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.     

Analytic phase structures and thermodynamic curvature for the charged AdS black hole in alternative phase space

In this paper, we visit the thermodynamic criticality and thermodynamic curvature of the charged AdS black hole in a new phase space. It is shown that when the square of the total charge of the charged black hole is considered as a thermodynamic quantity, the charged AdS black hole also admits a van der Waals-type critical behavior without the help of thermodynamic pressure and thermodynamic volume. Based on this, we study the fine phase structures of the charged AdS black hole with fixed AdS background in the new framework. On the one hand, we give the phase diagram structures of the charged AdS black hole accurately and analytically, which fills up the gap in dealing with the phase transition of the charged AdS black holes by taking the square of the charge as a thermodynamic quantity. On the other hand, we analyse the thermodynamic curvature of the black hole in two coordinate spaces. The thermodynamic curvatures obtained in two different coordinate spaces are equivalent to each other and are also positive. Based on an empirical conclusion under the framework of thermodynamic geometry, we speculate that when the square of charge is treated as an independent thermodynamic quantity, the charged AdS black hole is likely to present a repulsive between its molecules. More importantly, based on the thermodynamic curvature, we obtain a universal exponent at the critical point of phase transition.     

Temperature and thermodynamic geometry of the Kerr-Sen black hole

This paper studies the thermodynamic properties of the Kerr-Sen black hole from the viewpoint of geometry.It calculates the temperature and heat capacity of the black hole,Weinhold metric and Ruppeiner metric are also obtained respectively.It finds that they are both curved and the curvature scalar of Weinhold curvature implies no information about the phase transition while the Ruppeiner one does.But they both carry no information about the second-order phase transition point reproduced from the capacity.Besides,the Legendre invariant metric of the Kerr-Sen black hole has been discussed and its scalar curvature gives the information about the second-order phase transition point.     

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.     

Thermodynamics,phase transitions and Ruppeiner geometry for Einstein–dilaton–Lifshitz black holes in the presence of Maxwell and Born–Infeld electrodynamics

In this paper, we first obtain the higher-dimen-sional dilaton–Lifshitz black hole solutions in the presence of Born–Infeld (BI) electrodynamics. We find that there are two different solutions for the cases of \(z=n+1\) and \(z\ne n+1\) where z is the dynamical critical exponent and n is the number of spatial dimensions. Calculating the conserved and thermodynamical quantities, we show that the first law of thermodynamics is satisfied for both cases. Then we turn to the study of different phase transitions for our Lifshitz black holes. We start with the Hawking–Page phase transition and explore the effects of different parameters of our model on it for both linearly and BI charged cases. After that, we discuss the phase transitions inside the black holes. We present the improved Davies quantities and prove that the phase transition points shown by them are coincident with the Ruppeiner ones. We show that the zero temperature phase transitions are transitions in the radiance properties of black holes by using the Landau–Lifshitz theory of thermodynamic fluctuations. Next, we turn to the study of the Ruppeiner geometry (thermodynamic geometry) for our solutions. We investigate thermal stability, interaction type of possible black hole molecules and phase transitions of our solutions for linearly and BI charged cases separately. For the linearly charged case, we show that there are no phase transitions at finite temperature for the case \( z\ge 2\). For \(z<2\), it is found that the number of finite temperature phase transition points depends on the value of the black hole charge and there are not more than two. When we have two finite temperature phase transition points, there is no thermally stable black hole between these two points and we have discontinuous small/large black hole phase transitions. As expected, for small black holes, we observe finite magnitude for the Ruppeiner invariant, which shows the finite correlation between possible black hole molecules, while for large black holes, the correlation is very small. Finally, we study the Ruppeiner geometry and thermal stability of BI charged Lifshtiz black holes for different values of z. We observe that small black holes are thermally unstable in some situations. Also, the behavior of the correlation between possible black hole molecules for large black holes is the same as for the linearly charged case. In both the linearly and the BI charged cases, for some choices of the parameters, the black hole system behaves like a Van der Waals gas near the transition point.     

Thermodynamics and weak cosmic censorship conjecture of 4D Gauss-Bonnet-Maxwell black holes via charged particle absorption

Recently, the non-trivial solutions for 4-dimensional black holes of Einstein-Gauss-Bonnet gravity had been discovered. In this paper, considering a charged particle entering into a 4-dimensional Gauss-Bonnet-Maxwell black hole, we calculate the black hole thermodynamic properties using the Hamilton-Jacobi equation. In the normal phase space, the cosmological constant and Gauss-Bonnet parameter are fixed, the black hole satisfies the first and second laws of thermodynamics, and the weak cosmic censorship conjecture (WCCC) is valid. On the other hand, in the case of extended phase space, the cosmological constant and Gauss-Bonnet parameter are treated as thermodynamic variables. The black hole also satisfies the first law of thermodynamics. However, the increase or decrease in the black hole's entropy depends on some specific conditions. Finally, we observe that the WCCC is violated for the near-extremal black holes in the extended phase space.     

