Study of thermodynamical geometries of conformal gravity black hole

This work deals with the applications of thermodynamical geometries on conformal gravity black holes (CGBH) consisting of conformal parameters a and k. The stability of black hole (BH) addressed with the aid of small, middle, large and divergency roots, respectively. For this purpose, graphical behavior of heat capacity and temperature versus horizon radius is presented which help us to show the stability conditions. Further, studied the different geometries like Weinhold, Ruppeiner, Geometrothermodynamics (GTD) and Hendi-Panahiyah-Eslam-Momennia (HPEM), and found relationship between divergency of scalar curvature and zeros of heat capacity. As a result, it is noticed that Ruppeiner, HPEM and GTD metric exhibit more important information as compared to Weinhold.


Introduction
In the last century, Bekenstein and Hawking give the concept of gravitational temperature and heat capacity of BH by utilizing the concept of entropy [1,2]. Fu. [3] present another model of the universe, that is, the universe is gravitationally closed and topped off with harmonious radiation at 3K temperature. Its primary part is the tremendous sea and remaining is the genuine matter which hushes up a little fraction. The interaction within the BH draws in radiation from space is known as the energy concentrating process which is quite useful because it reduces entropy and it is also against the second law of thermodynamics. These ideas are the opposite of Hawking and Bekenstein. According to the Hawking theory, entropy of a BH could not be decreased. But it's not true but why? As we know that entropy increases in a system just because of irreversible processes that occur in the system where the external envia e-mail: imran.asjad@umt.edu.pk (corresponding author) ronment does not involve. Now, we know the area of BH increased by accreting the energy from external environment. If the accretion process could not start then it is not possible to increase the area of the event horizon of BH. Hence, both processes increased something and different from each other, as the entropy increased due to internal process and area of event horizon increases due to external energy. By the above discussion, we concluded that there is no similarity between an increasing area of BH and an increase in entropy of BH. One can say that this statement of Hawking and Bekenstein that the area of the BH is entropy false and their rough results are not acceptable. Bekenstein and Hawking also discussed that the radiation of BH is directly related to temperature by taking the reciprocal of radiated heat then it forms the temperature of BH which is also entropy. This argument is also not correct. The author proves that Hawking radiation in one direction is totally different from isotropic radiation of BH at a specific temperature [4]. One can argue that there are many ambiguities in Hawking-Bekenstein entropy which are partially covered in logarithmic corrected entropy, higherorder corrected entropy, etc. Many authors [5][6][7][8][9][10] discussed in detail the Hawking-Bekenstein area relationship and found various useful results. One can not completely deny the idea of the Hawking-Bekenstein area entropy relationship. Moreover, the thermodynamical quantities like temperature along with entropy is related to its geometrical structures like horizon area and surface gravity. According to statistical mechanics, BHs are considered to be unstable due to Hawking radiations emitted from it. Many different exciting facts were discovered on macroscopic thermodynamical characteristics of BHs and very well-known is the thermal stability of BHs. One of the most important thermodynamical property is considered to be thermal stability which suggests the small deviation of thermodynamical system occurs due to the variation in thermodynamical parameters. Several critical behaviors are proposed for analyzing the phase structure [11]. Thermal stability of a BH is analyzed by utilizing the heat capacities which must be positive for stable system [12][13][14]. The heat capacity of the BH is also analyzed the phase transition [13][14][15][16][17][18] and there are two types of phase transition, type I is the roots and type II is the divergency of heat capacity, respectively. One can analyze the reasons behind the irregularities of temperature along with heat capacity in normal thermodynamics of BH [5]. Many authors proposed different methods to construct thermodynamical geometries and phase transitions such as Hermann [19] developed the differential manifold to find thermodynamical geometries. Note that scalar curvature 'R' is directly connected with the divergency point of the system which also provide the information about the internal structure of BH. In this regard, scalar curvature analyzed the microscopic structures of the system, for detail see in [20,21]. Thermodynamical geometries give us important information about divergency as well as phase transition of BH [22][23][24][25][26][27][28][29][30][31]. Furthermore, one can find the BH solutions by utilizing the Einstein's gravity and conformal theory but it's thermodynamical quantities such as entropy and mass depend on the conformal theory instead of the metric. The thermodynamical characteristics of the conformal theory of BHs have been explored in the 4-D in the following references [32][33][34][35]. Most importantly, motion of test particles and the light rays can provide the information about material properties of conformal field equations. Moreover, the motion of fluid and light could be utilized to characterize the conformal spacetime and highlight its characteristics. One can get important information related to the space-time of the BHs by studying geodesic motion [36]. Geodesic equations of motion having solutions with the same mathematical method but taking different space-time which extensively being explored in [37,38]. The Jacobi reversal problems have been tackled [39,40] with Schwarzschild (SH) anti-de Sitter (AdS) spacetime and examined in [41,42]. Many authors investigated the significant properties of conformal gravity theory in [43][44][45]. The work related to the geodesics study in this theory has been discussed in [46]. Sultana et al. [47] have explored the utilization of geodesic solutions in conformal gravity theory. Due to the significant importance of conformal gravity theory, our aim is to study the important geometries like Weinhold, Ruppeiner, HPEM, and GTD to describe the internal structure of CGBH. Weinhold presented his mathematical technique in the year 1975 [48], in which he demonstrated that a metric of BH is characterized in space of balance conditions of thermodynamical systems. Weinhold utilized the idea of conformal theory from the Riemannian space to thermodynamical space. Weinhold's metric is characterized as the Hessian in the mass portrayal as shown below: where M represents the mass, S shows the entropy, and N r is the other extensive factors of the system. Furthermore, Ruppeiner (in 1979) [69] introduced another metric which is reduce to Hessian in the entropy portrayal as follows, The Ruppeiner metric is quiet similar to Weinhold's geometry which is given by [49,50].
where T represents the temperature of the thermodynamical system. Ruppeiner geometry completely provides information about microscopic characteristics of BHs from the thermodynamic perspective, see detail in [51][52][53][54][55]. The Ruppeiner geometry is a pure thermodynamical geometry [56,69] that is set up on the Riemannian thermodynamical geometry and it's applications are focused on the internal structures of simple thermodynamical system. Moreover, HPEM [59] and GTD are the new approaches in this direction [60][61][62][63][64]. The HPEM metric for n different variables (n ≥ 2) is, where ξ i = S and M S is the derivative of mass M w.r.t. entropy S, and the general metric of GTD geometry is, where, and E a , I b , and φ represent the extensive, intensive thermodynamical parameters and thermodynamical potential, respectively. Our aim is to analyze the roots as well as divergency points and thats provide the location of stability conditions of CGBH. For this purpose, we have found the scalar curvature of thermodynamical geometries, most importantly, if it coincides with the zeroes of heat capacity then we can have the important information about the internal structure of CGBH. The paper is planned as follows: In Sect. 2, the CGBH model and its thermodynamical properties are presented. In Sect. 3, the roots and divergencies as well as thermal stability is discussed. Next, in Sect. 4, we analyzed different thermodynamical geometries and discussed graphical behavior of scalar curvature and heat capacity. Conclusion is presented in last section.

