Scale dependent three-dimensional charged black holes in linear and non-linear electrodynamics
Angel Rincon, Ernesto Contreras, Pedro Bargueño, Benjamin Koch, Grigorios Panotopoulos, Alejandro Hernández-Arboleda
JJournal name manuscript No. (will be inserted by the editor)
Scale dependent three-dimensional charged black holes inlinear and non-linear electrodynamics ´Angel Rinc´on a,1 , Ernesto Contreras b,2,c , Pedro Bargue˜no , BenjaminKoch , Grigorios Panotopoulos , Alejandro Hern´andez-Arboleda Instituto de F´ısica, Pontificia Universidad Cat´olica de Chile,Av. Vicu˜na Mackenna 4860, Santiago, Chile Departamento de F´ısica, Universidad de los Andes,Apartado A´ereo 4976, Bogot´a, Distrito Capital, Colombia CENTRA, Instituto Superior T´ecnico, Universidade de Lisboa,Av. Rovisco Pa´ıs 1, Lisboa, PortugalReceived: date / Accepted: date
Abstract
In the present work we study the scale de-pendence at the level of the effective action of chargedblack holes in Einstein-Maxwell as well as in Einstein-power-Maxwell theories in (2+1)-dimensional spacetimeswithout a cosmological constant. We allow for scale de-pendence of the gravitational and electromagnetic cou-plings, and we solve the corresponding generalized fieldequations imposing the “null energy condition”. Cer-tain properties, such as horizon structure and thermo-dynamics, are discussed in detail.
Keywords
Black holes; Scale dependence; 2+1gravity.
PACS
PACS code1 · PACS code2 · more In recent years gravity in (2+1) dimensions has at-tracted a lot of interest for several reasons. The absenceof propagating degrees of freedom, its mathematicalsimplicity, the deep connection to Chern-Simons theory[1–3] are just a few of the reasons why to study three-dimensional gravity. In addition (2+1) dimensional blackholes are a good testing ground for the four-dimensionaltheory, because properties of (3+1)-dimensional blackholes, such as horizons, Hawking radiation and blackhole thermodynamics, are also present in their three-dimensional counterparts.On the other hand, the main motivation to studynon-linear electrodynamics (NLED) was to overcomecertain problems of the standard Maxwell theory. In a e-mail: [email protected] b e-mail: [email protected] c On leave from Universidad Central de Venezuela particular, non-linear electromagnetic models are intro-duced in order to describe situations in which this fieldis strong enough to invalidate the predictions providedby the linear theory. Originally the Born-Infeld non-linear electrodynamics was introduced in the 30’s in or-der to obtain a finite self-energy of point-like charges [4].During the last decades this type of action reappears inthe open sector of superstring theories [5] as it describesthe dynamics of D-branes [6]. Also, these kind of elec-trodynamics have been coupled to gravity in order toobtain, for example, regular black holes solutions [8–10],semiclassical corrections to the black hole entropy [11]and novel exact solutions with a cosmological constantacting as an effective Born–Infeld cut–off [12]. A partic-ularly interesting class of NLED theories is the so calledpower-Maxwell theory described by a Lagrangian den-sity of the form L ( F ) = F β , where F = F µν F µν / β is an arbitrary rationalnumber. When β = 1 one recovers the standard lin-ear electrodynamics, while for β = D/ D beingthe dimensionality of spacetime, the electromagneticenergy momentum tensor is traceless [13, 14]. In 3 di-mensions the generic black hole solution without impos-ing the traceless condition has been found in [15], whileblack hole solutions in linear Einstein-Maxwell theoryare given in [16, 17].Scale dependence at the level of the effective actionis a generic result of quantum field theory. Regardingquantum gravity it is well-known that its consistentformulation is still an open task. Although there areseveral approaches to quantum gravity (for an incom-plete list see e.g. [18–26] and references therein), most ofthem have something in common, namely that the basicparameters that enter into the action, such as the cos-mological constant or Newton’s constant, become scale a r X i v : . [ h e p - t h ] A p r dependent quantities. Therefore, the resulting effectiveaction of most quantum gravity theories acquires a scaledependence. Those scale dependent couplings are ex-pected to modify the properties of classical black holebackgrounds.It is the aim of this work to study the scale de-pendence at the level of the effective action of three-dimensional charged black holes in linear (Einstein -Maxwell) and non-linear (Einstein-power-Maxwell) elec-trodynamics. We use the formalism and notation of[27, 28] where the authors applied the same techniqueto the BTZ black hole [29,30]. Our work is organized asfollows: After this introduction, in the next section wepresent the action and the classical black hole solutionboth in Einstein-Maxwell and Einstein-power-Maxwelltheories. The framework and the “null” energy