PHYSICAL REVIEW D 101, 106007 (2020)

Thermodynamic geometry and interacting microstructures of BTZ black holes

† Aritra Ghosh * and Chandrasekhar Bhamidipati School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Jatni, Khurda, Odisha 752050, India

(Received 1 February 2020; accepted 23 April 2020; published 6 May 2020)

In this work, we present a study to probe the nature of interactions between black hole microstructures for the case of the Banados, Teitelboim, and Zanelli (BTZ) black holes. Even though BTZ black holes without any angular momentum or electric charge thermodynamically behave as an ideal gas, i.e., with noninteracting microstructures; in the presence of electric charge or angular momentum, BTZ black holes are associated with repulsive interactions among the microstructures. We extend the study j to the case of exotic BTZ black holes with mass M ¼ αm þ γ l and angular momentum J ¼ αj þ γlm,for 1 1 arbitrary values of ðα; γÞ ranging from purely exotic ðα ¼ 0; γ ¼ 1Þ to slightly exotic ðα > 2 ; γ < 2Þ to 1 1 highly exotic ðα < 2 ; γ > 2Þ. We find that, unlike the normal BTZ black holes [the case (α ¼ 1, γ ¼ 0)], there exist both attraction as well as repulsion dominated regions in all the cases of exotic BTZ black holes.

DOI: 10.1103/PhysRevD.101.106007

I. INTRODUCTION particular system the curvature of the Ruppeiner metric indicates the nature of interactions between the underlying Since the work of Bekenstein [1,2] and Hawking [3,4] molecules. For a system in which the microstructures and related developments [5,6], there have been active interact attractively as in a van der Waals fluid, the efforts to understand the microscopic degrees of freedom, curvature scalar of the Ruppeiner metric carries a negative thermodynamics, and phase transitions [7] of black holes, sign, whereas in a system in which the microstructures more recently, in extended thermodynamic phase space – interact in a repulsive manner, the curvature is positive. [8 26]. Further, there has been a considerable amount For a noninteracting system such as the ideal gas, the metric of interest in the study of thermodynamic geometry of – is flat. In the attempts to probe the nature of interactions black holes and their underlying microstructures [27 62]. between the microstructures, Ruppeiner geometry has Since it is possible to define a temperature for a black hole, therefore been applied quite extensively in black hole it is then the most natural to think of an associated thermodynamics. The investigation of the nature of inter- microscopic structure. These black hole microstructures actions between the black hole microstructures was first are in general interacting just like the molecules in a done in the context of Banados, Teitelboim and Zanelli nonideal fluid. The fact that asymptotically anti-de Sitter (BTZ) black holes in three dimensions in Ref. [50]. (AdS) black holes have thermodynamic behavior similar to Furthermore, in Ref. [56], Ruppeiner geometry for that of a van der Waals fluid [11,12,24] makes the picture of Reissner-Nordström, Kerr, and Reissner-Nordström-AdS fluidlike interacting microstructures quite natural. The black holes was explored in the nonextended thermody- commonly adapted technique to probe the nature of namic space, where the divergence of the scalar curvature interactions between these microstructures is to look at is consistent with the Davies phase transition point [57]. the thermodynamic geometry of the corresponding macro- For example, for the case of both Reissner-Nordström- scopic system. For example, the study of Ruppeiner AdS (RN-AdS) and electrically charged Gauss-Bonnet- geometry in standard thermodynamics reveals that for a AdS (GB-AdS) black holes [27,29–31,62], there is a competition between attractive and repulsive interactions *[email protected] between the microstructures of the black hole. In the case † [email protected] in which attraction exactly balances the repulsion, the black hole can be regarded as effectively noninteracting. Published by the American Physical Society under the terms of It has also been recently noted that the Schwarzschild- the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to AdS [32] and neutral GB-AdS black holes [61] are the author(s) and the published article’s title, journal citation, dominated by attractive interactions among the putative and DOI. Funded by SCOAP3. microstructures.

2470-0010=2020=101(10)=106007(13) 106007-1 Published by the American Physical Society ARITRA GHOSH and CHANDRASEKHAR BHAMIDIPATI PHYS. REV. D 101, 106007 (2020)

