Geometrothermodynamics as a singular conformal thermodynamic geometry

In this letter, we first redefine our formalism of the thermodynamic geometry introduced in [1, 2] by changing coordinates of the thermodynamic space by means of Jacobian matrices. We then show that the geometrothermodynamics (GTD) is conformally related to this new formalism of the thermodynamic geometry. This conformal transformation is singular at unphysical points were generated in GTD metric. Therefore, working with our metric neatly excludes all unphysical points without imposing any constraints.


I. INTRODUCTION
Hawking and Bekenstein were the first physicists to notice an analogy between black holes and common thermodynamic systems [3,4]. In fact, by considering the surface gravity and the horizon area, respectively, as the temperature and the entropy, one can interpret the four laws of thermodynamics for a black hole system [5]. However, the statistical origin of black hole thermodynamics is still a challenging question.
Several attempts have been made in order to describe thermodynamic behaviors of a black hole by making use of the Riemannian geometry [6][7][8]. In particular, Weinhold's metric [9] and Ruppeiner's metric [10,11] defined as the Hessian matrix of the internal energy and entropy, respectively, were used to find a direct correspondence between curvature singularities and phase transitions. However, the metrics introduced by Weinhold and Ruppeiner were not able to represent a general correspondence between thermodynamics and geometry [12].
In our earlier works [1,2], we proposed a new formulation of the thermodynamic geometry, by considering the thermodynamic potentials related to the mass (instead of the entropy) by Legendre transformations. It was proved that the thermodynamic geometry represents a one-toone correspondence between the divergences of the heat capacities and those of the curvature scalars.
On the other hand, Geometrothermodynamics (GTD) approach was introduced as a Riemannian thermodynamic structure obeying Legendre invariance [13]. However, in the special case of the phantom RN-AdS black hole, GTD fails to explain the one-to-one correspondence between phase transitions and singularities of the scalar curvature [14,15]. In fact, the indiscriminate use of the natural thermodynamic variables and modified variables in black hole thermodynamics, leads to reveal some ambiguities in GTD [16]. It was argued that this issue might be solved by applying the fact that any thermodynamic system must be represented by a fundamental equation * shossein@ipm.ir & shosseini@shahroodut.ac.ir † b.mirza@cc.iut.ac.ir which is mathematically determined by a homogeneous function [17,18]. We will discuss both homogeneous and nonhomogeneous cases and explain that the NTG is a well defined theory with no unphysical singular points that usually are generated by other approaches to thermodynamic geometry.
In this paper, we propose a general simple metric and study correspondence between phase transition points and curvature singularities. We prove that both the NTG and GTD metrics are conformally related to each other. However, the conformal transformation between both metrics is singular at unphysical points that are generated by the GTD metric. Finally, by considering the phantom RN-AdS black hole as an example, we apply both metrics to understand which one can provide better description of the phase transitions.
The outline of this paper is as follows. In Section II, we review our previous works about thermodynamic geometry and propose the new thermodynamic geometry (NTG). In Section III, we obtain a conformal transformation between the NTG and GTD. In Section III we study the thermodynamic geometry of the phantom RN-AdS black hole, and compare results of the NTG with the GTD. Finally, Section IV is devoted to discussions of our results.

II. THE NEW THERMODYNAMIC GEOMETRY
Let us first review our previous results [1,2,23] and then introduce a new formalism of the thermodynamic geometry.
