Physics Letters B 807 (2020) 135529

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Physics Letters B

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The correspondence between thermodynamic curvature and isoperimetric theorem from ultraspinning black hole ∗ Zhen-Ming Xu a,b,c,d, a Institute of Modern Physics, Northwest University, Xi’an 710127, China b School of Physics, Northwest University, Xi’an 710127, China c Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an 710127, China d Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China a r t i c l e i n f o a b s t r a c t

Article history: In this paper, a preliminary correspondence between the thermodynamic curvature and the isoperimetric Received 22 February 2020 theorem is established from a 4-dimensional ultraspinning black hole. We find that the thermodynamic Received in revised form 25 May 2020 curvature of ultraspinning black hole is negative which means the ultraspinning black hole is likely to Accepted 29 May 2020 present an attractive between its molecules phenomenologically if we accept the analogical observation Available online 1 June 2020 that the thermodynamic curvature reflects the interaction between molecules in a black hole system. Editor: N. Lambert Meanwhile we obtain a general conclusion that the thermodynamic curvature of the extreme black hole of the super-entropic black hole has a (positive or negative) remnant approximately proportional to the reciprocal of of the black hole. © 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

  1 −   1 1. Introduction (d − 1)V d 1 ωd−2 d−2 R = ≥ 1, (1) ωd−2 4S A very interesting and challenging problem in black hole ther- = (n+1)/2 + modynamics is the volume of black hole. Although there are var- where ωn 2π / [(n 1)/2] is the standard volume of the ious versions of black hole volume discussion [1–10], there is no round unit sphere, and the equality is attained for the (charged) Schwarzschild-AdS black hole. Physically, the above isoperimetric unified description yet. In the problem of understanding the vol- ratio indicates that the entropy of black holes is maximized for ume of black holes, especially in AdS black holes, the application the (charged) Schwarzschild-AdS black hole at a given thermody- of isoperimetric theorem deepens our mathematical understanding namic volume. Up to now, the ratio has been verified for a variety of black hole insofar as it places a constraint on of black holes with the horizon of spherical topology and black the thermodynamic volume and entropy of an AdS (or dS) black rings with the horizon of toroidal topology [16]. The black hole, hole [11,12]. Isoperimetric theorem is an ancient mathematical which violates the reverse isoperimetric inequality, i.e., R < 1, is problem, which simply means that in a simple closed curve of a called a super-entropic black hole [17]. To date, there are only given length on a plane, the area around the circumference is the two known super-entropic black holes. One is (2 + 1)-dimensional largest. With the proposal of black hole area entropy (in the natu- charged Banados-Teitelboim-Zanelli (BTZ) black hole which is the ral unit system, S = A/4, where S is the entropy of the black hole simplest [18–22]. Another important super-entropic black hole is a and A is the area of the ) [13,14] and the introduc- kind of ultraspinning black hole [23–25]. tion of extended phase space [15], Cvetic,ˇ Gibbons, Kubiznák,ˇ and Now turn to another important concept, thermodynamic curva- Pope creatively applied the theorem to AdS black hole system and ture. It is now the most important physical quantity in studying conjectured that in general for any d-dimensional asymptotic AdS the micro-mechanism of black holes from the axioms of thermo- black hole, its thermodynamic volume V and entropy S satisfy the dynamics phenomenologically. Its theoretical basis is mainly based on the thermodynamics geometry, which is mainly to use the Hes- reverse isoperimetric inequality [11], sian matrix structure to represent the thermodynamic fluctuation theory [26]. Hitherto without an underlying theory of quantum gravity, the exploration on the microscopic structure of black holes * Correspondence to: Institute of Modern Physics, Northwest University, Xi’an 710127, China. is bound to some speculative assumptions. Owing to the well- E-mail address: [email protected]. established black hole thermodynamics, as an analogy analysis and https://doi.org/10.1016/j.physletb.2020.135529 0370-2693/© 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 Z.