Black hole as a magnetic monopole within exponential nonlinear electrodynamics
aa r X i v : . [ g r- q c ] M a r Black hole as a magnetic monopole within exponential nonlinearelectrodynamics
S. I. Kruglov Department of Chemical and Physical Sciences, University of Toronto,3359 Mississauga Road North, Mississauga, Ontario L5L 1C6, Canada
Abstract
We perform the gauge covariant quantization of the exponentialmodel of nonlinear electrodynamics. Magnetically charged black holes,in the framework of our model are considered, and the regular blackhole solution is obtained in general relativity. The asymptotic blackhole solution at r → ∞ is found. We calculate the magnetic mass ofthe black hole and the metric function which are expressed via the pa-rameter β of the model and the magnetic charge. The thermodynamicproperties and thermal stability of regular black holes are analysed.We calculate the Hawking temperature of black holes and their heatcapacity at the constant magnetic charge. We find a point where thetemperature changes the sign that corresponds to the first-order phasetransition. It is shown that at critical point, where the heat capacitydiverges, there is a phase transition of the second-order. We obtainthe parameters of the model when the black hole is stable. Classical electrodynamics is modified due to quantum corrections and be-comes nonlinear electrodynamics (NLED). Thus, one-loop corrections in QEDlead to nonlinear Heisenberg-Euler electrodynamics [1] which admits thephenomenon of vacuum birefringence. This effect, when indexes of refrac-tion in the presence of the external magnetic field depend on polarizationstates, is now under experimental verification by PVLAS and BMV collab-orations. Therefore, viable models of NLED should describe the birefrin-gence phenomenon. In well-known Born-Infeld electrodynamics [2] the effectof birefringence is absent. But in the modified Born-Infeld electrodynam-ics, containing two parameters, the birefringence phenomenon takes place E-mail: [email protected] r → ∞ .In this paper we consider exponential nonlinear electrodynamics, pro-posed in [15], coupled to GR. We investigate magnetically charged blackholes and obtain solutions similar to RN solution with some corrections.The thermodynamics of such black holes is also studied. We demonstratethat there are the first-order and the second order phase transitions in blackholes.The structure of the paper is as follows. It was shown in Section 2 thatcausality and unitarity principles are satisfied in our model. We performthe gauge covariant Dirac quantization of the exponential electrodynamics.NLED coupled with GR is investigated in Section 3 and we obtain black holesolutions. We find corrections to RN solution at r → ∞ . In this Section themagnetic mass and metric function are obtained. We show that weak energycondition holds in exponential nonlinear electrodynamics. It is demonstratedthat only at b = 2 / √ β/ ( qG ) ≤ .
83 there are two or one horizons. We showthat there are not singularities of Ricci’s scalar at r → ∞ and at r → c = ¯ h = 1 and the metric tensor signature η = diag( − , , , , , ,
3, while for the spatial in-dexes, designated by the Latin letters, the values are 1 , , The gauge covariant Dirac quantization ofexponential electrodynamics
