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Super-entropic black hole: horizon embedding. The horizon geometry is embedded in E3 for the following choice of parameters: l = 1, r+=10 and μ=2π.
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We review recent developments on the thermodynamics of black holes in extended phase space, where the cosmological constant is interpreted as thermodynamic pressure and treated as a thermodynamic variable in its own right. In this approach, the mass of the black hole is no longer regarded as internal energy, rather it is identified with the chemica...
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Citations
... Furthermore, to consider black holes as quasi-homogeneous systems, an extended thermodynamic space is required. For example, in the case of asymptotically AdS black hole solutions, the cosmological constant has been interpreted as a new thermodynamic variable with properties consistent with an effective "pressure" [13][14][15]. According to these results, the mass of the black hole no longer has the meaning of internal energy. ...
... is fulfilled. Moreover, as we pointed out before, it is possible to further extend the thermodynamic space if we interpret the curvature radius l as an independent length-scale in the theory [13][14][15]. Hence, in the most general case, Eq. (14) is a quasi-homogeneous function of degree β M = (D + z − 2)/(D − 2)β S − (z + 1)β l . ...
... Notice that for D ≥ 4, z > 1, and b > −r 2(D+z−3) h the above volume is always positive. We might compare the thermodynamic volume with the geometric volume, which is essentially defined as the full D-dimensional volume element over a t = const slice [13,35]. For a stationary planar black hole, it yields ...
We study a particular Einstein-Maxwell-Dilaton black hole configuration with cosmological constant, expressed in terms of the curvature radius, from the point of view of quasi-homogeneous thermodynamics. In particular, we show that the curvature radius and the coupling constant of the matter fields can be treated as thermodynamic variables in the framework of extended thermodynamics, leading in both cases to a van der Waals-like behavior. We also investigate in detail the stability and critical properties of the black holes and obtain results, which are compatible with the mean field approach.
... It is not the first time that a definition for a chemical potential has been proposed for black hole thermodynamics, see for instance [37][38][39][40] in the context of AdS/CFT. However, to the best of my knowledge, this is the first time that the definition (4.4) is proposed. ...
The generalized second law states the total entropy of any closed system as the universe cannot decrease if we include black hole entropy. From the point of view of an asymptotic observer, a black hole can be described at late time as an open system at fixed temperature which can radiate energy and entropy to infinity. I argue that for massless free quantum fields propagating on a black hole background, we can define a black hole dynamical free energy using observables defined at future null infinity which decreases on successive cross sections. The proof of this spontaneous evolution law is similar to Wall's derivation of the generalized second law and relies on the monotonicity properties of the relative entropy. I discuss first the simpler case of the Schwarzschild background in which the grey body factor are neglected and show that in this case the free energy only depends on the Bondi mass, the Hawking temperature and the von Neumann entropy of the propagating quantum fields. Then I argue that taking into account the grey body factors adds a new term to the thermodynamic potential involving the number of particles detected at future null infinity conjugated to a chemical potential. Finally, I discuss the case of the Kerr black hole for which an angular momentum piece needs to be added to the free energy.
... Over the last century, many theories of gravity have been developed to extend GR, and most of them -if not all -contain black holes. These objects, despite their phenomenological differences, appear to obey the same laws of thermodynamics [28][29][30][31]. In this work, we will focus our attention only on the first law. ...
... It has been suggested that one should be able to recover the area law for the entropy in modified theories of gravity such as LL [31]. To do so, one needs to cancel the contribution to the entropy coming from higher-order terms ...
... It is known that, in Horndeski theories of gravity, the speed of gravitons is affected by the non-minimal coupling between the scalar field and the background metric, entailing the need to modify the temperature which enters in the first law. The authors of [31] suggests a similar mechanism as a reason to justify the needed modification of the temperature ensuing the entropy change. Due to the intrinsic nonlinearity of LL theories (remember that only in GR Lagrangian, metric second derivatives appear linearly [59]), one could expect that gravitons on a background propagate with respect to an effective metric which is not the background, and thus move at a speed which is different from the speed of light. ...
A bstract
This work introduces a novel prescription for the expression of the thermodynamic potentials associated with the couplings of a Lanczos-Lovelock theory. These potentials emerge in theories with multiple couplings, where the ratio between them provide intrinsic length scales that break scale invariance. Our prescription, derived from the covariant phase space formalism, differs from previous approaches by enabling the construction of finite potentials without reference to any background. To do so, we consistently work with finite-size systems with Dirichlet boundary conditions and rigorously take into account boundary and corner terms: including these terms is found to be crucial for relaxing the integrability conditions for phase space quantities that were required in previous works. We apply this prescription to the first law of (extended) thermodynamics for stationary black holes, and derive a version of the Smarr formula that holds for static black holes with arbitrary asymptotic behaviour.
... Since the global stability of the system is measured by Gibbs free energy, its global minimum is estimated to be the preferred state of the black hole 74 . We have presented the behaviour of Gibbs free energy G + vs horizon x + and temperature T + of regular AdS black hole in (cf. ...
