Geometric thermodynamics: black holes and the meaning of the scalar curvature

In this paper we show that the vanishing of the scalar curvature of Ruppeiner-like metrics does not characterize the ideal gas. Furthermore, we claim through an example that flatness is not a sufficient condition to establish the absence of interactions in the underlying microscopic model of a thermodynamic system, which poses a limitation on the usefulness of Ruppeiner's metric and conjecture. Finally, we address the problem of the choice of coordinates in black hole thermodynamics. We propose an alternative energy representation for Kerr-Newman black holes that mimics fully Weinhold's approach. The corresponding Ruppeiner's metrics become degenerate only at absolute zero and have non-vanishing scalar curvatures.


I. INTRODUCTION
Thermodynamics is considered one of the finest descriptions of real phenomena. The capability of its systems to predict real-life results in such an accurate fashion has settled it in a privileged place among the accepted physical theories. However, as Gibbs and Carathéodory noticed by the end of the 19th century and the beginning of the 20th, respectively, it lacks the mathematical precision of other areas, such as classical mechanics or electrodynamics.
This problem has inspired a line of research whose main goal is to describe thermodynamic systems geometrically. One of the first steps towards this aim were the pioneering works of Weinhold [1][2][3][4][5], where it was shown that Riemannian geometry arises naturally in the study of equilibrium thermodynamics.
A class of thermodynamic systems that has received special attention recently is that formed by black holes. Based upon the so-called first law of black hole thermodynamics [6], mass is commonly regarded as a thermodynamic potential whose variables are entropy, angular momentum and charge [7,8]. However, as we show in this paper, a different thermodynamic potential must be chosen in order to mimic exactly Weinhold's approach in this context (cf. [9,10]).
An important aspect of the Riemannian-geometrical approach to thermodynamics arises when studying mod-els that predict phase transitions. In the van der Waals gas, for instance, the scalar curvature of the space of equilibrium states of closed systems has a singularity at the (thermodynamic) critical point. In the context of black hole thermodynamics, many results link singularities of scalar curvature to divergences of heat capacities (which traditionally correspond to critical states of a thermodynamic system). Following these results, it has been conjectured [8,11] that the singularities of the scalar curvature are related to the critical points of the system, and that the value of the former gives certain information regarding the interactions of the underlying microscopic model. However, no proof of this assertion has been given. In this work, we show that scalar curvature is not a characteristic of thermodynamic systems, in the sense that it does not determine uniquely the system under consideration (this remains true even for families of systems with a fixed thermodynamical parameter, e. g., a heat capacity). Moreover, we display a simple closed hydrostatic system whose manifold of equilibrium states is flat but presents states where the isothermal compressibility diverges to infinity. This paper is organized as follows. In the second section we present a short review on Weinhold's metrics. In the third section, we show that scalar curvature is not a characteristic of thermodynamic systems by exhibiting a family of flat closed hydrostatic systems. In section IV we review the basic concepts of black hole thermodynamics. Section V is devoted to Weinhold's metrics in the setting of black hole thermodynamics; the point of view adopted in this section is different from the ones that can be found in the literature, since we place black hole thermodynamics in a mathematical situation similar to that of other more common thermodynamic systems. Further-more, we analyze the behavior of the scalar curvature in the Riemannian submanifolds of the space of equilibrium states of the Kerr-Newman black holes. Though alluring, we show in the sixth section that the correspondence between microscopic interaction and scalar curvature cannot be fully established, by setting out an example of a flat closed hydrostatic system whose isothermal compressibility goes to infinity on a set of "critical states." We present our conclusions in section VII.

