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

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## Abstract

We study the information-geometric properties of the Deser–Sarioglu–Tekin black hole, which is a higher derivative gravity solution with contributions from a non-polynomial term of the Weyl tensor to the Einstein–Hilbert Lagrangian. Our investigation is focused on deriving the relevant information metrics and their scalar curvatures on the space of equilibrium states. The analysis is conducted within the framework of thermodynamic information geometry and shows highly non-trivial statistical behavior. Furthermore, the quasilocal formalism, developed by Brown and York, was successfully implemented in order to derive the mass of the Deser–Sarioglu–Tekin black hole.

## 1 Introduction

In recent years the puzzling existence of dark matter and dark energy, which cannot be explained either by Einstein’s general theory of relativity (GR) or the Standard model of elementary particles, suggests that alternative models have to be kept in mind.

From gravitational perspective one can consider modified theories of gravity and in particular higher derivative theories (HDTs), which include contributions from polynomial or non-polynomial functions of the scalar curvature. The most prominent of them are the so called *f*(*R*) theories [1, 2, 3]. The *f*(*R*) gravity is a whole family of models with a number of predictions, which differ from those of GR. Therefore, there is a great deal of interest in understanding the possible phases and stability of such higher derivative theories and, thereof, their admissible black hole solutions [4, 5, 6, 7, 8, 9].

However, a consistent description of black holes necessarily invokes the full theory of quantum gravity. Unfortunately, at present day, our understanding of such theory is incomplete at best. This prompts one to resort to alternative approaches, which promise to uncover many important aspects of quantum gravity and black holes. One such example is called information geometry [10, 11, 12, 13].

The framework of information geometry is an essential tool for understanding how classical and quantum information can be encoded onto the degrees of freedom of any physical system. Since geometry studies mutual relations between elements, such as distance and curvature, it provides us with a set of powerful analytic tools to study previously inaccessible features of the systems under consideration. It has emerged from studies of invariant geometric structures arising in statistical inference, where one defines a Riemannian metric, known as Fisher information metric [10], together with dually coupled affine connections on the manifold of probability distributions. Information geometry already has important applications in many branches of modern physics, showing intriguing results. Some of them, relevant to our study, include condensed matter systems [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], black holes [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49] and string theory [31, 41, 50, 51]. Further applications can also be found in [11, 13].

When dealing with systems such as black holes, which seem to possess enormous amount of entropy [52, 53, 54, 55], one can consider their space of equilibrium states, equipped with a suitable Riemannian metric such as the Ruppeiner information metric [56]. The latter is a thermodynamic limit of the above-mentioned Fisher information metric. Although Ruppeiner developed his geometric approach within fluctuation theory, when utilized for black holes, it seems to capture many features of their phase structure, resulting from the dynamics of the underlying microstates. In this case one implements the entropy as a thermodynamic potential to define a Hessian metric structure on the state space statistical manifold with respect to the extensive parameters of the system.

Moreover, one can utilize the internal energy (the ADM mass in the case of black holes) as an alternative thermodynamic potential, which lies at the heart of Weinhold’s metric approach [14] to equilibrium thermodynamic states. The resulting Weinhold information metric is conformally related to Ruppeiner metric, with the temperature *T* as the conformal factor. Unfortunately, the resulting statistical geometries coming from both approaches do not often agree with each other. The reasons for this behavior are still unclear, although several attempts to resolve this issue have already been suggested [42, 44, 57, 58, 59].

In the current paper we are going to study the equilibrium thermodynamic state space of the Deser–Sarioglu–Tekin (DST) black hole [60] within the framework of thermodynamic information geometry. The DST black hole solution is a static, spherically symmetric black hole solution in higher derivative theory of gravity with contributions from a non-polynomial term of the Weyl tensor to Einstein–Hilbert Lagrangian.

