Information Geometry on the Space of Equilibrium States of Black Holes in Higher Derivative Theories

Eur. Phys. J. C (2019) 79:71 https://doi.org/10.1140/epjc/s10052-019-6553-6

Regular Article - Theoretical Physics

Information geometry on the space of equilibrium states of black holes in higher derivative theories

Tsvetan Vetsov1,2,a 1 Department of Physics, Soﬁa University, 5 J. Bourchier Blvd., 1164 Soﬁa, Bulgaria 2 The Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Moscow Region 141980, Russia

Received: 3 August 2018 / Accepted: 2 January 2019 / Published online: 28 January 2019 © The Author(s) 2019

Abstract We study the information-geometric properties 1 Introduction of the DeserÐSariogluÐTekin black hole, which is a higher derivative gravity solution with contributions from a non- In recent years the puzzling existence of dark matter and polynomial term of the Weyl tensor to the EinsteinÐHilbert dark energy, which cannot be explained either by Einstein’s Lagrangian. Our investigation is focused on deriving the rel- general theory of relativity (GR) or the Standard model of evant information metrics and their scalar curvatures on the elementary particles, suggests that alternative models have space of equilibrium states. The analysis is conducted within to be kept in mind. the framework of thermodynamic information geometry and From gravitational perspective one can consider modified shows highly non-trivial statistical behavior. Furthermore, theories of gravity and in particular higher derivative theo- the quasilocal formalism, developed by Brown and York, ries (HDTs), which include contributions from polynomial or was successfully implemented in order to derive the mass of non-polynomial functions of the scalar curvature. The most the DeserÐSariogluÐTekin black hole. prominent of them are the so called f (R) theories [1Ð3]. The f (R) gravity is a whole family of models with a number of Contents predictions, which differ from those of GR. Therefore, there is a great deal of interest in understanding the possible phases 1 Introduction ...... 1 and stability of such higher derivative theories and, thereof, 2 Information geometry on the space of equilibrium their admissible black hole solutions [4Ð9]. thermodynamic states ...... 2 However, a consistent description of black holes necessar- 3 The DST black hole ...... 4 ily invokes the full theory of quantum gravity. Unfortunately, 4 Thermodynamics of the DST black hole ...... 5 at present day, our understanding of such theory is incomplete 5 Hessian thermodynamic geometries on the equilib- at best. This prompts one to resort to alternative approaches, rium state space of the DST black hole solution ... 5 which promise to uncover many important aspects of quan- 5.1 Extended equilibrium state space ...... 5 tum gravity and black holes. One such example is called 5.2 Ruppeiner information metric ...... 6 information geometry [10Ð13]. The framework of information geometry is an essential 5.3 Weinhold information metric ...... 6 tool for understanding how classical and quantum informa- 6 Legendre invariant thermodynamic geometries on tion can be encoded onto the degrees of freedom of any the equilibrium state space of the DST black hole physical system. Since geometry studies mutual relations solution ...... 7 between elements, such as distance and curvature, it pro- 6.1 Quevedo information metric ...... 7 vides us with a set of powerful analytic tools to study pre- 6.2 HPEM information metric ...... 7 viously inaccessible features of the systems under consid- 6.3 MM information metric ...... 8 eration. It has emerged from studies of invariant geometric 7 Conclusion ...... 8 structures arising in statistical inference, where one defines References ...... 9 a Riemannian metric, known as Fisher information metric [10], together with dually coupled affine connections on the manifold of probability distributions. Information geometry a e-mail: [email protected]fia.bg already has important applications in many branches of mod- 123 71 Page 2 of 11 Eur. Phys. J. C (2019) 79 :71 ern physics, showing intriguing results. Some of them, rele- 2 Information geometry on the space of equilibrium vant to our study, include condensed matter systems [14Ð27], thermodynamic states black holes [28Ð49] and string theory [31,41,50,51]. Further applications can also be found in [11,13]. Due to the pioneering work of Bekenstein [52] and Hawk- When dealing with systems such as black holes, which ing [55] we know that any black hole represents a thermal seem to possess enormous amount of entropy [52Ð55], one system with well-defined temperature and entropy. Taking can consider their space of equilibrium states, equipped with into account that black holes may also possess charge Q and a suitable Riemannian metric such as the Ruppeiner infor- angular momentum J, one can formulate the analogue to the mation metric [56]. The latter is a thermodynamic limit of first law of thermodynamics for black holes as the above-mentioned Fisher information metric. Although Ruppeiner developed his geometric approach within fluctu- dM = TdS+ Q dQ + dJ. (2.1) ation theory, when utilized for black holes, it seems to cap- ture many features of their phase structure, resulting from Here Q is the electric potential and is the angular velocity the dynamics of the underlying microstates. In this case one of the event horizon. Equation (2.1) expresses the conserved implements the entropy as a thermodynamic potential to ADM mass M as a function of entropy and other extensive define a Hessian metric structure on the state space statis- parameters, describing the macrostates of the black hole. One tical manifold with respect to the extensive parameters of the can equivalently solve Eq. (2.1) with respect to the entropy S. system. In the framework of geometric thermodynamics all exten- Moreover, one can utilize the internal energy (the ADM sive parameters of the given black hole background can be mass in the case of black holes) as an alternative thermody- used in the construction of its equilibrium thermodynamic namic potential, which lies at the heart of Weinhold’s met- parameter space. The latter can be equipped with a Rieman- ric approach [14] to equilibrium thermodynamic states. The nian metric in several ways. In particular, one can introduce resulting Weinhold information metric is conformally related Hessian metrics, whose components are calculated as the to Ruppeiner metric, with the temperature T as the confor- Hessian of a given thermodynamic potential. For example, mal factor. Unfortunately, the resulting statistical geometries depending on which potential we have chosen for the descrip- coming from both approaches do not often agree with each tion of the thermodynamic states in equilibrium, we can write other. The reasons for this behavior are still unclear, although the two most popular thermodynamic metrics, namely the several attempts to resolve this issue have already been sug- Weinhold information metric [14], gested [42,44,57Ð59]. 2 = ∂ ∂ a b, In the current paper we are going to study the equi- dsW a b MdX dX (2.2) librium thermodynamic state space of the DeserÐSariogluÐ Tekin (DST) black hole [60] within the framework of ther- defined as the Hessian of the ADM mass M, or the Ruppeiner modynamic information geometry. The DST black hole solu- information metric [16], tion is a static, spherically symmetric black hole solution in 2 =−∂ ∂ a b, higher derivative theory of gravity with contributions from a dsR a b SdY dY (2.3) non-polynomial term of the Weyl tensor to EinsteinÐHilbert Lagrangian. defined as the Hessian of the entropy S.HereXa[Yb], a, b = The text is organized as follows. In Sect. 2 we shortly 1,..., n, collectively denote all of the system’s extensive discuss the basic concepts of geometrothermodynamics and variables except for M[S]. One can show that both metrics related approaches. In Sect. 3 we calculate the mass of the are conformally related to each other via the temperature: DST solution via the quasilocal formalism developed by 2 = 2 . Brown and York in [61]. In Sect. 4 we calculate the stan- dsW TdsR (2.4) dard thermodynamic quantities such as the entropy and the Hawking temperature of the DST black hole solution and The importance of using Hessian metrics on the equilibrium we show that the first law of thermodynamics is satisfied. manifold is best understood when one considers small fluctu- In Sects. 5 and 6 we study the Hessian information metrics ations of the thermodynamic potential. The latter is extremal and several Legendre invariant approaches, respectively. We at each equilibrium point, but the second moment of the fluc- show that the Hessian approaches of Ruppeiner and Weinhold tuation turns out to be directly related to the components of fail to produce viable state space metrics, while the Legen- the corresponding Hessian metric. From statistical point of dre invariant metrics successfully manage to incorporate the view one can define Hessian metrics on a statistical manifold Davies phase transition points. Finally, in Sect. 7,wemake spanned by any type or number of extensive (or intensive) a short summary of our results. parameters. In this case the first law of thermodynamics has to be properly generalized in order to include the chemical 123 Eur. Phys. J. C (2019) 79 :71 Page 3 of 11 71