In [1,2], we have proved that there is a one-to-one correspondence between divergences of the heat capacities at a fixed electric charge, C Q ≡ T ∂S ∂T Q , and those of the curvature scalar,R(S, Φ), by defining the following metric,, where X i = (S, Φ) and M (S, Φ) = M (S, Q(S, Φ)) − ΦQ(S, Φ) is the enthalpy potential associated with the arXiv:1905.01733v1 [gr-qc] 5 May 2019 mass potential M (S, Q) by Legendre transformation. Notice that, in two dimensional thermodynamic space, the denominator of R(S, Φ) is proportional to the square of the metric determinant, R ∝ (g) −2 . According to the first law of thermodynamics for Enthalpy potential, dM (S, Φ) = T dS − QdΦ, the metric matrix is given by, Thus its determinant reads Here we have used the identity matrix in the first line and defined the specific heat at the fixed entropy as, Notice that here the charge, Q, is a function of (S, Φ). The equality (3) indicates that the phase transitions of C Q (S, Q(S, Φ)) occur exactly at the singularities of R(S, Φ). Moreover, the curvature singularities of the free-energyM = M − T S potential are also located at the phase transitions of C Q [1]. It means that the line element of the free energy is associated with the line element of the Enthalpy potential [19]. In some black hole systems, writing Q as a function of (S, Φ) is usually hard or impossible. Therefore, it is convenient to work at the coordinate (S, Q). Thus allow us transfer from the coordinate (S, Φ) to (S, Q) by using below Jacobian matrix.
In the new coordinate, the metric elements can be changed by where, J T is the transpose of J. Under changing coordinate, the metric matrix takes, therefore, the following form.ḡ After using Maxwell relation, ∂T ∂Q | S = ∂Φ ∂S | Q , one gets from Eq.(6) to the following expression for the metric elements [23].
In which we have used the first law of thermodynamic, i.e. dM = T dS + ΦdQ. It is interesting that after changing coordinates from (S, Φ) to (S, Q), the metric in Eq.(1) converts to a new form which is defined by the mass potential M (S, Q).
Analogously to the previous procedure, for general black holes with (n + 1) thermodynamic variables, X i = (S, Φ 1 , ..., Φ n ), and Enthalpy potential, where the first law of thermodynamic dM = T dS − i Q i dΦ i have been used. Regarding the Jacobian matrix, under the coordinate change {S, Φ 1 , ..., Φ n } → {S, Q 1 , ..., Q n }, finally the metric tensor takes the below block-diagonal matrix form.
where G is a square matrix of order n which is given by Therefore, the metric determinant in the new coordinates isḡ where G = det(G). As a result, the correspondence between the singularities of the Ricci scalarR and the phase transition of C Q1,Q2,...,Qn is clear. Finally, it is worth pointing out that, while in this section we have introduced a metric based on the entalphy potentialM (S, Φ 1 , ..., Φ n ) in Eq.(1), after changing coordinates from (S, Φ 1 , ..., Φ n ) to (S, Q 1 , ..Q n ) we arrived at a new form of metric which is associated to the mass potential M (S, Q 1 , ..., Q n ). It is clear that by using the last equality in Eq. (11), we can rerwite Eq. (10) as Follows: where η j i = diag(−1, 1, ..., 1) and X i = (S, Q 1 , ..., Q n ) are extensive thermodynamic variables. This expression for the thermodynamic geometry shows that the singularities of the curvature made by thermodynamic potential M (S, Q 1 , ..., Q n ), correspond to the phase transitions of C Q1,Q2,...,Qn .
On the other hand, for charged black holes with the first law of thermodynamics, dM = T dS + ΦdQ, we have also demonstrated that the phase transitions of specific heat C Φ = T ∂S ∂T Φ occur exactly at the singularities of R(S, Q), which is the Ricci scalar curvature of the Ruppenier metric [1,2]. The Ruppeiner metric in the mass representation can be defied as: where X i = (S, Q) are extensive parameters. According to the first law of thermodynamics, the denominator of the scalar curvature R(S, Q) is given by: It is obvious that the phase transitions of C Φ (S, Φ(S, Q)) correspond to the singularities of R(S, Q), whereas the scalar curvature R(S, Q) is not able to explain the properties of the phase transitions of C Q at all. Expanding to the general case with (n + 1) extensive thermodynamic variables, X i = (S, Q 1 , ..., Q n ), one now gets the following form for metric.
As one considers the below Jacobian matrix, under the coordinate change {S, Q 1 , ..., Q n } → {S, Φ 1 , ..., Φ n }, the metric element in new coordinate can be written as follows.
