Extended Phase Space Research Articles

Is the Copenhagen Interpretation Inapplicable to Quantum Cosmology?

It is generally accepted that the Copenhagen interpretation is inapplicable to quantum cosmology, by contrast with the many worlds interpretation. I shall demonstrate that the two basic principles of the Copenhagen interpretation, the principle of wholeness and the principle of complementarity, do make sense in quantum gravity, since we can judge about quantum gravitational processes in the very early Universe by their vestiges in our macroscopic Universe. I shall present the extended phase space approach to quantum gravity and show that it can be interpreted in the spirit of the Everett’s “relative states” formulation, while there is no contradiction between the “relative states” formulation and the mentioned basic principles of the Copenhagen interpretation.

Revisiting black hole thermodynamics in massive gravity: charged particle absorption and infalling shell of dust * * Supported by the Natural Science Foundation of China (11705161, 11675139, 11875220). Xiao-Mei Kuang is also partly Supported by Natural Science Foundation of Jiangsu Province (BK20170481) and Fok Ying Tung Education Foundation (171006). Shi-Qian Hu is also Supported by the Postgraduate Reaearch & Practice

In this study, we apply two methods to consider the variation of massive black holes in both normal and extended thermodynamic phase spaces. The first method considers a charged particle being absorbed by the black hole, whereas the second considers a shell of dust falling into it. With the former method, the first and second laws of thermodynamics are always satisfied in the normal phase space; however, in the extended phase space, the first law is satisfied but the validity of the second law of thermodynamics depends upon the model parameters. With the latter method, both laws are valid. We argue that the former method's violation of the second law of thermodynamics may be attributable to the assumption that the change of internal energy of the black hole is equal to the energy of the particle. Finally, we demonstrate that the event horizon always ensures the validity of weak cosmic censorship in both phase spaces; this means that the violation of the second law of thermodynamics, arising under the aforementioned assumption, does not affect the weak cosmic censorship conjecture. This further supports our argument that the assumption in the first method is responsible for the violation and requires deeper treatment.

Thermodynamic criticality of d-dimensional charged AdS black holes surrounded by quintessence with a cloud of strings background

We focus on the study of the exact solution corresponding to charged AdS black holes surrounded by quintessence with a cloud of strings present in higher dimensional spacetime. We then investigate its corresponding thermodynamic criticality in the extended phase space and show that the spacetime dimension has no effect on the existence of small/large phase transition for such black holes. The heat capacity is evaluated and the geothermodynamics of Quevedo analyzed for different spacetime dimensions with the cloud of strings and quintessence parameters. We calculate the critical exponents describing the behavior of relevant thermodynamic quantities near the critical point. Finally, we also discuss the uncharged case, show how it is sensitive to the quintessence and strings cloud parameters, and when the thermodynamic behavior of the uncharged black holes is similar to Van der Waals fluid.

Testing the weak cosmic censorship conjecture in torus-like black hole under charged scalar field

We investigate weak cosmic censorship conjecture in charged torus-like black hole by the complex scalar field scattering. Using the relation between the conserved quantities of a black hole and the scalar field, we can calculate the change of the energy and charge within the infinitesimal time. The change of the enthalpy is connected to the change of energy, then we use those results to test whether the first law, the second law as well as the weak cosmic censorship conjecture are valid. In the normal phase–space, the first law of thermodynamics and the weak cosmic censorship conjecture are valid, and the second law of thermodynamics is not violated. For the specific black hole under scalar field scattering we consider, in the extended phase–space, the first law of thermodynamics and the weak cosmic censorship conjecture are valid. However, the second law of thermodynamics is violated when the black hole’s initial charge reaches a certain value.

Holographic phase transition from novel Gauss\u2013Bonnet AdS black holes

In the framework of holography, we discuss on the phase transition behavior from a novel Gauss–Bonnet AdS black hole discovered in [Phys. Rev. Lett. 124, 081301 (2020)] in both charged and neutral cases. First, we explore the thermodynamic behavior for the black hole entropy by investigating T-S diagram, specific heat capacity and free energy, and we find that, in the T-S plane, the black hole exhibits a van der Waals phase transition similar to that in P-V plane in extended phase space. Secondly, we detect thermodynamic phase transition for the two point correlation function, and find that, the van der Waals phase transition can also be displayed in the T-delta L plane which is completely similar to that of black hole entropy in the T-S plane. Our result turns out that, not only the charged but also neutral cases, we can observe holographic van der Waals phase transition for the novel Gauss–Bonnet AdS black hole by employing the black hole entropy and two point correlation function.