∂ potentials of all relevant fluctuating parameters. This is due to a βab E = β , (2.8) the fact that the Hessian metrics are not Legendre invariant, ∂ Eb thus they do not necessarily preserve the geometric properties of the system when a different thermodynamic potential where βab = diag(β1,β2,...,βN ). In the case βab = δab is chosen. However, for Legendre invariant metrics, the first one returns to the standard Euler’s identity. If we choose law of thermodynamics follows naturally. to work with βab = δab, for complicated systems this may In order to make things Legendre invariant, one can lead to non-trivial conformal factor, which is no longer pro- start from the (2n + 1)-dimensional thermodynamic phase portional to the potential . On the other hand, if we set I space F, spanned by the thermodynamic potential ,the χab = δab, the resulting metric g can be used to investigate set of extensive variables Ea, and the set of intensive vari- systems with at least one first-order phase transition. Alter- a ables I , a = 1,...,n. Now, consider a symmetric bilin- natively, the choice χab = ηab = diag(− 1, 1,...,1) leads ear form G = G(Z A) defining a non-degenerate metric on to a metric gII, which applies to systems with second-order F with Z A = (, Ea, I a), and the Gibbs 1-form = phase transitions. a b d−δab I dE , where δab is the identity matrix. If the con- Once the information metric for a given statistical system dition ∧ (d)n = 0 is satisfied, then the triple (F, G,) is constructed, one can proceed with calculating its algebraic defines a contact Riemannian manifold. The Gibbs 1-form is invariants, i.e. the information curvatures such as the Ricci invariant with respect to Legendre transformations by con- scalar, the Kretschmann invariant, etc. All curvature related struction, while the metric G is Legendre invariant only if its quantities are relevant for extracting information about the functional dependence on Z A does not change under a Leg- phase structure of the system. As suggested by Ruppeiner in endre transformation. Legendre invariance guarantees that Ref. [56], the Ricci information curvature RI is related to the the geometric properties of G do not depend on the choice of correlation volume of the system. This association follows thermodynamic potential. from the idea that it will be less probable to fluctuate from one On the other hand, one is interested in constructing a equilibrium thermodynamic state to the other, if the distance viable Riemannian metric g on the n-dimensional subspace between the points on the statistical manifold, which corre- of equilibrium thermodynamic states E ⊂ F. The space spond to these states, increases. Furthermore, the sign of RI E is defined by the smooth mapping φ : E → F or can be linked to the nature of the inter-particle interactions Ea → ((Ea), Ea, I a), and the condition φ∗() = 0. in composite thermodynamic systems [62]. Specifically, if The last restriction leads explicitly to the generalization of RI = 0, the interactions are absent, and we end up with a the first law of thermodynamics (2.1) free theory (uncorrelated bits of information). The latter sit- uation corresponds to flat information geometry. For positive a b curvature, R > 0, the interactions are repulsive, therefore d = δab I dE , (2.5) I we have an elliptic information geometry, while for negative curvature, R < 0, the interactions are of attractive nature and the condition for thermodynamic equilibrium, I and an information geometry of hyperbolic type is realized. Finally, the scalar curvature of the parameter manifold can ∂ = δ I b. (2.6) also be used to measure the stability of the physical system ∂ a ab E under consideration. In particular, the information curvature approaches infinity in the vicinity of critical points, where The natural choice for g is the pull-back of the phase space phase transition occurs [17]. Moreover, the curvature of the metric G onto E, g = φ∗(G). Here, the pull-back also imposes information metric tends to diverge not only at the critical the Legendre invariance of G onto g. However, there are points of phase transitions, but on whole regions of points