= block-diag 1 T ∂ 2M ∂S 2 , −Ḡ Note that we have used the Maxwell relations in the above relation [1]. Moreover,Ḡ is a square matrix of order n that defines as In analogy with the metric (13), we can write Eq.(18) as follows.
where X i = (S, Φ 1 , ..., Φ n ). Clearly, this formalism is an alternative formulation of the thermodynamic geometry which relies on the application of the enthalpy potential, M (S, Φ 1 , ..., Φ n ), to describe the phase transition points of C Φ1,...,Φn . Considering the metrics (13) and (20), so we introduce a new formalism of the thermodynamic geometry as: where η j i = diag(−1, 1, ..., 1) and Ξ is the thermodynamic potential. It is surprising that the above equation demystifies the thermodynamic geometry concept in order to understand better thermodynamic properties of a system. We have used the abbreviation NTG (New Thermodynamic Geometry ) here for this metric. For example, in two dimensions, if we use the ensemble associated with the mass of the black hole, Ξ = M , the curvature obtained from NTG metric diverges exactly at the phase transitions of C Q . Moreover, if we consider the ensemble associated with the enthalpy, Ξ =M , curvature singularities give us the phase transitions of C Φ . It means that the phase transition structure of black holes depends on the ensemble. In next sections, we first introduce Geometrothermodynamics (GTD) metric and then try to derive a conformal factor between NTG and GTD.

III. CONFORMAL TRANSFORMATION BETWEEN THE NTG AND GTD
Let us consider the thermodynamic phase space with (2n + 1) dimensions, T , with independent coordinates Z A = {Ξ, X i , I i }, i = 1...n, where Ξ represents the thermodynamic potential, and X i and I i are the extensive and intensive thermodynamic variables, respectively. Now, one can select on T a non-degenerate metric G = G(Z A ), and the Gibbs 1-form Θ = dΞ − δ ij I i dX j , in which δ ij is the kornecher delta. Moreover, the metric G is Legendre invariant if its functional dependence on Z A does not change under a Legendre transformation, Indeed, the Legendre invariance guarantees that the geometric properties of G do not depend on the thermodynamic potentials [13]. Also, the Gibbs 1-form Θ is Legendre invariant in the sense that according to a Legendre transformation it behaves like Θ →Θ = dΞ −Ĩ i dX i . The equilibrium space, E is then a subspace of T by means of the pullback φ * which is associated with the embedding map φ : E → T with constraint φ * (Θ) = 0 i.e. dΞ = I i dX j which shows Ξ = Ξ(X i ) and I i = dΞ dXi . Under these conditions, the general metrics form can be defined as where η ij = diag(−1, 1, ...., 1) and ξ ij is a diagonal constant matrix [20,21]. Note that metric G GT D is used to describe systems with second-order phase transitions like black holes. Moreover, pullback induces metric on E as where η j i = diag(−1, 1, ..., 1) and ξ j i = ξ il δ jl [20,21]. It is obvious that components of the above metric can be calculated explicitly once the thermodynamic potential Ξ(X i ) is given. By using Eulers identity, the conformal term can be put proportional to the potential Ξ. When we consider the case where Ξ is homogeneous in X i of order β, i.e. Ξ(λX i ) = λ β Ξ(X i ), then Eulers identity satisfies βΞ = X i ∂Ξ ∂X i . Moreover, for the generic case where Ξ is a generalized homogeneous function, i.e. Ξ(λ αi X i ) = λ αΞ Ξ(X i ), the Eulers identity reads α i X i ∂Ξ ∂Xi = α Ξ Ξ. Therefore, One can always choose the components of the diagonal matrix ξ ij proportional to α Ξ so that the conformal factor becomes proportional to Ξ as follows [17,22].
A comparison between Eqs. (24,25) and Eq. (21) clarifies the conformal transformation between GTD metrics and the NTG metric for the potential functions which are related to the mass (instead of the entropy) by Legendre transformations. More precisely, one can derive a conformally equivalent thermodynamic metrics as where and ∂X k ∂X j . It is also easy task to show that there is a relationship between scalar curvatures associated to those metrics by following relation (see appendix A).