The text is organized as follows. In Sect. 2 we shortly discuss the basic concepts of geometrothermodynamics and related approaches. In Sect. 3 we calculate the mass of the DST solution via the quasilocal formalism developed by Brown and York in [61]. In Sect. 4 we calculate the standard thermodynamic quantities such as the entropy and the Hawking temperature of the DST black hole solution and we show that the first law of thermodynamics is satisfied. In Sects. 5 and 6 we study the Hessian information metrics and several Legendre invariant approaches, respectively. We show that the Hessian approaches of Ruppeiner and Weinhold fail to produce viable state space metrics, while the Legendre invariant metrics successfully manage to incorporate the Davies phase transition points. Finally, in Sect. 7, we make a short summary of our results.

## 2 Information geometry on the space of equilibrium thermodynamic states

*Q*and angular momentum

*J*, one can formulate the analogue to the first law of thermodynamics for black holes as

*M*as a function of entropy and other extensive parameters, describing the macrostates of the black hole. One can equivalently solve Eq. (2.1) with respect to the entropy

*S*.

*M*, or the Ruppeiner information metric [16],

*S*. Here \(X_a [Y_b], a,\,b=1,\dots ,\,n,\) collectively denote all of the system’s extensive variables except for

*M*[

*S*]. One can show that both metrics are conformally related to each other via the temperature:

In order to make things Legendre invariant, one can start from the \((2n + 1)\)-dimensional thermodynamic phase space \(\mathcal {F}\), spanned by the thermodynamic potential \(\Phi \), the set of extensive variables \(E^a\), and the set of intensive variables \(I^a\), \(a = 1, \dots , n\). Now, consider a symmetric bilinear form \(\mathcal {G}=\mathcal {G}(Z^A)\) defining a non-degenerate metric on \(\mathcal {F}\) with \(Z^A=(\Phi ,\,E^a,\,I^a)\), and the Gibbs 1-form \(\Theta = d\Phi - {\delta _{ab}}\,{I^a}\,d{E^b}\), where \(\delta _{ab}\) is the identity matrix. If the condition \(\Theta \wedge (d\Theta )^n\ne 0\) is satisfied, then the triple (\(\mathcal {F},\,\mathcal {G},\,\Theta \)) defines a contact Riemannian manifold. The Gibbs 1-form is invariant with respect to Legendre transformations by construction, while the metric \(\mathcal {G}\) is Legendre invariant only if its functional dependence on \(Z^A\) does not change under a Legendre transformation. Legendre invariance guarantees that the geometric properties of \(\mathcal {G}\) do not depend on the choice of thermodynamic potential.

*g*on the

*n*-dimensional subspace of equilibrium thermodynamic states \(\mathcal {E}\subset \mathcal {F}\). The space \(\mathcal {E}\) is defined by the smooth mapping \(\phi :\mathcal {E}\rightarrow \mathcal {F}\) or \(E^a\rightarrow (\Phi (E^a),\,E^a,\,I^a)\), and the condition \(\phi ^*(\Theta )=0\). The last restriction leads explicitly to the generalization of the first law of thermodynamics (2.1)

*g*is the pull-back of the phase space metric \(\mathcal {G}\) onto \(\mathcal {E}\), \(g=\phi ^*(\mathcal {G})\). Here, the pull-back also imposes the Legendre invariance of \(\mathcal {G}\) onto

*g*. However, there are plenty of Legendre invariant metrics on \(\mathcal {F}\) to choose from. In Ref. [59] it was found that the general metric for the equilibrium state space can be written in the form

Once the information metric for a given statistical system is constructed, one can proceed with calculating its algebraic invariants, i.e. the information curvatures such as the Ricci scalar, the Kretschmann invariant, etc. All curvature related quantities are relevant for extracting information about the phase structure of the system. As suggested by Ruppeiner in Ref. [56], the Ricci information curvature \(R_I\) is related to the correlation volume of the system. This association follows from the idea that it will be less probable to fluctuate from one equilibrium thermodynamic state to the other, if the distance between the points on the statistical manifold, which correspond to these states, increases. Furthermore, the sign of \(R_I\) can be linked to the nature of the inter-particle interactions in composite thermodynamic systems [62]. Specifically, if \(R_I=0\), the interactions are absent, and we end up with a free theory (uncorrelated bits of information). The latter situation corresponds to flat information geometry. For positive curvature, \(R_I>0\), the interactions are repulsive, therefore we have an elliptic information geometry, while for negative curvature, \(R_I<0\), the interactions are of attractive nature and an information geometry of hyperbolic type is realized.