A. Motivation and results The paper is organized as follows. In Sec. II,wegivea In this work, with a motivation to probe the nature of very brief review of thermodynamic geometry that will be interactions among the black hole microstructures, we useful for the rest of the paper. We begin our study of study BTZ black holes in the extended phase space with Ruppeiner geometry for BTZ black holes in Sec. III, full generality including the effects due to electric charge starting with the general charged and rotating case, hence and angular momentum. In the nonextended thermody- discussing the special cases. The results are then physically namic phase space, Ruppeiner geometry has been applied interpreted from a nonideal fluid point of view. In Sec. IV, previously to study BTZ black holes in various approaches we explore the thermodynamic geometry of neutral exotic [50–55]. However, to decipher the true microstructures of BTZ black holes and show that there are both attraction and black holes in AdS and to know the regions of attraction repulsion dominated regions as compared to only repulsive and repulsive behavior, a thorough study in extended phase interactions for the ordinary BTZ black holes with charge space (where the fluidlike behavior is most clear due to the and/or angular momentum. We end the paper with remarks availability of an equation of state) is required, as noted in Sec. V. recently in Refs. [27,30]. In three dimensions, there are interesting possible connections with exact formulas avail- II. THERMODYNAMIC GEOMETRY able to study the statistical interpretations from the holo- In this section, we recall the essential aspects of the – graphically dual side [63 65]. Motivated by this, we study geometry of the thermodynamic phase space and how it 2 1 these ( þ )-dimensional black holes, having a thermo- naturally leads to Ruppeiner geometry. It is well known that dynamic behavior which is not only interesting in itself but the thermodynamic phase space is endowed with a contact also provides clues to the understanding of the higher- structure [86–91]; i.e., it assumes the structure of a contact dimensional counterparts. For example, the thermodynam- manifold. This means that the thermodynamic phase space ics of the rotating BTZ black hole is much more transparent is a pair ðM; ηÞ where M is a manifold of odd dimension as compared to that of a Kerr black hole in higher [say (2n þ 1)] that is smooth and that M is equipped with a 2 1 n dimensions. In other words, these ( þ )-dimensional one form η such that the volume form, η ∧ ðdηÞ is nonzero black holes, even though simpler to study, can be regarded everywhere on M. One can always locally find (Darboux) i i as toy models that can be used to shed light onto the coordinates ðs; q ;piÞ on M where q and pi are said to be behavior of black holes in d ≥ 4. We show that a charged a conjugate pairs such that and nonrotating BTZ black hole is dominated by repulsive i interactions. This is expected since it is now understood η ¼ ds − pidq : ð2:1Þ that charged microstructures interact in a repulsive manner [27,29–31,62]. We also note that a neutral but rotating BTZ Of interest in thermodynamics is a very special class of black hole behaves qualitatively in a similar manner as a submanifolds of M, known as the Legendre submanifolds. charged BTZ and is dominated by repulsive interactions. If Φ∶ L → M is a submanifold with Φ being an embed- This is because at the thermodynamic level the angular ding, then if Φ η ¼ 0 or equivalently ηL ¼ 0, one calls L an momentum of the black hole behaves very much like isotropic submanifold. From Eq. (2.1), it is clear that an electric charge and therefore from the phenomenological isotropic submanifold cannot include a conjugate pair of level it suggests associating microstructures to rotating local coordinates. Further, if L is an isotropic submanifold BTZ black holes that interact repulsively. For the general of maximal dimension, it is then a Legendre submanifold. It case which includes both rotation and electric charge, these is easy to verify that the maximal dimension is n and two repulsive effects can add up, and therefore there is no therefore all Legendre submanifolds are n dimensional. On attraction at all. an arbitrary Legendre submanifold L, one has We also study the case of exotic BTZ black holes i [66–68], in which the roles of mass and angular momentum ds − pidq ¼ 0: ð2:2Þ are reversed [69–72] and in the general case the solutions may emerge from a gravitational action [73–77], which is a One immediately identifies this statement as a first law of linear combination of the standard Einstein-Hilbert together thermodynamics which for black holes in the extended with the Chern-Simons action [73]. The study of holog- phase space is given by raphy with gravitational Chern-Simons terms has opened up interesting new avenues [78–85]. For the exotic BTZ dM − TdS − VdP − ΦdQ − ΩdJ ¼ 0; ð2:3Þ black hole case [66–68], we find that the Ruppeiner curvature scalar changes sign with the inner horizon radius. with the identifications s ¼ M, q1 ¼ S, q2 ¼ P, q3 ¼ Q, 4 Therefore, there are both attraction and repulsion domi- q ¼ Ω, p1 ¼ T, p2 ¼ V, p3 ¼ Φ, p4 ¼ J. Here, M is nated regions for exotic BTZ black holes, in general. the mass of the black hole, which in the extended