on plenty of Legendre invariant metrics on F to choose from. the statistical space, called spinodal curves. The latter can be In Ref. [59] it was found that the general metric for the equi- used to discern physical from non-physical situations. librium state space can be written in the form Furthermore, notice that in the case of Hessian metrics, in order to ensure global thermodynamic stability of a given ∂2 I, II c b a c macro configuration of the black hole, one requires that all g = β (E )χ dE dE , (2.7) a ∂ Eb ∂ Ec principal minors of the metric tensor be strictly positive definite, due to the probabilistic interpretation involved [16]. In χ b = χ δ fb where a af is a constant diagonal matrix any other cases (Quevedo, HPEM, etc) the physical interpre- and β ∈ R is the degree of generalized homogeneity, tation of the metric components is unclear and one can only β 1 β N β 1 N (λ 1 E ,...,λ N E ) = λ (E ,...,E ), βa ∈ R. impose the convexity condition on the thermodynamic poten- In this case the Euler’s identity for homogeneous functions tial, ∂a∂b ≥ 0, which is the second law of thermodynamics. can be generalized to the form Nevertheless, imposing positiveness of the black hole’s heat 123 71 Page 4 of 11 Eur. Phys. J. C (2019) 79 :71 capacity is mandatory in any case in order to ensure local is suitable for this task, thus thermodynamic stability. 2 σ+1 3 4 σ−1 λ(y) = (y − c) 4 + y (3.7) σ − 1 3 The DST black hole and One starts with the following action (in units κ = 1) σ−1 σ − 2 4σ−1 √ 2 4 1 1 4 n 1/n R (y) = y . (3.8) A = d x −g R + βn |Tr(C )| , (3.1) σ − 1 2 M We can now calculate the quasilocal mass, where C is the Weyl tensor and βn are some real constant coefficients. The spherically symmetric DeserÐSariogluÐ 2( ) Tekin solution [60,63], M( ) = 1 dR y λ1/2( ) λ1/2( ) − λ1/2( ) , y y 0 y y (3.9) 2 (1−p(σ)) c 2 dy ds2 =−k2 r p(σ) p(σ) − dt2 r 1/p(σ) derived in [65], where λ (y) is an arbitrary non-negative dr2 0 + + r 2 dθ 2 + sin2θ dφ2 , (3.2) function which determines the zero of the energy for a back- p(σ) − c r1/p(σ) ground spacetime. Because there is no cosmological horizon √ present, the large y limit of (3.9) determines the mass of the follows from (3.1) by setting n = 2 and σ = β2/ 3. Here, the integration constant k can be eliminated by a proper black hole. The explicit result for the DST solution is given rescaling of the time coordinate t. For convenience we have by defined the function p(σ) as 2 σ +1 √ 3 2(1−4)σ 1 − σ M(y) = c−y+ λ0(y) y − c + 4 y . p(σ) = . (3.3) σ − 1 1 − 4 σ (3.10) To preserve the signature of the metric, we have to exclude the interval 1/4 <σ <1. There is only one horizon of the The arbitrary function λ0(y) can be fixed as the first term in black hole, which is at the positive root of grr = 0: the large y asymptotic expansion of λ(y). The expression is given by σ− 1 p 3 4 σ−1 c r = c + 4 = . (3.4) 2 σ+1 h σ − 3 4 σ−1 1 p λ (y) = y 4 + y (3.11) 0 σ − 1 We also note that in general the metric (3.2) is not asymptotically flat, unless we consider the case σ = 0, for which Now, in the limit y →∞, one finds the quasilocal mass of the charge c > 0 can be interpreted as the ADM mass of a the DST black hole, Schwarzschild black hole (c = 2 M). Using the quasilocal 1 = / c formalism [61,65], we can support the claim that M c 2 M = lim M(y) = . (3.12) is the mass of the DST black hole for any σ<1/4 and σ>1. y→∞ 2 To show this, one has to bring the DST metric (3.2)inthe form The latter expression relates the unknown integration constant c from Eq. (3.2) to the quasilocal mass M of the DST σ+ dy2 black hole. Equation (3.12) is valid only when 2 1 ≥ 0, ds2 =−λ(y) dt2 + + R2(y) d2. (3.5) 4 σ−1 λ(y) 2 i.e. σ ≤−1/2orσ>1/4. One can impose further restrictions on the parameter σ The following change of variables by calculating the independent curvature invariants, i.e. the Ricci scalar, σ −1 4 σ − 1 4 σ −1 r = y (3.6) 6 σ(2 M (σ − 1) + 3 σ y) σ − 1 =− , R 2 σ+1 2 (3 σ−1) (3.13) (σ − 1) 4σ−1 ((4 σ − 1) y) 4 σ−1 1 Subtleties in defining the energy and when it matches the ADM mass can be found in [64]. and the Kretschmann invariant, 123 Eur. Phys. J. C (2019) 79 :71 Page 5 of 11 71

