Universal thermodynamic extremality relations for charged AdS black hole surrounded by quintessence
aa r X i v : . [ g r- q c ] M a y CTP-SCU/2020018
Universal thermodynamic extremality relations for charged AdSblack hole surrounded by quintessence
Qiuhong Chen a , ∗ Wei Hong a , † and Jun Tao a ‡ a Center for Theoretical Physics, College of Physics,Sichuan University, Chengdu, 610065, China
Abstract
The perturbation correction of general relativity has obtained the general relationship betweenentropy and extremality. The relation was found to exactly hold for the charged AdS black hole,and extended to the rotating and massive gravity black holes. In this paper, We confirm theuniversal relation for the four dimensional RN-AdS black hole surrounded by the quintessence, andfind a new extremality relation between black hole mass and the normalization factor a related tothe density of quintessence. Besides, we also find that when the black hole entropy increases, thecorrection constant is negative, the black hole mass and the mass-charge ratio decreases, hence thisblack hole displays WGC-like behavior. Moreover, we derive the general relationship on nonlinearelectrodynamic source couples to AdS spacetime with quintessence field. We find another newextremality relation between black hole mass and the magnetic monopole charge β . ∗ Electronic address: [email protected] † Electronic address: thphysics˙[email protected] ‡ Electronic address: [email protected] ontents I. Introduction II. RN-AdS black holes surrounded by the quintessence III. Regular Bardeen AdS black hole surrounded by quintessence field IV. Conclusion Acknowledgments References I. INTRODUCTION
Quantum gravity combines the principles of general relativity and quantum theory, whichexpected to be able to provide a satisfactory description of the microstructure of spacetimeon the Planck scale. This scale is so far from current experimental capabilities that it isalmost impossible to empirically test the theory of quantum gravity along the standardroute. Quantum gravity leads to the area dependence of these solutions, that cannot beclearly explained in terms of the original Einstein gravity [1]. Many aspects of quantumgravity can be analyzed within the framework of efficient field theory. String theory iswidely regarded as the complete description of quantum gravity. This theory is also thoughtto acknowledge an astronomical vacuum that behaves as the efficient field theory at lowenergies, and this consistent set of string vacuums is known as landscape. With so manylow energy descriptions, it may be difficult or impossible to find a vacuum to describe ourworld. Under this background, the swampland program [2–4] was proposed. The swamplandprogram aims to obtain consistent conditions for efficient field theory by requiring low energyefficiency field theory to always be coupled with quantum gravity.A convincing proposal for such a condition is the weak gravity conjecture (WGC) [4],which is a universal principle of string vacuums with uniform constraints. Various spec-ulations have been proposed, but the efficiency field theory that appears as a low-energydescription of quantum gravity theory must have a charge state greater than mass. If the2bove conjecture is not true, the extreme or near-extreme black hole will not be able toevaporate, because releasing the sub-extreme state will cause the remaining black hole tobecome a super-extreme, which violates the cosmic censorship. Various applications of theWGC and related quantum gravity conjectures have been discussed in some works [5–18].Among them, a method for judging WGC by the amount of change in entropy is mentioned[19]. This method points out: If ∆
S >