106007-2 THERMODYNAMIC GEOMETRY AND INTERACTING … PHYS. REV. D 101, 106007 (2020) 1 2 ∂T V thermodynamics framework is equated to the enthalpy, i.e., ds2 dS2 dSdP − dP2; : M H 1 R ¼ þ ð2 8Þ ¼ , while other symbols have their usual meanings CP T ∂P S TBS from black hole thermodynamics. A contact manifold can be associated with a Riemannian where CP is the specific heat at constant pressure and BS ¼ metric (see Refs. [90–93]) G, which is in a sense compat- −Vð∂P=∂VÞS is the adiabatic bulk modulus of the black ible with the contact structure. The metric is bilinear, hole. Notice that BS ¼ ∞ for black holes where V and S are symmetric, and nondegenerate. It is usually written down not independent and the last term drops from Eq. (2.8). in the coordinate form as Alternatively, the Ruppeiner metric is also often calculated on the ðT;VÞ plane in a representation in which the 2 i G ¼ η − dpidq : ð2:4Þ fundamental thermodynamic potential is naturally the Helmholtz potential. The line element for the general case However, the full metric is not particularly as important in is given by [32,62] thermodynamics as is its projections on various Legendre 1 ∂P 2 ∂P C submanifolds. Legendre submanifolds appearing in ds2 dV2 dTdV V dT2; : R ¼ T ∂V þ T ∂T þ T2 ð2 9Þ thermodynamics have the local form given by T V C ∂ϕ qi where V is the specific heat at constant volume, which turns s ϕ qi ; p ð Þ ; : out to be zero for black holes where the thermodynamic ¼ ð Þ i ¼ ∂qi ð2 5Þ volume and the entropy are not independent. The scalar curvature remains equivalent on both the ðS; PÞ and ðT;VÞ i where ϕ ¼ ϕðq Þ is a thermodynamic potential and is planes. This can be easily verified in the examples. We known as the generator of the Legendre submanifold remark that the singularities of the Ruppeiner curvature are (say L). It then follows that when projected on L the related to critical points in the thermodynamics (see, for metric takes the form example, the recent work [96] and references therein). We shall, however, not be concerned with the curvature singu- ∂2ϕ larities in the work and rather focus on probing the inter- G −dp dqi − dqidqj; i;j ∈ 1;2;…:;n : jL ¼ i jL ¼ ∂qi∂qj f g actions relying on the sign of the curvature. ð2:6Þ III. BTZ BLACK HOLES This exactly corresponds to the metric of Weinhold [94] Black hole solutions in (2 þ 1)-dimensional topological and Ruppeiner [95], whose lines elements are, respectively, gravity were found by BTZ [97,98]. These are the BTZ given as black holes with a negative cosmological constant which, from the black hole thermodynamics perspective, leads to a ∂2U ∂2S positive thermodynamic pressure and therefore a fluidlike ds2 − dxidxj; ds2 − dxidxj; : W ¼ ∂xi∂xj R ¼ ∂xi∂xj ð2 7Þ behavior in extended black hole thermodynamics. Even though in (2 þ 1) dimensions gravitational fields do not where fxig are independent thermodynamic variables. It is have dynamical degrees of freedom and general relativity 2 2 has no Newtonian limit, (2 þ 1)-dimensional theories are a simple exercise to show that dsR ¼ βdsW, where β ¼ 1=T is the inverse temperature, so the metrics differ only by a studied because they are often easier to work with yet share 3 1 conformal factor. In case of static black holes, however, similar properties with their ( þ )-dimensional counter- since entropy and volume are not independent thermody- parts. Despite certain similarities the BTZ black holes share namic variables, the study of metric structures should not with higher-dimensional black holes in extended black hole 2 1 be done taking the internal energy U ¼ UðS; VÞ as a thermodynamics, these ( þ )-dimensional black holes are associated with some peculiar thermodynamic behavior. For fundamental potential [32]. 2 In the extended thermodynamic phase space framework, instance, static BTZ black holes behave exactly like an ideal taking the enthalpy (equated to the mass of the black hole) gas if the specific volume of the black hole is understood as as the generator of the Legendre submanifold representing the fluid volume. This feature is unlike any other asymp- the black hole, it follows that on the ðS; PÞ plane the totically AdS black holes in higher dimensions which Ruppeiner line element after some manipulations can be approach this ideal gas behavior only when the limit of r written down in the form [32,62] large þ is taken. Another important feature is that BTZ black holes do not admit any critical behavior, unlike most higher- dimensional counterparts which show critical behavior just 1In this paper, we shall work in the fixed charge Q and angular momentum J ensemble; i.e., we set dQ ¼ dJ ¼ 0 in subsequent discussions. 2With static, we mean Q ¼ 0, J ¼ 0.

106007-3 ARITRA GHOSH and CHANDRASEKHAR BHAMIDIPATI PHYS. REV. D 101, 106007 (2020) like nonideal systems in thermodynamics. BTZ black holes R vs S: Charged–Rotating BTZ R are therefore interesting to study in their own right apart from being regarded as toy models for asymptotically AdS black 2.5 holes in higher dimensions. The thermodynamics of static BTZ black holes is 2.0 P=1,Q=1,J=1 described by their enthalpy being given as [99] 1.5

4PS2 1.0 M H S; P ; : ¼ ð Þ¼ π ð3 1Þ 0.5 S where the black hole entropy is defined from the horizon 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 radius r as þ FIG. 1. Ruppeiner curvature for a charged-rotating BTZ black A hole on the ðS; PÞ plane as a function of S for fixed P. S ;A2πr : : ¼ 4 ¼ þ ð3 2Þ one with interactions. The thermodynamics is, however, not This leads to the following equation of state: of the van der Waals type, as will be pointed out later. The pffiffiffi enthalpy is given by [67] pffiffiffiffi πT P V ¼ : ð3:3Þ 4 π2J2 1 32PS2 4PS2 H S; P − Q2 : : ð Þ¼128S2 32 log π þ π ð3 4Þ The equation of state exactly corresponds to the ideal gas limit of black holes in d ¼ 3 where the specific volume in pffiffiffiffiffiffiffiffiffi The thermodynamic volume now depends on charge and is this case is identified as v ¼ 4 V=π. All other black holes not directly related to entropy. As a result, CV is not zero for d ≥ 4 PV1=ðd−1Þ ∼ T in approach this ideal gas behavior in charged BTZ black holes. The volume is given as r the large þ limit. A straightforward calculation shows that the Ruppeiner metric for this case is flat; i.e., the scalar 4S2 Q2 V − : : curvature is zero as expected for the case of an ideal gas. ¼ π 32P ð3 5Þ We shall conclude from here that nonrotating and neutral BTZ black holes are associated with microstructures that do Specific heat CP for the charged and rotating case is not interact. given by

