Consequences of thermal geometries in Brane-World black holes

Abstract

The implications of thermal geometries on Brane-World black hole (consisting of cosmological and dark matter effects) will be investigated by extracting the stability conditions (small and large roots) and divergences. To achieve this, we develop the temperature and heat capacity with respect to horizon radius, cosmological and dark matter parameters, respectively, which help in analyzing the small and large roots (stability conditions) and divergences. Furthermore, we study various methods of thermodynamical geometry such as Weinhold, Ruppeiner, Hendi–Panahiyah–Eslam–Momennia (HPEM), and Geometrothermodynamics (GTD) formulations and derive corresponding scalar curvatures for Brane-World black hole. It is found that Ruppeiner and HPEM formulations provide more information as compared to Weinhold and GTD methods for Brane-World black holes.

Introduction

The black hole (BH) thermodynamics has been an exciting and active research field with fascinating results since the work of Bekenstein [1], [2], Hawking [3], [4], [5], and other scientists [6], [7]. As we know that the temperature and entropy are the thermodynamical quantities that are associated with its geometrical quantities such as the surface gravity and horizon area. Semiclassical analysis has claimed that BHs are thermally unstable due to the emission of Hawking radiations. Hence, the temperature of BH increases when its size decreases which is a complete thermal runway process. A lot of other interesting aspects have been revealed on the thermodynamical properties of BHs and one of them is the thermal stability of BHs. As the thermal stability is one of the most important thermodynamic property which shows the behavior of the system after small deviation in thermodynamic parameters. There are numerous techniques for studying the phase structure of a BH system near a critical point [8]. The specific heat is a well-known standard analysis of thermal stability. The heat capacity must be positive which is physically represented as the system is thermally stable [9], [10], [11]. The heat capacity of BH provides a direction to discuss the phase transition [10], [11], [12], [13], [14], [15]. There are two types of phase transition; roots of heat capacity represented as first type of phase transition while divergency of heat capacity is type two-phase transition. One can analyze the comprehensive reasons for irregularities of temperature and specific heat in the case of normal thermodynamics of BHs.

In a completely different context, a geometrical approach to thermodynamics and phase transitions has been developed by many authors. Hermann [16] presented a differential manifold as an entanglement of thermodynamic phase space with a natural contact structure of subspace in equilibrium states. In this context, Weinhold proposed another geometrical method [17] by defining a metric of equilibrium state in thermodynamic space and used an approach of conformal mapping from Riemannian to thermodynamic space which involves geometry of thermodynamic fluctuation. Interestingly, there are two equilibrium states connected with the positive definite line interval in this geometry and one of them is the probability distribution of thermodynamic fluctuation in Gaussian approximation. One can argue that the interactions in the underlying microscopic statistical basis are encoded in the scalar curvature of thermodynamic quantities arising from this geometry [18], [19]. Therefore, the scalar curvature is related to the divergency at the critical point and the correlation volume of the system.

It is visualized that a thermodynamical scalar curvature is a geometrical approach that describes the macroscopic structure of the system and connects it to the microscopic structure by utilizing Gaussian fluctuation. In this context, the thermodynamic scalar curvature provides information about nature of microscopic interaction as described in [20], [21]. This geometrical framework has provided important insight into divergency or phase structure and critical phenomena for BHs [22], [23], [24], [25], [26], [27], [28], [29], [30], [31].

Moreover, scalar curvature also provides thermal stability to the system (For direct illustration see [32], [33], [34]). For this purpose, Weinhold introduces a metric for curvature scalar which is interpreted as Hessian metric in mass representation as [21]:

$$g_{ij}^W={\partial }_i{\partial }_{j}M(S,N^r),$$

where $S$ is entropy, $M$ is the mass, and other extensive parameters of the system represented as $N^r$. Hereafter, Ruppeiner [33], [34] defined a new formalism with the negative value of Hessian metric in the entropy representation which is given by

$$g_{ij}^R=-{\partial }_i{\partial }_{j}S(M,N^r).$$

The metric of Weinhold geometry is conformally related to the metric of Ruppeiner formalism as follows [35].

$$d{s}_R^2=\frac{1}{T}d{s}_W^2,$$

here $T$ is represented as the temperature of the thermodynamic system.

Furthermore, HPEM [36], [37], [38] and GTD is the latest approach in this direction [39], [40], [41], [42], [43]. The generalized HPEM metric with “n” extensive variable $(n\ge 2)$ is given by

$$d{s}^2=\frac{S{M}_S}{{\left(\prod _{i=2}^n\frac{{\partial }^{2}M}{\partial {\xi }_i^2}\right)}^3}(-M_{SS}d{S}^2+{\Sigma }_{i=2}^n\left(\frac{{\partial }^{2}M}{\partial {\xi }_i^2}\right)d{\xi }_i^2),$$

in which, ${\xi }_i\ne S$ and $M_S$ represent the derivative of mass M w.r.t. entropy $S$. Moreover, the generic form of the metric in GTD formalism has the following form

$$g=\left(E^c\frac{\partial \varphi }{\partial E^c}\right)\left({\eta }_{ab}{\delta }^{bc}\frac{{\partial }^2\varphi }{\partial E^c\partial E^d}d{E}^{a}d{E}^d\right),$$

in which

$$\frac{\partial \varphi }{\partial E^c}={\delta }_{bc}{I}^b,$$

where $I^b$, $E^a$ and $\varphi$ are the intensive, extensive thermodynamic parameters and thermodynamic potential, respectively.

The main objective of this manuscript is to study the roots and divergencies of Brane-World BH. For this purpose, we analyze the scalar curvature of different thermodynamical geometries. Interestingly, the scalar curvature arising from the proposed geometrical structure and singular points matches well with the results of heat capacity which provides a useful physical information about the nature of microscopic interactions for Brane-World BHs. After the introduction, the rest of the paper is as follows: In Section 2, we briefly review a Brane-World BH and its thermodynamical properties. Thermal stability and divergency criteria for Brane-World BH are discussed in Section 3 and Section 4 is dedicated to studying the thermodynamical geometry for Brane-World BH and physical interpretation of the results are discussed in the same section. Concluding remarks of the manuscript are given in Section 5.