We consider four-derivative corrections to the bosonic sector of five-dimensional N=2 gauged supergravity. Since this theory includes the N=2 graviphoton, we consider both curvature and graviphoton field-strength terms that show up at the four-derivative level. We construct, to linear order, the higher-derivative corrections to the non-rotating R-charged AdS_5 black hole and demonstrate how this solution transforms under field redefinitions.
We carry out field redefinitions in ten-dimensional Type IIB supergravity and show that they do not give rise to any physical corrections to the holographic renormalization group structure in the AdS/CFT correspondence. We in particular show that the holographic Weyl anomaly of the N=4 SU(N) super Yang-Mills theory does not change under the field redefinition of the ten-dimensional metric of the form G_{MN} -> G_{MN}+αR G_{MN}+βR_{MN}. These results are consistent with the fact that classical supergravity represents the on-shell structure of massless modes of superstrings, which should not cha…
Using AdS/CFT correspondence and the Euclidean action formalism for black hole entropy Kraus and Larsen have argued that the entropy of a BTZ black hole in three dimensional supergravity with (0,4) supersymmetry does not receive any correction from higher derivative terms in the action. We argue that as a consequence of AdS/CFT correspondence the action of a three dimensional supergravity with (0,4) supersymmetry cannot receive any higher derivative correction except for those which can be removed by field redefinition. The non-renormalization of the entropy then follows as a consequence of th…
We claim that there is a one to one correspondence between $NAdS_2$ spacetimes and dilatonic $AdS_2$ black holes. We show that these have the same thermodynamics and can be described by the same states of 1D CFTs with the same central charge. In addition, we show that these are actually the same set of solutions since a field redefinition in the Schwarzian theory transforms its equation of motion into the time independent part of the bulk Einstein equation. The deformations of the $AdS_2$ boundary as a function of the boundary time are exactly the conformal factor of the $AdS_2$ black hole met…
We discuss the quantization of a system of slowly-moving extreme Reissner-Nordstrom black holes. In the near-horizon limit, this system has been shown to possess an SL(2,R) conformal symmetry. However, the Hamiltonian appears to have no well-defined ground state. This problem can be circumvented by a redefinition of the Hamiltonian due to de Alfaro, Fubini and Furlan (DFF). We apply the Faddeev-Popov quantization procedure to show that the Hamiltonian with no ground state corresponds to a gauge in which there is an obstruction at the singularities of moduli space requiring a modification of th…
We show that the solutions describing charged rotating black holes in five-dimensional gauged supergravities found recently by Cvetic, Lu and Pope [hep-th/0406196,hep-th/0407058] are completely specified by the mass, charges and angular momentum. The additional parameter appearing in these solutions is removed by a coordinate transformation and redefinition of parameters. Thus, the apparent hair in these solutions is unphysical.
We find the full interacting Lagrangian and Hamiltonian for quantum strings in a near plane wave limit of AdS_4 x CP^3. The leading curvature corrections give rise to cubic and quartic terms in the Lagrangian and Hamiltonian that we compute in full. The Lagrangian is found as the type IIA Green-Schwarz superstring in the light-cone gauge employing a superspace construction with 32 grassmann-odd coordinates. The light-cone gauge for the fermions is non-trivial since it should commute with the supersymmetry condition. We provide a prescription to properly fix the kappa-symmetry gauge condition t…
We study the hybrid formulation of Green-Schwarz superstrings on AdS_3 with NS flux and the boundary N=4 superconformal algebra. We show the equivalence between the NSR and GS superstrings by a field redefinition. The boundary N=4 superconformal algebra is realized by the free fields of the affine Lie superalgebra A(1|1)^{(1)}. We also consider the light-cone gauge and obtain the N=4 super-Liouville theory which describes the effective theory of the single long string near the singularities of the D1-D5 system.
We compute logarithmic corrections to the entropy of a magnetically charged extremal black hole in AdS4 x S7 using the quantum entropy function and discuss the possibility of matching against recently derived microscopic expressions.
