We report on an investigation of Ruppeiner curvature $R_U$ for $(d+1)$-dimensional hyperbolic black holes in anti-de Sitter spacetimes and its connection with Renormalization Group (RG) flows in the dual d-dimensional conformal field theories (CFTs) in Minkowski spacetimes. A repulsive type interaction among microstructures is found, which is weaker for positive mass black holes and grows stronger for negative mass black holes at low temperatures. In particular, we show that $R_U$ evaluated along a zero mass curve is a universal constant, depending only on the dimension of space-time. The extremal black holes are pointed out to have a positive Ruppeiner curvature irrespective of their horizon topology.
Taking the quantum electrodynamics (QED) effect into account, we study the black hole phase transition and Ruppeiner geometry for the Euler-Heisenberg anti-de Sitter black hole in the extended phase space. For negative and small positive QED parameter, we observe a small/large black hole phase transition and reentrant phase transition, respectively. While a large positive value of the QED parameter ruins the phase transition. The phase diagrams for each case are explicitly exhibited. Then we construct the Ruppeiner geometry in the thermodynamic parameter space. Different features of the corres…
Due to the non-independence of entropy and thermodynamic volume for spherically symmetric black holes in the AdS spacetime, when applying the Ruppeiner thermodynamic geometry theory to these black holes, we often encounter an unavoidable problem of the singularity about the line element of thermodynamic geometry. In this paper, we propose a basic and natural scheme for dealing with the thermodynamic geometry of spherically symmetric AdS black holes. We point out that enthalpy, not internal energy, is the fundamental thermodynamic characteristic function for the Ruppeiner thermodynamic geometry…
Originally considered for van der Waals fluids and charged black holes [Phys. Rev. Lett. 123, 071103 (2019)], we extend and generalize our approach to higher-dimensional charged AdS black holes. Beginning with thermodynamic fluctuations, we construct the line element of the Ruppeiner geometry and obtain a universal formula for the scalar curvature $R$. We first review the thermodynamics of a van der Waals fluid and calculate the coexistence and spinodal curves. From this we are able to clearly display the phase diagram. Notwithstanding the invalidity of the equation of state in the coexistence…
The connection between the shadow radius and the Ruppeiner geometry of a charged static spherically symmetric black hole is investigated. The normalized curvature scalar is adopted, and its close relation to the Van der Waals-like and Hawking-Page phase transition of Reissner-Nordström AdS black hole is studied. The results show that the shadow radius is a useful tool to reveal the correct information of the phase structure and the underlying microstructure of the black hole, which opens a new window to investigate the strong gravity system from the observational point of view.
Thermodynamic fluctuation metrics in Ruppeiner's formalism are worked out for Kerr-AdS black holes in the extended state space. The implications of constraints upon the state space geometry and their correspondence with thermodynamical ensembles are explicitly worked out in the most general setting. The state space scalar curvature for a given ensemble is found to be sensitive to the instabilities/phase transitions therein. In particular, it is found that the appropriate Ruppeiner scalar curvature does encode critical phenomena in the Kerr-AdS black holes. A detailed study is undertaken of the…