condi-tion are introduced in sections 3 and 4. The scale de-pendence for linear electrodynamics is presented in thesection 5, while the corresponding solutions for non-linear theory are given in section 6. The discussion ofour results and remarks are shown in section 7 whereasin section 8 we summarize the main ideas and conclude.Finally, we present a brief appendix in which we showthe effective Einstein field equations for an arbitraryindex β in the last section. In this section we present the classical theories of lin-ear and non-linear electrodynamics. Those theories willthen be investigated in the context of scale dependentcouplings. The starting point is the so-called Einstein-power-Maxwell action without cosmological constant( Λ = 0), assuming a generalized electrodynamics i.e. L ( F ) = C | F | β which reads S = (cid:90) d x √− g (cid:20) πG R − e β L ( F ) (cid:21) , (1)where G is Einstein’s constant, e is the electromag-netic coupling constant, R is the Ricci scalar, L ( F )is the electromagnetic Lagrangian density where C isa constant, F is the Maxwell invariant defined in theusual way i.e. F = (1 / F µν F µν and F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic field strength tensor. Weuse the metric signature ( − , + , +), and natural units( c = (cid:126) = k B = 1) such that the action is dimensionless.Note that β is an arbitrary rational number, which alsoappears in the exponent of the electromagnetic cou-pling in order to maintain the action dimensionless. Itis easy to check that the special case β = 1 repro-duces the classical Einstein-Maxwell action, and thus the standard electrodynamics is recovered. For β (cid:54) = 1one can obtain Maxwell-like solutions. In the followingwe shall consider both cases: first when β = 3 /
4, sinceit is this value that allows us to obtain a trace-free elec-trodynamic tensor, precisely as in the four-dimensionalstandard Maxwell theory, and second when β = 1 be-cause is the usual electrodynamics in 2+1 dimensions.In both cases one obtains the same classical equationsof motion, which are given by Einstein’s field equations G µν = 8 πG e β T µν . (2)The energy momentum tensor T µν is associated to theelectromagnetic field strength F µν through T µν = L F g µν − L ( F ) F µγ F ν γ , (3)remembering that L F = d L /dF . In addition, for staticcircularly symmetric solutions the electric field E ( r ) isgiven by F µν = ( δ rµ δ tν − δ rν δ tµ ) E ( r ) . (4)For the metric circular symmetry implies ds = − f ( r ) dt + g ( r ) dr + r dφ . (5)Note that in the classical case one finds that g ( r ) = f ( r ) − . Finally, the equation of motion for the Maxwellfield A µ ( x ) reads D µ (cid:32) L F F µν e β (cid:33) = 0 . (6)With the above in mind, for charged black holes oneonly needs to determine the set of functions { f ( r ) , E ( r ) } .Using Einstein’s field equations 2 and Eq. 6 combinedwith Eq 4 and the definition of L F , one obtains theclassical electric field as well as the lapse function f ( r ).It is possible to determine the electric field as well as thelapse function without assuming a particular value for β for classical solutions, however we will focus on twoof them. First, the Einstein-Maxwell case is in itself in-teresting due to its relation with the four-dimensionalcase. On the other hand, the Einstein-power-Maxwellcase with β = 3 / β can be found in the appendix 9. β = 1) is given by f ( r ) = − M G − Q e ln (cid:18) r ˜ r (cid:19) , (7) E ( r ) = Q r e , (8)where M is the mass and Q is the electric charge ofthe black hole and ˜ r stands for the radius where theelectrostatic potential vanishes. The apparent horizon r is obtained by demanding that f ( r ) = 0, whichreads r = ˜ r e − M G e Q , (9)and rewriting the lapse function using the apparenthorizon one gets f ( r ) = − Q e ln (cid:18) rr (cid:19) . (10)Black holes have thermodynamic behaviour. Here, theHawking temperature T , the Bekenstein-Hawking en-tropy S , as well as the heat capacity C are found tobe T ( r ) = 14 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q e r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (11) S ( r ) = A H ( r )4 G , (12) C ( r ) = T ∂S∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q = − S ( r ) . (13)Note that A H ( r ) is the horizon area which is given by A H ( r ) = (cid:73) d x √ h = 2 πr , (14)2.2 Einstein-power-Maxwell caseSolving Einstein’s field equations for β = 3 /