( 2 ( σ 2 + )(σ − )2 + σ(σ3 − ) + σ 2 (σ ( σ − ) + ) 2) = 12 4 M 5 1 1 12 M 1 y 3 7 2 4 y . K 2 (2 σ+1) 6 (2 σ−1) (3.14) (σ − 1) 4 σ−1 ((4 σ − 1)y) 4 σ−1

Both quantities are singular at σ = 1 and σ = 1/4. At σ → 0 of the DST black hole diverges at σ = 1/4 and σ =−1/2 one recovers the Schwarzschild case. and tends to zero at σ = 1. However the points σ = 1/4 and σ = 1 are not physical due to the divergences of the physical curvature (3.13), while σ =−1/2 corresponds to 4 Thermodynamics of the DST black hole Davies type phase transition. In the limit σ → 0 one recovers General relativity, where the heat capacity reduces to the We proceed with Wald’s proposal [66] to calculate the Schwarzschild case, C =−8 π M2 < 0, which is known to entropy, be thermodynamically unstable. Our subsequent considerations will also discard this limit. δL (0) One can also check that the first law of thermodynamics, =− π ( ) 2, S 8 = R y d (4.1) = y yh δR dM TdS, is satisfied. t=const ytyt of the DST black hole with metric in the form (3.5). The variational derivative of the Lagrangian is given by [63]: 5 Hessian thermodynamic geometries on the √ equilibrium state space of the DST black hole solution δL (gαγ gβδ − gαδ gβγ ) 3 σ R = 1 + √ δRαβγ δ 32 π 3 C2 5.1 Extended equilibrium state space √ 3 σ αβγ δ αγ βδ σ + √ [2 R − (g R If we consider thermal fluctuations of the parameter ,we 32 π C2 have to take into account its contribution to the first law βδ αγ αδ βγ βγ αδ + g R − g R − g R )]. (4.2) of thermodynamics. In Ruppeiner’s approach one takes the entropy as a thermodynamic potential, thus In Eq. (4.1) the superscript (0) indicates that the variational derivative is calculated on the solution. After some lengthy 1 dS = dM + dσ, (5.1) calculations one arrives at the explicit formula for Wald’s T entropy of the DST black hole: which is the generalized first law from Eq. (2.5). Here 2 (σ−1) plays the role of the chemical potential for σ (considered as 3 4σ−1 S = π 2 M 4 + , (4.3) a new extensive parameter). The explicit form of is given σ − 1 by which is positive for σ<1/4 and σ>1. The Hawking 2(σ−1) temperature yields 1−2 σ 6 σ M 4σ−1 = 3 π 8 1−4 σ (4 σ − 1) 1−4 σ σ − 1 1−2 σ 1+2 σ 1 dλ 8 4 σ−1 3 4 σ−1 3 T = (yh) = M 4 + , (4.4) × ln 2 M + 4 − 1 . (5.2) 4 π dy π σ − 1 σ − 1 = where yh 2 M is the location of the event horizon given by The equilibrium thermodynamic state space of the DST solu- λ( ) = the zeros of y 0. The singularities of the temperature tion (3.2) is now considered as a two-dimensional manifold σ = / σ = occur at 1 4 and 1. The temperature has one local equipped with a suitable Riemannian metric extremal point at (M = 1/4,σ=−1/2), which is a saddle point. At σ →−1/2 one has T → 1/(4π) and the DST (I ) ds2 = g dEa dEb, (5.3) temperature doesn’t depend on the mass M. Moreover, the I ab heat capacity, where Ea, a = 1, 2, are the extensive parameters such as the ∂ S σ C = T mass, the entropy or the parameter . Depending on the cho- ∂T sen thermodynamic potential, one discerns several possibil- 1−2 σ 2 σ+1 2 (σ−1) π 8 1−4 σ (σ − 1) 4 σ−1 (M (4 σ − 1)) 4 σ−1 ities for the information thermodynamic metric as discussed = , (4.5) 2 σ + 1 in the Introduction section. 123 71 Page 6 of 11 Eur. Phys. J. C (2019) 79 :71