A. BTZ black holes with electric charge and angular π3J2S − 512PS5 4πQ2S3 C − þ : : momentum P ¼ 3 2 4 2 2 ð3 6Þ 3π J þ 512PS þ 4πQ S We now turn to BTZ black holes which are associated with both electric charge and angular momentum. The From Eq. (3.4), it is possible to calculate the Ruppeiner thermodynamics is much more interesting and the corre- metric on the ðS; PÞ plane analytically, with the Ruppeiner sponding behavior is akin to that of a nonideal fluid, i.e., curvature obtained as

−π A A A R ð 1 þ 2 þ 3Þ ¼ B 9 6 2 2 2 6 4 2 4 2 2 A1 ¼ 3π J Q ð1280PS − 3πQ Þþ13824π J PQ S ðπQ − 128PS Þ 2 8 2 2 3 A2 ¼ 24576PQ S ðπQ − 128PS Þ 2 2 4 2 2 2 2 4 2 2 2 4 A3 ¼ 16π J S ðπQ − 128PS Þ ð−262144P S þ 3072πPQ S þ π Q Þ B ¼ðπ3J2S − 512PS5 þ 4πQ2S3Þð3π4J2Q2 þ 4S2ðπQ2 − 128PS2Þð256PS2 þ πQ2ÞÞ2: ð3:7Þ

The curvature for the metric on the ðS; PÞ plane is plotted in free energy representation. The corresponding curvature is Fig. 1 as a function of S for fixed P. As seen from Fig. 6, plotted as a function of V in Fig. 2. It is clearly noted that temperature is positive only beyond S ¼ 0.5, and hence the curvatures carry a positive sign in either of the planes, only that region in Fig. 1 is shown, signifying the presence clearly indicating repulsive interactions. For the case with of repulsive interactions. We may alternatively calculate the Q ≠ 0 and J ¼ 0, the specific heat at constant volume CV is Ruppeiner metric on the ðT;VÞ plane using the Helmholtz nonzero and is expressed as

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R vs V: Charged–Rotating BTZ Q R v ¼ pffiffiffiffiffiffipffiffiffiffi : ð3:11Þ min 2π P

0.15 We now consider the neutral (Q ¼ 0) and rotating cases (J ≠ 0). We note that, since thevolume does not depend on J, 0.10 J=1,T=1,Q=0.5 in this case, the entropy and the volume are independent. Consequently, CV ¼ 0 for neutral rotating BTZ black holes. 0.05 From Eq. (3.7), we note the Ruppeiner curvature on the ðS; PÞ plane for this case is V 5 10 15 20 4π3J2 RQ 0 ¼ : ð3:12Þ FIG. 2. Ruppeiner curvature for a charged-rotating BTZ black ¼ 512PS5 − π3J2S hole on the ðT;VÞ plane as a function of V for fixed T. The Ruppeiner curvature is always positive if the temperature 4S3 − πSV T of the black hole is taken to be positive. On the ðT;VÞ C ; : V ¼ πV − 12S2 ð3 8Þ plane, the Ruppeiner curvature is even simpler, given as J2 and the Ruppeiner curvature on the ðS; PÞ plane can be R0 : 3:13 Q¼0 ¼ TV2 ð Þ obtained from Eq. (3.7) as

384πPQ2S A few simple manipulations will show that the Ruppeiner R : : curvatures on either of the planes are equivalent and are J¼0 ¼ 256PS2 πQ2 2 ð3 9Þ ð þ Þ positive definite if we demand positivity of the temperature. The interactions are therefore repulsive. Since the neutral and We note that charged black holes can be associated with nonrotating BTZ black holes are associated with micro- – electrically charged microstructures [27,29 31,62] which structures that are noninteracting, it is then natural to think of are repulsive. This explains the origin of the positive scalar the rotating cases as being associated with additional micro- curvature of the Ruppeiner metric and hence repulsive structures which interact repulsively and carry the degrees of interactions between black hole microstructures from a freedom of the total angular momentum. On the other hand, if phenomenological level. It should also be remarked that a black hole is associated with both electric charge and charged BTZ black holes can be thought of as a particular angular momentum, both classes of these repulsive micro- case of a general fluid with the equation of state, structures are present, thereby leading to an overall repulsion or equivalently a positive sign of Ruppeiner curvature, as T a P ; : obtained from Eq. (3.7). ¼ þ 2 ð3 10Þ v v The thermodynamic equation of state for the rotating BTZ black holes is given by where a>0 with the specific choice of the constant, a ¼ 2 Q =2π in Eq. (3.10) corresponding to the case of charged T 8J2 P ; : BTZ black hole. The equation of state resembles that of the ¼ v þ πv4 ð3 14Þ van der Waals fluid. The second term in the right-hand side, however, carries a positive sign, as opposed to the negative where v 4r is the specific volume. The second term in sign for the case of the van der Waals fluid. This signifies the ¼ þ the right-hand side implies repulsive interactions and presence of repulsive interactions unlike the attractive interactions that are present in the van der Waals case. consequently no phase transitions associated with the Moreover, the presence of such a repulsive term in Eq. (3.10) rotating BTZ black hole. If the black hole were a fluid, imposes a lower bound in the specific volume, which for the such a term would account for repulsions between four case of the charged BTZ black hole is given by3 molecules and appear in the virial expansion form of the equation of state for nonideal fluids. In fact, Eq. (3.14) corresponds to a particular case of the RN-AdS fluid 3 V In the present case, the thermodynamic volume and specific proposed recently in Ref. [100] with the absence of the volume v are not linearly related, as seen from Eq. (3.5), πv2 Q2 bimolecular attraction term. One notes the presence of a i.e., V ¼ 16 − 32P. To get the correct vmin, one has to choose V 0 minimum volume being given by min ¼ . There is, however, an alternative possibility of retain- V v ing a linear relation between and , at the cost of introducing pffiffiffi a new renormalization length scale [67], in which case 23=4 J v V V v ffiffiffi ffiffiffiffi : 3:15 min ¼ min. We, however, continue using the definition of min ¼ p4 p4 ð Þ as obtained in Eq. (3.5) from enthalpy. π P