Higher derivative bulk gravity (without Riemann tensor square term) admits AdS-Schwarzschild black hole as exact solution. It is shown that induced brane geometry on such background is open, flat or closed FRW radiation dominated Universe. Higher derivative terms contributions appear in the Hawking temperature, entropy and Hubble parameter via the redefinition of 5-dimensional gravitational constant and AdS scale parameter. These higher derivative terms do not destroy the AdS-dual description of radiation represented by strongly-coupled CFT. Cardy-Verlinde formula which expresses cosmological …
We study analytical solutions of charged black holes and thermally charged AdS with generalized warped factors in Einstein-Maxwell-Dilaton system. We calculate Euclidean action for charged AdS and thermally charged AdS. The actions in both backgrounds are regularized by the method of background subtraction. The study of phase transition between charged black hole and thermally charged AdS gives an insight to the confinement/deconfinement transition. The plots of grand potential vs temperature and chemical potential vs transition temperature are obtained.
Motivated by an open question raised in recent times regarding the phase transition during the collapse of a neutron star to form a black hole and related stability issues, we have constructed charged neutron stars in $AdS_5$ and show that these stars become unstable at a particular value of their radius, regarded as the Chandrasekhar radius. We reproduced the calculations recently done in [20] in our $AdS_5$ charged star. The analysis shows that the non-Fermi liquid behavior found there in $AdS_4$ is still true in this higher dimensional case with the presence of Kosevich-Lifshitz oscillation…
In the background of a charged AdS dilaton black hole, we investigate the movement of a self-graviting 3-brane and relevant holographic effects as the brane move close to the AdS boundary. The induced metric on brane corresponds to an exact FLRW geometry, while the evolution of brane is determined by Israel junction condition and the effective Einstein field equation on brane together. When the brane approaches the AdS boundary, AdS/CFT correspondence implies that a radiation dominated FLRW-universe ($P=\frac{1}{3}ρ$) should be given. According to the holographic renormalization procedure, we …
We discuss some properties of higher derivative (HD) bulk gravity without Riemann tensor square term. Such a kind of gravity admits Schwarzschild Anti de Sitter (SAdS) black hole as exact solution. It is shown that induced brane geometry on such background is Friedmann-Robertson-Walker (FRW) radiation dominated Universe. We show that HD terms contributions appear in the Hawking temperature, entropy and Hubble parameter via the redefinition of 5-dimensional gravitational constant and AdS scale parameter. These HD terms do not destroy the AdS-dual description of radiation represented by strongly…
Six dimensional supergravities on $ADS_3 \times S^3$ present interest due to the role they play in the $AdS/CFT$ correspondence. The correspondence in this case states the equivalence between supergravity on the given background and a still unknown conformal field theory. The conformal field theory in question is expected to appear by deforming of the free conformal field theory on $S^N(T^4)$ in a way which preserves the superconformal symmetry. The purpose of this paper is to compute the first nontrivial corrections to the equations of motion for the chiral primary fields coming from supergra…
Rotating charged Gauss-Bonnet black hole solutions in higher dimensional ($d >4$), asymptotically anti-de Sitter spacetime are obtained in the small angular momentum limit. The angular momentum, magnetic dipole moment, and the gyromagnetic ratio of the black holes are calculated and it turns out that the Gauss-Bonnet term does not affect to the gyromagnetic ratio.
We compute the shear viscosity in the supersymmetric Yang-Mills theory dual to the STU background. This is a thermal gauge theory with a chemical potential. The quotient of the shear viscosity over the entropy density exhibits no deviation from the well known result 1/4π.
In this work, we study the effect of $κ$-deformed space on the thermodynamic quantities, this are find through the holographic renormalization that provide the free energy, which is fundamental to derive the another thermodynamic quantities. For this scenario we consider an charged AdS$_{4}$ black hole for an Einstein-Maxwell model where the derivative quadrivector is replaced by a deformed version inspired in Kaniadakis statistics. Besides, we extract the transport coefficient know as electrical conductivity.
We study the mixed phase of charged AdS black hole and radiation when the total energy is fixed below the threshold to produce a stable charged black hole branch. The coexistence conditions for the charged AdS black hole and radiation are derived for the generic case when radiation particles carry charge. The phase diagram of the mixed phase is demonstrated for both fixed potential and charge ensemble. In the dual gauge picture, they correspond to the mixed phase of quark-gluon plasma~(QGP) and hadron gas in the fixed chemical potential and density ensemble respectively. In the nuclei and heav…
We derive a class of charged rotating dilaton black string solutions in the background of anti-de-Sitter spaces with an appropriate combination of three Liouville-type dilaton potentials. We also present the suitable counterterm which removes the divergences of the action in the presence of dilaton potential. The solutions are analyzed and their thermodynamics is discussed by using the counterterm method.