Let us consider exponential nonlinear electrodynamics proposed in [15]. TheLagrangian density of exponential electrodynamics reads L = −F exp( − β F ) , (1)where F = (1 / F µν F µν = ( B − E ) / F µν = ∂ µ A ν − ∂ ν A µ . The parameter β possesses the dimension of the (length) and the upper bound on the β ( β ≤ × − T − ) was found from PVLAS experiment [15]. The fieldequations follow from the Lagrangian density (1) [15] ∇ · D = 0 , ∂ D ∂t − ∇ × H = 0 , (2)where the electric displacement field is given by D = E (1 − β F ) exp( − β F ) , (3)and the magnetic field is H = B (1 − β F ) exp( − β F ) . (4)Equations of vacuum nonlinear electrodynamics correspond to the continuousmedia electrodynamics equations with the specific constitutive relations (3)and (4). The second pair of the Maxwell equations, that is the consequenceof the Bianchi identity, reads ∇ · B = 0 , ∂ B ∂t + ∇ × E = 0 . (5)The theory is viable if general principles of causality and unitarity are sat-isfied. The causality principle tells us that the group velocity of excitationsover the background should be less than the light speed. This gives therequirement L F ≡ ∂ L /∂ F ≤ L F = − (1 − β F ) exp( − β F ) . (6)Thus, at β F ≤ E = 0), which will be considered, this requirement3s B ≤ √ / √ β . The unitarity principle requires L F + 2 F L FF ≤ L FF ≥ L FF = β (2 − β F ) exp( − β F ) , L F + 2 F L FF = ( − β F − β F ) exp( − β F ) . (7)We find from Eqs. (7) that the unitarity principle gives the restriction β F ≤ (5 − √ / ≃ . β F ≤ (5 − √ /
4. For the case E = 0 this gives B ≤ q (5 − √ / (2 β ) ≃ . / √ β .The Lagrangian density (1) is invariant under gauge transformations de-scribed by the U (1) group. The phase space, as in any field theory, is infinitedimensional. The Lagrangian, corresponding to Lagrangian density (1), isgiven by L = R d x L , and the action is I = R dtL . We will study the timeevolution of fields, and therefore, the formalism looks like non-relativistic al-though the theory is Lorentz covariant. The “coordinates” and “velocities”here are A µ and ∂A µ /∂t ≡ ∂ A µ , respectively. From Eq. (1) according tothe Dirac procedure [16] we find the momenta π i = ∂ L ∂ ( ∂ A i ) = − E i (1 − β F ) exp( − β F ) = − D i , π = ∂ L ∂ ( ∂ A ) = 0 . (8)Thus, the spatial part of the momentum π equals the displacement field D with the opposite sign. From second equation in (8) one finds the primaryconstraint ϕ ( x ) ≡ π , ϕ ( x ) ≈ . (9)We use Dirac’s symbol ≈ for the equation that holds only weakly, i.e. ϕ ( x )can possess nonzero Poisson brackets with some variables. Equations (9)represent an infinite set of constraints for every spatial coordinate x . Withthe help of the Poisson brackets { A i , π j } = δ ij δ ( x − y ), and using equation π j = − D j , we arrive at { A i ( x , t ) , D j ( y , t ) } = − δ ij δ ( x − y ) . (10)4ultiplying Eq. (10) by the the operator ǫ mki ∂/∂x k ( ǫ mki is the antisym-metric Levi-Civita symbol) and performing a summation over repeated in-dexes, one obtains the Poisson brackets between the magnetic induction field B = ∇ × A and the electric displacement field D { B m ( x , t ) , D j ( y , t ) } = ǫ mjk ∂∂x k δ ( x − y ) . (11)Equation (11) also takes place in Born-Infeld electrodynamics [17], [16]. Inthe quantum field theory, the Poisson brackets should be replaced by thequantum commutator, { B, D } → − i [ B, D ], where [
B, D ] = BD − DB .With the help of Eqs. (1) and (8), and the relation H = π µ ∂ A µ − L , we findthe Hamiltonian density: H = D i E i + F exp( − β F ) + π m ∂ m A . (12)Because the primary constraint (9) should be a constant of motion, we arriveat the equation ∂ π = { π , H } = ∂ m π m = 0 . (13)Here H = R d x H is the Hamiltonian. Equation (13) guarantees that theprimary constraint (9) is conserved and represents the Gauss law as π i = − D i . From Eq. (13) we arrive at the secondary constraint ϕ ( x ) ≡ ∂ m π m , ϕ ( x ) ≈ . (14)Both constraints (9) and (14) can be considered on the same footing. Wenote that the weak equality ≈ is not compatible with the Poisson brackets[16]. The time evolution of the second constraint is given by ∂ ϕ = { ϕ , H } ≡ . (15)Equation (15) shows that there are no additional constraints. Because { ϕ , ϕ } =0 there are not second class constraints. In Maxwell’s electrodynamics andNLED [16], [27], [8] second class constraints are absent. To obtain the to-tal Hamiltonian density, according to the Dirac method [16], we add to thedensity of the Hamiltonian the Lagrange multiplier terms v ( x ) π , u ( x ) ∂ m π m , H T = D i E i + F exp( − β F ) + π m ∂ m A + v ( x ) π + u ( x ) ∂ m π m , (16)where the functions v ( x ), u ( x ) do not possess physical meaning and are aux-iliary variables that are connected with gauge degrees of freedom. The first5lass constraints in Eq. (16) generate the gauge transformations. Thus, Eq.