Extended Phase Space in the Framework of Holography

We consider a holographic extended phase space in the presence of Reissner-Nordstrom-Anti-de Sitter(RNAdS) and Born-Infeld-Anti-de Sitter(BI-AdS) black holes in the bulk. In this extended phase space the cosmological constant is investigated as pressure and volume is defined as the codimension one-time slice in the bulk geometry enclosed by the minimal area appearing in the computation of the holographic entanglement entropy. These thermodynamics quantities can serve as probes of the underlying phase transition dictated by black hole thermodynamics, but do not describe different structures. We find that the isocharges on the pressure-volume plane exhibit a Van der Waals-like structure, for RN-AdS black holes in the background. For BI-AdS black holes, we observe the analogy with a Van der Waals liquid-gas system for βQ 1/2 and Reentrant phase transition for βQ 1/2 in the holographic extended phase space. The same holographic thermodynamic behavior is observed when we use the fidelity susceptibility as the volume and the cosmological constant as the pressure for RN-AdS black hole in the background.     

Phase transition and thermodynamic stability in extended phase space and charged Hořava–Lifshitz black holes

For charged black holes in Ho?ava–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 electric charge on thermodynamic stability of Ho?ava–Lifshitz black hole. We also use thermodynamic geometry to study phase transition in extended phase space. We investigate the behavior of scalar curvature of Weinhold, Ruppeiner, and Quevedo metric in extended phase space of charged Ho?ava–Lifshitz black holes. It is checked if these curvatures could reproduce the result of specific heat for the phase transition.     

Observing dynamic oscillatory behavior of triple points among black hole thermodynamic phase transitions

Understanding the dynamic process of black hole thermodynamic phase transitions at a triple point is a huge challenge. In this paper, we conduct the first investigation of dynamic phase behavior at a black hole triple point. By numerically solving the Smoluchowski equation near the triple point for a six-dimensional charged Gauss-Bonnet anti-de Sitter black hole, we report that initial small, intermediate, or large black holes can transit to the other two coexistent phases at the triple point, indicating that thermodynamic phase transitions can indeed occur dynamically. More significantly, we observe characteristic weak and strong oscillatory behavior in this dynamic process, which can be understood from an investigation of the rate of first passage from one phase to another. Our results further an understanding of the dynamic process of black hole thermodynamic phase transitions.     

Geometrothermodynamics of five dimensional black holes in Einstein–Gauss–Bonnet theory

We investigate the thermodynamic properties of 5D static and spherically symmetric black holes in (i) Einstein–Maxwell–Gauss–Bonnet theory, (ii) Einstein–Maxwell–Gauss–Bonnet theory with negative cosmological constant, and in (iii) Einstein–Yang–Mills–Gauss–Bonnet theory. To formulate the thermodynamics of these black holes we use the Bekenstein–Hawking entropy relation and, alternatively, a modified entropy formula which follows from the first law of thermodynamics of black holes. The results of both approaches are not equivalent. Using the formalism of geometrothermodynamics, we introduce in the manifold of equilibrium states a Legendre invariant metric for each black hole and for each thermodynamic approach, and show that the thermodynamic curvature diverges at those points where the temperature vanishes and the heat capacity diverges.     

QED effects on phase transition and Ruppeiner geometry of Euler-Heisenberg-AdS black holes

Considering the quantum electrodynamics (QED) effect, we study the phase transition and Ruppeiner geometry of Euler-Heisenberg anti-de Sitter black holes in the extended phase space. For negative and small positive QED parameters, we observe a small/large black hole phase transition and reentrant phase transition, respectively, whereas a large positive value of the QED parameter ruins the phase transition. Phase diagrams for each case are explicitly shown. Then, we construct the Ruppeiner geometry in thermodynamic parameter space. Different features of the corresponding scalar curvature are shown for both the small/large black hole phase transition and reentrant phase transition cases. Of particular interest is the additional region of positive scalar curvature, indicating a dominant repulsive interaction among black hole microstructures, for the black hole with a small positive QED parameter. Furthermore, universal critical phenomena are observed for the scalar curvature of Ruppeiner geometry. These results indicate that the QED parameter has a crucial influence on the black hole phase transition and microstructure.     

f(R) Black Holes as Heat Engines

With the cosmological constant considered as a thermodynamic variable in the extended phase space, it is natural to study the thermodynamic cycles of the black hole, which is conjectured to be performed using renormalization group flow. We first investigate the thermodynamic cycles of a 4-dimensional asymptotically AdS f(R) black hole. Then we study the thermodynamic cycles of higher dimensional asymptotically AdS f(R) black holes. It is found that when ΔV ? ΔP, the efficiency of isobar-isochore cycles running between high temperature T_H and low temperature T_C will increase to its maximum value, which is exactly the Carnot cycles’ efficiency both in 4-dimensional and in higher dimensional cases. We speculate that this property is universal for AdS black holes, if there is no phase transition in the thermodynamic cycle. This result may deepen our understanding of the thermodynamics of the AdS black holes.     