We construct R-charged adS bubbles in $D =5$, ${\cal N} =8$ supergravity. These bubbles are charecterised by four parameters. The asymptotic boundary of these solutions are deSitter times a circle. By comparing boundary energies, we study the possibility of a transition from certain class of black holes to these bubbles below a critical radius of the boundary circle. We argue that this may occur when four parameters of the bubble satisfy a constraint among themselves.
We investigate the phase structure of a charged AdS dilaton black hole in the extended phase space which takes the cosmological constant, i.e. the AdS-$Λ$ parameter as pressures. Through both thermal ensemble and quasinormal mode analysis, we find that stable phase of the black hole with non-trivial dilaton profiles always exists for both large and small couplings when the AdS-$Λ$ is considered dynamical degrees of freedom. This forms a somewhat contrast with previous works which does not do so. Our results provide new examples for the parallelism or equivalences between thermal ensemble metho…
We study certain features of a strongly coupled theory whose gravitational dual is given by two-charge dilatonic black hole in AdS5 which has recently been used to study holographic Fermi liquids. By making use of the gravity description, we have studied conductivity, holographic entanglement entropy and dynamics of a charged scalar field. In particular at low energy we find that the temperature dependence of the real part of the conductivity goes as T^3 and the background is stable against scalar condensations.
We obtain some exact results for black holes in the Randall-Sundrum model with a single brane. We consider an extreme black hole charged with respect to a Maxwell field on the brane. The near-horizon geometry is determined. The induced metric on the brane and the black hole entropy are compared with the predictions of 4d General Relativity. There is good agreement for large black holes, with calculable subleading corrections. As a separate application, the bulk solution provides a gravitational dual for (strongly coupled, large N) N=4 SYM in AdS(2)xS(2) for arbitrary relative size of AdS(2) an…
In the previous paper we studied the transport coefficients of Quark-Gluon Plasma in finite temperature and finite density in vector and tensor modes. In this paper, we extend it to the scalar modes. We work out the decoupling problem and hydrodynamic analysis for the sound mode in charged AdS black hole and calculate the sound velocity, the charge susceptibility and the electrical conductivity. We find that Einstein relation among the conductivity, the diffusion constant and the susceptibility holds exactly.
We study a set of CFT operators suitable for reconstructing a charged bulk scalar field $ϕ$ in AdS${}_3$ (dual to an operator ${\cal O}$ of dimension $Δ$ in the CFT) in the presence of a conserved spin-$n$ current in the CFT. One has to sum a tower of smeared non-primary scalars $\partial_{+}^{m} J^{(m)}$, where $J^{(m)}$ are primaries with twist $Δ$ and spin $m$ built from ${\cal O}$ and the current. The coefficients of these operators can be fixed by demanding that bulk correlators are well-defined: with a simple ansatz this requirement allows us to calculate bulk correlators directly from t…
We study the thermodynamic properties and critical behaviors of the topological charged black hole in AdS space under the consideration of the generalized uncertainty principle (GUP). It is found that only in the spherical horizon case there are Van der Waals-like first-order phase transitions and reentrant phase transitions. From the equation of state we find that the GUP-corrected black hole can have one, two and three apparent critical points under different conditions. However, it is verified by the Gibbs free energy that in either case there is at most one physical critical point.
