# Quasi-homogeneous black hole thermodynamics

## Abstract

Although the fundamental equations of ordinary thermodynamic systems are known to correspond to first-degree homogeneous functions, in the case of non-ordinary systems like black holes the corresponding fundamental equations are not homogeneous. We present several arguments, indicating that black holes should be described by means of quasi-homogeneous functions of degree different from one. In particular, we show that imposing the first-degree condition leads to contradictory results in thermodynamics and geometrothermodynamics of black holes. As a consequence, we show that in generalized gravity theories the coupling constants like the cosmological constant, the Born–Infeld parameter or the Gauss–Bonnet constant must be considered as thermodynamic variables.

## 1 Introduction

An important property of thermodynamic laboratory systems is that their fundamental equations are given in terms of homogeneous functions of first degree. Recall that a fundamental equation is a function that relates an extensive thermodynamic potential (entropy or energy) with the extensive thermodynamic variables necessary to describe the system. Then, the homogeneity condition is a consequence of the fact that extensive variables are additive [1]. Generalizations of the extensivity property have been also considered in the literature and concepts like sub-extensive and supra-extensive variables have been introduced to correctly describe the behavior of certain thermodynamic systems. Recently in [2], we proposed to classify thermodynamic systems into ordinary and non-ordinary by using an exact mathematical concept, namely, the concept of homogeneous and generalized homogeneous functions.

*n*degrees of freedom. Then, a system described by the fundamental equation \(\Phi (E^a)\) is called ordinary if \(\Phi \) is a homogeneous function

As pointed out in [5, 6, 7], black holes and other non-ordinary systems should be considered as quasi-homogeneous systems for different reasons and, in fact, it is a consequence of the well known non-extensivity property of certain thermodynamic systems. In particular, quasi-homogeneity is important to correctly describe the behavior of non-ordinary systems near the critical points [5]. In a different context, by investigating the thermodynamics of AdS black holes [8], it has been long suggested [9, 10] that the cosmological constant can be considered as a thermodynamic variable. It turns out that in this case the cosmological constant can naturally be thought of as a pressure and the mass of the black hole as the enthalpy of the spacetime. By following this idea, it has been recently established that AdS black holes can be investigated from the point of view of chemistry and that there exists an intriguing physical analogy between van der Waals fluids and black holes [11]. This is a remarkable result that deserves further investigation and will certainly contribute to the understanding of the physical properties of black holes. It is therefore important to find out why the cosmological constant can be assumed to be an additional thermodynamic variable for black holes. In this work, we will see that this assumption is not only possible, but also necessary in order for black holes to be quasi-homogeneous systems. Although black hole thermodynamics is perfectly well defined without cosmological constant, the recently proposed extended thermodynamics with cosmological constant reveals new physical aspects of black holes, hitherto unseen.

On the other hand, differential geometric methods have been applied during the past few decades in classical thermodynamics to investigate the stability and the critical points of thermodynamic systems [12, 13, 14, 15]. The main goal of these studies consists in finding connections between the thermodynamic properties of the system and the geometric properties of the corresponding equilibrium space. The approach of thermodynamic geometry consists in introducing a completely fixed Riemannian metric into the space of equilibrium states of a given system. In this case, the components of the metric are associated with the second moment of the fluctuations of a particular thermodynamic potential [14]. A second approach consists in demanding that the metric of the equilibrium space be invariant with respect to Legendre transformations, i.e., with respect to the choice of thermodynamic potential. This approach is known as geometrothermodynamics (GTD) [15]. In this case, the metric of the equilibrium space is derived by using Legendre invariance and turns out to contain certain degree of arbitrariness. We will see below that this arbitrariness becomes completely fixed as a consequence of imposing the quasi-homogeneity condition.

Thus, in this work, we will explore the consequences of demanding quasi-homogeneity for black holes in two different contexts. First, we will see how the quasi-homogeneity condition fixes the thermodynamic metric which is used in GTD to describe black holes and, moreover, that GTD is able to detect the non-correct use of this condition. Second, we explore black hole thermodynamics from the point of view of quasi-homogeneity and show that it dictates the thermodynamic properties of the parameters that enter the fundamental equation of black holes. If quasi-homogeneity is not handled correctly, it turns out that the thermodynamic properties of a black hole configuration can change drastically.

This work is organized as follows. In Sect. 2, we review the main physical consequences of imposing homogeneity in ordinary thermodynamic systems. In Sect. 3, we explore thermodynamic quasi-homogeneity in the context of GTD, and show that systems with intrinsic thermodynamic interaction can lead to contradictory results for the corresponding equilibrium space, when the quasi-homogeneity condition is not implemented properly. In Sect. 4, we analyze the fundamental thermodynamic equation of black hole configurations in several gravity theories, and show that the physical parameters, such as the coupling constants, that enter the action in a field theoretical approach must be considered as thermodynamic variables as a consequence of the quasi-homogeneity condition. Moreover, we show the thermodynamic inconsistencies that can arise when the quasi-homogeneity condition is not applied appropriately. Finally, in Sect. 5, we review our results, and propose some tasks for future investigations.