0, then the single U (1) form of the WGC followsimmediately, the black hole systems display WGC-like behavior [20].Recently, Goon and Penco proposed a universal thermodynamic extremality relation bythe perturbative corrections of generic thermodynamic systems [20] ∂M ext ( ~ Q , ǫ ) ∂ǫ = lim M → M ext ( ~ Q ,ǫ ) − T ∂S ( M, ~ Q , ǫ ) ∂ǫ ! M, ~ Q . (1)Where ǫ is the control parameter of the correction, M ext , T , and S are the extremalitymass, temperature and entropy of the black hole respectively after the correction. And Q denotes the extensive quantities of the black hole thermodynamics such as charge, angularmomentum or other quantities. This relation was verified in the four-dimension RN-AdSblack hole and extended to the rotating and massive gravity black holes by rescaling thecosmological constant as a perturbative correction [20–22].In this paper, we will calculate the universal thermodynamic relations of black holessurrounded by quintessence. Nowadays, the nature of dark energy is driving the observedaccelerated development of the universe, and this is undoubtedly one of the most excitingand challenging issues facing physicists and astronomers. It is one of the cores of currentastronomical observations and proposals and is driving the way particle theorists try to un-derstand the nature of the early and late universes. Astronomical observations indicate thatthe universe is accelerating, which implies the existence of a state of negative pressure. Thecause of negative pressure can be twofold. One side is called quintessence [23–27], with apossible explanation coming from the anti-gravity nature of dark energy. Quintessence canbe described by an ordinary scalar field ϕ that is minimally coupled with gravity, whichhas particular potentials that lead to late time inflation. We introduce a perturbative cor-rection into the action and derive the extremality relations between the mass and pressure,entropy, charge, and quintessence factor, respectively. Furthermore, we can make certainmodifications to the black hole surrounded by the quintessence, such as considering theblack hole under the nonlinear electrodynamics background [28, 29], which is called regular3ardeen AdS black hole. A black hole has no singularity at the origin and has an eventhorizon. Bardeen first constructed a black hole solution with regular non-singular geometryand event horizon [30].The rest of this paper is organized as follows. In the next section, the AdS black holesurrounded by the quintessence is given and its perturbative correction is discussed. Wewill use numerical analysis to calculate the extremality black hole mass M ext and entropy S after small constant correction to the cosmological constant. In Sec. III, we will showthe universal thermodynamic extremality relation to the regular Bardeen AdS black holesurrounded by quintessence field. In Sec. IV, we will summarize our research work. II. RN-ADS BLACK HOLES SURROUNDED BY THE QUINTESSENCE
We start with a simpler case and derive our hypothesis. We place the RN-AdS black holein the quintessence considered as barotropic perfect fluid, and do not add other physicalcorrections. The bulk action for a RN-AdS black hole surrounded by quintessence darkenergy in four dimensional is described as follows [31, 32] S = 116 πG Z d x (cid:2) √− g ( R − − F µν F µν ) + L q (cid:3) . (2)In this action, R is the Ricci scalar, the cosmological constant is related to the AdS spaceradius l by Λ = − /l and F µν is the electromagnetic field tensor. The last term L q in theaction is the Lagrangian of quintessence as a barotropic perfect fluid, which is given by [33] L q = − ρ q (cid:20) ω q ln (cid:18) ρ q ρ (cid:19)(cid:21) , (3)where ρ q is energy density, ρ is the constant of integral, and the barotropic index ω q .Quintessence with the state equation given by the relation between the pressure p q andthe energy density ρ q , so that p q = ω q ρ q at ω q in the range of − < ω q < − / ω q < − ω q = − ds = − f ( r ) dt + 1 f ( r ) dr + r ( r ) (cid:0) dθ + sin θdϕ (cid:1) ,f ( r ) = 1 − Mr + Q r + r l − ar ω q +1 . (4)4here M and Q are the mass and electric charge of the black hole, respectively, and a isthe factor related to the