(16) represents the set of Hamiltonians. The physical space is the constantsurface and one can get the energy density from the Hamiltonian on the con-straint surface. As a result, the density of energy, obtained from Eq. (16), isgiven by ρ = D i E i + F exp( − β F ) . (17)The energy density (17) can be obtained also from the energy-momentumtensor T µν by the relation ρ = T [15]. One must represent the total densityof Hamiltonian (16) in terms of fields A µ and momenta π µ to obtain equationsof motion. Then we find the Hamiltonian equations ∂ A i = { A i , H } = δHδπ i = − E i + ∂ i A − ∂ i u ( x ) , (18) ∂ π i = { π i , H } = − δHδA i = − ǫ ijk ∂ j H k , (19) ∂ A = { A , H } = δHδπ = v ( x ) , ∂ π = { π , H } = − δHδA = ∂ m π m , (20)where H = R d x H T . Equation (18) represents the gauge covariant formof equation for the electric field. Equation (19) is equivalent to the secondequation in (2), and Gauss’s law is the second constraint in this Hamiltonianformalism. As the function u ( x ) is arbitrary, we may introduce new function u ′ ( x ) = u ( x ) − A and the Hamiltonian (16), after the integration by partsto get the term A ∂ m π m , will not contain the A . Thus, the component A is not the physical degree of freedom. For a particular case v ( x ) = ∂ u ′ ( x ),one finds from Eqs. (18), (20) the relativistic form of gauge transformations A ′ µ ( x ) = A µ ( x ) − ∂ µ Λ( x ), where Λ( x ) = R dtu ′ ( x ). There are two arbitraryfunctions u ′ ( x ), v ( x ) in the general case. The Hamiltonian equations (18),(19) give the time evolution of physical fields that are equivalent to theEuler-Lagrange equations, and Eqs. (20) represent the time evolution ofnon-physical fields A and π which are connected with the gauge degrees offreedom, and the variables π , ∂ m π m equal zero as constraints.The dynamical variables ˆ A i and ˆ π i = − ˆ D i have in quantum theory thecommutator h ˆ A i ( x , t ) , ˆ D j ( y , t ) i = − iδ ij δ ( x − y ) (21)and the wave function | Ψ i obeys the Schr¨odinger equation i d | Ψ i dt = H | Ψ i , (22)6nd the equations as follows [16]:ˆ D | Ψ i = 0 , ∂ m ˆ D m | Ψ i = 0 , (23)where ˆ D = − ˆ π . As a result, the physical state is invariant under the gaugetransformations. Eqs. (23) give restrictions on the physical state | Ψ i whichis gauge invariant. The physical fields E , B , D , H are represented by theHermitian operators and do not depend on A and they are invariants of thegauge transformations. One can apply for exponential electrodynamics thegauge fixing method which is beyond the Dirac’s approach [28], [29]. The action of our model of exponential electrodynamics in general relativityis I = Z d x √− g (cid:20) κ R + L (cid:21) , (24)where R is the Ricci scalar, κ = 8 πG ≡ M − P l , G is Newton’s constant, and M P l is the reduced Planck mass. The Einstein and NLED equations followfrom action (24) R µν − g µν R = − κ T µν , (25) ∂ µ h √− gF µν (1 − β F ) exp( − β F i = 0 , (26)where the symmetric energy-momentum tensor of our model is given by [15]: T µν = exp( − β F ) h ( β F − F µλ F νλ + g µν F i . (27)It should be mentioned that Dirac-type magnetic monopole solution existswithout sources in (26) [30]. Electrically charged black holes and correspond-ing solutions were considered in [31]. Our goal here is to obtain the staticmagnetic black hole solutions to Eqs. (25) and (26). It was shown in [30] thatthe invariant F compatible with the spherical symmetry, for pure magneticfield, is given by F = q / (2 r ), where