Conformal gravity black hole and its thermodynamical properties
CGBH is a very important outcome of conformal invariant theory as its action does not change w.r.t coordinate and general conformal transformations, respectively. In a reference frame, a regular quantity is singular but not regular in an alternative frame due to conformal transformation. Singularity is not a physical quantity but quite unique to corresponding reference frame. Now, we have conformal singularity which is related to the conformal factor but not a naked singularity of BH. One can notice that it could not follow the same mathematical procedure to study spacetime singularity in Einstein and conformal gravity. Here, we have scalar curvature and Kretschmann scalar that do not change w.r.t conformal transformation as they are not connected under conformal gravity. So, the action of conformal theory could be constructed by following [65].
• In general relativity, it is a covariant advancement theory.
• Symmetric principle exists when it prevents the cosmological term and Einstein Hilbert action. • Conformal theory has conformal transformation i.e. g μ ν → 2 (x) g μ ν • The action of conformal gravity is defined in terms of coupling constant α g and Weyl tensor C η λμν. It gives the equation of motion of fourth-order with more constants of integration and parameters in the solutions, respectively.
The line element of spherically symmetric space-time is as follows [65], where, M represents the mass of CGBH and a and k are constants of integration. If we choose a = 0, the result is the Schwarzschild-de-sitter solution. If we choose a = k = 0 then the result is the simple Schwarzschild solution. Here k is a cosmological constant and takes a very small value. Constant a measures all departures of the Weyl tensor from that of Einstein with a cosmological term. The Weyl theory experimentally successful like Einstein theory for very small values of a. The significance of parameter a is to relate the properties of background geometry or interior dynamics of static source of interest. The exterior and interior solutions could be matched by utilizing these properties [67]. The Values of conformal gravity parameters determined by Mannheim for the very first time [66]. There has been some discussion about the nature of conformal gravity at a short distance. It was shown by Kazanas and Mannheim that fourth-order con-formal gravity recovered both structures in the appropriate limit [67,68]. The detail analysis of the roots of CGBH are presented by finding the limits of important parameter a and k [65]. Also, one can observe that by considering large values of a and k, the roots of CGBH vanished. We discuss the important thermodynamical quantities of CGBH. First, one can find the mass of CGBH given by which satisfies the first law of BH thermodynamics, Hence, the corresponding Smarr relation is given by, Now, we can show the mass of BH in terms of S, a and k as follows, The BH entropy is given by, Next, the temperature of the CGBH is given by, where, r is the event horizon of CGBH. These equations will help us to calculate some important thermodynamical properties of CGBH. Figure 1 presented the behavior of mass (left panel) and temperature (right panel) versus horizon radius for various values of cosmological term. We observe that both mass and temperature is positive which shows the physical behavior for small values of cosmological term. One can determine the limits of cosmological term k from Fig. 1. We can conclude that mass and temperature become more physical (positive) for lower values of cosmological term and a = 1.