4, the lapsefunction f ( r ) and the electric field E ( r ) are found tobe f ( r ) = 4 G Q r − G M , (15) E ( r ) = Q r . (16)It is worth mentioning that, unlike in the previous sec-tion, the solutions here considered do not contain theelectromagnetic coupling. This is due to the fact thata dimensional analysis on the action (1) for β = 3 / f ( r ) = 0, which reads r = 43 Q M . (17)Expressing the mass M in terms of the horizon oneobtains f ( r ) = 43 G Q (cid:34) r − r (cid:35) . (18)Classical thermodynamics plays a crucial role since itprovides us with valuable information about the under-lying black hole physics. The Hawking temperature T ,the Bekenstein-Hawking entropy S as well as the heatcapacity C are given by T ( r ) = 14 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M G r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (19) S ( r ) = A H ( r )4 G , (20) C ( r ) = − A H ( r )4 G . (21)In agreement with the notation in the previous section, A H ( r ) is the so-called horizon area. This section summarizes the equations of motion for thescale dependent Einstein-Maxwell and Einstein-power-Maxwell theories. The notation follows closely [49] aswell as [27, 28].The scale dependent couplings of the theories are i) thegravitational coupling G k , and ii) the electromagneticcoupling 1 /e k . Furthermore, there are three indepen-dent fields, which are the metric g µν ( x ), the electro-magnetic four-potential A µ ( x ), and the scale field k ( x ).The effective action for the non-linear electrodynamicsreads Γ [ g µν , k ] = (cid:90) d x √− g (cid:20) κ k R − e βk L ( F ) (cid:21) , (22)The equations of motion for the metric g µν ( x ) are givenby G µν = κ k e βk T effec µν , (23) with T effec µν = T EM µν − e βk κ k ∆t µν . (24)Note that T EM µν is given by Eq. 3, κ k = 8 πG k is the Ein-stein constant and the additional object ∆t µν is definedas follows ∆t µν = G k (cid:16) g µν (cid:3) − ∇ µ ∇ ν (cid:17) G − k . (25)The equations of motion for the four-potential A µ ( x )taking into account the running of e k are D µ (cid:32) L F F µν e βk (cid:33) = 0 . (26)It is important to note that since the renormalizationscale k is actually not constant any more, this set ofequations of motion do not close consistently by itself.This implies that the stress energy tensor is most likelynot conserved for almost any choice of functional de-pendence k = k ( r ). This type of scenario has largelybeen explored in the context of renormalization groupimprovement of black holes in asymptotic safety scenar-ios [31–45]. The loss of a conservation laws comes fromthe fact that there is one consistency equation missing.This missing equation can be obtained from varying theeffective action (22) with respect to the scale field k ( r ),i.e. ddk Γ [ g µν , k ] = 0 , (27)which can thus be understood as variational scale set-ting procedure [46–50]. The combination of (27) withthe above equations of motion guarantees the conser-vation of the stress energy tensors. A detailed analysisof the split symmetry within the functional renormal-ization group equations, supports this approach of dy-namic scale setting [51].The variational procedure (27), however, requiresthe knowledge of the exact beta functions of the prob-lem. Since in many cases the precise form of the betafunctions is unknown (or at least unsure) one can, forthe case of simple black holes, impose a null energycondition and solve for the couplings G ( r ) , Λ ( r ) , e ( r )directly [27, 28, 52, 53]. This philosophy of assuring theconsistency of the equations by imposing a null energycondition will also be applied in the following studyon Einstein-Maxwell and Einstein-power-Maxwell blackholes. The so-called Null Energy Condition (hereafter NEC) isthe less restrictive of the usual energy conditions (dom-inant, weak, strong, and null), and it helps us to obtaindesirable solutions of Einstein’s field equations [54, 55].Considering a null vector (cid:96) µ , the NEC is applied on thematter stress energy tensor such as T mµν (cid:96) µ (cid:96) ν ≥ . (28)The application of such a condition was appropriatelyimplemented in Ref. [27] inspired by the Jacobson idea[56]. Note that in proving fundamental black hole the-orems, such as the no hair theorem [57] and the sec-ond law of black hole thermodynamics [58], the NECis, indeed, required. For scale dependent couplings, onerequires that the aforementioned condition is not vi-olated and, therefore, the NEC is applied on the ef-fective stress energy tensor for a special null vector (cid:96) µ = { f − / , f / , } such as T effec µν (cid:96) µ (cid:96) ν = (cid:18) T EM µν − e βk κ k ∆t µν (cid:19) (cid:96) µ (cid:96) ν ≥ . (29)In addition, the left hand side (LHS) is null as well as T EM µν (cid:96) µ (cid:96) ν = 0 and the condition reads ∆t µν (cid:96) µ (cid:96) ν = 0 . (30)One should note that Eq. 30 allows us to obtain thegravitational coupling G ( r ) easily by solving the differ-ential equation G ( r ) d G ( r ) dr − (cid:18) dG ( r ) dr (cid:19) = 0 , (31)which leads to G ( r ) = G (cid:15)r . (32)The NEC allows us to decrease the number of degreesof freedom, and thus it becomes an important tool forscale dependent black hole problems. In order to get insight into non-linear electrodynamicsregarding the running of couplings, one first has to dis-cuss the effects of scale dependence in linear electrody-namics. With this in mind, one also needs to determinethe set of four functions { G ( r ) , E ( r ) , f ( r ) , e ( r ) } whichare obtained by