It is also well-known that in two dimensions all the rele- (R)( ,σ) lim RI M vant information about the phase structure is encoded only σ→− 1 2 in the information metric and its scalar curvature. The latter 2 ln 29+7ln2M3+ln(32 M) − 4 ln(2 M) + ln 2ln 2+3ln2−4 + 4 is proportional to the (only one independent) component of = , 8 π M ln2[2ln 2(2 M)ln(8 M)] the Riemann curvature tensor, (5.9)

2 RI,1212 thus the Davies critical point cannot be covered by this par- RI = , (5.4) ( (I )) ticular thermodynamic geometry. The latter mismatch shows det gab that the Ruppeiner information approach is not an appropriate choice for the description of the equilibrium state space ( (I )) where det gab is the determinant of the information met- of the DST black hole solution. ric (5.3). Once the scalar information curvature is obtained, Although Ruppeiner metric fails to produce a viable ther- we can identify its singularities as phase transition points, modynamic description, one can always impose only local which should be compared to the resulting divergences of thermodynamic stability deﬁned by the positive values of the the heat capacity. If a complete match is found one can rely heat capacity C > 0. The latter condition leads to the param- on the considered information metric as suitable for describ- eter region − 1 <σ<−1/2, together with σ>1 and ing the space of equilibrium states for the given black hole arbitrary large mass M > 0. solution. 5.3 Weinhold information metric 5.2 Ruppeiner information metric The Weinhold metric is deﬁned as the Hessian of the mass We begin by calculating the Ruppeiner information metric, of the black hole with respect to the entropy and the other extensive parameters. In the DST case one has