R I,II ∝ (Ω I,II ) −2 R N T G It is worth emphasizing that the above equation implies that the scalar curvature R I,II has some extra singularity points coming from Ω I,II = 0 in comparison with the scalar curvature R N T G . Therefore, it is necessary to impose some physical constraints to get rid of these singularities. However, it turns out that the conformal transformation (27) and (28) are singular conformal transformation with physically different properties. In fact, because these transformations are singular at the points of extra (unphysical) roots, i.e. Ω I,II = 0, the NTG formalism physically exclude these extra (unphysical) roots form the singularities of the curvature. In the light of the above discussion, it should be noted that the conformal transformations are not invertible since the Jacobians of the transformation, (Ω I,II ) 2 , vanish at the (unphysical) roots.
In next section, we consider a phantom RN-AdS black hole as an extraordinary example to illustrate the important distinctions between NTG and GTD.

IV. THE PHANTOM RN-ADS BLACK HOLE
The mass of a phantom RN-AdS black hole [14] is expressed as a function of the thermodynamic variables (S, Q) as where, Λ is the cosmological constant, and η = ±1. As η = 1, we have RN-AdS black hole solutions, while Phantom RN-AdS black hole solutions are obtained by choice of η = −1 [14]. Moreover, S = πr 2 + is the Bekenstein-Hawking entropy. According to the first law of thermodynamics, dM = T dS + ΦdQ, the Hawking temperature, T , the electric potential, Φ, and the specific heat capacity, C Q , are given by It is obvious that the heat capacity C Q diverges at the values of the entropy S 1 = −(π/2Λ)(1 + 1 + 12ηΛQ 2 ) and S 2 = −(π/2Λ)(−1 + 1 + 12ηΛQ 2 ). More precisely, in the phantom RN-AdS black hole case, the value of the S 1 is only positive. Thus this case possesses just one point of phase transition. It is also straightforward to check that M is not homogeneous in (S, Q) because one can not find a real β such that M (λS, λQ) = λ β M (S, Q). Now allow us, first, to apply the NTG approach in order to take into account the second order of the phase transition points of C Q . As mentioned, one needs to put the thermodynamic potential, Ξ = M (S, Q) in Eq. (21) to check the phase transitions of C Q . Thus we have and the denominator of R T G is obtained by It is clear that, the first part of the denominator is zero only at the extremal limit (T = 0) that it is forbidden by the third law of thermodynamics. Moreover, the roots of the second part of the curvature denominator gives us all the phase transitions of C Q . This point is also confirmed in Fig. (1), where we plot the curvature scalar R N T G and the heat capacity C Q as a function of the entropy for a phantom RN-AdS black hole by choice of Λ = −1, η = −1, and Q = 0.25. As a consequence of the NTG method, the curvature diverges exactly at the phase transitions with no other additional roots.  On the other hand, by substituting Ξ = M and X i = (S, Q) in Eqs. (26) and (27), the line element of the GTD approach is given by Making use of Eq. (A5), the denominator of the scalar curvature R I can be calculated as Clearly, the term in the leading parentheses of Eq. (36) presents phase transitions, whereas the term in the last parentheses describes the zero of the conformal factor, i.e. ST + ΦQ. The same issue has been reperted in [24] for RN black holes in presence of quintessence. As shown in Fig. (2), it is obvious that the GTD approach gives us some extra singularity points which don't coincide with phase transition points. In other words, in the non-homogeneous potential case, Eq. (29); the GTD metric is not able to provide a one-to-one correspondence between singularities and phase transitions.
