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Banados, Teitelboim, and Zanelli (BTZ) black holes are excellent laboratories for studying black hole thermodynamics, which is a bridge between classical general relativity and the quantum nature of gravitation. In addition, three-dimensional gravity could have equipped us for exploring some of the ideas behind the two-dimensional conformal field theory based on the AdS3/CFT2. Considering the significant interest in these regards, we examine charged BTZ black holes. We consider the system contains massive gravity with energy dependent spacetime to enrich the results. In order to make high curvature (energy) BTZ black holes more realistic, we modify the theory by energy dependent constants. We investigate thermodynamic properties of the solutions by calculating heat capacity and free energy. We also analyze thermal stability and study the possibility of the Hawking-Page phase transition. At last, we study the geometrical thermodynamics of these black holes and compare the results of various approaches. © 2017 American Physical Society., Research Institute for Astronomy and Astrophysics of Maragha

In this paper, we consider linearly charged dilatonic black holes and study their thermodynamical behavior in the context of phase transitions and thermodynamic geometry. We show that, depending on the values of the parameters, these black holes can undergo two types of phase transition. We also find that there are three critical behaviors near the critical points for these black holes: nonphysical unstable to physical stable, large to small, and small to large black-hole phase transitions. Next, we employ a thermodynamical metric to study the thermodynamical geometry of these black holes. We show that the characteristic behavior of the Ricci scalar of this metric enables one to recognize the type of phase transition and critical behavior of the black holes near phase-transition points. Finally, we will extend thermodynamical space by considering the dilaton parameter as an extensive parameter. With this consideration, we will show that the Weinhold, Ruppeiner, and Quevedo metrics provide extra divergencies that are not related to any phase-transition point, whereas our new method provides an effective mechanism to study phase transitions. © 2015 American Physical Society., Research Institute for Astronomy and Astrophysics of Maragha

In this paper, we will generalize the Gauss-Bonnet gravity to an energy dependent Gauss-Bonnet theory of gravity, which we shall call as the Gauss-Bonnet gravity's rainbow. We will also couple this theory to a Maxwell's theory. We will analyze black hole solutions in this energy dependent Gauss-Bonnet gravity's rainbow. We will calculate the modifications to the thermodynamics of black holes in the Gauss-Bonnet's gravity's rainbow. We will demonstrate that even though the thermodynamics of the black holes get modified in the Gauss-Bonnet gravity's rainbow, the first law of thermodynamics still holds for this modified thermodynamics. We will also comment on the thermal stability of the black hole solutions in this theory., Comment: 12 pages with 26 figures, Accepted for publication in PRD

In this paper, we consider logarithmic and exponential forms of nonlinear electrodynamics as a source and obtain magnetic brane solutions of the Lovelock gravity. Although these solutions have no curvature singularity and no horizon, they have a conic singularity with a deficit angle. We investigate the effects of nonlinear electrodynamics and the Lovelock gravity on the value of deficit angle and find that various terms of Lovelock gravity do not affect deficit angle. Next, we generalize our solutions to spinning cases with maximum rotating parameters in arbitrary dimensions and calculate the conserved quantities of the solutions. Finally, we consider nonlinear electrodynamics as a correction of the Maxwell theory and investigate the properties of the solutions., 17 pages, 11 figures. arXiv admin note: text overlap with arXiv:1103.1774 by other authors

The objective of the present paper is to analyze the phase transition of asymptotically anti-de Sitter (AdS) black hole solutions in Lovelock gravity in the presence of nonlinear electrodynamics. First, we present the asymptotically AdS black hole solutions for two classes of the Born-Infeld type of nonlinear electrodynamics coupled with Einstein, Gauss-Bonnet and third order Lovelock gravity, separately. Then, in order to discuss the phase transition, we calculate both the heat capacity and the Ricci scalar of the thermodynamical line element. We present a comparison between the singular points of the Ricci scalar using Geometrothermodynamics method and the corresponding vanishing points of the heat capacity in the canonical ensemble. In addition, we discuss the effects of both Lovelock and nonlinear electrodynamics on the phase transition points., 13 pages with 9 figures

Employing higher order perturbation theory, we obtain five-dimensional rotating black holes in the Einstein gravity with nonlinear electrodynamics. We consider asymptotically flat charged rotating black hole solutions with two equal angular momenta and introduce the electric charge, q, as the perturbative parameter. We construct the extremal rapidly rotating black holes in the Einstein-nonlinear Maxwell theory in five dimensions, where we solve the equations of motion up to fourth order in q. We show that the perturbative parameters q and the nonlinearity parameter ß may modify the values of the physical quantities of the black holes. In particular, we find that ß may change the sign of the magnetic moment and the gyromagnetic ratio of the black hole spacetime when this parameter starts to depart strongly from the Maxwell limit. Finally, we obtain a generalized Smarr formula for the rotating black hole solutions. © 2014 American Physical Society.