Finally, the scalar curvature of the parameter manifold can also be used to measure the stability of the physical system under consideration. In particular, the information curvature approaches infinity in the vicinity of critical points, where phase transition occurs [17]. Moreover, the curvature of the information metric tends to diverge not only at the critical points of phase transitions, but on whole regions of points on the statistical space, called spinodal curves. The latter can be used to discern physical from non-physical situations.

Furthermore, notice that in the case of Hessian metrics, in order to ensure global thermodynamic stability of a given macro configuration of the black hole, one requires that all principal minors of the metric tensor be strictly positive definite, due to the probabilistic interpretation involved [16]. In any other cases (Quevedo, HPEM, etc) the physical interpretation of the metric components is unclear and one can only impose the convexity condition on the thermodynamic potential, \(\partial _a\partial _b\Phi \ge 0\), which is the second law of thermodynamics. Nevertheless, imposing positiveness of the black hole’s heat capacity is mandatory in any case in order to ensure local thermodynamic stability.

## 3 The DST black hole

*C*is the Weyl tensor and \(\beta _n\) are some real constant coefficients. The spherically symmetric Deser–Sarioglu–Tekin solution [60, 63],

*k*can be eliminated by a proper rescaling of the time coordinate

*t*. For convenience we have defined the function \(p(\sigma )\) as

^{1}[61, 65], we can support the claim that \(M=c/2\) is the mass of the DST black hole for any \(\sigma <1/4\) and \(\sigma >1\). To show this, one has to bring the DST metric (3.2) in the form

*y*limit of (3.9) determines the mass of the black hole. The explicit result for the DST solution is given by

*y*asymptotic expansion of \(\lambda (y)\). The expression is given by

*c*from Eq. (3.2) to the quasilocal mass

*M*of the DST black hole. Equation (3.12) is valid only when \(\frac{{2\,\sigma + 1}}{{4\,\sigma - 1}}\ge 0 \), i.e. \(\sigma \le -1/2\) or \(\sigma >1/4\).

## 4 Thermodynamics of the DST black hole

*M*. Moreover, the heat capacity,

One can also check that the first law of thermodynamics, \(dM=T\,dS\), is satisfied.

## 5 Hessian thermodynamic geometries on the equilibrium state space of the DST black hole solution

### 5.1 Extended equilibrium state space

### 5.2 Ruppeiner information metric

Although Ruppeiner metric fails to produce a viable thermodynamic description, one can always impose only local thermodynamic stability defined by the positive values of the heat capacity \(C>0\). The latter condition leads to the parameter region \(- \ 1<\sigma <-1/2\), together with \(\sigma >1\) and arbitrary large mass \(M>0\).

### 5.3 Weinhold information metric

*M*of the DST black hole is given in terms of the entropy

*S*and the parameter \(\sigma \) such as

## 6 Legendre invariant thermodynamic geometries on the equilibrium state space of the DST black hole solution

### 6.1 Quevedo information metric

^{2}

### 6.2 HPEM information metric

^{3}

### 6.3 MM information metric

The final geometric approach, which we are going to consider, was proposed by Mansoori and Mirza in [46]. In the article the authors define a conjugate thermodynamic potential 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 thermodynamic curvature.