combining Einstein’s effective equationsof motion with the NEC taking into account the EOMfor the four-potential A µ . G ( r ) = G (cid:15)r ,E ( r ) = Q r e ( r ) ,f ( r ) = − G M ( r(cid:15) + 1) − Q e (ln( r/ ˜ r ) + r(cid:15) )( r(cid:15) + 1) , (33) e ( r ) = e (cid:34) r(cid:15) ) + 4 r(cid:15) (1 + r(cid:15) ) − (cid:32) M G − Q + 2 Q ln (cid:18) r ˜ r (cid:19)(cid:33) r (cid:15) (1 + r(cid:15) ) (cid:35) . where the integration constants are chosen such as theclassical Einstein-Maxwell (2+1)-dimensional black holeis recovered according to [59]. It is relevant to say thatthe gravitational coupling G ( r ) is obtained by takingadvantage of NEC, while the electric field E ( r ) is givenby the covariant derivative 26, which depends on theelectromagnetic coupling constant e ( r ). Besides, the lapsefunction f ( r ) and the coupling e ( r ) are directly ob-tained by using Einstein’s effective field equations com-bined with the solutions for E ( r ) and G ( r ). In addition,our solution reproduces the results of the classical the-ory in the limit (cid:15) →
0, i.e.lim (cid:15) → G ( r ) = G , lim (cid:15) → E ( r ) = Q r e , lim (cid:15) → f ( r ) = − G M − Q e ln (cid:18) r ˜ r (cid:19) , (34)lim (cid:15) → e ( r ) = e . which justifies the naming of the constants aforemen-tioned { G , M , Q , e } in terms of their meaning in theabsence of scale dependence [27], as it should be. Be-sides, the parameter (cid:15) controls the strength of the newscale dependence effects, and therefore it is useful totreat it as a small expansion parameter as follows G ( r ) ≈ G (cid:104) − r(cid:15) + O ( (cid:15) ) (cid:105) , (35) E ( r ) ≈ Q r e (cid:20) (cid:15)r + O ( (cid:15) ) (cid:21) , (36) f ( r ) ≈ f ( r ) + (cid:20) G M − Q e (37)+ Q e ln (cid:18) r ˜ r (cid:19)(cid:21) r(cid:15) + O ( (cid:15) ) ,e ( r ) ≈ e (cid:20) (cid:15)r + O ( (cid:15) ) (cid:21) . (38) In figure 1 the lapse function f ( r ) is shown for dif- - - r f ( r ) Fig. 1
Lapse function f ( r ) for (cid:15) = 0 (black solid line), (cid:15) = 0 .
04 (blue dashed line), (cid:15) = 0 .
15 (dotted red line) and (cid:15) = 1 (dotted dashed green line). The values for the rest ofthe parameters have been taken as unity. ferent values of (cid:15) in comparison to the classical (2+1)-dimensional Einstein-Maxwell solution. The figure showsthat the scale dependent solution for small (cid:15) · r valuesis consistent with the classical case. However, when (cid:15) · r becomes sufficiently large, a deviation from the classi-cal solution appears. The electromagnetic coupling e ( r )is shown in Figure 2 for different values of (cid:15) . Note thatwhen (cid:15) is small the classical case is recovered, but when (cid:15) increases the electromagnetic coupling tends to de-crease until it is stabilized. - - r e ( r ) Fig. 2
Electromagnetic coupling e ( r ) for (cid:15) = 0 (black solidline), (cid:15) = 0 . (cid:15) = 0 .
007 (dottedred line), (cid:15) = 0 .
02 (dotted dashed green line), (cid:15) = 0 .
08 (longdashed orange line) and (cid:15) = 0 . R andthe Kretschmann scalar K . Both of them are relevantin order to check if some additional divergences appear.For the static and circularly symmetric metric we haveconsidered, the Ricci scalar is given by R = − f (cid:48)(cid:48) ( r ) − f (cid:48) ( r ) r , (39)or more precisely R = Q r e (1 + r(cid:15) ) − M G e + 4 Q ln( r/ ˜ r )2 r (1 + r(cid:15) ) e r(cid:15) (40)+ 4 M G e − Q + 2 Q ln( r/ ˜ r )2 r (1 + r(cid:15) ) e ( r(cid:15) ) . We require that classically the Ricci scalar reads R = 12 Q e r . (41)Considering r values close to zero one obtains R ≈ R (cid:34) − (cid:34) M G e Q + 4 ln (cid:18) e r ˜ r (cid:19)(cid:35) r(cid:15) + · · · (cid:35) . (42)Thus, upon comparing Eq.40 with Eq.41 we observethat the scale dependent effect strongly distorts this in-variant. Despite that, for small values of r the standardcase R is recovered. In the same way, one expects that (cid:15) should be small, therefore one can expand the Ricciscalar around (cid:15) = 0 but the solution is exactly the samereported for r (cid:28)
1. Regarding the Kretschmann scalar,it is computed to be K = R µναβ R µναβ . (43)Thus, when (cid:15) is small the Kretschmann scalar reads K ≈ K (cid:34) − (cid:34) M G e Q + 83 ln (cid:18) r ˜ r (cid:19)(cid:35) r(cid:15) (cid:35) + · · · (44)Note that the classical result for this invariant is indeed K = 3 Q / r , which coincides with our solution when (cid:15) → r → ∞ . In this limit thelapse function f ( r ) decays as r − which disagrees withthe classical result shown in Eq.15. On the other hand,the electromagnetic coupling e ( r ) also tends to zero as r − in contrast with the expected result, e . Finally,one obtains that E ( r ) ∼ r − , R ∼ r − and K ∼ r − ,all of them going to zero as expected. However, it canbe shown that these functions decay faster than thosecorresponding to the classical solutions. In fact, in ab-sence of running coupling, a straightforward calculationreveals that E ( r ) ∼ r − , R ∼ r − and K ∼ r − . 