# Geometric Thermodynamics: Black Holes and the Meaning of the Scalar Curvature

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Hessian Structures in Thermodynamics

^{k}

^{+}

^{n}, since ε is thereby endowed with a smooth structure. When k = 1, the system is called simple. We consider only this class of systems.

^{1},…, x

^{n}constitutes a global chart on ε. The former is called the energy representation of Σ, whilst the latter is known as the entropy representation.

^{0},…, a

^{n}, and any x ∈ ε,

^{0}= U. Hence, −∂

_{α}∂

_{β}S are the components of a positive semidefinite tensor, denoted by g

_{R}, which we shall refer to as Ruppeiner-like metrics [8].

_{W}—known as Weinhold’s metric tensor—which is conformally equivalent to g

_{R}, viz. [9,10]

_{W}is positive semidefinite in ε. Hence, any integral distribution transversal to ker g

_{W}is the tangent space of a Riemannian submanifold of ε. One can easily find such a submanifold in the following way: let r = rank g

_{W}. Then, by relabeling if necessary,

^{*}g

_{W}, where ı : $\mathcal{D}$ → ε is the inclusion. In all known cases, every submanifold given by x

^{i}= const. is a Riemannian submanifold of ε, for i ∈ {1,…, n}, provided that dim ker g

_{W}= 1 (observe that g

_{R}is also a Riemannian metric tensor on these submanifolds).

^{i}= const., ${g}_{W}=-\tilde{\nabla}\mathrm{d}\tilde{U}$, where $\tilde{U}$ is the thermodynamic potential y

_{i}x

^{i}(no summation) and $\tilde{\nabla}$ is the flat connection whose affine coordinate system is (T, y

_{k}), with k ∈ {1,…, n} \ {i}, where y

_{1},…, y

_{n}represent the intensive variables of the system. For instance, Gibbs’ Free Energy is the dual potential of internal energy in the submanifold given by constant particle number. This, in turn, makes evident the usefulness of Weinhold’s metrics: since thermodynamic systems undergo first-order phase transition whenever the potentials with intensive “natural variables” are not concave [23,25], any such system would have states x ∈ $\mathcal{D}$ such that $\tilde{\nabla}\mathrm{d}{\tilde{U}}_{x}$ is not negative definite, which amounts to g

_{Wx}not being positive definite. Hence, the behavior of Weinhold’s metrics is sensitive to phase transitions, needless of any other auxiliary structure whatsoever (cf. Bravetti et al. [18] and Liu et al. [26]).

## 3. Flat Hydrostatic Closed Systems

_{R}has the form

_{V}= const. Defining

_{R}takes the form

^{t}κ

_{T})

^{−}

^{1}

^{/}

^{2}/∂t is a function of v only. Therefore, flat closed systems with constant C

_{V}are given by

_{1}and f

_{2}are functions of v only (a similar result holds for Weinhold’s metrics, with ${\kappa}_{T}^{-1}={(\tilde{t}{\tilde{f}}_{1}+{\tilde{f}}_{2})}^{2}$, where $\tilde{t}:=2\sqrt{{C}_{V}T}$, and ${\tilde{f}}_{1}$ and ${\tilde{f}}_{2}$ are functions of v only). In consequence, the ideal gas is only a particular case of closed hydrostatic system with C

_{V}= const. and identically vanishing scalar curvature, given by f

_{2}∝ v

^{−}

^{1}and f

_{1}= 0 ( ${\tilde{f}}_{1}\propto {v}^{-1}$ and ${\tilde{f}}_{2}=0$, respectively).

_{1}and F

_{2}are functions of N only. The ideal gas is only the particular case given by F

_{1}= 0 and F

_{2}= const.

## 4. Non-Interacting Systems with First-Order Phase Transitions?

_{1}= c/v

^{2}and f

_{2}= −ct

_{0}/(v

_{0}v), where c, v

_{0}, and t

_{0}are constants.

_{R}|

_{C}is degenerate, where

_{W}and g

_{R}are conformally equivalent with a non-vanishing conformal factor (temperature), the degeneracy of g

_{R}renders g

_{W}degenerate. Hence, C is a set of critical states and the system presents first-order phase transitions, yet has non-interacting microscopic constituents (according to Ruppeiner’s conjecture).