*F*, which is related to the mass

*M*by the following Legendre transformation

## 7 Conclusion

Our current investigation is instigated by the intriguing existence of dark matter and dark energy in the Universe, which cannot be explained by traditional approaches. This motivates us to consider alternative models, which can include effects related to these dark phenomena. Highly promising alternatives are the so called higher derivative theories of gravity, which include contributions from higher powers of the Ricci scalar or other geometric invariants. In particular, our focus is on the thermodynamic properties of their admissible black hole solutions, which will allow us to constrain the possible dark matter/energy contributions, at least when thermodynamics is concerned.

In this paper we consider one known four-dimensional higher derivative black hole solution, namely the Deser–Sarioglu–Tekin black hole. The latter being a static, spherically symmetric gravitational solution of a theory with contributions from a non-polynomial term of the Weyl tensor to the Einstein–Hilbert Lagrangian. In order to study any implications for the black hole thermodynamics, we take advantage of two different geometric formulations, namely those of the Hessian information metrics (geometric thermodynamics) and the formalism of Legendre invariant thermodynamic metrics (geometrothermodynamics) on the space of equilibrium states of the DST black hole.

In general, the formalism of thermodynamic information geometry identifies the phase transition points of the system with the singularities of the corresponding thermodynamic information curvature \(R_I\). Near the critical points the underlying inter-particle interactions become strongly correlated and the equilibrium thermodynamic considerations are no longer applicable. In this case one expects that a more general approach should hold.

In the Hessian formulation we analyzed the Ruppeiner and the Weinhold thermodynamic metrics and showed that they are inadequate for the description of the DST black hole equilibrium state space. This is due to the occurring mismatch between the singularities of the heat capacity and the singularities of the corresponding thermodynamic curvatures. Therefore the Hessian thermodynamic geometries are unable to reproduce the Davies type transition points of the DST black hole heat capacity.

On the other hand, in the Legendre invariant case, all considered thermodynamic metrics successfully manage to incorporate the relevant phase transition points. Consequently they can be taken as viable metrics on the equilibrium state space of the DST black hole. However, some of them, such as the Quevedo metric, encounter additional singularities in their thermodynamic curvatures, the latter having obscure physical meaning at best. On the contrary, the HPEM and the MM metric seem to deal well with this problem and manage to get rid of the redundant spinodal curves in the case of the DST black hole.

Finally, let us address the problem of thermodynamic stability of the DST solution. For global stability one refers to Sylvester criterion for positive definite information metric, together with the positivity of the black hole heat capacity (local thermodynamic stability). Unfortunately, in the framework of geometric thermodynamics, both conditions can be interpreted as global thermodynamic stability only within the context of the Hessian metrics, due to their probabilistic interpretation. For the Legendre invariant metrics, imposing Sylvester criterion together with positive heat capacity does not necessarily guarantee global thermodynamic stability. The latter is caused by the current lack of physical interpretation of the components of the corresponding information metrics. Therefore, due to the failure of Hessian geometries, the DST black hole is only stable locally from a thermodynamic standpoint. The condition for local thermodynamic stability, together with the divergences of the physical DST metric curvature, constrain the values of the unknown parameter \(\sigma \) in the regions \(\sigma <- \ 1/2\) and \(\sigma >1\). The latter is also confirmed by imposing the Sylvester criterion for Quevedo and HPEM thermodynamic metrics.

## Footnotes

## Notes

### Acknowledgements

The author would like to thank R. C. Rashkov, H. Dimov, S. Yazadjiev, D. Arnaudov and P. Fiziev for insightful discussions and for careful reading of the draft. This work was supported by the Bulgarian NSF Grant no. DM18/1 and Sofia University Research Fund under Grant no. 80-10-104. The support from BLTP in JINR, is also gratefully acknowledged.

### Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: There is no associated data or any other codes related to this manuscript, and no such data will be deposited in the future.]