q is a magnetic charge. The sphericallysymmetric line element in this case is given by ds = − A ( r ) dt + 1 A ( r ) dr + r ( dϑ + sin ϑdφ ) , (28)7here the metric function A ( r ) reads A ( r ) = 1 − GM ( r ) r , (29)and the mass function M ( r ) is M ( r ) = Z r ρ ( r ) r dr = m − Z ∞ r ρ ( r ) r dr. (30)Here m = R ∞ ρ ( r ) r dr is the magnetic mass of the black hole, and the energydensity for the case E = 0, found from Eq. (17), is as follows: ρ = q r exp − βq r ! . (31)From Eqs. (30) and (31) we obtain the mass function M ( r ) = q Z r drr exp − βq r ! = q / / β / Γ , βq r ! , (32)where Γ( s, x ) is incomplete gamma function given by the expressionΓ( s, x ) = Z ∞ x t s − e − t dt. (33)From Eq. (32) we find the magnetic mass of the black hole m = M ( ∞ ) = q / Γ(1 / / β / ≃ . q / β / . (34)Taking into account Eqs. (29) and (32) one obtains the metric function A ( r ) = 1 − Gq / / β / r Γ , βq r ! . (35)From Eq. (35) we find the asymptotic value of the metric function at r → ∞ A ( r ) = 1 − Gmr + Gq r − βGq r + O ( r − ) . (36)The solution (36) is similar to the RN solution with some corrections in theorder of O ( r − ). At the limit r → ∞ the spacetime asymptotically becomes8he Minkowski spacetime. If β = 0 we arrive at Maxwell’s electrodynamicsand solution (36) is the RN solution. The asymptotic value of the metricfunction at r →
0, obtained from Eq. (35), is given by A ( r ) = 1 − exp − βq r ! " Gr β − Gr β q + O ( r ) . (37)Equation (37) shows that we have the regular black hole solution at r → x = (2 / ( βq )) / r and b = 2 / √ β/ ( qG ), we canrepresent the metric function (35) as follows: A ( x ) = 1 − Γ (cid:16) , x (cid:17) bx . (38)One can find horizons by solving the equation A ( r ) = 0. Internal Cauchy x − and event x + horizons for different parameters b are given in Table 1. TheTable 1: Internal Cauchy x − and event x + horizons b x − x +
35 17 11 7.76 5.89 5.894 3.5 2.6593 2.406 2.1453plot of the function Γ (cid:16) , x (cid:17) /x is represented in Fig. 1. According to Fig. 1there can be one, two or no horizons. At b > .
83 there are not horizons thatlead to a particle-like solution. The function (38) for b = 1 , . , . b ≃ .
83 there is one solution whichcorresponds to the extremal black hole. At b < .
83 we have two horizons ofthe regular black hole.The trace of the energy-momentum tensor, obtained from (27), reads
T ≡ T µµ = 4 β F exp( − β F ) = βq r exp − βq r ! . (39)We can obtain the Ricci scalar from Einstein’s equation (25) R = κ T = κ βq r exp − βq r ! . (40)9 b Figure 1: The plot of the function b = Γ (cid:16) , x (cid:17) /x .It follows from Eq. (40) that at r → ∞ and at r → R →
0, i.e. there are no singularities of Ricci’s curvature.However the Kretschmann scalar possesses the singularity only at r = 0 (seeAppendix). Therefore, at r → ∞ spacetime becomes flat. It should bementioned that regular black hole solution in GR coupled to NLED wasobtained in [32]. Let us consider the weak energy condition (WEC) [33]which guarantees that the energy density is positive for any local observer.For a system which obeys the spherical symmetry the radial magnetic field is B ( r ) = F = − F and the components of the energy-momentum tensor are ρ = T = T rr = − p r , where p r is a radial pressure. As a result ρ + p r = 0.WEC reads: ρ ≥ ρ + p r ≥ ρ + p ⊥ ≥
0, where p ⊥ = − ρ − rρ ′ / ρ ′ = dρ/dr [34], [35]. The first two conditions are satisfied. Let us verify thethird condition. From Eq. (31) we obtain ρ ′ ( r ) = q ( βq − r ) r exp − βq r ! . (41)Therefore, if ρ ′ ( r ) ≤ p ⊥ + ρ = − rρ ′ / ≥ ρ ′ ( r ) ≤ β F = βq / (2 r ) ≤
1. Thus, at r ≥ ( βq / / WEC is satisfied. The10 A ( x ) Figure 2: The plot of the function A ( x ). The dash-dot curve corresponds to b = 1, the solid (thick) curve is for b = 0 .
83, and the dashed curve correspondsto b = 0 . β F ≤
1, was made from the causalityprinciple.