density of quintessence as [34, 35] ρ q = − a ω q r ( ω q + 1) . (5)We use the black hole entropy S = πr + and combine the black hole horizon equation, themass M and temperature T of this black hole can be expressed as T = 34 aπ ωq ω q S − ωq − + 3 √ S π / l − √ πQ S / + 14 √ π √ S , (6) M = − aπ ωq S − ωq + S / π / l + √ πQ √ S + √ S √ π . (7)Moreover, we can write those physical quantity η = − π ωq S − ωq conjugate to the pa-rameter a , electric potential Φ = √ πQ/ √ S and thermodynamic volume V = 4 S / / √ π .For comparing with the modified mass M and entropy S , we fixed some parameters here todraw a graph of the unmodified mass M and entropy S as a function of the charge Q whenthe black hole near-extreme. From the picture on the left of FIG. (1), we draw the criticalstate where the mass-to-charge ratio is equal to one, which is represented by a dashed line.We can find that the ratio of the unmodified black hole mass to the charge amount is greaterthan one. From the picture on the right of FIG. (1), we can find that the unmodified blackhole entropy increases as the black hole charge increases. l = = = M (a)unmodified M as a function of charge Q l = = = S (b)unmodified S as a function of charge Q FIG. 1: The parameters of the black hole vary with the black hole charge Q . We fixed density ofquintessence ρ q as 1.0, ω q as − / l as 0.1, 0.2, 0.3 in order. The solid linein the figure is the curve of the black hole mass varies with the black hole charge, and the blackhole entropy varies with the black hole charge. And the dashed line in the figure is the criticalcurve where the ratio of the black hole mass to the charge is one. S = 116 πG Z d x (cid:8) √− g [ R − ǫ )Λ − F µν F µν ] + L q (cid:9) , (8)and the modified mass M and temperature TT = 34 aπ ωq ω q S − ωq − + 3 √ S ( ǫ + 1)4 π / l − √ πQ S / + 14 √ π √ S , (9) M = − aπ ωq S − ωq + S / ( ǫ + 1)2 π / l + √ πQ √ S + √ S √ π . (10)With these expressions, we can do a numerical analysis of the black hole mass and black holeentropy corrected with a small constant ǫ . From the picture on the left of FIG. (2), we findthe mass of black hole has been some small perturbations if we compare them unmodifiedblack hole mass. When ǫ = 0 .
1, the mass of the black hole increases, and when ǫ = − .
1, themass of the black hole decreases. From the picture on the right of FIG. (2), we can obtainthat the black hole entropy also has been some small perturbation similar to the black holemass. However, unlike the black hole mass perturbation, the condition of the black holeentropy increases and decreases in the opposite. When ǫ = 0 .
1, the entropy of the black holedecreases, and when ǫ = − .
1, the entropy of the black hole increases. From here, we canpreliminarily see the clue of WGC. When our constant correction ǫ to be negative, the massof the black hole can be lower, the mass-charge ratio of the black hole gradually decreasesand approaches to one. The change of the entropy ∆ S > = unmodifiedl = ϵ = = ϵ =- M (a)modified M as a function of charge Q compareswith unmodified M as a function of charge Q . l = unmodifiedl = ϵ = = ϵ =- S (b)modified S as a function of charge Q compareswith unmodified S as a function of charge Q . FIG. 2: The parameters of the black hole vary with the black hole charge Q . We fixed thenormalization factor a as 1.0, AdS space radius l as 0.1 and ω q as − /
3. Besides, we fixed thesmall constant ǫ as -0.1, 0.0, and 0.1, which are represented by green, red and blue solid linesrespectively. And, the dashed line represents the critical state which means the mass-to-chargeratio is equal to one. From Eq. (10), we can obtain the small correction ǫǫ = al π ωq +32 S − ωq +32 + 2 π / l MS / − π l Q S − πl S − . (11)Taking the derivative of it with respect to S, one can get (cid:18) ∂ǫ∂S (cid:19) M,Q,l,a = − al π ωq +32 ω q S − ωq − + π l Q S − πl S − ǫ + 1)2 S . (12)Combining equations (9) and (12), we can obtain − T (cid:18) ∂S∂ǫ (cid:19) M,Q,l,a = S / π / l . (13)Let’s consider (cid:0) ∂M ext ∂ǫ (cid:1) a,l,Q now. The new extremality can be obtained as T = 0,it’s very difficult to solve for entropy directly. We can express extremal mass as M ext ( a, l, Q, S ( a, l, Q, ǫ ) , ǫ ), then use the chain rule of derivation ∂M ext ∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) a,l,Q = ∂M ext ∂S (cid:12)(cid:12)(cid:12)(cid:12) a,l,Q,ǫ ∂S∂ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a,l,Q,ext + ∂M ext ∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) a,l,Q,S , (14)where ∂S∂ǫ (cid:12)(cid:12) a,l,Q,ext can be obtained by extremality condition. And ∂M ext ∂S (cid:12)(cid:12) a,l,Q,ǫ and ∂M ext ∂ǫ (cid:12)(cid:12) a,l,Q,S are partial derivatives of Eq. (10). When Eq. (9) equals to zero, we have ∂ǫ∂S (cid:12)(cid:12)(cid:12)(cid:12) a,l,Q,ext = 32 al ( ω q + 1) π ωq + ω q S − ωq − − π l Q S + πl S . (15)7ombining these results, we can obtain ∂M ext ∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) a,l,Q = S / π / l = − T (cid:18) ∂S∂ǫ (cid:19) M,Q,l,a . (16)Similarly, we continue to investigate the extremality relation between the mass and charge.From Eq. (11), the differential of ǫ to Q takes the form (cid:18) ∂ǫ∂Q (cid:19) M,S,l,a = − πl QS , . (17)Multiplying by the electric potential Φ = √ πQ/ √ S and taking the extremality, we have − Φ (cid:18) ∂Q∂ǫ (cid:19) M,S,l,a = S / π / l = ∂M ext ∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) a,l,Q . (18)Next, we would like to regard the cosmological constant as a variable related to pressure,and investigate the extremality relation between the mass and pressure. Bring the relation P = 3 / πl to Eq. (11), the differential of P to ǫ takes the form (cid:18) ∂P∂ǫ (cid:19) M,S,Q,a = 3 (cid:16) − aπ ωq +12 S − ωq − √ πM √ S + πQ + S (cid:17) S ( ǫ + 1) . (19)Multiplying by the thermodynamic volume V = 4 S / / √ π and taking the extremality, wehave − V (cid:18) ∂P∂ǫ (cid:19) M,S,Q,a = S / π / l = ∂M ext ∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) a,l,Q . (20)Last, we can consider the extremality relation between the mass and conjugate quantity η . From Eq. (11), the differential of quintessence parameter a to ǫ takes the form (cid:18) ∂a∂ǫ (cid:19) M,S,Q,l = π − ω − S ω + l . (21)Multiplying by η = − π ωq S − ωq and taking the extremality, we have − η (cid:18) ∂a∂ǫ (cid:19) M,S,Q,l = S / π / l = ∂M ext ∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) a,l,Q . (22)Therefore, we confirm the Goon-Penco extremality relation for the charged RN-AdS blackhole surrounded by quintessence under the constant correction of the action. In order toshow how the parameter ǫ modifies the black hole mass and black hole entropy, we performnumerical analysis on corrected mass and entropy. In such a case, when ǫ = 0, the system8egenerates to the situation without correction. From the figure on the left of FIG. (3), wefind that the mass of the black hole increases when the correction constant ǫ is positive, anddecreases when the correction constant ǫ is negative. In addition, the larger the radius l ofthe AdS space, the easier it is to correct the black hole mass-charge ratio M/Q to less thanone, which is what we expect. From the figure on the right of FIG. (3), we can obtain that theentropy of the black hole decreases when the correction constant ǫ is positive, and increaseswhen the correction constant ǫ is negative. In other word, when the black hole entropyincreases, the correction constant is negative, the black hole mass and the mass-charge ratiodecreases, hence this black hole displays WGC-like behavior. l = = l = - - ϵ M e x t (a)modified M as a function of parameter ǫ l = = = - - ϵ S e x t (b)modified S as a function of parameter ǫ FIG. 3: The parameters of the black hole vary with the small correction parameter ǫ . We fixednormalization factor a as 1.0, ω q as − /