Using the new normalized thermodynamic scalar curvature, we investigate the microstructures and phase transitions of black holes in massive gravity for horizons of various topologies. We find that the graviton mass enhances the repulsive interactions of small black holes and weakens the attractive interactions of large black holes, with possiblity of new repulsive regions for microstructures in phase space. In addition, the repulsive interactions of small black hole are strong for spherical topology, followed by flat and hyperbolic topology; while, the attractive interactions of large black ho…
In this paper, we analytically study the phase structure and construct the Ruppeiner geometry in the extended phase space for the five-dimensional neutral Gauss-Bonnet AdS black hole. Through calculating the scalar curvature of the Ruppeiner geometry and combining the phase transition, we show that the attractive interaction is dominant in the microstructure of the black hole system. More significantly, there is an intriguing property that the normalized scalar curvature has the same expression for the saturated small and large black hole curves. This implies that although the microstructure i…
Topological Signatures and Geometrothermodynamics of Critical Phenomena in Regularized Maxwell Black Holes
Khazar University（哈萨尔大学）；Chitkara University Institute of Engineering and Technology, Chitkara University（奇特卡拉大学工程与技术学院，奇特卡拉大学）；Institute of Nuclear Physics（核物理研究所）；Department of Mathematical and Physical Sciences, College of Arts and Sciences, University of Nizwa（尼兹瓦大学文理学院数学与物理科学系）；Departamento de Química, Física y Ciencias Ambientales y del Suelo, Escuela Politécnica Superior – Lleida, Universidad de Lleida（莱里达大学高等理工学院化学、物理与环境及土壤科学系）；Urgench State University（乌尔根奇国立大学）；University of Tashkent for Applied Sciences（塔什干应用科学大学）；National University of Uzbekistan（乌兹别克斯坦国立大学）；Tashkent State Technical University（塔什干国立技术大学）
Universal relations and constants have important applications in understanding a physical theory. In this article, we explore this issue for Hawking-Page phase transitions in Schwarzschild anti-de Sitter black holes. We find a novel exact dual relation between the minimum temperature of the ($d$+1)-dimensional black hole and the Hawking-Page phase transition temperature in $d$ dimensions, reminiscent of the holographic principle. Furthermore, we find that the normalized Ruppeiner scalar curvature is a universal constant at the Hawking-Page transition point. Since the Ruppeiner curvature can be…
Two-phase transition branches of the Euler-Heisenberg (EH) anti-de Sitter (AdS) black hole (BH) were derived from its phase transition critical behavior by Magos et al. [Phys. Rev. D. 102, 084011 (2020)]. We found that the phase transition is unstable. Considering the high-order quantum electrodynamics (QED) correction, we re-derive the EHAdS BH solution and investigate its critical thermodynamic quantities. It is found that the corrected EHAdS BH has only one stable phase transition branch, and its critical exponents are equivalent to that of the vdW system. From the microscopic point of view…
In this paper, we study the small-large black hole phase transition and construct the Ruppeiner geometry for the five-dimensional charged Gauss-Bonnet-AdS black hole in the grand canonical ensemble. By making use of the equal area law, we obtain the analytical coexistence curve of the small and large black holes. Then the phase diagrams are examined. We also calculate the change of the thermodynamic volume during the small-large phase transition, which indicates that there exists a sudden change among the black hole microstructures. The corresponding normalized scalar curvature of the Ruppeine…
Thermodynamic geometry analysis of interesting Letter [Phys. Rev. Lett. 115, 111302 (2015)] for charged AdS black holes, which is based on studying the Ruppeiner invariant behavior is not correct and the authors made a mistake in calculating this quantity. In the present Letter, we address the correct Ruppeiner scalar curvature and reveal the correct possible microscopic properties of 4-dimensional charged AdS black holes arise from it. Some of these properties have not been discussed in pointed out Letter.
