Difference between anti-de Sitter and de Sitter spaces: wave equation approach

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 frame. The obtained results further support that the QNM can be a dynamic probe of the thermodynamic properties in black holes.

We investigate the stability of $D$ dimensional singly rotating Myers-Perry-AdS black holes under superradiance against scalar field perturbations. It is well known that small four dimensional rotating or charged Anti-de Sitter (AdS) black holes are unstable against superradiance instability of a scalar field. Recent works extended the existence of this instability to five dimensional rotating charged AdS black holes or static charged AdS black holes in arbitrary dimensions. In this work we analytically prove that, rotating small AdS black holes in arbitrary dimensions also show superradiance instability irrespective of the value of the (positive) angular momentum quantum number. To do this we solve the Klein-Gordon equation in the slow rotation, low frequency limit. By using the asymptotic matching technique, we are able to calculate the real and imaginary parts of the correction terms to the frequency of the scalar field due to the presence of the black hole, confirming the presence of superradiance instability. We see that, unlike in the case of static AdS black holes, the analytical method is valid for rotating AdS black holes for any value of angular momentum number and space-time dimensions. For comparison we derive the corresponding correction terms for Myers-Perry black holes in the black hole bomb formalism in Appendix and see that the results are in agreement.

The anti-de Sitter (AdS) black hole plays an important role in the holographic principle. In this study, the upper bound in energy corresponding to the mass of the Schwarzschild black hole is modified to be that of the AdS black hole. Via the correspondence between the ultraviolet (UV) and infrared (IR) cutoffs, the constant term in the energy density of the holographic dark energy (HDE) can be obtained from the negative cosmological constant from the black hole. Interestingly, the proposed dark energy model could drive the late-time expansion of the Universe without the causality violation. The cosmic evolution is investigated by choosing the Hubble and particle horizons as the IR length scales. It is found that the accelerated expansion at late time can be obtained for both cases. This result may shed light on the connection between the AdS black hole and the de Sitter (dS) spacetime in the context of cosmology.

Reissner-Nordstr\"om Anti-de Sitter (RNAdS) black holes are unstable against the charged scalar field perturbations due to the well-known superradiance phenomenon. We present the time domain analysis of charged scalar field perturbations in the RNAdS black hole background in general dimensions. We show that the instabilities of charged scalar field can be explicitly illustrated from the time profiles of evolving scalar field. By using the Prony method to fit the time evolution data, we confirm the mode that dominates the long time behavior of scalar field is in accordance with the quasinormal mode from the frequency domain analysis. The superradiance origin of the instability can also be demonstrated by comparing the real part of the dominant mode with the superradiant condition of charged scalar field. It is shown that all the unstable modes are superradiant, which is consistent with the analytical result in the frequency domain analysis. Furthermore, we also confirm there exists the rapid exponential growing modes in the RNAdS case, which makes the RNAdS black hole a good test ground to investigate the nonlinear evolution of superradiant instability.

In this study, we analyzed the impact of a perfect fluid on the phase transition of Anti-de Sitter (AdS) black holes within the Rastall gravitational background. Compared to similar studies in the literature, the findings of this work are highlighted by the determination of analytical expressions for critical points for charged and Kerr-Newman AdS black holes using approximated formulas for the horizon radius. An accurate analysis of these new analytical expressions allowed us to discover a new viable condition that relates the Rastall parameter κ λ to the equation of state parameter ω, expressed as: κλ=ω1+ω . Thanks to this new condition, we were able to reproduce all the analytical expressions of critical points calculated within the framework of Einsteinâs general relativity for two cases: a charged AdS black hole and a rotating AdS black hole. These findings suggest that Rastall gravity, considering this new condition, could serve as an alternative theory of gravitation to general relativity. The approximate expression of the horizon radius also enabled the exploration of the distinctiveness of fractional-order phase transitions in these AdS black holes. Furthermore, we calculated the critical exponents, offering insights into the behavior of crucial thermodynamic quantities near the inflection point. Examining how a perfect fluid influences phase transition reveals various critical behaviors, demonstrating that the variation in the phase transition depends on the intensity of the perfect fluid. Notably, this variation is portrayed by a linearly increasing trajectory with the escalation of this intensity.

