Entropy of nonlinear black holes in quadratic gravity
aa r X i v : . [ g r- q c ] F e b Entropy of nonlinear black holesin quadratic gravity
Jerzy Matyjasek ∗ Institute of Physics, Maria Curie-Sk lodowska University,pl. Marii Curie-Sk lodowskiej 1, 20-031 Lublin, Poland
Abstract
Employing the Noether charge technique and Visser’s Euclidean approach the en-tropy of the nonlinear black hole described by the perturbative solution of the system ofcoupled equations of the quadratic gravity and nonlinear electrodynamics is constructed.The solution is parametrized by the exact location of the event horizon and charge. Spe-cial emphasis in put on the extremal configuration. Consequences of the second choiceof the boundary conditions, in which the solution is paramerized by the charge and thetotal mass as seen by a distant observer is briefly examined.
Recently, a great deal of efforts have been devoted to the important issue of regular blackholes. One of the most intriguing solutions of this type have been constructed by Ay´on-Beato and Garc´ıa [1] and by Bronnikov [2]. In both cases, the line element is a solution ofthe coupled system of equations of nonlinear electrodynamics and gravity. (We shall referto the solutions of this type as ABGB geometries). The former solution describes a regular,static and spherically symmetric configuration with the electric charge, Q e , whereas the latterone describes a similar geometry characterized by the mass and the magnetic charge Q . Forcertain values of the parameters both solutions describe black holes. On the other hand,the no-go theorem proved in Ref. [3] (see also [2, 4]) forbids, for the class of electromagneticLagrangians with a Maxwell asymptotic in a weak field limit, existence of the electricallycharged, static and spherically- symmetric solutions with the regular center. It should benoted, however, that the electric solution is not in conflict with the non existence theorem,as the formulation of the nonlinear electrodynamics [5] employed by Ay´on-Beato and Garc´ıa(P framework in the nomenclature of Ref. [2] ) differs from the one to which one refers in theassumptions of the no-go theorem. Indeed, the solution of Ay´on-Beato and Garc´ıa has beenconstructed in a formulation of the nonlinear electrodynamics obtained from the original one(F framework) by means of a Legendre transformation (see Ref. [2] for details). Moreover,the no-go theorem does not forbid existence of the solutions with magnetic charge as well assome hybrid configurations in which the electric field does not extend to the central region.The status of the nonlinear electrodynamics in the model considered here is to providea static matter source, perhaps the exotic one, to the field equations. That means thatthe casual structure of the spacetime is still governed by the null geodesics or “ordinary”photons rather than the photons of the nonlinear theory. Actually, the latter move alongthe geodesics of the effective space [6, 7]. Outside the event horizon the solution of the ∗ [email protected], [email protected] r = 0 nowwe have the regular interior.An attractive feature of the ABGB solutions is possibility to express the location of thehorizons in terms of the Lambert special functions [8, 9]. Similarly, the Lambert functions [10,11] may be used in the discussion of the extremal configurations [12].According to our present understanding a proper description of the gravitational phenom-ena should be given by the quantum gravity, being perhaps a part of a more fundamentaltheory. And although at the present stage we have no clear idea how this theory lookslike, we expect that the action functional describing its low-energy approximation shouldconsist of the higher order terms constructed from the curvature tensor, its contractionsand covariant derivatives to some required order. Among various generalizations of theEinstein- Hilbert action a special role is played by the quadratic gravity (see for exampleRefs. [13, 14, 15, 16, 17, 18, 19, 20, 21]). Motivations for introducing