## References

- 1.A. Starobinsky, A new type of isotropic cosmological models without singularity. Phys. Lett. B
**91**(1), 99–102 (1980)ADSCrossRefGoogle Scholar - 2.A. De Felice, S. Tsujikawa, f(R) theories. Living Rev. Relativ.
**13**, 3 (2010). arXiv:1002.4928 [gr-qc]ADSCrossRefGoogle Scholar - 3.J.A.R. Cembranos, Dark matter from R2-gravity. Phys. Rev. Lett.
**102**, 141301 (2009). arXiv:0809.1653 [hep-ph]ADSCrossRefGoogle Scholar - 4.G. Cognola, E. Elizalde, S. Nojiri, S.D. Odintsov, S. Zerbini, One-loop f(R) gravity in de Sitter universe. JCAP
**0502**, 010 (2005). arXiv:hep-th/0501096 ADSMathSciNetCrossRefGoogle Scholar - 5.G. Cognola, E. Elizalde, S. Nojiri, S.D. Odintsov, S. Zerbini, Dark energy in modified Gauss–Bonnet gravity: late-time acceleration and the hierarchy problem. Phys. Rev. D
**73**, 084007 (2006). arXiv:hep-th/0601008 ADSCrossRefGoogle Scholar - 6.J. Oliva, S. Ray, A new cubic theory of gravity in five dimensions: black hole, Birkhoff’s theorem and C-function. Class. Quant. Grav.
**27**, 225002 (2010). arXiv:1003.4773 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 7.J. Oliva, S. Ray, Classification of six derivative Lagrangians of gravity and static spherically symmetric solutions. Phys. Rev. D
**82**, 124030 (2010). arXiv:1004.0737 [gr-qc]ADSCrossRefGoogle Scholar - 8.Y.-F. Cai, D.A. Easson, Black holes in an asymptotically safe gravity theory with higher derivatives. JCAP
**1009**, 002 (2010). arXiv:1007.1317 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 9.Z. Berezhiani, D. Comelli, F. Nesti, L. Pilo, Exact spherically symmetric solutions in massive gravity. JHEP
**07**, 130 (2008). arXiv:0803.1687 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 10.S. Amari, H. Nagaoka,
*Methods of Information Geometry. Translations of Mathematical Monographs*(American Mathematical Society, Providence, 2007)CrossRefGoogle Scholar - 11.S-i Amari,
*Information Geometry and its Applications*, 1st edn. (Springer, Berlin, 2016)CrossRefGoogle Scholar - 12.S. Amari,
*Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics*(Springer, New York, 2012)Google Scholar - 13.N. Ay, J. Jost, H. Lê, L. Schwachhöfer,
*Information Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics*(Springer International Publishing, New York, 2017)Google Scholar - 14.F. Weinhold, Metric geometry of equilibrium thermodynamics. J. Chem. Phys.
**63**(6), 2479–2483 (1975)ADSMathSciNetCrossRefGoogle Scholar - 15.F. Weinhold, Metric geometry of equilibrium thermodynamics. v. aspects of heterogeneous equilibrium. J. Chem. Phys.
**65**(2), 559–564 (1976)ADSCrossRefGoogle Scholar - 16.G. Ruppeiner, Thermodynamics: a Riemannian geometric model. Phys. Rev. A
**20**, 1608–1613 (1979)ADSCrossRefGoogle Scholar - 17.H. Janyszek, R. Mrugala, Riemannian geometry and the thermodynamics of model magnetic systems. Phys. Rev. A
**39**, 6515–6523 (1989)ADSMathSciNetCrossRefGoogle Scholar - 18.D.A. Johnston, W. Janke, R. Kenna, Information geometry, one, two, three (and four). Acta Phys. Polon. B
**34**, 4923–4937 (2003). arXiv:cond-mat/0308316 [cond-mat.stat-mech]ADSMathSciNetzbMATHGoogle Scholar - 19.B.P. Dolan, Geometry and thermodynamic fluctuations of the ising model on a bethe lattice. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci.
**454**(1978), 2655–2665 (1998)MathSciNetCrossRefGoogle Scholar - 20.B.P. Dolan, D.A. Johnston, R. Kenna, The Information geometry of the one-dimensional Potts model. J. Phys. A
**35**, 9025–9036 (2002). arXiv:cond-mat/0207180 ADSMathSciNetCrossRefGoogle Scholar - 21.W. Janke, D. Johnston, R. Malmini, The information geometry of the ising model on planar random graphs. Phys. Rev. E
**66**, 056119 (2002)ADSCrossRefGoogle Scholar - 22.W. Janke, D.A. Johnston, R. Kenna, The information geometry of the spherical model. Phys. Rev. E
**67**, 046106 (2003). arXiv:cond-mat/0210571 ADSCrossRefGoogle Scholar - 23.H. Quevedo, S.A. Zaldivar, A geometrothermodynamic approach to ideal quantum gases and Bose–Einstein condensates. arXiv:1512.08755 [gr-qc]
- 24.H. Quevedo, A. Sanchez, A. Vazquez, Thermodynamic systems as bosonic strings. arXiv:0805.4819 [hep-th]
- 25.H. Quevedo, F. Nettel, C.S. Lopez-Monsalvo, A. Bravetti, Representation invariant Geometrothermodynamics: applications to ordinary thermodynamic systems. J. Geom. Phys.
**81**, 1–9 (2014). arXiv:1303.1428 [math-ph]ADSMathSciNetCrossRefGoogle Scholar - 26.H. Quevedo, A. Sanchez, S. Taj, A. Vazquez, Phase transitions in geometrothermodynamics. Gen. Relativ. Gravit.
**43**, 1153–1165 (2011). arXiv:1010.5599 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 27.T. Vetsov, R. Rashkov, Fisher metric for diagonalizable quadratic Hamiltonians and application to phase transitions, in
*Proceedings of the Nineteenth International Conference on Geometry, Integrability and Quantization, Avangard Prima, Sofia, Bulgaria*, (2018). https://doi.org/10.7546/giq-19-2018-225-233, pp. 225–233 - 28.J.E. Aman, I. Bengtsson, N. Pidokrajt, Geometry of black hole thermodynamics. Gen. Relativ. Gravit.
**35**, 1733 (2003). arXiv:gr-qc/0304015 ADSMathSciNetCrossRefGoogle Scholar - 29.J.-Y. Shen, R.-G. Cai, B. Wang, R.-K. Su, Thermodynamic geometry and critical behavior of black holes. Int. J. Mod. Phys.
**A22**, 11–27 (2007). arXiv:gr-qc/0512035 ADSMathSciNetCrossRefGoogle Scholar - 30.W. Janke, D.A. Johnston, R. Kenna, Geometrothermodynamics of the Kehagias–Sfetsos black hole. J. Phys. A Math. Theor.
**43**(42), 425206 (2010)ADSMathSciNetCrossRefGoogle Scholar - 31.S. Ferrara, G.W. Gibbons, R. Kallosh, Black holes and critical points in moduli space. Nucl. Phys. B
**500**, 75–93 (1997). arXiv:hep-th/9702103 ADSMathSciNetCrossRefGoogle Scholar - 32.R.-G. Cai, J.-H. Cho, Thermodynamic curvature of the BTZ black hole. Phys. Rev. D
**60**, 067502 (1999). arXiv:hep-th/9803261 ADSMathSciNetCrossRefGoogle Scholar - 33.J.E. Aman, N. Pidokrajt, Geometry of higher-dimensional black hole thermodynamics. Phys. Rev. D
**73**, 024017 (2006). arXiv:hep-th/0510139 ADSMathSciNetCrossRefGoogle Scholar - 34.J.E. Åman, N. Pidokrajt, Ruppeiner geometry of black hole thermodynamics. EAS Publ. Ser.
