Extremal black holes in the Hořava–Lifshitz gravity

We study the near-horizon geometry of extremal black holes in the z=3 Ho?ava–Lifshitz gravity with a flow parameter λ. For λ>1/2, near-horizon geometry of extremal black holes are AdS ₂×S ² with different radii, depending on the (modified) Ho?ava–Lifshitz gravity. For 1/3≤λ≤1/2, the radius v ₂ of S ² is negative, which means that the near-horizon geometry is ill-defined and the corresponding Bekenstein–Hawking entropy is zero. We show explicitly that the entropy function approach does not work for obtaining the Bekenstein–Hawking entropy of extremal black holes.     

Entropy function approach to charged BTZ black hole

We find solution to the metric function f(r) = 0 of charged BTZ black hole making use of the Lambert function. The condition of extremal charged BTZ black hole is determined by a non-linear relation of M _e (Q) = Q ²(1 − ln Q ²). Then, we study the entropy of extremal charged BTZ black hole using the entropy function approach. It is shown that this formalism works with a proper normalization of charge Q for charged BTZ black hole because AdS₂ × S¹ represents near-horizon geometry of the extremal charged BTZ black hole. Finally, we introduce the Wald’s Noether formalism to reproduce the entropy of the extremal charged BTZ black hole without normalization when using the dilaton gravity approach.     

Holography at an extremal De Sitter horizon

Rotating maximal black holes in four-dimensional de Sitter space, for which the outer event horizon coincides with the cosmological horizon, have an infinite near-horizon region described by the rotating Nariai metric. We show that the asymptotic symmetry group at the spacelike future boundary of the near-horizon region contains a Virasoro algebra with a real, positive central charge. This is evidence that quantum gravity in a rotating Nariai background is dual to a two-dimensional Euclidean conformal field theory. These results are related to the Kerr/CFT correspondence for extremal black holes, but have two key differences: one of the black hole event horizons has been traded for the cosmological horizon, and the near-horizon geometry is a fiber over dS₂ rather than AdS₂.     

Entropy of an extremal regular black hole

We introduce a magnetically charged extremal regular black hole in the coupled system of Einstein gravity and nonlinear electrodynamics. Its near horizon geometry is given by AdS₂×S²${\mathit{AdS}}_2\times S^2$. It turns out that the entropy function approach does not automatically lead to a correct entropy of the Bekenstein–Hawking entropy. This contrasts to the case of the extremal Reissner–Norström black hole in the Einstein–Maxwell theory. We conclude that the entropy function approach does not work for a magnetically charged extremal regular black hole without singularity, because of the nonlinearity of the entropy function.     

Moduli and (un)attractor black hole thermodynamics

We investigate four-dimensional spherically symmetric black hole solutions in gravity theories with massless, neutral scalars non-minimally coupled to gauge fields. In the non-extremal case, we explicitly show that, under the variation of the moduli, the scalar charges appear in the first law of black hole thermodynamics. In the extremal limit, the near horizon geometry is AdS ₂ × S ² and the entropy does not depend on the values of moduli at infinity. We discuss the attractor behaviour by using Sen’s entropy function formalism as well as the effective potential approach and their relation with the results previously obtained through special geometry method. We also argue that the attractor mechanism is at the basis of the matching between the microscopic and macroscopic entropies for the extremal non-BPS Kaluza–Klein black hole.     

Thermodynamics in rotating anti-de Sitter black holes with massive scalar field in three dimensions

We investigated the tendency in the variations of CFT₂ when a rotating AdS₃ black hole changes because of the fluxes transferred by the scattering of a massive scalar field according to the anti-de Sitter (AdS)/conformal field theory (CFT) correspondence. The conserved quantities of the black hole are definitely constrained by the extremal condition. Moreover, the laws of thermodynamics provide a direction for the changes in the conserved quantities. Therefore, the black hole cannot be extremal under the scattering; this is naturally preferred. According to the relationship between the rotating AdS₃ black hole and dual CFT₂, we find that such changes in the black hole constrain the variations in the eigenstates of dual CFT₂. Furthermore, the tendency in the variations is closely related to the laws of thermodynamics.     

General Rotating Black Hole with Four Charges and Liouville Theory

By considering the dual Liouville theory emerging in the near-horizon limit, we study the thermodynamics of general rotating black hole with four charges in four dimensions. Both the black hole entropy and temperature are found to agree with the gravitational expectations. The relations between the new Liouville formalism and the anomaly approach are also discussed.     