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IV. GENERAL EXOTIC BTZ BLACK HOLES α þ γ ¼ 1: ð4:5Þ General exotic BTZ black holes [66–68] which originate α 1 from gravitational actions following from purely Chern- The case with ¼ corresponds to the standard rotating α 0 Simons terms in three dimensions have generated a lot of BTZ black hole [70], whereas ¼ leads to the purely interest, as being examples of systems which may be exotic BTZ black hole [66]. More general situations are – M superentropic.4 Let us start from the (2 þ 1)-dimensional noted in other models [69 72]. Mass , angular momen- J S rotating BTZ black hole with metric given as tum , and entropy for general exotic BTZ black holes are given as [68] 1 4j 2 ds2 ¼ −VðrÞdt2 þ dr2 þ r2 dθ − dt ; ð4:1Þ α r2 r2 γr r VðrÞ r2 M ð − þ þÞ − þ ; : ¼ 8l2 þ 4l2 ð4 6Þ with αr r γ r2 r2 J − þ ð − þ þÞ ; : r2 16j2 ¼ 4l þ 8l ð4 7Þ V r −8m : : ð Þ¼ þ l2 þ r2 ð4 2Þ 1 S παr πγr : : The inner and outer horizon radii are given as ¼ 2 ð þ þ −Þ ð4 8Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 l lm l2m2 − j2 ¼ ð Þ. This form of the metric 3r r V απr2 γπr2 þ − − : : actually solves the Einstein equations with a negative ¼ þ þ − 2r 2r ð4 9Þ cosmological constant in three dimensions in quite different − þ cases [75]. More generally, if one takes the form of I αI γI The temperature and angular velocity are given as gravitational action to be of the form ¼ EH þ GCS, I where EH stands for the Einstein-Hilbert (EH) action and r2 − r2 r I stands for the gravitational Chern-Simons (GCS) T þ − ; Ω − ; : GCS ¼ 2πl2r ¼ r l ð4 10Þ action [73,83], then it is possible to obtain more general þ þ situations in which the form of the black hole metric does α 1 not change, but the parameters of the black hole satisfy which are the same for either normal BTZ ( ¼ ) or exotic α 0 novel relations [66–68]. For the case of an exotic black BTZ ( ¼ ). Together, the above thermodynamic varia- dM TdS VdP ΩdJ hole, the conserved mass M and angular momentum J are bles satisfy the first law, ¼ þ þ , and TS − 2PV ΩJ 0 related to the parameters in the metric (4.1) as also respect the Smarr relation, þ ¼ . The specific choice α ¼ 8=π2 reproduces the results of normal j BTZ black holes discussed in Secs. I, II, and III. In this M αm γ ; : ¼ þ l ð4 3Þ section, we concentrate on the cases in which both α and γ take different values restricted by the constraint, which J ¼ αj þ γlm: ð4:4Þ corresponds to the general exotic BTZ black holes. The enthalpy can be obtained from mass M in Eq. (4.6) by α γ r r Here, and are constant coupling functions with limits: eliminating − and þ, in favor of thermodynamic varia- α ∈ ½0; 1 and bles, to be

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffipffiffiffiffi ! 2 P ð1 − 2αÞ P π3=2ðα − 1ÞJ þ 2 PS2 pffiffiffi pffiffiffiffi HðS;PÞ¼ 223=4ðα − 1ÞS − þ 2π3=2ðα − 1ÞαJ þ 4S2 Pα ; ð4:11Þ πðα − 1Þ2 α − 1 α − 1 where for later convenience γ was eliminated using Eq. (4.5), without loss of generality, and thus we have α ≤ 1. The temperature T and volume V, resulting from enthalpy in Eq. (4.11), are given, respectively, as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ffiffi ffiffiffi 3=4 3=2 pffiffiffi p p 2 2 pπ ffiffiffiffiffiffiffiffiffiffiffið1−2αÞJP 4 ð1−2αÞ P 3=2 2 PS 4 pffiffi þ 4PS 2S − α −π J − ð1−2αÞ P α−1 α−1 T − α−1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ pffiffipffiffiffi ð4 12Þ 2 3=2 2 PS2 πðα − 1Þ −π J − α−1

4They violate the reverse isoperimetric (RI) inequality [20], by having more entropy than what is allowed by RI inequality. For these systems, new instability conjectures are being actively pursued [101–104] and depend on the signs of CP as well as CV .