II. HESSIAN STRUCTURES IN THERMODYNAMICS
In this section we review the concepts of geometrical thermodynamics introduced by Weinhold [1].
Let Σ be a thermodynamic system and E the set of all its equilibrium states. Each element of E is uniquely determined by the values of n deformation parameters plus k non-deformation parameters, i. e., there exists an injective function It is convenient to assume that ϕ(E) is open in R k+n , since E is thereby endowed with a smooth structure. When k = 1, the system is called simple. We consider only this class of systems.
As a consequence of both the first and second laws of thermodynamics, one can choose either the internal energy U or the entropy S as the non-deformation parameter that together with x 1 , . . . , x n constitutes a global chart on E. The former is called the energy representation of Σ, whilst the latter is known as the entropy representation.
Another essential postulate for the geometrical approach to thermodynamics is the entropy maximum principle. It asserts that all spontaneous processes involving composite systems (processes for which total work and heat are zero, but the starting and ending points are not the same) reach a final state in which entropy attains a maximum (observe that this processes occur at constant total energy). This yields as a result that for any numbers a 0 , . . . , a n , and any x ∈ E, where α and β run through {0, . . . , n} and x 0 = U . This means that ∂ α ∂ β S are the components of a negative semidefinite tensor, which will be denoted by h [12]. To avoid reference to any particular coordinate system, one may rely on concepts of the geometry of Hessian structures to write h (see, for example, [13]). From the fact (following from the Gibbs-Duhem equation) that the entropy and the energy representations are related by an affine coordinate transformation, there exists a flat con-nection∇ on E having these two charts as affine coordinates. Thus, h =∇dS. (1) The tensor h in the last equation is known as Ruppeiner's metric tensor. Equivalently, it can be proven that the Hessian of U with respect to the flat connection∇ -known as Weinhold's metric tensor-is conformally equivalent to h, viz. [14] g :=∇dU = −T∇dS. ( Notice that since T is positive, g is positive semidefinite in E. Hence, any integral distribution transversal to ker g is the tangent space of a Riemannian submanifold of E. One can easily find such a submanifold in the following way: let r = rank g. Then, by relabeling if necessary, is a Riemannian manifold with the metric ı * g, where ı : D → E is the inclusion.Yet more, (ı * ∇ , ı * g) is a Hessian structure having ı * U as potential. In all known cases, every submanifold given by x i = const. is a Riemannian submanifold of E, for i ∈ {1, . . . , n}, provided that dim ker g = 1.
One appealing feature of this geometric approach to thermodynamics is that it portrays thermodynamic potentials as natural geometrical objects: they are potentials of the corresponding dual Hessian structure.

III. FLAT HYDROSTATIC CLOSED SYSTEMS
In this section we consider simple hydrostatic systems with fundamental equation where U , T , S, p, V , µ, and N denote the system's internal energy, temperature, entropy, pressure, volume, chemical potential, and number of particles, respectively. The submanifold N defined by N = const. is Riemannian; in terms of coordinates (T, V ), the metric tensor g has the form where C V is the heat capacity at constant volume and κ T is the isothermal compressibility. It is well known that a closed ideal gas has a flat manifold of equilibrium states [12,15]. We will prove that this is not a characteristic of an ideal gas: there exist infinitely many flat closed systems, even with C V = const. Defining the metric g takes the form The scalar curvature of N is thus given by It can readily be seen that for an ideal gas, the scalar curvature of this manifold is equal to zero. However, substituting R = 0 in eq. (6) and solving for κ −1 T yields where f and h are functions of v only. In consequence, one can see that the ideal gas is only a particular case of hydrostatic system with C V = const. and R = 0, given by h = 0 and f ∝ v −1 .
The role that eq. (7) plays in the ostensible relationship between curvature and microscopic interactions will be revealed below.

IV. BASIC CONCEPTS OF BLACK HOLE THERMODYNAMICS
The most general known solutions to the Einstein-Maxwell field equations that contain black holes are the solutions of the so-called Kerr-Newman family [6]. These are rotating, charged black holes, each one with a well defined value of its mass M , its angular momentum L, and its charge q. Since the interior of a black hole is invisible and inaccessible from the exterior universe, we cannot tell the difference between two black holes with the same M , L, and q. Thus, only these three global parameters have physical significance [6]. Hence, we can say that these features define the "thermodynamic state" of a black hole. More precisely, we can regard the set of all possible thermodynamic states of a black hole as a differentiable manifold E, the triad (M, L, q) as a coordinate system on this manifold, and each x ∈ E as a black hole of a solution contained in the Kerr-Newman family ("macroscopically" indistinguishable, in the sense of physical significance, from any other sharing the same global parameters).
In this context, the first law of black hole thermodynamics is postulated to yield ϑ = dM − ωdL − Φdq (8) (commonly, this equation is known as the first law), where ω, Φ ∈ C ∞ (E). The 1-form ϑ plays the role of the heat form in classical thermodynamics; the functions ω and Φ represent the angular velocity and the electric potential of the black hole, respectively [16].
Using the definition of ω and Φ for Kerr-Newman black holes [17], it can readily be seen that the heat form is integrable. In classical thermodynamics, the functions involved in the integrability of ϑ represent the temperature and the entropy of the system. In this context, the surface area A of the event horizon plays the role of the entropy. The black hole uniqueness results ensure that we can regard A ∈ C ∞ (E), since there is a well defined value of the surface area for given M , J, and q [18]; in terms of these parameters, for the Kerr-Newman black holes, A is given by [6,19] It is easily verified, using equations (9), (10), and (11), that whence, comparing with equation (8), it follows that dM = (8π) −1 κdA+ωdL+Φdq. The integrating factor of the heat 1-form, κ, which plays the role of temperature, is known as the surface gravity of the black hole. The relationship between the surface gravity and temperature goes beyond a mere analogy. Actually, a black hole can be assigned a physical temperature, related to its surface gravity by means of the quantum field theory in curved spacetime through the equation T = (2π) −1 κ (similarily, the entropy of the black hole is given by S = A/4) [18,19]. Hence, equation (8) takes the familiar form dM = T dS + ωdL + Φdq.