Thermal stability of the solutions
Here, we will discuss thermal stability by analyzing the roots and divergencies of the CGBH. The heat capacity is given by [62], The heat capacity of CGBH is given by, Next, we have plotted the heat capacity and temperature in Fig. 2 Moreover, the roots of temperature or zeros of heat capacity refers to bound points (BP). Here, positive and negative values of temperature show physical and non-physical solutions of BH, respectively. Note that BH has a physical solution between small and large roots. The roots are as follows [5], where, subscript BP shows bound points, to have real and positive roots of heat capacity then it must fulfill the following conditions − a 4 + 144k 2 > 0, a 2 ± 12k > 0. (20) Similarly, by taking the denominator of heat capacity equal to zero, one can calculate divergency roots referring to phase transition (r +,D P ) where, DP shows divergence points. The expression of these roots are very large, for simplicity we present the behavior of Divergency roots graphically in Fig. 7.
ss The behavior of conformal parameters a and k on the roots and divergency of heat capacity are presented in Figs. 3, 4, 5, 6 and 7. From Fig. 3, we observe that small and large roots show decreasing behavior w.r.t cosmological parameter k. It is clear that small and large root shows decreasing and increasing behavior w.r.t conformal parameter a, respectively in Fig. 4. Moreover, the behavior of roots in Eq. (19) are presented in Figs. 5 and 6. One can see that small and large root show decreasing behavior and middle root shows increasing behavior w.r.t 'k' and vice versa w.r.t 'a' as shown in Figs. 5 and 6, respectively.   Finally, we plotted the divergency root in Fig. 7 versus a (left panel) and k (right panel). One can observe that divergency shows decreasing behavior w.r.t a and k, respectively. Hence, conformal parameters play an important role in the stability of CGBH.