The entropic force and the imaginary part of potential experienced by a moving quarkonia in a plasma, are computed in a charged deformed AdS background by holography approaches. In that case, the axis of the moving quarkonia has an arbitrary angle with wind. We show that the charge in deformed AdS background deeply effect on the increasing entropic force and decreasing thermal width. So, the dissociation length of the meson becomes short when charge is increasing. In addition, the entropic force and thermal width are increased by the deformation of the space. Therefore the deformation paramete…
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 $α$, while for large $α$ the system has an unstable phase, where $α$ is a coupling constant…
We consider the brane universe in the bulk background of the charged topological AdS black holes. The evolution of the brane universe is described by the Friedmann equations for a flat or an open FRW-universe containing radiation and stiff matter. We find that the temperature and entropy of the dual CFT are simply expressed in terms of the Hubble parameter and its time derivative, and the Friedmann equations coincide with thermodynamic formulas of the dual CFT at the moment when the brane crosses the black hole horizon. We obtain the generalized Cardy-Verlinde formula for the CFT with an R-cha…
We present the metric for the $(n+1)$-dimensional charged rotating dilaton black branes with cylindrical or toroidal horizons in the background of anti-de Sitter spacetime. We find the suitable counterterm which removes the divergences of the action in the presence of the dilaton potential in all higher dimensions. We plot the Penrose diagrams of the spacetime and reveal that the spacetime geometry crucially modifies in the presence of the dilaton field. The conserved and thermodynamic quantities of the black branes are also computed.
We study the leading $α^\prime$ corrections to the entropy of certain black holes with AdS$_5\times S^5$ asymptotics. We find that, in the supersymmetric limit, the entropy does not receive $α^\prime$ corrections. This result strengthens recent calculations that match the index of $\mathcal{N}=4$ Super-Yang-Mills with the corresponding partition function in the supersymmetric limit. In the small temperature regime, we find that the entropy corrections are concordant with the weak gravity conjecture.
The coexistence curve and molecule number density of a 4-dimensional AdS topological charged black hole in massive gravity is investigated. We find that the analytic expression of the coexistence curve in the reduced parameter space is dependent on theory parameters. This is very different from the previous results obtained in other modified gravity such as $f(R)$ gravity and Gauss-Bonnet gravity. Besides, we derive the explicit expression of the physical quantity which describes the difference of the number densities of AdS topological charged black hole molecules between the small and large …
In this work we use the quasinormal mode of a massless scalar perturbation to probe the phase transition of the charged-AdS black hole in time profile. The signature of the critical behavior of this black hole solution is detected in the isobaric process. This paper is a natural extension of [1, 2] to the time domain analysis. More precisely, our study shows a clear signal in term of the damping rate and the oscillation frequencies of the scalar field perturbation. We conclude that the quasinormal modes can be an efficient tool to detect the signature of thermodynamic phase transition in the i…
It is well known that the thermodynamics of certain near-extremal black holes in asymptotically flat space can be lifted to an effective string description created from the intersection of D-branes. In this paper we present evidence that the semiclassical thermodynamics of near-extremal R-charged black holes in AdS(5)xS(5) is described in a similar manner by effective strings created from the intersection of giant gravitons on the S(5). We also present a free fermion description of the supersymmetric limit of the one-charge black hole, and we give a crude catalog of the microstates of the two …
We investigate phase transitions and critical phenomena of four dimensional dyonic charged AdS black holes in the framework of thermodynamic geometry. In a mixed canonical grand canonical ensemble with a fixed electric charge and varying magnetic charge these black holes exhibit liquid gas like first order phase transition culminating in a second order critical point similar to the Van der Waals gas. We show that the thermodynamic scalar curvature R for these black holes follow our proposed geometrical characterization of the R-crossing Method for the first order liquid gas like phase transiti…
We present a new class of ${\cal N}=4$ supersymmetric quiver matrix models and argue that it describes the stringy low-energy dynamics of internally wrapped D-branes in four-dimensional anti-de Sitter (AdS) flux compactifications. The Lagrangians of these models differ from previously studied quiver matrix models by the presence of mass terms, associated with the AdS gravitational potential, as well as additional terms dictated by supersymmetry. These give rise to dynamical phenomena typically associated with the presence of fluxes, such as fuzzy membranes, internal cyclotron motion and the ap…
We study the possible dynamical emergence of IR conformal invariance describing the low energy excitations of near-extremal R-charged global AdS5 black holes. We find interesting behavior especially when we tune parameters in such a way that the relevant extremal black holes have classically vanishing horizon area, i.e. no classical ground-state entropy, and when we combine the low energy limit with a large N limit of the dual gauge theory. We consider both near-BPS and non-BPS regimes and their near horizon limits, emphasize the differences between the local AdS3 throats emerging in either ca…