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V V V qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ q2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ffiffiffiffi pffiffiffi pffiffipffiffiffi 3p ð1−2αÞ P π3=2ðα−1ÞJþ 2 PS2 πðα − 1Þ P α−1 − α−1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffipffiffiffiffi ! pffiffiffiffi 1 − 2α P π3=2 α − 1 J 2 PS2 pffiffiffi pffiffiffiffi V 8 PS2 α − 1 α ð Þ − ð Þ þ 4 2 2α − 1 PS 1 ¼ ð Þ α − 1 α − 1 þ ð Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffipffiffiffiffi ! pffiffiffi 1 − 2α P π3=2 α − 1 J 2 PS2 pffiffiffi pffiffiffiffi V 2π3=2 α − 1 J α − 1 α ð Þ − ð Þ þ 3 4 2 2α − 1 PS : : 2 ¼ ð Þ ð Þ α − 1 α − 1 þ ð Þ ð4 13Þ

Writing back in appropriate variables, it can of course be also phase structure of BTZ black holes in these two broad checked that these are same as the ones given in Eqs. (4.9) classes are different and hence need to be treated separately, and (4.10). Now, we note that it is in general difficult to as noted in Ref. [68]. Figures 3 and 4 summarize the results write an equation of state P ¼ PðV;TÞ in the case of exotic of the computation of curvature for various values of α and BTZ black holes, due to the highly nonlinear nature of γ. The most important feature we find from these figures is Eqs. (4.12) and (4.13), as indicated in Ref. [68]. Now, using that the Ruppeiner curvature crosses over from positive to the form of the metric in the ðS; PÞ plane in Eq. (2.8), the negative at a valid point (meaning the temperature is Ruppeiner curvature can be calculated in the most general positive) of entropy and represents the presence of both case analytically. The special case of purely exotic black attraction and repulsion dominated regions in exotic BTZ hole (α ¼ 0, γ ¼ 1) is treated separately, and more details black holes. This generic feature is quite contrasting to the are given later in this section. For general exotic black case α ¼ 1, γ ¼ 0 of normal BTZ black holes (charged and/ holes, since the expressions are quite cumbersome, we or rotating cases) discussed in Sec. III A, in which the directly give the plots for Ruppeiner curvature and classify microstructures are only repulsive. the results in two broad classes, namely, α > γ (slightly We now discuss two special cases, corresponding to exotic) and α < γ (highly exotic). The thermodynamics and normal and purely exotic BTZ black holes, which are the extreme cases, coming from making the choices γ ¼ 0 and α ¼ 0, respectively. The purely exotic case is quite inter- R vs S: Slightly Exotic Rotating BTZ esting, as it was one of the first studied cases of exotic black R 0.006 holes in which mass and angular momentum are essentially interchanged, as seen from Eq. (4.3). In these extreme 0.004 J=1,P=1, =0.55, =0.45 cases, the enthalpy and related thermodynamic quantities J=1,P=1, =0.62, =0.38 cannot be obtained from Eq. (4.11), as the limits α ¼ 0 and 0.002 α 1 γ 0 J=1,P=1, =0.75, =0.25 ¼ ( ¼ ) are singular. However, the enthalpy for each of these cases can be obtained directly from (4.6) as5 0.000 S 5 10 15 20 π2αJ2 4PS2 −0.002 Hnormal S; P 4:14 γ¼0 ð Þ¼ 2S2 þ πα ð Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffi FIG. 3. Ruppeiner curvature on ðS; PÞ plane for slightly exotic 2 2π3=2γJ 2 4PS pffiffiffi − 4S rotating BTZ black holes as a function of S for fixed P and J. P Hpurely exotic S; P : 4:15 α¼0 ð Þ¼ πγ ð Þ

For the case of normal rotating BTZ black holes, thermo- R vs S: Highly Exotic Rotating BTZ dynamic quantities can be derived from Eq. (4.14) and are R 15 noted to be the same as in Sec. III A (obtained by setting Q ¼ 0). For the case of normal BTZ black holes (γ ¼ 0), 10 J=1,P=1, =0.1, =0.9 the Ruppeiner curvature is shown explicitly in Eq. (3.9) and 5 J=1,P=1, =0.25, =0.75 was found to be only positive. For the case of purely exotic

0 S J=1,P=1, =0.4, =0.6 rotating BTZ, the temperature and volume resulting from 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Eq. (4.15) are −5

5 −10 The enthalpy in Eq. (4.14) corresponding to normal BTZ black hole can be seen to be same as the one obtained from FIG. 4. Ruppeiner curvature on ðS; PÞ plane for highly exotic Eq. (3.4), for the choice α ¼ 1, Q ¼ 0 and scaling J by a factor rotating BTZ black holes as a function of S for fixed P and J. of 64.