-M. Xu / Physics Letters B 807 (2020) 135529 a primary description, it can be said that the thermodynamic ge- where ometry should yet be regarded as probe kits to phenomenologi-   2 2 cally or qualitatively extract certain information about interactions r  = r2 + l2 cos2 θ,  = l + − 2mr, (5) of black holes. In this scene, one can regard that an empirical l observation in ordinary thermodynamics that negative (positive) and the horizon r defined by (r ) = 0. In addition, due to the thermodynamic curvature is associated with attractive (repulsive) h h new azimuthal coordinate ψ is noncompact, Refs. [23,24] choose microscopic interactions, is also applicable to black hole systems ∼ + [27]. Based on this empirical analogy analysis, the primary micro- to compactify by requiring that ψ ψ μ with a dimensionless scopic information of the BTZ black hole, (charged) Schwarzschild parameter μ. For this black hole, in order to make the horizon (-AdS) black hole, Gauss-Bonnet (-AdS) black hole, higher dimen- exist, the mass of the black hole is required to have a minimum, sional black holes and other black holes are explored [28–59]. that is, an extreme black hole, In this paper, we shall calculate the thermodynamic curvature 8 l of 4-dimensional ultraspinning black hole and explore the corre- m ≥ m0 = √ l, r0 = √ . (6) spondence between thermodynamic curvature and isoperimetric 3 3 3 theorem of super-entropic black hole. First, the thermodynamic Correspondingly, the first law of ultraspinning black hole thermo- curvature of ultraspinning black hole has never been analyzed, so dynamics is [23,24] we want to fill this gap. Second, the isoperimetric ratio (1) has been simply an observation made in the literature, but no phys- dM = TdS + VdP + dJ, (7) ical reason has been given for the bound. Hence we want to try to understand this isoperimetric ratio from the point of view of where the basic thermodynamic properties, i.e., enthalpy M, tem- thermodynamics geometry. Third, in our previous work [22]about perature T , entropy S, thermodynamic pressure P , thermodynamic the thermodynamic curvature of (2 + 1)-dimensional charged BTZ volume V , angular momentum J and angular velocity , of ultra- black hole, we give a preliminary conjecture that when the isoperi- spinning black hole associated with horizon radius rh are [23,24] metric ratio is saturated (R = 1), the thermodynamic curvature of an μm l extreme black hole tends to be infinity while for super-entropic black M = , J = Ml, = , holes (R < 1), the thermodynamic curvature of the extreme black hole 2π r2 + l2 h goes to a finite value. In present paper, through the analysis of the μ(r2 + l2) 1 3r2 thermodynamic curvature of the only second super-entropic black S = h , T = h − 1 , 2 hole, we want to verify and perfect the previous conjecture and 2 4πrh l establish a new correspondence, that is, the correspondence of 2 2 3 2μrh(r + l ) thermodynamics curvature and isoperimetric theorem of AdS black P = , V = h . (8) 2 holes. 8πl 3 Meanwhile authors in Refs. [23,24]find the above ultraspinning 2. Thermodynamic properties of ultraspinning black hole black hole is super-entropic, i.e., the relation between the entropy S and thermodynamic volume V in Eq. (8)violates the reverse We start to demonstrate this procedure with the 4-dimensional isoperimetric inequality (1). Kerr-AdS black hole and write its metric in the standard Boyer- We notice that the above first law (7)is mathematically prob- = Lindquist form [8,23] lematic, like as the Maxwell relation (∂ T /∂ P)S, J (∂ V /∂ S)P, J . Be-   cause angular momentum J = Ml (it’s also known in the Ref. [23] 2  a sin2 θ   as chirality condition), it renders the enthalpy M of a black hole 2 =− a − + a 2 + a 2 ds dt dφ dr dθ just a function of entropy S and pressure P . Hence we need to a  a  find a more suitable expression of the first law and the derived 2 2 + 2 2 expressions of temperature and volume. By inserting the chirality + sin θ − r a adt dφ (2) condition into the Eq. (7), we can get the right form of the first a  law of ultraspinning black hole where = ˜ + ˜ 2 2 dM TdS VdP, (9) 2 2 2 a a 2 a = r + a cos θ,  = 1 − , = 1 − cos θ, l2 l2 where   2 2 2 r 2 + 2  = (r + a ) 1 + − 2mr, (3) ˜ rh l 3 1 a 2 T = − , (10) l 4πr l2 2 h rh here m is related to black hole mass, l is the AdS radius which and is connected with the negative cosmological constant via = −1/l2 and a is rotation parameter. 2 2 + 2 2 ˜ μl (rh l ) 2 1 To avoid a singular metric in limit a → l, Refs. [23,24]define V = − . (11) 4r l2 r2 a new azimuthal coordinate ψ = φ/ and identify it with period h h 2 / to prevent a conical singularity. After these coordinate trans- π Of course, naturally, we can verify the Maxwell relation (∂ T˜ /∂ P) = formations and then taking the limit a → l, one can get the metric S (∂ V˜ /∂ S) . Meanwhile we can write the corresponding Smarr rela- of the ultraspinning black hole [23,24] P tion

 2   ds2 =− dt − l sin2 θdφ + dr2 + dθ 2 M = 2TS˜ − 2VP˜ , (12)   sin2 θ

4 2 which can also be derived from a scaling (dimensional) argu- sin θ 2 2 + ldt − (r + l )dφ (4) ment [60]. Next let’s check whether the ultraspinning black hole is  Z.-M. Xu / Physics Letters B 807 (2020) 135529 3