General Rotating Black Hole with Four Charges and Liouville Theory

By considering the dual Liouville theory emerging in the near-horizon limit, we study the thermodynamics of general rotating black hole with four charges in four dimensions. Both the black hole entropy and temperature are found to agree with the gravitational expectations. The relations between the new Liouville formalism and the anomaly approach are also discussed.     

Double-horizon limit, AdS geometry and entropy function

We start from a generic metric which describes four-dimensional stationary black holes in an arbitrary theory of gravity and show that the AdS₂ part of the near-horizon geometry is a consequence of the double-horizon limit and finiteness. We also show that the field configurations of the near horizon are determined if the same conditions are applied to the equations of motion. This is done by showing that in the double-horizon limit field equations at the horizon decouple from the bulk of the space. Solving these equations gives the near-horizon field configurations. It is shown that these decoupled equations can be obtained from an action derived from the original action by applying the double-horizon condition. Our results agree with the entropy function method.     

Holographic Entanglement Entropy of the BTZ Black Hole

We investigate quantum entanglement of gravitational configurations in 3D AdS gravity using the AdS/CFT correspondence. We derive explicit formulas for the holographic entanglement entropy (EE) of the BTZ black hole, conical singularities and regularized AdS₃. The leading term in the large temperature expansion of the holographic EE of the BTZ black hole reproduces exactly its Bekenstein-Hawking entropy S _BH , whereas the subleading term behaves as ln S _BH . We also show that the leading term of the holographic EE for the BTZ black hole can be obtained from the large temperature expansion of the partition function of a broad class of 2D CFTs on the torus. This result indicates that black hole EE is not a fundamental feature of the underlying theory of quantum gravity but emerges when the semiclassical notion of spacetime geometry is used to describe the black hole.     

Thermodynamics of the black holes under the extended generalized uncertainty principle with linear terms

In this paper, we employ the extended generalized uncertainty principle with linear terms (LEGUP) to investigate the thermodynamics properties of the Schwarzschild and Reissner–Nordström (RN) black holes. Firstly, by constructing the theoretical framework of LEGUP, the minimal temperature of the Schwarzschild black hole and the modified mass–temperature function for the black hole are calculated. Furthermore, the heat capacity function for the Schwarzschild black hole is obtained. After that, we compare LEGUP black hole thermodynamics with EGUP black hole and with the usual forms. Besides, the modification of black hole entropy is discussed, which involves a heuristic analysis of particles absorbed by the black hole. Finally, we derive the LEGUP-corrected temperature, heat capacity and entropy functions of the RN black hole.     

Quantum Corrections for Regular Black Holes with Charge and Coupling Constant

In this paper, we use semi-classical tunneling approach to calculate the quantum corrections to the Hawking temperature as well as entropy of the Kehagias-Sftesos asymptotically flat black hole solution and charged regular black hole with Fermi-Dirac distribution. For this purpose, we apply the first law of black hole thermodynamics to investigate the semi-classical entropy of both black holes having mass as well as charge or coupling constant. For both black holes, the entropy corrections contain the logarithmic term as a leading order correction term. For Kehagias-Sftesos asymptotically flat black hole, the semi-classical Hawking temperature and black hole entropy will behave asymptotically by considering the vanishing coupling constant b = 0. We have obtained the same analysis for the corrected thermodynamical quantities for this BH. For charged regular black hole with Fermi-Dirac distribution, if we neglect the charged effects in our analysis, i.e., q = 0, then these corrections approximately leads to the Schwarzschild black hole which is already given in the literature.     

Thermodynamics and phase transition of the Reissner–Nordstr?m black hole surrounded by quintessence

We investigate thermodynamics and phase transition of the Reissner–Nordstr?m black hole surrounded by quintessence. Using thermodynamical laws of black holes, we derive the expressions of some thermodynamics quantities for the Reissner–Nordstr?m black hole surrounded by quintessence. The variations of the temperature and heat capacity with the entropy were plotted for different values of the state parameter related to the quintessence, ω _q , and the normalization constant related to the density of quintessence c. We show that when varying the entropy of the black hole a phase transition is observed in the black hole. Moreover, when increasing the density of quintessence, the transition point is shifted to lower entropy and the temperature of the black hole decreases.     