5.3 HorizonsThe apparent horizon occurs when the lapse functionvanishes, i.e. f ( r H ) = 0. Thus, this Einstein-Maxwellblack hole solution represents a non trivial deviationfrom the classical solution which is manifest when wecompare our solution with the corresponding black holesolution without the scale dependence. Here, the hori-zon read r H = 1 (cid:15) W (cid:32) (cid:15)e − G M e Q (cid:33) , (45)where W ( · ) is the so-called Lambert- W function, whichis a set of functions, namely the branches of the inverserelation of the function Y ( r(cid:15) ) = r(cid:15)e r(cid:15) with r(cid:15) being acomplex number. In particular, Eq 45 is also the prin-cipal solution for r(cid:15) . In Figure 3 the scale dependenteffect on horizon is shown. We can see that the devia-tion from the classical case is also evident for small M values. M r H Fig. 3
Black hole horizons r H as a function of the mass M for (cid:15) = 0 (black solid line), (cid:15) = 1 (blue dashed line), (cid:15) = 2 . (cid:15) = 6 (dotted dashed green line). Thevalues of the rest of the parameters have been taken as unity. In addition, one can expand the horizon around (cid:15) = 0obtaining the classical solution plus corrections i.e. r H ≈ r (cid:34) − (cid:15)r + O ( (cid:15) ) (cid:35) . (46)5.4 Thermodynamic propertiesAfter having gained experience on the horizon structureone can now move towards the usual thermodynamicproperties associated with our solution shown at Eq.
33. Thus, the Hawking temperature of the black holeassuming the ansatz 5 is given by T H ( r H ) = 14 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim r → r H ∂ r g tt √− g tt g rr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (47)i.e. T H ( r H ) = 14 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q r H (1 + (cid:15)r H ) e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (48)Taking advantage of the fact that the integration con-stant (cid:15) should be small, one can expand around (cid:15) = 0to get the well-known Hawking temperature (at leaderorder) i.e. T H ( r H ) ≈ T ( r ) (cid:12)(cid:12)(cid:12) (cid:15)r + O ( (cid:15) ) (cid:12)(cid:12)(cid:12) . (49)In Figure 4 we show the effective temperature whichtakes into account the running coupling effect. M T H Fig. 4
The Hawking temperature T H as function of theclassical mass M for (cid:15) = 0 (black solid line), (cid:15) = 750 (bluedashed line), (cid:15) = 1800 (dotted red line) and (cid:15) = 3000 (dotteddashed green line). The other values of the rest of the param-eters have been taken as unity. Note that the vertical axis isscaled 1 : 10 Moreover, the Bekenstein-Hawking entropy for his blackhole is S = A H ( r H )4 G ( r H ) = S ( r H )(1 + (cid:15)r H ) , (50)and assuming small values of (cid:15) one can expand to get S ≈ S ( r ) (cid:34) −
12 ( (cid:15)r ) + O ( (cid:15) ) (cid:35) . (51)In Figure 5 below we show the entropy for our (2+1)-dimensional Einstein-Maxwell scale dependent black hole.It is clear that the running effect is dominant when (cid:15) M S Fig. 5
The Bekenstein-Hawking entropy S as function ofclassical mass M for (cid:15) = 0 (black solid line), (cid:15) = 200 (bluedashed line), (cid:15) = 600 (dotted red line) and (cid:15) = 1000 (dotteddashed green line). The other values have been taken as unity. is not small, while for large values of M the effect ispractically zero.Finally, the heat capacity is computed in the usual wayi.e.: C Q = T ∂S∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q , (52)which read C Q = − S ( r H )(1 + (cid:15)r H ) . (53)The classical case is, of course, recovered in the (cid:15) → (cid:15) dependence it was necessary toplot the figure with very large values of (cid:15) in order togenerate a visible effect. The scale dependent effect isnotoriously small for those quantities.5.5 Total chargeThe electric field is parametrized through the total char-ge Q , but in our previous discussion Q only denotes anintegration constant which coincides with the charge ofthe classical theory. In general, we need to compute thetotal charge by the following relation [61] Q = (cid:90) √− g d Ω (cid:32) L F F µν e βk (cid:33) n µ σ ν , (54)where n µ and σ ν are the unit spacelike and timelikevectors normal to the hypersurface of radius r , and theyare given by n µ = ( f − / , ,
0) and σ ν = (0 , f / ,