Thermodynamic geometries
The study of thermodynamical geometry explains the internal structure of a BH. Mostly we discuss the thermodynamics of different black holes under some kind of limitations. We do not discuss the irregularities of temperature, mass and heat capacity, etc. Although plenty of work has been done now for the study of different aspects of the geometry of BH under ordinary circumstances like Weinhold, Ruppeiner, HPEM, and GTD [65]. It is not necessary that a BH must show all geometries. Some BHs show all geometries like, Kiselve BH and some BHs show flat geometries. For example, both RN, BTZ BHs show flat Ruppeiner geometry. Also, Kerr BH shows flat Weinhold geometry. Flat Ruppeiner geometry shows that our mechanical model must be non-interacting. Firstly, we discuss the Weinhold metric of CGBH as follows [49], The corresponding line element of CGBH is represented by, and its matrix form, Using the above equations, one can find scalar curvature for CGBH but its mathematical expression is very large. So we presented it's graphical representation in Fig. 8. One can observe that the scalar curvature of Weinhold geometry of CGBH has no singularity so it provides no physical information. Interestingly, we also observed that the zeros of heat capacity and scalar curvature meets at r = 0.78. Hence, one can conclude that the Weinhold geometry is compatible with CGBH. Next, Ruppeiner geometry is quite related to Weinhold geometry which is given by, and the relevant matrix is [69][70][71][72].
One can calculate the scalar curvature for Ruppeiner geometry which is quite lengthy. So we presented the graphical behavior of Ruppeiner geometry in Fig. 9. We can see that the scalar curvature of Ruppeiner geometry for CGBH has only one singularity which coincide with the zeros of heat capacity, so it provides very useful information. Furthermore, we discuss the HPEM geometry for CGBH and it's metric can be represented as [72].
We have plotted the scalar curvature of HPEM geometry and heat capacity of CGBH in Fig. 10. One can analyze the divergence of scalar curvature of CGBH which has three singular points approximately at r + = 1.28, 2, 3. We observe that the scalar curvature of HPEM geometry exactly coincide with the zeros of heat capacity. Hence, HPEM geometry provide very important physical information and this scalar curvature is related to the divergency at the critical point and the correlation volume of the system. Finally, the GTD metric for the CGBH can be written as [73] The graphical behavior of scalar curvature of GTD geometry and heat capacity of CGBH is presented in Fig. 11. The divergency of scalar curvature of GTD geometry also coincide with zero of heat capacity. Hence, GTD geometry gives useful information aboout the internal structure of CGBH.

Conclusion
In this paper, the thermodynamical properties of the CGBH were discussed and observed that mass and temperature is physical in small horizon radius for lower values of cosmological term. For the system remain physical, one could find the limits of conformal parameters a = 1 and k = 0.1. Moreover, we studied the thermal stability of the CGBH and also discussed that the effect of conformal parameters on the bound points and divergency points. The behavior of small, middle and large roots were discussed. The small and large roots were decreasing function of k and increasing function of a, respectively. The middle roots were increasing function of a and k. The divergency roots were decreasing function of a and k. We analyzed the temperature and heat capacity and observed that for higher values of cosmological term (k = 0.5, 1), the CGBH was unstable for small horizon and it was stable for large horizon. For k = 0.1, one could notice that small BH was unstable at r < 1.6, intermediate BH was stable at 1.6 < r < 4.2 and large BH was stable for r > 4.2. The zero of temperature and heat capacity coin- Finally, the thermal geometries of CGBH by using Weinhold, HPEM, Ruppeiner and GTD formalism were presented. There was no divergency in Weinhold geometry, so it provided no useful information but the compatibility with CGBH were shown as the zero of heat capacity and scalar curvature coincided at r = 0.78. Moreover, the important information about the structure of CGBH was provided by Ruppeiner, HPEM and GTD geometries as the divergencies of scalar curvatures coincided with the zeros of heat capacities. Hence, the scalar curvature provided vital information about the nature of microscopic interaction as described above. Hence, these geometrical structures had provided the important insight into critical phenomena and phase structure or divergency of CGBH.

Data Availability Statement
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