After obtaining an exact regular-AdS black hole resulting from the coupling of general relativity with nonlinear electrodynamics (NED), we explore the thermodynamics of the extended phase space, treating the cosmological constant ( $$\Lambda$$ Λ ) as the pressure ( P ) of the black holes and its conjugate as thermodynamic volume ( V ). Considering the NED parameter ( g ), we investigate the Hawking temperature, entropy, Gibb’s free energy and specific heat at the horizon radius. Due to the presence of NED charge, the black hole exhibits van der Waals-like phase transition instead of Hawking-Page phase transition, which could be observed through the $$G-T$$ G - T plots, which display a swallowtail pattern below the critical pressure, and it gives rise to second-order phase transitions when pressure attains its critical value. The first-order phase transition shares similarities with the liquid-gas phase transition. We determine the exact critical points and explore the influence of NED on $$P-V$$ P - V criticality, revealing that the isotherms undergo a liquid-gas-like phase transition for temperatures below its critical value $$T_C$$ T C , especially at lower $$T_C$$ T C . The identical critical exponent to that of the van der Waals fluid suggests that the NED does not alter the critical exponents, as observed in other arbitrary AdS black holes.
... This relation can also be derived using the scaling method in Ref. [47]. From the energy expression (36), one can obtain the scaling relations: r h → αr h , L → αL, P → α −2 P , E → αE, S → α 2 S. The Euler's homogeneous function theorem along with the energy (36) tells us that ...
We study the conserved energy of an accelerating AdS black hole by employing the quasilocal formalism, which links the off-shell ADT formalism with the off-shell covariant phase space formalism. In the presence of conical singularities in the accelerating black hole, the expression for the energy depends on the surface term derived from the variation of the action. The essential ingredient of our computations for the quasilocal energy is that the surface contributions come from the conical singularities as well as the ordinary radial boundary. Consequently, the resulting quasilocal energy naturally satisfies the thermodynamic first law for the black hole without any modifications to the thermodynamic variables. Additionally, we obtain the Smarr relation for the black hole using the differential operator method and the scaling argument of the relevant thermodynamic quantities.
... [45][46][47][48][49][50] Afterwards, a series of interesting phase transitions were discovered in the extended phase space, which include the reentrant phase transition and the triple point. [51][52][53][54][55][56][57][58][59][60][61][62][63][64] For example, Wei et al. demonstrated that there are triple points in 6-dimensional GB AdS BHs, [53][54][55] and Frassino et al. found the reentrant phase transition in 3rd-order Lovelock gravity. [54] In the case of higher-order gravity theories, there are also multicritical phase transitions in AdS BHs, i.e., in higher-order Lovelock gravity. ...
By considering the negative cosmological constant Λ as a thermodynamic pressure, we study the thermodynamics and phase transitions of the D-dimensional dyonic AdS black holes (BHs) with quasitopological electromagnetism in Einstein–Gauss–Bonnet (EGB) gravity. The results indicate that the small/large BH phase transition that is similar to the van der Waals (vdW) liquid/gas phase transition always exists for any spacetime dimensions. Interestingly, we then find that this BH system exhibits a more complex phase structure in 6-dimensional case that is missed in other dimensions. Specifically, it shows for D = 6 that we observed the small/intermediate/large BH phase transitions in a specific parameter region with the triple point naturally appeared. Moreover, when the magnetic charge turned off, we still observed the small/intermediate/large BH phase transitions and triple point only in 6-dimensional spacetime, which is consistent with the previous results. However, for the dyonic AdS BHs with quasitopological electromagnetism in Einstein–Born–Infeld (EBI) gravity, the novel phase structure composed of two separate coexistence curves observed by Li et al. [Phys. Rev. D 105 104048 (2022)] disappeared in EGB gravity. This implies that this novel phase structure is closely related to gravity theories, and seems to have nothing to do with the effect of quasitopological electromagnetism. In addition, it is also true that the critical exponents calculated near the critical points possess identical values as mean field theory. Finally, we conclude that these findings shall provide some deep insights into the intriguing thermodynamic properties of the dyonic AdS BHs with quasitopological electromagnetism in EGB gravity.
... The thermodynamics of each EOW brane is described by a JT black hole 1 with a tension parameter changing the cosmological constant effectively. This is reminiscent of the "thermodynamic volume" study of black holes [41][42][43]. Therefore, we may extend the thermodynamics with the thermodynamic volume and pressure. ...
... Since L 2 (1 + τ 2 ) provides an effective cosmological constant, the variation of the tension is equivalent to the variation of the cosmological constant for the JT black hole. This consideration is reminiscent of studies of the thermodynamic volume [41][42][43]. The variation of τ leads to ...
... The conjugate volume V τ is called the thermodynamic volume in literature. See [42,43,47,48] for instance. Interestingly, this is related to the volume inside the horizon as follows: ...