_{R}to the interactions of the underlying microscopic model.

## 5. Riemannian-Geometrical Approach to Black Hole Thermodynamics

^{2}and y := q

^{2}/2 (which is not a global chart on the space of equilibrium states of Kerr-Newmann black holes), as the energy representation of the system, since according to Smarr’s formula [34]

^{2}< x

^{2}[35]. We will further restrict our analysis to real values of S.

_{R}is degenerate on ε; moreover, any submanifold defined by x

^{i}= const., for some i ∈ {1, 2, 3}, is Riemannian with metric ı

^{*}g

_{R}, where ı is the inclusion.

## 6. Concluding Remarks

## Acknowledgments

## Conflicts of Interest

**PACS Classifications:**02.40.Ky, 04.70.Dy, 05.70.-a 1. Introduction

## References

- Gibbs, W.J. Graphical Methods in the Thermodynamics of Fluids. In The Collected Works of J. Willard Gibbs; Longmans, Green and Co.: New York, NY, USA, 1928; Volume 1, Chapter 1; pp. 33–54. [Google Scholar]
- Carathéodory, C. Untersuchungen über die Grundlagen der Thermodynamik. Math. Ann.
**1909**, 67, 355–386. [Google Scholar] - Weinhold, F. Metric geometry of equilibrium thermodynamics. J. Chem. Phys.
**1975**, 63, 2479–2483. [Google Scholar] - Weinhold, F. Metric geometry of equilibrium thermodynamics. II. Scaling, homogeneity, and generalized Gibbs-Duhem relations. J. Chem. Phys.
**1975**, 63, 2484–2487. [Google Scholar] - Weinhold, F. Metric geometry of equilibrium thermodynamics. III. Elementary formal structure of a vector algebraic representation of equilibrium thermodynamics. J. Chem. Phys.
**1975**, 63, 2488–2495. [Google Scholar] - Weinhold, F. Metric geometry of equilibrium thermodynamics. IV. Vector-algebraic evaluation of thermodynamic derivatives. J. Chem. Phys.
**1975**, 63, 2496–2501. [Google Scholar] - Weinhold, F. Metric geometry of equilibrium thermodynamics. V. Aspects of heterogeneous equilibrium. J. Chem. Phys.
**1976**, 65, 559–564. [Google Scholar] - Ruppeiner, G. Thermodynamics: A Riemannian geometric model. Phys. Rev. A
**1979**. [Google Scholar] - Mrugała, R. On equivalence of two metrics in classical thermodynamics. Phys. A
**1984**, 125, 631–639. [Google Scholar] - Torres del Castillo, G.F.; Montesinos Velásquez, M. Riemannian Structure of the Thermodynamic Phase Space. Rev. Mex. Fís.
**1993**, 39, 194–202. [Google Scholar] - Ruppeiner, G. Riemannian geometry in thermodynamic fluctuation theory. Rev. Mod. Phys.
**1995**, 67, 605–659. [Google Scholar] - Ruppeiner, G. Thermodynamic curvature measures interactions. Am. J. Phys
**2010**, 78, 1170. [Google Scholar] - Ruppeiner, G. Thermodynamic curvature from the critical point to the triple point. Phys. Rev. E
**2012**, 86, 021130. [Google Scholar] - Ruppeiner, G.; Sahay, A.; Sarkar, T.; Sengupta, G. Thermodynamic geometry, phase transitions, and the Widom line. Phys. Rev. E
**2012**, 86, 052103. [Google Scholar] - Ruppeiner, G. Thermodynamic curvature and black holes. In Breaking of Supersymmetry and Ultraviolet Divergences in Extended Supergravity; Springer: Berlin, Germany, 2014; pp. 179–203. [Google Scholar]
- Cai, R.G.; Cho, J.H. Thermodynamic Curvature of the BTZ Black Hole. Phys. Rev. D
