Black Hole Entropy Research Articles

Phase transition of modified thermodynamics of the Kerr-AdS black hole

Abstract We investigate the critical phenomena of Kerr-AdS black holes in the modified first law of thermodynamics.
The modified black hole thermodynamics exhibits the van der Waals-like phase
structure. All the critical exponents are calculated, and the swallowtail diagram of free energy is
plotted. Comparing with existing results, the main difference is the correspondence between thermal
quantities of the Kerr-AdS black holes and the van der Waals system. In previous work[16],
the correspondence was $(\Omega_H, J)\to (V, P)$. In our work, the correspondence is $(J, {\hat\Omega}_H)\to (V, P)$.
This difference results from rotating effect. The modified black hole thermodynamics is associated
with rotating observers. The free energy in such reference contains an extra rotating energy. This
extra part induces a Legendre transformation in $({\hat\Omega}_H, J)$ cross-section, which yields the different
correspondence.

Gauss-Bonnet AdS planar and spherical black hole thermodynamics and holography

Abstract In this work, we extend the study in \cite{Bilic:2022psx} incorporating the AdS/CFT duality to establish a relationship between the local temperatures (Tolman temperatures) of a large (AdS) spherical and a (AdS) planar Schwarzschild black hole near the AdS boundary considering Gauss-Bonnet curvature correction in the gravitational action. We have shown that the higher curvature corrections appear in the local temperature relationship due to the inclusion of Gauss-Bonnet term in the bulk. By transforming the metric into Fefferman-Graham form, we have calculated the energy density of the conformal fluid at the boundary. The obtained result contains finite coupling corrections which are holographically induced by the Gauss-Bonnet curvature correction in the bulk theory. Following the well known approach of fluid/gravity duality, the energy density of the conformal fluid at the boundary is then compared with the black body radiation energy density. This comparison shows that the energy density is proportional to the temperature of the conformal fluid. The temperature of the conformal fluid is then shown to be related to the Tolman temperature of the black hole which then eventually helps us to establish both the Hawking temperature and Tolman temperature relationship between large spherically symmetric and planar Schwarzschild black holes in Gauss-Bonnet gravity near the AdS boundary.

Restricted Phase Space Thermodynamics of NED-AdS Black Holes

Abstract We study the Restricted Phase Space Thermodynamics (RPST) of magnetically charged Anti de Sitter (AdS) black holes sourced by nonlinear electrodynamics(NED). The first law and the corresponding Euler relation are examined using the scaling properties. While the mass is homogeneous in the first order, the intensive variables are observed to follow zeroth order homogeneity. We use numerical and graphical techniques to find the critical points of the various thermodynamic quantities. By utilizing the re-scaling properties of the equation of states, we study the thermodynamic processes using different pairs of variables. From our analysis, we infer that although the RPS thermodynamics of NED-AdS black hole resembles those of RN-AdS, Kerr-AdS, Kerr-Sen-Ads black holes in most of its aspects, hinting at a possible universality, there exists one particular μ − C process that differs in its behaviour from its counterparts in earlier reported works.

Refined thermodynamics of black holes with proper conserved charges

Refined thermodynamics of black holes with proper conserved charges

Magnetic black holes in 4D Einstein–Gauss–Bonnet massive gravity coupled to nonlinear electrodynamics

We investigated Einstein–Gauss–Bonnet (EGB) 4D massive gravity coupled to nonlinear electrodynamics (NED) in an anti-de Sitter (AdS) background and found an exact magnetically charged black hole solution. The metric function was analyzed for different values of massive gravity parameters. The first law of black hole thermodynamics and generalized Smarr formula were verified, where we treated the cosmological constant as thermodynamic pressure. We defined vacuum polarization as the conjugate to NED parameter. To analyze the local stability of the black hole, we computed specific heat. We investigated the van der Waals-like/re-entrant phase transition of the black holes and estimated the critical points. We observed small black hole (SBH)/large black hole (LBH) and SBH/intermediate black hole (IBH)/LBH phase transitions. The Joule–Thomson coefficient, inversion, and isenthalpic curves were discussed. Finally, the minimum inversion temperature and the corresponding event horizon radius were obtained using numerical techniques.