Nonquasinormal modes and black hole physics

The near-horizon geometry of a large class of extremal and near-extremal black holes in string and M-theory contains three-dimensional asymptotically anti-de Sitter space. Motivated by this structure, we are led naturally to a discrete set of complex frequencies defined in terms of the monodromy at the inner and outer horizons of the black hole. We show that the correspondence principle, whereby the real part of these "nonquasinormal frequencies" is identified with certain fundamental quanta, leads directly to the correct quantum behavior of the near-horizon Virasoro algebra, and thus the black hole entropy. Remarkably, for the rotating black hole in five dimensions we also reproduce the fractionization of conformal weights predicted in string theory.     

Validity of thermodynamic laws and weak cosmic censorship for AdS black holes and black holes in a cavity

By throwing a test charged particle into a Reissner-Nordstrom (RN) black hole, we test the validity of the first and second laws of thermodynamics and the weak cosmic censorship conjecture (WCCC) with two types of boundary conditions: the asymptotically anti-de Sitter (AdS) space and a Dirichlet cavity wall placed in an asymptotically flat space. For the RN-AdS black hole, the second law of thermodynamics is satisfied, and the WCCC is violated for both extremal and near-extremal black holes. For the RN black hole in a cavity, the entropy can either increase or decrease depending on the change in the charge, and the WCCC is satisfied/violated for the extremal/near-extremal black hole. Our results indicate that there may be a connection between the black hole thermodynamics and the boundary condition imposed on the black hole.     

Angular Momentum-Free of the Entropy Relations for Rotating Kaluza-Klein Black Holes

Based on a mathematical lemma related to the Vandermonde determinant and two theorems derived from the first law of black hole thermodynamics, we investigate the angular momentum independence of the entropy sum as well as the entropy product of general rotating Kaluza-Klein black holes in higher dimensions. We show that for both non-charged rotating Kaluza-Klein black holes and non-charged rotating Kaluza-Klein-AdS black holes, the angular momentum of the black holes will not be present in entropy sum relation in dimensions d≥4, while the independence of angular momentum of the entropy product holds provided that the black holes possess at least one zero rotation parameter a _j = 0 in higher dimensions d≥5, which means that the cosmological constant does not affect the angular momentum-free property of entropy sum and entropy product under the circumstances that charge δ=0. For the reason that the entropy relations of charged rotating Kaluza-Klein black holes as well as the non-charged rotating Kaluza-Klein black holes in asymptotically flat spacetime act the same way, it is found that the charge has no effect in the angular momentum-independence of entropy sum and product in asymptotically flat spactime.     

Corrections to the Hawking Tunneling Radiation from MDR

We investigate some aspects of black hole (BH) thermodynamics in the framework of a modified dispersion relation. We calculate a minimal length and a maximal momentum to find a relation between spacetime dimensions and the presence of logarithmic prefactor in the black hole entropy relation. We show that the logarithmic prefactor appears not only in an even number of dimensions but also in an odd number of dimensions. In addition, the sign of the logarithmic factor is different for positive values of α in all dimensions. Using the corrected entropy, the black hole radiation probability is calculated in the tunneling formalism, which is corrected up to the same order of the Planck length and shows a more probable quantum tunneling.     

Unitarity and Page Curve for Evaporation of 2D AdS Black Holes

We explore the Hawking evaporation of two-dimensional anti-de Sitter (AdS2), dilatonic black hole coupled with conformal matter, and derive the Page curve for the entanglement entropy of radiation. We first work in a semiclassical approximation with backreaction. We show that the end-point of the evaporation process is AdS2 with a vanishing dilaton, i.e., a regular, singularity-free, zero-entropy state. We explicitly compute the entanglement entropies of the black hole and the radiation as functions of the horizon radius, using the conformal field theory (CFT) dual to AdS2 gravity. We use a simplified toy model, in which evaporation is described by the forming and growing of a negative mass configuration in the positive-mass black hole interior. This is similar to the “islands” proposal, recently put forward to explain the Page curve for evaporating black holes. The resulting Page curve for AdS2 black holes is in agreement with unitary evolution. The entanglement entropy of the radiation initially grows, closely following a thermal behavior, reaches a maximum at half-way of the evaporation process, and then goes down to zero, following the Bekenstein–Hawking entropy of the black hole. Consistency of our simplified model requires a non-trivial identification of the central charge of the CFT describing AdS2 gravity with the number of species of fields describing Hawking radiation.