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R vs S: Rotating BTZ R we can use the positivity of temperature from the relevant 30 curves in Fig. 5 to find the physically meaningful range of

20 entropy, as the temperature becomes negative for higher =1, =0: Normal values of S (purely exotic case) and there are also further 10 =0, =1: Purely Exotic thermodynamic instabilities [102]. We note the following 0 S from Fig. 5: for the normal BTZ black holes, the curvature 0.5 1.0 1.5 2.0 2.5 3.0 is always positive, and the plots are valid only beyond S ¼ −10 1.403 (the point before which temperature is negative, as −20 seen from case of normal BTZ in Fig. 7). On the other hand, S S; P P 1 J for the purely exotic case, the plots beyond the point ¼ FIG. 5. Ruppeiner curvature on ð Þ plane for ¼ and ¼ 1.403 are not valid, as the temperature becomes negative 1 for (a) thick red curve, normal rotating BTZ black holes (α ¼ 1, γ ¼ 0) and (b) blue dashed curve, purely exotic rotating BTZ after this (as seen from purely exotic case in Fig. 7). black holes (α ¼ 0, γ ¼ 1). Furthermore, the crossing of Ruppeiner curvature from positive to negative at S ¼ 1.04 (for the particular choice of parameters shown in the figure) represents the shifting of pffiffiffiffi pffiffiffi dominant interactions from being repulsive to attractive. It 8π3=2γJ P − 16 2PS2 is, of course, important to remember that in the case of a T ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffi ð4:16Þ 2π3=2γJ 2 πγ pffiffiffi − 2S purely exotic BTZ black hole in which α ¼ 0, γ ¼ 1 the P entropy is given in terms of the inner, rather than the outer, pffiffiffipffiffiffiffi horizon [66], providing a novel viewpoint of the question of 6π3=2γJS − 8 2 PS3 location of microstructures in black holes. V ¼ ffiffiffiffiqpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð4:17Þ p 2π3=2γJ 2 πγ P pffiffiffi − 2S The sign change of the Ruppeiner curvature indicates P that both attraction and repulsion dominated regions exist for all the cases of general exotic BTZ black holes. The Now, using the above thermodynamic quantities in the physics of such shifting of interactions at the point where form of the metric in the ðS; PÞ plane in Eq. (2.8), the 6 the curvature scalar has a zero crossing can be understood Ruppeiner curvature can be calculated analytically as qualitatively from the two fluid model [29,62], which ffiffiffiffi 3=2 p essentially considers such an interacting system as a binary 4π γJ PSðA1 þ A2Þ R ; : mixture of two fluids, respectively, with purely repulsive purely exotic ¼ B B ð4 18Þ 1 2 and attractive interactions among the molecules which share the degrees of freedom of the total entropy of the where system. The crossing point is then determined from the pffiffiffi relative number densities of the fluid molecules of the two A −50320π6γ4J4P2S8 18336 2π9=2γ3J3P5=2S10 1 ¼ þ types which compete with one another, deciding which pffiffiffi þ 15744π3γ2J2P3S12 − 18304 2π3=2γJP7=2S14 kind of interaction has an overall dominance at a particular thermodynamic point. The number density has previously 9216P4S16; : þ ð4 19Þ been used to understand microstructures and phase tran- ffiffiffi ffiffiffiffi sitions in charged black holes in AdS [27]. It should also be 12 8 8 p 21=2 7 7p 2 A2 ¼ −189π γ J þ 1827 2π γ J PS remarked that a change of sign of the Ruppeiner curvature pffiffiffi does not imply a phase transition and is consistent with the − 13734π9γ6J6PS4 þ 25938 2π15=2γ5J5P3=2S6; previously noted result that exotic BTZ black holes do not ð4:20Þ admit a critical behavior [68]. The nature of microstructure ffiffiffi ffiffiffiffi ffiffiffi interactions can some times be modeled in the mean field p 3 2 2 3=2 p 2 p 4 B1 ¼ 2π γ J − 6π γJ PS þ 4 2PS ; ð4:21Þ approximation using an interaction potential between two neighboring fluid molecules. For instance, for the van der ffiffiffi ffiffiffiffi 6 4 4 p 9=2 3 3p 2 3 2 2 4 Waals fluid, interaction potential is taken to be of the B2 ¼ð9π γ J − 81 2π γ J PS þ 408π γ J PS pffiffiffi Lennard-Jones type, which effectively provides a short- − 392 2π3=2γJP3=2S6 þ 256P2S8Þ2: ð4:22Þ range repulsive and a long-range attractive interaction. The attractive part is known to dominate in this system. But R purely exotic is plotted in Fig. 5, together with the Ruppeiner repulsive interactions could still be there [105] due to curvature for normal rotating BTZ black holes. Note that thermal fluctuation related effects or also intermolecular collisions (which is very much the case for charged black 6 holes in AdS [30]). It would be interesting to explore the Ruppeiner curvature using a different three-dimensional metric was used in Ref. [55] to study the critical points and aforementioned issues to gain a better understanding of phase transitions in exotic BTZ black holes. microstructures of black holes.