(R) ( ) g =−∂a∂b S(M,σ), a, b = 1, 2, (5.5) W = ∂ ∂ ( ,σ), , = , , ab gab a b M S a b 1 2 (5.10) with components where the mass M of the DST black hole is given in terms of the entropy S and the parameter σ such as 2 σ−1 2 σ+1 σ− σ− (R) π 8 4 1 (σ − 1) 4 1 (2 σ + 1) σ − 3 3 = , 1 ( −σ)−2 (σ− ) +2 gMM 6 σ (5.6) M(S,σ)= π 2 1 S 2 1 . (5.11) (M (4 σ − 1)) 4 σ−1 2 (4 σ − 1) (R) (R) g σ = gσ M M The heat capacity now looks quite simple, 2 σ−1 σ− π 4 σ−1 (σ − ) 4 1 + σ + 3 8 2 1 ln 2 M σ−1 2 1 =− , ∂ M 3 2 σ+1 2 (σ−1) 10 σ−1 = S = − , 4σ −1 (σ − ) 4 σ−1 ( σ − ) 4 σ−1 C 1 S (5.12) M 1 4 1 ∂2 M 2 σ + 1 (5.7) S 2 σ−1 2 (σ−1) σ− σ− σ =− / (R) 3 π 8 4 1 M 4 1 and the Davies transition point at 1 2 is present. The gσσ = 2 (2 σ−1) 2(7 σ−1) metric components are given explicitly by (σ − 1) 4 σ−1 (4 σ − 1) 4 σ−1 ( σ 2− σ+ ) 3 −2 3 × 46 σ − 32 σ 2 − 5 + ln 24 8 7 1 ( ) π 2−2 σ (2 σ + 1) S 2 (σ−1) g W = , (5.13) SS 8 (σ − 1) σ+1 − ln 8 ln 4 + 2 (σ − 1) 3 3 + − σ −2 (σ− ) 1 1−4σ 4σ−1 (W) (W) 3 π 2 2 S 2 1 ln(π S ) 4 σ − 1 g σ = gσ =− , × 2 + 16 σ − 3ln 2 M S S 8 (σ − 1)2 (4 σ − 1) σ − 1 (5.14) σ − 4 1 3 3 + × ln M . (5.8) − σ −2 (σ− ) 2 σ − (W) 3π 2 2 S 2 1 1 gσσ = (σ − )3( σ − )3 8⎡ 1 4 1 Critical points of phase transitions can be identiﬁed by the × ⎣ + (σ − )( (σ − )2 + ( − σ) π) singularities of the Ruppeiner information curvature (5.4). 32 4 1 8 1 3 1 4 ln ln 2 The resulting expression is lengthy, but one can check that + σ 2 − σ 3 − σ + (σ − )2 2 at σ =−1/2, which is the relevant divergence for the heat 96 32 96 12 1 ln 2 capacity (4.5), the Ruppeiner curvature is ﬁnite, + 3 (1 − 4 σ)2 ln2π − 16 (σ − 1)2 (4 σ − 1) ln π 123 Eur. Phys. J. C (2019) 79 :71 Page 7 of 11 71

− 4 (σ − 1) ln(σ − 1) (2 (σ − 1)(4 σ − 4 + ln 8) ( − σ) ( −σ) + ln(π3 1 4 (σ − 1)3 1 )) 4 8 + 2 (4 σ − 1) 2 σ 4 σ − 8 + ln π6 ⎛ ⎞ 2 (σ− ) 3 2 1 π (σ− ) σ − 1 4 σ−1 − ln (σ − 1)6 1 + 8 ln ⎝ S⎠ 64 2 0 + 3 (1 − 4 σ)2 ln2 ⎛ ⎞⎤ 2 (σ−1) σ − 1 4 σ−1 –2 × ⎝ S⎠⎦ . (5.15) 2

The Weinhold approach also fails to reproduce the Davies –4 transition point due to the fact that the Weinhold curvature is finite at σ =−1/2, 0.0 0.1 0.2 0.3 0.4 0.5 0.6 (W)( ,σ) lim RI M σ →−1/2 Fig. 1 The regions of positive definite information metric together with 6.28319 (ln S − 3.14473)(ln2 S − 3.28946 ln S + 4.45514) C > 0 (the shaded regions) for the DST black hole with respect to =− , < S (ln S − 1.14473)4 the Quevedo information approach. The upper region lies within 1 σ<∞ and 0 < M < 1/3, while the lower region is defined within (5.16) −∞ <σ <−1/2and1/3 < M < ∞ taking into account that S > 0 is everywhere assumed. are shown on Fig. 1. The upper region is constrained within σ>3/2 and 0 < M < 1/3, while the lower region lies 6 Legendre invariant thermodynamic geometries on the within σ<− 1/2 and M > 1/3. One should have in mind equilibrium state space of the DST black hole solution that contrary to the Ruppeiner’s case, in Quevedo’s case we do not have clear physical interpretation of the components of 6.1 Quevedo information metric the information metric, thus one is not compelled to impose the Sylvester criterion. The latter will not necessarily give The Quevedo information metric on the equilibrium state the regions of global thermodynamic stability. On the other space of the DST solution is given by ∂ ∂ ≥ hand, one can check that the convexity condition, a b S 0, 2 = β ∂2 σ 2 − ∂2 2 = (Q) 2 + (Q) σ 2. cannot be satisfied here. dsQ S S σ Sd M SdM gMM dM gσσ d The Quevedo thermodynamic curvature on the two- (6.1) dimensional manifold (M,σ),2