3, black hole charge Q as 1.0, and AdS space radius l as0.1, 0.2, 0.3 in order. The color solid lines in the figure is the curve of black hole mass and entropyvaries with small correction parameter ǫ . The dashed line in the figure is the critical curve withthe black hole mass equaling to charge. The intersection of the black vertical line and the colorsolid lines at ǫ = 0 in the figure indicates the uncorrected mass and entropy of the black hole when Q = 1 . III. REGULAR BARDEEN ADS BLACK HOLE SURROUNDED BYQUINTESSENCE FIELD
Based on the Sec. II, we consider the universal thermodynamic extremality relationsunder a nonlinear electrodynamic source coupled to AdS spacetime with quintessence field.This type of black hole can be called as regular Bardeen AdS black hole.9he action for regular Bardeen AdS black hole surrounded by quintessence field is givenby [28, 29] S = Z d x √− g (cid:20) π R − π Λ − π L ( F ) (cid:21) . (23)Where R is the Ricci scalar, the cosmological constant is related to the AdS space radius l by Λ = − /l , and L ( F ) represents the Lagrangian for a nonlinear electrodynamic source L ( F ) = 3 Mβ p β F p β F . (24)Where F = F µν F µν , β is the magnetic monopole charge of a self gravitating magnetic fielddescribed by nonlinear electromagnetic source. The solution for this action is given by ds = − f ( r ) dt + 1 f ( r ) dr + r ( r ) (cid:0) dθ + sin θdϕ (cid:1) ,f ( r ) = 1 − M r ( β + r ) / + r l − ar ω q +1 . (25)The following work is the same as in the Section II. We make a small correction to thecosmological constant term in the action S = Z d x √− g (cid:20) π R − (1 + ǫ )8 π Λ − π L ( F ) (cid:21) . (26)We can obtain the modified mass M and temperature T , which are described by entropy S ,magnetic monopole charge β and normalization factor aT = 34 aπ ωq p πβ + SS − ωq − (cid:2) πβ ( ω q + 1) + Sω q (cid:3) + 3( ǫ + 1) p πβ + S π / l + ( S − πβ ) p πβ + S π / S , (27) M = ( πβ + S ) / (cid:16) S ωq +12 − aπ ωq +12 (cid:17) π / S ωq +32 + ( ǫ + 1) ( πβ + S ) / π / l . (28)In order to calculate the left side of Eq.(1), we need to insert the entropy at the extremalityinto the mass (28) to obtain the differential relationship. It’s very difficult to do that,however, we can use the chain rule of derivation ∂M ext ∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) a,l,β = ∂M ext ∂S (cid:12)(cid:12)(cid:12)(cid:12) a,l,β,ǫ ∂S∂ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a,l,β,ext + ∂M ext ∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) a,l,β,S , (29)where ∂S∂ǫ (cid:12)(cid:12) a,l,β,ext can be obtained by extremality condition. And ∂M ext ∂S (cid:12)(cid:12) a,l,β,ǫ and ∂M ext ∂ǫ (cid:12)(cid:12) a,l,β,S are partial derivatives of Eq. (27). When Eq. (27) equals to zero, we have ∂ǫ∂S (cid:12)(cid:12)(cid:12)(cid:12) a,l,β,ext = πl S ωq +72 h aβ π ωq +32 (cid:0) ω q + 8 ω q + 5 (cid:1) + 9 aSπ ωq +12 ω q ( ω q + 1) − πβ S ωq +12 + 2 S ωq +32 i . (30)10herefore, we can get ∂M ext ∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) a,l,β = ( πβ + S ) / π / l . (31)The next process is similar to the black hole which we derived in Sec. II. We continue tocalculate the extremality relation between the mass and thermodynamic quantities such asentropy etc. From Eq. (28), we can obtain the small correction ǫ as ǫ = al π ωq +32 S − ωq +32 + 2 π / M l (cid:0) πβ + S (cid:1) − / − πl S − − . (32)Taking the derivative of it with respect to S , one can get (cid:18) ∂ǫ∂S (cid:19) M,a,l,β = −
32 ( ω q + 1) al π ωq +32 S − ωq +52 − π / M l (cid:0) πβ + S (cid:1) − / + πl S − . (33)And, taking the derivative of it with respect to a , one can obtain (cid:18) ∂ǫ∂a (cid:19) M,S,β,l = l π ωq +32 S − ω +32 . (34)Then, we using the relations η = − π ωq ( πβ + S ) / S − ωq +32 and the temperature T toobtain the following relations ∂M ext ∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) a,l,β = ( πβ + S ) / π / l = − T (cid:18) ∂S∂ǫ (cid:19) M,a,l,β = − η (cid:18) ∂a∂ǫ (cid:19) M,S,l,β . (35)Consider the effective potential B which conjugates to β with small correction parameter ǫ , which can be obtained from the first law of thermodynamics as B = 3 p πβ + S π / l (cid:16) πl + Sǫ + S − al π ωq + S − ω q − (cid:17) . (36)Combining the Eq. (32) and Eq. (36), we can find that −B (cid:18) ∂β ∂ǫ (cid:19) M,S,a,l = ( πβ + S ) / π / l = ∂M ext ∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) a,l,β . (37)Thus, we find a new form of the extremality relation (1). From above, we confirm theGoon-Penco relation for the regular Bardeen AdS black hole surrounded by quintessencefield. 11 V. CONCLUSION