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k T a d V. REMARKS P B ; : ¼ v þ v2 þ v4 ð5 2Þ In this work, we have used Ruppeiner geometry to probe f f f the nature of interactions between microstructures in the which resembles the charged and rotating BTZ black hole, case of normal and exotic BTZ black holes. For the normal 2 2 only for the specific choice a ¼ Q and d ¼ J . Here, vf is BTZ black hole, for which (α ¼ 1, γ ¼ 0), it was recorded the volume per molecule, and if a and d are arbitrary that the interactions are always repulsive for black holes positive constants, Eq. (5.2) stands for a general BTZ fluid carrying charge and/or angular momentum, whereas there with Cv ≠ 0 in general. There is, in fact, a minimum are no interactions between the microstructures for BTZ f black holes without electric charge and/or rotation. In the volume for the fluid molecules, which can be found out by case without charge or angular momentum, the BTZ black imposing the positivity of temperature in Eq. (5.2) as holes admit the thermodynamics of the ideal gas, and as a 1 result, the underlying microstructures are noninteracting. v ffiffiffiffiffiffi a2 4Pd 1=2 a 1=2; : f;min ¼ p ½ð þ Þ þ ð5 3Þ Moreover, at the thermodynamic level, the similarity 2P between angular momentum and electric charge of a black hole is well known in black hole thermodynamics. For which differs from that of the RN-AdS fluid [100] in the a instance, plotting the expression for temperature of non- sign of . The minimum volumes noted in the special cases exotic BTZ black holes, for only charged and only rotating BTZ black holes given in Eqs. (3.15) and (3.11), respectively, match with the π2J2 8PS Q2 expression given in Eq. (5.3), in appropriate limits. There T − − ; : ¼ 64S3 þ π 16S ð5 1Þ is, of course, an important difference when compared to RN-AdS fluids [100]. In the limit P → ∞, minimum S πr =2 where ¼ þ is the entropy of the black hole horizon, volume in Eq. (5.3) goes to zero, similar to the RN-AdS shows that, qualitatively, rotating and charged BTZ fluids. But at P ¼ 0, minimum volume in Eq. (5.3) goes to black holes behave in a similar fashion. The plot is infinity. The thermodynamics is therefore, as one could shown in Fig. 6. Therefore, similar to electrically charged appropriately call it, the anti-van der Waals type. An microstructures for charged black holes as discussed in immediate consequence of this fact is that there are no Refs. [27,29–31,62], it is suggested to associate rotating phase transitions because such an anti-van der Waals fluid black holes with microstructures of another type, which can never undergo a liquefaction due to presence of only carry the rotating degrees of freedom and interact with each repulsive interactions between molecules. It would be other in a repulsive manner. For black holes with both interesting to compute the Ruppeiner curvature for the C electric charge and rotation, the interactions can add up BTZ fluid taking vf as a constant [30,31,62,100] and since they are both repulsive. This also means that compare it with the one for BTZ black hole. The Ruppeiner nonexotic BTZ black holes do not admit any critical curvatures in both cases may in general be different, but behavior, as has been noted earlier [25]. It is also interesting features, such as the presence of only repulsive interactions, to make contact with the fluid picture proposed in are not expected to change for BTZ fluids. Ref. [100], with a novel equation state in a virial expansion For the general exotic BTZ black holes, the Ruppeiner (analogous to, yet different from, van der Waals fluids), curvature was calculated exactly for various values of the which resembles the charged fluids in AdS and, in parameters α and γ, and the results were presented by particular, has nonzero heat capacity at constant volume classifying them into two broad classes, namely, α > γ [106]. This BTZ fluid picture is important, as it can lead to (slightly exotic) and α < γ (highly exotic), apart from a statistical mechanical understanding of the microstruc- the case α ¼ 0, γ ¼ 1, corresponding to purely exotic tures [107]. In fact, it is possible to have such a fluid black holes. For all three cases, it was noted that there picture, by proposing an equation of state as are in general attraction and repulsion dominated regions,

T Temp. vs Entropy: BTZ 6

4 P=1,Q=0,J=0 (Neutral Nonrotating BTZ) P=1,Q=0,J=1 (Neutral Rotating BTZ) 2 P=1,Q=1,J=0 (Charged Nonrotating BTZ) 0 S 0.5 1.0 1.5 2.0 P=1,Q=1,J=1 (Charged Rotating BTZ) −2

−4

FIG. 6. Variation of the black hole temperature T vs entropy S for BTZ black holes.

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Temp. vs Entropy: Limits of Exotic BTZ T For the slightly exotic BTZ case, although, the temperature 8 variation with respect to entropy is similar to that of normal =1, =0 (normal) 6 BTZ (as noted from Fig. 7), the behavior of Ruppeiner =0.75, =0.25 (slightly exotic) curvature is different. A change in sign of curvature 4 happens for slightly exotic BTZ black holes as well, as =0.25, =0.75 (highly exotic) 2 seen from Fig. 3, which should be compared with the =0, =1 (purely exotic) appropriate curve in Fig. 5 for the standard rotating BTZ 0 S 0.5 1.0 1.5 2.0 2.5 3.0 black hole, in which such a sign change does not happen. It −2 would be interesting to explore other black hole systems in three dimensions, with the inclusion of dilaton and other FIG. 7. Temperature T with respect to entropy in several limits of nonlinear couplings involving nonlinear electromagnetic exotic BTZ black holes. The behavior of curves for slightly exotic fields, in which further interesting behavior of micro- and highly exotic cases is similar for any other values of α and γ. structures may exist, including novel critical behavior. More importantly, a statistical mechanical understanding as the Ruppeiner curvature crosses zero, as noted from of the microscopic degrees of freedom may be explored Figs. 3, 4, and 5. We also found that the behavior of with phenomenological models for normal [50] as well as Ruppeiner curvature in Figs. 4 and 5 for the highly exotic exotic BTZ black holes [67]. and purely exotic cases is similar, in which cases the nature of temperature curves is also identical, as shown in Fig. 7.7 ACKNOWLEDGMENTS 7J in the curves used in Fig. 7 is scaled by a factor of 64, as We thank the anonymous referee for valuable comments. compared to the one in Fig. 6.

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