Imprints of Barrow–Tsallis cosmology in primordial gravitational waves

Both the Barrow and Tsallis δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} entropies are one-parameter generalizations of the black-hole entropy, with the same microcanonical functional form. The ensuing deformation is quantified by a dimensionless parameter Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document}, which in the case of Barrow entropy represents the anomalous dimension caused by quantum fluctuations over the horizon surface, while in Tsallis’ case, it describes the deviation of the holographic scaling from extensivity. Here, we utilize the gravity-thermodynamics conjecture with the Barrow–Tsallis entropy to investigate the implications of the related modified Friedmann equations on the spectrum of primordial gravitational waves. We show that, with the experimental sensitivity of the next generation of gravitational wave detectors, such as the Big Bang Observer, it will be possible to discriminate deviations from the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM model up to Δ≲O(10-3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta \\lesssim \\mathcal {O}(10^{-3})$$\\end{document}. In this limit, Barrow–Tsallis entropy reduces to a logarithmic correction to holographic scaling, which is nearly universally predicted by both entanglement entropy calculations in the UV regime and by several candidate theories of quantum gravity. Hence, our considerations and results are expected to have general validity in the quantum gravity framework.

Superconformal indices of 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFTs and holography

We study the superconformal index of 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 superconformal field theories on S1 ×ωS2 in the Cardy-like limit where the radius of the S1 is much smaller than that of the S2. We show that the first two leading terms in this Cardy-like expansion are dictated by the Bethe Ansatz formulation of the topologically twisted index of the same theory. We apply this relation to 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 holographic superconformal field theories describing the low-energy dynamics of N M2-branes and derive closed form expressions, valid to all orders in the 1/N expansion, for the two leading terms in the Cardy-like expansion of the superconformal index. We also discuss the implications of our results for the entropy of supersymmetric Kerr-Newman black holes in AdS4 and the four-derivative corrections to 4d gauged supergravity.

Corrected Thermodynamics of Black Holes in f(R) Gravity with Electrodynamic Field and Cosmological Constant.

The thermodynamics of black holes (BHs) and their corrections have become a hot topic in the study of gravitational physics, with significant progress made in recent decades. In this paper, we study the thermodynamics and corrections of spherically symmetric BHs in models f(R)=R+αR2 and f(R)=R+2γR+8Λ under the f(R) theory, which includes the electrodynamic field and the cosmological constant. Considering thermal fluctuations around equilibrium states, we find that, for both f(R) models, the corrected entropy is meaningful in the case of a negative cosmological constant (anti-de Sitter-RN spacetime) with Λ=-1. It is shown that when the BHs' horizon radius is small, thermal fluctuations have a more significant effect on the corrected entropy. Using the corrected entropy, we derive expressions for the relevant corrected thermodynamic quantities (such as Helmholtz free energy, internal energy, Gibbs free energy, and specific heat) and calculate the effects of the correction terms. The results indicate that the corrections to Helmholtz free energy and Gibbs free energy, caused by thermal fluctuations, are remarkable for small BHs. In addition, we explore the stability of BHs using specific heat. The study reveals that the corrected BH thermodynamics exhibit locally stable for both models, and corrected systems undergo a Hawking-Page phase transition. Considering the requirement on the non-negative volume of BHs, we also investigate the constraint on the EH radius of BHs.

