Smarr Relation Research Articles

Quantum charged black holes

Within the framework of braneworld holography, we construct a quantum charged black hole localized on a three-dimensional anti-de Sitter (AdS) brane that intersects the asymptotic boundary of the four-dimensional AdS spacetime at the conformal defects and incorporates quantum backreaction effects from the conformal field theory (CFT) on the brane. This quantum charged black hole is an exact solution of the semiclassical gravitational equation corresponding to a theory with higher curvature gravity and nonminimally coupled nonlinear gauge field. In the framework of double holography, we investigate the thermodynamics of the quantum charged black hole from three perspectives: a pure bulk perspective, in which four-dimensional classical Einstein gravity couples to Maxwell electrodynamics and a codimension-one tensional brane; a brane perspective, where semiclassical higher curvature gravity is subject to quantum backreaction from the holographic CFT on the brane, yielding a quantum charged black hole; and a boundary perspective, where the defect CFT is coupled to a boundary CFT at the asymptotic boundary and the degrees of freedom for defect quantum conformal matter is considered. In so doing, we obtain doubly holographic formulations of both the first law of thermodynamics and the Smarr (energy) relations for the quantum charged black holes.

Thermodynamics of Magnetic Black Holes with Nonlinear Electrodynamics in Extended Phase Space

We study Einstein’s gravity in AdS space coupled to nonlinear electrodynamics. Thermodynamics in extended phase space of magnetically charged black holes is investigated. We compute the metric and mass functions and their asymptotics, showing that black holes may have one or two horizons. The metric function is regular, f(0)=1, and corrections to the Reissner–Nordström solution are in the order of O(r−3) when the Schwarzschild mass is zero. We prove that the first law of black hole thermodynamics and the generalized Smarr relation hold. The magnetic potential and vacuum polarization conjugated to coupling are computed and depicted. We calculate the Gibbs free energy and the heat capacity showing that first-order and second-order phase transitions take place.

Extended phase space thermodynamics of magnetized black holes with nonlinear electrodynamics

Einstein’s gravity in AdS space coupled to nonlinear electrodynamics (NED) with two parameters is studied. We investigate magnetically charged black holes. The metric and mass functions and their asymptotic are obtained showing that black holes may have one or two horizons. Thermodynamics in extended phase space was studied and it was proven that the first law of black hole thermodynamics and the generalized Smarr relation hold. The magnetic potential and the vacuum polarization conjugated to coupling (NED parameter), are computed and depicted. We calculate the Gibbs free energy and heat capacity.

Thermodynamics of Taub-NUT-AdS spacetimes

We apply the generalised Komar method proposed in [arXiv:2208.05494] to Taub-NUT-AdS spacetime in the theory of Einstein gravity plus a cosmological constant. Based on a generalised closed 2-form, we derive the total mass and NUT charge of the Taub-NUT-AdS spacetime. Together with other thermodynamic quantities calculated through standard method, we conform the first law and Smarr relation. Then, we consider charged AdS NUT spacetimes in Einstein–Maxwell theory with a cosmological constant, and show that the generalised Komar method works, too. We obtain all the thermodynamic quantities, and the first law and Smarr relation are checked to be satisfied automatically.

Bulk-boundary thermodynamics of charged black holes in higher-derivative theory

The connection between bulk and boundary thermodynamics in Einstein–Maxwell theory is well established using AdS/CFT correspondence. In the context of general higher-derivative gravity coupled to a U(1) gauge field, we examine the resemblance of the first law of thermodynamics between bulk and boundary, followed by an extended phase space description on both sides. Higher-derivative terms related to different powers of the string theory parameter α′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha '$$\\end{document} emerged from a consistent truncation in the bulk supergravity action. We demonstrate that one must include the fluctuation of α′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha '$$\\end{document} in the bulk thermodynamics as a bookkeeping tool to match the bulk first law and Smarr relation with the boundary side. Consequently, the Euler relation and the boundary first law are altered by adding two central charges (a,c).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\ exttt{a}}, {\ exttt{c}}).$$\\end{document} To support our general conclusion, we consider the black hole in Gauss–Bonnet gravity and the general four-derivative theory. Finally, we examine the bulk and boundary aspects of the extended phase space description for higher-derivative-corrected black holes.