In this paper, we study the thermodynamic geometry of charged Gauss-Bonnet black holes (and Reissner-Nordström black holes, for the sake of comparison) in AdS: in both $(T,V)$- and $(S,P)$-planes. The thermodynamic phase space is known to have an underlying contact and metric structure; Ruppeiner geometry then naturally arises in this framework. Sign of Ruppeiner curvature can be used to probe the nature of interactions between the black hole microstructures. It is found that there are both attraction and repulsion dominated regions which are in general determined by the electric charge, Gauss…
The Ruppeiner geometry of thermodynamic fluctuations provides a powerful diagnostic of black hole microstructures. We investigate this for charged AdS black holes and find that while an attractive microstructure interaction dominates for most parameter ranges, a weak repulsive interaction dominates for small black holes of high temperature. This unique property distinguishes the black hole system from that of a Van der Waals fluid, where only attractive microstructure interactions are found. We also find two other novel universal properties for charged black holes. One is that the repulsive in…
In this paper, we study thermodynamics and thermodynamic geometry of a black hole surrounded by the perfect fluid in Rastall theory. In particular, we calculate the physical quantity like mass, temperature and heat capacity of the system for two different cases. From the resulting heat capacity, we emphasize stability of the system. Following Weinhold, Ruppiner, Quevedo and HPEM formalism, thermodynamic geometry of this black hole in Rastall gravity is also analyzed. We find that the singular points of the curvature scalar of Ruppeiner and HPEM metrics entirely coincides with zero points of th…
In this paper, we consider a static black hole in $f(R)$ gravity. We recapitulate the expression for corrected thermodynamic entropy of this black hole due to small fluctuations around equilibrium. Also, we study the geometrothermodynamics (GTD) of this black hole and investigate the adaptability of the curvature scalar of geothermodynamic methods with phase transition points of this black hole. Moreover, we study the effect of correction parameter on thermodynamic behaviour of this black hole. We observe that the singular point of the curvature scalar of Ruppeiner metric coincides completely …
In the AdS/CFT correspondence, a dynamical cosmological constant $Λ$ in the bulk corresponds to varying the number of colors $N$ in the boundary gauge theory with a chemical potential $μ$ as its thermodynamic conjugate. In this work, within the context of Schwarzschild black holes in $AdS_5 \times S^5$ and its dual finite temperature $\mathcal{N}=4$ superconformal Yang-Mills theory at large $N$, we investigate thermodynamic geometry through the behavior of the Ruppeiner scalar $R$. The sign of $R$ is an empirical indicator of the nature of microscopic interactions and is found to be negative f…
In this paper, we study the effect of dark energy on the extended thermodynamic structure and interacting microstructures of black holes in AdS, through an analysis of thermodynamic geometry. Considering various limiting cases of the novel equation of state obtained in charged rotating black holes with quintessence, and taking enthalpy $H$ as the key potential in the extended phase space, we scrutinize the behavior of the Ruppeiner curvature scalar $R$ in the entropy-pressure $(S,P)$-plane (or equivalently in the temperature-volume ($T,V$)-plane). Analysis of $R$ empirically reveals that dark …
We implement a universal method for renormalizing AdS gravity actions applicable to arbitrary higher curvature theories in up to five dimensions. The renormalization procedure considers the extrinsic counterterm for Einstein-AdS gravity given by the Kounterterms scheme, but with a theory-dependent coupling constant that is fixed by the requirement of renormalization for the vacuum solution. This method is shown to work for a generic higher curvature gravity with arbitrary couplings except for a zero measure subset, which includes well-known examples where the asymptotic behavior is modified an…
We investigate the relationship between shadow radii (photosphere radii) and black hole microstructures for a static spherically symmetric black hole and confirm their close connection. As a concrete analysis, we take the Reissner-Nordstr$\mathrm{\ddot{o}}$m AdS black hole surrounded by perfect fluid dark matter (RN AdS black hole surrounded by PFDM) as an example. We calculate the Ruppeiner thermodynamic scalar curvature and shadow radius (photosphere radius) for the specific model. On the one hand, we find that the greater density of the perfect fluid dark matter makes black hole molecules m…
Motivated by a recent work on asymptotically Ad$S_4$ black holes in M-theory, we investigate both thermodynamics and thermodynamical geometry of Raissner-Nordstrom-AdS black holes from M2-branes. More precisely, we study AdS black holes in $AdS_{4}\times S^{7}$, with the number of M2-branes interpreted as a thermodynamical variable. In this context, we calculate various thermodynamical quantities including the chemical potential, and examine their phase transitions along with the corresponding stability behaviors. In addition, we also evaluate the thermodynamical curvatures of the Weinhold, Ru…