Static spherically symmetric Einstein-Yang-Mills-dilaton black hole and its thermodynamics

A static black hole with spherical symmetry is obtained and examined in the framework of Einstein-Yang-Mills-dilaton theory. The obtained black hole solution allowed us to derive and investigate entropy, temperature, and heat capacity. To better examine the thermodynamics of the black hole, extended phase space is also used. On this ground, the equation of state is obtained and studied. We also investigate the Gibbs free energy and it is shown that below the critical temperature the system demonstrates phase transitions of the first as well as of the zeroth order, which is a notable feature for other types of dilaton black holes. At the end, critical exponents for the black hole are calculated.

Overcharging problem and thermodynamics in extended phase spaces

In this paper, the overcharging problem and thermodynamics in the extended phase spaces of the five-dimensional spherically symmetric topological black holes are investigated by absorptions of scalar particles and fermions. The cosmological constant is regarded as a variable related to pressure and its conjugate quantity is a thermodynamic volume. The first law of thermodynamics is recovered. The second law is violated in the extended phase space of the extremal and near-extremal black holes. The overcharging problem is tested by the existence of the event horizons. The event horizon is determined by the metric component f(r). The minimal values of the metric component at the final stage show that the extremal and near-extremal black holes can not be overcharged.

Newman–Penrose scalars and black hole equations of state

In this work we explore the connections between Newman-Penrose scalars, including the Penrose-Rindler $\mathcal{K}$-curvature, with the equation of state of asymptotically Anti-de Sitter Reissner-Nordstr\"om black holes. After briefly reviewing the equation of state for these black holes from the point of view of both the Extended Phase Space and the Horizon Thermodynamics approaches, a geometric splitting is given for such an equation in terms of the non vanishing Newman-Penrose scalars which define the $\mathcal{K}$-curvature at the horizon. From this splitting, a possible thermodynamical interpretation is developed for such scalars in the context of the black hole thermodynamics approaches initially discussed. Afterwards, the square root of the Bel-Robinson tensor is employed to propose conditions at the horizons in terms of pressures or energy densities, which can be understood as alternative thermodynamical definitions of these surfaces.

Two θμν -deformed covariant relativistic quantum phase spaces as Poincaré-Hopf algebroids

We consider two quantum phase spaces which can be described by two Hopf algebroids linked with the well-known $\theta_{\mu \nu }$-deformed $D=4$ Poincare-Hopf algebra $\mathbb{H}$. The first algebroid describes $\theta_{\mu \nu }$-deformed relativistic phase space with canonical NC space-time (constant $\theta_{\mu \nu }$ parameters) and the second one incorporates dual to $\mathbb{H}$ quantum $\theta_{\mu \nu }$-deformed Poincare-Hopf group algebra $\mathbb{G}$, which contains noncommutative space-time translations given by $\Lambda $-dependent $\Theta_{\mu \nu }$ parameters ($% \Lambda $ $\equiv \Lambda_{\mu \nu }$ parametrize classical Lorentz group). The canonical $\theta_{\mu \nu }$-deformed space-time algebra and its quantum phase space extension is covariant under the quantum Poincare transformations described by $\mathbb{G}$. We will also comment on the use of Hopf algebroids for the description of multiparticle structures in quantum phase spaces.

Extended thermodynamics and microstructures of four-dimensional charged Gauss-Bonnet black hole in AdS space

The discovery of new four-dimensional black hole solutions presents a new approach to understand the Gauss-Bonnet gravity in low dimensions. In this paper, we test the Gauss-Bonnet gravity by studying the phase transition and microstructures for the four-dimensional charged AdS black hole. In the extended phase space, where the cosmological constant and the Gauss-Bonnet coupling parameter are treated as thermodynamic variables, we find that the thermodynamic first law and the corresponding Smarr formula are satisfied. Both in the canonical ensemble and grand canonical ensemble, we observe the small-large black hole phase transition, which is similar to the case of the van der Walls fluid. This phase transition can also appear in the neutral black hole system. Furthermore, we construct the Ruppeiner geometry, and find that besides the attractive interaction, the repulsive interaction can also dominate among the microstructures for the small black hole with high temperature in a charged or neutral black hole system. This is quite different from the five-dimensional neutral black hole, for which only dominant attractive interaction can be found. The critical behaviors of the normalized scalar curvature are also examined. These results will shed new light into the characteristic property of four-dimensional Gauss-Bonnet gravity.

Joule–Thomson expansion of the Bardeen-AdS black holes

The Joule-Thomson expansion process is studied for Bardeen-AdS black holes in the extended phase space. Firstly, we get Joule-Thomson coefficient and find that the divergent point of Joule-Thomson coefficient coincides with the zero point of temperature. The inversion curves are also obtained from the zero point of Joule-Thomson coefficient. Then the minimum inversion temperature and the corresponding mass are obtained. In addition, the ratio between minimum inversion and critical temperature for Bardeen-AdS black holes is also calculated. We obtain the isenthalpic curve in $T$-$P$ graph and demonstrate the cooling-heating region by the inversion curve. An interesting phenomenon we get is that black hole is always in heating process. The same phenomenon can be also obtained from the charged AdS black holes.