**30**, 269–273 (2008). arXiv:0801.0016 [gr-qc]CrossRefGoogle Scholar - 35.J.E. Aman, N. Pidokrajt, J. Ward, On geometro-thermodynamics of dilaton black holes. EAS Publ. Ser.
**30**, 279–283 (2008). arXiv:0711.2201 [hep-th]CrossRefGoogle Scholar - 36.J. Suresh, C.P. Masroor, G. Prabhakar, V.C. Kuriakose, Thermodynamics and geometrothermodynamics of charged black holes in massive gravity. arXiv:1603.00981 [gr-qc]
- 37.J. Suresh, V.C. Kuriakose, Geometrothermodynamics of BTZ black hole in new massive gravity. arXiv:1606.06098 [gr-qc]
- 38.H. Quevedo, M.N. Quevedo, A. Sánchez, Einstein–Maxwell-dilaton phantom black holes: thermodynamics and geometrothermodynamics. Phys. Rev. D
**94**(2), 024057 (2016). arXiv:1606.02048 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 39.P. Channuie, D. Momeni, On the scalar-vector-tensor gravity: black hole, thermodynamics and geometrothermodynamics. Phys. Lett. B
**785**, 309 (2018). https://doi.org/10.1016/j.physletb.2018.08.052. arXiv:1802.03672 [gr-qc]ADSCrossRefzbMATHGoogle Scholar - 40.H. Quevedo, M.N. Quevedo, A. Sanchez, Geometrothermodynamics of phantom AdS black holes. Eur. Phys. J. C
**76**(3), 110 (2016). arXiv:1601.07120 [gr-qc]ADSCrossRefGoogle Scholar - 41.A. Larranaga, S. Mojica, Geometrothermodynamics of a charged black hole of string theory. Braz. J. Phys.
**41**, 154–158 (2011). arXiv:1012.2070 [gr-qc]ADSCrossRefGoogle Scholar - 42.T. Sarkar, G. Sengupta, B. Nath Tiwari, On the thermodynamic geometry of BTZ black holes. JHEP
**11**, 015 (2006). arXiv:hep-th/0606084 ADSMathSciNetCrossRefGoogle Scholar - 43.D. Astefanesei, M.J. Rodriguez, S. Theisen, Thermodynamic instability of doubly spinning black objects. JHEP
**08**, 046 (2010). arXiv:1003.2421 [hep-th]ADSCrossRefGoogle Scholar - 44.S.A.H. Mansoori, B. Mirza, E. Sharifian, Extrinsic and intrinsic curvatures in thermodynamic geometry. Phys. Lett. B
**759**, 298–305 (2016). arXiv:1602.03066 [gr-qc]ADSCrossRefGoogle Scholar - 45.S.A.H. Mansoori, B. Mirza, M. Fazel, Hessian matrix, specific heats, Nambu brackets, and thermodynamic geometry. JHEP
**04**, 115 (2015). arXiv:1411.2582 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 46.S.A.H. Mansoori, B. Mirza, Correspondence of phase transition points and singularities of thermodynamic geometry of black holes. Eur. Phys. J. C
**74**(99), 2681 (2014). arXiv:1308.1543 [gr-qc]ADSCrossRefGoogle Scholar - 47.Y.-H. Wei, Thermodynamic properties of a regular black hole in gravity coupling to nonlinear electrodynamics. Entropy
**20**(3), 192 (2018)ADSMathSciNetCrossRefGoogle Scholar - 48.Y.-G. Miao, Z.-M. Xu, Parametric phase transition for Gauss–Bonnet AdS black hole. Phys. Rev. D
**98**(8), 084051 (2018). https://doi.org/10.1103/PhysRevD.98.084051. arXiv:1806.10393 [hep-th]ADSCrossRefGoogle Scholar - 49.G. Ruppeiner, Thermodynamic black holes. Entropy
**20**(6), 460 (2018). arXiv:1803.08990 [gr-qc]ADSCrossRefGoogle Scholar - 50.H. Dimov, S. Mladenov, R.C. Rashkov, T. Vetsov, Entanglement entropy and Fisher information metric for closed bosonic strings in homogeneous plane wave background. Phys. Rev. D
**96**(12), 126004 (2017). arXiv:1705.01873 