In this work, we study the real-time dynamics of the charged hairy black hole with the time-dependent source of scalar field in asymptotically anti-de Sitter (AdS) spacetime. The numerical results reveal a novel descalarization mechanism. In order to obtain the hairy black hole as the initial data for the quench process, we first analyze the quasi-normal modes of the massive complex scalar field on the Reissner-Nordström anti-de Sitter (RN-AdS) black hole background. We find the dominant unstable modes for large and small RN-AdS black holes come from the zero-damped modes and AdS modes, respec…
A general framework for applying the pp-wave approximation to holographic calculations in the AdS/CFT correspondence is proposed. By assuming the existence and some properties of string field theory (SFT) on $AdS_5 \times S^5$ background, we extend the holographic ansatz proposed by Gubser, Klebanov, Polyakov and Witten to SFT level. We extract relevant information of assumed SFT on $AdS_5 \times S^5$ from its approximation, pp-wave SFT. As an explicit example, we perform string theoretic calculations of the conformal three point functions of the BMN operators. The results agree with the previ…
We show how to construct correlators for the $CFT_1$ which is dual to non-commutative $AdS_2$ ($ncAdS_2$). We do it explicitly for the example of the massless scalar field on Euclidean $ncAdS_2$. $ncAdS_2$ is the quantization of $AdS_2$ that preserves all the isometries. It is described in terms of the unitary irreducible representations, more specifically discrete series representations, of $so(2,1)$. We write down symmetric differential representations for the discrete series, and then map them to functions on the Moyal-Weyl plane. The Moyal-Weyl plane has a large distance limit which can be…
We study phase transition between electrically charged Ricci-flat black holes and AdS soliton spacetime of Horowitz and Myers in five dimensions. Boundary topology for both of them is $S^1 \times S^1 \times R^2$. We consider Reissner-Nordstrom black hole and R-charged black holes and find that phase transition of these black holes to AdS soliton spacetime depends on the relative size of two boundary circles. We also perform the stability analysis for these black holes. In order to use the AdS/CFT correspondence, we work in the grand canonical ensemble.
In this note, we consider the equations of motion for charged branes in AdS space and show that they can be cast into one-dimensional coupled Toda type. Then we solve the equations of motion to construct static charged AdS brane solutions which are invariant under translation along $p$ directions and have toroidal symmetry. The solutions are described by mass, charges and a dilaton coupling constant.
We numerically construct charged boson stars in asymptotically AdS spacetime. We find an intricate phase diagram dominated by solutions whose main matter contribution are alternately provided by the scalar field or by the gauge field.
In this work we study the phase transitions between the planar charged AdS black hole and the planar charged soliton. The planar soliton is obtained as a double analytic continuation of the charged black hole metric, which also involves analytically continuing the electric charge. We show that there are phase transitions between both solutions depending on the electric potential, magnetic flux and temperature. The analysis is carried out in the Grand-Canonical ensemble.
In this paper, we consider a charged rotating AdS black holes in four dimensions and study the effects of leading-order thermal corrections on the thermodynamics of such system explicitly. The first-order corrected thermodynamical quantities also satisfy the first-law of thermodynamics of the black holes. The holographic duality between the charged rotating AdS black holes and Van der Waals fluid is also emphasized through the $P-v$ diagram. Finally, we study the effects of the leading-order thermal corrections on the stability of the charged rotating black holes.
We present a new family of topological charged hairy black hole solutions in asymptotically AdS space in $D$-dimensions. We solve the coupled Einstein-Maxwell-Scalar gravity system and obtain exact charged hairy black hole solutions with planar, spherical and hyperbolic horizon topologies. The scalar field is regular everywhere. We discuss the thermodynamics of the hairy black hole and find drastic changes in its thermodynamic structure due to the scalar field. For the case of planar and spherical horizons, we find charged hairy/RN-AdS black hole phase transition, with thermodynamically prefer…
As an intriguing topological defect, global monopole's influence on behaviors of black holes has always been anticipated but still remains less clear. Analyzing the thermodynamics of charged AdS black hole incorporating a global monopole manifests that the black hole undergoes a first-order phase transition at critical point, and of special interest, the critical behaviors qualitatively resemble a Van der Waals liquid-gas system. This paper concentrates on further investigating the charged AdS black hole with a global monopole, aiming at clarifying the significant effects of the global monopol…
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 an 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…
We study electromagnetically charged dilatonic black brane solutions in arbitrary dimensions with flat transverse spaces, that are asymptotically AdS. This class of solutions includes spacetimes which possess a bulk region where the metric is approximately invariant under Lifshitz scalings. Given fixed asymptotic boundary conditions, we analyze how the behavior of the bulk up to the horizon varies with the charges and derive the extremality conditions for these spacetimes.