Information entropies associated with the energy density in position and momentum spaces are build for an anti-de Sitter (AdS) black hole. These quantities, that satisfy an entropic uncertainty relation, vary with the temperature. The higher is the black hole temperature, the greater/smaller is the information encoded respectively in the position/momentum distributions of energy. On the other hand, as it is well known, AdS black holes are subject to the Hawking Page phase transition. The amplitude for dominance of the black hole phase over the thermal AdS phase increases with the temperature. So, as the system becomes more stable, there is a change in the way that information is stored. In particular: information stored in the spatial energy density increases while information stored in the energy density in momentum space decreases.

We study a close relationship between the topological anti-de Sitter (TAdS) black holes and topological de Sitter (TdS) spaces including the Schwarzschild–de Sitter (SdS) black hole in five dimensions. We show that all thermal properties of the TdS spaces can be found from those of the TAdS black holes by replacing k by −k. Also we find that all thermal information for the cosmological horizon of the SdS black hole is obtained from either the hyperbolic-AdS black hole or the Schwarzschild–TdS space by substituting m with −m. For this purpose we calculate thermal quantities of bulk (Euclidean) conformal field theory (ECFT) and moving domain wall by using the A(dS)/(E)CFT correspondences. Further, we compute logarithmic corrections to the Bekenstein–Hawking entropy, Cardy–Verlinde formula and Friedmann equation due to thermal fluctuations. It implies that in the thermal relation between the TdS spaces and TAdS black holes, the cosmological horizon plays the same role as the horizon of TAdS black holes. Finally we note that the dS/ECFT correspondence is valid for the TdS spaces in conjunction with the AdS/CFT correspondence for the TAdS black holes.

Exploring Tsallis thermodynamics for boundary conformal field theories in gauge/gravity duality

In this paper, we relate the complexity for a holographic state to a simple gravitational object of which the growth rate at late times is equal to temperature times black hole entropy. We show that if this is correct, the thermodynamics of anti-de Sitter (AdS) black holes implies that for generic holographic states dual to static AdS black holes the complexity growth rate at late times will saturate the Lloyd bound at the high-temperature limit. In particular, for AdS planar black holes, the result holds at lower temperatures as well. We conjecture that the complexity growth is bounded above as $d\mathcal{C}/dt\ensuremath{\le}\ensuremath{\alpha}TS/\ensuremath{\pi}\ensuremath{\hbar}$ or $d\mathcal{C}/dt\ensuremath{\le}\ensuremath{\alpha}({T}_{+}{S}_{+}\ensuremath{-}{T}_{\ensuremath{-}}{S}_{\ensuremath{-}})/\ensuremath{\pi}\ensuremath{\hbar}$ for black holes with an inner horizon, where $\ensuremath{\alpha}$ is an overall coefficient for our new proposal. The conjecture passes a number of nontrivial tests for black holes in Einstein's gravity. However, we also find that the bound may be violated in the presence of stringy corrections.

The energy (due to matter plus fields including gravity) distribution of the Reissner–Nordstrom anti-de Sitter (RN AdS) black holes is studied by using the Moller energy-momentum definition in general relativity. This result is compared with the energy expression obtained by using the Einstein and Tolman complexes. The total energy depends on the black hole mass M and charge Q and the cosmological constant Λ. The energy distribution of the RN AdS is also calculated by using the Moller prescription in teleparallel gravity. We get the same result for both of these different gravitation theories. The energy obtained is also independent of the teleparallel dimensionless coupling constant, which means that it is valid not only in the teleparallel equivalent of general relativity, but also in any teleparallel model. In special cases of our model, we also discuss the energy distributions associated with the Schwarzschild AdS, RN and Schwarzschild black holes, respectively.

We study Hawking radiation and P−v criticality of charged dynamical (Vaidya) black hole in Anti-de Sitter (AdS) space. By investigating the near horizon properties of scalar field, we derive Hawking temperature of dynamical charged (Vaidya) AdS black hole. Based on this result, by regarding the cosmological constant as the thermodynamic pressure, we investigate the analogy between the charged dynamical (Vaidya) AdS black holes in the ensemble with the fixed charge and van der Waals liquid-gas system in detail, including equation of state, P−v diagram, critical point, heat capacities and critical exponents near the critical point. It is shown that the relationship among the critical pressure, critical volume, and critical temperature gets modified while the critical exponents are not affected by the dynamical nature of the black hole. We also find that, when the rate of change of the black hole horizon exceeds the critical value, the P−v criticality of the charged dynamical (Vaidya) AdS black hole will disappear.