μ still super-entropic in our new thermodynamic framework. Keep- S0 at x0 . Hence at the vicinity of the local maximum, we can ing in mind that the space is compactified due to ψ ∼ ψ + μ, we expand the entropy S of the system to a series form about the have ω2 = 2μ [23]. For convenience, we set a dimensionless pa- equilibrium state = 2 2 rameter x l /rh . Consequently, the isoperimetric ratio reads 2 = + ∂ S B μ + ∂ S E μ + 1 ∂ S B μ ν   S S0 μ x μ x μ x xB x 1/3 ∂x B ∂x E 2 ∂x ∂xν B R = (1 + x)1/6 1 − . (13) B E B B 2 2 + 1 ∂ S E μ ν +··· μ x xE , (15) Now let’s analyze the situation of the extreme black hole in our 2 ∂x ∂xν E new thermodynamic framework. E E where xμ stand for some independent thermodynamic variables. • For the black hole , the temperature Due the conservation of the entropy of the equilibrium isolated and thermodynamic volume of the system should be non- system and the condition S B S E ∼ S, the above formula approx- negative (we mainly focus on these two physical quantities imately becomes and the others are positive). For the case of negative tem- 2 perature and negative thermodynamic volume, this is beyond 1 ∂ S B μ ν S = S0 − S ≈− x x , (16) the scope of this paper, so we have to exclude this situation. 2 μ ν B B ∂xB ∂xB Especially for negative thermodynamic volume, it is not well defined in thermodynamics. where the so-called Ruppeiner metric is (here we omit subscript • For the ultraspinning black hole, the original extreme black B) hole corresponds to Eq. (6). There is a lower bound for the ∂2 S mass of the black hole. In short, the original black hole sat- 2 =− μ ν = S μ ν l μ ν x x gμνx x . (17) isfies the condition 0 ≤ x ≤ 3. Under this condition, the tem- ∂x ∂x perature and thermodynamic volume are not negative, and the Now focus on the system of the ultraspinning black hole and its extreme black hole is at x = 3. But unfortunately, as mentioned surrounding infinite environment. Black hole itself can be regarded earlier, the first law of thermodynamics Eq. (7)for the black as the small subsystem mentioned above. In the light of the right hole is mathematically problematic. form of the first law of thermodynamics Eq. (9), we can get the • In our new thermodynamic framework, see Eqs. (9), (10), general form of the Ruppeiner metric for the ultraspinning black and (11), we guarantee the right form of the first law of ther- holes modynamics by introducing new expressions of black hole 2 1 ˜ 1 ˜ temperature and thermodynamic volume. In order to ensure l = T S + V P . (18) T˜ T˜ the non-negativity of these two thermodynamics quantities, we must require 0 ≤ x ≤ 2. Under this new condition, the In principle, according to the first law Eq. (9), the phase space { ˜ ˜ } first law of thermodynamics of the ultraspinning black hole is of the ultraspinning black hole is T , P, S, V . For the thermo- mathematically reasonable, but the cost is to change the orig- dynamics geometry, it is carried out in the space of generalized { } { ˜ } { ˜ ˜ } { ˜ } inal extreme configuration of the black hole. Specifically, the coordinates, like as S, P , S, V , T , V and T , P . There is Leg- new extreme black hole is at x = 2or corresponds to the new endre transformation between the thermodynamic potential func- lower bound tions corresponding to these coordinate spaces. Hence the thermo- dynamic curvatures obtained in these coordinate spaces are same. 9 l m ≥ m˜ 0 = √ l, r˜0 = √ . (14) For avoiding the technique complexity, we take the coordinate 4 2 2 space {S, P} as an example for detailed calculation. The line ele- This is different from the original extreme black hole structure ment of thermodynamic geometry becomes [22,44] Eq. (6). 1 ∂ T˜ 2 ∂ T˜ 1 ∂ V˜ 2 = 2 + + 2 ≤ R ≤ l S SP P At 0 < x 2, we can easily prove that 1, which implies T˜ ∂ S T˜ ∂ P T˜ ∂ P that the ultraspinning black hole is still super-entropic in our new P S S 2 thermodynamic framework. When the value of x exceeds 2, the 1 ∂ M μ ν μ ν = X X = gμνX X ,(μ, ν = 1, 2) thermodynamic volume of black hole becomes negative, and the T˜ ∂ Xμ∂ Xν isoperimetric ratio is no longer applicable, so it is impossible to (19) determine whether the ultraspinning black hole is super-entropic 1 2 = or not. where (X , X ) (S, P) and in the right part of the second equal sign, we have used the first law of thermodynamics Eq. (9). The S 3. Thermodynamic curvature of ultraspinning black hole above thermodynamic metric gμν is equivalent to the metric gμν in Eq. (17), but they have different representation forms. The met- S Now we start to calculate the thermodynamic curvature of the ric gμν is in the entropy representation, while the metric gμν ultraspinning black hole, so as to verify the corresponding relation- is in the enthalpy representation. Next according to the specific ship proposed by Ref. [22] between the thermodynamic curvature form of the metric gμν , we start to calculate the thermodynamic and the isoperimetric theorem, and extract the possible micro- curvature, which is the “thermodynamic analog” of the geomet- scopic information of the ultraspinning black hole completely from ric curvature in general relativity. By using the Christoffel symbols α = μα + − a thermodynamic point of view. βγ g ∂γ gμβ ∂β gμγ ∂μ gβγ /2 and the Riemannian cur- α = α − α + μ α − μ α Considering an isolated thermodynamic system with entropy S vature tensors R βγ δ ∂δβγ ∂γ βδ βγ μδ βδμγ , we = μν ξ in equilibrium, the author Ruppeiner in Refs. [26–28]divided it can obtain the thermodynamic curvature R SP g R μξν . into a small subsystem S B and a large subsystem S E with require- With the help of Eqs. (10), (11) and the expressions of entropy ment of S B S E ∼ S. We have known that in equilibrium state, S and thermodynamic pressure P in Eq. (8), the thermodynamic the isolated thermodynamic system has a local maximum entropy curvature can be directly read as 4 Z.-M. Xu / Physics Letters B 807 (2020) 135529