V. RIEMANNIAN STRUCTURES IN BLACK HOLE THERMODYNAMICS
In this section we analyze Weinhold's metrics on the space of equilibrium states of the Kerr-Newmann black hole family. A naïve approach would lead us to define the metric tensor's components in the coordinates (S, L, q) as the components of the Hessian of M . However, strictly speaking, there is an essential part of that construction that does not hold in this case, making it impossible to mimic it directly in this context. In hydrostatic systems, entropy is a homogeneous function of degree one of U , V , and N . In this case, S is a degree-one homogeneous function of M 2 , L, and q 2 . The homogeneity of S of degree one is needed to prove that the components of the Hessian of S define a negative semi-definite tensor. Moreover, this property of entropy ensures that the manifold of equilibrium states of a system always admits a flat connection, allowing a straightforward coordinate-free expression for Ruppeiner's or Weinhold's metrics.
To achieve a full analogy to the aforementioned construction, we use the global chart (M 2 , J, q 2 ). Then, in terms of these variables, S is a homogeneous function of degree one, and we have that g := ∂ i ∂ j M 2 dx i dx j is positive semi-definite on E (cf. [9,10,19]), with x 1 = S, x 2 = L, and x 3 = q 2 (on the region where κ ≥ 0). Besides, M 2 can be written as with Θ := Φ/(2q). The homogeneity of M 2 implies the existence of a Gibbs-Duhem equation, which means in turn that g is degenerate on E. However, any submanifold defined by x i = const., for some i ∈ {1, 2, 3}, is Riemannian with metric ı * g, where ı is the inclusion.
To begin, we consider the submanifold defined by q = const. It can readily be seen that the matrix representation of Weinhold's metrics in terms of the chart (S, L) is given by Notice that the metric becomes degenerate whenever Indeed, the scalar curvature of this submanifold of E is given by The metric tensor in this submanifold of E has a singularity at L = q 2 /(2 √ 3) (cf. [17]). Similarly, in the surface given by L = const., in coordinates (Z, q 2 ) g = (15) In this case Since Z(x) = 0 for all black holes of the Kerr-Newmann family, no rotating black hole has a curvature singularity (referring to the space of equilibrium states). As can be seen, the critical points of the scalar curvature we have found using M 2 as potential and (S, L, q 2 ) as the entropy representation do not agree with the thermodynamical critical points that are reported in the literature. This does not fully contradict the conjecture relating thermodynamic and geometric properties of systems, though. The path we have followed in this section rather illustrates another ambiguity in the geometric treatment of thermodynamics: what conditions validate a function on E as a potential for the Hessian structure? Even further, in the context of other Riemannian approaches to thermodynamics where potentials play a fundamental role in the construction of metrics [20], what mathematical requirements must a function meet to be considered as a suitable potential?
On the other hand, regarding the common association of scalar curvature to thermodynamic interaction, the fact that the two thermodynamically relevant [22] Riemannian submanifolds of the space of equilibrium states in this case are not flat does not imply that interaction is predicted by the underlying microscopic model of this black hole family (regardless of how these models might appear in this context). As we shall show in the next section, to assume that the relationship between the singularities and critical points were true for all thermodynamic systems could lead to erroneous physical conclusions.

VI. FLAT SYSTEMS WITH CRITICAL STATES
As we mentioned before, one can construct thermodynamic systems whose space of equilibrium states contains a Riemannian submanifold that is both flat and has critical states.
For the sake of simplicity, we will consider a simple closed hydrostatic system. We have shown that the isothermal compressibility of one such system with flat space of equilibrium states is given by eq. (7). In particular, let f = 1/v and h = −t 0 /(v 0 v). Then, up to a function of temperature, where t 0 = 2 √ C V T 0 and v 0 = 2 √ V 0 , for some prescribed values T 0 and V 0 of temperature and volume, respectively. Observe that κ −1 The states in C present the typical behavior of phase transitions. Supposing that scalar curvature measures thermodynamic interaction, this thermodynamic system is one whose microscopic components do not interact with each other, yet it has a critical point.

VII. CONCLUDING REMARKS
In this paper, we have shown that scalar curvature is not a characteristic of thermodynamics systems.
Furthermore, we have pointed out two fundamental issues of the geometric approach to thermodynamics that have not been thoroughly addressed yet. The first one is related to the problem of describing thermodynamic systems from a geometrical viewpoint, consistent with but independent from any physical information. This regards, for instance, the precise mathematical definition of a representation (like the energy and the entropy representations), which requires a more accurate definition of thermodynamic potentials and deformation coordinates.
The second issue is the relation that has commonly been conjectured between certain geometric and thermodynamic features, such as scalar curvature and microscopic interaction. We have shown that the scope of this alleged relation is limited and must be further reviewed.