In the remaining of this section, let us build a firstdegree generalized homogeneous potential function from the fundamental mass (29) and then give a brief explanation of the general characteristics of the GTD and NTG methods, respectively. As discussed in [17], it is necessary to consider the cosmological constant as a thermodynamic variable [25] in order to make a homogeneous function. By rescaling S → λ α S S, Q → λ α Q Q, and Λ → λ αΛ Λ and assuming the conditions α Λ = −α S and α q = 1 2 α S , the fundamental mass (29) is a generalized homogeneous function of degree β = α S /2. For example, by replacing the cosmological constant Λ by the AdS radius l via l 2 = −3/Λ and choosing α S = 1, the mass formula will be a generalized homogeneous function of degree 1/2, i.e., M (λS, λ 1/2 l, λ 1/2 Q) = λ 1/2 M (S, l, Q) . (37) Note that one can reduce the degree of any generalized homogeneous function to one, by selecting the appropriate variables [26]. By introducing the new entropy , the mass (29) converts to It is easy to check that Eq. (38) is a first-degree homogeneous function according to the Euler's identity. Now, form the first law of thermodynamic, dm = T ds + Ldl + ΦdQ; the heat capacity is given by which indicates a second-order phase transition occurs at 3s 4 = −ηq 2 l 2 . In the homogeneous case, one should apply the GTD metric, Eq. (25), for explanation of the phase transition behaviors. By selecting the potential function Ξ = m, the metric (25) can be written as it becomes manifest that, considering the cosmological constant as a thermodynamic variable, the equilibrium thermodynamic space must be extended by one dimension. From this definition and using Eq.(38) in the above relation, the denominator of the scalar curvature is calculated by D(R II ) = 3(3s 4 + ηQ 2 l 2 ) 2 (s 4 + s 2 l 2 + ηQ 2 l 2 ) 3 (41) The term appearing in the first parenthesis in Eq. (41) clarifies the phase transitions of C Q,l , whereas the roots of the other parenthesis comes from m = 0, which is unphysical constraint, because in this case the first law of thermodynamics breaks down and the GTD metric can not defined at all [17]. Such a constraint, which emerges in this framework, is not present in the NTG formalism. In other words, the NTG formalism physically excludes completely unphysical points m = 0 which appear in the GTD metric. Fig. (3) illustrates the comparison between phase transitions of C Q,l and singularities of the R II for a phantom RN-AdS with η = −1, l = 1.73, and Q = 0.25. In order to have a better understanding of the above mentioned point, let us construct the NTG metric for the recent case. Starting from NTG metric (21), and considering Ξ = m and X i = (s, l, Q), we get: Thus the denominator of the scalar curvature reads It is interesting that the curvature singularity, 3s 4 = −ηq 2 l 2 , give us the phase transition point without any assumption for unphysical constraints like m = 0 (See Fig. (4)). Therefore, compared to other techniques like GTD, the NTG geometry provides a powerful tool to achieve a one-to-one correspondence between singularities and phase transitions.

V. CONCLUSION
In this paper we expressed our new formalism of the thermodynamics geometry, previously introduced in [ in the language of the general potential function Ξ via Eq. (21) that is called the NTG metric. It is worth mentioning that second-order phase transition points of a black hole coincide with the curvature singularity of the NTG metric. We found the conformal transformations that connect different GTD metrics and the NTG metric to each other. Moreover, we have shown that in the non-homogeneous case, the zeros of the conformal factor, ξ j i X i ∂Ξ ∂X j , lead to create some singularity points that do not coincide with phase transition points in GTD method. Whereas in the homogeneous case, by ignoring unphysical constraint Ξ = 0 roots, curvature singularities obtained by both metrics exist at those places in the equilibrium space, where second-order phase transitions occur. We investigated thermodynamic properties of a spherically symmetric AdS charged phantom black hole by using the NTG metric and the GTD metric for homogeneous and non-homogeneous potential functions.
In case of the non-homogeneous potential, the GTD approach fails to explain phase behaviors through the singularities. Contrary to this, the NTG metric is able to establish a one-to-one correspondence between singularities and phase transition points.
In the case that the fundamental equation is a firstdegree homogeneous function, by considering the cosmological constant as a thermodynamic variable, both metrics predict the same result for phase transitions. In fact, the zeros of the unphysical constraint, m = 0, must be ignored from GTD curvature singularities, while these unphysical points disappear physically form curvature singularities by using the NTG formalism.

VI. ACKNOWLEDGMENTS
We are grateful to Mohamad Ali Gorji and Mustapha Azreg-Aïnou for extremely helpful discussions and com-ments about this work. We thank Mohamad Mahdi Davari Esfahani, Matteo Baggioli and Tsvetan Vetsov for reading a preliminary version of the draft.