Thermodynamic properties of charged three-dimensional black holes in the scalar-tensor gravity theory

The action of three-dimensional charged Einstein-dilaton gravity theory has been obtained from that of scalar-tensor modified gravity theory by utilizing the suitable conformal transformations. The field equations of the Einstein-dilaton gravity coupled to the power Maxwell nonlinear electrodynamics have been solved and two new classes of static and spherically symmetric charged dilatonic black holes, as the exact solutions to the coupled scalar, electromagnetic and gravitational field equations, have been obtained. Also, the dilaton potential has been written as the linear combination of two Liouville-type potentials. The black hole conserved charges and thermodynamic quantities have been calculated by utilizing the geometrical and thermodynamical methods, separately. The compatibility of the results obtained from these two alternative approaches confirms the validity of the first law of black hole thermodynamics for both of the new black hole solutions in the Einstein frame. A black hole stability or phase transition analysis has been performed in the context of the canonical ensemble. By calculating the black hole heat capacity, with the black hole charge as a constant, the type one and type two phase transition points have been determined. Also, the ranges of the black hole horizon radii at which the Einstein black holes are thermally stable have been identified for both of the new black hole solutions. Then making use of the inverse conformal transformations, two new classes of the scalar-tensor black holes have been obtained from their Einstein frame counterparts. The thermodynamic properties and thermal stability of the new scalar-tensor black holes have been investigated. It has been found that the new charged black holes have the same thermodynamic behaviors in both of the Einstein and Jordan frames.

By applying conformal transformations on the action of scalar–tensor-Euler–Heisenberg theory, we obtain the exact black hole (BH) solutions in its conformal related, the well-known Einstein frame. Through imposing the conditions of (a) vanishing the electric potential at large distance from the source and (b) validity of the first law of BH thermodynamics, we obtain a set of three requirements which are not consistent, mathematically. For solving this problem we assume that under conformal transformations the nonlinearity parameter of Euler–Heisenberg (EH) electrodynamics must transform as a→ae4αϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\rightarrow ae^{4\\alpha \\phi }$$\\end{document}. Then, we obtain the exact solutions of this theory, in both of Einstein and Jordan frames, without any mathematical problems. After calculating thermodynamic quantities, we investigate validity of the thermodynamical first law (TFL) and thermal stability of the EH-BTZ BHs in both of Jordan and Einstein frames, separately.

Starting from the action of the scalar-tensor modified gravity theory coupled to the power law nonlinear electrodynamics and making use of the suitable transformation relations, the corresponding action of the Einstein-power-Maxwell-dilaton gravity theory has been obtained. Thermodynamical properties of the new charged dilatonic black holes have been investigated in the presence of a power Maxwell field as the nonlinear electrodynamics. Making use of a special form of a scalar field, it has been found that the scalar potential can be written as the linear combination of three Liouville-type potentials. Three new classes of spherically symmetric charged dilatonic black hole solutions have been obtained as the exact solutions to the field equations of the Einstein-power-Maxwell-dilaton theory. The conserved and thermodynamical quantities, related to the new black hole solutions, have been calculated from geometrical and thermodynamical approaches separately. Through comparison of the results obtained from these two alternative approaches, the validity of the first law of black hole thermodynamics has been confirmed for either of three new black hole solutions. Thermal stability or phase transition of the new black hole solutions has been analyzed, making use of the canonical ensemble method, and by calculating the black hole heat capacity at the fixed black hole charge. The points of type one and type two phase transitions as well as the ranges at which the black holes are locally stable have been determined precisely. The nonlinearly charged string black hole solutions have been obtained from their Einstein's counterparts. The thermodynamical properties as well as the thermal stability of the new charged string black holes have been investigated by utilizing the canonical ensemble method.

In this paper, we present the charged dilaton solutions and black hole formation in three dimensions. Firstly we revisit the famous Chan–Mann charged dilaton black hole, describing one parameter family of static charged black hole solution in three dimensional Einstein–Maxwell–Dilaton gravity. This solution with a special parameter choice b=4a can lead to the three dimensional string solution. Then we give another class of charged dilaton black hole solution with bne 4a. We discuss their geometrical properties, the horizon structure and the causal structure. The time-dependent solution is also presented, which can characterize the three dimensional charged black hole formation in Einstein–Dilaton gravity. Especially, there is no exact time-dependent solution describing the gravitational collapse to the Chan–Mann charged dilaton black hole. Finally, we discuss the gravitational collapse of a dilaton field in the context of the Cosmic Censorship Conjecture.