**1999**, 60, 067502. [Google Scholar] - Mansoori, S.A.H.; Mirza, B. Correspondence of phase transition points and singularities of thermodynamic geometry of black holes. Eur. Phys. J. C
**2014**, 74, 2681. [Google Scholar] - Bravetti, A.; Nettel, F. Thermodynamic curvature and ensemble nonequivalence. Phys. Rev. D
**2014**, 90, 044064. [Google Scholar] - Medved, A.J.M. A Commentary on Ruppeiner Metrics for Black Holes. Mod. Phys. Lett. A
**2008**, 23, 2149–2161. [Google Scholar] - Shen, J.; Cai, R.G.; Wang, B.; Su, R.K. Thermodynamic Geometry and Critical Behavior of Black Holes. Int. J. Mod. Phys. A
**2007**, 22, 11–27. [Google Scholar] - Åman, J.E.; Pidokrajt, N. Ruppeiner Geometry of Black Hole Thermodynamics. EAS Publ. Ser.
**2008**. [Google Scholar] [CrossRef] - Mirza, B.; Zamani-Nasab, M. Ruppeiner Geometry of RN Black Holes: Flat or Curved? JHEP
**2007**, 2007. [Google Scholar] [CrossRef] - Callen, H.B. Thermodynamics and an Introduction to Thermostatistics; John Wiley & Sons: Hoboken, NJ, USA, 1985. [Google Scholar]
- Shima, H. The Geometry of Hessian Structures; World Scientific: Singapore, Singapore, 2007. [Google Scholar]
- Linder, B. Thermodynamics and Introductory Statistical Mechanics; Wiley-Interscience: Hoboken, NJ, USA, 2004. [Google Scholar]
- Liu, H.; Lü, H.; Luo, M.; Shao, K.N. Thermodynamical Metrics and Black Hole Phase Transitions. JHEP. [CrossRef]
- Nulton, J.D.; Salamon, P. Geometry of the ideal gas. Phys. Rev. A
**1985**. [Google Scholar] - Banerjee, R.; Modak, S.K.; Samanta, S. Second Order Phase Transition and Thermodynamic Geometry in Kerr-AdS Black Hole. Phys. Rev. D
**2011**, 84, 064024. [Google Scholar] - Pidokrajt, N.; Ward, J. Thermodynamic Geometry and Type 0A Black Holes
**2011**, arXiv, 1106.2831. - Ruppeiner, G. Thermodynamic curvature and phase transitions in Kerr-Newman black holes. Phys. Rev. D
**2008**, 78, 024016. [Google Scholar] - Sahay, A.; Sarkar, T.; Sengupta, G. Thermodynamic Geometry and Phase Transitions in Kerr-Newman-AdS Black Holes. J. High Energy Phys.
**2010**. [Google Scholar] [CrossRef] - Tiwari, B.N. Sur les corrections de la géométrie thermodynamique des trous noirs
**2008**, arXiv, 0801.4087. (In French) - Belgiorno, F. Black Hole Thermodynamics in Carathéodory’s Approach. Phys. Lett. A
**2003**, 312, 324–330. [Google Scholar] - Davies, P.C.W. The thermodynamic theory of black holes. Proc. R. Soc. Lond. A
**1977**, 353, 499–521. [Google Scholar] - Davies, P.C.W. Thermodynamics of Black Holes. Rep. Prog. Phys.
**1978**, 41, 1313–1355. [Google Scholar] - Sokołowski, L.M.; Mazur, P. Second-order phase transitions in black-hole thermodynamics. J. Phys. A Math. Gen.
**1980**, 13, 1113–1120. [Google Scholar]

© 2014 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

García-Ariza, M.Á.; Montesinos, M.; Torres del Castillo, G.F.
Geometric Thermodynamics: Black Holes and the Meaning of the Scalar Curvature. *Entropy* **2014**, *16*, 6515-6523.
https://doi.org/10.3390/e16126515

**AMA Style**

García-Ariza MÁ, Montesinos M, Torres del Castillo GF.
Geometric Thermodynamics: Black Holes and the Meaning of the Scalar Curvature. *Entropy*. 2014; 16(12):6515-6523.
https://doi.org/10.3390/e16126515

**Chicago/Turabian Style**

García-Ariza, Miguel Ángel, Merced Montesinos, and Gerardo F. Torres del Castillo.
2014. "Geometric Thermodynamics: Black Holes and the Meaning of the Scalar Curvature" *Entropy* 16, no. 12: 6515-6523.
https://doi.org/10.3390/e16126515