In this paper, we investigated the thermodynamic relation when a constant correction tothe action is included in the charged black holes surrounded by quintessence. We confirmedthe extremality relations between the mass and entropy, charge, pressure which derived by[20–22] first. We proposed a new effective parameter η and potential B for the charged blackholes surrounded by quintessence, and confirmed it for both the two black holes. In fourdimensional RN-AdS black hole, we obtain extremality relations between black hole massand the normalization factor a related to the density of quintessence (cid:18) ∂M ext ∂ǫ (cid:19) a,l,Q = lim M → M ext − η (cid:18) ∂a∂ǫ (cid:19) M,S,l,Q . (38)And, in four dimensional regular Bardeen black hole, we obtain the extremality relationbetween black hole mass and the magnetic monopole charge β (cid:18) ∂M ext ∂ǫ (cid:19) a,l,β = lim M → M ext −B (cid:18) ∂β ∂ǫ (cid:19) M,S,a,l . (39)In Ref. [21], the researches assumed a general formula of the extremality relation existed inblack holes. Our work gives a partial verification to this formula.By using this new method of proving the WGC, we make small constant correction ǫ to thecosmological constant Λ term in the action of the black hole. This causes the perturbationon the mass and entropy of the black hole. When the correction constant ǫ is less than zero,the extremality mass of the black hole can be reduced, and the entropy of the black holecan increase, which has been shown in FIG. (3). In this case, the mass-charge ratio M/Q of the black hole will gradually decreases and approaches to one, and ∆
S > l of the AdS spacetime, the easier it is for the black hole to be corrected to the M < Q stateby a small constant correction parameter. 12 cknowledgments
We are grateful to Deyou Chen, Yuzhou Tao and Peng Wang for useful discussions. Thiswork is supported by NSFC (Grant No. 11947408). [1] A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett.B (1996), 99-104.[2] C. Vafa, The String landscape and the swampland, [arXiv:hep-th/0509212 [hep-th]].[3] H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland, Nucl.Phys. B (2007), 21-33.[4] N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The String landscape, black holes andgravity as the weakest force, JHEP (2007), 060.[5] W. M. Chen, Y. T. Huang, T. Noumi and C. Wen, Unitarity bounds on charged/neutral statemass ratios, Phys. Rev. D (2019) no.2, 025016.[6] L. Aalsma, A. Cole and G. Shiu, Weak Gravity Conjecture, Black Hole Entropy, and ModularInvariance, JHEP (2019) 022.[7] S. J. Lee, W. Lerche and T. Weigand, A Stringy Test of the Scalar Weak Gravity Conjecture,Nucl. Phys. B (2019) 321.[8] S. J. Lee, W. Lerche and T. Weigand, Modular Fluxes, Elliptic Genera, and Weak GravityConjectures in Four Dimensions, JHEP (2019) 104.[9] Y. Kats, L. Motl and M. Padi, Higher-order corrections to mass-charge relation of extremalblack holes, JHEP (2007) 068.[10] G. J. Loges, T. Noumi and G. Shiu, Thermodynamics of 4D Dilatonic Black Holes and theWeak Gravity Conjecture, arXiv:1909.01352 [hep-th].[11] M. Montero, T. Van Riet and G. Venken, Festina Lente: EFT Constraints from Charged BlackHole Evaporation in de Sitter, JHEP (2020) 039.[12] P. A. Cano, S. Chimento, R. Linares, T. Ortn and P. F. Ramrez, α ′ corrections of Reissner-Nordstrm black holes, JHEP (2020) 031.[13] P. A. Cano, T. Ortn and P. F. Ramirez, On the extremality bound of stringy black holes,JHEP (2020) 175.