Scaling symmetry, Smarr relation, and the extended first law in lower-dimensional Lovelock gravity

Recently, it was discovered that lower-dimensional versions of Lovelock gravity exist as scalar-tensor theories that are examples of Horndeski gravity. We study the thermodynamics of the static black hole solutions in these theories up to cubic order through Euclidean methods. Considering solutions with spherical, planar and hyperbolic event horizons (k=+1,0,−1), we show that the universality of the thermodynamics for planar black holes (k=0) and the extended first law that include the variation of the couplings together with their associated potentials hold also in lower dimensions. We find that in D=4,6 where the 2nd- and the 3rd-order Lovelock Lagrangians are boundary terms respectively, the Smarr relation is modified since the entropy is not a homogenous function in these dimensions. We also present a derivation of the Smarr relation and its modified version based on the global scaling properties of the reduced action that is used to obtain the solutions consistently. Unlike the other hairy black hole solutions that have been analyzed before, despite the terms in the reduced action that break the scaling symmetry, the derivation still follows from a conserved Noether charge.

Barrow entropy and AdS black holes in RPS thermodynamics

In this paper, we examine the restricted phase space (RPS) thermodynamics for charged AdS black holes by considering the impact of quantum gravity on the event horizon area. The primary aim of this work is to elucidate the influence of quantum gravitational effects on thermodynamic behaviors, critical phenomena, phase transitions, and the stability of black holes. We observe that charged AdS black holes exhibit thermodynamic behavior similar to that of Van der Waals fluids when influenced by quantum gravity. Furthermore, we introduce a novel black hole thermodynamic phenomenon, which we term “resistance of phase transitions”. Our study uncovers a violation of the homogeneity property of the Smarr relation in RPS thermodynamics due to the effects of quantum gravity.

Aspects of three-dimensional C-metric

In this work, we present an extensive analysis of the thermodynamics and holographic properties of three-dimensional C-metrics in the FG gauge, where we find that the free energy is equal to the Euclidean on-shell action with a generic conformal factor. For the black hole solutions we find that Smarr relation and the first law of thermodynamics can be formulated when the contributions of the boundary entropy are considered. We also compute holographic entanglement entropy following the AdS/BCFT formalism. By comparing the free energies of different bulk solutions with a fixed flat torus boundary geometry, we find that a specific type of accelerating black hole is dominant in the high temperature regime.

Magnetic Black Hole Thermodynamics in an Extended Phase Space with Nonlinear Electrodynamics.

We study Einstein's gravity coupled to nonlinear electrodynamics with two parameters in anti-de Sitter spacetime. Magnetically charged black holes in an extended phase space are investigated. We obtain the mass and metric functions and the asymptotic and corrections to the Reissner-Nordström metric function when the cosmological constant vanishes. The first law of black hole thermodynamics in an extended phase space is formulated and the magnetic potential and the thermodynamic conjugate to the coupling are obtained. We prove the generalized Smarr relation. The heat capacity and the Gibbs free energy are computed and the phase transitions are studied. It is shown that the electric fields of charged objects at the origin and the electrostatic self-energy are finite within the nonlinear electrodynamics proposed.