To study the thermal stability of charged black holes we will calculate theHawking temperature and heat capacity of the black hole. If the Hawkingtemperature and heat capacity change the sign, this will indicate on thefirst-order phase transition. The point where the heat capacity is singularcorresponds to the second-order phase transition [36]. The unstable state ofthe black hole holds in the region of the negative temperature. The Hawkingtemperature is defined as follows: T H = κ S π = A ′ ( r + )4 π , (42)11here κ S is the surface gravity and r + is the event horizon. From Eqs. (28)and (29), one can find the relations A ′ ( r ) = 2 GM ( r ) r − GM ′ ( r ) r , M ′ ( r ) = r ρ, M ( r + ) = r + G . (43)From Eqs. (30),(42), and (43) we obtain the Hawking temperature T H = 12 / π √ qβ / x + − − /x ) x Γ (cid:18) , x (cid:19) , (44)where we took into account the relations x + = βq ! / r + , bx + = Γ , x ! , b ≡ / √ βGq . (45)The plot of the function T H √ qβ / vs x + is represented in Fig. 3. At x + ≃ + T H ( q β ) ( / ) Figure 3: The plot of the function T H √ qβ / vs x + .2 .
145 ( r + ≃ . √ qβ / ) the temperature is zero, T H = 0 and, therefore,there is the black hole phase transition of the first-order. If x + < . x + ≃ . r + ≃ . √ qβ / ). At this value ∂T H /∂r + = 0 and the heat capacitydiverges which tells us that there is the phase transition of the second-order.To study the heat capacity we use the entropy which satisfies the Hawkingarea law S = Area/ (4 G ) = πr /G . Then we explore the heat capacity atthe constant charge C q = T H ∂S∂T H | q = T H ∂S/∂r + ∂T H /∂r + = 2 πr + T H G∂T H /∂r + . (46)From Eqs. (44) and (46) we arrive at heat capacity Gπq √ β C q = x Γ (cid:18) , x (cid:19) e x − x Γ (cid:18) , x (cid:19) (8 x − (cid:18) , x (cid:19) − x Γ (cid:18) , x (cid:19) e x + 16 x e − x . (47)The plot of the function C q is represented by Fig. 4. Heat capacity C q is + G C q / ( π q ( β ) . ) Figure 4: The plot of the function GC q / ( πq √ β ) vs x + .singular because the denominator of C q becomes zero at the value x + ≃ .
733 as we mentioned before. As a result, we have the second-order phasetransition at x + ≃ . b ≃ . x + ≥ .
733 ( r + ≥ . √ qβ / ), the heat capacity is negative, C q <
0, and the black holebecomes unstable. One can obtain the critical values of the parameters β , m , and T H which correspond to the horizon, x + ≃ . β = ( bqG ) ≃ . q G , m = 0 . q / β / ≃ . q √ G , T H ≃ . q √ G . (48)For these values we obtain the critical horizon at r + = β / √ qx + / / ≃ . q √ G . Thus, the parameter of the model β , the magnetic mass of theblack hole, and the Hawking temperature, corresponding to the second-orderphase transition, are expressed through the magnetic charge of the black hole q and the Newton constant G . As a result, if the horizon is greater than thecritical value r + ≃ . q √ G the black hole becomes unstable. The first-orderphase transition occurs at x + ≃ .
145 ( r + ≃ . √ qβ / ). If r + < . √ qβ / the Hawking temperature is negative and the black hole is unstable. We findthe parameters corresponding to first-order phase transition b = Γ(1 / , /x ) x + ≃ . , r + ≃ . q √ G,β ≃ . q G , m ≃ q √ G . (49)For these parameters the Hawking temperature is zero, T H ≃
0. As a result,the black hole, in the framework of our model, is stable for the range of theevent horizons 0 . q √ G < r + < . q √ G . We have considered exponential electrodynamics which is converted to Maxwell’selectrodynamics for weak fields. In this model the birefringence phenomenontakes place [15] similar to quantum electrodynamics with loop corrections[1]. We have shown that WEC is satisfied if B ≤ √ / √ β that guaranteesthat the energy density is non-negative for any local observer. The causalityand unitarity principles are satisfied in our model at B ≤ q (5 − √ / (2 β ).Because the parameter of the model β ≤ × − T − is very small [15]14he causality principle and the unitarity of the theory take place up to verystrong electromagnetic fields. Thus, the model is of definite theoretical in-terest. The gauge covariant Dirac’s quantization of the exponential modelof nonlinear electrodynamics was performed. This procedure of quantizationis similar to classical electrodynamics and BI electrodynamics quantization.We have investigated black holes possessing a magnetic charge in the frame-work of our model. The regular black hole solution was obtained in generalrelativity and the asymptotic black hole solution at r → ∞ was found. Wehave calculated the magnetic mass of the black hole and the metric func-tion. It was demonstrated that the Ricci scalar does not have singularities at r → ∞ and at r →
0. We shown that only at b = 2 / √ β/ ( qG ) ≤ .
83 thereare two or one horizons. If b > .
