Ruppeiner Curvature Research Articles

Attractive interactions in the microstructures of asymptotically flat black holes

In this work, we investigate the microstructure of asymptotically flat black holes with Ruppeiner curvature. Specially, the cosmological constant is considered to have a fluctuation around 0. Under such consideration, both repulsive and attractive interactions are found in the Reissner–Nordström and Kerr black holes, while the Schwarzschild black hole has dominant attractive interaction. The result obtained is quite different from that of excluding the fluctuation of cosmological constant, where these black holes are found to be always characterized by repulsive interaction.

Novel relations in massive gravity at Hawking-Page transition

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 this point again approaches a universal negative constant, independent of all parameters of the theory. The geometry of black holes in AdS close to the HP transition point is also studied via novel near horizon scaling limits, and reveals the emergence of fully decoupled Rindler spacetimes.

Ruppeiner curvature along a renormalization group flow

We report on an investigation of Ruppeiner curvature RU 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 RU 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.

Thermodynamic Curvature of AdS Black Holes with Dark Energy

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 energy parameterized by α, significantly alters the dominant interactions of neutral, charged and slowly rotating black hole microstructures. In the Schwarzschild-AdS case: black holes smaller than a certain size continue to have attractive interactions whereas larger black holes are completely dominated by repulsive interactions which arise to due dark energy. For charged or rotating AdS black holes with quintessence, R can change sign at multiple points depending upon the relation between α and charge q or angular momentum J. In particular, above a threshold value of α, R is never negative at all, suggesting heuristically that the repulsive interactions due to quintessence are long ranged as opposed to the previously known short ranged repulsion in charged AdS black holes. A mean field interaction potential is proposed whose extrema effectively capture the points where the curvature R changes sign.

Novel dual relation and constant in Hawking-Page phase transitions

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 treated as an indicator of the intensity of the interactions amongst black hole microstructures, we conjecture that this universal constant denotes an interaction threshold, beyond which a virtual black hole becomes a real one. This new dual relation and universal constant are fundamental in understanding Hawking-Page phase transitions, and might have new important applications in the black hole physics in the near future.

Thermodynamic geometry for charged Gauss-Bonnet black holes in AdS spacetimes

In this paper, we study the thermodynamic geometry of charged Gauss-Bonnet black holes (and Reissner-Nordstr\"om black holes, for the sake of comparison) in anti--de Sitter spacetimes 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-Bonnet coupling, and horizon radius of the black hole. The results are physically explained by considering that these black hole systems consist of charged as well as neutral microstructures much like a binary mixture of fluids.

Repulsive Interactions and Universal Properties of Charged Anti–de Sitter Black Hole Microstructures

The Ruppeiner geometry of thermodynamic fluctuations provides a powerful diagnostic of black hole microstructures. We investigate this for charged anti-de Sitter 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 interaction is independent of the black hole charge and temperature. The other is that the behavior of the Ruppeiner curvature scalar near criticality is characterized by a dimensionless constant that is identical to that for a van der Waals fluid, providing us with new insight into black hole microstructures.

The Thermodynamic Geometry and Phase Transition of the Plane Symmetric Black Hole

In this paper, we investigate the thermodynamic properties and the thermodynamic geometry of the plane symmetric black hole. We obtain the thermodynamic curvature based on the Weinhold geometry curvature, Ruppeiner geometry curvature and the Quevedo curvature. We find the Weinhold curvature always equals to zero and there is a phase transition point for the Ruppeiner curvature. The Quevedo curvature produces a same phase structure as the heat capacity.

Perfect Fluid Dark Matter Influence on Thermodynamics and Phase Transition for a Reissner-Nordstrom-Anti-de Sitter Black Hole

Based on Reissner-Nordstrom-anti-de Sitter(RN-AdS) black hole surrounded by perfect fluid dark matter, we study the thermodynamics and phase transition by extending the phase space defined by the charge square Q2 and the conjugate quantity ψ, where ψ is a function of horizon radius. The first law of thermodynamics and the equation of state are derived in the form Q2=Q2(T,ψ). By investigating the critical behaviour of perfect fluid dark matter around Reissner-Nordstrom-anti-de Sitter black hole, we find that these thermodynamics system are similar to Van der Waals system and can be explained by mean field theory. We also explore the Ruppeiner thermodynamic geometry feature and their connection with microscopic structure. We find that in extended phase space there are existence singularity points of Ruppeiner curvature and they could explained as phase transitions.