We study the thermodynamics of spherically symmetric, neutral and non-rotating black holes in conformal (Weyl) gravity. To this end, we apply different methods: (i) the evaluation of the specific heat; (ii) the study of the entropy concavity; (iii) the geometrical approach to thermodynamics known as thermodynamic geometry; (iv) the Poincaré method that relates equilibrium and out-of-equilibrium thermodynamics. We show that the thermodynamic geometry approach can be applied to conformal gravity too, because all the key thermodynamic variables are insensitive to Weyl scaling. The first two methods, (i) and (ii), indicate that the entropy of a de Sitter black hole is always in the interval 2/3le Sle 1, whereas thermodynamic geometry suggests that, at S=1, there is a second order phase transition to an Anti de Sitter black hole. On the other hand, we obtain from the Poincaré method (iv) that black holes whose entropy is S < 4/3 are stable or in a saddle-point, whereas when S>4/3 they are always unstable, hence there is no definite answer on whether such transition occurs. Since thermodynamics geometry takes the view that the entropy is an extensive quantity, while the Poincaré method does not require extensiveness, it is valuable to present here the analysis based on both approaches, and so we do.

In this paper we discuss a formulation of extended phase space thermodynamics of black holes in Anti de Sitter (AdS) spacetimes from the contact geometry point of view. Thermodynamics of black holes can be understood within the framework of contact geometry as flows of vector fields generated by Hamiltonian functions on equilibrium submanifolds in the extended phase space that naturally incorporates the structure of a contact manifold. Deformations induced by the contact vector fields are used to construct various maps among thermodynamic quantities. Thermodynamic variables and equations of state of Schwarzschild black holes are mapped to that of Reissner-Nordstr\"{o}m black holes in AdS, with charge as the deformation parameter. In addition, the equations of state of general black holes in AdS are shown to emerge from the high-temperature ideal gas limit equations via suitable deformations induced by contact vector fields. The Hamilton-Jacobi formalism analogous to mechanics is set up, and the corresponding characteristic curves of contact vector fields are explicitly obtained to model thermodynamic processes of black holes. Extension to thermodynamic cycles in this framework is also discussed.

It has been argued that charged Anti-de Sitter (AdS) black holes have similar thermodynamic behavior as the Van der Waals fluid system, provided one treats the cosmological constant as a thermodynamic variable (pressure) in an extended phase space. In this paper, we disclose the deep connection between charged AdS black holes and Van der Waals fluid system from an alternative point of view. We consider the mass of an AdS black hole as a function of square of the charge Q2 instead of the standard Q, i.e. M=M(S,Q2,P). We first justify such a change of view mathematically and then ask if a phase transition can occur as a function of Q2 for fixed P. Therefore, we write the equation of state as Q2=Q2(T,Ψ) where Ψ (conjugate of Q2) is the inverse of the specific volume, Ψ=1/v. This allows us to complete the analogy of charged AdS black holes with Van der Waals fluid system and derive the phase transition as well as critical exponents of the system. We identify a thermodynamic instability in this new picture with real analogy to Van der Waals fluid with physically relevant Maxwell construction. We therefore study the critical behavior of isotherms in Q2–Ψ diagram and deduce all the critical exponents of the system and determine that the system exhibits a small–large black hole phase transition at the critical point (Tc,Qc2,Ψc). This alternative view is important as one can imagine such a change for a given single black hole i.e. acquiring charge which induces the phase transition. Finally, we disclose the microscopic properties of charged AdS black holes by using thermodynamic geometry. Interestingly, we find that scalar curvature has a gap between small and large black holes, and this gap becomes exceedingly large as one moves away from the critical point along the transition line. Therefore, we are able to attribute the sudden enlargement of the black hole to the strong repulsive nature of the internal constituents at the phase transition.

Black holes in anti-de Sitter (AdS) backgrounds play a pivotal role in the gauge/gravity duality where they determine, among other things, the approach to equilibrium of the dual field theory. We undertake a detailed analysis of perturbed Kerr-AdS black holes in four- and five-dimensional spacetimes, including the computation of its quasinormal modes, hydrodynamic modes and superradiantly unstable modes. Our results shed light on the possibility of new black hole phases with a single Killing field, possible new holographic phenomena and phases in the presence of a rotating chemical potential, and close a crucial gap in our understanding of linearized perturbations of black holes in anti-de Sitter scenarios.