For charged black holes in Horava-Lifshitz gravity, a second order phase transition takes place in extended phase space where the cosmological constant is taken as thermodynamic pressure. We relate the second order nature of phase transition to the fact that the phase transition occurs at a sharp temperature and not over a temperature interval. Once we know the continuity of the first derivatives of the Gibbs free energy, we show that all the Ehrenfest equations are readily satisfied. We study the effect of the perturbation of the cosmological constant as well as the perturbation of the electr…
Combining with the small-large black hole phase transition, the thermodynamic geometry has been well applied to study the microstructure for the charged AdS black hole. In this paper, we extend the geometric approach to the rotating Kerr-AdS black hole and aim to develop a general approach for the Kerr-AdS black hole. Treating the entropy and pressure as the fluctuation coordinates, we construct the Ruppeiner geometry for the Kerr-AdS black hole by making the use of the Christodoulou-Ruffini-like squared-mass formula, which is quite different from the charged case. Employing the empirical obse…
Considering the unexpected similarity between the thermodynamic features of charged AdS black holes and that of the van der Waals fluid system, we calculate the number densities of black hole micromolecules and derive the thermodynamic scalar curvature for the small and large black holes on the co-existence curve based on the so-called Ruppeiner thermodynamic geometry. We reveal that the microscopic feature of the small black hole perfectly matches that of the ideal anyon gas, and that the microscopic feature of the large black hole matches that of the ideal Bose gas. More importantly, we inve…
Real-Space renormalization group techniques are developed for tackling large curvature fluctuations in quantum gravity. Within cells of invariant volume $a^4$, only certain types of fluctuations are allowed. Normal coordinates are used to avoid redundancy of the degrees of freedom. The relevant integration measure is read off from the metric on metrics. All fluctuations in a group of cells are averaged over to get an effective action for the larger cell. In this paper the simplest type of fluctuations are kept. The measure is simply an integration over independent components of the curvature t…
The phase transition and critical behaviours of charged AdS black holes in $f(R)$ gravity with a conformally invariant Maxwell (CIM) source and constant curvature are further investigated. As a highlight, this research is carried out by employing new state parameters $(T,Q,Φ)$ and contributes to deeper understanding the thermodynamics and phase structure of black holes. Our analyses manifest that the charged $f(R)$-CIM AdS black hole undergoes a first order small-large black hole phase transition, and the critical behaviours qualitatively behave like a Van der Waals liquid-vapor system. Howeve…
We take a bottom-up approach to find the interaction potential between the $AdS$ black hole molecules under mean-field approximation. We start with the equation of state of dyonic $AdS$ black holes in fixed charge ensemble and use the method of classical cluster expansion to find the mean-field potential. We show that the Lennard-Jones (LJ) potential is a feasible choice to describe the equation of state. The LJ potential describes a two-body interaction. There exists a critical distance $r_0$ such that two interacting particles repel (attract) each other for $r < r_0$ ($r > r_0$). We compute …
We study non-linear sigma models on target manifolds with constant (positive or negative) curvature using the functional renormalization group and the background field method. We pay particular attention to the splitting Ward identities associated to the invariance under reparametrization of the background field. Implementing these Ward identities imposes to use the curvature as a formal expansion parameter, which allows us to close the flow equation of the (scale-dependent) effective action consistently to first order in the curvature. We shed new light on previous work using the background f…
Sign and magnitude of Ruppeiner's curvature $R_N$ is an empirical indicator of the respective nature and strength of microstructures of a thermodynamic system. For d-dimensional Schwarzschild black holes in AdS, $R_N$ at the Hawking-Page (HP) transition point is a universal constant. In this article we study the nature of $R_N$ at HP transition in a massive gravity theory and show that its constancy is broken due to the dependence on graviton mass as well as horizon topology. Although, when the graviton mass reaches a critical value the HP transition is driven to zero temperature, and $R_N$ at…
With the help of the parametric solution of the Maxwell equal area law for the Gauss-Bonnet AdS black hole in five dimensions, we find the second analytical solution to the first order phase transition. We analyze the asymptotic behaviors of some characteristic thermodynamic properties for the small and large black holes at the critical and zero temperatures and also calculate the critical exponents and the corresponding critical amplitudes in detail. Moreover, we give the general form of the thermodynamic scalar curvature based on the Ruppeiner geometry and point out that the attractive inter…