Static black hole in minimal Horndeski gravity with Maxwell and Yang-Mills fields and some aspects of its thermodynamics

In this work we obtain a static spherically symmetric charged black hole solution in the framework of minimal Horndeski gravity with additional Maxwell and Yang-Mills fields. The obtained solution is examined; in particular, its asymptotics are studied. Thermodynamics of the black hole is investigated, namely, we use an effective surface gravity to derive black hole temperature. To obtain the first law of black hole thermodynamics, the Wald method is applied. We also use the extended thermodynamics approach, namely, it allows us to derive the Smarr relation, Gibbs free energy, and the thermal equation of state. The study of thermal values in the extended space shows rich phase behavior, in particular, the domain where the first-order phase transition takes place and the critical point with the second-order phase transition. We also study thermal behavior near the critical point, obtain critical exponents, and analyze the Ehrenfest equations at the critical point. Finally, we calculate the Prigogine-Defay ratio confirming the conclusion about the second-order phase transition at the critical point. Published by the American Physical Society 2024

Thermodynamics and Phase Transitions of Dyonic AdS Black Holes in Gauss-Bonnet-Scalar Gravity

In this paper, by treating the cosmological constant as a thermodynamic pressure, we study the thermodynamics and phase transitions of the dyonic AdS black holes in Gauss-Bonnet-Scalar gravity, where the conformal scalar field is considered. In a more general extended phase space, we first verified the first law of black hole thermodynamics, and find that it is always true. Meanwhile, the corresponding Smarr relation is also obtained. Then, we found that this black hole exhibits interesting critical behaviors in six dimensions, i.e., two swallowtails can be observed simultaneously. Interestingly, in a specific parameter space, we observed the small/intermediate/large black hole phase transitions, with the triple point naturally appearing. Additionally, the small/large black hole phase transition, similar to the liquid/gas phase transition of the van der Waals fluids, can also be found in other parameter regions. Moreover, we note that the novel phase structure composed of two separate coexistence curves discovered in the dyonic AdS black holes in Einstein-Born-Infeld gravity disappears in Gauss-Bonnet-Scalar gravity. This suggests that this novel phase structure may be related to gravity theory, and importantly, it is generally observed that the triple point is a universal property of dyonic AdS black holes. On the other hand, we calculated the critical exponents near the critical points and found that they share the same values as in mean field theory. Finally, it is true that these results will provide some deep insights into the interesting thermodynamic properties of the dyonic AdS black holes in the background of conformal scalar fields.

Einstein-AdS Gravity Coupled to Nonlinear Electrodynamics, Magnetic Black Holes, Thermodynamics in an Extended Phase Space and Joule–Thomson Expansion

We studied Einstein’s gravity with negative cosmological constant coupled to nonlinear electrodynamics proposed earlier. The metric and mass functions and corrections to the Reissner–Nordström solution are obtained. Black hole solutions can have one or two horizons. Thermodynamics and phase transitions of magnetically charged black holes in Anti-de Sitter spacetime are investigated. The first law of black hole thermodynamics is formulated and the generalized Smarr relation is proofed. By calculating the Gibbs free energy and heat capacity we study the black hole stability. Zero-order (reentrant), first-order, and second-order phase transitions are analyzed. The Joule–Thomson expansion is considered, showing the cooling and heating phase transitions. It was shown that the principles of causality and unitarity are satisfied in the model under consideration.

Einstein–AdS gravity coupled to modified arctan-electrodynamics, magnetic black hole thermodynamics and Joule–Thomson expansion

The metric and mass functions of Einstein–AdS gravity coupled to modified arctan- electrodynamics (proposed here) are obtained. We find corrections to the Reissner–Nordström solution. It is demonstrated that if the Schwarzschild mass is zero and as r→0 the asymptotic has the de Sitter core. Thermodynamics of magnetically charged black holes in Anti-de Sitter spacetime in an extended phase space, where the cosmological constant is treated as a pressure and the black hole mass is considered as the chemical enthalpy, is studied. It is shown that the black hole thermodynamics mimics the Van der Waals liquid–gas thermodynamics. The magnetic potential and vacuum polarization are calculated. We formulate the first law of black hole thermodynamics and proof that the generalized Smarr relation holds. The critical specific volume, temperature and pressure are evaluated and the critical behaviour of black holes is investigated. We calculate and study the Gibbs free energy and show that phase transitions take place. The local stability are studied by calculating the heat capacity. The black hole Joule–Thomson adiabatic expansion, cooling and heating phase transitions are investigated. We evaluate the Joule–Thomson coefficient and the inversion temperature.