83 we have only a particle-like solution andno horizons. The thermodynamic properties and thermal stability of regularblack holes were analysed. We have calculated the Hawking temperature ofblack holes and their heat capacity at the constant magnetic charge. It wasshown that at the horizon r + ≃ . √ qβ / the first-order phase transitionoccurs. If r + < . √ qβ / the black hole is unstable. When heat capacitydiverges, there is a phase transition of the second-order, which takes placeat r + ≃ . √ qβ / . If the horizon r + > . √ qβ / ≃ . q √ G the blackhole becomes unstable. We have estimated the parameters β , m and T H corresponding to first-order and second-order phase transitions. We foundthe range, 0 . q √ G < r + < . q √ G , when the black hole is stable.The results obtained in this paper show that the model of exponentialelectrodynamics possesses attractive characteristics, may be tested in cos-mology, and could describe physics of black holes. Let us estimate the Kretschmann scalar K ( r ) which can be calculated fromthe relation [21] K ( r ) ≡ R µναβ R µναβ = A ′′ ( r ) + A ′ ( r ) r ! + A ( r ) r ! , (50)where A ′ ( r ) = ∂A ( r ) /∂r , x = (2 / ( βq )) / r and the metric function A ( x ) isgiven by A ( x ) = 1 − Γ (cid:16) , x (cid:17) bx . (51)15aking use of the equality [37] ∂ Γ( s, z ) ∂z = − z s − exp( − z ) , (52)we obtain from Eqs. (51) and (52) A ′ ( x ) = Γ (cid:16) , x (cid:17) bx − bx exp (cid:18) − x (cid:19) , (53) A ′′ ( x ) = − (cid:16) , x (cid:17) bx + 16(1 + x ) bx exp (cid:18) − x (cid:19) . (54)With aid of Eqs. (53) and (54) and the equality x = (2 / ( βq )) / r one canwrite down the Kretschmann scalar (50). It is important to investigate theasymptotic of the Kretschmann scalar at r → r → ∞ . For this purposewe explore the relations [37]Γ( s, z ) = Γ( s ) − z s " s − zs + 1 + z s + 2) + O ( z ) z → , (55)Γ( s, z ) = exp( − z ) z s " z + s − z + s − s + 2 z + O ( z − ) z → ∞ . (56)From Eqs. (55) and (56) we obtain the asymptoticΓ (cid:18) , x (cid:19) = exp (cid:18) − x (cid:19) (cid:20) x − x + O ( x ) (cid:21) x → , (57)Γ (cid:18) , x (cid:19) = Γ (cid:18) (cid:19) − x (cid:20) − x + O ( x − ) (cid:21) z → ∞ . (58)By virtue of Eqs. (51), (53), (54) and (58) one findslim x →∞ A ( x ) x = lim x →∞ A ′ ( x ) = lim x →∞ A ′′ ( x ) = 0 . (59)Then from Eq. (50) and the relation x = (2 / ( βq )) / r we obtainlim r →∞ K ( r ) = 0 . (60)Thus, Eq. (60) shows that there is no the singularity of the Kretschmannscalar at r → ∞ . Making use of Eqs. (51), (53), (54) and (57) one can findlim x → A ( x ) x = ∞ , lim x → A ′ ( x ) x = lim x → A ′′ ( x ) = 0 . (61)16s a result, lim r → K ( r ) = ∞ , (62)and the Kretschmann scalar has the singularity at r = 0. This also occursin other models of NLED [21]. References [1] W. Heisenberg and H. Euler, Z. Physik, , 714 (1936)(arXiv:physics/0605038).[2] M. Born and L. Infeld, Proc. Royal Soc. (London) A , 425 (1934).[3] S. I. Kruglov, J. Phys. 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