The duality of gravitational dynamics (projected on a null hypersurface) and of fluid dynamics is investigated for the scalar tensor (ST) theory of gravity. The description of ST gravity, in both Einstein and Jordan frames, is analyzed from fluid-gravity viewpoint. In the Einstein frame the dynamical equation for the metric leads to the Damour-Navier- Stokes (DNS) equation with an external forcing term, coming from the scalar field in ST gravity. In the Jordan frame the situation is more subtle. We observe that finding the DNS equation in this frame can lead to two pictures. In one picture, the usual DNS equation is modified by a Coriolis-like force term, which originates completely from the presence of a non-minimally coupled scalar field (ϕ) on the gravity side. Moreover, the identified fluid variables are no longer conformally equivalent with those in the Einstein frame. However, this picture is consistent with the saturation of Kovtun-Son-Starinets (KSS) bound. In the other picture, we find the standard DNS equation (i.e. without the Coriolis-like force), with the fluid variables conformally equivalent with those in Einstein frame. But, the second picture, may not agree with the KSS bound for some values of ϕ. We conclude by rewriting the Raychaudhuri equation and the tidal force equation in terms of the relevant parameters to demonstrate how the expansion scalar and the shear-tensor evolve in the spacetime. Although, the area law of entropy is broken in ST gravity, we show that the rewritten form of Raychaudhuri’s equation correctly results in the generalized second law of black hole thermodynamics.

In this work, the charged black hole solution to the Brans-Dicke gravity theory in the presence of the nonlinear electrodynamics has been investigated. To simplify the field equations, a suitable conformal transformation has been used which transforms the Brans-Dicke-Born-Infeld Lagrangian to that of Einstein-dilaton theory with new nonlinear electrodynamics field. A new class of 4-dimensional black hole solution has been constructed out as the exact solution to the Brans-Dicke theory in the presence of the Born-Infeld nonlinear electrodynamics. The physical properties of the solutions have been studied. The black hole charge and temperature have been calculated making use of the Gauss’s law and the concept of surface gravity, respectively. Also, the black hole mass and entropy have been obtained from geometrical methods. Through a Smarr-type mass formula as a function of the black hole charge and entropy the black hole temperature and electric potential, as the intensive parameters conjugate to the black hole entropy and charge, have been calculated.

Thermodynamics and thermal stability of novel charged dilaton black holes have been analyzed in the presence of the exponential nonlinear electrodynamics. It has been found that the scalar field can be written as the linear combination of two Liouville-type potentials. Three new classes of nonlinearly charged dilatonic black holes have been obtained as the exact solutions to the coupled scalar, electromagnetic, and gravitational field equations. The impacts of nonlinear electrodynamics on the conserved and thermodynamic quantities of the new black hole solutions have been calculated. It has been illustrated that the first law of black hole thermodynamics is valid for either of the new black hole solutions. A thermal stability or phase transition analysis has been performed making use of the canonical ensemble method. The points of type-1 and type-2 phase transitions as well as the ranges at which the new nonlinearly charged dilatonic black holes are locally stable have been determined.

In this work, the charged black hole solution to the Brans–Dicke gravity theory in the presence of the nonlinear electrodynamics has been investigated. To simplify the field equations, a conformal transformation has been introduced which transforms the Brans–Dicke–Born–Infeld Lagrangian to that of Einstein-dilaton–Born–Infeld theory. A new class of (n+1)-dimensional black hole solution has been constructed out as the exact solution to the Brans–Dicke theory in the presence of the Born–Infeld nonlinear electrodynamics. The physical properties of the solutions have been studied. The black hole charge and temperature have been calculated making use of the Gauss’s law and the concept of surface gravity, respectively. Also, the black hole mass and entropy have been obtained from geometrical methods. Trough a Smarr-type mass formula as a function of the black hole charge and entropy the black hole temperature and electric potential, as the intensive parameters conjugate to the black hole entropy and charge, have been calculated. The consistency of results of the geometrical and thermodynamical approaches confirms the validity of the first law of black hole thermodynamics for this new black hole solution. Finally, making use of the ensemble canonical method, the local stability or phase transition of the new (n+1)-dimensional Brans–Dicke–Born–Infeld black hole solution has been analyzed.

We investigate the classical stability of Schwarzschild black hole in Jordan and Einstein frames that are related by the conformal transformations. For this purpose, we introduce two models of the Brans-Dicke theory and Brans-Dicke-Weyl gravity in the Jordan frame and two corresponding models in the Einstein frame. The former model is suitable for studying the massless spin-2 graviton propagating around the Schwarzschild black hole, while the latter is designed for the massive spin-2 graviton propagating around the black hole. It turns out that the black hole (in)stability is independent of the frame which shows that the two frames are equivalent to each other.