We study the thermodynamics in the grand canonical ensemble of D-dimensional charged Gauss-Bonnet-AdS black holes in the extended phase space. We find that the usual small-large black hole phase transition, which exhibits analogy with the Van de Waals liquid-gas system holds in five-dimensional spherical charged Gauss-Bonnet-AdS black holes when its potential is fixed within the range $0<Φ<\frac{\sqrt{3}π}{4}$. For the other higher dimensional and topological charged Gauss-Bonnet-AdS black holes, there is no such phase transition. In the limiting case, Reissner-Nordstrom-AdS black holes, with …
We generalize the vacuum static black brane solutions of Einstein's equations with negative cosmological constant recently discussed in literature, by introducing rotations and an electromagnetic field. We investigate numerically the thermodynamical properties of the charged and of the rotating $AdS$ black brane and we provide evidences for the existence of the charged and rotating case. In particular, we study the influence of the rotation and charge on the tension and mass. We find that the rotation essentially influences the tensions while the charge essentially influences the mass.
We investigate the holographic entanglement entropy in the Rindler-AdS space-time to obtain an exact solution for the corresponding minimal surface. Moreover, the holographic entanglement entropy of the charged single accelerated AdS Black holes in four dimensions is investigated. We obtain the volume of the codimension one-time slice in the bulk geometry enclosed by the minimal surface for both the RindlerAdS space-time and the charged accelerated AdS Black holes in the bulk. It is shown that the holographic entanglement entropy and the volume enclosed by the minimal hyper-surface in both the…
We study the thermodynamic properties of the RN-AdS black hole in full phase space and propose a generalized thermodynamic identity. As an example, we use it to find relations of thermodynamical coefficients between the grand canonical and canonical ensembles. We also show, for the first order phase transition, that the usual Maxwell's equal area law should be extended to a new form for the RN-AdS black hole.
We discuss an enhancement of the Brown-Henneaux boundary conditions in three-dimensional AdS General Relativity to encompass Weyl transformations of the boundary metric. The resulting asymptotic symmetry algebra, after a field-dependent redefinition of the generators, is a direct sum of two copies of the Witt algebra and the Weyl abelian sector. The charges associated to Weyl transformations are non-vanishing, integrable but not conserved due to a flux driven by the Weyl anomaly coefficient. The charge algebra admits an additional non-trivial central extension in the Weyl sector, related to th…
Treating the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we reconsider the critical behaviour of charged AdS black holes. We complete the analogy of this system with the liquid-gas system and study its critical point, which occurs at the point of divergence of specific heat at constant pressure. We calculate the critical exponents and show that they coincide with those of the Van der Waals system.
In this letter we investigate the throttling process of the charged Anti-de Sitter (AdS) black holes in the rainbow gravity. In the extended phase space of these black holes, the cosmological constant plays the role of a varying thermodynamic pressure and the black hole mass is identified with the thermodynamic enthalpy. We derive exact expressions for the Joule-Thomson coefficient and the inversion temperature in terms of black hole parameters and constants of rainbow gravity analytically, and then perform a numerical analysis for the isenthalpic and inversion curves of charged AdS black hole…
We study separatively the quasinormal modes (QNM) of electromagnetic perturbations around three-dimensional anti-de Sitter(AdS) black holes in Jordan and Einstein frames, which are related by the conformal transformations and a redefinition of a scalar field. We find that, in the Jordan frame, the imaginary parts of QNM frequencies can reflect the thermodynamical stabilities of hairy black holes, including the possible phase transition between the hairy black hole and BTZ black hole, disclosed by examining the corresponding free energies. Similar results are also uncovered in the Einstein fram…