14] H. S. Reall and J. E. Santos, Higher derivative corrections to Kerr black hole thermodynamics,JHEP (2019) 021.[15] B. Heidenreich, M. Reece and T. Rudelius, Repulsive Forces and the Weak Gravity Conjecture,JHEP (2019) 055.[16] S. Brahma and M. W. Hossain, Relating the scalar weak gravity conjecture and the swamplanddistance conjecture for an accelerating universe, Phys. Rev. D (2019) no.8, 086017.[17] E. Gonzalo and L. E. Ibez, A Strong Scalar Weak Gravity Conjecture and Some Implications,JHEP (2019) 118.[18] Y. Hamada, T. Noumi and G. Shiu, Weak Gravity Conjecture from Unitarity and Causality,Phys. Rev. Lett. (2019) no.5, 051601.[19] C. Cheung, J. Liu and G. N. Remmen, Proof of the Weak Gravity Conjecture from Black HoleEntropy, JHEP (2018) 004.[20] G. Goon and R. Penco, A Universal Relation Between Corrections to Entropy and Extremality,Phys. Rev. Lett. (2020) no.10, 101103.[21] S. W. Wei, K. Yang and Y. X. Liu, Universal thermodynamic relations with constant correc-tions for rotating AdS black holes, arXiv:2003.06785 [gr-qc].[22] D. Chen, J. Tao and P. Wang, Thermodynamic extremality relations in the massive gravity,arXiv:2004.10459 [gr-qc].[23] S. M. Carroll, W. H. Press and E. L. Turner, The Cosmological constant, Ann. Rev. Astron.Astrophys. (1992) 499.[24] B. Ratra and P. J. E. Peebles, Cosmological Consequences of a Rolling Homogeneous ScalarField, Phys. Rev. D , 3406 (1988).[25] S. M. Carroll, Quintessence and the rest of the world, Phys. Rev. Lett. (1998) 3067.[26] I. Zlatev, L. M. Wang and P. J. Steinhardt, Quintessence, cosmic coincidence, and the cos-mological constant, Phys. Rev. Lett. (1999) 896[27] P. J. E. Peebles and A. Vilenkin, Quintessential inflation,Phys. Rev. D (1999) 063505.[28] E. Ayon-Beato and A. Garcia, Regular black hole in general relativity coupled to nonlinearelectrodynamics, Phys. Rev. Lett. (1998), 5056-5059.[29] E. Ayon-Beato and A. Garcia, New regular black hole solution from nonlinear electrodynamics,Phys. Lett. B (1999) 25.[30] J. Bardeen, in Proceedings of GR5, Tbilisi, U.S.S.R. (1968).
31] V.V. Kiselev, Quintessence and black holes, Class.Quant.Grav. 20 (2003) 11871198[32] H. Ghaffarnejad, M. Farsam and E. Yaraie, Effects of quintessence dark energy on the actiongrowth and butterfly velocity, Adv. High Energy Phys. (2020), 9529356.[33] O. Minazzoli and T. Harko, New derivation of the Lagrangian of a perfect fluid with abarotropic equation of state, Phys. Rev. D (2012) 087502.[34] M. Azreg-Anou and M. E. Rodrigues, Thermodynamical, geometrical and Poincar methodsfor charged black holes in presence of quintessence, JHEP (2013) 146.[35] M. Azreg-Anou, Charged de Sitter-like black holes: quintessence-dependent enthalpy and newextreme solutions, Eur. Phys. J. C (2015) no.1, 34.(2015) no.1, 34.