Geometric and topological corrections to Schwarzschild black hole

In this paper, we compute departures in the black hole thermodynamics induced by either geometric or topological corrections to general relativity. Specifically, we analyze the spherically symmetric spacetime solutions of two modified gravity scenarios with Lagrangians L∼R1+ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {L}\\sim R^{1+\\epsilon }$$\\end{document} and L∼R+ϵG2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {L}\\sim R+\\epsilon \\, \\mathcal {G}^2$$\\end{document}, where G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document} is the Euler density in four dimensions, while 0<ϵ≪1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ 0<\\epsilon \\ll 1$$\\end{document} measures the perturbation around the Hilbert–Einstein action. Accordingly, we find the expressions of the Bekenstein–Hawking entropy by the Penrose formula, and the black hole temperature and horizon of the obtained solutions. We then investigate the heat capacities in terms of the free parameters of the theories under study. In doing so, we show that healing the problem of negative heat capacities can be possible under particular choices of the free constants, albeit with limitations on the masses allowed for the black hole solutions.

Scalarization of Taub-NUT black holes in extended scalar-tensor-Gauss-Bonnet theory

Recently, the scalarization of the Schwarzschild black hole has been extensively studied. In this work, we explore the scalarization of the Taub-NUT black hole within the context of the extended scalar-tensor-Gauss-Bonnet theory, which admits a Ricci-flat Taub-NUT black hole as a solution. We carried out an analysis of the probe scalar field to identify the mass parameter and NUT parameter (m, n) where hairy black holes begin to emerge. Subsequently, we used the shooting method to construct the scalarized Taub-NUT black hole numerically. Unlike the Schwarzschild case, there are two branches of new hairy black holes that are smoothly connected. We calculated the entropy of the scalarized black holes and compared these entropies with those of scalar-free Taub-NUT black holes, finding that the entropies of the new hairy black holes are larger. A novel phenomenon emerges in this system: the entropy of the black holes at the bifurcation point is constant for a positive mass parameter. We then conjecture a maximal entropy bound for all scalarized black holes whose mass parameter at the bifurcation point is greater than zero.

Study of first-order quantum corrections of thermodynamics to a Dyonic black hole solution surrounded by a perfect fluid

In this work, we review the metric for a dyonic global monopole with a perfect fluid. To calculate the standard temperature of a Dyonic black hole surrounded by a perfect fluid, we analyze the graphical interpretation of the Hawking temperature concerning the horizon under the effects of the black hole's surrounding field and electric-magnetic charges. For this purpose, we follow the semi-classical method, the Wentzel-Kramers-Brillouin (WKB) approximation, and the Lagrangian equation in the presence of quantum gravity as seen in the generalized uncertainty principle (GUP). We calculate the improved temperature of a Dyonic black hole using a bosonic tunneling strategy based on the Hamilton-Jacobi technique. We observe that the physical state of the Dyonic black hole is surrounded by a perfect fluid under the effects of the black hole solution and the gravity parameter. Further, we examine the improved entropy to study the influences of quantum gravity and black hole geometry on entropy. We explore the graphical behavior of entropy based on the black hole's horizon structure and analyze the influence of perfect fluid parameters, electric charge, magnetic charge, and quantum gravity on entropy. Finally, we investigate the unstable and stable conditions of a black hole via graphical results.

The light we can see: extracting black holes from weak Jacobi forms

We quantify how constraints on light states affect the asymptotic growth of heavy states in weak Jacobi forms. The constraints we consider are sparseness conditions on the Fourier coefficients of these forms, which are necessary to interpret them as gravitational path integrals. Using crossing kernels, we extract the leading and subleading behavior of these coefficients and show that the leading Cardy-like growth is robust in a wide regime of validity. On the other hand, we find that subleading corrections are sensitive to the constraints placed on the light states, and we quantify their imprint on the asymptotic growth of states. Our approach is tested against the generating function of symmetric product orbifolds, where we provide new insights into the factors contributing to the asymptotic growth of their Fourier coefficients. Finally, we use our methods to revisit the UV/IR connection that relates black hole microstate counting to modular forms. We provide a microscopic interpretation of the logarithmic corrections to the entropy of BPS black holes in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2, 4 ungauged supergravity in four and five dimensions, and tie it to consistency conditions in AdS3/CFT2.