Thermodynamic extended phase space and P-V criticality of black holes at Pure Lovelock gravity

In this work the chemistry of asymptotically AdS black hole, for the charged and uncharged solutions of Pure Lovelock gravity, is discussed. The charged case behaves as a Van der Waals fluid and whose first order phase transitions, between small stable/large stable black holes, are analogous to the liquid/gas phase transitions as in AdS black hole for Einstein Hilbert theory. However, the thermodynamics behavior differs from the generic Lovelock theory, because there is a unique critical point, unlike the generic case where there may be more than one critiackcal point. Also, it is shown that the thermodynamics behavior of the Pure Lovelock black holes (in the extended phase space) can be represented by variables that are analytic functions of n and d, where n corresponds to the highest power of the Riemann tensor in the Lagrangian and d corresponds to the number of dimensions. This allows to obtain several results. For instance, the critical compressibility factor Z is a function of n and d that satisfies Z<1 strictly, matching the behaviour of a real gas, but the new values computed differ from the 3/8 value of a Van der Waals gas except for d=4 and n=1. New versions of the Smarr formula and equation of state and its behavior near the critical points are computed, which are also functions of n, d and Z. For all the cases the critical exponent are similar to those of the Van der Waals fluid. The first law of thermodynamics, in the extended space, is deduced by the variation of parameters of the Pure Lovelock solution. The entropy, volume and electric potential are consistent with the previously known results in the literature.

Hamiltonian thermodynamics in the extended phase space: a unifying theory for non-linear molecular dynamics and classical thermodynamics

The Hamiltonian theory of thermodynamics for reversible and irreversible processes is presented in detail. Particularly, it is demonstrated that non-linear molecular dynamics and thermodynamics of equilibrium and non-equilibrium processes can merge in a single dynamical theory by constructing a homogeneous of first degree in momenta Hamiltonian function on the extended thermodynamic state space and in the entropy representation. The geometrical properties of the physical equilibrium state manifold are discussed and results of numerical experiments with the Henon–Heiles model that dissipates in a simple thermodynamic system with multiple friction parameters are presented. It is well known that this classical paradigm carries most of the non-linear dynamical characteristics of generic molecular systems. The construction of homogeneous Hamiltonians for composite thermodynamic systems is also outlined in the appendixes, where the theory of Hamiltonian thermodynamics is developed in current differential geometry terms.

Thermodynamics of Gauss–Bonnet–de Sitter black holes

We investigate the thermodynamics of Gauss-Bonnet black holes in asymptotically de Sitter spacetimes embedded in an isothermal cavity, via a Euclidean action approach. We consider both charged and uncharged black holes, working in the extended phase space where the cosmological constant is treated as a thermodynamic pressure. We examine the phase structure of these black holes through their free energy. In the uncharged case, we find both Hawking-Page and small-to-large black hole phase transitions, whose character depends on the sign of the Gauss-Bonnet coupling. In the charged case, we demonstrate the presence of a swallowtube, signaling a compact region in phase space where a small-to-large black hole transition occurs.

Three dimensional Lifshitz-like black hole in a special class of F(R) gravity

Regarding a special class of pure F(R) gravity in three dimensions, we obtain, analytically, Lifshitz-like black hole solutions. We check the geometrical properties of the solutions which behave such as charged BTZ black holes in special limit. We also investigate the thermodynamic properties of the solutions and examine the first law of thermodynamics and Smarr formula. In addition, we study thermal stability via the heat capacity and discuss the possibility of criticality in the extended phase space.