0) aswell as √− g d Ω = r d φ . Making use of these we obtain Q = 2 πQ , (55)which is proportional to the classical value and has no (cid:15) dependence. This section is devoted to the study of a (2+1) scale de-pendent gravity coupled to a power-Maxwell source. Asmentioned before, the case β = 3 / { G ( r ) , E ( r ) , f ( r ) , e ( r ) } , which areobtained by combining Einstein’s effective equations ofmotion with the NEC taking advantage of the EOM forthe four-potential A µ . In what follows we shall obtainthe solutions of the system in terms of the functionsmentioned above.6.1 SolutionThe integration constants have been chosen such as thescale dependent solution reduces to the classical NLEDcase when the appropriate limit is taken. Thus, our so-lution reads G ( r ) = G (cid:15)r ,E ( r ) = Q r (cid:18) e ( r ) e (cid:19) , (56) f ( r ) = 4 G Q r ( r(cid:15) + 1) − M G (cid:0) r (cid:15) + 3 r (cid:15) + 3 r (cid:1) r ( r(cid:15) + 1) ,e ( r ) = e (cid:34) (2 r(cid:15) (3 r(cid:15) + 2) + 1)( r(cid:15) + 1) − M r (cid:15) ( r(cid:15) + 4)4 Q ( r(cid:15) + 1) (cid:35) . In the limit (cid:15) → (cid:15) → G ( r ) = G , lim (cid:15) → E ( r ) = Q r , (57)lim (cid:15) → f ( r ) = 4 G Q r − G M , lim (cid:15) → e ( r ) = e . Note that if we set e = 1, the classical solution insection 2.2 is recovered. Even more, if one demands that G = 1 (which is the standard lore) then we are incomplete agreement with the classical solution given atRef. [60]. 6.2 Asymptotic behaviourThe asymptotic behaviour of this solution can be stud-ied by computing geometrical invariants i.e. the Ricciscalar, which for our solution is R = − G (cid:15) (cid:34) M + 4 Q (cid:15)r ( r(cid:15) + 1) (cid:35) , (58)where the classical case (with a null cosmological con-stant) is clearly R = 0. For r → R ≈ − G (cid:15) (cid:34) M + 4 Q (cid:15)r (cid:35) + O ( r ) . (59)We observe that the Ricci scalar is altered in presence ofscale dependent coupling. In addition, one note that anunexpected r divergence appears, which is controlledby (cid:15) . Another geometrical invariant is the Kretschmannscalar K which is given by K = R µναβ R µναβ . (60)For r → K ≈ G Q r (cid:34) − (cid:32) M Q (cid:15) + 4 (cid:15) (cid:33) r (cid:35) + O ( r − ) . (61)Taking into account that the (cid:15) should be small we have K ≈ G Q r (cid:34) − M r Q (cid:15) + O ( (cid:15) ) (cid:35) , (62)where the standard value K has been obtained de-manding that (cid:15) goes to zero. Classically, the Ricci scalarfor null cosmological constant is identically zero, how-ever in presence of scale dependent couplings it exhibitsa singularity. The Kretschmann scalar exhibits a singu-larity at r → r → ∞ both for the Ricci and the Kretschmannscalar. The Ricci scalar as well as the Kretschmannscalar are asymptotically close to zero.Regarding the limit r → ∞ the lapse function goesas r − in agreement with the asymptotic behaviour ofthe classical solution. In addition, note the unusual be-haviour of the electromagnetic coupling in the light ofscale dependent framework in Fig. 8. Starting from e the electromagnetic coupling decays softly and it stabi-lizes whenlim r →∞ e ( r ) = − (cid:18) r (cid:15) (cid:19) e , (63)instead of reach the classical value. The electric fieldtends to zero as expected but slowly compared with the - - r f ( r ) Fig. 6
Lapse function f ( r ) for (cid:15) = 0 (black solid line), (cid:15) = 0 .
04 (blue dashed line), (cid:15) = 0 .
15 (dotted red line) and (cid:15) = 1 (dotted dashed green line). The values of the rest of theparameters have been taken as unity. classical case. In fact, E ( r ) behaves as r − in clearlydeviation with respect to the result shown in Eq.16.Finally, the curvature and Kretschmann scalars holdthe same asymptotic behaviour of the results obtainedin absence of running, i.e. R ∼ r − and K ∼ r − .6.3 HorizonsApplying the condition f ( r H ) = 0 one obtains the scaledependent horizon which reads r H = − (cid:15) (cid:20) − (cid:104) (cid:15)r (cid:105) / (cid:21) , (64) r ± = − (cid:15) (cid:34) ± i √ (cid:20) (cid:15)r (cid:21) / (cid:35) . (65)where r is the classical value given by Eq. 17. Notethat one obtains three horizons, out of which one isreal (physical horizon) and two r ± are complex (non-physical).In addition, since the scale dependence of coupling con-stants is usually assumed to be weak, it is reasonableto consider the dimensionful parameter (cid:15) as small com-pared to the other scales and, therefore, one can expandaround (cid:15) close to zero, which gives us r H ∼ = r (cid:20) − (cid:15)r + 53 ( (cid:15)r ) + · · · (cid:21) . (66)One should note that when (cid:15) tends to zero the classicalcase is recovered. Besides, although (cid:15) could take posi-tive or negative values, here in order to obtain desirablephysical results we require that (cid:15) >
0. In our set of so-lutions { G ( r ) , E ( r ) , f ( r ) , e ( r ) } we can expand around M r H Fig. 7
Black hole horizons r H as a function of the mass M for (cid:15) = 0 (black solid line), (cid:15) = 0 . (cid:15) = 1(dotted red line) and (cid:15) = 2 (dotted dashed green line). Thevalues of the rest of the parameters have been taken as unity. zero for small values of (cid:15) , i.e. G ( r ) ≈ G (cid:104) − r(cid:15) + O ( (cid:15) ) (cid:105) , (67) E ( r ) ≈ E ( r ) + O ( (cid:15) ) , (68) f ( r ) ≈ f ( r ) + (cid:20) G M − G Q r (cid:21) r(cid:15) + O ( (cid:15) ) , (69) e ( r ) ≈ e (cid:20) O ( (cid:15) ) (cid:21) . (70) - - r e ( r ) Fig. 8
Electromagnetic coupling e ( r ) for (cid:15) = 0 (black solidline), (cid:15) = 0 .