such terms into theaction functional are numerous. When invented, for example, the equations of quadraticgravity have been treated as an exact formulation of the theory of gravitation. On the otherhand, it may be considered, quite naturally, as truncation of series expansion of the actionof the more general theory. Such terms appear generically in the one-loop calculations of thequantum field theory in curved background [22]. Moreover, from the point of view of thesemi-classical gravity, the quadratic terms in the field equations might be treated as some sortof the simplified stress-energy tensor. Such a toy model of the renormalized stress- energytensor allows to mimic the fairly more complex sources in a relatively simple way. This ap-proach is especially useful when the general pattern that lies behind the calculations of bothtypes is essentially the same. Thus, some general features of the full semi- classical solutionscan be analyzed and understand without referring to otherwise intractable equations.It should be noted that any higher curvature theory contain solutions which are unavail-able to the theory based on the classical Einstein- Hilbert Lagrangian. This can most easilybe seen by counting the degrees of freedom: the quadratic gravity is known to posses 8 de-grees of freedom whereas the General Relativity has only 2. Moreover, there are solutionsthat are not analytic in the coupling constants, i. e., they do not reduce to solutions of theclassical Einstein field equations. (For a comprehensive discussion see for example [23] andthe references cited therein). Unfortunately, because of complexity of the equations of thequadratic gravity it is practically impossible to construct their exact solutions and one isforced to refer either to approximations or to numerical methods. The natural method toobtain reasonable results consists of treating the higher curvature contributions perturba-tively. This approach also guarantees that the black hole exists as the perturbative solutionof the higher-order solution provided it exists classically [24]. Finally, observe that in theperturbative approach the casual structure is determined by the classical metric, however,the equations of motion of test particles and various characteristics of the solution acquirethe first order correction.Analyses of the spherically-symmetric and static solutions to the higher derivative theoryhas been carried out in [14, 25, 26, 27, 28, 29, 30]. Specifically, in Ref. [30] the perturbativesolutions of the ABGB-type to the equations of the effective quadratic gravity have beenconstructed and discussed. In this paper we shall calculate the entropy of such black holesusing Wald’s approach [31, 32, 33] and confirm the final results employing computationallyindependent but closely related Euclidean techniques propounded by Visser [34, 35, 36]. S = 116 π Z (cid:0) R + αR + βR ab R ab + γR abcd R abcd − L ( F ) (cid:1) √− g d x, (1)where L ( F ) is some functional of F = F ab F ab (its exact form will be given later) andall symbols have their usual meaning. The cosmological constant is assumed to be zero.To simplify our discussion from the very beginning we shall relegate the term involving theKretschmann scalar, R abcd R abcd , from the total action employing the Gauss-Bonnet invariant.The coupling constants α and β have the dimension of length squared and throughout thepaper we shall assume αL ∼ βL ≪ , (2)where L is the local curvature scale. Assumption that the mass scales associated with thelinearized equations are real may place additional constrains [20, 37, 38] on α and β. Here,however, we shall treat them as small and of comparable order but arbitrary.The entropy of the black hole may be calculated using various methods. It seems, however,that Wald’s technique is especially well suited for calculations in the higher curvature theories.Here we shall follow this very approach. Other competing techniques are the method basedon the field redefinition [33, 39] and Visser’s Euclidean approach.For the Lagrangian