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 51.H. Dimov, S. Mladenov, R.C. Rashkov, T. Vetsov, Entanglement of higher-derivative oscillators in holographic systems. Nucl. Phys. B
**918**, 317–336 (2017). arXiv:1607.07807 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 52.J.D. Bekenstein, Black holes and entropy. Phys. Rev. D
**7**, 2333–2346 (1973)ADSMathSciNetCrossRefGoogle Scholar - 53.J.M. Bardeen, B. Carter, S.W. Hawking, The four laws of black hole mechanics. Commun. Math. Phys.
**31**(2), 161–170 (1973)ADSMathSciNetCrossRefGoogle Scholar - 54.J.D. Bekenstein, Generalized second law of thermodynamics in black-hole physics. Phys. Rev. D
**9**, 3292–3300 (1974)ADSCrossRefGoogle Scholar - 55.S.W. Hawking, Black holes in general relativity. Commun. Math. Phys.
**25**(2), 152–166 (1972)ADSMathSciNetCrossRefGoogle Scholar - 56.G. Ruppeiner, Riemannian geometry in thermodynamic fluctuation theory. Rev. Mod. Phys.
**67**, 605–659 (1995)ADSMathSciNetCrossRefGoogle Scholar - 57.B. Mirza, M. Zamani-Nasab, Ruppeiner geometry of RN black holes: flat or curved? JHEP
**06**, 059 (2007). arXiv:0706.3450 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 58.H. Quevedo, Geometrothermodynamics. J. Math. Phys.
**48**(1), 013506 (2007)ADSMathSciNetCrossRefGoogle Scholar - 59.H. Quevedo, M.N. Quevedo, A. Sanchez, Homogeneity and thermodynamic identities in geometrothermodynamics. Eur. Phys. J. C
**77**(3), 158 (2017). arXiv:1701.06702 [gr-qc]ADSCrossRefGoogle Scholar - 60.S. Deser, O. Sarioglu, B. Tekin, Spherically symmetric solutions of Einstein + non-polynomial gravities. Gen. Relativ. Gravit.
**40**, 1–7 (2008). arXiv:0705.1669 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 61.J.D. Brown, J.W. York Jr., Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev. D
**47**, 1407–1419 (1993). arXiv:gr-qc/9209012 ADSMathSciNetCrossRefGoogle Scholar - 62.G. Ruppeiner, Thermodynamic curvature measures interactions. Am. J. Phys.
**78**, 1170–1180 (2010). arXiv:1007.2160 [cond-mat.stat-mech]ADSCrossRefGoogle Scholar - 63.E. Bellini, R. Di Criscienzo, L. Sebastiani, S. Zerbini, Black hole entropy for two higher derivative theories of gravity. Entropy
**12**, 2186 (2010). arXiv:1009.4816 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 64.D. Astefanesei, R. Ballesteros, D. Choque, R. Rojas, Scalar charges and the first law of black hole thermodynamics. Phys. Lett. B
**782**, 47–54 (2018). arXiv:1803.11317 [hep-th]ADSCrossRefGoogle Scholar - 65.S.S. Yazadjiev, Non-asymptotically flat, non-dS/AdS dyonic black holes in dilaton gravity. Class. Quant. Grav.
**22**, 3875–3890 (2005). arXiv:gr-qc/0502024 ADSMathSciNetCrossRefGoogle Scholar - 66.R.M. Wald, Black hole entropy is the Noether charge. Phys. Rev. D
**48**(8), R3427–R3431 (1993). arXiv:gr-qc/9307038 ADSMathSciNetCrossRefGoogle Scholar - 67.S .H. Hendi, S. Panahiyan, B. Eslam Panah, M. Momennia, A new approach toward geometrical concept of black hole thermodynamics. Eur. Phys. J. C
**75**(10), 507 (2015). arXiv:1506.08092 [gr-qc]ADSCrossRefGoogle Scholar

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