25 (dashed blue line), (cid:15) = 0 .
45 (dotted red line)and (cid:15) = 1 (dotted dashed green line). The values of the restof the parameters have been taken as unity. hole. At the outer horizon this temperature is givenby the simple formula T H = 14 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim r → r H ∂ r g tt √− g tt g rr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (71)which reads in term of the horizon radius T H = 14 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M G r H (1 + (cid:15)r H ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (72) M T H Fig. 9
Hawking temperature T H as a function of the classi-cal mass M for (cid:15) = 0 (black solid line), (cid:15) = 20 (blue dashedline), (cid:15) = 50 (dotted red line) and (cid:15) = 100 (dotted dashedgreen line). The values of the rest of the parameters have beentaken as unity. In order to recover the classical result we expand around (cid:15) = 0 and upon evaluating at the classical horizon weobtain T H ( r H ) ≈ T ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:15)r ) + O ( (cid:15) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (73)where it is clear that (cid:15) → S = 14 (cid:73) d x √ hG ( x ) , (74)where h ij is the induced metric at the horizon. For thepresent circularly symmetric solution this integral istrivial because the induced metric for constant t and r slices is d s = r d φ and moreover G ( x ) = G ( r H ) isconstant along the horizon. Using these facts, the en-tropy for this solution is found to be [27, 28] S = A H G ( r H ) = S ( r H )(1 + (cid:15)r H ) , (75) while for small values of (cid:15) one obtains S ≈ S ( r ) (cid:34) −
13 ( (cid:15)r ) + O ( (cid:15) ) (cid:35) , (76)which, of course, coincides with the classical results inthe limit (cid:15) →
0. In addition, the heat capacity (at con- M S Fig. 10
The Bekenstein-Hawking entropy S as a function ofthe classical mass M for (cid:15) = 0 (black solid line), (cid:15) = 20 (bluedashed line), (cid:15) = 50 (dotted red line) and (cid:15) = 100 (dotteddashed green line). The other values have been taken as unity. stant charge) C Q can be calculated by C Q = T ∂S∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q . (77)Combining Eq. 72 with 75 one obtains the simple rela-tion C Q = − M T H = − S ( r H )(1 + (cid:15)r H ) . (78)Note that the black hole is unstable since C Q <
0, andit coincides with the classical result in the limit (cid:15) → Q needs to becomputed by the relation [61] Q = (cid:90) √− g d Ω (cid:32) L F F µν e βk (cid:33) n µ σ ν . (79)In this case we obtain Q = Q e / , (80)which also is proportional to the classical value anddoes not have (cid:15) dependence. Scale dependent gravitational couplings can induce non-trivial deviations from classical Black Holes solutions.We have studied two cases, first Einstein-Maxwell andsecond Einstein-power-Maxwell case. Both of them havea common feature: the lapse function tends to zerowhen r → ∞ , characteristic which is absent in the clas-sical solutions. In addition, the total charge is modifiedas a consequence of our scale dependent framework.Moreover, we have found that, for the same value ofthe classical black hole mass, the apparent horizon ra-dius (and the Bekenstein-Hawking entropy) decreaseswhen the strength of the scale dependence increases.This is in agreement with the findings in [31–45]. On theother hand, the Hawking temperature increases with (cid:15) .Please, note that the effect of the scale dependence inthe Einstein-power-Maxwell case is stronger than theEintein-Maxwell case. The behaviour of the electromag-netic coupling e ( r ) depends on the choice of the electro-magnetic Lagrangian density. While e ( r ) goes to zeroin the limit r → ∞ for a Maxwell Lagrangian density,it approaches a constant value for the power-Maxwellcase. Finally, it is well known that a black hole (as athermodynamical system) is locally stable if its heat ca-pacity is positive [62]. In both scale dependent cases itis found that these black holes are unstable ( C Q < In this article we have studied the scale dependenceof charged black holes in three-dimensional spacetimeboth in linear (Einstein-Maxwell) and non-linear (Eins-tein-power-Maxwell) electrodynamics. In the second ca-se we have considered the case where the electromag-netic energy momentum tensor is traceless, which hap-pens for β = 3 /
4. After presenting the models and theclassical black hole solutions, we have allowed for a scaledependence of the electromagnetic as well as the grav-itational coupling, and we have solved the correspond-ing generalized field equations by imposing the ”nullenergy condition” in three-dimensional spacetimes withstatic circular symmetry. Horizon structure, asymptoticspacetimes and thermodynamics have been discussed indetail.