Unification of gravity with other interactions, achieving the ultimate framework of quantum gravity, and fundamental problems in particle physics and cosmology motivate to consider extra spatial dimensions. The impact of these extra dimensions on the modified theories of gravity has attracted a lot of attention. One way to examine how extra dimensions affect the modified gravitational theories is to analytically investigate astrophysical phenomena, such as black hole shadows. In this study, we aim to investigate the behavior of the shadow shapes of higher-dimensional charged black hole solutions including asymptotically locally flat (ALF) and asymptotically locally AdS (ALAdS) in Einstein–Horndeski–Maxwell (EHM) gravitational theory. We utilize the Hamilton–Jacobi method to find photon orbits around these black holes as well as the Carter approach to formulate the geodesic equations. We examine how extra dimensions, negative cosmological constant, electric charge, and coupling constants of the EHM gravity affect the shadow size of the black hole. Then, we constrain these parameters by comparing the shadow radius of these black holes with the shadow size of M87* supermassive black hole captured by the Event Horizon Telescope (EHT) collaborations. We discover that generally the presence of extra dimensions within the EHM gravity results in reducing the shadow size of higher-dimensional ALF and ALAdS charged black holes, whereas the impact of electric charge on the shadow of these black holes is suppressible. Interestingly, we observe that decreasing the negative cosmological constant, i.e., increasing its absolute value, leads to increase the shadow size of the ALAdS charged higher-dimensional black hole in the EHM gravity. Surprisingly, based on the constraints from EHT observations, we discover that only the shadow size of the four dimensional ALF charged black hole lies in the confidence levels of EHT data, whereas owing to the presence of the negative cosmological constant, the shadow radius of the four, five, and seven dimensional ALAdS charged black holes lie within the EHT data confidence levels.

We disclose a novel phase transition in black hole physics by investigating thermodynamics of charged dilaton black holes in an extended phase space where the charge of the black hole is regarded as a fixed quantity. Along with the usual critical (second-order) as well as the first-order phase transitions in charged black holes, we find that a finite jump in Gibbs free energy is generated by dilaton-electromagnetic coupling constant, $\alpha$, for a certain range of pressure. This novel behavior indicates a small/large black hole \emph{zeroth-order} phase transition in which the response functions of black holes thermodynamics diverge e.g. isothermal compressibility. Such zeroth-order transition separates the usual critical point and the standard first-order transition curve. We show that increasing the dilaton parameter($\alpha$) increases the zeroth-order portion of the transition curve. Additionally, we find that the second-order (critical) phase transition exponents are unaffected by the dilaton parameter, however, the condition of positive critical temperature puts an upper bound on the dilaton parameter ($\alpha<1$).

We study thermodynamics and thermal stability of three-dimensional (3D) charged black holes (BHs) in the Brans–Dicke (BD) gravity theory. The BD field equations, which are written in the Jordan frame, are nonlinear and too complex for direct solving. Thus, by utilizing the conformal transformations (CT), we translated them to the Einstein frame, where the equations are solved easier. In the Einstein–Maxwell-dilaton (EMd) theory, the number of unknowns is one more than the unique equations. Thus, we solved this problem by using an exponential ansatz function and introduced four novel classes of 3D EMd BHs with unusual asymptotic behavior. We proved validity of the first law of BH thermodynamics (FLT), and explored thermal stability of the EMd BHs by use of the canonical ensemble method. We introduced 3D BD-Maxwell BHs by applying the inverse CT on the Einstein frame counterparts. By using the plots, we showed that our solutions can produce BD BHs without horizon, with one or two horizons. By direct calculations, we proved that thermodynamic quantities are conformal-invariant and FLT is valid for the BD BHs too. Moreover, stability properties of the BD BHs are just like those of EMd ones.

A new theory of gravity called Eddington-inspired Born-Infeld (EiBI) gravity was recently proposed by Ba\~{n}ados and Ferreira. This theory leads to some exciting new features, such as free of cosmological singularities. In this paper, we first obtain a charged EiBI black hole solution with a nonvanishing cosmological constant when the electromagnetic field is included in. Then based on it, we study the strong gravitational lensing by the asymptotic flat charged EiBI black hole. The strong deflection limit coefficients and observables are shown to closely depend on the additional coupling parameter $\kappa$ in the EiBI gravity. It is found that, compared with the corresponding charged black hole in general relativity, the positive coupling parameter $\kappa$ will shrink the black hole horizon and photon sphere. Moreover, the coupling parameter will decrease the angular position and relative magnitudes of the relativistic images, while increase the angular separation, which may shine new light on testing such gravity theory in near future by the astronomical instruments.

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 zne 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 zge 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.

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 phase regions, we find that the scalar curvature is always negative for the van der Waals fluid, indicating that attractive interactions dominate amongst the fluid microstructures. Along the coexistence curve, the scalar curvature $R$ decreases with temperature, and goes to negative infinity at a critical temperature. We then numerically study the critical phenomena associated with the scalar curvature. We next consider four-dimensional charged AdS black holes. Vanishing of the heat capacity at constant volume yields a divergent scalar curvature. In order to extract the corresponding information, we define a new scalar curvature that has behaviour similar to that of a van der Waals fluid. We analytically confirm that at the critical point of the small/large black hole phase transition, the scalar curvature has a critical exponent 2, and $R(1-\tilde{T})^{2}C_{v}=1/8$, the same as that of a van der Waals fluid. However we also find that the scalar curvature can be positive for the small charged AdS black hole, implying that repulsive interactions dominate among the black hole microstructures. We then generalize our study to higher-dimensional charged AdS black holes.