Charged solution with equal metric ansatz in Gauss–Bonnet theory coupled to scalar field

In this study, we apply the Gauss–Bonnet equation of motions coupled with the scalar field and potential and derive a new charged spherically symmetric black hole (BH) solution. In this study, we derive this BH using an equal metric potential, i.e., gtt=grr. Among many things, we succeed in determining the form of the scalar field, the function of the scalar, and the Gauss–Bonnet term. We investigate the physics of this BH solution and show that we have additional terms than the Riessner Nordström BH solution, i.e., O(1r6) and O(1r9). The existence of these additional terms is due to the contribution of the scalar field ξ. These additional terms are responsible for reproducing the multi-horizon BH solution. Due to the existence of these additional terms, the modified first law of thermodynamics and the corresponding Smarr relation are derived.

Thermodynamic quantities and phase transitions of five-dimensional de Sitter hairy spacetime* *Supported by the National Natural Science Foundation of China (12075143)

In this study, we take the mass, electric charge, hair parameter, and cosmological constant of five-dimensional de Sitter hairy spacetime as the state parameters of the thermodynamic system, and when these state parameters satisfy the first law of thermodynamics, the equivalent thermodynamic quantities of spacetime and the Smarr relation of five-dimensional de Sitter hairy spacetime are obtained. Then, we study the thermodynamic characteristics of the spacetime described by these equivalent thermodynamic quantities and find that de Sitter hairy spacetime has a phase transition and critical phenomena similar to those of van de Waals systems or charged AdS black holes. It is shown that the phase transition point of de Sitter hairy spacetime is determined by the ratio of two event horizon positions and the cosmic event horizon position. We discuss the influence of the hair parameter and electric charge on the critical point. We also find that the isochoric heat capacity of the spacetime is not zero, which is consistent with the ordinary thermodynamic system but differs from the isochoric heat capacity of AdS black holes, which is zero. Using the Ehrenfest equations, we prove that the critical phase transition is a second order equilibrium phase transition. Research on the thermodynamic properties of five-dimensional de Sitter hairy spacetime lays a foundation for finding a universal de Sitter spacetime thermodynamic system and studying its thermodynamic properties. Our universe is an asymptotically dS spacetime, and the thermodynamic characteristics of de Sitter hairy spacetime will help us understand the evolution of spacetime and provide a theoretical basis to explore the physical mechanism of the accelerated expansion of the universe.

Magnetic black holes within Einstein–AdS gravity coupled to nonlinear electrodynamics, extended phase space thermodynamics and Joule–Thomson expansion

Einstein’s gravity in Anti-de Sitter space coupled to nonlinear electrodynamics is studied. We analyse the metric, mass functions, and corrections to the Reissner–Nordström solution. Magnetic black holes thermodynamics in extended phase space is investigated. We formulate the first law of black hole thermodynamics showing that the generalized Smarr relation holds. The black hole stability is studied by evaluating the Gibbs free energy and heat capacity. To study the cooling and heating phase transitions of black holes, we consider the Joule–Thomson isenthalpic expansion. The Joule–Thomson coefficient and the inversion temperature are calculated.