In this appendix we study some features of the scaledependent (2 + 1) gravity coupled to a power-Maxwell source for an arbitraty β . For this system the action isgiven by S = (cid:90) d x √− g (cid:20) πG ( r ) R − e ( r ) β L ( F ) (cid:21) , (81)where G ( r ) and e ( r ) are the gravitational and the theelectromagnetic scale-dependent couplings, R is the Ri-cci scalar, L ( F ) = C β | F | β is the electromagnetic La-grangian density, F = (1 / F µν F µν is the Maxwell in-variant, and C is a dimensionless constant which de-pends on the choice of β . Metric signature ( − , + , +)and natural units ( c = (cid:126) = k B = 1) are used in ourcomputations.Variations of the Eq.81 with respect to the metric fieldlead to the modified Einstein’s equations R µν − g µν R = 8 π G ( r ) e β ( r ) T µν − ∆t µν , (82)where T µν stands for the power-Maxwell energy mo-mentum tensor and ∆t µν = G ( r ) (cid:16) g µν (cid:3) − ∇ µ ∇ ν (cid:17) G ( r ) − , (83)is the non-material energy momentum tensor which ari-ses as a consecuence of the scale dependence of the gra-vitational coupling. On the other hand, after variationsof the action Eq.81 with respect to the electromagneticfour–potential, A µ , one obtains the modified Maxwellequations D µ (cid:32) L F F µν e ( r ) β (cid:33) = 0 . (84)Henceforth, only static and circularly symmetric solu-tions will be considered. Therefore we shall assume theansatz ds = − f ( r ) dt + f ( r ) − dr + r dΩ , (85) F µν = ( δ tµ δ rν − δ rµ δ tν ) E ( r ) , (86)for the metric and the electromagnetic tensor, respec-tively. With the former prescription is straightforwardto prove, from Eq.84, that the electric field is given interms of the electromagnetic coupling by E ( r ) = 2 β − β − C − β β − Q β − e ( r ) β β − β β − r β − , (87)or, in a more convenient way E ( r ) = (cid:34)(cid:32) β − C − β β (cid:33)(cid:32) Q r e ( r ) β (cid:33)(cid:35) β − . (88)Please, note that setting β = 1 and C = 1 the electricfield reported in Eq.33 is recovered E ( r ) = Q r e ( r ) . (89) In the same way, for β = 3 / C / = 2 / − e Q / one obtain E ( r ) = Q r (cid:18) e ( r ) e (cid:19) , (90)in complete agreement with Eq.56. It is worth notingthat, even in the general case the electric field dependson an specific power of the charge as a consequence ofthe non–linear electrodynamics, in the cases β = 1 and β = 3 /
4, this behaviour is not observed due to a par-ticular setting of C .If the null energy condition is used as an additional con-dition, we obtain that the scale-dependent gravitationalcoupling reads G ( r ) = G (cid:15)r , (91)where G is Newton’s constant and (cid:15) is the runningparameter. Note that the classical limit is recovered inthe limit (cid:15) →
0. Finally, Eq.82 reduces to a pair ofdifferential equations for { f ( r ) , e ( r ) α } given by2 α κ C − α Q α (2 β − re ( r ) α + β α r α (cid:16) (2 r(cid:15) + 1) f (cid:48) ( r ) + 2 (cid:15)f ( r ) (cid:17) = 0 , (92)2 α κ C − α Q α e ( r ) α − β α r α (cid:16) ( r(cid:15) + 1) f (cid:48)(cid:48) ( r ) + 2 (cid:15)f (cid:48) ( r ) (cid:17) = 0 , (93)where α = β β − and κ = 8 πG . It can be checkedby the reader that, in the case β = 3 /
4, the solutionsof the set of equations 92, 93, 91 and 87 coincide withthose listed in Eq.56 after an appropriate choice of theintegration constants.
Acknowledgements
The author A.R. was supported bythe CONICYT-PCHA/ Doctorado Nacional/2015-21151658.The author P.B. was supported by the Faculty of Science andVicerrector´ıa de Investigaciones of Universidad de los Andes,Bogot´a, Colombia.The author B.K. was supported by the Fondecyt 1161150.The author G.P. acknowledges the support from ”Funda¸c˜aopara a Ciˆencia e Tecnologia”.
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