involving the Riemann tensor and its symmetric derivatives up somefinite order, say n, Wald’s Noether charge entropy may be compactly written in the form[31, 32, 33] S = − π Z d x ( h ) / n X m =0 ( − m ∇ ( e ... ∇ e m ) Z e ...e m ; abcd ǫ ab ǫ cd , (3)where Z e ...e m ; abcd = ∂ L ∂ ∇ ( e ... ∇ e m ) R abcd , (4) h is the determinant of the induced metric, ǫ ab is the binormal to the bifurcation sphere, andthe integration is carried out across the bifurcation surface. Actually S can be evaluated notonly on the bifurcation surface but on an arbitrary cross-section of the Killing horizon. Since ǫ ab ǫ cd = ˆ g ad ˆ g bc − ˆ g ac ˆ g bd , where ˆ g ac is the metric in the subspace normal to cross section onwhich the entropy is calculated, one can rewrite Eq. (3) in the form S = 4 π Z d x h / n X m =0 ( − m ∇ ( e ... ∇ e m ) Z e ...e m ; abcd ˆ g ac ˆ g bd . (5)The tensor ˆ g ab is related to V a = K a / || K || ( K a is the timelike Killing vector) and the unitnormal n a by the formula ˆ g ab = V a V b + n a n b . The general expression describing entropy (5) has been applied in numerous cases, mostlyfor the Lagrangians that are independent of covariant derivatives of the Riemann tensor andits contractions. In Ref. [40], however, Eq. (5) has been employed in calculations of theentropy of the quantum-corrected black hole when the source term is described by the stress-energy tensor of the quantized fields in a large mass limit. Such a tensor is purely geometricaland besides ordinary higher curvature terms it involves also R ∇ a ∇ a R and R ab ∇ c ∇ c R ab . On the other hand, one can follow an approach propounded by Visser [34, 35, 36]. Thegeneral formula for the entropy of the stationary black hole with the Hawking temperature T H is given by S = A T H Z Σ ( ρ L − L E ) K a d Σ a + Z Σ s V a d Σ a , (6)3here A is the area of the event horizon, s is the entropy density associated with the fluctu-ations (ignored in this paper) and finally ρ L and L E are, respectively, the Lorentzian energydensity and the Euclideanized Lagrangian of the matter fields surrounding the black hole.(All higher curvature terms have been inserted into the Lagrangian describing matter fields.)For the specific case of the Einstein-Hilbert action augmented with the higher curvature terms(but not covariant derivatives of curvature) Visser’s result is equivalent to Wald’s formula.The coupled system of differential equations describing nonlinear electrodynamics inquadratic gravity can be obtained from the variational principle.Simple calculations indicate that the tensor F ab and its dual ∗ F ab , satisfy the equations ∇ a (cid:18) d L ( F ) dF F ab (cid:19) = 0 , (7) ∇ a ∗ F ab = 0 , (8)respectively. Differentiating functionally the total action S with respect to the metric tensorone obtains equations of the quadratic gravity in the form L ab ≡ G ab − αI ab − βJ ab = 8 πT ab , (9)where I ab = 2 ∇ b ∇ a R − RR ab + 12 g ab (cid:0) R − ∇ c ∇ c R (cid:1) , (10) J ab = ∇ b ∇ a R − ∇ c ∇ c R ab − R cd R cbda + 12 g ab (cid:0) R cd R cd − ∇ c ∇ c R (cid:1) (11)and T ba = 14 π (cid:18) d L ( F ) dF F ca F cb − δ ba L ( F ) (cid:19) . (12)In this paper we shall concentrate on the static and spherically-symmetric configurationsdescribed by the line element of the form ds = − e ψ ( r ) f ( r ) dt + dr f ( r ) + r d Ω , (13)where f ( r ) = 1 − M ( r ) r . (14)The spherical symmetry places restrictions on the components of F ab tensor and, conse-quently, its only nonvanishing components compatible with the assumed symmetry are F and F . Simple calculations yield F = Q sin θ (15)and r e − ψ d L ( F ) dF F = Q e , (16)where Q and Q e are the integration constants interpreted as the magnetic and electric charge,respectively.Since the no-go theorem forbids existence of the regular solutions with Q e = 0 in thelatter we shall assume that the electric charge vanishes. Now, since F = 2 F F , one has F = 2 Q r . (17)The stress-energy tensor (12) calculated for this configuration is T tt = T rr = − π L ( F ) (18)4nd T θθ = T φφ = 14 