Barrow’s nonlinear charged anti-de Sitter black hole and stability

AbstractAs we know, the horizon area of a black hole will increase when it absorbs matter. Based on Barrow’s concept of a fractal black hole horizon, it has been proposed (Phys Lett B 831:137181, 2022) that for a spherically fractal structure, the minimum increase in the horizon area is the area of the smallest bubble sphere. The corresponding black hole entropy is of a logarithmic form, which is similar to that of Boltzmann entropy under a certain condition. Based on this, we re-derive the entropy of Barrow’s Einstein–power-Yang–Mills (EPYM) anti-de Sitter (AdS) black hole, and calculate its temperature and heat capacity. There exists an interesting phenomenon wherein the ratio between Barrow’s temperature and the Hawking temperature of the EPYM AdS black hole is fully consistent with that of other Schwarzschild-like black holes. The Barrow and Hawking temperatures within a certain range of $$\Lambda $$ Λ are found to increase monotonically, and the corresponding heat capacities are all positive, which means that these black holes are thermodynamically stable. In addition, for Barrow’s EPYM AdS black hole, its heat capacity has a Schottky anomaly-like behavior, which may reflect the existence of a discrete energy level and microscopic degrees of freedom.

Prigogine’s Second Law and Determination of the EUP and GUP Parameters in Small Black Hole Thermodynamics

In 1974, Stephen Hawking made the groundbreaking discovery that black holes emit thermal radiation, characterized by a specific temperature now known as the Hawking temperature. While his original derivation is intricate, retrieving the exact expressions for black hole temperature and entropy in a simpler, more intuitive way without losing the core physical principles behind Hawking’s assumptions is possible. This is obtained by employing the Heisenberg Uncertainty Principle, which is known to be connected to thenvacuum fluctuation. This exercise allows us to easily perform more complex calculations involving the effects of quantum gravity. This work aims to answer the following question: Is it possible to reconcile Prigogine’s second law of thermodynamics for open systems and the second law of black hole dynamics with Hawking radiation? Due to quantum gravity effects, the Heisenberg Uncertainty Principle has been extended to the Generalized Uncertainty Principle (GUP) and successively to the Extended Uncertainty Principle (EUP). The expression for the EUP parameter is obtained by conjecturing that Prigogine’s second law of thermodynamics and the second law of black holes are not violated by the Hawking thermal radiation mechanism. The modified expression for the entropy of a Schwarzschild black hole is also derived.

Properties of dynamical black hole entropy

We study the first law for non-stationary perturbations of a stationary black hole whose event horizon is a Killing horizon, that relates the first-order change in the mass and angular momentum to the change in the entropy of an arbitrary horizon cross-section. Recently, Hollands, Wald and Zhang [1] have shown that the dynamical black hole entropy that satisfies this first law, for general relativity, is Sdyn = (1 − v∂v)SBH, where v is the affine parameter of the null horizon generators and SBH is the Bekenstein-Hawking entropy, and for general diffeomorphism covariant theories of gravity Sdyn = (1 − v∂v)SWall, where SWall is the Wall entropy. They obtained the first law by applying the Noether charge method to non-stationary perturbations and arbitrary cross-sections. In this formalism, the dynamical black hole entropy is defined as an “improved” Noether charge, which is unambiguous to first order in the perturbation. In the present article we provide a pedagogical derivation of the physical process version of the non-stationary first law for general relativity by integrating the linearised Raychaudhuri equation between two arbitrary horizon cross-sections. Moreover, we generalise the derivation of the first law in [1] to non-minimally coupled matter fields that are smooth on the horizon, using boost weight arguments rather than Killing field arguments, and we relax some of the gauge conditions on the perturbations by allowing for non-zero variations of the horizon Killing field and surface gravity. Finally, for f(Riemann) theories of gravity we show explicitly using Gaussian null coordinates that the improved Noether charge is Sdyn = (1 − v∂v)SWall, which is a non-trivial check of [1].