A new type of relaxation oscillation in a model with rate-and-state friction

In this paper, we prove the existence of a new type of relaxation oscillation occurring in a one-block Burridge–Knopoff model with Ruina rate-and-state friction law. In the relevant parameter regime, the system is a slow-fast ordinary differential equation with two slow variables and one fast. The oscillation is special for several reasons: firstly, its singular limit is unbounded, the amplitude of the cycle growing like as . As this estimate reflects, the unboundedness of the cycle—for this non-polynomial system—cannot be captured by a simple -dependent scaling of the variables, in contrast to e.g. Gucwa and Szmolyan (2009 Discrete Continuous Dyn. Syst. S 2 783–806). We therefore obtain its limit on the Poincaré sphere. Here we find that the singular limit consists of a slow part on an attracting critical manifold, and a fast part on the equator (i.e. at ) of the Poincaré sphere, which includes motion along a center manifold. The reduced flow on this center manifold runs out along the manifold’s boundary, in a special way, leading to a complex return to the slow manifold. We prove the existence of the limit cycle by showing that a return map is a contraction. The main technical difficulty lies in the fact that the critical manifold loses hyperbolicity at an exponential rate at infinity. We therefore use the method in Kristiansen (2017 Nonlinearity 30 2138–84), applying the standard blowup technique in an extended phase space. In this way, we identify a singular cycle, consisting of 12 pieces, all with desirable hyperbolicity properties, that enables the perturbation into an actual limit cycle for . The result proves a conjecture in Bossolini et al (2017 Nonlinearity 30 2805–34). The Bossolini et al (2017 Nonlinearity 30 2805–34) also includes a preliminary analysis based on the approach in Kristiansen (2017 Nonlinearity 30 2138–84) but several details were missing. We provide all the details in the present manuscript and lay out the geometry of the problem, detailing all of the many blowup steps.

Thermodynamics and weak cosmic censorship conjecture of an AdS black hole with a monopole in the extended phase space * * Supported by National Natural Science Foundation of China (11875095) and Basic Research Project of Science and Technology Committee of Chongqing (cstc2018jcyjA2480)

The first law of black hole thermodynamics has been shown to be valid in the extended phase space. However, the second law and the weak cosmic censorship conjecture have not been investigated extensively. We investigate the laws of thermodynamics and the weak cosmic censorship conjecture of an AdS black hole with a global monopole in the extended phase space in the case of charged particle absorption. It is shown that the first law of thermodynamics is valid, while the second law is violated for the extremal and near-extremal black holes. Moreover, we find that the weak cosmic censorship conjecture is valid only for the extremal black hole, and that it can be violated for the near-extremal black holes, which is different from the previous results.

The Hawking\u2013Page phase transitions in the extended phase space in the Gauss\u2013Bonnet gravity

In this paper, the Hawking–Page phase transitions between the black holes and thermal anti-de Sitter (AdS) space are studied with the Gauss–Bonnet term in the extended phase space, in which the varying cosmological constant plays the role of an effective thermodynamic pressure. The Gauss–Bonnet term exhibits its effects via introducing the corrections to the black hole entropy and Gibbs free energy. The global phase structures, especially the phase transition temperature T_{mathrm{HP}} and the Gibbs free energy G, are systematically investigated, first for the Schwarzschild–AdS black holes and then for the charged and rotating AdS black holes in the grand canonical ensembles, with both analytical and numerical methods. It is found that there are terminal points in the coexistence lines, and T_{mathrm{HP}} decreases at large electric potentials and angular velocities and also decreases with the Gauss–Bonnet coupling constant alpha .

Thermodynamics and weak cosmic censorship conjecture of charged AdS black hole in the Rastall gravity with pressure**Project supported by the National Natural Science Foundation of China (Grant Nos. 11875095 and 11903025), the Basic Research Project of Science and Technology Committee of Chongqing (Grant No. cstc2018jcyjA2480), and the Fundamental Research Funds of ChinaWest Normal University (Grant No. 18Q062).

Treating the cosmological constant as a dynamical variable, we investigate the thermodynamics and weak cosmic censorship conjecture (WCCC) of a charged AdS black hole (BH) in the Rastall gravity. We determine the energy momentum relation of charged fermion at the horizon of the BH using the Dirac equation. Based on this relation, it is shown that the first law of thermodynamics still holds as a fermion is absorbed by the BH. However, the entropy of both the extremal and near-extremal BH decreases in the irreversible process, which means that the second law of thermodynamics is violated. Furthermore, we verify the validity of the WCCC by the minimum values of the metric function h(r) at its final state. For the extremal charged AdS BH in the Rastall gravity, we find that the WCCC is always valid since the BH is extreme. While for the case of near-extremal BH, we find that the WCCC could be violable in the extended phase space (EPS), depending on the value of the parameters of the BH and their variations.

Thermal chaos of a charged dilaton-AdS black hole in the extended phase space

We have studied thermal chaotic behavior in the extended phase space for a charged dilaton-AdS black hole by Melnikov method and present the effect of dilaton parameter on the thermal chaos. Our result show that for the temporal perturbation the thermal chaos in the charged dilaton-AdS black hole occurs only if the perturbation amplitude is larger than certain a critical value, but for the spatially perturbation, the chaos always exists irrespective of perturbation amplitude. These behaviors are similar to those in other AdS black hole, which can be regarded as the common features of the static AdS black holes. Moreover, we also find that the critical temporal perturbation amplitude leading to chaos increases with the dilaton parameter and decreases with the charge. This means that under the temporal perturbation the presence of dilaton parameter makes the onset of chaos more difficult, which differs from that of the charge parameter.