## 2 Homogeneity of ordinary systems

*S*imply that it can be inverted with respect to the energy

*U*which is, in turn, a homogeneous function of first degree [1]. For concreteness, let us consider as a particular example the simple case of an ideal gas with a fixed number of particles

*N*, whose fundamental equation is given by [1]

*V*is the volume of the gas. This is a first-degree homogeneous function, i.e, \(S(\lambda U, \lambda V,\lambda N)= \lambda S(U,V,N)\) which can be inverted with respect to

*U*and yields

## 3 Quasi-homogeneity in geometrothermodynamics

*U*or

*S*by means of a Legendre transformation [17].

*G*and a canonical contact 1-form \(\Theta = d\Phi - I_a dE^a\). Whereas the contact 1-form is uniquely defined modulo a conformal function, there are three classes of Legendre invariant metrics [18, 19], namely,

In GTD, the equilibrium space \(\mathcal{E}\), with the set of coordinates \(\{E^a\}\), is considered as a subspace of the phase space \(\mathcal{T}\) and is defined by the embedding map \(\varphi : \mathcal E \rightarrow \mathcal T\) with \(\varphi : \{E^a\} \mapsto \{\Phi (E^a), E^a, I^a(E^a)\}\) and \(\varphi ^*(\Theta )=0\). Then, any metric *G* in \(\mathcal T\) induces a metric *g* in \(\mathcal E\) by means of the pullback \(g=\varphi ^*(G)\). This means that in GTD there can be also three different classes of metrics \(g^{^I}\), \(g^{^{II}}\) and \(g^{^{III}}\) for the equilibrium space.

*d*dimensions can be computed by using the formula [21, 22],

*d*dimensions. It is easy to see that it corresponds to a non-ordinary system because the degree of homogeneity is \(\frac{d-2}{d-3}\). In this particular case, the fundamental equation can be inverted, yielding in the mass representation the equation

*T*is the temperature and \(\phi \) the electric potential. From the fundamental equation and the first law it is, therefore, possible to derive the complete set of thermodynamic variables of the system. Analogously, in GTD all the geometric information about the equilibrium space can be obtained from the fundamental equation. Indeed, the metric \(g^{^{II}}\) with \(\Phi =M\) and \(E^a=\{S,Q\}\) leads to

*M*and

*Q*. We conclude that in GTD it is not allowed to perform a transformation of variables at the level of the fundamental equation with the aim of describing a black hole configuration by means of a homogeneous function of first degree. GTD detects such transformations by changing the geometric properties of the equilibrium space.

## 4 Quasi-homogeneity in black hole thermodynamics

*S*with its horizon area

*A*. This is the first fact that indicates a non-standard thermodynamic behavior in black holes. The horizon area, in turn, is a geometric quantity that can be calculated by using the metric of the corresponding spacetime and depends on the physical parameters of the black hole. In the case of the Einstein-Maxwell theory, the most general black hole is described by the Kerr-Newman spacetime which contains only three independent parameters, namely, the mass

*M*, angular momentum

*J*and electric charge

*Q*. A straightforward computation of the horizon area leads to the fundamental equation [23]

*T*, \(\Psi \) and \(\Omega \) are the corresponding intensive variables, which are interpreted as the temperature, electric potential and angular velocity at the horizon, respectively. Then, we obtain

*T*and \(\Omega \) are of degree \(-\beta _S/2\). In particular, the Hawking temperature

*T*will not behave as the temperature of an ordinary system. Since the constant \(\beta _S\) remains arbitrary, one is tempted to fix it by introducing new thermodynamic variables. In fact, this is possible because the degree of any quasi-homogeneous function can always be set equal to one by choosing the variables appropriately [4]. For instance, the change of variables

*T*, \(\Psi \) and \(\Omega \) that follow from the quasi-homogeneous fundamental equation (31). This is illustrated in Fig. 1.

*C*shows clearly a second-order phase transition which is lacking in the analysis of the capacity

*c*. This shows that a change of thermodynamic variables in order to get a first degree homogeneous functions can drastically change the thermodynamic properties of the system (Fig. 3).

This simple example shows the importance of correctly handling the homogeneous or quasi-homogeneous character of fundamental equations. In the next subsections, we will present several examples that illustrate the way we propose to handle quasi-homogeneous systems.

### 4.1 Einstein–Maxwell gravity with cosmological constant

*m*, specific angular momentum \(a=j/m\), electric charge

*q*and cosmological constant \(\Lambda \). Since these parameters are usually defined for asymptotically flat metrics, in the case of asymptotically AdS spacetimes the problem appears that several definitions are possible. In particular, the physical angular velocity can be defined as the difference between the angular velocity at infinity and on the horizon [8]. In the context of the formalism of isolated horizons [25, 26], it is also possible to address this problem and it has been shown that the intrinsic physical parameters in this case are given by [27]

*S*by means of the Smarr formula

*M*of the black hole with the extensive variables

*S*,

*Q*, and

*J*. It is easy to see that this equation cannot be inverted; this is one of the first signals indicating that it corresponds to a non-ordinary system.