π d L ( F ) dF Q r − π L ( F ) . (19)Further considerations require specification of the Lagrangian L ( F ) . Following Ay´on-Beato, Garc´ıa and Bronnikov let us chose it in the form L ( F ) = F " − tanh s r Q F ! , (20)where s = | Q | b , (21)and b is a free parameter. Inserting Eq. (17) into (20) and making use of Eq. (21) one obtains L ( F ) = 2 Q r (cid:18) − tanh Q br (cid:19) . (22)The system of coupled differential equations of the quadratic gravity with the sourceterm given by (18) and (19) with (22) is rather complicated and cannot be solved exactly.Fortunately, since the coupling constants α and β are expected to be small in a sense ofEq. 2, one can treat the system of the differential equations perturbatively, with the classicalsolution of the Einstein field equation taken as the zeroth- order approximation. Successiveperturbations are therefore solutions of the chain of the differential equations of ascendingcomplexity [41, 42, 43, 44]. It should be noted, however, that the higher order equations areprobably intractable analytically and the technical difficulties may limit the calculations tothe first order.In the next section, we shall employ perturbative techniques to construct the approximatesolution to the equations of the quadratic gravity with the source term being the stress-energytensor of the Bronnikov type. Such an approach is expected to yield reasonable results andbecause of complexity of the differential equations, it may be the only way to deal with thisproblem. To keep control of the order of terms in complicated series expansions we shall introduce adimensionless parameter ε substituting α → εα and β → εβ . We shall put ε = 1 at the finalstage of calculations. Of functions M ( r ) and ψ ( r ) we assume that they can be expanded inpowers of the auxiliary parameter as M ( r ) = M ( r ) + εM ( r ) + O (cid:0) ε (cid:1) (23)and ψ ( r ) = εψ ( r ) + O (cid:0) ε (cid:1) . (24)First, consider the left hand side of Eq. (9) calculated for the line element (13) withthe functions M ( r ) and ψ ( r ) given by (23) and (24), respectively. Making use of the aboveexpansions and subsequently collecting the terms with the like powers of ε, after some rear-rangements, one obtains [30] L tt = − r ( M ′ + εM ′ − εS tt ) , (25)5here S tt = β M ′ r − M M ′ r + 2 M ′ r − M ′′ r + 5 M M ′′ r − M ′ M ′′ r + M ′′ M (3)0 − M M (3)0 r − M ′ M (3)0 + r M (4)0 − M M (4)0 ! − α M M ′ r − M ′ r − M ′ r + 8 M ′′ r − M M ′′ r − M ′′ + 2 M ′ M ′′ r − M (3)0 + 6 M M (3)0 r + 2 M ′ M (3)0 − r M (4)0 + 4 M M (4)0 ! (26)and M ′ , M ′′ and M ( i )0 for i ≥ i − th derivatives with respect to theradial coordinate. On the other hand, a simple combination of the components of L ba tensor L rr − L tt = 0 (27)can be easily integrated to yield [30] ψ ( r ) = (2 α + β ) M (3)0 − r (3 α + β ) M ′ + C , (28)where C is the integration constant. It should be noted that contrary to the case of coupledsystem of the Maxwell equations and quadratic gravity considered in Refs. [25, 26, 27, 29],now we have explicit dependence on the parameter α. A comment is in order here regardingthe independence of the final result calculated for the Maxwell source on the parameter α. First, observe that the stress-energy tensor of the electromagnetic field for the spherically-symmetric an static configuration with a total charge e assumes simple form T ba = − e πr diag[1 , , − , − . (29)Therefore, the zeroth-order solution to the (0,0) -component of the equation (9) can bewritten in the form M ( r ) = − e r + C, (30)where C is the integration constant. Now, substituting (30) into (26) and (28) it can easilybe demonstrated that the expression in the second bracket in its right hand side of Eq. (26)as well as the expression M − M ′ /r in (28) vanish.One expects that all characteristics of the black hole, such as the location of the horizonsand temperature could also be calculated perturbatively. In the latter, for simplicity, we shallrefer to the perturbative solutions of the quadratic gravity using the names of their classicalcounterparts (the zeroth-order solutions) whenever it will not lead to confusion.To develop the model further one has to determine the integration constants and the freeparameter b . There are, in general, two interesting and physically motivated choices. Onecan relate the integration constant with the exact location of the event horizon, r + , and thiscan easily be done with the aid of the equation M ( r + ) = r + . (31)On the other hand it is possible to express solutions of the system of differential equationsconsisting of (0 ,