In general, the field equations of $f(R)$ theory coupled to a matter field are very complicated and hence it is not easy to find exact analytical solutions. However, if one considers traceless energy-momentum tensor for the matter source as well as constant scalar curvature, one can derive some exact analytical solutions from $f(R)$ theory coupled to a matter field. In this paper, by assuming constant curvature scalar, we construct a class of charged rotating black string solutions in $f(R)$-Maxwell theory. We study the physical properties and obtain the conserved quantities of the solutions. The conserved and thermodynamic quantities computed here depend on function $f'(R_{0})$ and differ completely from those of Einstein theory in AdS spaces. Besides, unlike Einstein gravity, the entropy does not obey the area law. We also investigate the validity of the first law of thermodynamics as well as the stability analysis in the canonical ensemble, and show that the black string solutions are always thermodynamically stable in $f(R)$-Maxwell theory with constant curvature scalar. Finally, we extend the study to the case where the Ricci scalar is not a constant and in particular $R=R(r)$. In this case, by using the Lagrangian multipliers method, we derive an analytical black string solution from $f(R)$ gravity and reconstructed the function $R(r)$. We find that this class of solutions has an additional logaritmic term in the metric function which incorporates the effect of the $f(R)$ theory in the solutions.

We study thermodynamic instability of a class of $(n+1)$-dimensional charged dilaton black holes in the background of anti-de Sitter universe. We calculate the quasilocal mass of the AdS dilaton black hole through the use of the subtraction method of Brown and York. We find a Smarr-type formula and perform a stability analysis in the canonical ensemble and disclose the effect of the dilaton field on the thermal stability of the solutions. Our study shows that the solutions are thermally stable for small $\alpha$, while for large $\alpha$ the system has an unstable phase, where $\alpha $ is a coupling constant between the dilaton and matter field., Comment: 14 pages, 6 figures

Considering both the nonlinear invariant terms constructed by the electromagnetic field and the Riemann tensor in gravity action, we obtain a new class of $(n+1)$-dimensional magnetic brane solutions in third order Lovelock-Born-Infeld gravity. This class of solutions yields a spacetime with a longitudinal nonlinear magnetic field generated by a static source. These solutions have no curvature singularity and no horizons but have a conic geometry with a deficit angle $\delta$. We find that, as the Born-Infeld parameter decreases, which is a measure of the increase of the nonlinearity of the electromagnetic field, the deficit angle increases. We generalize this class of solutions to the case of spinning magnetic solutions and find that, when one or more rotation parameters are nonzero, the brane has a net electric charge which is proportional to the magnitude of the rotation parameters. Finally, we use the counterterm method in third order Lovelock gravity and compute the conserved quantities of these spacetimes. We found that the conserved quantities do not depend on the Born-Infeld parameter, which is evident from the fact that the effects of the nonlinearity of the electromagnetic fields on the boundary at infinity are wiped away. We also find that the properties of our solution, such as deficit angle, are independent of Lovelock coefficients., Comment: 15 pages, one figure, references added, a subsection added

In this paper, we present topological black holes of third order Lovelock gravity in the presence of cosmological constant and nonlinear electromagnetic Born-Infeld field. Depending on the metric parameters, these solutions may be interpreted as black hole solutions with inner and outer event horizons, an extreme black hole or naked singularity. We investigate the thermodynamics of asymptotically flat solutions and show that the thermodynamic and conserved quantities of these black holes satisfy the first law of thermodynamic. We also endow the Ricci flat solutions with a global rotation and calculate the finite action and conserved quantities of these class of solutions by using the counterterm method. We compute the entropy through the use of the Gibbs-Duhem relation and find that the entropy obeys the area law. We obtain a Smarr-type formula for the mass as a function of the entropy, the angular momenta, and the charge, and compute temperature, angular velocities, and electric potential and show that these thermodynamic quantities coincide with their values which are computed through the use of geometry. Finally, we perform a stability analysis for this class of solutions in both the canonical and the grand-canonical ensemble and show that the presence of a nonlinear electromagnetic field and higher curvature terms has no effect on the stability of the black branes, and they are stable in the whole phase space., Comment: 14 pages