Although the cosmological constant is originally not interpreted as a thermodynamic variable, we see that if the Kerr-Newman-AdS black hole is to be considered as a quasi-homogeneous system, then the requirement appears that the cosmological constant must be a thermodynamic variable. Although the coefficient \(\beta _S\) remains arbitrary, one can consider it as positive to take into account the sub or supra extensive character of the entropy of the entropy. It then follows that \(\Lambda \) should be interpreted as an intensive variable. In fact, by using a completely different approach, it was shown that \(\Lambda \) can be interpreted as the pressure of the system [9, 10, 11].

### 4.2 Einstein–Born–Infeld gravity

*F*is the electromagnetic invariant defined as \(F=\frac{1}{4}F_{\mu \nu } F^{\mu \nu }\), and

*b*is known as the Born-Infeld parameter, which in string theory is related to the string tension \(\alpha ^\prime \) as \(b=\frac{1}{2\pi \alpha ^\prime }\).

*M*is the ADM mass and

*Q*the electric charge. The horizons of this \(3+1\) dimensional black hole are determined by the roots of the lapse function

*f*(

*r*). In terms of the outer horizon radius \(r_+\) and the electric charge

*Q*, the black hole mass is given by [28, 29]

*S*and

*Q*, we perform the transformation \(S\rightarrow \lambda ^{\beta _S} S\) and \(Q\rightarrow \lambda ^{\beta _Q} Q\). Then, the resulting function does not satisfy the quasi-homogeneous condition. However, if we also perform the transformations \(b \rightarrow \lambda ^{\beta _b} b\) and \(\Lambda \rightarrow \lambda ^{\beta _\Lambda } \Lambda \) , then the fundamental equation (54) becomes quasi-homogeneous \(M \rightarrow \lambda ^{\beta _M} M\) under the conditions

### 4.3 Einstein–Maxwell–Gauss–Bonnet gravity

*M*and

*Q*are identified as the mass and electric charge of the system. The above solution describes an asymptotically anti-de-Sitter black hole only if the expression inside the square root is positive and the function \(f(r_H)=0\) on the horizon radius, i. e.,

## 5 Conclusions

In this work, we argue that black holes are thermodynamic systems described by fundamental equations that should correspond to quasi-homogeneous functions. This means that the concept of extensivity and intensivity of black hole thermodynamic variables is not as clear and concrete as in the case of ordinary systems, which are described by homogeneous fundamental equations. Essentially, the origin of the quasi-homogeneity of black holes is already contained in the Hawking–Bekenstein entropy, which is proportional to the area and not to the volume, as in the case of ordinary systems.

From the condition of quasi-homogeneity of black holes, we derive the important property that coupling constants of gravity theories must be considered as thermodynamic variables. We prove this for the cosmological constant, the Born–Infeld parameter and the Gauss–Bonnet constant. The cosmological constant can indeed be interpreted as the coupling constant between the gravitational field and the matter described by the vacuum energy. In turn, the Born–Infeld parameter and the Gauss–Bonnet constant are coupling constants between gravity and, respectively, the non-linear electromagnetic field and the effective field represented by the topological term. In fact, the cosmological constant has been interpreted previously as a thermodynamic variable with properties consistent with an effective “pressure” [11]. In this context, it would be interesting to investigate the interpretation of the Born-Infeld parameter and the Gauss–Bonnet constant in the framework of black hole thermodynamics. Finally, since the explicit application of the quasi-homogeneity condition is quite simple, we can conjecture that our results hold for all the coupling constants of any generalization of Einstein gravity.

Since the degree and the coefficients of quasi-homogeneity are defined up to a multiplicative constant factor, one is tempted to use this freedom to fix the degree to 1, by transforming the thermodynamic variables appropriately. We have shown that this procedure can lead to contradictory results. In black hole thermodynamics, the phase transition structure can be modified by the transformation of variables. The free parameter that appears in the degree of quasi-homogeneity turns out to correspond to a multiplicative constant of the metric used in GTD to describe black holes so that it does not affect the geometric properties of the equilibrium space. However, GTD is very sensitive to the transformations of variables at the level of the fundamental equation, which can completely modify the thermodynamic curvature of the system under consideration.

According to our results, quasi-homogeneity is a property of non-ordinary systems which must be handled correctly in order to avoid unphysical and contradictory results. It also leads to a deep modification of the way we interpret coupling constants in gravity theories. It would be interesting to further explore the physical consequences of these modifications.

## Notes

### Acknowledgements

This work was carried out within the scope of the project CIAS 2312 supported by the Vicerrectoría de Investigaciones de la Universidad Militar Nueva Granada-Vigencia 2017. This work was partially supported by UNAM-DGAPA-PAPIIT, Grant no. 111617, and by the Ministry of Education and Science of RK, Grant nos. BR05236322 and AP05133630. We thank an anonymous referee for very useful comments and suggestions.

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