Quasi-normal modes of loop quantum black holes formed from gravitational collapse

In this paper, we study the quasi-normal modes (QNMs) of a scalar field in the background of a large class of quantum black holes that can be formed from gravitational collapse of a dust fluid in the framework of effective loop quantum gravity. The loop quantum black holes (LQBHs) are characterized by three free parameters, one of which is the mass parameter, while the other two are purely due to quantum geometric effects. Among these two quantum parameters, one is completely fixed by black hole thermodynamics and its effects are negligible for macroscopic black holes, while the second parameter is completely free (in principle). In the studies of the QNMs of such LQBHs, we pay particular attention to the difference of the QNMs between LQBHs and classical ones, so that they can be observed for the current and forthcoming gravitational wave observations, whereby place the LQBH theory directly under the test of observations.

Vector superstrata. Part II

Microstate geometries are proposed microstates of black holes which can be described within supergravity. Even though their number may not reproduce the full entropy of black holes with finite-sized horizons, they still offer a glimpse into the microscopic structure of black holes. In this paper we construct a new set of microstate geometries of the supersymmetric D1-D5-P black hole, where the momentum charge is carried by a vector field, as seen from the perspective of six-dimensional supergravity. To aid our construction, we develop an algorithm which solves a complicated partial differential equation using the regularity of the geometries. The new solutions are asymptotically AdS3 × S3, and have a long, but finite AdS2 throat that caps off without ever developing a horizon. These microstate geometries have a holographic interpretation as coherent superpositions of heavy states in the boundary D1-D5 CFT. We identify the states which are dual to our newly constructed solutions and carry out some basic consistency checks to support our identification.

Traces of Quantum Gravity Effects at Late-time Cosmological Dynamics via Distance Measures

Inspired by the entropy–area relation of black hole thermodynamics, we study the thermodynamics of the cosmological apparent horizon in a spatially flat Friedmann–Robertson–Walker universe in the framework of an extended uncertainty principle (EUP). The adopted EUP naturally admits a minimal measurable momentum (equivalently a maximal measurable length), as an infrared cutoff in the theory. We derive the modified Friedmann equations in this setup and explore some predictions of these equations for the late-time universe via distance measures. We show that in this framework it is possible to realize the late-time cosmic speedup and transition to the phantom phase of the equation-of-state parameter of the effective cosmic fluid without recourse to any dark energy component or modified gravity. Inspection of various distance measures in this framework shows that an EUP with a negative deformation parameter suffices for the interpretation of the late-time asymptotically de Sitter universe with standard nonrelativistic matter.

Probing the thermodynamics of charged Gauss Bonnet AdS black holes with the Lyapunov exponent

In this paper, we investigate the thermodynamic properties of the charged AdS Gauss–Bonnet black holes and their associations with the Lyapunov exponent. The chaotic features of the black holes and the isobaric heat capacity characterized by the Lyapunov exponent are studied to reveal the thermodynamic stability of the black hole phases. By considering both the timelike and null geodesics, we find that the relationship between the Lyapunov exponent and the Hawking temperature can accurately represent the features of the Small/Large phase transition and even the triple point. We also reveal the properties of the difference in the Lyapunov exponent as an order parameter. It is demonstrated that there is a negative correlation between the Lyapunov exponent and the size of the black hole shadow, which can be used to bridge the thermodynamic properties and the shadow of black holes.

Look Beyond Additivity and Extensivity of Entropy for Black Hole and Cosmological Horizons.

We present a comparative analysis of the plethora of nonextensive and/or nonadditive entropies which go beyond the standard Boltzmann-Gibbs formulation. After defining the basic notions of additivity, extensivity, and composability, we discuss the properties of these entropies and their mutual relations, if they exist. The results are presented in two informative tables that are of strong interest to the gravity and cosmology community in the context of the recently intensively explored horizon entropies for black hole and cosmological models. Gravitational systems admit long-range interactions, which usually lead to a break of the standard additivity rule for thermodynamic systems composed of subsystems in Boltzmann-Gibbs thermodynamics. The features of additivity, extensivity, and composability are listed systematically. A brief discussion on the validity of the notion of equilibrium temperature for nonextensive systems is also presented.