In this work, we study the end-of-the-world (EOW) branes anchored to the boundaries of BCFT $${}_2$$ 2 dual to the BTZ black hole. First, we explore the thermodynamics of the boundary system consisting of the conformal boundary and two EOW branes. This thermodynamics is extended by the tension appearing as the effective cosmological constant of JT black holes on the EOW branes. The tension contribution is identified with the shadow entropy equivalent to the boundary entropy of the BCFT $${}_2$$ 2 . The thermodynamics of the JT black holes and the bulk of BCFT $${}_2$$ 2 can be combined into a novel grafted thermodynamics based on the first law. Second, we focus on the observer’s view of the EOW branes by lowering the temperature. We show that the EOW branes generate a scale called “reefs” inside the horizon. This scale also appears in the grafted thermodynamics. At high temperatures, observers on the EOW branes see their respective event horizons. The reef starts to grow relatively to the horizon size at the temperature, $$T_{grow}$$ T grow . As the temperature cools down the reef area fills the entire interior of the JT black holes at the temperature $$T_{out}$$ T out . Then, the observers recognize their horizons disappear and see the large density of the energy flux. At this temperature, the two JT regions become causally connected. This connected spacetime has two asymptotic $$AdS_2$$ A d S 2 boundaries with a conformal matter. Also, we comment on the grafted thermodynamics to higher dimensions in Appendix B.
... is related to the pressure P of the system [79][80][81], and modifying the first law as ...
In the present work, considering critical gravity as a gravity model, an electrically charged topological anti–de Sitter black hole with a matter source characterized by a nonlinear electrodynamics framework is obtained. This configuration is defined by an integration constant, three key structural constants, and a constant that represents the topology of the event horizon. Additionally, based on the Wald formalism, we explore the possibility that this configuration enjoys nontrivial thermodynamic quantities, establishing the corresponding first law of black hole thermodynamics, as well as local stability under thermal and electrical fluctuations. Additionally, via the Gibbs free energy, we note that the topology of the base manifold allows us to compare this charged configuration with respect to the thermal AdS spacetime, allowing us to obtain a Phase Transition. The quasinormal modes and the greybody factor are also calculated by considering the spherical situation. We find that the quasinormal modes exhibit a straightforward change for variations of one of the structural constants.
... This conceptual shift has led to the development of extended phase thermodynamics, altering the conventional first law of black hole mechanics through the addition of a new pdV term [11][12][13][14][15][16][17][18][19][20]. By scrutinizing the P − V critical behavior across various black hole configurations, researchers have uncovered a compelling correspondence between the small/large phase transitions of black holes and the behavior exhibited in the Van der Waals liquid/gas system [21][22][23][24][25][26][27][28]. This interdisciplinary fusion underscores the intricate interplay between gravitational physics, holography, and the fundamental principles governing highenergy phenomena, offering profound insights into the nature of spacetime and its intricate relationship with thermodynamic concepts at the forefront of physical understanding. ...
... There are other good reasons why the regime of slow time evolution is particularly interesting. Our work is mainly motivated by an attempt to generalize the usual black hole thermodynamics [13][14][15][16] to a non-equilibrium setting, where quasi-stationary evolution and the corrections to it play a fundamental role. This new approach to black hole thermodynamics is essential because, for example, Schwarzschild black holes have a negative heat capacity [17], which is incompatible with equilibrium thermodynamics. ...
... Since r * → −∞ when r → r b , one sees that U = π 2 and V = − π 2 correspond to the black hole horizon, while from r → r c corresponding to r * → ∞, on sees that U = − π 2 and V = π 2 correspond to the cosmological horizon. Using the formula (13) and (14), one can then plot the lines of constant t s in the Penrose diagram, see the left diagram in figure 1. Additionally combining (9) with (13) and (14) allows one to plot the lines of constant t as well, which is done in the right diagram in figure 1. One sees there explicitly that these lines of constant t end perpendicularly on both the future black hole and cosmic horizon. ...
... Since r * → −∞ when r → r b , one sees that U = π 2 and V = − π 2 correspond to the black hole horizon, while from r → r c corresponding to r * → ∞, on sees that U = − π 2 and V = π 2 correspond to the cosmological horizon. Using the formula (13) and (14), one can then plot the lines of constant t s in the Penrose diagram, see the left diagram in figure 1. Additionally combining (9) with (13) and (14) allows one to plot the lines of constant t as well, which is done in the right diagram in figure 1. One sees there explicitly that these lines of constant t end perpendicularly on both the future black hole and cosmic horizon. ...
We revisit and improve the analytic study [1] of spherically symmetric but dynamical black holes in Einstein’s gravity coupled to a real scalar field. We introduce a series expansion in a small parameter ε that implements slow time dependence. At the leading order, the generic solution is a quasi-stationary Schwarzschild-de Sitter (SdS) metric, i.e. one where time-dependence enters only through the mass and cosmological constant parameters of SdS. The two coupled ODEs describing the leading order time dependence are solved up to quadrature for an arbitrary scalar potential. Higher order corrections can be consistently computed, as we show by explicitly solving the Einstein equations at the next to leading order as well. We comment on how the quasi-stationary expansion we introduce here is equivalent to the non-relativistic 1/c expansion.