0) component of Eqs. ( 9) and Eq. (27) in terms of the total mass M asseen by a distant observer lim r →∞ M ( r ) = M . (32)6or the function ψ ( r ) we shall always adopt the natural conditionlim r →∞ ψ ( r ) = 0 . (33)Inspection of Eqs. (25) and (27) reveals their different status. Indeed, Eq. (27) caneasily be integrated for a general function M ( r ) and the final solutions is to be obtained bydifferentiation of the zeroth-order solution and making use of the boundary conditions. Onthe other hand, the first integral of the differential equation for M ( r ) cannot be constructedand one has to know the zeroth-order solution to determine M . The assumed expansions of the functions M ( r ) and ψ ( r ) as given by Eqs. (23) and (24),respectively, suggests that one can rewrite the boundary conditions of the first type in thefollowing form: M ( r + ) = r + , M ( r + ) = 0 , ψ ( ∞ ) = 0 , (34)whereas for the boundary conditions of the second type one has M ( ∞ ) = M , M ( ∞ ) = 0 , ψ ( ∞ ) = 0 . (35)Now, let us concentrate on the zeroth-order equations supplemented with the conditionsof the first type. Putting ε = 0 in Eq. (25), form (22) and (18) one obtains dM dr = Q r (cid:18) − tanh Q br (cid:19) , (36)which can be easily integrated to yied M ( r ) = − b tanh Q br + C . (37)Finally, making use of the conditions (34) one arrives at the desired result M ( r ) = r + b tanh Q br + − b tanh Q br . (38)The thus obtained solution reduces to the Schwarzschild solution for Q = 0 and it can beeasily demonstrated that, by (26)and the boundary conditions (34) it remains so in thehigher-order calculations.To specify the solution further we shall make use of the well-known relation [45] M = κA H π − Z Σ (cid:0) T ba − T δ ba (cid:1) K a d Σ b , (39)where Σ is a constant time hypersurface and K a is a timelike Killing vector and apply it tothe zeroth-order solution. Making use of the explicit form of the stress-energy tensor of thenonlinear electrodynamics one obtains M H = r + b tanh Q br + , (40)where M H is the mass connected with the zeroth-order solution. We shall refer to M H as tothe horizon defined mass of the black hole.To develop the model further one has to determine the free parameter b. Our choice,which guarantees regularity of the zeroth-order line element at the center, is b = M H , andhence Eq. (38) becomes M ( r ) = M H (cid:18) − tanh Q M H r (cid:19) . (41)7nfortunately, the regularity of the zeroth-order solution does not guarantee regularity ofthe higher-order perturbative solutions [30].It should be noted that the M H = M H ( Q, r + ) and for fixed Q and r + one has to determine M H numerically. On the other hand it is possible to employ the equation M ( r + ) = r + / Q, M H ) parametrization insteadof ( Q, r + ) . One can, therefore, construct solutions of this equation in terms of the Lambertfunction. Simple manipulations yield r + = − M H Q W + ( − ρe ρ ) M H − Q , (42)where W + is a principal branch of the Lambert function and ρ = Q / M H . Analogoussolution for the inner horizon can be written in the form r − = − M H Q W − ( − ρe ρ ) M H − Q , (43)where W − is the second real branch of the Lambert function. (In fact, W + and W − are theonly real branches.) Making use of the elementary properties of the Lambert functions onecan demonstrate that the principal branch has the expansion W + ( x ) = x − x + 32 x − x + O ( x ) . (44)On the other hand, W − ( x ) → −∞ as x → , and, consquently, the location of the eventhorizon tends to the Schwarzschild