2 2 ( ) gMM(g ,σ gσσ,σ + gσσ, ) + gσσ(g ,σ + g , gσσ, − 2g (g ,σ,σ + gσσ, , )) R Q = MM M MM MM M M MM MM M M , (6.4) I 2 2 2 gMM gσσ

One can ﬁnd the degree of generalized homogeneity, βS, is singular at M = 1/3, and it is also singular at the Davies directly from Euler’s theorem for homogeneous funct- transition point σ →−1/2, suggesting that Quevedo infor- ions (2.8): mation metric is an appropriate metric for the description of the equilibrium state space of the DST solution. However, ∂ S ∂ S M + σ = β S. (6.2) one can check that there are also additional spinodal curves. ∂ M ∂σ S The latter equation for β leads to the following components S 6.2 HPEM information metric of the information metric: (Q) =−( ∂ + σ∂ )∂2 , gMM M M S σ S M S In order to avoid extra singular points in the Quevedo thermo- (Q) 2 dynamic curvature, which do not coincide with phase transi- gσσ = (M ∂M S + σ∂σ S)∂σ S. (6.3)

The regions of positive deﬁnite information metric, together 2 For clarity we have omitted the superscript (Q) from the metric com- with local thermodynamic stability C > 0, in Quevedo’s case ponents. 123 71 Page 8 of 11 Eur. Phys. J. C (2019) 79 :71

article the authors deﬁne a conjugate thermodynamic poten- 4 tial via an appropriate Legendre transformation. In the MM information approach the divergent points of the speciﬁc heat turn out to correspond exactly to the singularities of the ther- 2 modynamic curvature. The conjugate potential we choose to work with is the Helmholtz free energy F, which is related to the mass M by 0 the following Legendre transformation

( ,σ)= ( ,σ)− ( ,σ). –2 F T M T TST (6.8)

The latter yields –4 σ + 2 1 2− 6 2− 3 2− 3 F(T,σ)=− 2 σ+1 π 2 σ+1 T 2 σ+1 . σ − 2 (6.9) 0 2 4 6 8 10 4 1 The components of the MM thermodynamic metric are now Fig. 2 The regions of positive definite information metric together with given by C > 0 (the shaded regions) for the DST black hole with respect to the 2 2 HPEM information metric. The upper region lies within 1 <σ <∞, ( ) 1 ∂ F ( ) ( ) 1 ∂ F g MM = , g MM = g MM = , while the lower region is defined within −∞<σ <−1/2 TT T ∂ T 2 T σ σ T T ∂ T ∂σ 2 ( ) 1 ∂ F g MM = . (6.10) σσ ∂σ2 tions of any type, in [67] the authors proposed an alternative T information metric with different conformal factor, One can show that there are no regions in the (T,σ) parameter space, where the Sylvester’s criterion holds together with ∂ > 2 S M 2 2 2 2 C 0. More importantly, the MM information curvature, ds = S (− ∂ MdS + ∂σ Mdσ ), (6.5) HPEM 2 3 S (∂σ M) ( ) 1 R MM (T,σ)= ×{g [g g I 2(−ˆ) TT TT,T σσ,σ with components 2 det g − 2 g (g − 2 g ) ∂ ∂ T σ,σ TT,σ T σ,T (HPEM) 2 S M (HPEM) 2 S M g =−S ∂ M , gσσ = S ∂σ M . − ( + )] SS S 2 3 2 3 gσσ, g ,σ 2 g σ, (∂σ M) (∂σ M) T TT T T + ( ( − ) + 2 ) (6.6) gTT gσσ,σ gTT,σ 2 gT σ,T gσσ,T 2 + 2 g σ (g ,σ,σ − 2 g σ, ,σ + gσσ, , ) One can find the regions where the Sylvester’s criterion holds T TT T T T T + [ 2 + ( − ) together with C > 0 as shown on Fig. 2. gσσ gTT,σ gTT,T gσσ,T 2 gT σ,σ 3 − ( − + )]}, The HPEM information curvature, 2 gTT gTT,σ,σ 2 gT σ,T,σ gσσ,T,T