x(x + 1)[x2(x − 3)(x2 + 12) + 27x − 9] R =− . (20) super-entropic in our new thermodynamic framework, which is SP 2S(x − 3)[x2(x − 3) + 3x − 3]2 consistent with the result obtained in [23,24]. Meanwhile the ob- tained thermodynamic curvature is negative which means the ul- In view of the thermodynamic curvature obtained above, some traspinning black hole is likely to present a attractive between explanations are made. its molecules phenomenologically or qualitatively if we accept the analogical observation that the thermodynamic curvature reflects • For the extreme black hole, i.e., x = 2, we can observe the interaction between molecules in a black hole system. Through clearly that thermodynamic curvature is finite negative value the analysis of the extreme behavior of the thermodynamic cur- R | =−57/S. SP extreme vature, we can get a general conclusion that the thermodynamic • Due to 0 < x ≤ 2, with a little calculation, we can obtain curvature of the extreme black hole of the super-entropic black R < 0. We can speculate that the ultraspinning black hole SP hole has a (positive or negative) remnant approximately propor- is likely to present a attractive between its molecules phe- tional to 1/S. This is a very interesting result. nomenologically or qualitatively. In our previous work [44], we analyze the thermodynamic • Look at the original extreme black hole, i.e., x = 3, you might curvature of Schwarzschild black hole and obtain R = intuitively get that the thermodynamic curvature tends to be Schwarzschild ±1/S . This one is very similar to what we’ve got in infinite at this time according to Eq. (20). In fact, in this case, Schwarzschild present paper. Maybe it’s a coincidence? Maybe it suggests that the basic thermodynamic metric (19)is no longer valid, be- the excess entropy in the super-entropic black hole comes from cause the first law (7)is pathological. the Schwarzschild black hole? This unexpected problem needs fur- ther analysis and discussion. At present, the known super-entropic black holes are only Furthermore, we need to in the future confirm the conjecture (2+1)-dimensional charged BTZ black hole and the ultraspinning about the sub-entropic black hole, such as the Kerr-AdS black hole black hole. According to our current analysis and the calculation [11,61], STU black holes [61,62], Taub-NUT/Bolt black hole [63], of charged BTZ black hole in the previous paper [22], we have for generalized exotic BTZ black hole [20], noncommutative black hole ultraspinning black hole R | =−57 S and for charged BTZ SP extreme / [64] and accelerating black holes [65]. The verification of this con- | = black hole R SP extreme 1/(3S). Hence, a universal relationship is jecture will help us to improve the correspondence between the 1 thermodynamic curvature and the isoperimetric theorem, which is R SP| ∝ . (21) a very meaningful research content. extreme S We know that the reverse isoperimetric inequality physically in- Declaration of competing interest dicates that at a given thermodynamic volume, the (charged) Schwarzschild-AdS black holes are maximally entropic. The super- The authors declare that they have no known competing finan- entropic black hole means that the entropy of black hole exceeds cial interests or personal relationships that could have appeared to the maximal bound. For the (charged) Schwarzschild-AdS black influence the work reported in this paper. hole, the thermodynamic curvature of the corresponding extreme black hole tends to be infinity, which is verified by various simple Acknowledgements static black hole solutions of the pure Einstein gravity or higher- derivative generalizations thereof. Therefore, we can have the fol- The financial support from National Natural Science Founda- lowing corresponding relations: tion of China (Grant Nos. 11947208 and 11947301) is gratefully acknowledged. This research is also supported by The Double First- • R = For the black holes with 1, the thermodynamic curvature class University Construction Project of Northwest University. 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