value whereas r − → . A typical run of M H as a function of ξ for a few exemplary values of Q is showed in Fig. 1.For a given Q a line of M H = const. intersects Q = const. curve at one or two points or it hasno intersection points at all. The smaller one gives location of the inner horizon whereas thegreater is to be identified with the event horizon. The minimum of M H = M H ( Q = const, ξ )function represents extremal configurations when the two horizons merge. Ξ M H Figure 1: This graph shows solutions of the equation M H = ξ + M H tanh Q M H ξ for a fewexemplary values of the charge Q. From bottom to top the curves correspond to Q = 0 . i, for i = 1 , ..., . For M H = const., the greater solution represents location of the event horizon, r + whereas the smaller one represents the inner horizon, r − . The minimum of each curvecorresponds to the extremal configuration with r + = r − . It should be noted that the mass M H is not the mass that would be measured for theperturbed black hole by an observer at infinity. Indeed, even for the zeroth-order solutions8he meaning of M H and M is different and the substantial differences are transparent in thefirst order calculations. This can be easily seen by studying the limit M = lim r →∞ M ( r ) = M H + εM ( ∞ ) (45)and Eq. (32). Identical result can be obtained form Eq. (39). Indeed, in order to apply (39)for the perturbed black hole one has to move the higher curvature terms of Eq. (9 ) intoits right hand side and treat them as a contribution to the total stress-energy tensor. It canbe demonstrated explicitly, that making use of Eq. (39) in the first-order calculations oneobtains precisely (45).The function M ( r ) can be expressed in terms of the polylogarithms. Unfortunately,it is rather complicated and to avoid unnecessary proliferation of long formulas it will notbe displayed here. The first-order solution can be constructed employing the algorithmpresented in Appendix of Ref. [30]. It should be noted that the function M ( r ) presentedin [30] is calculated for the boundary conditions of the second type. Now, let us return to our main theme and calculate the entropy of the ABGB black hole.In doing so we shall put special emphasis on comparison of the results constructed for thenonlinear black hole with the analogous results obtained for the Reissner-Nordstr¨om solution.Such a comparison is especially interesting as the geometries of their classical counterpartsare practically indistinguishable in two important regimes. To demonstrate this it suffices toexpand the metric potentials in powers of | Q | /r + and r + /r, respectively. Since the expansiontakes the form f ( r ) = 1 − M H r + Q r − Q M H r + ..., (46)the differences in the metric structure between ABGB and RN geometries in the exteriorregion for | Q | /r + ≪ r ≫ r + .The higher curvature terms in the action functional lead to the appearance of additionalterms in the final expression describing entropy, which spoil area/entropy relation. Simplecalculations carried out within the Noether charge framework indicate that the contributionof the quadratic part of the action to the entropy is given by δ S = 2 πr (cid:20) αR + 12 β (cid:0) R tt + R rr (cid:1)(cid:21) | r + . (47)Now, substituting the line element (13) with (14) and ( 23), into the general expression (47),expanding the right hand side of Eqs. (9) with respect to ε, and, finally, retaining the linearterms only, one gets S = πr + 2 πεr (cid:20) αr M ′ ( r + ) + 2 α + βr + M ′′ ( r + ) (cid:21) + O (cid:0) ε (cid:1) . (48)For the nonextreme black hole with the boundary conditions of the first type (34) one has S = πr + 2 πQ M H r ε cosh − (cid:18) Q M H r + (cid:19) (cid:26) αQ tanh (cid:18) Q M H r + (cid:19) − β (cid:20) M H r + − Q (cid:18) Q M H r + (cid:19)(cid:21)(cid:27) , (49)9here M H = M H ( Q, r + ). In this approach the zeroth-order solution (41) determines the firstorder correction to the entropy of the nonextreme black hole completely. Having established M H for given Q and r + one can rewrite Eq. (49) putting ˜ S = S /M H , q = | Q | /M H , x + = r + /M H , ˜ α = α/M H and ˜ β = β/M H . Simple manipulations yield˜ S = πx + ε πq x cosh − q x + (cid:20) ˜ αQ tanh q x + − ˜ β (cid:18) x + − q q x + (cid:19)(cid:21) . (50)This results can be contrasted with the analogous result constructed for the Reissner-Nordstr¨omblack hole S = πr − β πQ r (51)or ˜ S = π (1 + p − q ) − β πq (1 + p − q ) . (52)To investigate the entropy S as given by Eq. (49) let us observe that for | Q | /r + ≪ r + ≈ M H . Now, expanding hyperbolic functions in powers of | Q | /r + one obtains S = πr − πβ Q r + O (cid:18) ( Qr + ) (cid:19) . (53)A comparison of Eqs. (53) and (51) shows that for | Q | /r + ≪ α andfor | Q | /r + ≪ ∼ ( Q/r + ) . The analysis of the extremal configuration is more involved. First, let us return to thezeroth-order solution. It should be emphasized that although we do not ascribe any particularmeaning to the zeroth-order solution, some of its features do possess clear and unambiguousmeaning. For the boundary conditions (34) such a solution is described by the exact r + and Q. The extremality condition places additional relation between the elements of the pair(
Q, r + ) or ( Q, M ). Here we shall confine ourselves to the first pair. Simple considerationsyield | Q | = 2 w / M H (54)and r + = 4 w w M H , (55)where w = W + (1 /e ), and consequently | Q | /r + = 1 + w w / . (56)Returning to the first-order solution we recall the relation valid for the extremal configurationin the Reissner-Nordstr¨om geometry r + = | Q | . (57)In Ref. [12] we have ascribed this simple relation to tracelessness of the stress-energy tensorof the matter fields. As the stress-energy tensor of the nonlinear electrodynamics consideredin this paper has a nonzero trace, one expects that the analogous relation between Q and r + in the ABGB geometry is more complicated. Indeed, after some algebra, one has r + = 2 w / | Q | (1 + w ) (cid:20) ε β + 2 α Q w ( w + 3) (cid:0) w − (cid:1)(cid:21) . (58)10ow, making use of (58) in (49) gives S extr = 4 πwQ (1 + w ) − πε w ) (cid:2) (2 α + β ) w + 2 (2 α − β ) w + 5 α + 2 β (cid:3) , (59)and the first term of the right hand side coincides with the Bekenstein- Hawking entropy [46].Numerically, one has S extr = πQ × . − πε × (0 . α + 0 . β ) , (60)where a common factor 2 π has been singled out for convenience. Analogous relation for theextremal Reissner-Nordstr¨om black hole reads S extr = πQ − πεβ. (61)Now, let us calculate the entropy of the ABGB black hole employing the Euclidean tech-niques propounded by Visser. First, observe that if the Lagrangian is arbitrary function ofthe Riemann tensor (and its contractions) but is independent of its covariant derivatives,both methods, i. e. Wald’s approach and Visser’s method are equivalent. One may wonder,therefore, why we intend to carry out such a calculation. The answer is simple: althoughboth methods should yield identical results, the calculational steps necessary to obtain thefinal result are quite different and consequently one can consider the calculations carried outwithin the framework of the one method as the useful check of the other. It is especiallyimportant in situations when the computational complexity of the considered problem maylead to numerous errors.The calculations proceed in a few steps. First, incorporate the Euclidean action functionalof the quadratic gravity into the matter part of the action. Similarly, the (Lorentzian) energydensity is given by ρ = − T tt = 116 π L ( F ) − ε (cid:18) α π I tt + β π J tt (cid:19) . (62)It could easily be demonstrated that ρ L − L E ∼ O ( ε ) and consequently it suffices to knowthe Hawking temperature to the zeroth-order. Moreover, due to subtle cancellations in theintegrand of Eq. (6) the final result of the quadratures does not contain polylogarithmfunctions. Now, substituting T H = 14 πr + (cid:18) − Q M r + + Q M H (cid:19) (63)and (62) into (6), after some algebra, one has δ S = εr ( η + 1) [ αs α + βs β ] , (64)where η = exp (cid:0) Q / M H r + (cid:1) ,s α = (cid:18) Q