2 2 ( ) g (g ,σ gσσ,σ + gσσ, ) + gσσ(g ,σ + g , gσσ, − 2g (g ,σ,σ + gσσ, , )) R HPEM (S,σ)= SS SS S SS SS S S SS SS S S , (6.7) I 2 2 2 gSS gσσ is singular only at the Davies point σ =−1/2, thus HPEM is singular exactly at the Davies transition point σ →−1/2 metric is also an appropriate Riemannian metric on the equi- without any extra critical points. librium state space of the DST black hole.

6.3 MM information metric 7 Conclusion

The final geometric approach, which we are going to con- Our current investigation is instigated by the intriguing exis- sider, was proposed by Mansoori and Mirza in [46]. In the tence of dark matter and dark energy in the Universe, which cannot be explained by traditional approaches. This moti- 3 For clarity we have omitted the superscript (HPEM) from the metric vates us to consider alternative models, which can include components. effects related to these dark phenomena. Highly promising 123 Eur. Phys. J. C (2019) 79 :71 Page 9 of 11 71 alternatives are the so called higher derivative theories of interpreted as global thermodynamic stability only within gravity, which include contributions from higher powers of the context of the Hessian metrics, due to their probabilistic the Ricci scalar or other geometric invariants. In particular, interpretation. For the Legendre invariant metrics, imposing our focus is on the thermodynamic properties of their admis- Sylvester criterion together with positive heat capacity does sible black hole solutions, which will allow us to constrain not necessarily guarantee global thermodynamic stability. the possible dark matter/energy contributions, at least when The latter is caused by the current lack of physical interpre- thermodynamics is concerned. tation of the components of the corresponding information In this paper we consider one known four-dimensional metrics. Therefore, due to the failure of Hessian geometries, higher derivative black hole solution, namely the DeserÐ the DST black hole is only stable locally from a thermo- SariogluÐTekin black hole. The latter being a static, spheri- dynamic standpoint. The condition for local thermodynamic cally symmetric gravitational solution of a theory with con- stability, together with the divergences of the physical DST tributions from a non-polynomial term of the Weyl tensor to metric curvature, constrain the values of the unknown param- the EinsteinÐHilbert Lagrangian. In order to study any impli- eter σ in the regions σ<− 1/2 and σ>1. The latter is also cations for the black hole thermodynamics, we take advan- confirmed by imposing the Sylvester criterion for Quevedo tage of two different geometric formulations, namely those of and HPEM thermodynamic metrics. the Hessian information metrics (geometric thermodynamics) and the formalism of Legendre invariant thermodynamic Acknowledgements The author would like to thank R. C. Rashkov, H. metrics (geometrothermodynamics) on the space of equilib- Dimov, S. Yazadjiev, D. Arnaudov and P. Fiziev for insightful discus- sions and for careful reading of the draft. This work was supported by rium states of the DST black hole. the Bulgarian NSF Grant no. DM18/1 and Sofia University Research In general, the formalism of thermodynamic information Fund under Grant no. 80-10-104. The support from BLTP in JINR, is geometry identifies the phase transition points of the system also gratefully acknowledged. with the singularities of the corresponding thermodynamic Data Availability Statement This manuscript has no associated data information curvature RI . Near the critical points the under- or the data will not be deposited. [Authors’ comment: There is no asso- lying inter-particle interactions become strongly correlated ciated data or any other codes related to this manuscript, and no such and the equilibrium thermodynamic considerations are no data will be deposited in the future.] longer applicable. In this case one expects that a more gen- Open Access This article is distributed under the terms of the Creative eral approach should hold. Commons Attribution 4.0 International License (http://creativecomm In the Hessian formulation we analyzed the Ruppeiner ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and the Weinhold thermodynamic metrics and showed that and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative they are inadequate for the description of the DST black Commons license, and indicate if changes were made. hole equilibrium state space. This is due to the occurring Funded by SCOAP3. mismatch between the singularities of the heat capacity and the singularities of the corresponding thermodynamic curvatures. Therefore the Hessian thermodynamic geometries are References unable to reproduce the Davies type transition points of the DST black hole heat capacity. 1. A. Starobinsky, A new type of isotropic cosmological models with- On the other hand, in the Legendre invariant case, all out singularity. 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