Breaking of the universal nature of the central charge criticality in AdS black holes in Gauss-Bonnet gravity

In this paper, we have studied the thermodynamics of Gauss-Bonnet black holes in D-dimensional $AdS$ spacetime. Here, the cosmological constant ($\Lambda$), Newton's gravitational constant ($G$) and the Gauss-Bonnet parameter ($\alpha$) are varied in the bulk, and a mixed first law is rewritten considering central charge ($C$) (of dual boundary conformal theory) and its conjugate variable utilising the gauge-gravity duality. A novel universal nature of central charge near the critical point of black hole phase transition in Einstein's gravity has been observed in \cite{mann1}. We observe that this universal nature breaks when such phase transition is considered for black holes in the Gauss-Bonnet gravity. Apart from this, treating the Gauss-Bonnet parameter as a thermodynamic variable as suggested in \cite{kastori} in light of the consistency between first law and the Smarr relation leads to modified thermodynamic volume (conjugate to variable cosmological constant), adding to a new understanding of the Van der Waals gas like behaviour of the black holes in higher dimensional and higher curvature gravity theories. Our analysis considers a general $D$ dimensional background. We have then imposed a greater focus in the analysis of the phase structure of the five dimensional Gauss-Bonnet spacetime. Our analysis also shows that the general universal nature of the critical value of the central charge (which was present in four dimensional $AdS$ spacetime), breaks down in case of five dimensional $AdS$ spacetime even in the absence of Gauss-Bonnet gravity. This finding indicates the universal nature of the central charge may be a special feature of the four dimensional $AdS$ spacetime only.

R2 corrections to the black string instability and the boosted black string

We consider a perturbative Gauss-Bonnet term supplementing the Einstein-Hilbert action, and evaluate its effect on the spectrum of the scalar mode that triggers the Gregory-Laflamme instability of black strings in five dimensional General Relativity. After studying some properties of the static black string, we provide the correction to the Lichnerowicz operator up to $\mathcal{O}(\alpha^2)$. For the scalar mode of the gravitational perturbation, we find a master variable and study its spectrum, providing an analysis of the regime of validity of our scheme. We show that the instability persists under the inclusion of the $R^2$-correction, and that the critical wavelength increases with the value of $\alpha/r_+^2<<1$, providing new evidence of the phenomenon which was already observed in other approaches. We also construct the boosted black strings and compute the correction to the mass, the momentum and the tension due to the higher curvature term. The presence of the dimensionful coupling $\alpha$ spoils the validity of the Smarr relation, which is the gravitational version of the Euler relation which must hold for every homogeneous, thermodynamic system. We show that this identity can be restored by working in an extended thermodynamic setup that includes variations of the Gauss-Bonnet coupling.

Thermodynamics of Taub-NUT and Plebanski solutions

We observe the parallel between the null Killing vector on the horizon and degenerate Killing vectors at both north and south poles in Kerr-Taub-NUT and general Plebanski solutions. This suggests a correspondence between the pairs of the angular momentum/velocity and the NUT charge/potential. We treat the time as a real line such that the Misner strings are physical. We find that the NUT charge spreads along the Misner strings, analogous to that the mass in the Schwarzschild black hole sits at its spacetime singularity. We develop procedures to calculate all the thermodynamic quantities and we find that the results are consistent with the first law (Wald formalism), the Euclidean action and the Smarr relation. We also apply the Wald formalism, the Euclidean action approach, and the (generalized) Komar integration to the electric and magnetic black holes in a class of EMD theories, and also to boosted black strings and Kaluza-Klein monopoles in five dimensions, to gain better understandings of how to deal with the subtleties associated with Dirac and Misner strings.

Magnetic black holes in AdS space with nonlinear electrodynamics, extended phase space thermodynamics and Joule–Thomson expansion

This paper studies thermodynamics of magnetically charged black holes in Anti-de Sitter space in an extended phase space. The cosmological constant is considered as a pressure and the black hole mass is treated as the chemical enthalpy. The black hole thermodynamics is similar to the Van der Walls liquid–gas thermodynamics. Quantities conjugated to the nonlinear electrodynamics parameter and a magnetic charge are obtained. The first law of thermodynamics and the generalized Smarr relation take place. We investigate critical behavior of black holes and Joule–Thomson expansion. The Gibbs free energy, the Joule–Thomson coefficient and the inversion temperature are calculated.