More insight into microscopic properties of RN-AdS black hole surrounded by quintessence via an alternative extended phase space

In this work, we study the phase transition of the charged-AdS black hole surrounded by quintessence via an alternative extended phase space defined by the charge square [Formula: see text] and her conjugate [Formula: see text], a quantity proportional to the inverse of horizon radius, while the cosmological constant is kept fixed. The equation of state is derived under the form [Formula: see text] and the critical behavior of such black hole analyzed. In addition, we examine the role of the quintessence parameter and its effects on phase transitions. Besides that, we explore the connection between the microscopic structure and Ruppeiner geothermodynamics. We also find that, at certain points of the phase space, the Ruppeiner curvature is characterised by the presence of singularities that are interpreted as a signal of the occurrence of the phase transitions.

Q − Φ criticality and microstructure of charged AdS black holes in f(R) gravity

The phase transition and critical behaviors of charged AdS black holes in [Formula: see text] 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 [Formula: see text] and contributes to deeper understanding of the thermodynamics and phase structure of black holes. Our analyses manifest that the charged [Formula: see text]-CIM AdS black hole undergoes a first-order small–large black hole phase transition, and the critical behaviors qualitatively behave like a Van der Waals liquid–vapor system. However, differing from the case in Einstein’s gravity, phase structures of the black holes in [Formula: see text] theory exhibit an interesting dependence on gravity modification parameters. Moreover, we adopt the thermodynamic geometry to probe the black hole microscopic properties. The results show that, on the one hand, both the Ruppeiner curvature and heat capacity diverge exactly at the critical point, on the other hand, the [Formula: see text]-CIM AdS black hole possesses the property as ideal Fermi gases. Of special interest, we discover a microscopic similarity between the black holes and a Van der Waals liquid–vapor system.

Thermodynamics of a two-dimensional charged black hole in the geometric framework

In this paper, we study the thermodynamic features of a two-dimensional charged black hole. Weinhold curvature and Ruppeiner curvature are explored as information geometry, respectively. Moreover, based on the Legendre invariant proposed by Hernando Quevedo, the geometro-thermodynamics behavior of this black hole is investigated.

Phase transition, critical behavior, and critical exponents of Myers-Perry black holes

The critical behavior of Myers-Perry black holes with equal angular momenta in even dimensions are studied. We include the corrections beyond the semiclassical approximation on Hawking temperature in the grand canonical ensemble. Having done so, we find that the critical behavior and critical exponents of Myers-Perry black holes correspond to those of a Van der Waals liquid-gas where this analogy holds in any dimension. Also, using Ehrenfest's equations, we calculate the order of the phase transition in the semiclassical approximation for the canonical ensemble and beyond the semiclassical approximation for the grand canonical ensemble near the critical point. Finally, the Ruppeiner curvature formula is used to investigate the thermodynamic geometry of Myers-Perry black holes.

Critical phenomena and thermodynamic geometry of charged Gauss-Bonnet AdS black holes

In this paper, we study the phase structure and equilibrium state space geometry of charged topological Gauss-Bonnet black holes in $d$-dimensional anti-de Sitter spacetime. Several critical points are obtained in the canonical ensemble, and the critical phenomena and critical exponents near them are examined. We find that the phase structures and critical phenomena drastically depend on the cosmological constant $\Lambda$ and dimensionality $d$. The result also shows that there exists an analogy between the black hole and the van der Waals liquid gas system. Moreover, we explore the phase transition and possible property of the microstructure using the state space geometry. It is found that the Ruppeiner curvature diverges exactly at the points where the heat capacity at constant charge of the black hole diverges. This black hole is also found to be a multiple system, i.e., it is similar to the ideal gas of fermions in some range of the parameters, while to the ideal gas of bosons in another range.

