg effective Lagrangian, Eq. (108) givesthe solution m ( r ) = M − Q / p r − Q m / p r for the Reissner-Nordstr¨om black hole with electricand magnetic charges. Performing the integration in Eq. (108), we approximately obtain m ( r ) = M − Q p r h − a p E Q i − Q m p r h − a p B Q i + a p Q m p r E Q . (109)19n the limit Q ≫ Q m , Eq. (109) becomes Eq. (56) of the electrically charged black hole. On thecontrary, in the limit Q m ≫ Q , Eq. (109) becomes Eq. (78) of the magnetically charged blackhole. In order to study the effect of the P term in the Euler-Heisenberg effective Lagrangian, weconsider the case with large P and small S , i.e., Q m ≈ Q . In this situation, Eq. (109) becomes m ( r ) = M − Q p r (cid:20) − a p E Q (cid:21) , (110)for Q m = Q , i.e., S = P . Comparing to the cases of electrically/magnetically chargedblack holes, the QED correction to the black hole energy becomes larger, which results fromthe combination effects of the vacuum polarization on electric and magnetic charges of blackholes in the Einstein-Euler-Heisenberg theory.In the same way that has been discussed in previous sections, up to the leading order of a ,we obtain the horizon radii r H + and r H − from Eq. (110), r H + = GM + s G M − GQ p (cid:20) − a p E Q + (cid:21) , (111) r H − = GM − s G M − GQ p (cid:20) − a p E Q − (cid:21) , (112)and the Christodoulou-Ruffini mass formula M = M ir + Q p GM ir (cid:20) − a p E Q + (cid:21) , (113)as well as the maximal energy extractable from a black hole e ex = Q p r H + (cid:20) − a p E Q + (cid:21) . (114)Analogously, we obtain the irreducible mass M ir by substituting Eq. (111) into Eq. (66), andthe black hole entropy S en by substituting Eq. (111) into Eq. (68). The irreducible mass M ir , theblack hole entropy S en , and the maximal energy extractable from a black hole receive the sameQED correction, but a factor of 7 / In this article, in addition to the Maxwell Lagrangian, we consider the contribution from theQED Euler-Heisenberg effective Lagrangian to formulate the Einstein-Euler-Heisenberg the-ory. On the basis of this theory, we study the horizon radius, area, total energy, entropy, andirreducible mass as well as the maximally extractable energy of spherically symmetric (non-rotating) black holes with electric and magnetic charges. Our calculations are made up to theleading order of the QED corrections in the limits of strong and weak fields. Our results show20hat the QED correction of the vacuum polarization results in the increase of the black holehorizon area, entropy and irreducible mass, as well as the decrease of the black hole total en-ergy and maximally extractable energy. The reason is that the QED vacuum polarization givesrise to the screening effect on the black hole electric charge and the paramagnetic effect on theblack hole magnetic charge. The condition of the extremely charged black hole M = Q / p or M = Q m / p is modified [ see Eqs. (65), (84), and (98)], which results from the screening andparamagnetic effects.To end this article, we would like to mention that in the Einstein-Euler-Heisenberg theory,it is worthwhile to study Kerr-Newman black holes, whose electric field E and magnetic field B are determined by the black hole mass M , charge Q , and angular momentum a [67]. Inaddition, it will be interesting to study the QED corrections in black hole physics by taking intoaccount the one-loop photon-graviton amplitudes of the effective Lagrangian (34) [48] and itsgeneralizations [49–53]. We leave these studies for future work. Acknowledgements
Yuan-Bin Wu is supported by the Erasmus Mundus Joint Doctorate Program by Grant Num-ber 2011-1640 from the EACEA of the European Commission. We thank the anonymous ref-eree for drawing our attention to the one-loop photon-graviton amplitudes.
References