Phase transition of quantum-corrected Schwarzschild black hole

We study the thermodynamic phase transition of a quantum-corrected Schwarzschild black hole. The modified metric affects the critical temperature which is slightly less than the conventional one. The space without black holes is not the hot flat space but the hot curved space due to vacuum fluctuations so that there appears a type of Gross–Perry–Yaffe phase transition even for the very small size of black hole, which is impossible for the thermodynamics of the conventional Schwarzschild black hole. We discuss physical consequences of the new phase transition in this framework.     

Thermodynamics and geometrothermodynamics of regular black hole with nonlinear electrodynamics

In this paper we investigate the phase transition and geometrothermodynamics of regular electrically charged black hole in nonlinear electrodynamics theory coupled to general relativity. We analyze the types of phase transition of the thermodynamic system by calculating its temperature, heat capacity, and free energy, etc. We find that there are secondorder phase transitions from the heat capacity for a large value of S. In addition, employing the geometrothermodynamics,we obtain a Legendre invariance metric and find the relationship between the thermodynamical phase transition and the singularity of the curvature scalar in the regular black hole with the nonlinear electrodynamics.     

Chaotic shadows of black holes: a short review

We give a brief review on the formation and the calculation of black hole shadows. Firstly, we introduce the concept of a black hole shadow and the current works on a variety of black hole shadows. Secondly, we present the main methods of calculating photon sphere radius and shadow radius, and then explain how the photon sphere affects the boundary of black hole shadows. We review the analytical calculation for black hole shadows which have analytic expressions for shadow boundary due to the integrable photon motion system. And we introduce the fundamental photon orbits which can explain the patterns of black hole shadow shape. Finally, we review the numerical calculation of black hole shadows with the backward ray-tracing method and introduce some chaotic black hole shadows with self-similar fractal structures. Since the gravitational waves from the merger of binary black holes have been detected, we introduce a couple of shadows of binary black holes, which all have eyebrowlike shadows around the main shadows with the fractal structures. We discuss the invariant phase space structures of the photon motion system in black hole space-time, and explain the formation of black hole shadow is dominated by the invariant manifolds of certain Lyapunov orbits near the fixed points.     

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.

     

Three-dimensional massive Kiselev AdS black hole and its thermodynamics

We present an exact three-dimensional massive Kiselev AdS black hole solution. This Kiselev black hole is neither perfectly fluid, nor is it the quintessential solution, but the BTZ black hole modified by the anisotropic matter. This black hole possesses an essential singularity at its radial origin and a single horizon whose radius will increase monotonically when the parameter of the anisotropic matter field ω decreases. We calculate all thermodynamic quantities and find that the first law of thermodynamics of this massive Kiselev AdS black hole can be protected, while the consistent Smarr formula is only held in the extended thermodynamic phase space. After examining the sign of free energy, we conclude that there is no Hawking-Page transition since the massive Kiselev AdS black hole phase is always thermodynamically favored. Moreover, we study the phase transition between the Kiselev AdS black hole and BTZ black hole by considering the matchings for their temperature. We find that the Kiselev AdS black hole is still a thermodynamically more preferred phase, because it always has a smaller amount of free energy than the BTZ black hole, which seems to indicate that the anisotropic matter field may emerge naturally in BTZ black hole spacetime under some thermal fluctuations. We also show a first order phase transition between the Kiselev AdS black hole phase with -1w -1/2 and the black hole phase with -1/2w0. As the Kiselev AdS black hole has some notable features on the phase transition of black holes in three dimensions, it provides important clues to further investigate these both surprising and similar behaviors in four and higher dimensions.     

Maxwell’s equal area law for black holes in power Maxwell invariant

In this paper, we consider the phase transition of black hole in power Maxwell invariant by means of Maxwell’s equal area law. First, we review and study the analogy of nonlinear charged black hole solutions with the Van der Waals gas–liquid system in the extended phase space, and obtain isothermal P-v diagram. Then, using the Maxwell’s equal area law we study the phase transition of AdS black hole with different temperatures. Finally, we extend the method to the black hole in the canonical (grand canonical) ensemble in which charge (potential) is fixed at infinity. Interestingly, we find the phase transition occurs in the both ensembles. We also study the effect of the parameters of the black hole on the two-phase coexistence. The results show that the black hole may go through a small-large phase transition similar to those of usual non-gravity thermodynamic systems.     

“Charged” particle’s tunneling from rotating black holes

The behavior of a scalar field theory near the event horizon in a rotating black hole background can be effectively described by a two dimensional field theory in a gauge field background. Based on this fact, we proposal that the quantum tunneling from rotating black hole can be treated as “charged” particle’s tunneling process in its effectively two dimensional metric. Using this viewpoint and considering the corresponding “gauge charge” conservation, we calculate the non-thermal tunneling rate of Kerr black hole and Myers–Perry black hole, and results are consistent with Parikh–Wilczek’s original result for spherically symmetric black holes. Especially for Myers–Perry black hole which has multi-rotation parameters, our calculation fills in the gap existing in the literature applying Parikh–Wilczek’s tunneling method to various types black holes. Our derivation further illuminates the essential role of effective gauge symmetry in Hawking radiation from rotating black holes.