r + M − Q M H (cid:19) η − (cid:18) Q M H − Q r + + 12 Q r + M + 8 Q r + M H (cid:19) η + (cid:18) Q M H − Q r + M + 56 Q r + M H (cid:19) η + (cid:18) Q r + M − Q r + + 20 Q M H − Q r + M H (cid:19) η (65)11nd s β = (cid:18) Q r + + 2 Q r + M − Q M H (cid:19) η − (cid:18) M H Q + 4 Q r + M H + 12 Q M H − Q r + + 6 Q r + M − Q r + (cid:19) η + (cid:18) Q M H − Q r + M − Q r + + Q r + + 28 Q r + M H − M H Q (cid:19) η − (cid:18) M H Q − Q r + − Q M H − Q r + M + 44 Q r + + 8 Q r + M H (cid:19) η (66)At first glance this result does not resemble Eq. (49). However, making use of the identity η = 4 M H r + − , (67)one can easily demonstrate that Eqs. (64-66) reduce precisely to Eq. (49). In this paper we have constructed the entropy of the nonlinear ABGB-type black holes usingthe boundary conditions (34). The zeroth-order solution coincides, as expected, with theABGB line element whereas the first-order correction can be elegantly expressed in termsof the polylogarithm functions. Now, let us briefly discuss the consequences of the secondchoice, in which the results are expressed in terms of the total mass of the system as measuredby a distant observer. To calculate the location of the event horizon to the required order in( Q, M ) parametrization one has to solve the first-order equations for M ( r ) and ψ ( r ) , and,subsequently, perturbatively solve the equation g tt ( r + ) = 0 assuming that the event horizoncan be expanded as r + = r + εr + O ( ε ) . (68)Unfortunately, the function M ( r ) is rather complicated (it can be expressed in terms of thepolylogarithms) and, once again, to avoid unnecessary proliferation of long formulas it willnot be presented here. Interested reader is referred to [30].Generally, for the nonexterme black hole one has S = πr + 2 πr ε + 32 πεr (cid:20) αr M ′ ( r ) + 2 α + βr M ′′ ( r ) (cid:21) + O (cid:0) ε (cid:1) . (69)On the other hand, making use of (68), the equation (69) can be rewritten in the equivalentform S = πr + ε πM ( r )1 − M ′ ( r ) + 32 πεr (cid:20) αr M ′ ( r ) + 2 α + βr M ′′ ( r ) (cid:21) + O (cid:0) ε (cid:1) . (70)The extremal case should be analyzed separately. The extremal configuration of theABGB black hole being the solution of the Einstein gravity is described by | Q c | = 2 w / M (71)and r c = 4 w w M . (72)12ne expects, that the higher-order curvature terms modify these relations, shifting (in a spaceof the parameters) extremal solution into a slightly different position. Indeed, treating M as a function of Q and r, after some algebra, one concludes that the extremal configurationis still possible and is described by the relations Q = Q c + ε ∆ , r + = r c + εδ, (73)where ∆ = − (cid:18) ∂∂Q M (cid:19) − M . (74)and δ = − (cid:18) ∂ ∂r M (cid:19) − (cid:18) ∂∂r M (cid:19) + (cid:18) ∂∂Q M ∂ ∂r M (cid:19) − (cid:18) M ∂ ∂r∂Q M (cid:19) . (75)Both δ and ∆ are to be calculated for the parameters describing extremal zeroth-ordersolution. Numerically, one has ∆ = 1 . M α + 0 . M β (76)and δ = − . M α − . M β. (77)Since the calculations of the entropy follow the general scheme sketched in previous sectionthey will not be presented here.The purpose of the present paper (besides importance of the quadratic gravity in its ownand the natural curiosity) is twofold. First, one can treat the calculations presented in thispaper as the first step in understanding of the influence of the higher curvature terms on theentropy of black holes in a more complex setting than Maxwell electrodynamics. The nextstep would involve, for example, inclusion of the all curvature invariants of the order 4 and 6and degree 2 and 3 [47, 48, 49]. Moreover, it would be interesting to extend this analysis togeneral D-dimensional manifolds. The natural candidate for a higher-curvature theory is theLovelock gravity [50]. Moreover, one may consider the more general curvature terms, witharbitrary coefficients rather than those inspired by particular theory.(See, for example [51, 47]and references cited therein.) 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