Thermodynamic and geometric framework of a (2+1)-dimensional black hole with non-linear electrodynamics

The thermodynamic properties of a (2 + 1)-dimensional black hole with non-linear electrodynamics from the viewpoint of geometry is studied and some kinds of temperatures of the black hole have been obtained. Weinhold curvature and Ruppeiner curvature are explored as information geometry. Moreover, based on Quevedo's theory, the Legendre invariant geometry is investigated for the black hole. We also study the relationship between the scalar curvatures of the above several metrics and the phase transitions produced from the heat capacity.

On the extremality and ultraspinning instability of Myers-Perry black holes

We use a contact geometric method to study Myers-Perry (MP) black holes in arbitrary dimensions with arbitrary angular momenta. We have shown that the MP black holes of dimension d with n equal nonzero spins and 2n ≥ d − 3 all have extremal limits as expected and that we should classify MP black holes in three series depending on whether the value of 2n − d + 3 is 0, 1 or 2. For black holes with 2n < d − 3 the Ruppeiner curvature diverges although they have no extremal limits. In order to have an ultraspinning mode at least one spin of the MP black hole must be set to zero. Our result agrees with others in the literature where the authors are able to establish the minimum temperature surface on which the membrane phase of ultraspinning MP black holes occurs. We conjecture that the membrane phase of ultraspinning MP black holes is reached around the minimum temperature in the case 2n < d − 3 which is where the Ruppeiner curvature diverges.

Ruppeiner Geometry of (2 + 1)-Dimensional Spinning Dilaton Black Hole

In this paper, we study the geometrothermodynamics of(2 + 1)-dimensional spinning dilaton black hole. We show that the Ruppeiner curvature vanishes, which implies that there exist no phase transitions and thermodynamic interactions. However when the thermodynamics fluctuation is included, the geometry structure is reconsidered. The non-vanishing Ruppeiner curvature is obtained, which means the phase space is non-flat. We also study the phase transitions and show that it can indeed take place at some points.

Ruppeiner geometry and 2D dilaton gravity in the thermodynamics of black holes

We study the geometric approach to the black hole thermodynamics. The geometric description of the equilibrium thermodynamics comes from Ruppeiner geometry based on a metric on the thermodynamic state space. For this purpose, we consider the Reissner–Nordström-AdS (RN-AdS) black hole which provides two different ensembles: canonical ensemble for fixed-charge case and grand canonical ensemble for fixed-potential case. Two cases are independent and cannot be mixed into each other. Hence, we calculate different Ruppeiner curvatures for two ensembles. However, we could not find the consistent behaviors of Ruppeiner curvature corresponding to those of heat capacity. Alternatively, we propose the curvature scalar in the 2D dilaton gravity approach which shows the features of extremal, Davies and minimum temperature points of RN-AdS black hole, clearly.

Ruppeiner geometry of RN black holes: flat or curved?

In some recent studies [1]–[3], Aman et al. used the Ruppeiner scalar as a measure of underlying interactions of Reissner-Nordström black holes, indicating that it is a non-interacting statistical system for which classical thermodynamics could be used at any scale. Here, we show that if we use the complete set of thermodynamic variables, a non-flat state space will be produced. Furthermore, the Ruppeiner curvature diverges at extremal limits, as it would for other types of black holes.

Geometry of higher-dimensional black hole thermodynamics

We investigate thermodynamic curvatures of the Kerr and Reissner-Nordstr\"om (RN) black holes in spacetime dimensions higher than four. These black holes possess thermodynamic geometries similar to those in four dimensional spacetime. The thermodynamic geometries are the Ruppeiner geometry and the conformally related Weinhold geometry. The Ruppeiner geometry for $d=5$ Kerr black hole is curved and divergent in the extremal limit. For $d \geq 6$ Kerr black hole there is no extremality but the Ruppeiner curvature diverges where one suspects that the black hole becomes unstable. The Weinhold geometry of the Kerr black hole in arbitrary dimension is a flat geometry. For RN black hole the Ruppeiner geometry is flat in all spacetime dimensions, whereas its Weinhold geometry is curved. In $d \geq 5$ the Kerr black hole can possess more than one angular momentum. Finally we discuss the Ruppeiner geometry for the Kerr black hole in $d=5$ with double angular momenta.